Controlling alkaline earth metal atoms for quantum computing and metrology applications

Abstract

A device for individually trapping atoms, individually imaging atoms, and individually cooling atoms to prevent loss of atoms from the trap caused by imaging. The device can be implemented in various quantum computing, sensing, and metrology applications (e.g., in atomic clocks).

Description

[0001] Cross references to related applications

[0002] This application claims the benefit under 35 U.S.C. Section 119(e) of the following concurrently pending and commonly assigned U.S. provisional patent applications:

[0003] Serial No. 62 / 745,198, filed on October 12, 2018, by Manuel Endres, Alexandre Cooper-Roy, Jacob P. Covey, and Ivaylo S. Madjarov, titled “Controlling Alkaline Earth Atoms in Tweezer Arrays for Quantum Computing Applications” (CIT-8113-P);

[0004] Serial No. 62 / 767,619, filed Nov. 15, 2018, by Manuel Endres, Alexandre Cooper-Roy, Jacob P. Covey, and Ivaylo S. Madjarov, and titled “Controlling Alkaline Earth Atoms in Tweezer Arrays for Quantum Computing Applications” (CIT-8113-P2);

[0005] Serial No. 62 / 896,438, filed on September 5, 2019, by Manuel Endres; Alexandre Cooper-Roy; Jacob P. Covey; Ivaylo S. Madjarov; Adam L. Shaw; Tai Hyun Yoon; Vladimir Schkolnik; and Jason R. Williams, titled “Controlling Alkaline Earth Atoms InTweezer Arrays For Quantum Computing Applications” (CIT-8113-P3);

[0006] Serial No. 62 / 889,371, filed on August 20, 2019, by Manuel Endres; Alexandre Cooper-Roy; Jacob P. Covey; Ivaylo S. Madjarov; Adam L. Shaw; Tai Hyun Yoon; Vladimir Schkolnik; and Jason R. Williams, titled “Controlling Alkaline-Earth Atoms InTweezer Arrays For Quantum Metrology Applications” (CIT-8333-P);

[0007] These applications are incorporated herein by reference.

[0008] Statement Regarding Federally Funded Research and Development

[0009] This invention was made with government support under Grant Nos. PHY1733907 & PHY1753386 awarded by the National Science Foundation and FA9550-19-1-0044 awarded by the U.S. Air Force. The government has certain rights in this invention. Technical Field

[0010] The present invention relates to devices for trapping, imaging and manipulating atoms and methods of making and using the same. Background Art

[0011] (Note: This application references a number of different publications as indicated in the specification by one or more reference numbers within parentheses, such as [x]. A list of these different publications sorted by these reference numbers can be found in the section entitled "References". Each of these publications is incorporated herein by reference).

[0012] Optical tweezers (OT) and the related optical micropotential technique (OT) have matured into powerful tools for quantum science experiments on individually controlled atoms, as illustrated by a variety of recent results spanning quantum simulations using Rydberg atoms [1-3], entanglement manipulation [4-7], bottom-up assembly of Hubbard models [8,9], and cavity QED realizations [10-12]. In these experiments, individual atoms are directly trapped from laser-cooled clouds using tweezers or long-wavelength optical lattices [13-15]. Some more recent technical advances include, for example, sideband cooling to near the motional ground state in the tweezers [16,17], which has enabled experiments based on coherent collisions [7] and trapping near nanophotonic structures

[10] . Furthermore, recently developed atom-by-atom assembly techniques [18–23] provide a means for producing defect-free arrays of up to ∼60 atoms from initially randomly loaded OTs [7, 8, 15, 24–26], which has led to recent Rydberg quantum simulation applications [1–3].

[0013] In terms of key properties such as effective coherence time, scalability and controllability, these experiments are now comparable to and in many ways complementary to other quantum science platforms with local control, such as quantum gas microscopes

[27] , ion traps [28,29], or superconducting qubits

[30] . However, an open question is how non-alkali species with very different properties can be combined via OT and combined with single-atom control for new and improved implementations. Of particular interest are alkaline earth (-like) atoms (AEAs), which offer important features such as narrow and ultra-narrow optical transitions, which have had a strong impact in a variety of scientific fields ranging from quantum metrology [31-33] and simulation [34-39] to new methods for atomic and molecular control [40,41]. Summary of the invention

[0014] The present disclosure describes a device for trapping, imaging, and cooling one or more atoms. The device can be embodied in a variety of ways including but not limited to the following.

[0015] 1. A laser emits one or more first laser beams that generate one or more trapping potentials; one or more atoms, wherein:

[0016] The trapping potentials each trap only one (a single one) of the atoms, and

[0017] The atoms each have three energy levels including:

[0018] First energy level;

[0019] A second energy level having an energy higher than the first energy level; and

[0020] The third energy level;

[0021] One or more second laser beams illuminate the one or more atoms to produce fluorescence from each of the atoms, wherein the second laser beam has a frequency and polarization that is tuned to excite a first (e.g., optical) transition between a first energy level and a second energy level such that the fluorescence includes spontaneous emission from the second energy level back to the first energy level.

[0022] a detector receiving the fluorescent light to generate respective images of the atoms from the fluorescent light; and

[0023] One or more third laser beams irradiate the one or more atoms to cool each of the atoms.

[0024] 2. The device of Example 1,

[0025] The first objective lens focuses the first laser beam received from the laser source at one or more focal points to generate the trapping potentials at the focal points.

[0026] 3. The device of example 1, wherein:

[0027] The atom comprises an alkaline earth metal atom or an alkaline earth metal-like atom including two valence electrons in an s shell forming a spin singlet state,

[0028] The second energy level includes 1 valence electron in the s shell, 1 valence electron in the p shell, which forms a spin singlet state, and

[0029] The third energy level includes 1 valence electron in the s-shell and 1 valence electron in the p-shell, which forms one of three spin triplets.

[0030] In one example, in the ground state, the atoms each include two valence electrons at a first energy level including the s-shell, which forms a spin singlet state; in the first excited state, the atoms each include 1 valence electron at a first energy level including the s-shell and 1 valence electron at a second energy level including the p-shell, which forms a spin singlet state; and in the second excited state, the atoms each include 1 valence electron at a first energy level including the s-shell and 1 valence electron at a third energy level including the p-shell, which forms one of three spin triplet states.

[0031] 4. The apparatus of examples 1-4, wherein the third laser beam has a wavelength tuned to induce a second (eg, optical) transition between the first energy level and the third energy level to laser cool the atoms by transferring the atoms to a lower energy state of motion.

[0032] 5. The apparatus of example 4, wherein the laser cooling comprises Sisyphus cooling or resolved sideband cooling.

[0033] 6. The device of example 4, wherein:

[0034] The third laser beam does not include or provide magic trapping conditions associated with the second (e.g., optical) transition such that the trapping potential experienced by atoms in the first energy level is different from the trapping potential experienced by atoms in the third energy level, and

[0035] The atoms were cooled using Sisyphus cooling.

[0036] 7. The device of example 6, wherein:

[0037] The capture potential of an atom in the ground state (electrons in the first energy level) is higher than the capture potential of an atom in an excited state where the electrons are in the third energy level,

[0038] The third laser beam is blue-detuned (frequency is greater than the transition frequency of non-trapped atoms in free space), and

[0039] The cooling is repulsive Sisyphus cooling.

[0040] 8. The device of example 6, wherein:

[0041] The capture potential for an atom in the ground state (electrons in the first energy level) is lower than the capture potential for an atom in an excited state in which the electrons are in the third energy level,

[0042] The third laser beam is red-detuned (the frequency of the third laser beam is less than the transition frequency of the non-trapped atoms in free space), and

[0043] The cooling is attractive Sisyphus cooling.

[0044] 9. The device of example 4, wherein:

[0045] (1) the third laser beam is tuned to a magic trapping condition associated with the second (e.g., optical) transition such that the trapping potential experienced by atoms in the ground state (electrons in the first energy level) is the same as the trapping potential experienced by atoms in an excited state in which the electrons are in the third energy level,

[0046] (2) the atom further comprises a first set of motional energy levels indexed by an integer n for an electron in the first energy level and a second set of motional energy levels indexed by an integer m for an electron in the third energy level, the third laser beam excites the atom from the nth state in the first energy level to the m=(n-1)th state in the third energy level, so that the atom decays by emitting spontaneous emission from the mth state to the (n-1)th state in the first energy level,

[0047] (3) Repeat step (2) (irradiating the atom with a third laser beam) until the atom is in the n=0th motion state in the first energy level.

[0048] 10. The apparatus of any of the preceding examples, further comprising an array of first laser beams and / or an array of third laser beams (cooling beams) and / or an array of second laser beams (imaging beams) forming (e.g., optical) tweezers, wherein each of the tweezers captures one of the atoms, each of the cooling beams cools one of the atoms, and each of the imaging beams images one of the atoms.

[0049] 11. The apparatus of any of the preceding examples, wherein the third laser beam comprises:

[0050] a laser beam propagating perpendicularly to the first laser beam to cool the atoms in a radial direction, and

[0051] A fifth laser beam propagates parallel to the first laser beam to cool the atoms in a longitudinal direction.

[0052] 12. The apparatus of any preceding example, wherein:

[0053] The atoms each have a fourth energy level that is higher than the first energy level and lower than the third energy level;

[0054] The first laser beam is tuned to have a wavelength that is magic for the first and fourth energy levels but not magic for the third energy level, and

[0055] Cooling using the third laser beam is Sisyphus cooling.

[0056] 13. The device of example 12, wherein the fourth energy level is a clock state, and the transition from the first energy level to the fourth energy level is used to create a (e.g., optical) qubit in a quantum computing configuration (the ground state is the first energy level and the excited state is the clock state), and

[0057] Images using the fluorescence are used to read out the state of the qubit (as well as to image / determine the occupancy of the well).

[0058] 14. The apparatus of any preceding example, wherein:

[0059] The atoms each have a fifth energy level that is higher than the first energy level and lower than the second energy level; wherein after transitioning from the first energy level to the second energy level, the electron transfers to the fifth energy level; and

[0060] The first laser beam has a frequency that causes atoms containing electrons in a fifth energy level to experience a trapping potential so that the atoms can transfer the electrons to a third energy level that experiences an anti-trapping potential, wherein the atoms will be transferred out of a (eg, optical) well or trapping potential.

[0061] 15. The apparatus of any of the preceding examples, further comprising performing repeated imaging of said atoms (at least 2000 imaging steps), which shows a long lifetime of said atoms under the imaging conditions.

[0062] 16. The apparatus of any preceding example, comprising simultaneously or alternately performing imaging and cooling, wherein the cooling ensures that the atoms are not lost from the light trap by the imaging process.

[0063] 17. The apparatus of any of the preceding examples, further comprising: atoms each comprising a fourth energy level, the fourth energy level having an energy higher than the first energy level and an energy lower than the third energy level; and one or more fourth laser beams (e.g., a clock laser beam output from a clock laser) tuned to excite a clock transition between the first energy level and the fourth energy level; a detector that detects the presence or absence of the fluorescence to generate a signal indicating the presence or absence of each of the atoms in a ground state (electrons in the first energy level) or a clock state (electrons in the fourth energy level), imaging each atom individually; the detector detecting the signal multiple times as follows:

[0064] (1) after preparing the atom to the ground state, the absence of the fluorescence indicates that the atom does not occupy the well (first signal);

[0065] (2) after being excited from the first energy level to the second energy level using a clock laser beam that is red-detuned from the clock transition (having a frequency lower than the clock transition), such that the presence of the fluorescence indicates that the atom is in an excited state in which the electron is in the first energy level (second signal);

[0066] (3) after preparing the atom to the ground state after step (2), such that the absence of the fluorescence indicates that the atom does not occupy the well (third signal);

[0067] (4) after being excited from the first energy level to the second energy level using a clock laser beam that is red-detuned from the clock transition (having a frequency higher than the clock transition), such that the presence of the fluorescence indicates that the atom is in an excited state in which the electron is in the first energy level (fourth signal); and

[0068] a computer / processor that generates an error signal using the signal; and a modulator that modulates a fourth laser beam with the error-corrected frequency to stimulate a clock transition between the first energy level and the fourth energy level using the fourth laser beam having the one or more error-corrected frequencies.

[0069] 18. The apparatus of example 17, comprising wherein generating the error signal comprises determining which of the wells are occupied; and for each of the occupied wells:

[0070] determining a first occupancy number of atoms in an excited state in which electrons are in a first energy level after excitation with a red-detuned fourth laser beam, and

[0071] determining a second occupancy number of atoms in an excited state in which electrons are in the first energy level after excitation with a blue detuned fourth laser beam,

[0072] determining an error signal for each of the wells, comprising a difference between a first occupancy count and a second occupancy count in each of the wells;

[0073] converting the error signal to one or more error-corrected frequencies; and

[0074] 19. The apparatus of example 18, wherein the computer averages the error signal over time and / or for each of the wells to obtain an average error signal for generating an error-corrected frequency including an average frequency.

[0075] 20. The apparatus of embodiment 17, comprising said computer / processor generating an error signal using said signal, comprising:

[0076] determining which of the wells are occupied; and

[0077] For each of the occupied wells:

[0078] If the signal after red detuning is higher than the signal after blue detuning, indicating that the frequency of the fourth laser beam should be increased to resonantly excite the clock transition, a first error is assigned,

[0079] If the signal after red detuning is lower than the signal after blue detuning, indicating that the frequency of the fourth laser beam should be reduced to resonantly excite the clock transition, a second error is assigned,

[0080] If the signals after red detuning and blue detuning are the same, indicating that the frequency of the fourth laser beam does not need to be corrected, then a zero error is assigned.

[0081] converting the error signal to one or more error-corrected frequencies; and

[0082] The modulator modulates the fourth laser beam with the plurality of error-corrected frequencies to resonantly excite a clock transition between the first energy level and the fourth energy level using the fourth laser beam having the one or more error-corrected frequencies.

[0083] 21. The apparatus of examples 17-20, wherein an error signal is generated for each atom in the trap using the imaging of each atom so that the frequency of the fourth laser beam that excites the clock transition is corrected for each atom.

[0084] 22. The apparatus of examples 17-20, further comprising generating an average error signal comprising an average of the error signals of the atoms, such that a frequency of the fourth laser beam that excites the clock transition is generated from the average error signal.

[0085] 23. A spatially resolved sensor / diagnostic comprising the apparatus of examples 1-22.

[0086] The present disclosure further describes a method for trapping atoms. The method can be embodied in a variety of ways including but not limited to the following:

[0087] 24. A method for trapping atoms, comprising:

[0088] (a) capturing the atom;

[0089] (b) imaging the atoms; and

[0090] (c) cooling the atoms, wherein the atoms are cooled to prevent loss of atoms from the trap caused by the imaging.

[0091] 25. The method of example 24, wherein the cooling counteracts multiple heating mechanisms.

[0092] 26. The method of example 24 or 25, wherein the imaging reads out the occupancy of atoms in the ground state without destroying the clock state (depopulating it) (because the cooling does not affect the clock state (1-3 transition is detuned to 1-4).

[0093] 27. The apparatus or method of examples 1-26, wherein the (eg, optical) trap comprises (eg, optical) tweezers or an optical lattice or a laser trap.

[0094] The present disclosure further describes a computer-implemented method. The method is implemented in a variety of ways including but not limited to the following:

[0095] 28. The method comprises numerically simulating the dynamics of a plurality of atoms, the evolution of which is described by an interaction between a laser field and a plurality of atoms trapped in an array (trapped atoms), each of which atoms comprises at least two energy levels and wherein the interaction comprises a transition between the two energy levels excited by the laser field, the method comprising:

[0096] Obtaining a noise spectrum of the laser field;

[0097] performing a numerical calculation of an error signal representing the detuning between the frequency of the laser field and the resonant frequency required to resonantly excite the transition, including solving the Schrödinger equation describing the interaction of trapped atoms with the laser field; and

[0098] A response of the trapped atoms to the laser field is calculated as a function of time, wherein the response includes the dynamics of the trapped atoms interacting with the laser field whose frequency fluctuates with time.

[0099] 29. The method of example 28, further comprising using the response to stabilize the frequency of coherent radiation (including the laser field) to a resonant frequency of the transition.

[0100] The present disclosure further describes devices implemented in a variety of ways including but not limited to the following:

[0101] 30. An apparatus comprising:

[0102] An array of physical systems (such as, but not limited to, one or more impurities, atoms, electrons, or superconductors in a solid state material) each having two energy level transitions;

[0103] a source of coherent radiation for exciting said transition;

[0104] a detector that measures the excitation probability of each physical system in the array (wherein the excitation probability determines how well the coherent radiation excites or drives the transition);

[0105] a computer that converts the excitation probability into a detuning between the frequency of the coherent radiation and a resonant frequency of the transition;

[0106] A modulator provides feedback to the coherent radiation including the detuning such that for each of the physical systems in the array, the frequency of the coherent radiation is tuned to a resonant frequency, for example such that an oscillator is stabilized to a transition.

[0107] 31. A sensor comprising the device of example 30.

[0108] 32. The sensor of example 31, comprising a gravity sensor.

[0109] 33. A method of operating the apparatus or method of examples 1-32, comprising changing the environment (e.g., but not limited to, magnetic field / environment, polarization, power, or temperature) of the array of physical systems; and

[0110] Changes in the frequency of the laser are measured to quantify changes in the environment or frequency changes in response to environmental perturbations.

[0111] 34. The method or apparatus of any preceding example, wherein the laser beam comprises electromagnetic radiation having a plurality of wavelengths.

[0112] In one example, we demonstrate single shot imaging and narrow-line cooling of individual alkaline earth metal atoms (specifically strontium trapped in 515.2 nm light) in optical tweezers. Our approach enables high-fidelity detection of single atoms by imaging photons from broad singlet transitions while cooling on a narrow interstate combination line, and we extend the technique to a highly uniform two-dimensional tweezer array with 121 sites. Cooling during imaging is based on a previously unobserved narrow-line Sisyphus mechanism that we predict can be used in a wide variety of experimental situations. Furthermore, we demonstrate optically resolved sideband cooling of single atoms to near the tweezers' motional ground state, tuned to a magic trapping configuration achieved by elliptical polarization. Finally, we present computational results that agree well with experimental results, predicting linear polarization and polarization-independent magic crossings at 520(2) nm and 500.65(50) nm, respectively. Our results pave the way for a wide range of new experimental avenues based on individually controlled alkaline earth metal atoms in tweezers—from fundamental experiments in atomic physics to quantum computing, simulations, and metrology.

[0113] In another example, we demonstrate single-atom-resolved imaging with a survival probability of 0.99932(8) and a fidelity of 0.99991(1), which enables us to repeatedly image single atoms in the tweezers with high fidelity thousands of times. We further observe lifetimes of more than 7 minutes under laser cooling, an order of magnitude longer than in previous tweezers studies. The experiments were performed with strontium atoms in an array of 813.4 tweezers, the magic wavelength for clock transitions. Tuning to this wavelength was achieved by off-magic Sisyphus cooling based on interstate combination lines, which allows us to choose the tweezer wavelength almost arbitrarily. We find that a single non-retroreflected cooling beam in the radial direction is sufficient to mitigate recoil heating during imaging. Furthermore, this cooling technique produces temperatures below 5 μK, as measured by release and recapture. Finally, we demonstrate clock state-resolved detection with an average survival probability of 0.996(1) and an average state detection fidelity of 0.981. Our work paves the way for the atom-by-atom assembly of large defect-free arrays of alkaline earth metal atoms, where repeated interrogation of clock transitions is an imminent possibility.

[0114] Measuring time is at the heart of all science. Currently, the most accurate and stable clocks are based on optical interrogation of ensembles of neutral atoms or single ions constrained in optical lattices. In yet another example, we demonstrate a new optical clock system based on arrays of trapped atoms read out at the single-particle level, incorporating many of the benefits of ion and lattice clocks and building a bridge to recently developed techniques in quantum simulation and computation with neutral atoms. We apply this approach to the evaluation of unit-site resolved frequency shifts (frequency shifts) and systematics, as well as for atom-by-atom statistical analysis and feedback control. The system is also characterized by strongly suppressed interaction shifts and short dead times, all in a relatively simple experimental setup. This opens a new path for advancing fixed and mobile clock systems, and provides a new starting point for entanglement-enhanced metrology and quantum clock networks, as well as for single-atom-based thermometry. The demonstrated technology can also enable applications in quantum computing and communications using individual neutral atoms that require optical clock state control. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] Referring now to the drawings, wherein like reference numerals designate corresponding parts throughout:

[0116] Figure 1 : Tweezers trapping of strontium. (ab) Single strontium atoms are trapped in a 515.2 nm laser beam focused through a microscope objective with NA = 0.5 (bottom objective) (along the The atoms are imaged by scattering photons at a broad blue transition (461 nm) by the lateral imaging beam and are simultaneously cooled at a narrow red transition (689 nm) by three red MOT beams (one red beam overlapping the imaging beam is not shown). The fluorescence photons are collected with the bottom objective, while the top objective is mainly used to monitor the tweezer light. (c) The applied narrow-line cooling mechanisms, sideband cooling and Sisyphus cooling, depend crucially on the ground state and 3 The relative trapping potential between the three excited sublevels of P1. In linearly polarized tweezers, these sublevels can be projected with the angular momentum quantum number m j = -1, 0, 1. In elliptical light, the rotational symmetry is broken and the sublevels are no longer eigenstates of angular momentum. Therefore, we use different symbols to mark these states: |φ C >、|φ A >、|φ B > (from left to right). The two states are displaced as a function of the ellipticity (dashed line compared to the solid line). (d) tweezers ellipticity as a function of the tweezers ellipticity angle γ 3 The differential well depths of the three sublevels of P1 (proportional to the differential polarizability) are measured using excitation-depletion spectroscopy (Appendix 2 and Section 5). The tweezer polarizability is given by The solid line is a fit to the eigenvalues ​​of the AC Stark Hamiltonian (Appendix 2). at 1 S0 and 3 P1|φ A >The differential polarizability between becomes zero (short dash-dash line). For all γ, the other two sublevels experience weaker trapping potentials (positive differential well depths).

[0117] Figure 2 : Imaging in a single tweezer. (a) Histogram of detected photons obtained under typical imaging conditions, showing good discrimination between zero-atom and single-atom peaks. Results are for tweezers with magic polarization. Inset: Average fluorescence image of a single atom (see Section 3 for details). (b) Imaging fidelity and loss probability as a function of imaging time. The fidelity (defined as the accuracy of the image classification) reaches a maximum value of F = 99.3 (9)% for sufficiently long imaging times. However, the losses also increase with imaging time. The fidelity is ultimately limited by the estimated number of atoms that are lost before they can emit enough photons to be detected. (c) The loss coefficient as a function of the detuning of the cooling light where p sis the survival probability and N is the number of scattered photons. The narrow regime of cooling towards the red detuning side is interpreted as sideband cooling, while the much broader regime towards the blue detuning side is interpreted as Sisyphus cooling. Both regimes achieve the same optimal value of χ. (d) χ as a function of the scattering rate estimated for a fixed imaging time of 200 ms. The data shown are for a 1.4 mK well under Sisyphus cooling. Below 60 kHz, χ approaches a constant minimum, indicating that the losses are dominated by population reduction (white area) rather than heating. When the scattering rate increases to above 60 kHz, cooling can no longer mitigate the heating losses (red area). Inset: χ versus imaging time, which is taken at the scattering rate indicated by the arrow (~27 kHz). χ remains approximately constant even for very long times.

[0118] Figure 3 : Tweezers array. (a) We created a two-dimensional tweezers array using two perpendicular acousto-optic deflectors (AODs). A 4f telescope (not shown) maps the light between the two AODs. Each AOD is driven by a multicolor RF waveform with tones evenly spaced in frequency. (b) Average fluorescence image (of 6000 experimental runs) of each individual strontium atom in a square array of 11×11 tweezers. The interatomic distance is ~9μm. (c) Single excitation image of each individual strontium atom in a square array of 11×11 tweezers. The filling fraction is close to 0.5. (d) The well depth of all 121 tweezers, as measured by the distance from the tweezers to the atoms. (e) Loss coefficient χ as a function of cooling frequency, averaged over an 11 × 11 array of linearly polarized light. Features are similar to those seen in a single magic tweezer, but are pushed further away from the free-space resonance due to the larger differential polarizability in the linear light. Inset: χ versus blue scattering rate under Sisyphus cooling, averaged over the array.

[0119] Figure 4 : Sisyphus cooling. (a) Illustration of a regime in which excited states are less trapped than the ground state. Diagram of the mechanism of Sisyphus cooling of transitions. The red cooling beam of frequency v is detuned away from the free-space resonance by the blue one, effectively becoming a beam with energy E 帽resonant conditions are created for the ground state atoms. During fluorescence imaging, the atoms are heated until their energy reaches the Sisyphus cap, at which point they are excited and preferentially decay back to the ground state with lower kinetic energy. (b) Average equilibrium energy of atoms after fluorescence imaging as a function of Sisyphus detuning. The solid line is a linear fit to the experimental data. The shaded area represents the equilibrium energy after fluorescence imaging with sideband cooling instead of Sisyphus cooling. (c) Average equilibrium energy of atoms versus imaging time for a Sisyphus detuning of 1.2 MHz. The energy initially increases linearly (solid line, t≤15 ms) and saturates later. (d) Survival probability versus normalized final well depth after adiabatically ramping down the well depth for various Sisyphus cap energies (Appendix VI).

[0120] Figure 5 .Sideband cooling. (a) Based on magic tuning Figure 1. Diagram of the method for resolved sideband cooling of transitions. The optical excitation of the red sideband is spectrally resolved and the subsequent decay preserves the quantum of motion with high probability. (b) Measurement protocol for sideband spectra. 1 The atoms in the S0 ground state are excited by the 689 nm excitation pulse (solid double arrows) to 3 P1 excited state. 3 The atoms in the P1 excited state are then excited by a 688 nm depletion pulse (solid double arrow) to 3 S1 state, where they decay radiatively to 3 P0 and 3 P2 metastable dark state. (c) Radial sideband spectra before (inset) and after sideband cooling. Overlaid is a simulated spectrum with a ground state fraction of 0.80 (solid grey line). The bulge visible in this simulated spectrum is the Fourier peak due to the finite 74 μs excitation pulse. The first radial sideband is separated from the carrier frequency by 211 (4) kHz. After cooling, the amplitude of the red sideband is highly suppressed, as is the width of the blue sideband—both cases indicating a larger ground state fraction. (d) Axial sideband spectra before (inset) and after the second stage of axial cooling. Overlaid is a simulated spectrum with a ground state fraction of 0.50 (solid grey line). The first axial sideband is separated from the carrier frequency by 32.2 (8) kHz. The suppression of the red sideband and the enhancement of the carrier both indicate a larger ground state fraction.

[0121] Figure 6 .5s of Sr under linear well polarization 21 S0 and 5s5p 3 Polarizability of the P1 state. Calculations using both ab initio (dashed line) and recommended (solid line) values ​​predict The same magic wavelength at 520(2)nm for the transition and α=880(25)au. Calculations using the recommended values ​​predict Another magic wavelength for the transition is at λ = 5006.5 (50) nm and α = 1230 (13) au. We note that the latter crossing is valid even for elliptical well polarization since it belongs to an exciton level with a polarization-insensitive polarizability.

[0122] Figure 7 . Differential well depth spectroscopy. (a) For non-magic ellipse angle γ = 28° Spectral signal of the transition measurement. The signal is fitted to the line shape of pure thermal broadening (blue dashed curve) and pure power broadening (red dash-dash-dotted curve). The vertical lines indicate the position of the edge frequency (blue dashed line) and the center frequency (red dash-dash-dotted line). We expect that the true rescaled differential well depth lies between these two values, because the combination of power and thermal broadening determines the true line shape. (b) The linear relationship between the three ellipticity angles for various The differential well depth of the transition measurement, which shows the line shape of thermal broadening (dark marks) and power broadening (light marks). The measured values ​​are simultaneously fitted to the analytical solution of the eigenvalues ​​of the Stark Hamiltonian operator with three free parameters. At the magic ellipse angle |γ|=24° (black dash-line), about The differential well depth of the transition becomes zero.

[0123] Figure 8 .Preparation of single atoms via parity projection. (a) The probability of detecting an occupied tweezer after a 60 ms parity projection (PP) pulse varies with the detuning of the addressing frequency. The highest step on the left corresponds to the situation when the PP pulse is detuned away from any atomic or molecular resonance, so that many atoms remain in the well. The lowest step on the right corresponds to heating on the blue detuned motion sideband, which expels atoms from the well. The middle plateau corresponds to the PP region where the occupancy probability is 0.5. Inset: As the duration of the PP pulse increases, the probability of detecting an occupied tweezer decreases monotonically and saturates to 0.5. (b) As the blue MOT loading time increases, the initial number of atoms in the tweezers increases, so that the probability of detecting an occupied tweezer is close to 1.0 before the PP (blue squares), but saturates to 0.5 after the PP pulse (red circles). The frequency detuning from the free-space resonance here is -226 kHz.

[0124] Fig. 9.Sideband thermometry. (ab) Ratio of the red to blue sideband amplitudes as a function of the ground state fraction, obtained by fitting simulated spectra to (a) the radial spectrum and (b) the axial spectrum. The dependence on the ground state fraction is fitted to a quadratic function (red curve). The solid blue line is the fitted sideband ratio to our experimental data, and the dashed lines indicate 1σ confidence intervals. We quote a range of consistent ground state fractions where this confidence interval intersects the fitted quadratic function.

[0125] Fig.10 . 88 Low-loss imaging scheme for Sr. (a) Our single-atom imaging scheme requires only a single non-retroreflected cooling beam at 689 nm for narrow-line Sisyphus cooling (red single arrow) to offset the recoil heating from the fluorescence generated by exciting the atoms with a single retroreflected 461 nm beam (blue double arrow). A NA=0.5 microscope objective is used to generate 813.4 nm tweezers and collect the fluorescence light. (b) Previous studies have found that the single-atom imaging scheme can be achieved by using a branching ratio of 1:20,000. 1 P1 to 1 The imaging loss channel of the attenuation of D2, where atoms 1 12 is strongly resistant to trapping [21,22] and leaves the tweezers. Crucially, it is now trapped at 813.4 nm 1 D2 state, and our results show that the atom recovers to 3 P J Multiple clusters. Two lasers (679nm and 707nm) pump the atoms back to 3 P1, which decays back to the ground state, thus blocking 1 D2 loss channel. (c) We use the m-based J = ±1 states

[23] , which was originally proposed in references [24, 25]. The mechanism is based on the fact that the excited state is trapped more strongly than the ground state (in contrast to the repulsive Sisyphus cooling we show in this paper

[21] ). The atoms at the bottom of the well are excited and have to climb a steeper potential than they are in the ground state, resulting in an average reduction in energy after spontaneous emission. The cooling arises from the trapping potential mismatch rather than from photon recoil, so only a single cooling beam is required. (d) Average image (top) and single excitation image (bottom) of atomic fluorescence from 25 homogenized tweezers with an imaging time of 1 s.

[0126] Fig.11.Low-loss high-fidelity imaging. (a) Histogram of fluorescence counts from a single representative tweezer. We find a detection fidelity of 0.99991(1) and an average survival probability of 0.99932(8), demonstrating simultaneous high-fidelity and low-loss imaging. Results are for an imaging time of t = 50 ms with simultaneous repumping and Sisyphus cooling. (b) Survival fraction as a function of hold time in minutes under these imaging conditions (blue squares and fitted line). Importantly, we find a lifetime of τ = 126(3) s, while only τ = 126(3) s is required to achieve high detection fidelity. imaging time, resulting in a small loss fraction consistent with exp(-t / τ). In addition, we find a lifetime of 434(13)s under Sisyphus cooling alone (without 461nm), which demonstrates a vacuum-limited lifetime of greater than 7 minutes (red circles and fit line). (c) Survival fraction versus number of images for 2000 repeated images. The dark red line represents the average for 40 realizations, and the lighter red line represents the standard error of the mean. Atoms were imaged with high fidelity for 50ms, followed by a 29ms cooling block. (d) Representative realization of atom detection over the course of 2000 images. The detected atoms are plotted in red versus the number of images, with one row representing 25 tweezers. Note that we find no atomic returns following the loss of an atom.

[0127] Fig.12 .Sisyphus cooling during imaging. Fig.12 (a): Survival probability versus detuning of the free-space resonance relative to the 689 nm cooling beam. Fig.12 (b): Survival fraction of atoms versus scattering rate from the 461 nm imaging beam under simultaneous repumping and Sisyphus cooling for an imaging time of 50 ms. Sisyphus Cooling to Low Temperatures (a) Survival fraction in the array in a release-recapture measurement performed by adiabatically shutting down the trap for a variable time and then suddenly opening it

[37] . We show the data after imaging (blue squares) and after adding a dedicated cooling block with Sisyphus cooling alone (red circles). The results are compared with classical Monte Carlo simulations of the three-dimensional thermal distribution at 5 μK (dashed lines). Note that the release-and-recapture method is mainly sensitive to the energy distribution in the radial direction. (b) Survival fraction in the array after release and recapture versus intensity I / I for an off time of 60 μs. s Frequency in red during a 25 ms Sisyphus cooling of ≈ 200. The dashed line shows the case without a dedicated cooling block. We find that for an appropriately chosen detuning in red, the atoms are cooled, and for further detuning towards blue, the atoms are heated. This is consistent with the interpretation of Sisyphus cooling as an attractor in energy space [24,25]. The data in (a) are at a detuning of -2.6 MHz.

[0128] Fig.13 . Low-loss state-resolved detection. (a) The statistical mixture of optical clock qubit states is represented as a circle, where 1 The green part at S0≡|g> indicates the number of particles in |g>, and 3 The purple part at P0≡|e> indicates the number of particles in |e>. To measure the number of particles in |g>, we imaged without the 679 nm repump, during which |e> remained dark. The accuracy of the measurement of the number of particles in |g> is limited by the off-resonance scattering of the tweezer light pumping |e> back to |g> and by the path of pumping |g> to |e>, such as the scattering from the tweezer light during cooling. 3 Off-resonance scattering of P1 and 1 The D2 attenuation channel is limited. As a result, the average state detection fidelity is 0.981(1). (b) We perform a second image that includes a 679 nm re-pump (which transfers |e> via 3 The S1 state is pumped to |g>) and re-pumped at 707nm so that both states are detected. The pumping process is illustrated by the purple arrows. This image measures the number of particles in |g> and |e> and tells whether an atom was lost due to the first image. We find that the average survival probability for this double imaging sequence is 0.996(1).

[0129] Fig.14 . Atomic array optical clock. (A) We interrogate ≈40 atoms trapped in an 81-site tweezer array. 88Ultranarrow clock transitions of strontium atoms at 698 nm, and high-resolution fluorescence imaging at 461 nm is used to detect population changes in the clock states (labeled |g> and |e>) with single-atom resolution. This information is processed by a central processing unit (CPU), and a feedback signal is applied to the clock laser frequency using an acousto-optic modulator (AOM). (B) Average probability of the tweezers remaining in |g> (circles) as a function of the frequency shift measured with the in-loop probe sequence. The horizontal dashed line represents the state-resolved detection fidelity

[32] . To generate the error signal, we interrogate twice: once below resonance (A) and once above resonance (B). (C) Average tweezers error signal (circles) as a function of frequency shift. The shaded areas in B and C show results from MC simulations. (D) Simplified experimental sequence consisting of tweezer loading and N repeated AB feedback blocks, followed by an optional probe block, where N=10 throughout. (E) To detect the clock state population in block A, we take a first image before interrogation to identify which tweezers are occupied, and a second image after interrogation to detect which atoms remain as |g> (images 1 and 2). The same procedure is repeated for block B (images 3 and 4). We show the fluorescence images with the identified atoms (circles)

[32] , and the single tweezers error signal e j Instance of .

[0130] Fig.15 : Site-resolved error signal. (A) Single-tweezers error signal averaged over repeated events as a function of frequency offset measured with in-loop sequence <e j >. (B) For our usual U1=245(31)E r The fitted zero-crossing of the interrogation well depth as a function of the tweezer index, where E r = h × 3.43 kHz (circles). The solid line corresponds to the theoretical prediction, and the shaded area is derived from the systematic uncertainty of the well depth

[32] . (C) For the selected tweezers, e j The vertical dashed line represents the mean. (D) Variance of the error signal as a function of the number of atoms calculated by post-selection. The solid line is the variance of the error signal calculated by post-selection using 1 / N A The fit of the function plus the offset. The purple area is the MC simulation. (E) Plot of the correlation between the error signals for even and odd sites.

[0131] Fig.16: Operational magic tuning and site-resolved systematics. (A) Illustration of an interleaved self-comparison where two independent AOM frequencies (f1 and f2) are updated in an alternating manner. The respective interrogation blocks are set to two independent tweezer depths U1 and U2. (B) Average frequency difference f2-f1 as a function of U2 / U1 for multiple frequency shifts of the trapping laser, where U1 is fixed to our usual interrogation depth (see legend for color coding). Taking into account the unknown frequency shifts, we fit the data (simultaneously for all data) with a model for light displacements in optical tweezers (colored lines) with only a single free parameter

[32] . The operational magic intensity is found at the minimum of these curves (grey squares and connecting lines). The trap laser frequency is tuned so that the minimum coincides with our nominal depth. (C) Comparison of this technique with the single-tweezers resolution error <e j >Combined, we can extract the frequency dependence of each tweezer (colored squares) on the well depth. The solid lines show the expected dependence for the outermost and central tweezers. The data correspond to the -7MHz setting in B. Inset: Local frequency shift for U2 / U1=10. The color coding of the inset defines the color coding of the sub-graphs it contains.

[0132] Fig.17 : Stability results. (A) Fractional Allan deviation σ obtained via self-comparison as a function of integration time τ y (circle). For those who exceed the initial lock start time Behavior is fitted (red solid line), we find The shaded area indicates the MC result. The purple solid line shows the quantum projection noise limit obtained from MC by turning off all other noise sources. (B) Based on atom-by-atom feedback control, we A A series of self-comparisons are performed under the condition of A The Allan variance of the function at one second (From Fitting). Inset: As 1 / N A Allan variance of a function of The solid line shows a functional form The fitting of The scale is

[0133] Fig.18 : Rabi oscillations on clock transitions. Rabi oscillations on clock transitions using a π-pulse length of 110 ms. The points are directly probed after stabilizing the clock laser with a feedback sequence as described in the text. The shaded area represents the Monte Carlo results.

[0134] Fig.19: Clock sideband thermometry. Array-averaged radial sideband spectrum of an optical clock transition, taken using a carrier Rabi frequency of ≈360 Hz. The narrow carrier lies between two wider sidebands on the red and blue detuned sides. The sideband broadening is mainly due to small inhomogeneities in the array. The suppressed red sideband indicates a significant number of moving ground-state particles. The solid line is a simultaneous fit to two skewed Gaussians. From the ratio of the area under the red sideband to the area under the blue sideband, we obtain The carrier is detected with an interrogation time of 1.4 ms, while the sidebands are detected with an interrogation time of 3.3 ms.

[0135] Fig. 20 : Spatially resolved clock comparison. Fractional Allan deviation from asynchronous clock comparison between the left and right halves of our array. For clocks beyond the initial lock start time Behavior fitting, we found This is slightly higher than the result obtained by self-comparison of the entire array ( Figure 4 ). Importantly, for values ​​close to 10 4 s time and in 10 -16 Below the level, we see no rise, indicating that slowly varying drifts in the gradient across the array do not contribute to instabilities up to our sensitivity.

[0136] Fig.21 : Frequency noise spectrum of a clock laser. Power spectral density of the frequency noise of our clock laser, measured from the beat signal with a reference laser over a 42 hour period (red trace). Our theoretical estimate of the thermal noise contribution is plotted in yellow. Our best (purple) and worst (blue) case models for the total frequency noise as used in the Monte Carlo simulations are also plotted.

[0137] Fig. 22 . Flowchart illustrating the method of trapping, imaging and cooling atoms

[0138] Fig.23 .A flowchart illustrating a method for manufacturing a device according to a first example.

[0139] Fig.24 .A flowchart illustrating a method for manufacturing a device according to a first example.

[0140] Fig.25 .A schematic diagram illustrating an apparatus for locking an oscillator to a physical system.

[0141] Fig.26 .A flow chart illustrating a method for simulating the response of a physical system.

[0142] Fig. 27. is the hardware environment used to perform the calculation and control functions described in this article DETAILED DESCRIPTION

[0143] In the following description of the preferred embodiment, reference is made to the accompanying drawings, which form a part thereof and in which are shown by way of example specific embodiments in which the present invention may be practiced. It should be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention.

[0144] Technical Description

[0145] Part A: Trapping, imaging and cooling of atoms.

[0146] Equipment structure

[0147] Figure 1 A shows an apparatus 100 for trapping, imaging and cooling one or more atoms 102. The apparatus comprises one or more lasers 104a, 104b, 104c (or sources of coherent electromagnetic radiation, for example) that emit one or more first laser beams 106 comprising a first electromagnetic radiation, one or more second laser beams 108 comprising a second electromagnetic radiation, and one or more third laser beams 110 comprising a third electromagnetic radiation.

[0148] The one or more first laser beams generate one or more trapping potentials 112 (or one or more wells 112a, each of which includes a trapping potential). The device further includes one or more atoms 102, wherein the trapping potentials each trap only one of the atoms. The one or more second laser beams irradiate the one or more atoms to generate fluorescence 114 from each of the atoms. The device includes a detector 116 that receives the fluorescence to generate an image of each of the atoms from the fluorescence. The one or more third laser beams irradiate the one or more atoms to cool each of the atoms.

[0149] In one or more examples, the laser 104a comprises a laser system including one or more lasers, optical devices and diffraction elements. In one or more examples, the apparatus further comprises a first objective lens 118 that focuses the first laser beam at one or more focal points to generate each of the trapping potentials at each of the focal points.

[0150] Figure 1B shows that the atom includes a first energy level 122; a second energy level 122 having an energy higher than the first energy level; and a third energy level 126. In one or more examples, the atom includes an alkaline earth metal atom or an alkaline earth metal-like atom including two valence electrons in an s shell forming a spin singlet state, the second energy level includes 1 valence electron in the s shell, 1 valence electron in the p shell forming a spin singlet state, and the third energy level includes 1 valence electron in the s shell and 1 valence electron in the p shell forming one of three spin triplet states. In one or more instances, in the ground state, the atoms each include two valence electrons at a first energy level including the s-shell, which forms a spin singlet state; in the first excited state, the atoms each include 1 valence electron at a first energy level including the s-shell and 1 valence electron at a second energy level including the p-shell, which forms a spin singlet state; and in the second excited state, the atoms each include 1 valence electron at a first energy level including the s-shell and 1 valence electron at a third energy level including the p-shell, which forms one of three spin triplet states.

[0151] In one or more examples, the second laser beam includes second electromagnetic radiation having a frequency and polarization tuned to excite a first (e.g., optical) transition between a first energy level and a second energy level such that the fluorescence includes spontaneous emission from the second energy level back to the first energy level.

[0152] In one or more examples, the third laser beam includes third electromagnetic radiation having a wavelength tuned to induce a second (eg, optical) transition between the first energy level and a third energy level to laser cool the atoms by transferring the atoms to a lower energy state of motion.

[0153] Laser Cooling Examples

[0154] Examples of laser cooling include, but are not limited to, Sisyphus cooling or sideband cooling. Figure 1 C shows an example in which the atoms are each cooled using Sisyphus cooling and the third laser beam does not include or provides magic trapping conditions associated with the second (e.g., optical) transition such that the trapping potential experienced by atoms in the ground state (e.g., where the electrons are at the first energy level) is different from the trapping potential experienced by atoms in the excited state (e.g., where at least one of the electrons is at the third energy level).

[0155] Examples of Sisyphus cooling include repulsive and attractive Sisyphus cooling. In the repulsive Sisyphus cooling example, the trapping potential for atoms in the ground state (electrons in the first energy level) is higher than the trapping potential for atoms in the excited state (e.g., where at least one of the electrons is in the third energy level), and the third laser beam is blue detuned to have a frequency greater than the transition frequency of the second transition used to excite atoms in free space (non-trapped atoms).

[0156] In an example of attractive Sisyphus cooling, the trapping potential for atoms in a ground state (e.g., electrons at a first energy level) is lower than the trapping potential for atoms in an excited state (e.g., where at least one of the electrons is at a third energy level), and the third laser beam is red-detuned so that the frequency of the third laser beam is less than the transition frequency of the second transition used to excite atoms in free space (non-trapped atoms).

[0157] Figure 1 C further shows an example of laser cooling including sideband cooling, where:

[0158] (1) the third laser beam is tuned to a magic trapping condition associated with the second (e.g., optical) transition such that the trapping potential experienced by atoms in the ground state (where the electrons are in the first energy level) is the same as the trapping potential experienced by atoms in the excited state (where at least one of the electrons from the first energy level is transferred to the third energy level),

[0159] (2) The atom further includes a first set of motion energy levels indexed by an integer n for electrons in the first energy level and a second set of motion energy levels indexed by an integer m for electrons in the third energy level, and a third laser beam excites the atom from the nth state in the first energy level to the m=(n-1)th state in the third energy level, so that the atom decays by emitting spontaneous radiation from the mth state to the (n-1)th state in the first energy level.

[0160] (3) Repeat step (2) (irradiating the atom with a third laser beam) until the atom is in the n=0th motion state in the first energy level.

[0161] In one or more examples, the atoms are cooled in multiple directions. In one example, the third laser beam includes a laser beam propagating perpendicular to the first laser beam to cool the atoms in a radial direction, and a fifth laser beam propagating parallel to the first laser beam to cool the atoms in a longitudinal direction.

[0162] Array Instance

[0163] Figure 3The following example is shown: wherein the device further includes an array of first laser beams and / or an array of third laser beams (cooling beams) and / or an array of second laser beams (imaging beams) forming (e.g., optical) tweezers, each of which captures one of the atoms, each of which cools one of the atoms, and each of which images one of the atoms.

[0164] Operation Example

[0165] In yet another example, the atoms each have a fourth energy level that is higher than the first energy level and lower than the third energy level; the first laser beam is tuned to have a wavelength that is magic for the first energy level and the fourth energy level, but not magic for the third energy level, and cooling using the third laser beam is Sisyphus cooling or another form of laser cooling.

[0166] The fourth energy level is the clock state, and the transition from the first energy level to the fourth energy level is used to create a (e.g., optical) qubit in a quantum computing configuration (the ground state is the first energy level, and the excited state is the clock state). Images using the fluorescence are used to read out the state of the qubit (as well as to image / determine the occupancy of the well).

[0167] In yet another further example, the atoms each have a fifth energy level that is higher than the first energy level and lower than the second energy level. After transitioning from the first energy level to the second energy level, the electron is transferred to the fifth energy level. The first laser beam has a frequency that causes the atom including the electron in the fifth energy level to experience a trapping potential, so that the atom can transfer the electron to a third energy level that experiences an anti-trapping potential, wherein the atom will be transferred out of a (e.g., optical) well or trapping potential.

[0168] In one or more instances, the apparatus is used to perform repeated imaging of the atoms (at least 2000 imaging steps), which shows a long lifetime of the atoms under the imaging conditions. For example, the one or more second laser beams repeatedly generate fluorescence for imaging the one or more atoms, which shows that each of the atoms remains in their respective trapping potentials after at least 2000 imaging steps. In one or more instances, the imaging and cooling are performed simultaneously or alternately, and the cooling ensures that the atoms are not lost from the light trap through the imaging process.

[0169] Further Examples

[0170] 1. Example of tweezer capture of strontium

[0171] Tweezer trapping exploits the AC Stark shift

[47] , which attracts atoms to the point of maximum intensity in a tightly focused beam

[14] . We use a high-resolution objective ( Figure 1a, Appendix III) creates a single tweezer with a beam waist of w0≈500 nm in the center of an ultrahigh vacuum cell. The generation of tweezer arrays is discussed in Section 2 and here we restrict the discussion to a single tweezer. To load the tweezer, we connect it to a laser-cooled 88 Clouds of Sr atoms overlap in a narrow-line magneto-optical trap (MOT) [48, 49]. Specifically, we load the tweezers for 12 ms with a red MOT beam that is detuned toward the red by a few hundred kilohertz from the frequency used in the final stage of the red MOT. After the MOT cloud disperses, at least one atom remains in the tweezers with a probability greater than 99.95%, which corresponds to an average number of at least 7 atoms assuming a Poisson distribution for the loading statistics. Subsequently, we induce light-assisted collisions that effectively remove atomic pairs [14, 50]. Thus, the tweezers are populated with at most one atom and the observed occupancy probability is ∼50% (Appendix IV and Section 2).

[0172] For single-atom detection, we collect blue fluorescence photons while applying narrow linewidth cooling to mitigate recoil heating ( Figure 1 b, c). To this end, we implement a special type of Sisyphus cooling mechanism [45,46] that relies on the fact that excited states of narrow optical transitions are less trapped than the ground state. In contrast, resolved sideband cooling requires “magic” conditions, i.e., situations in which the ground and excited states experience the same trapping potential [32,51,52].

[0173] In our direction 3 In the narrow transitions of the P1 multicluster, we were able to achieve both conditions for different sublevels simultaneously, allowing us to study both Sisyphus cooling and sideband cooling in a single experimental setup. Specifically, we tuned 3 The polarizability of the P1 sublevel ( Figure 1 d, Appendix II). For one of these sublevels, we found a “magic angle” that equalizes the ground-state and excited-state polarizabilities, allowing sideband cooling [53,54]. The other two sublevels experience significantly weaker trapping for all polarizations, allowing Sisyphus cooling without the need for fine-tuning.

[0174] We compare our differential polarizability measurements at 515.2 nm with the theoretical model in Appendix I. We find good agreement for the ratio of the differential polarizabilities under linear polarization. This quantity provides a new benchmark for the theoretical model - being sensitive to even small changes in several matrix elements. Our theoretical model further predicts a magic crossover in linearly polarized light at a wavelength of 520(2) nm and a polarization-insensitive magic crossover at 500.65(50) nm.

[0175] 2. Example: Imaging with a single tweezer

[0176] Under typical conditions, the fluorescence signal observed on an electron multiplying charge coupled device (EMCCD) camera enables single-shot single-atom resolution detection with high fidelity. Specifically, the histogram of photons detected in a 7×7 pixel box is divided into two resolved peaks of roughly equal area: a zero-atom background peak and a single-atom peak ( Figure 2 a) These results are consistent with single atoms occupying the well in -50% of the repetition events (see also Appendix IV).

[0177] We calculated the single-shot imaging fidelity F via the accuracy of image classification. Images were classified as positive (atoms detected) or negative (atoms not detected) by choosing a threshold for the number of detected photons. The classification accuracy was defined as the fraction of images that were correctly identified. By estimating the number of false positives and false negatives, we calculated the probability of a single-shot image being correctly identified at a long imaging time limit ( Figure 2 b, Appendix V) this quantity reaches F = 99.3 (9)%. These values ​​are for a well depth of 1.4 mK. We have briefly studied imaging in shallower wells and can achieve fidelities better than 98% for wells at least as shallow as 300 μK.

[0178] While we were able to correctly identify the presence or absence of atoms with high fidelity, we found that a small fraction of atoms were lost during the imaging period. In the histogram, the losses appear as small, roughly flat distributions bridging the single-atom peak and the no-atom peak. This bridge arises from atoms that were lost before the end of the imaging period and therefore resulted in fewer scattered photons. However, we emphasize that losses during imaging do not mean that atoms were not detected, as most lost atoms still emit enough photons to reach above the classification threshold. Nonetheless, long-term imaging fidelity is ultimately limited by atoms that are lost before they can emit enough photons to be detected (Appendix V).

[0179] To quantify the loss, we take two consecutive images and assign the survival probability p of the detected atoms to s is defined as the probability of detecting an atom in the second image conditional on the fact that it was detected in the first image. Since the loss grows with imaging time, there is a trade-off between fidelity and survival fraction. As a typical number, we cite a value of p for an imaging time of ∼20 ms. s ~97% survival probability F~99% ( Figure 2 b).

[0180] Under optimized imaging conditions, we found that the experimentally observed survival probability p s Consistent with the exponential loss of scattered photons, p s≈exp(-χ·N), where N is an estimate of the number of blue photons scattered ( Figure 1 c, d and Appendix V). For example, we observe that the loss coefficient χ, defined as As a function of the imaging time during which N increases, it is constant (insert Figure 2 d). For optimized cooling parameters, we find that for blue scattering rates below ~60 kHz, χ is roughly independent of the scattering rate ( Figure 2 d). Furthermore, within this limit of low blue scattering rates, we find roughly the same χ( Figure 2 c).

[0181] These observations are consistent with the excited state 1 P1 via weak attenuation channel 1 P1→ 1 D2( Figure 1 b) The loss mechanism of the number of particles is the same. In our capture wavelength, 1 D2 is strongly resistant to confinement, so we expect atoms to be ejected faster than they can decay into triplet multiplets. Assume that all decays are 1 D2 leads to loss, χ -1 Provides attenuation feedback 1 S0 and decay are 1 The lower limit of the branching ratio between D2. We find that χ -1 In from 17(3)×10 3 to 24(4)×10 3 This lower limit agrees well with the ab initio prediction of the branching ratio of 20.5(3)×10 3 (Appendix 1). Note that, in contrast, the commonly cited branching ratio is 50×10 3

[55] . We discuss strategies for mitigating this population reduction penalty in Section 6.

[0182] We find the lowest loss coefficient χ( Figure 2 c) We simultaneously cool the atoms with 689 nm light while driving the blue transition. We observe narrow cooling features on the red-detuned side of the 689 nm free-space resonance, which we interpret as a magic-tuned transition to |φ A> sideband cooling. On the blue detuning side (where we excite non-magic transitions), there is a much broader feature, which we interpret as Sisyphus cooling (Section 4). For detuning far from the cooling feature, the loss coefficient increases as heating losses become dominant. The cooling light is provided by three counter-propagating red MOT beams, although we have observed that a single non-counter-propagating beam achieves similar fidelity in the Sisyphus regime, consistent with the interpretation that the cooling in this regime is not provided by photon recoil but by the differential potential between the ground and excited states.

[0183] 3. Example Tweezer Array

[0184] We now extend this imaging strategy to 2D tweezer arrays. At the same time, this serves as a proof of principle for generating larger 2D tweezer arrays using acousto-optic deflectors (AODs), which have previously been used for 1D arrays of up to 100 sites

[19] and 2D arrays of four sites

[56] or 16 sites

[57] . To this end, we generated a square array of 11 × 11 = 121 tweezers ( Figure 3 ac), each of which is driven by a polychromatic radio frequency (RF) signal (Appendix III). Having shown effective cooling in magic-tuned tweezers, we here instead opted for linear tweezer polarization. This choice helps maintain polarization uniformity across the array and lets us explore how the cooling signature varies with modified differential polarizability. We achieve uniform well depths across the array with a peak-to-peak variation of <5% and a standard deviation of 2% ( Figure 3 d) To achieve this level of uniformity, we start by roughly homogenizing the well depth by imaging the captured light onto a CMOS camera and feeding back the amplitude of the RF frequency. Spectroscopy of the transition achieves fine homogenization, which provides an accurate measure of the well depth due to its large differential polarizability and narrow linewidth. We ultimately use this signal as feedback to calibrate our shortcomings in imaging on CMOS, and measure uniformity after iterations are complete.

[0185] Our measured trap depth and radial trap frequency (see Section 5) are consistent with a nearly diffraction-limited tweezer waist of ∼500 nm. We additionally confirmed this value by imaging the focal plane of the trapped light with an ultra-high resolution objective. However, the observed size of our single-atom point spread function ( Figure 2 a. Figure 3 b) greater than the theoretical diffraction limited value. We suspect that thermospatial broadening, pixelation effects, color shifts between the green well and blue fluorescence, and / or aberrations in the imaging system are responsible for this. We leave this for further study as it does not directly affect the results presented in this paper.

[0186] We observe cooling signatures over the entire linearly polarized array similar to those of a single tweezer with magic polarization ( Figure 3 e). We again find a narrow red-detuned cooling feature, but closer to the red than in the magic polaritons. We expect this feature to be a combination of sideband cooling and Sisyphus cooling in the regime of more strongly trapped excited states [45,46]. The blue-detuned Sisyphus feature is also still present, although extending even further into the blue. These observations are consistent with how we expect the excited-state polarizability to shift with the ellipticity of the tweezer polaritons ( Figure 1 d). For the optimum cooling condition, we again see that the loss coefficient χ reaches the same minimum value over a wide range of settings ( Figure 3 e), albeit with values ​​higher than those observed in a single magic tweezer. We leave this observation for further investigation and, at this point, only hypothesize that it may be due in part to an altered fluorescence emission pattern due to differences in the polarization of the tweezers (Appendix V).

[0187] 4 Examples of Sisyphus Cooling

[0188] We now investigate the mechanism behind the broad, blue-detuned cooling feature observed during fluorescence imaging. This feature spans the frequency range for which there is a non-magic The local resonance condition of the transition ( Figure 4 a). Since the red transition is much narrower than the differential well depth Selective excitation of narrow equipotential multiclusters in the trap is thus possible. By appropriate choice of the detuning, atoms can lose energy by excitation on the multiclusters, where the energy of the absorbed photon is less than the energy of the photon emitted after oscillating in the excited state. This is only valid if the time the atom spends in the excited state is at least comparable to the trapping period, so the condition must also hold. Such a cooling scheme is reminiscent of Sisyphus cooling between ground-state hyperfine multiclusters of alkali metal atoms

[58] . Sisyphus cooling of the narrow-linewidth form has been discussed theoretically in the literature [45,46], albeit in the case where the excited states experience stronger trapping, which—as we describe in detail below—leads to different behavior than the case studied here, where the excited states experience weaker trapping.

[0189] We measure the equilibrium energy reached during fluorescence imaging with simultaneous Sisyphus cooling and observe a linear dependence on the detuning ( Figure 4 b). We confirmed that equilibrium was reached by also measuring the average energy as a function of imaging time and finding that it saturates after an initial linear increase ( Figure 4c). These measurements are made by probing the energy distribution via an adiabatic ramp down of the well

[59] ( Figure 4 d and Appendix VI). We quote mean energies rather than temperatures because it is not clear a priori whether the equilibrium state reached corresponds to a thermal distribution.

[0190] Our interpretation of the linear behavior of the mean energy versus detuning is as follows: When atoms scatter blue photons, they heat up, eventually reaching an energy multicluster that is resonant with the red cooling light. Here, Sisyphus cooling counteracts recoil heating. Equilibrium is reached when recoil heating pushes the energy upward, overcoming the "Sisyphus cap." Detuning that is closer to the free-space resonance, resonant with the equipotentials near the top of the well, results in a higher energy cap. Detuning that is resonant with the equipotentials deep in the well, further toward the blue of free space, results in a lower energy cap. Consistent with this interpretation, the observed mean energy is slightly below the calculated cap energy and follows it in a linear fashion.

[0191] We further observe that if the Sisyphus detuning is suddenly changed to a value closer to the blue at energy equilibrium, rapid heating and atom loss occur even when the blue fluorescence is turned off (not shown). These observations, supported by numerical simulations, paint a broader picture of the Sisyphus mechanism as a repulsive body in energy space. That is, atoms with energies below the resonant multicluster are pushed to lower energies, while atoms with energies above the resonant multicluster are heated to even higher energies. We note that we drive the transitions so that the excited state experiences weaker trapping (α e <α g ). Previous proposals for narrow-line Sisyphus cooling [45,46] have focused on the opposite regime (α e >α g ), where the Sisyphus mechanism instead acts as an attractor in energy space. The latter regime has been proposed as a mechanism for ground-state cooling, but our scheme does not follow this, since cooling stops when the atoms have been cooled to a certain energy where they are no longer resonant with the repeller; however, dynamic scanning (frequency sweeping) detuning can be achieved at very low energies, which we leave for further study.

[0192] 5 Examples of sideband cooling in a single tweezer

[0193] In this example, we show proof of principle of resolved sideband cooling in optical tweezers, thereby demonstrating direct optical control of the motional degrees of freedom of tightly trapped single atoms. Related work on Raman sideband cooling has been performed with alkali metal atoms [16,17], and narrow-line resolved sideband cooling has been previously observed with (quasi-)alkaline earth metal atoms [32,42] and trapped ions [60,61]. Here, we exploit the power of elliptically polarized tweezers tuned to the magic angle. transition. The zero differential polarizability of this transition simplifies sideband cooling and spectroscopy, since the sideband transition frequencies (up to anharmonic effects) do not depend on the state of motion. However, we do not discount the possibility of high-fidelity sideband cooling in finite-difference polarizability and leave this for future research.

[0194] because The line width of the transition (7.4kHz) is smaller than our trap frequency, and we can selectively drive the red sideband transition, which reduces the number of motional quantum states by ( Figure 5 a). Specifically, for our well depth of 1.4 mK (29 MHz), the radial (axial) well frequency is ν r =211(4)kHz(ν a = 32.2(8)kHz). Cooling depends on the tendency of the atoms to retain their quantum numbers of motion when decaying from excited states, which is the case when the Lamb-Dicke parameter η is small, i.e. For us, the radial direction has η r =0.15, and the axial direction has η a =0.39.

[0195] Before the cooling sequence begins, the atoms are imaged with Sisyphus cooling and have equilibrated at an average energy, where we expect the ground state population to be negligible (Section 4). To cool close to the moving ground state, we perform sideband cooling by alternating three beams of 100 μs pulses (two orthogonal beams in the radial plane and one beam in the axial direction, which are collimated by our objective). None of the beams are retroreflected. We divide the cooling into two stages: the first stage targets the fifth red axial sideband, while the second stage targets the first red axial sideband. Both stages target the first red radial sideband. The first stage is repeated for 100 consecutive cycles, while the second stage is repeated for 50.

[0196] To extract information about the final motion state, we Transition excitation-depletion spectroscopy ( Figure 5 b) to detect the sideband spectrum after cooling. We first use a 74μs excitation pulse to excite the ground state atoms based on Then, we used a 10 μs depletion pulse at 688 nm to 3 The S1 state will be 3 The atomic pump in P1 is 3 P0 and 3 P2 metastable dark state. This excitation-depletion cycle is repeated 3 times to increase the signal. Therefore, 3 The number of particles of P1 was measured as apparent loss during the second fluorescence imaging.

[0197] We observed sideband asymmetry after cooling ( Figure 5 c, d), which is absent before cooling (inset), thus directly demonstrating the reduction in motion energy. Similar levels of asymmetry are observed in the orthogonal radial spectra (not shown). To quantify the final motion state, we fit our data to simulate probe spectroscopy including finite decay effects (Appendix VII). We find that our data are consistent with the hot ground state fraction in the interval [0.69, 0.96] in the radial direction and [0.45, 0.59] in the axial direction. These values ​​refer to the motion state just after sideband cooling and before the probe is applied.

[0198] We finally note that we observed a small loss probability during sideband cooling and hypothesize that this may be due to the fact that when the atoms are 3 P1 is caused by off-resonance excitation from the trapped light. Such excitations can lead to losses by populating states outside our imaging and cooling cycles. Longer wavelength traps have the potential to reduce these losses by further detuning from higher states.

[0199] 6 Possible modifications and variations

[0200] We have demonstrated the trapping, high-fidelity detection, and narrow-line cooling of individual AEAs in optical tweezers. Our imaging technique is based on fluorescence imaging, while cooling is performed using a novel narrow-linewidth Sisyphus scheme.

[0201] Robust operation of the Sisyphus mechanism away from finely tuned magic conditions opens up the possibility of aiding single-atom imaging in countless situations. Specifically, this provides a viable option for cooling any atomic species with sufficiently narrow optical lines, such as other atoms or dipolar atoms, during imaging [62,63]. As a reference point, we have demonstrated high-fidelity imaging in well depths as low as 300 μK and expect to extend to even shallower depths with further optimization. We note that Sisyphus cooling can be achieved with a single beam because it relies on energy transfer from differential trapping, rather than photon momentum. This is often an advantage in such imaging applications because stray light can be minimized.

[0202] With regard to strontium itself, Sisyphus cooling could enable imaging at a variety of useful wavelengths. For example, in the presence of high-power lasers, a quantum gas microscope could operate with 1064nm light. Another interesting possibility is imaging at 813.4nm (which is Importantly, for these wavelengths, we expect 1The D2 state will be trapped so that the imaging loss caused by population reduction can be further mitigated by re-pumping in triplet and / or singlet multiplet clusters.

[0203] More broadly, the results presented open the door to a wide range of experimental possibilities that can be realized by combining OT-based single-atom control techniques with the interesting features of AEAs. For example, the unique spectral properties of AEAs are currently exploited in optical lattice clocks

[31] . Here, combining single-atom control with such high spectral resolution can be used to explore systematic shifts introduced by dipole-dipole interactions

[64] , or to achieve single-experiment staggered clock operation

[65] . Furthermore, the combination of OT with long-range interactions mediated by Rydberg states [66,67] or cavity modes

[68] can be used to controllably introduce and detect entanglement in clock transitions—a possible route to quantum-enhanced clock operation.

[0204] We further note new avenues for quantum simulation and computation. Previously, the combination of high-precision spectral control, unique spin properties [35, 69], and orbital spin exchange interactions [36, 37] has been experimentally explored and proposed in a range of AEA quantum simulation applications, including the generation of spin-orbit coupling in synthetic (composite) dimensions [38, 39] or work on Kondo-like systems [34, 70, 71]. Related ideas appear in a whole range of quantum computing protocols for AEAs [72–75]. Specifically, such quantum computing architectures require specialized single-atom control techniques, which can be implemented as originally envisioned by replacing the optical lattice with OT

[76] . In modifications to these protocols, Kondo-type models [34, 69, 71, 77] can be explored in a bottom-up manner similar to the Hubbard model [8], using either OT alone or by combining OT with a degenerate quantum gas to introduce impurities.

[0205] Furthermore, our experiments will allow control of the AEA Rydberg interaction at the single-atom level [66, 78–83], which could lead to an increase in the effective coherence time (compared to alkali metals) through the use of metastable intermediate states [66, 83]—an important aspect for further progress in Rydberg-based quantum simulations and computations.

[0206] Finally, we consider the application of OT-based strategies to fundamental atomic physics experiments. For example, we envision the controlled ionization of alkaline earth metal atoms trapped in tweezers, thus providing new avenues for the optical trapping and control of ions

[84] . In addition, we note the possibility of using optical tweezers to produce cold molecules involving AEA in an atom-by-atom manner

[86] .

[0207] Appendix to Part A

[0208] I. Example calculations of polarizability, magic wavelength, and branching ratio

[0209] I.1 Overview

[0210] The trapping potential experienced by an atom prepared in its internal state i is equal to the state-dependent polarizability of the optical tweezers The product of the intensity distribution I(r, z) makes

[0211]

[0212] where ∈0 is the vacuum dielectric constant and c is the speed of light in vacuum

[88] . Depends on the wavelength λ and polarization vector of the captured light 1 The polarizability of the S0 ground state is independent of polarization, but 3 The polarizabilities of the three sublevels of the P1 excited state depend on the polarization due to the vector and tensor components of the polarizability.

[0213] We use both recommended values ​​for the transition wavelength and dipole matrix elements and ab initio calculations. 1 S0 and 3 The polarizability of the P1 state ( Figure 6 ) (A breakdown of the calculated and recommended values ​​and their contributions to the polarizability is given in Table 1). The recommended values ​​combine theoretical calculations with experimental measurements to calculate estimates of the electric dipole matrix elements and polarizability. Under linear well polarization, we use both ab initio calculations and the recommended values ​​to predict The magic wavelength of the transition is 520(2)nm. We use the recommended value to predict Another magic wavelength of the transition is λ=500.65(50)nm.

[0214] The wavelength of our tweezers is 515.2 nm, so that for linear polarization, The capture potential ratio in the excited state 1 The trapping potential in the S0 ground state is 5% (30%) larger (smaller). We achieved the magic trapping conditions by tuning to elliptical polarization as detailed in Appendix II.

[0215] I.2 Example calculation of Sr polarizability and magic wavelength

[0216] The frequency-dependent scalar polarizability α(ω) of an atom in state i can be divided into the nuclear polarizability α 核 and the contribution from the valence electrons α v (ω). The nuclear polarizability is the ion Sr 2+The sum of the nuclear polarizability and a countervailing term that compensates for the violation of the Pauli principle for core-valence excitations from the core to the valence shell. The ionic nuclear polarizability is small and the static value calculated using the random phase approximation (RPA) gives sufficient accuracy

[89] .

[0217] The total polarizability for linear polarization is given by

[0218]

[0219] Among them J i is the total angular momentum quantum number of state i, m j is the angular momentum along the polarization axis of the tweezers The magnetic quantum number associated with the projection of s and α t are the scalar polarizability and tensor polarizability respectively. i = 1 state is given by:

[0220] For m j =0, α=α s -2α t (3)

[0221] and

[0222] For m j = ±1, α = α s +α t (4)

[0223] We calculate the valence polarizability using a hybrid method [CI+full order] that combines configuration iteration and the linearized coupled cluster method

[90] . The application of this method to polarizability calculations is described in Ref.

[91] . In short, with a total angular momentum J i and projection m j The valence part of the polarizability of state i is determined by solving the inhomogeneous equation of perturbation theory in valence space, which is approximately as follows

[92]

[0224]

[0225] Then use the wave function Ψ(v, m′ j ) has J′ i =J i ,J i The fraction of angular momentum of ±1 determines the scalar and tensor polarizabilities. Includes full-order corrections calculated using the linearized coupled cluster method with single and double excitations. The effective dipole operator D eff Includes RPA correction. This approach automatically includes contributions from all possible states.

[0226] Table 1: For 5s at 520nm and 515.2nm 2 1 S0 and 5s5p 3 Sr scalar α in P1 state s and α s The contribution to the polarizability is given in au. The corresponding energy difference ΔE (in cm -1 ) and the reduced electric dipole matrix element D (in au).

[0227]

[0228] To improve accuracy, we extract several contributions to the valence polarizability using the sum-of-states formula

[93] :

[0229]

[0230]

[0231]

[0232] where C is given by

[0233]

[0234] We used ab initio energy and matrix element calculations to calculate 1 Two such contributions to the S0 polarizability and to 3 15 contributions to the P1 polarizability, which are completely consistent with our calculations using the inhomogeneous equation (5) and determine the remaining contributions of all other states. We then perform the same calculations using the experimental energies and, when available, the recommended values ​​of the matrix elements from reference

[91] . For The recommended values ​​for the matrix elements come from 1 P1 lifetime measurement

[94] . We add the core and the remaining contributions from other states (labeled “Other” in Table 1) to these values ​​to obtain the final result.

[0235] The calculated results for 520 nm and 515.2 nm are expressed in atomic units (au), and the energy difference ΔE=E k -E i In cm -1 The absolute values ​​of the reduced electric dipole matrix elements D are listed in Table 1 as a0|e|(au), where a0 is the Bohr radius and e is the elementary charge. The nuclear and residual contributions are also listed. 1 We perform the same calculation for other wavelengths to determine 1 S0 and 3The P1 polarizability has the same value of magic wavelength. The results of the ab initio calculation and the calculation corresponding to Table 1 (recommended) are Figure 6 Shown in.

[0236] 1 We use the conventional atomic unit (au) system, where e, the electron mass, m e , and the reduced Planck constant h has a value of 1, and the electric constant ∈0 has a value of 1 / (4π). The atomic unit of α can be obtained via α / h [Hz / (V / m) 2 ]=2.48832ⅹ10 -8 α[au] is converted to SI units, where the conversion factor is 4π∈0a0 3 / h and does not take Planck's constant h into account.

[0237] I.3 Example of calculating Q value

[0238] Table 2: For Sr at 515.2 nm for 5s 2 1 S0 and 5s5p 3 Polarizability and Q values ​​(in au) for the P1 state. The recommended Q values ​​were obtained using the polarizability values ​​provided in Table 1. The Q values ​​listed in the row labeled "Experimental Energy" were obtained using experimental energies and theoretical matrix elements.

[0239]

[0240] We use the polarizability results to calculate the Q value, which is defined as the ratio of the differential polarizabilities

[0241]

[0242] Our results are summarized in Table 2. We note that varying the proposed matrix elements D by their estimated uncertainty ΔD, i.e., using the values ​​of D+ΔD and D-ΔD of the matrix elements, gives values ​​of Q=-4 and Q=-10, although 3 The P1 polarizability changes by only 2%. Q is therefore an excellent new benchmark for theoretical methodology, since it is extremely sensitive to even small changes in several matrix elements. We note that only five matrix elements, 5s5d 3 D 1,2 and 5p 2 3 P 0,1,2 The uncertainty in the value of makes a significant contribution to the uncertainty in the value of Q. We compare the theoretical Q values ​​with experimental measurements in Appendix 2C.

[0243] I.4 Example of calculating branch ratio

[0244] For the effective dipole operator, we obtain < 1 D2||D|| 1 P1>=1.956au. Including other minor corrections described in reference

[89] yields a final value of < 1 D2||D|| 1 P1>=1.92(4)au. The E1 transition rate A is determined using

[0245]

[0246] The transition wavelength λ is The line intensity S is in atomic units. 1 S0||D|| 1 P1>=5.248(2)au, we obtain

[0247] A( 1 P1→ 1 D2)=9.25(40)×10 3 s -1 (9)

[0248] A( 1 P1→ 1 S0)=1.9003(15)×10 8 s -1 (10)

[0249] The resulting ratio is

[0250]

[0251] II. Example Experiment Tuning and Polarizability Measurement

[0252] II.1 Susceptibility tuning using elliptical polarization

[0253] The dependence of the polarizability (and hence the trap depth) on the trap polarization can be obtained analytically by solving the eigenvalues ​​of the AC Stark Hamiltonian [88,95]. We start by writing the optical trapping field at a particular point in space as follows:

[0254]

[0255] in yes The complex conjugate of is the complex unit polarization vector. We will The ellipticity of is quantified by the ellipticity angle γ[54,96], which is written as

[0256]

[0257] Here, we use the unit vector A Cartesian coordinate system is defined, where Along with the optical tweezers Vector orientation. We neglect the axial component and spatial variations of the polarization due to non-paraxial effects near the focal plane

[17] . Linear polarization is given by γ = 0 and circular polarization is given by γ = π / 4.

[0258] The trapped fields act as perturbations on the bare atom Hamiltonian, resulting in energy shifts (often called AC Stark shifts or optical shifts) and mixing of electronic energy levels. Using second-order time-dependent perturbation theory, and after organizing the terms into scalar, vector, and tensor contributions, we can write the perturbations on a particular sublevel multicluster as a time-independent AC Stark Hamiltonian [88,95]:

[0259]

[0260]

[0261] where {·,·} is the transposition position, α s , α v , and α t are scalar, vector and tensor polarizabilities, g J is the Landé g factor, is the effective magnetic field (discussed below), and is a vector whose components are the angular momentum operators. Here, we restrict ourselves to 3 The P1 sublevel has multiple clusters. Therefore, in our case is a 3×3 matrix.

[0262] We define the effective magnetic field in equation (14) as

[0263]

[0264] This term (which is non-zero when the polarization has any ellipticity) causes the ) is the same perturbation of the magnetic field. Writing the Stark Hamiltonian in this way makes it easy to write Replace with To add some external real magnetic field contribution.

[0265] In the absence of an external magnetic field In the case of , the eigenvalues ​​of the Stark Hamiltonian are given by

[0266]

[0267]

[0268]

[0269] in

[0270]

[0271] is a mixing factor that depends on the vector polarizability, tensor polarizability, and ellipticity angle. The analytical formula for the corresponding eigenvector is A quantized axis of is possible and is given in non-canonical form by

[0272]

[0273]

[0274]

[0275] in

[0276]

[0277] The eigenstates are independent of the ellipticity angle, and so are their corresponding eigenvalues, while |φ B (γ)> and |φ A (γ)>The eigenstate depends on the naked light caused by the light field The polarization ellipticity caused by the mixing of sub-energy levels.

[0278] For the special case of linear polarization (γ=0), we have f(α v ,α t ; 0) = 3α t , so that the eigenvalues ​​are given by

[0279]

[0280]

[0281]

[0282] Along the propagation axis of the tweezers The unnormalized eigenvectors for the chosen quantization axis are given by

[0283]

[0284]

[0285]

[0286] A more common choice of the quantization axis (used in Appendix I) is along the tweezer polarization For this choice, in the degenerate |φ B > and |φ C It is also more convenient to choose different bases in the subspace of states, so that we can equivalently write (up to degeneracy)

[0287]

[0288]

[0289]

[0290] In the presence of an external longitudinal magnetic field In the case of , the eigenvalues ​​and eigenvectors of the Stark Hamiltonian can be obtained by replacing the vector polarizability with

[0291]

[0292] This will be observed as an asymmetry in the energy spectrum between left-handed and right-handed elliptics. We measured this asymmetry in our spectra and found it to be consistent with a longitudinal magnetic field of the order of ~15 mG. It can also be observed in the presence of a transverse magnetic field (i.e., in or The Stark Hamiltonian in the equation (in the figure) can be diagonalized, although the resulting formula is cumbersome. The transverse field leads to the otherwise degenerate |φ under linear polarization (γ = 0) B > and |φ C > Splitting of eigenstates. Within our precision we observe no such splitting and conclude that the external transverse field is sufficiently well ineffective.

[0293] II.2 Example of measuring differential well depth

[0294] We Excitation-depletion spectroscopy ( Figure 4 b) and fitting the spectral signal to the spectral lines of thermal broadening and power broadening ( Figure 7 a) to measure the differential well depth as a function of the ellipticity γ. Specifically, we assume that the spectral signal measured after n repetitions of the excitation-depletion cycle is given by S n (v) = S0·(1-p t (v)) n , where S0 is the baseline signal measured in the absence of an excitation-depletion pulse, and p t(v) is the probability of pumping an atom from the ground state into the metastable dark state after a single excitation-depletion cycle. We further assume that the transition probability is 1 The thermal energy distribution in the S0 ground state is proportional to p t (v)∝f(E(v))Θ(E(v)), where is the Boltzmann energy distribution of the three-dimensional harmonic oscillator, and Θ(E) is the Heaviside function which restricts the evaluation of this number to positive energy values.

[0295] The resonance condition of an atom at energy E can be written as Here, α e and α g are the polarizabilities of the excited and ground states, respectively. The differential well depth is ΔU, and the detuning from the free-space resonance is Δν. Importantly, when the detuning matches the differential well depth, E is zero. Therefore, the edge of the thermal profile yields the differential well depth. Using this approach, we fit the spectral signal to the thermally broadened spectral lines and extract the differential well depth ( Figure 7 ). To account for possible estimation errors associated with power broadening, we further fit the spectral signal to pure power broadened spectral lines, where g(ν) is the normalized Lorentzian function. The average of the Lorentzian fits provides a bound on the differential well depth extracted from the cutoff edge, which we Figure 1 d are used as systematic error bars. (Even in the limit of extreme power broadening, we expect the true value to be between the edge and center frequencies of the Lorentz fit.) If the saturation parameter is known precisely from independent measurements, then Fit the signal to a composite line shape.

[0296] II.3 Example comparison between measured and calculated values ​​of polarizability

[0297] We use the analytical form of the light displacement from Eqs. (17-19) to use three free parameters {α s ,α v ,α t} while fitting our experimental measurements of differential well depth ( Figure 7 b) In the absence of any or α g Under the assumption that , we can estimate the Q value defined in equation (7) as follows:

[0298]

[0299] where α c (0) = α s +α t And αA (0) = α s -2α t The measured value of Q = -5.1 (3) is consistent with the value of Q∈[-5.8, -5.1] estimated from our calculations of the polarizability (Table 2).

[0300] In addition, in the absence of or α g Without making any assumptions, we can get from Δv BA (γ) / Δv BA (0) Extraction amount |α v | / |α t |=0.10(4), where

[0301] III. Example Experiment System

[0302] Our scientific facility has two ultra-high vacuum areas: the first is a high-flux atomic beam furnace and Zeeman moderator for strontium (AOSense, Inc.) with integrated lateral cooling in a two-dimensional magneto-optical trap (MOT); the second is a large stainless steel chamber connected to a glass cell (Japan Cell) where the experiments are performed. 3 A vacuum lifetime of up to 60 s was observed in the magnetic trap loaded in the P2 state.

[0303] We used four laser systems: a blue laser system, a red laser system, a re-pumping laser system, and a green trapping laser system. The blue laser system (Toptica Photonics, TA-SHG Pro System) is a 922 nm diode laser amplified by a tapered amplifier (TA) and frequency doubled in a bow-tie second harmonic generation (SHG) cavity. The red laser system is a 689 nm diode laser (Toptica Photonics, DL pro) locked to a high finesse optical cavity (Stable Laser Systems) and amplified with a homemade TA with a maximum output power of 500 mW. The green trapping laser system has a 10 W fiber laser (Azur Light Systems) at 515.2 nm operating in free space without any additional fiber. The re-pumping laser system has a 10 W fiber laser (Azur Light Systems) for driving the green trapping laser. The transitions were performed with three diode lasers stabilized by a wave meter (HighFinesse, WS / 7).

[0304] We further split the red laser beam into three red MOT beams and three red cooling beams. The vertical and horizontal MOT beams are angled at 65° relative to the vertical axis of the glass cell to get out of the way of two vertically mounted microscope objectives, while the lateral MOT beam is aligned with the strong axis of the magnetic field gradient. The red cooling beam is oriented along the radial (R1, R2) and axial (A) directions. The two orthogonal radial cooling beams are angled at 45° relative to the horizontal axis of the glass cell. The axial cooling beam is focused at the rear exit port of the bottom objective to collimate it at the output of the objective.

[0305] We cool the atoms in a 3D MOT based on The wide dipole-allowed blue transition (λ = 460.9 nm, Γ / 2π = 30.2 MHz) is then operated based on We operate on the narrow spin-forbidden red transition (λ = 689.5 nm, Γ / 2π = 7.4 kHz). We have generated 50×10 6 The blue MOT of the atoms, which we then convert to about 10 6 To load atoms into the magic tweezers, we hold the red MOT at a frequency of 220kHz detuned from the free-space resonance to the red for 25ms, and then load the atoms in the tweezers at a frequency of 500kHz detuned from the free-space resonance to the red for 12ms. Two pairs of three counter-propagating blue and red MOT beams are overlapped with a dichroic mirror.

[0306] We used excitation-depletion spectroscopy of the red MOT (see Figure 4 b) Calibrate 7.4kHz The free-space resonant frequency of the transition. We used an excitation-depletion cycle consisting of a 40μs 689nm excitation pulse and a 10μs 688nm depletion pulse. We repeated this cycle up to five times to increase the depletion fraction without significantly disturbing the resonant feature. By sweeping the frequency of the excitation pulse in a low saturation regime, we determined the free-space resonance with a statistical error at the kHZ level. We also used this technique to eliminate stray magnetic fields by minimizing the Zeeman splitting observed in this feature.

[0307] We created a two-dimensional array of optical tweezers using two acousto-optic deflectors (Opto-Electronic, DTSX-400-515) driven by polychromatic RF waveforms generated by two independent channels of an arbitrary waveform generator (Spectrum Instrumentation Corp., M4i6622-x8). We imaged the first AOD onto the second AOD using a series of one-to-one telescopes (f=300 mm), which then imaged the second AOD onto the rear exit port of the bottom microscope objective. We stabilized the intensity of a single tweezer by monitoring the optical power after the first AOD and feeding the output signal of a servo controller (NewFocus, LB1005) into a voltage variable attenuator (VVA) that modulated the amplitude of the RF signal driving the first AOD. We used the same VVA to vary the well depth of the tweezers.

[0308] We use a transverse imaging beam oriented in the radial plane of the tweezers to scatter The atoms are imaged by using photons from the transitions. This imaging beam is not retroreflected to avoid standing waves or polarization gradients. We use two microscope objectives to collect the photons scattered by the atoms. The bottom objective (which is also used for focusing the tweezers) images the scattered photons on a single-photon sensitive EMCCD camera (ANDOR, iXon888), while the top objective collects additional photons retroreflected back by the bottom objective to improve the photon collection efficiency.

[0309] We use a combination of four possible beam paths of red laser light for Sisyphus cooling and resolved sideband cooling: a red MOT beam, a radial cooling beam (R1, R2), and an axial cooling beam (A). Although cooling can be achieved using several different beam geometries, we typically use a red MOT beam, which allows us to cool in 3D and provide essentially all polarization components; however, no retroreflected cooling beam is required for Sisyphus cooling or sideband cooling. In particular, effective Sisyphus cooling is possible using only a single beam.

[0310] IV. Example Parity Projection

[0311] We prepare single atoms in the tweezers using parity projection (PP). The initial number N of atoms loaded into the tweezers from the red MOT is assumed to follow a Poisson distribution. This is projected onto a binary distribution in such a way that even values ​​of N are projected as N=0 and odd values ​​of N are projected as N=1, resulting in pairwise losses between atoms. This method of performing PP (which is ubiquitous in experiments with alkali atoms such as quantum gas microscopy [97,98] and tweezers [14,50]) is induced by photoassociation (PA) via diatomic molecular resonances

[50] . For atom states that asymptotically correspond to 3 Such molecular resonances have been confirmed in the molecular potential of strontium for electronic excitations in the P1 state [41, 85]. The first vibrationally bound state in this potential has a binding energy of -400 kHz relative to the bare atomic resonance [41, 85].

[0312] We are The transition induces a parity projection with a 60 ms excitation pulse, which is detuned from the free space resonance by -226 kHz ( Figure 8 a). For standard loading parameters, the probability of detecting an occupied tweezer before PP is greater than 99.95%, indicating that a large number of atoms are loaded into the well on average. For long PP pulses the occupancy probability decreases and stabilizes to 0.5 ( Figure 8 a, inset), which is a feature lost in a pairwise manner. The reliable single-atom preparation is further demonstrated by the observation that the occupation probability of 0.5 after PP is robust to the initial number of loaded atoms ( Figure 8 b) We can vary said quantity by loading our MOT for a variable amount of time, resulting in variable cloud density.

[0313] A quantitative understanding of the position and width of the PA feature is beyond the scope of this work, but the resonance appears to lie between the binding energy of the molecular state at -400 kHz and the red radial motion sideband of the atoms in the trap at -211 kHz. We note that the internuclear spacing of the bound state of the molecule in free space is 27 nm

[41] and can be reduced in the tweezers due to the strong harmonic confinement. 1 S0 state and 3 The Franck-Condon overlap between bound molecular states in P1 depends strongly on the internuclear spacing between atoms in the tweezers. This spacing decreases as the atoms are cooled, so the PA rate may increase due to cooling, skewing this feature closer to the red radial motion sideband.

[0314] V. Example Fluorescence Imaging

[0315] V.1 Example Imaging Fidelity

[0316] We define imaging fidelity as the fraction of images that are correctly identified (a measure also called classification accuracy). Images are identified as positive or negative by counting the number of photons detected in a certain region of interest and comparing that number to a fixed classification threshold. We calculate fidelity by estimating the proportion of false positive and false negative identifications. These quantities depend on the choice of classification threshold, and different imaging conditions typically have different optimal choices for the threshold. For our Figure 2 For the imaging fidelity cited in b, we chose a fixed threshold for all times that was optimal for long times.

[0317] False positives are easily estimated by measuring the number of false positives in an image region near the region on which the atom is imaged. By also measuring the false positives for the atomic region when atomic loading is turned off, we confirm that this nearby region produces the same number of false positives as the atomic region.

[0318] False negatives occur when atoms do not scatter enough photons to be detected. This can occur for two obvious reasons: (1) the imaging time is too short, or (2) the atom is lost before it can scatter enough photons. False negatives due to (1) are estimated by fitting the single-atom histogram peak to a Gaussian and calculating the fitted area below the classification threshold. These types of false negatives tend to zero as the imaging time increases.

[0319] Estimating type (2) false negatives requires understanding the loss mechanisms at work. We show in the main paper that we can reach a regime where the losses are dominated by population reduction, so that the probability of loss is given by p s (N) = e -χ· N is given. In the case where χ has been measured, we can calculate χ·p s Type (2) false negatives are estimated by integrating (N) (appropriately normalized to a probability distribution) from zero to N (corresponding to our classification threshold). These false negatives depend only on the position of the threshold and are independent of the imaging time for sufficiently long times. Therefore, in a regime of long imaging times that makes type (1) false negatives negligible, optimal imaging fidelity is achieved due to the choice of a threshold that is a balance between minimizing false positives (requiring a higher threshold) and minimizing type (2) false negatives (requiring a lower threshold). If imaging is non-destructive, a fidelity of 1 can be achieved by imaging for a long time and setting the threshold high enough.

[0320] Finally, we note that imaging fidelity can be improved in post-processing by weighting the photons detected at each pixel by the relative weight of that pixel in the average point spread function. We use this technique in all the guarantees cited.

[0321] V.2 Example Collection Efficiency and Radiation Pattern

[0322] We estimated the number of scattered photons by counting the photons detected on our camera and estimating the collection efficiency of our imaging system. This estimate takes into account the 0.84 sr solid angle of our NA = 0.5 objective, the measured transmission through all optical components (0.47), the quoted quantum efficiency of our camera (0.76 at 461 nm), and the calibration of the camera gain (390) via characterization of the dark image

[99] . We measured the number of detected photons by multiplying the number of photoelectron counts by a conversion factor proportional to the gain.

[0323] The radiation pattern of the fluorescing atoms leaves a large systematic error. A naive guess is that it is a dipole pattern oriented along the polarization of the imaging beam (f(θ) = sin 2 (θ)). In this case, the collection efficiency varies by up to 7.3 times between polarization in the radial plane (best case) and polarization along the tweezer axis (worst case).

[0324] We observe that the dependence of the collection efficiency on the imaging polarization is consistent with a dipole pattern, as collection is maximal when the polarization is in the radial plane and minimal when the polarization is axial. We find that radial polarization not only maximizes detected photons, but also minimizes losses relative to detected photons, confirming that it indeed increases the collection efficiency, not just the scattering rate.

[0325] However, a complete analysis of the radiation pattern will require consideration of the projection of the imaging beam polarization onto the coordinate frame defined by the tweezers polarization and the 1 The scattering rate is estimated for each of the three non-degenerate states of P1 (each with a different radiation pattern). We abandon such an analysis and instead assume that the radiation pattern is between the spherically symmetric and the dipole pattern along the radial plane. We think this is a reasonable assumption since our imaging polarization is in the radial plane and we have confirmed that this indeed yields the best collection efficiency. The collection efficiency of the radial dipole pattern is 1.4 times higher than that of the spherically symmetric pattern. This factor is χ -1 The main source of error.

[0326] VI Example Sisyphus Cooling

[0327] We measure the energy distribution of atoms after Sisyphus cooling using the adiabatic ramp-down method [59,100]. Specifically, we measure the probability that an atom remains in the tweezers after adiabatically ramping down the tweezer depth from its nominal value U0 to some target value U≤U0. The cumulative energy distribution of the atom before ramping down, F(E / U0), is the survival probability p of the atom in the well.s (U / U0), obtained after converting the well depth U / U0 to the initial energy E / U0 of the atom using the law of conservation of action

[59] . The average energy of the atom is calculated by integrating the cumulative energy distribution.

[0328] VII Example: Sideband Temperature Measurement

[0329] Unlike Raman sideband transitions that can be driven coherently without attenuation [16,17], direct excitation to 3 The sideband transitions of P1 are inherently attenuated. This complicates the analysis because the detected sideband spectra are inevitably perturbed. Since the detected red sidebands are cooler and the detected blue sidebands are hotter, the measured spectrum exhibits an exaggerated asymmetry and a naive analysis will underestimate the temperature.

[0330] We therefore fit our measured sideband spectra to numerical simulations to extract the ground state fraction. We simulate a driven 1D quantum harmonic oscillator where the decay is achieved via quantum jumps

[101] . The Hilbert space is defined as the product space of 20 kinematic states and 2 electronic states (|g> and |e>). The non-Hermitian effective Hamiltonian is given by

[0331]

[0332]

[0333]

[0334]

[0335] Where ω is the well angular frequency, δ is the detuning, Ω is the Rabi frequency, η is the Lamb-Dicke parameter, and Γ = 2π × 7.4 kHz is 3 The decay rate of the P1 state.

[0336] The simulation is performed with a time step of Δt = 1μ. At each time step, the evolution operator is applied to the state |ψ〉. |ψ〉 is then normalized and the probability of the quantum transition is calculated as p QJ =p e ΓΔt, where p e =|<e|ψ> | 2 is the number of excited state particles. The quantum transition converts the operator Applied to |ψ>, where is the wave vector corresponding to the 689nm light in the direction sampled from the dipole pattern. Although the quantum jump operator is defined in 3 real dimensions, only its projection onto the relevant dimension is used.

[0337] We run this simulation up to the same amount of time as used for the experimental probing (74 μs). In the experiment, we use 3 such probing cycles, where at the end of each cycle we project the electronic state to the ground state or to one of the two 3 P J In the simulation, this is achieved by running the probing cycle up to 3 times, where at the end of each cycle the quantum state is replaced with probability α·p e Project to excited states, where α = 0.7 is the projection fidelity factor, which we found to be necessary to fit our data well. If the state is projected to an excited state, the simulation ends (representing losses, as measured in the experiment). If the state is instead projected to the ground state, the simulation either completes one more cycle, or ends if 3 cycles have been completed. Since there is still some baseline loss in our data, we achieve this in a post-simulation by projecting the ground state population to excited states with a probability given by our measured baseline loss.

[0338] We compare the number of excited state particles calculated in the simulation with the loss fraction measured in the experiment. Since quantum transitions are a stochastic process, we averaged over 2000 trials to obtain the final density matrix for each δ in our spectrum. The Ω used in the simulation was chosen to fit the width of the carrier peak observed in the experiment, and ω was chosen to fit the sideband frequencies.

[0339] We simulated spectra for various ground state fractions. The ground state fraction is initialized by sampling an initial quantum state |ψ(t=0)〉 from the thermal distribution of motional eigenstates. For each ground state fraction, we fit the amplitudes of the red and blue sidebands and calculate the ratio. We compare this with the ratio obtained by performing the same fit to our experimentally measured spectra and find a range of ground state fractions ( Fig. 9 ).

[0340] Measuring time is central to all science. Currently, the most precise and stable clocks are based on optical interrogation of ensembles of neutral atoms or single ions confined in optical lattices. Here we demonstrate a new optical clock system based on arrays of trapped atoms read out at the single-particle level, incorporating many of the benefits of ion and lattice clocks and building a bridge to recently developed techniques in quantum simulation and computation with neutral atoms. We use this approach to evaluate unit-site-resolved frequency shifts and systematics, as well as for atom-by-atom statistical analysis and feedback control. The system is also characterized by strongly suppressed interaction shifts and short dead times, all in a relatively simple experimental setup. This opens a new avenue for advancing stationary and mobile clock systems and provides a new starting point for entanglement-enhanced metrology and quantum clock networks, as well as for single-atom-based thermometry. The demonstrated technique also enables applications in quantum computation and communication with individual neutral atoms that require control of the optical clock state.

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[0444] Example: 2000 repetitive imaging of strontium atoms in the magic tweezers array

[0445] Optical lattice clocks of (quasi) alkaline earth atoms (AEAs) have achieved record accuracy [1,2], thus making it possible to explore fundamental physics such as geodesy [3], gravitational waves [3], and even dark matter [5]. However, while precise optical control of AEAs has been demonstrated in low-entropy arrays [2], the ability to address and detect individual atoms is currently lacking. Such single-atom control techniques would provide new avenues for optical clock systems. Specifically, they are needed to implement a myriad of quantum computing protocols for AEAs using clock states [6–10], and could provide a basis for the generation and detection of entanglement in quantum-enhanced metrology [11,12]. Optical tweezers (OT) techniques have matured into powerful tools for single-atom control; for example, they offer the versatility required for defect-free arrays, atom-by-atom assembly [13–17], and they automatically position individual atoms at distances such that interaction shifts at clock transitions are expected to be strongly reduced

[18] . Furthermore, OT experiments typically have fast experimental repetition rates and, as demonstrated below, enable repeated low-loss readout of the clock state without reloading the atoms. This technique can provide a way to tame the Dick effect for quasi-continuous interleaved clock operation

[19] . Since clock operation requires state-insensitive “magic” trapping conditions

[20] , tweezers operating at the clock’s magic wavelength are highly suitable for these directions.

[0446] Two-dimensional arrays of AEAs, particularly Sr[21,22] and Yb

[26] , have been demonstrated in optical tweezers, including single-atom-resolution imaging. Cooling during imaging has been based on narrow The inter-state combination line is carried out (see Fig.10 ), see also

[27] . For this reason, the trapping wavelength has been chosen so that the differential polarizability about this transition is small, thus enabling spectral resolution of the moving sidebands in the case of Sr [21,22], but precluding the possibility of achieving the magic trapping conditions for optical clock transitions. It is worth noting that the more versatile Sisyphus cooling mechanism

[28] has been observed for Sr atoms [21,23], providing a general approach to cooling based on narrow lines with strong polarizability mismatches. This observation combined with the description in [24,25] should allow tweezer trapping and cooling of AEAs - and more generally atoms with narrow transitions - over a wide wavelength range.

[0447] Here, we demonstrate the ability of a single clock in a magic optical tweezers array at a wavelength of 813.4 nm. 88Detection and cooling of Sr atoms [2, 12, 29-32], where losses during imaging are suppressed by two orders of magnitude compared to work on Sr [21, 22]. Specifically, for single-atom detection, we find a survival probability of 0.99932(8) and a fidelity of 0.99991(1), allowing us to perform thousands of repeated high-fidelity detections. We also observe lifetimes of more than 7 minutes under laser cooling.

[0448] These values ​​provide a benchmark for simultaneous low-loss and high-fidelity imaging in addition to the trapping lifetime of single neutral atoms, including work with alkali metals [13–17, 33–35]. We anticipate that this development will be important for improved scalability of atom-by-atom assembly schemes [13–17, 33] and for demonstrating high-fidelity quantum manipulation using neutral atoms [34, 36]. For example, the probability of success in atom-by-atom assembly is substantially governed by Limit, where p s is the combined survival probability for two images and the reordering retention time, and M is the final array size

[14] . Our work will This basic limitation has been improved to This in principle enables the assembly of arrays with thousands of atoms in terms of imaging- and vacuum-limited lifetimes. Finally, we demonstrate single-shot clock state-resolved detection with an average fidelity of 0.981(1) and an average atom survival probability of 0.996(1), which can be used for repeated clock interrogation without reloading.

[0449] Experimental technique - Single atoms are randomly loaded from a narrow-line magneto-optical trap into a tweezer array as described in detail in Ref.

[21] . In contrast to

[21] , we use 813.4 nm light to generate the tweezers ( Fig.10 a). While providing the magic wavelength for the clock transition, this wavelength also closes the loss channel observed previously, thus providing the basis for the low-loss detection demonstrated here ( Fig.10 b) [21,22]. Furthermore, the imaging scheme is simplified to a single non-retroreflected cooling beam at 689 nm and a retroreflected imaging beam at 461 nm. Both beams propagate in a plane orthogonal to the tweezer propagation axis. The cooling (imaging) beam is polarized parallel (perpendicular) to the tweezer propagation axis. We modulate the retro-mirror of the imaging beam to wash out the interference pattern

[14] . The tweezers are linearly polarized and have a depth of ≈450 μK and a waist of ≈700 nm. The array of 25 tweezers has a pitch of ≈7.4 μm and is homogenized to within ≈2%

[21] . The tweezer array is generated using a bottom objective, while a second top objective is used to image the fluorescence light on an electron multiplying charge coupled device (EMCCD) camera.

[0450] Cooling during imaging is based on a narrow-line attractive Sisyphus cooling scheme at the 7.4 kHz transition at 689 nm ( Figure 1 c), which follows the original proposal in Refs. [24,25], which has been observed in CW beam deceleration

[23] . In contrast to the repulsive Sisyphus cooling scheme demonstrated here by our group

[21] , the attractive scheme relies on the excited state experiencing a significantly stronger trapping potential compared to the ground state. For linearly polarized trapped light, this is the case for 3 P1m J = ±1 sublevels are achieved at wavelengths ranging from ≈700 nm to ≈900 nm. (For longer wavelengths including 1064 nm, repulsive Sisyphus cooling can be used.) This allows us to fine-tune the wavelength to 813.4 nm, which is very useful for 3 The clock transition of P0 is magical, and 3 The transition of P1 provides cooling conditions.

[0451] Imaging Results - Our results show simultaneous high fidelity and low loss detection of single atoms. First, we observe the histogram of photons collected within 50 ms, which has clearly resolved count distributions corresponding to the cases without atoms and with single atoms ( Fig.11 a). Taking a second image, after a 29 ms hold time under Sisyphus cooling alone, we find a survival probability of 0.99932(8). At the same time, the fidelity of the scheme (defined by the accuracy of distinguishing between no atoms and single atoms

[21] ) reaches a value of 0.99991(1), demonstrating simultaneous low-loss and high-fidelity imaging. These values ​​are achieved with lifetimes exceeding two minutes under imaging conditions ( Fig.11 b), although it only takes tens of milliseconds to obtain enough photons. We also found that the lifetime under Sisyphus cooling (without 461nm light) exceeds 7 minutes.

[0452] These results allow us to image single atoms repeatedly thousands of times. Specifically, we alternate 2000 times between 50 ms long imaging blocks and 29 ms pure cooling blocks and collect photons on the EMCCD camera in each imaging block. Recording the survival probability as a function of the number of images N, we find that even after 2000 high-fidelity images the survival fraction remains above ≈ 0.5. The decay follows the same Approximately, the exponential trend is observed, with a single image survival probability of p1≈0.9997, slightly higher than the above-cited value measured with only two images. We note that the success probability in the atom-by-atom assembly scheme is also affected by the (The factor of 2 arises due to the need for two coherent images with staggered hold times for a successful rearrangement.) Our results suggest that assembly of systems with thousands of atoms is possible in terms of imaging fidelity, loss probability, and lifetime during the assembly step.

[0453] Cooling Results - These low-loss high-fidelity results were achieved by optimizing the Sisyphus cooling frequency and picking a conservative 461nm scattering rate of ≈41kHz, as reported in the Supplementary Material (SM) and Fig.12 (a), as detailed in 12(b).

[0454] Furthermore, the absence of attractive Sisyphus cooling of the 461 nm beam leads to radial temperatures below 5 μK ( Fig.12 (c), (d)), which we cite as a conservative upper bound based on the release and recapture technique [37, 38]. This technique is primarily sensitive to the radial temperature, and it is compared to classical Monte Carlo simulations to extract the temperature. However, the comparison with classical simulations overestimates the actual temperature at energy scales close to or below the radial trapping frequency. For us, it is about 2.4 μK. More precise measurements of lower temperatures can be done via resolved sideband spectroscopy, which we leave for further work. An open question in this case is whether cooling to the moving ground state is achievable in the strongly deviating from the magic cooling configuration used here.

[0455] Since Sisyphus cooling occurs due to the trapping mismatch between the ground and excited states, it is expected that cooling occurs in all directions even for a single radial cooling beam. The low losses we observed during imaging already provide evidence for this mechanism, since fluorescence recoil heating must be mitigated in all directions. Determination of the axial temperature after cooling can be achieved via techniques such as adiabatic ramping [37,39] or spectroscopy of thermally broadened optical shifts

[21] . Our preliminary results using this technique are consistent with a three-dimensional temperature similar to the radial temperature we cite; however, we leave a thorough investigation to future work. We note that we did not explicitly attempt to cool the axial direction further, and that it might be feasible to do so by applying a beam in this direction. Finally, we note that the clock magic condition of our tweezers opens the door to well-resolved sideband thermometry based on clock transitions, which would require the sight of resolved axial sidebands that are otherwise poorly resolved on the interstate combination line at our trapping frequency.

[0456] Clock state-resolved detection - Finally, as an overview, we characterize our implementation of optical clock states 1 S0≡|g> and 3The ability to perform low-loss state-resolved readout of P0≡|e>. Our scheme relies on shelving techniques that are commonly used in ion trap experiments to achieve low-loss, high-fidelity state-resolved detection [40–42]. They are also popular in optical lattice clocks using alkaline earth metal atoms, but have not yet been extended to single-atom resolved imaging [31, 32]. More generally, low-loss state-resolved detection of single neutral atoms has only recently been achieved using alkali metal atoms [34, 35, 43–45]. Since hyperfine states are used in this case, simultaneous cooling during state-resolved detection is challenging and deep wells are therefore required. This will limit scalability and, so far, the approach has only been demonstrated for up to five wells

[35] . Note that Stern–Gerlach detection of hyperfine spins via spatial separation in the lattice has been performed [34, 44], which provides an alternative route to high-fidelity lossless state-resolved detection.

[0457] Our scheme consists of two consecutive images ( Fig.13 ). In the first image, our goal is to detect atoms in |g>. To do this, we turn off the 679nm repumping laser so that the atoms in |e> do not scatter any photons in principle. Therefore, if we find a signal in the first image, we identify the state as |g>. In the second image, we turn the 679nm repumping laser back on to detect atoms in both |g> and |e>. Therefore, if no atoms were detected in the first image, but atoms appear in the second image, we can identify it as |e>. If both images show no signal, we identify the state as "no atom".

[0458] We find that the inaccuracies of this scheme are dominated by off-resonance scattering of the tweezer light when the atoms are held in |e〉 during the first image. Specifically, by pumping the atoms into |e〉 before imaging, we observe that, at our well depth of ≈450 μK, they are radiated with a speed of τ p =470(30)ms, the decay back to |g>. This results in events in the first image where |e> atoms are misidentified as |g> atoms. To minimize the probability of misidentification, the first imaging time should be as short as possible. To reduce the imaging time, we compromise slightly on the survival probability in order to work with a higher 461 nm scattering rate in the first image (see SM). Specifically, we use t = 15 ms, a scattering rate of ≈72 kHz. The second image is taken with the same Fig.11 The same settings are performed in .

[0459] Furthermore, if atoms in |g> were pumped into |e> in the first image, they might be misidentified as |e>. This can be done by1 D2 leakage channel and subsequent scattering of 717nm photons, or via the cooling period from 3 This occurs due to off-resonance scattering of the trapping light from P1. We determine this probability of misidentification by initializing atoms in |g〉 and counting how often we identify them as |e〉 in the state-resolved imaging scheme.

[0460] In short, we We place a conservative upper bound on the probability of misidentifying |e> as |g> and directly measure the probability of misidentifying |g> as |e>, obtaining 0.008(1). We define the average state detection fidelity for a general initial state as the average of these probabilities

[34] , obtaining an average state detection fidelity of 0.981(1). In addition, we similarly define the average survival probability of the dual imaging scheme based on the measured survival probabilities of |g> and |e>, for which we obtain 0.996(1).

[0461] Our fidelity is comparable to recent measurements using alkali metal atoms in tweezers [35, 43, 45], however, to our knowledge, our survival probability is significantly higher than any tweezers- or lattice-based scheme [34, 35, 43-45]. These results constitute a good setup for continuous measurements with an optical clock. However, we emphasize that this study is not exhaustive and we expect that further optimization of these imaging parameters is possible. In general, these values ​​can be further improved by imaging in shallower tweezers or in tweezers at wavelengths that are further detuned from higher states. For example, tweezers operating at 1064 nm are a promising possibility and would be a convenient choice for operating quantum gas microscopes. Furthermore, it is possible to switch between 813.4 nm tweezers / lattice for clock interrogation (during which the well depth can be reduced by several orders of magnitude) and 1064 nm tweezers / lattice for imaging.

[0462] In summary, we have addressed two major limitations of tweezer arrays for quantum information processing and metrology based on optical clocks. By operating at the magic wavelength for clock operation, we observed imaging fidelity of 0.99991(1) and survival probability of 0.99932(8), and lifetimes of more than 7 minutes under cooling. By employing a dual imaging technique with a specific combination of repumping lasers, we investigated low-loss state-resolved detection and observed an average fidelity of 0.981(1) with an average survival probability of 0.996(1). This work provides a setup for continuous measurements with optical clocks that can suppress laser fluctuations due to the Dick effect [19,46]. The AEA-based bosonic isotopes used in this work (e.g. 88Clock operations on Sr) have been performed with Sr

[47] and Yb[48,49]. In addition, the tools developed in this work can 3 P0 is excited to a highly excited Rydberg state in the 3S1 series. Engineering long-range Rydberg-mediated interactions will facilitate the generation of entanglement between optical clock qubits, which can be used for quantum information processing

[50] , quantum simulation [51,52], and quantum enhanced metrology via spin compression

[11] .

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[0517] Part B: Devices for quantum metrology and locking lasers to atomic transitions

[0518] Instance structure

[0519] Figure 1 and Fig.10 An apparatus is shown that includes: a laser that emits one or more first laser beams that produce an array of (e.g., optical) traps that each include a trapping potential; a plurality of atoms, wherein the trapping potentials each trap only one of the atoms; and one or more second laser beams that illuminate the atoms to produce fluorescence; a detector that detects the presence or absence of the fluorescence and outputs one or more signals in response thereto; one or more third laser beams that illuminate the atoms to cool each of the atoms; one or more fourth laser beams (e.g., clock lasers) that are tuned to excite a clock transition between a first energy level and a fourth energy level; a computer / processor that uses the signal to produce an error signal; and a modulator that modulates the fourth laser beam with the error-corrected frequency to excite a clock transition between the first energy level and the fourth energy level using the fourth laser beam having the one or more error-corrected frequencies.

[0520] The atoms each have energy levels including: a first energy level; a second energy level having energy higher than the first energy level; a third energy level; and a fourth energy level higher than the first energy level and lower than the third energy level.

[0521] The second laser beam has a frequency and polarization that is tuned to excite a first (e.g., optical) transition between a first energy level and a second energy level such that the fluorescence comprises spontaneous emission from the second energy level back to the first energy level (imaged with single-atom resolution to identify atoms in each well).

[0522] Detection mechanism

[0523] The detector detects the presence or absence of the fluorescence to generate a signal representing the presence or absence of each of the atoms in the ground state (electrons in the first energy level) or the clock state (electrons in the fourth energy level), which images each of the atoms individually.

[0524] The detector detects the signal multiple times as follows:

[0525] (1) after preparing the atom to the ground state, the absence of the fluorescence indicates that the atom does not occupy the well (first signal);

[0526] (2) after being excited from the first energy level to the second energy level using a clock laser detuned from the clock transition red, such that the presence of the fluorescence indicates that the atom is in an excited state in which the electron is in the first energy level (second signal);

[0527] (3) after preparing the atom to the ground state after step (2), such that the absence of the fluorescence indicates that the atom does not occupy the well (third signal);

[0528] (4) After being excited from the first energy level to the second energy level using a clock laser detuned from the clock transition blue, the presence of the fluorescence indicates that the atom is in an excited state in which the electron is in the first energy level (fourth signal).

[0529] Error Correction

[0530] The computer / processor generates an error signal using the signal, including determining which of the wells are occupied; and for each of the occupied wells:

[0531] (1) If the signal after red detuning is higher than the signal after blue detuning, indicating that the frequency of the fourth laser beam should be increased to stimulate the clock transition, a first error is assigned,

[0532] (2) if the signal after red detuning is lower than the signal after blue detuning, indicating that the frequency of the fourth laser beam should be reduced to stimulate the clock transition, then assign a second error, and

[0533] (3) If the signals after red and blue detuning are the same, indicating that the frequency of the fourth laser beam does not need to be corrected, then zero error is assigned.

[0534] (4) Converting the error signal into one or more error-corrected frequencies.

[0535] In another example, the computer / processor generates an error signal using the signal, including determining which of the wells are occupied; and for each of the occupied wells:

[0536] (1) determining the first occupancy number of an atom in an excited state in which the electron is in the first energy level after detuning with the red, and

[0537] (2) determining the second occupation number of the atom in the excited state in which the electron is in the first energy level after excitation with blue detuning,

[0538] (3) for each of the wells, determining an error signal comprising a difference between a first occupancy number and a second occupancy number in each of the wells;

[0539] (4) Converting the error signal into one or more error-corrected frequencies.

[0540] As described herein, the modulator modulates the fourth laser beam with the error-corrected frequencies to stimulate a clock transition between the first energy level and the fourth energy level using the fourth laser beam having the one or more error-corrected frequencies.

[0541] In one or more examples, the computer averages the error signal over time and / or for each of the wells to obtain an average error signal for use in generating an error-corrected frequency including an average frequency.

[0542] In one or more examples, for each atom in the trap, an error signal is generated using the imaging of each atom, so that the frequency of the fourth laser beam that excites the clock transition is corrected for each atom.

[0543] In one or more examples, the apparatus generates an average error signal including an average of error signals of the atoms, such that a frequency of the fourth laser beam that excites the clock transition is generated by the average error signal.

[0544] In one or more examples, the device comprises a spatially resolved sensor / diagnostic. In one or more examples, the trap comprises (eg, optical) tweezers or an optical lattice or a laser trap.

[0545] Example: Atomic array optical clock with single-atom readout

[0546] Optical clocks—based on the interrogation of ultranarrow optical transitions in ions or neutral atoms—have surpassed conventional microwave clocks in both relative frequency stability and precision [1,2,3,4]. In addition to the prospective redefinition of the SI second [9], they have enabled new experiments in geodesy [5,2], fundamental physics [6,7], and quantum many-body physics [8]. At the same time, single-atom detection and control techniques have advanced quantum simulation and computing applications based on arrays of trapped atoms; in particular, ion traps

[10] , optical lattices

[11] , and optical tweezers [12,13]. Integrating such techniques into optical clocks would provide atom-by-atom error evaluation, feedback, and thermometry

[14] ; facilitate quantum metrology applications such as quantum-enhanced clocks [15,16] and clock networks

[17] ; and enable novel quantum computing, simulation, and communication architectures that require control of optical clock states in combination with single-atom trapping [18,19,20]. As for current optical clock platforms, ion clocks have introduced single-particle detection and control

[21] , but they are typically operated using only a single ion

[22] . In contrast, optical lattice clocks (OLCs) [1, 2, 4] interrogate thousands of atoms to improve short-term stability, but single-atom detection remains an outstanding challenge. In this context, an ideal clock system would therefore combine the advantages of ion clocks and lattice clocks; that is, large arrays of isolated atoms that can be individually read out and controlled.

[0547] In this example, as a major advance in this direction, we present an atomic array optical clock with single-atom resolved readout of ≈40 individually trapped neutral atoms. We exploited a magic wavelength 81-site tweezer array randomly populated with single strontium-88 ( 88 Sr) atoms

[23] , and their respective optical clock transitions are probed simultaneously for frequency stabilization [24,25]. With single-atom and unit-site resolution, we define an error signal from an arbitrary subset of the tweezers, which we use to measure position-resolved frequency shifts and systematic errors. Furthermore, atom-by-atom feedback control and statistical analysis allow us to isolate the effects of atom number dependence on clock stability. By self-comparison, we find that with 1 s integration time, the error signal is as low as 10 -15 The fractional instability in the regime is limited by the frequency noise of our portable local oscillator and is comparable to OLC using similar laser systems

[26] .

[0548] The new approach also leads to suppressed interaction displacements and short dead times—important features for precision [1] and stability [27, 28, 4], respectively. The larger interatomic spacing compared to three-dimensional (3d) OLC [25, 28] greatly reduces dipole

[29] and hopping-induced

[30] interactions, similar to tweezers, which provides immunity to on-site collisions

[31] . This fundamental absence of interaction effects enables detailed modeling using newly developed ab initio Monte Carlo (MC) simulations

[32] , which directly incorporate laser noise, projection readout, finite temperature, and feedback dynamics, leading to higher predictive power compared to traditionally used analytical methods [1]. Finally, repeated interrogation

[23] , similar to that used in ion experiments, provides a short dead time of ≈100 ms between clock interrogation blocks (much shorter than in 3d OLC). We also note recent complementary results from Ref.

[33] , which show that in a system populated with ≈4 88 Seconds-long coherence in a tweezer array of Sr atoms.

[0549] The basic functional principle is as follows. We use an acousto-optic deflector (AOD) and a high-resolution imaging system ( Fig.15 A)

[23] Tweezer arrays with linear polarization and 2.5 μm site-to-site spacing were created in an ultrahigh vacuum glass cell. As described below, the tweezer array wavelength was tuned to a magic trapping configuration close to 813.4 nm. We loaded the array from a cold atomic cloud and subsequently induced light-assisted collisions to eliminate high trap occupancies [34, 23]. As a result, ≈40 of the tweezers were randomly populated with single atoms. We cooled the atoms to The average transverse motion occupation number of is measured by clock sideband spectroscopy

[32] . The atom is then interrogated twice during the clock transition, once below the (A) resonance and once above the (B) resonance, to obtain an error signal ( ) that quantifies the frequency offset from the resonance center. Figure 1 B, C). We use this error signal to feed back to the frequency shifter to stabilize the frequency of the interrogation laser—acting as a local oscillator—to the atomic clock transition. Since our imaging scheme has a survival fraction > 0.998

[23] , we perform multiple feedback cycles, each consisting of a series of cooling, interrogation, and readout blocks, before reloading the array ( Fig.15 D).

[0550] For state-resolved readout with single-shot, single-atom resolution, we use a detection scheme consisting of two high-resolution images for each of the A and B interrogation blocks ( Fig.15E)

[23] . The first image determines whether the tweezers are occupied, followed by a clock interrogation. After the interrogation, the second image determines whether the atom has remained in the ground state |g>. This results in an error signal for all tweezers that are occupied at the beginning of both interrogation blocks, while unoccupied tweezers are ignored. For the occupied tweezers, we record the |g> occupancy number s in the images after interrogation with A and B, respectively. A,j ={0,1} and s B,j = {0,1}, where j is the tweezer index. Difference e j =s A,j -s B,j A single tweezer error variable is defined with three possible values ​​e j ={-1,0,+1}, where the values ​​indicate queries below, at, or above resonance, respectively. Note that in many queries, e j The average <e j > is just an estimate of the difference in transition probabilities between blocks A and B.

[0551] For feedback to the clock laser, e j The array average error is obtained by averaging over all occupied sites in a single AB interrogation cycle. where the sum is calculated for all occupied tweezers, and N A is the number of atoms present. We will It is added to the frequency shifter by a multiplication factor, where the magnitude of the factor is optimized to minimize in-loop noise.

[0552] We begin by describing the results of an in-loop detection sequence. Here, feedback is applied to the clock laser (as described previously), and a probe block is added after each feedback cycle that interrogates the frequency change. Using a single probe block with an interrogation time of 110 ms (corresponding to a π pulse on resonance) shows a full width at half maximum of ≈7 Hz ( Figure 1 B). We also use these parameters for the feedback interrogation module, where the A and B interrogation frequencies are separated by a total of 7.6 Hz. Using the same in-loop detection sequence, we can also directly reveal the shape of the error signal by using two subsequent detection blocks separated by this frequency difference and sweeping the common frequency offset ( Fig.15 C) The experimental results agree with the MC simulations with a systematic error, represented by the entire shaded area, arising from the uncertainty in the noise properties of the interrogating laser

[32] .

[0553] Importantly, these data also exist at the level of individual tweezers, both in terms of average and statistical fluctuations. As a first example, we show the average error signal of the repeat events for all 81 wells as a function of the frequency offset <e j >Tweezers resolution measurement results ( Fig.16 A) Fitting <e j The zero crossings of > allow us to detect differences in resonant frequencies with sub-Hz resolution ( Fig.16 B). We show that small gradients across the array due to the use of AOD:tweezers spaced 500 kHz apart in optical frequency lead to approximately linear variations in the clock transition frequency. This effect can be avoided by using spatial light modulators to generate the tweezer array

[35] . We note that the total frequency variation is smaller than the width of our interrogation signal. Such “sub-bandwidth” gradients can still cause noise through random occupation of sites with slightly different frequencies; in our case we predict an effect of 10 -17 We propose to use the local feedback correction factor from

[32] to remove this type of noise in future clock iterations.

[0554] Before we proceed, we note that e j is a three-variable probability distribution defined for each tweezer ( Figure 2 C) a random variable. Fig.16 The result in A is the average of this distribution as a function of frequency offset. In addition to such an average, a signal with perfect position resolution enables valuable statistical analysis. As an example, for an in-loop probing sequence in which the probing block is centered on a resonance, we extract Variance Changing the number of atoms considered (via post-selection) shows 1 / N A Scaling, which has an offset mainly from laser noise ( Figure 2 D) The above prefactor is dominated by quantum projection noise (QPN) [1]. A more detailed analysis reveals that for our atomic population, QPN has a significant effect on The relative noise contribution of is only ≈26%

[32] . Similar conclusions can be drawn at a qualitative level by evaluating the correlation between the tweezers resolution errors from odd and even sites, which show a strong common mode contribution ( Fig.16 E).

[0555] We now turn to the interleaved self-comparison [36,37], which we use for stability assessment and system studies. The self-comparison consists of running two feedback loops in parallel, where the feedback is given in an alternating manner to update two independent AOM frequencies f1 and f2 ( Fig.17 A). This is used for a lock-in type evaluation of the clock frequency variation under changing parameters. As a specific example, we operate the clock at our usual interrogation well depth U1 during the block fed back to f1 and at a different well depth U2 during the block fed back to f2. The average frequency difference f2-f1 now reveals the shift in the clock operating frequency depending on U2 ( Fig.17 B) For optimal clock operation, a two-lock comparison is performed on different wavelengths ( Fig.17 B)

[32] , we found an “operationally magical” condition [38,39,40] that minimizes sensitivity to well depth fluctuations. We note that this condition only holds for the average level of the tweezers.

[0556] In this context, an important question is how this lock-in technique can be extended to reveal the site-resolved systematic error as a function of varying external parameters. To this end, we use the tweezer-resolved error signal <e j > combined with interleaved self-comparison ( Fig.17 C) <e j > Convert to frequency (using a measurement error function, e.g. in Fig.16 A) generates the frequency estimates δf of each tweezer during the f1 and f2 feedback blocks, respectively. 1,j and δf 2,j These estimates correspond to the relative resonant frequency of each tweezer relative to the center frequency of the respective lock. 2,j -δf 1,j The f2-f1 plot shows the absolute frequency change of each tweezer as a function of well depth ( Fig.17 C).

[0557] We use the same self-comparison sequence to evaluate fractional clock instability by operating two lock peaks with the same conditions ( Fig.18 A) This approach follows previous clock studies where a true comparison with a second, completely independent clock system could not be obtained [36,37]. Figure 4 A middle pair The Allan deviation σ y Plot

[41] , where v0 is the clock transition frequency and The factor is introduced to take into account the addition of noise from two identical sources. The results are shown after the peak lock start time. behavior, where τ is the average time in seconds. Fitting this behavior yields It is similar to MC simulation ( Figure 4A) Very consistent. We further used our simulations to predict for single-clock operation

[0558] Self-comparison evaluates how quickly averaging can be performed for system studies - e.g. Fig.17 - and reveals the effects of various noise sources on short-term stability; however, by design, this technique suppresses slow drifts common to the f1 and f2 interrogation blocks. We performed a separate stability analysis by locking f1 to the left half of the array and f2 to the right half of the array (a method that is sensitive to slow drifts in the gradient)

[28] and found no long-term drift in the gradient within our sensitivity range

[32] .

[0559] In general, clock stability increases with the number of atoms (as ) is improved by a decrease in the readout noise. However, in the presence of laser noise—a common pattern for all atoms—there are limits to the stability even for an infinite number of atoms [1]. Interestingly, we can extract these contributions directly by performing a series of self-comparisons in which we adjust the number of atoms one by one ( Fig.17 B) To do this, we restrict the feedback operation to a subset of atoms of desired size at the center of the array, ignoring the rest. We can use typical feedback parameters for N A ≥3 to achieve a stable locking condition. We evaluate as N A The Allan variance of the function at 1 second, and the result is used to calculate the Allan variance of the function Fitting, where Convert to We found and σ ∞ =2.3×10 -15 , the latter being an estimate of the limit of our clock set by the laser noise, consistent with MC simulations.

[0560] Our results essentially combine single-particle readout and control techniques for neutral atom arrays with optical clocks based on ultranarrow spectroscopy. By increasing the interrogation time and number of atoms, this atom array optical clock (ACO) can approach the sub-1000 MHz performance of OLC. The AOC is a highly efficient and reliable optical lattice with a very low power density and a very low power density. The AOC is a very powerful lattice with a very low power density and a very low power density. The AOC is a very powerful lattice with a very low power density and a very low power density. The AOC is a very powerful lattice with a very low power density and a very low power density. The AOC is a very powerful lattice with a very low power density and a very low power density. The AOC is a very powerful lattice with a very high ...

[0561] Regarding systematics, AOC provides fully position-resolved evaluations combined with fundamental mitigation of interaction shifts, while being ready for implementation of local thermometry using Rydberg states

[14] for more precise determination of blackbody-induced shifts [1]. Furthermore, AOC provides an advanced toolset for generating and detecting entanglement to achieve operations beyond the standard quantum limit—either via cavities

[16] or via Rydberg excitations

[15] —and for realizing quantum clock networks

[17] . Furthermore, the demonstrated techniques provide a pathway for quantum computation and communication using neutral alkaline-earth metal-like atoms [18, 8, 20]. Finally, the characteristics of atomic array clocks, such as experimental simplicity, short dead time, and three-dimensional confinement, make these systems attractive candidates for robust portable clock systems and space-based missions.

[0562] Example: Supplementary material for an example atomic array optical clock with single-atom readout

[0563] 1. Experimental Details

[0564] 1.1 Experimental system

[0565] Our strontium device is described in detail in Refs. [23,34]. Sr-88 atoms from an atom beam furnace are slowed down and cooled to a few microkelvins by a 3D magneto-optical trap, which is firstly operated at 461 nm based on a broad dipole-allowed The transition operation is then based on the narrow spin-forbidden Transition operation. Strontium atoms are filled into λ T =813.4nm (which is double forbidden The magic wavelength of the optical clock transition) in a 1D array of 81 optical tweezers. The tweezers have a Gaussian beam waist radius of 800(50)nm and an array spacing of 2.5μm. During filling, cooling and imaging (state detection), the well depth is 2447(306)Er Here E r is the tweezer photon recoil energy, which is given by Given, where h is Planck's constant, and m is 88 The mass of Sr. The tweezer depth is determined from the measured beam waist and the radial trapping frequency found from sideband measurements of the clock transitions (discussed in more detail in Section 1.7). After parity projection, each tweezer has a probability of 0.5 of containing a single atom or being empty. Therefore, after each filling cycle of the experiment the total number of atoms N A follows a binomial distribution and averages the number of atoms

[0566] 1.2 Example Clock Laser System

[0567] The clock laser in this example is based on a modified portable clock laser system (StableLaser Systems) consisting of: an external cavity diode laser (Moglabs) stabilized to an isolated high-finesse optical cavity using a Pound-Drever-Hall scheme and electronic feedback of the laser diode current; and a piezoelectric transducer. The optical cavity is a 50 mm cubic cavity made of ultra-low expansion glass maintained at a zero-crossing temperature of 40.53°C with a mirror substrate made of fused quartz with a finesse of F>300,000 at 698 nm

[43] . The clock laser light passes through a first AOM in a double-pass configuration, injected into an anti-reflection coated laser diode (Sacher Lasertechnik GmbH, SAL-0705-020), through a second AOM, and through a 10 m long optical fiber to the main experiment, with a maximum output optical power of 20 mW. The first AOM is used to shift and stabilize the frequency of the clock laser, while the second AOM is used for intensity noise and fiber noise elimination. The clock laser light has a Gaussian beam waist radius of 600 μm along the tweezers array. This large width was chosen to minimize clock intensity gradients across the array caused by slight beam angle misalignments.

[0568] 1.3 Example Bosonic Clock Transition

[0569] By applying a bias magnetic field B, the boson-like alkaline earth metal atoms are promoted Optical excitation of clock transmission

[24] . This field generates 3 P1 small amount mixed into 3 P0, and leads to The Rabi frequency is , where I is the intensity of the clock-probe beam and α is the coupling constant. 88 Sr, α=198Hz / T(mW / cm 2 )1 / 2

[24] This probe beam causes an AC Stark shift Δv P = kI, where 88 Sr, k = -18mHz / (mW / cm 2 )

[24] . This magnetic field produces a second-order Zeeman shift Δv B =βB 2 , among which for 88 Sr, β = -23.3MHz / T

[24] . We choose B = 0.9mT, for which Δv B ≈-19Hz, we choose I≈2000mW / cm 2 , for which Δv P ≈-36Hz.

[0570] 1.4 Example Query Sequence

[0571] Using fluorescence imaging based on the 461nm transition for 30ms, while cooling and re-pumping the atoms out of the metastable state based on the 689nm transition 3 P 0,2 state, we confirm the presence of each atom in the tweezers. The imaging procedure is initialized in 1 We then further cooled the atoms for 10 ms using attractive Sisyphus cooling based on the 689 nm transition

[23] and ramped adiabatically down to 245(31)E r We applied a weak bias magnetic field of B = 0.9 mT along the lateral direction of the tweezer array to enable direct optical excitation of this doubly forbidden clock transition at 689 nm [24, 44]. Fig.18 ) we ramp the well depth adiabatically back to 2447(306)E r , so that no longer pumping to In the case above, 30 ms of fluorescence imaging was used to detect the number of atoms in |g>. This interrogation sequence was repeated several times before the array was refilled with atoms.

[0572] 1.5 Example Clock State Detection Fidelity

[0573] Based on the method presented in Ref.

[23] , we analyze the detection of 1 S0(|g>) and 3 The fidelity of our state detection is shown in Figure 1. We use two consecutive images to diagnose the fidelity of our state detection. In the first image, we detect atoms in |g> as follows: We pump the laser again so that the atoms in |e> remain in principle in |e> and do not scatter photons

[22] . Therefore, if we see a signal in the first image, we identify the state as |g>. In the second image, we see The pump laser is then turned back on to detect atoms in both |g> and |e>. Therefore, if no atom is detected in the first image, but an atom appears in the second image, we can identify it as |e>. If both images show no signal, we identify the state as "no atom".

[0574] The inaccuracies of this scheme are dominated by off-resonance scattering of the tweezer light when the atoms are held in |e> during the first image. By pumping the atoms into |e> before imaging, we observed that they r The depth of the imaging well is τ p =370(4)ms. This results in events in the first image where |e> atoms are misidentified as |e> atoms. Furthermore, if atoms in |g> were pumped to |e> in the first image, they could be misidentified as |e>. We measure this misidentification probability by initializing atoms in |g> and counting how often we identify them as |e>. Using this method, we set a lower bound on the probability of correctly identifying |e> as And directly measured the probability of correctly identifying |g> as f g =0.977(2). These values ​​are Figure 1 Shown as a dash in B.

[0575] 1.6 Example Stabilization to Atomic Signals

[0576] The clock laser is actively stabilized to the atomic signal using a signal control system. The frequency deviation of the clock laser from the atomic transition is determined by the frequency of the interrogation time of 110 ms in δ o It is estimated from two-point measurements of the Rabi spectral signal at / 2π = ±3.8 Hz (which corresponds to the experimentally measured spectral line shape with a full width at half maximum of 7 Hz). This is converted to a frequency correction value by multiplying it by a factor of κ = 3 Hz. We choose κ to be the largest value possible before the variance of the error signal in the in-loop detection sequence starts to grow. Feedback is done by adding this frequency correction value to the frequency of the RF synthesizer (Moglabs ARF421) driving the first AOM along the clock beam path.

[0577] 1.7 Example of Sideband Thermometry for Clock Transitions

[0578] We perform sideband thermometry on the clock transitions using the same beam that we use to interrogate atoms for clock operation. Fig.19 Using the standard technique of taking the ratio of the integrated areas under the first red and blue sidebands

[45] , we obtain the direction along the interrogating beam (oriented along one of the tight radial axes of our tweezers). By separating the sidebands, we measure the trap frequency to be ω≈2π×24.5kHz. These values ​​are obtained at a depth of 2447(306)E r Based on the narrow The transition cools down for 10 ms

[23] and ramps adiabatically down to our 245 (31) E. r The depth is measured after the clock queries it.

[0579] We note that the clock transitions are narrow enough to observe sub-kHz inhomogeneities in the trap frequency between the tweezers. This precision provided by the clock transitions allows detailed insight into inhomogeneities in the array, and we envision using it in the future for fine correction and homogenization of the array. However, for the purposes of thermometry, we broaden the clock lines to the point where these inhomogeneities are unresolved at the array average level, so that we obtain spectra that can be easily fitted and integrated. Specifically, we use a much higher magnetic field of ≈75 mT to obtain a carrier Rabi frequency of ≈360 Hz at the same light intensity.

[0580] 1.8 Example Evaluation of Allen Deviation

[0581] Repeated interrogation introduces a bimodal distribution in the times between feedback events due to the periodic refilling of the array. To account for this variation, we roughly estimate that all feedbacks are equally spaced in time by Δt ≈ 835 ms. This introduces a slight error Δτ ≈ 100 ms for all τ, although this error is insignificant for fitting the long-term Allan deviation behavior. We use Fitting is performed from τ = 10s to τ = 100s, where the free parameter A = σ y (τ=1s).

[0582] 1.9 Example Statistical Properties of Error Signals

[0583] 1.9.A Example Probability Distribution Function

[0584] In the absence of additional noise and given N A In the case of atoms, N is found after a single clock query block g The probability that an atom is in the ground state is given by the binomial distribution Given, where p is the probability of detecting an atom in its ground state after interrogation by the clock. The probability is thus determined by measuring the difference in the number of atoms The probability is given by is the number of atoms detected in the ground state after the A(B) interrogation block. It can be seen that ΔN g The probability distribution of is the convolution of two binomial distributions This discrete distribution is given in {-N A ,-N A +1,…,N A} to get 2N A +1 non-zero value supported. Therefore, The probability distribution of In the absence of a statistical correlation between the A and B query blocks, this distribution has mean and variance

[0585] 1.9.B Example of additional noise processing

[0586] In the presence of noise (such as laser noise or finite temperature), the excitation probability p A and p B These fluctuations can be explained by introducing a joint probability density function π(p A ,p B ) to explain,

[0587]

[0588] Where <·> represents the relationship between π(p A ,p B ). Assume The mean value of <p A >= <p B >≡ , and p A and p B The variances are equal, It can be seen The variance of is given by:

[0589]

[0590] Where C is p A and p B The correlation function between <p A p B >- <p A > <p B >.

[0591] 1.9.C Example Further Experimental Data

[0592] We can use the results of (2) and (4) of the effective tweezers to Figure 1 E) Direct extraction of the correlation function C. We explicitly determine that C is related to the number of atoms used in each AB interrogation cycle and extract C = -0.025. The anticorrelation is an indication of laser noise. Note that, in contrast to C, is not easily accessible experimentally because it is masked by the QPN. Figure 2 D) is fitted Therefore, we can use the fitted offset of 0.169 combined with knowledge of C to extract Instead we can use 1 / N which is 0.379 A The fitted coefficients of the terms are similar to the measured =0.41 to extract To determine the contribution of QPN relative to other noise sources in the standard deviation of the error signal, we use For N A =40.5 to get As cited in the text.

[0593] 2. Example of using single point resolution signal

[0594] Example 2.1 Stability depending on the number of atoms

[0595] To study the performance of our clock as a function of the number of atoms, we can choose to use only a portion of our entire array for clock operations ( Figure 4 B). We preferentially select atoms near the center of the array to minimize errors due to gradients in the array (e.g. from the AOD). Due to the stochastic nature of array filling, we typically use different tweezers during each filling cycle, e.g. to always count signals from a fixed number of atoms. For high target atom numbers, this sometimes results in not having enough atoms in the array, resulting in small error bars for the atom number.

[0596] Example 2.2. Clock comparison between the two halves of the array

[0597] We use the ability to lock to a subset of occupied wells to perform stability analysis that is sensitive to slow drifts in gradients in the array (e.g., from spatial variations in well uniformity or external fields). In this case, we lock f1 to wells 1-40 and f2 to wells 42-81, so that noise sources that vary across the array will show divergence in Allan deviations over sufficiently long times. Fig. 20 As shown in , we have 4 s time and as low as σ y =1×10 -16 The analysis was performed at the level and no violation was observed behavior. We therefore conclude that this temporal variation in the gradient is not a resolvable system for our current experiments. However, when using y =10 -17 This analysis will prove useful when upgrading systems where stability at the horizontal or lower level becomes an issue. In principle, this locking can be done at a single well position at a time, which would allow a well-by-well systematics analysis.

[0598] 3. Example in-situ error correction

[0599] Single-site resolution provides the opportunity to analyze single-atom signals, as discussed in the main text, and to modify these signals before they are used for feedback. As an example, the AOD introduces a spatial gradient in the trap frequency across the array, resulting in spatial variations in the zero crossings of the error signal (e.g., Figure 2 B), and subsequently leads to the y ≈10 -17 Although this effect is not currently significant in our experiments, it and other array inhomogeneities may be visible for future experiments with improved stability.

[0600] We therefore propose that this problem can be corrected (for inhomogeneities within the detection bandwidth) by: Before, the error signal e of each tweezer j This is adjusted by a correction factor, which will generate feedback to the local oscillator. For example, consider the following modification: in is the average error of the tweezers, in Hz, ζ j is the conversion factor for tweezer resolution, which can be obtained, for example, from Figure 2 A obtains, and f 0,j is the zero crossing of the error signal resolved by the tweezers. This new formulation mitigates the inhomogeneity without any physical changes to the array. While it would be ideal to physically enforce array uniformity, this is a tool that can simplify the complexity of making corrections for experimental systematics.

[0601] 4. Example Monte Carlo Simulation

[0602] 4.1 Example Operation

[0603] We compare the performance of our clock with Monte Carlo (MC) simulations that include the effects of laser frequency noise, dead time during loading and between interrogations, quantum projection noise, finite temperature, random filling of the tweezers, and experimental imperfections such as state detection distortion and atom loss.

[0604] The Rabi query is done by converting the initial state |g> into a time-dependent Hamiltonian The time evolution is simulated, where Ω is the Rabi frequency, δ o is the query offset, and δ(t) is defined such that The instantaneous frequency noise of is , where φ(t) is the optical phase in the rotating frame. The frequency noise δ(t) for each Rabi interrogation is sampled from a pre-generated noise trace (Sections 4.2, 4.3) with discrete time steps of 10 ms. The dead time between interrogations and between array refills is simulated by sampling at time-separated intervals from this noise trace. Random filling is performed by sampling the number of atoms N from a binomial distribution for each filling cycle A is achieved, and atomic loss is achieved by probabilistically reducing N between queries A To achieve.

[0605] To simulate finite temperature, N is added before each query. A Each atom is assigned a quantum number of motion n, where n is measured using our experimental (Section 1.7) sampled from a 1d thermal distribution. Here, n represents the number of quanta of motion along the axis of the interrogating clock beam. For each unique value of n sampled, The modified Rabi frequency given by

[46] is subjected to a separate Hamiltonian evolution, where is the Lamb-Dicke parameter, L n is a Laguerre polynomial of order n, and Ω is the valid bare Rabi frequency within the bounds of infinitely tight constraints.

[0606] At the end of each query, the excitation probability p is calculated for each n from the final state e (n)=| <e|ψ n >| 2 The state detection distortion is obtained by defining the excitation probability adjusted To simulate, where f g and f e are the ground state and excited state detection fidelity, respectively (Section 1.5). To simulate the readout of the jth atom at the ith interrogation, we perform a A Bernoulli test of , resulting in a binary readout s j,i By alternating δ on alternating interrogation cycles o The symbol generates an error signal every two interrogation cycles. This error signal generates a control signal (using the same gain factor as used in the experiments) which is summed with the noise trace generated for the next interrogation cycle, thus closing the feedback loop.

[0607] 4.2 Example Frequency Noise Model

[0608] The power spectral density of the frequency noise of our clock laser is calculated by -2 ), flicker frequency modulation (FFM) noise (f -1 ) and white frequency modulation (WFM) noise (f 0 ) to model the sum of the contributions of v (f) = αf -2 +βf -1 +γf 0 We obtain these parameters by an estimate of the thermal noise of our reference cavity and a fit to the partially specified frequency noise power spectral density obtained by striking our laser with the reference laser ( Fig.21 ). Since there is still large uncertainty in the white noise floor of our laser, we define a worst-case and best-case noise model. The range between these models is the main source of uncertainty in our Monte Carlo simulations.

[0609] The FFM noise is derived from the thermomechanical fluctuations of the reference cavity [48,49]. By estimating the noise contribution from the ultra-low expansion spacer, the fused quartz mirror and its reflective coating, we estimate σ at 1 s. y =1.6×10 -15 The fractional frequency instability corresponds to βf when f = 1 Hz. -1 =0.34Hz 2 / Hz frequency noise power spectral density.

[0610] As a worst-case noise model, we assume that the crossover frequency from FFM to WFM noise is at 1 Hz ( Fig.21 ), so that γ=βf -1 =0.34Hz 2 / Hz, and we estimate that the frequency noise power spectral density at 1Hz for RWFM noise is αf -2 =0.05Hz 2 / Hz. As a best-case noise model, assume that there is no crossover from FFM to WFM noise (making γ = 0.00 Hz 2 / Hz), we estimate the frequency noise power spectral density of RWFM noise at f = 1Hz to be αf -2 =0.08Hz 2 / Hz.

[0611] 5 Examples of Light Displacement Caused by Tweezers

[0612] Several previous studies have analyzed 88 The polarizabilities and hyperpolarizabilities of alkaline earth metal atoms, including Sr, in magic wavelength optical lattices [38,39,40,53]. In their analysis, these studies used the Taylor expansion of the lattice potential to express the (I is the lattice strength) to the power of 1 and include the effects of the finite-atom temperature

[53] . We repeat the derivation for optical tweezers instead of optical lattices.

[0613] The Gaussian tweezers strength (assuming azimuthal symmetry) is given by Given, where w0 is the beam waist, is the maximum intensity, P0 is the beam power, and is the Rayleigh range. The trapping potential is derived from the intensity I(ρ,z) through the electric dipole polarizability α E1 , electric quadrupole and magnetic dipole polarizability α qm =α E2 +α M1 , and the hyperpolarizability effect βI 2 Sure.

[0614] By considering the harmonic approximation in the x- and y-directions and the harmonic and anharmonic terms in the z-direction, we obtain the following expression for the differential optical displacement of the clock transition in optical tweezers, where And n ρ (=n x +n y ) and n z are the vibration quantum numbers along the radial and axial directions respectively:

[0615]

[0616] in is the differential E1 polarizability; where Δα qm is the differential E2 and M1 polarizabilities; Where Δβ is the differential hyperpolarizability; u = I / (E R / α E1 ) is the tweezers depth.

[0617] We use this formula to predict the light displacement studied in the main text ( Figure 3 ). Since we found that the results are mostly insensitive to temperature for low temperatures, we assume zero temperature for simplicity. We allow for a single fitting parameter, which is the overall frequency shift due to the uncertainty in the optical frequency of the trapped light. Other factors are taken from previous studies, as summarized in Table S1.

[0618] 88 Optical shift of the Sr clock. The fit and prediction based on Eq. 3 used the following values ​​from previous studies.

[0619]

[0620] References for Part B (Atomic Clocks)

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[0678] Method steps

[0679] Methods for trapping atoms

[0680] Fig. 22 is a flow chart illustrating a method for trapping atoms. The method comprises the following steps.

[0681] Block 2200 represents individually trapping one or more atoms, each of which is trapped in a different trapping potential or well.

[0682] Block 2202 represents imaging the one or more atoms individually. In one or more instances, the imaging reads out the occupancy of each atom in the ground state without destroying the clock state (depopulating it) (because the cooling does not affect the clock state (1-3 transition is detuned from 1-4).

[0683] Block 2204 represents cooling the atoms, wherein the atoms are each cooled to prevent loss (of each of the atoms from their respective traps) caused by the imaging. In one or more examples, the cooling counteracts multiple heating mechanisms.

[0684] 1. In one example, the method of trapping atoms includes individually trapping one or more atoms, each of the atoms being trapped in a different trapping potential or well; individually imaging each of the atoms; and individually cooling each of the atoms, wherein each of the atoms is cooled to prevent loss (of each of the atoms from their respective wells) caused by the imaging.

[0685] 2. In one embodiment of Example 1, the cooling counteracts multiple heating mechanisms.

[0686] 3. In another embodiment of example 1 or 2, the occupancy of atoms in the ground state is read out by imaging without destroying the clock state (or reducing its population) because the cooling does not affect the clock state.

[0687] Method of making a device, for example, for capturing, detecting and / or controlling the state of one or more atoms .

[0688] Fig.23 is a flow chart illustrating a method of manufacturing a device.

[0689] Block 2300 represents providing one or more lasers.

[0690] The one or more lasers emit one or more first laser beams that generate one or more trapping potentials that each trap only one of the atoms having a first energy level, a second energy level, a third energy level, and optionally a fourth energy level.

[0691] The one or more lasers emit one or more second laser beams that illuminate the one or more atoms to produce fluorescence from each of the atoms, wherein the second laser beam has a frequency and polarization that is tuned to excite a first (e.g., optical) transition between a first energy level and a second energy level such that the fluorescence includes spontaneous emission from the second energy level back to the first energy level.

[0692] The one or more lasers emit one or more third laser beams that irradiate the one or more atoms to cool each of the atoms.

[0693] Block 2302 represents operably coupling (eg, electromagnetically connecting or coupling) one or more detectors that receive the fluorescence to generate respective images of the atoms from the fluorescence.

[0694] Block 2304 represents optionally operably connecting a computer to the detector and / or the one or more lasers.

[0695] Block 2306 represents optionally connecting a modulator to the one or more lasers and / or the computer.

[0696] Block 2308 represents the end result, namely the device. In one or more examples, the laser beam includes electromagnetic radiation having multiple wavelengths. In one or more examples, an atomic (time) clock includes the device.

[0697] Block 2310 represents optionally connecting the device to an application.

[0698] Examples include, but are not limited to, the following:

[0699] 1. An apparatus for capturing, imaging and cooling one or more atoms, comprising:

[0700] one or more lasers that emit one or more first laser beams, one or more second laser beams, and one or more third laser beams;

[0701] One or more atoms, where:

[0702] The one or more first laser beams generate one or more wells each comprising a trapping potential, the trapping potentials each trapping only one of the atoms, and

[0703] The atoms each have three energy levels including:

[0704] First energy level;

[0705] A second energy level having an energy higher than the first energy level; and

[0706] The third energy level;

[0707] The one or more second laser beams illuminate the one or more atoms to generate fluorescence from each of the atoms, and the one or more second laser beams have a frequency and a polarization that are tuned to excite a first transition between a first energy level and a second energy level such that the fluorescence includes spontaneous emission from the second energy level back to the first energy level;

[0708] the one or more third laser beams irradiating the one or more atoms to cool each of the atoms; and

[0709] A detector receives the fluorescent light to generate respective images of the atoms from the fluorescent light.

[0710] 2. The device of example 1, comprising:

[0711] A first objective lens focuses the first laser beam at one or more focal points to generate each of the trapping potentials at each of the focal points.

[0712] 3. The device of example 1, wherein:

[0713] The atoms include alkaline earth metal atoms or alkaline earth metal-like atoms,

[0714] In the ground state, the atoms each include two valence electrons in a first energy level including an s-shell, which forms a spin singlet state,

[0715] In a first excited state, the atoms each include 1 valence electron at a first energy level including an s-shell and 1 valence electron at a second energy level including a p-shell, which forms a spin singlet state, and

[0716] In the second excited state, the atoms each include 1 valence electron at a first energy level of an s-shell and 1 valence electron at a third energy level including a p-shell, which forms one of three spin triplet states.

[0717] 4. The apparatus of example 1, wherein the one or more third laser beams have a wavelength tuned to induce a second transition between the first energy level and the third energy level to laser cool the atoms by transferring the atoms to a lower energy state of motion.

[0718] 5. The apparatus of example 4, wherein the laser cooling comprises Sisyphus cooling or resolved sideband cooling.

[0719] 6. The apparatus of claim 4, wherein:

[0720] The one or more third laser beams do not provide magic trapping conditions associated with the second transition such that the trapping potential experienced by atoms in a ground state in which the electrons are in the first energy level is different from the trapping potential experienced by atoms in an excited state in which at least one of the electrons is transferred to the third energy level, and

[0721] The atoms were cooled using Sisyphus cooling.

[0722] 7. The device of example 6, wherein:

[0723] The trapping potential for atoms in the ground state is higher than that for atoms in the excited state,

[0724] The one or more third laser beams are blue detuned and have a frequency greater than a transition frequency for exciting a second transition of atoms in free space (non-trapped atoms), and

[0725] The cooling is repulsive Sisyphus cooling.

[0726] 8. The device of example 6, wherein:

[0727] The capture potential for an atom in the ground state (electrons in the first energy level) is lower than the capture potential for an atom in an excited state in which one of the electrons is in the third energy level,

[0728] The one or more third laser beams are red detuned and have a frequency less than a transition frequency of a second transition for exciting atoms in free space (non-trapped atoms), and

[0729] The cooling is attractive Sisyphus cooling.

[0730] 9. The device of example 4, wherein:

[0731] (1) the one or more third laser beams are tuned to provide magic trapping conditions associated with the second transition such that the trapping potential experienced by atoms in a ground state (where the electrons are in the first energy level) is the same as the trapping potential experienced by atoms in an excited state where at least one of the electrons is transferred to the third energy level,

[0732] (2) the atom further comprises a first set of motional energy levels indexed by an integer n for an electron in the first energy level and a second set of motional energy levels indexed by an integer m for an electron in the third energy level, the third laser beam excites the atom from the nth state in the first energy level to the m=(n-1)th state in the third energy level, so that the atom decays by emitting spontaneous emission from the mth state to the (n-1)th state in the first energy level,

[0733] (3) Repeat step (2) (irradiating the atom with a third laser beam) until the atom is in the n=0th motion state in the first energy level.

[0734] 10. The device of Example 1, further comprising an array of first laser beams and / or an array of third laser beams (cooling beams) and / or an array of second laser beams (imaging beams) forming tweezers, wherein each of the tweezers captures one of the atoms, each of the cooling beams cools one of the atoms, and each of the imaging beams images one of the atoms.

[0735] 11. The apparatus of example 1, wherein the third laser beam comprises:

[0736] a laser beam propagating perpendicularly to the first laser beam to cool the atoms in a radial direction, and

[0737] A fifth laser beam propagates parallel to the first laser beam to cool the atoms in a longitudinal direction.

[0738] 12. The device of example 1, wherein:

[0739] The atoms each have a fourth energy level that is higher than the first energy level and lower than the third energy level;

[0740] The one or more first laser beams are tuned to have a wavelength that is magic for the first energy level and the fourth energy level, but not magic for the third energy level, and

[0741] The cooling using the one or more third laser beams is Sisyphus cooling.

[0742] 13. The device of example 12, wherein the fourth energy level is a clock state and the transition from the first energy level to the fourth energy level is used to create a qubit in a quantum computing configuration, and

[0743] Images obtained using the fluorescence are used to read out the state of the qubits and to determine the respective occupancy of the wells.

[0744] 14. The apparatus of any preceding example, wherein:

[0745] The atoms each have a fifth energy level that is higher than the first energy level and lower than the second energy level; wherein after transitioning from the first energy level to the second energy level, the electron transfers to the fifth energy level; and

[0746] The first laser beam has a frequency that causes an atom containing an electron in a fifth energy level to experience a trapping potential so that the atom can transfer the electron to a third energy level that experiences an anti-trapping potential, wherein the atom will be transferred out of the well or trapping potential.

[0747] 15. The apparatus of example 1, wherein the one or more second laser beams repeatedly image the one or more atoms, showing that each of the atoms remains in their respective trapping potentials after at least 2000 imaging steps.

[0748] 16. The apparatus of any of the preceding examples, wherein said one or more second laser beams and said one or more third laser beams perform said imaging and cooling simultaneously or alternately, and said cooling ensures that said atoms are not lost from the optical trap by the imaging process.

[0749] 20. An apparatus comprising:

[0750] one or more lasers that emit one or more first laser beams, one or more second laser beams, one or more third laser beams, and one or more fourth laser beams;

[0751] A number of atoms, where:

[0752] The one or more first laser beams generate an array of electromagnetic traps each comprising a trapping potential, the trapping potentials each trapping only one of the atoms, and

[0753] The atoms each have energy levels including:

[0754] First energy level;

[0755] A second energy level having a higher energy than the first energy level;

[0756] The third energy level;

[0757] a fourth energy level having an energy higher than the first energy level and an energy lower than the third energy level; and

[0758] The one or more second laser beams illuminate the atoms to produce fluorescence, and the one or more second laser beams have a frequency and a polarization that are tuned to excite a first transition between a first energy level and a second energy level, such that the fluorescence includes spontaneous emission from the second energy level back to the first energy level;

[0759] the one or more third laser beams irradiating the atoms to cool each of the atoms;

[0760] The one or more fourth laser beams (clock laser beams) are tuned to excite a clock transition between the first energy level and the fourth energy level;

[0761] a detector that detects the presence or absence of the fluorescence to generate a signal representing the presence or absence of each of the atoms in a ground state (electrons in a first energy level) or a clock state (at least one of the electrons in a fourth energy level), the detector imaging each of the atoms individually;

[0762] The detector detects the signal multiple times as follows:

[0763] (5) after preparing the atom to the ground state, causing the absence of the fluorescence to indicate that the atom does not occupy the well (first signal);

[0764] (6) after excitation from the first energy level to the second energy level using one or more fourth laser beams detuned from the clock transition red, such that the presence of the fluorescence indicates that the atom is in an excited state in which at least one of the electrons is in the first energy level (second signal);

[0765] (7) after preparing the atom to the ground state after step (2), such that the absence of the fluorescence indicates that the atom is not occupying its corresponding well (third signal);

[0766] (8) after excitation from the first energy level to the second energy level using one or more fourth laser beams detuned from clock transition blue, such that the presence of the fluorescence indicates that the atom is in an excited state in which at least one of the electrons is in the first energy level (fourth signal); and

[0767] A computer / processor that uses the signal to generate an error signal, comprising:

[0768] determining which of the wells are occupied; and

[0769] For each of the occupied wells:

[0770] determining a first occupancy number of atoms in an excited state in which electrons are in a first energy level after excitation with one or more fourth laser beams having a red detuning, and

[0771] determining a second occupancy number of atoms in an excited state in which the electrons are in the first energy level after excitation with one or more fourth laser beams having a blue detuning,

[0772] determining an error signal for each of the wells comprising a difference between a first occupancy number and a second occupancy number in each of the wells;

[0773] converting the error signal to one or more error-corrected frequencies; and

[0774] One or more modulators modulate the one or more fourth laser beams with the error-corrected frequencies to stimulate a clock transition between a first energy level and a fourth energy level using the one or more fourth laser beams having the one or more error-corrected frequencies.

[0775] 21. The apparatus of example 20, wherein the computer averages the error signal over time and / or for each of the wells to obtain an average error signal for generating an error-corrected frequency including an average frequency.

[0776] 22. An apparatus comprising:

[0777] one or more lasers that emit one or more first laser beams, one or more second laser beams, one or more third laser beams, and one or more fourth laser beams;

[0778] A number of atoms, where:

[0779] The one or more first laser beams produce an array of wells each comprising a trapping potential, the trapping potentials each trapping only one of the atoms, and

[0780] The atoms each have energy levels including:

[0781] First energy level;

[0782] A second energy level having a higher energy than the first energy level;

[0783] The third energy level;

[0784] a fourth energy level that is higher than the first energy level and lower than the third energy level; and

[0785] The one or more second laser beams illuminate the atoms to produce fluorescence, and the second laser beams have a frequency and polarization that are tuned to excite a first transition between a first energy level and a second energy level, so that the fluorescence includes spontaneous emission from the second energy level back to the first energy level, which is imaged with single-atom resolution to identify one of the atoms in each well, and

[0786] the one or more third laser beams irradiating the atoms to cool each of the atoms; and

[0787] The one or more fourth laser beams (clock laser beams) are tuned to excite a clock transition between the first energy level and the fourth energy level;

[0788] a detector that detects the presence or absence of the fluorescence to generate a signal representing the presence or absence of each of the atoms in a ground state (where the electron is in a first energy level) or a clock state (where one of the electrons is in a fourth energy level), the detector imaging each of the atoms individually;

[0789] The detector detects the signal multiple times as follows:

[0790] (1) after preparing the atom to the ground state, the absence of the fluorescence indicates that the atom does not occupy the well (first signal);

[0791] (2) after excitation from the first energy level to the second energy level using one or more fourth laser beams detuned from the clock transition red, such that the presence of the fluorescence indicates that the atom is in an excited state in which at least one of the electrons is in the first energy level (second signal);

[0792] (3) after preparing the atom to the ground state after step (2), such that the absence of the fluorescence indicates that the atom is not occupying its corresponding well (third signal);

[0793] (4) after excitation from the first energy level to the second energy level using one or more fourth laser beams detuned from the clock transition blue, such that the presence of the fluorescence indicates that the atom is in an excited state in which one of the electrons is in the first energy level (fourth signal); and

[0794] A computer / processor that uses the signal to generate an error signal, comprising:

[0795] determining which of the wells are occupied; and

[0796] For each of the occupied wells:

[0797] assigning a first error if the signal after illumination with the one or more fourth laser beams having a red detuning is higher than the signal after illumination with the one or more fourth laser beams having a blue detuning, indicating that the frequency of the one or more fourth laser beams should be increased to resonantly excite the clock transition,

[0798] assigning a second error if the signal after illumination with the one or more fourth laser beams having a red detuning is lower than the signal after illumination with the one or more fourth laser beams having a blue detuning, indicating that the frequency of the one or more fourth laser beams should be reduced to resonantly excite the clock transition,

[0799] assigning a zero error if the signals after illumination with the one or more fourth laser beams having red detuning and blue detuning are the same, indicating that the frequency of the one or more fourth laser beams does not need to be corrected;

[0800] converting the error signal to one or more error-corrected frequencies; and

[0801] A modulator modulates the one or more fourth laser beams with the error-corrected frequencies to resonantly excite a clock transition between a first energy level and a fourth energy level using the one or more fourth laser beams having the one or more error-corrected frequencies.

[0802] 23. The apparatus of example 22, wherein an error signal is generated for each atom in its respective well using respective imaging of said atom so that the frequency of the fourth laser beam exciting the clock transition is corrected for each atom.

[0803] 24. The apparatus of example 23, further comprising generating an average error signal comprising an average of the error signals of the atoms, such that a frequency of the fourth laser beam that excites the clock transition is generated from the average error signal.

[0804] 25. A spatially resolved sensor comprising the device of example 22.

[0805] 26. The device of example 22, wherein the trap comprises tweezers or an optical lattice or a laser trap.

[0806] 27. The apparatus of example 1, wherein the laser beam comprises electromagnetic radiation having a plurality of wavelengths.

[0807] 28. An atomic clock comprising the apparatus of example 1.

[0808] Method of manufacturing the device according to the second embodiment

[0809] Fig.24 A method of manufacturing a device is shown

[0810] Block 2400 represents obtaining an array of physical systems (eg, but not limited to, one or more impurities, atoms, electrons, or superconductors in a solid state) each having two energy levels that can be coupled via coherent radiation.

[0811] Block 2402 represents operatively coupling an oscillator or coherent radiation source to excite / couple a transition (eg, electrically, electromagnetically, or magnetically connect / couple the output of the oscillator to the transition). In one example, the output includes laser radiation and the oscillator is a laser.

[0812] Block 2404 represents operatively coupling (electrically, electromagnetically, or magnetically connecting / coupling) a detector that measures the probability of excitation of each physical system in the array (wherein the probability of excitation determines how well the coherent radiation excites or drives the transition or couples the two energy levels). In one or more examples, the detector includes a camera that measures absorption of laser radiation emitted from a laser.

[0813] Block 2406 represents connecting a computer to the detector. The computer converts the excitation probability into a detuning between the frequency of the coherent radiation and the resonant frequency of the transition.

[0814] Block 2408 represents connecting a modulator to the oscillator or coherent radiation source, the modulator providing feedback including the detuning to the coherent radiation, oscillator or electromagnetic radiation source so that the frequency of the coherent radiation is tuned to the respective resonant frequencies of the physical systems in the array, for example, so that the oscillator is stabilized to the transition.

[0815] Block 2410 represents the final result, such as Fig.25 The device shown in , which includes an array of physical systems (e.g., one or more impurities, atoms, electrons, superconductors in a solid state) each having two energy levels; a coherent radiation source (e.g., coherent electromagnetic radiation) that excites transitions or couples the two energy levels;

[0816] The detector measures the probability of excitation of each physical system in the array; the computer converts the excitation probability into a detuning between the frequency of coherent radiation and a resonant frequency of a transition; and the modulator provides feedback including the detuning.

[0817] In one or more instances, a sensor includes the device (eg, a gravity sensor).

[0818] Block 2412 represents a method of operating the device, which includes changing the environment of the array of the physical system (e.g., magnetic field environment, polarization environment or field, power, temperature); and measuring frequency changes in coherent radiation (e.g., from a laser) to quantify the changing environment or frequency changes relative to environmental perturbations. Methods for determining the response of trapped atoms or physical systems

[0819] Fig.26 is a flow chart illustrating a computer-implemented method. The method includes the following steps.

[0820] Block 2600 represents numerically simulating the dynamics of a plurality of atoms, the evolution of which is described by an interaction between a laser field and a plurality of atoms trapped in an array (trapped atoms), the atoms each comprising at least two energy levels, and wherein the interaction comprises a transition between the two energy levels excited by the laser field, comprising:

[0821] Obtaining a noise spectrum of the laser field;

[0822] numerically computing an error signal representing the detuning between the frequency of the laser field and the resonant frequency required to resonantly excite the transition, including solving the Schrödinger equation describing the interaction of trapped atoms with the laser field; and

[0823] A response of the trapped atoms to the laser field is calculated as a function of time, wherein the response includes the dynamics of the trapped atoms interacting with the laser field whose frequency fluctuates with time.

[0824] Block 2602 represents using the response in an application (eg, in a sensor application). In one example, the method includes using the response to stabilize the frequency of coherent radiation (including a laser field) to a resonant frequency of the transition.

[0825] Examples of computer-implemented methods include, but are not limited to, the following.

[0826] 1. A computer-implemented method comprising:

[0827] Numerically simulating the dynamics of a plurality of atoms, the evolution of which is described by an interaction between a laser field and a plurality of atoms trapped in an array (trapped atoms), the atoms each comprising at least two energy levels, and wherein the interaction comprises a transition between the two energy levels excited by the laser field, the method comprising:

[0828] Obtaining a noise spectrum of the laser field;

[0829] numerically computing an error signal representing the detuning between the frequency of the laser field and the resonant frequency required to resonantly excite the transition, including solving the Schrödinger equation describing the interaction of trapped atoms with the laser field; and

[0830] A response of the trapped atoms to the laser field is calculated as a function of time, wherein the response includes the dynamics of the trapped atoms interacting with the laser field whose frequency fluctuates with time.

[0831] 2. The method of embodiment 1, further comprising using the response to stabilize the frequency of coherent radiation (including a laser field) to a resonant frequency of the transition.

[0832] Advantages and improvements

[0833] Capture and Imaging Implementation

[0834] Embodiments of the invention described herein include new systems for imaging and cooling individual alkaline earth metal atoms trapped in optical tweezers. The system of the invention has been demonstrated by imaging and cooling individual strontium atoms in one-dimensional and two-dimensional arrays of up to 121 tweezers. The system can be used to prepare ensembles of individual alkaline earth metal atoms for a wide range of applications in quantum science and technology.

[0835] Embodiments of the present invention may be applied to the fields of quantum simulation, quantum metrology, and quantum computing using atoms, more particularly alkaline earth metal atoms. Alkaline earth metal atoms are used in the world's most accurate clocks and may be used in quantum computing. Quantum computing is one of the most important technologies currently being pursued. The purpose of quantum computing is to use quantum mechanical objects such as atoms to process information and improve performance compared to classical computers.

[0836] Embodiments of the system described herein include ultra-high vacuum systems in which alkaline earth metal atoms are captured, cooled and imaged using a combination of multiple laser beams. Individual alkaline earth metal atoms are trapped in multiple optical tweezers formed by focusing multiple laser beams through a microscope objective. Multiple laser beams are generated by sending a single laser beam through a crystal modulated by a microwave field. Individual alkaline earth metal atoms are imaged by collecting light emitted by the atoms with a microscope objective and a sensitive camera. Individual alkaline earth metal atoms are cooled using the mechanism of Sisyphus cooling and resolution sideband cooling.

[0837] Improvements over existing methods include imaging of multiple single alkaline earth metal atoms, precise determination of atomic parameters including magic wavelength and branching ratio, demonstration of a technique to tune the trapping potential using elliptical polarization, creation of a two-dimensional array of tweezers, and demonstration of two mechanisms for cooling atoms: Sisyphus cooling and resolved sideband cooling. We further demonstrate repeated low-loss, high-fidelity state-resolved detection of single atoms at the magic wavelength used for optical clock transitions.

[0838] The embodiments described herein can be used to implement a quantum computer. The present invention realizes the first important step of preparing the state of a quantum system before using quantum operations to manipulate its quantum state. Capturing alkaline earth metal atoms in a tweezer array enables the preparation of a quantum system with a large number of basic components (qubits), the positions of which can vary in space and time. Imaging single atoms provides information about their positions, which can be used to identify lost atoms and replace them with atoms from reserves. Cooling single atoms during imaging using Sisyphus cooling offsets heating and prevents the loss of atoms, which enables multiple images of bad quantum systems to be taken without destroying them. Cooling single atoms after imaging using resolved sideband cooling reduces the temperature of atoms, reducing their movement during quantum operations, so that quantum operations can be implemented with higher precision. Using elliptical polarization to adjust the capture potential enables the realization of magic capture conditions, which is necessary for suppressing inefficiency sources during quantum operations.

[0839] Quantum Metrology Applications

[0840] Optical clocks based on the interrogation of ultranarrow optical transitions in ions or neutral atoms have surpassed conventional microwave clocks in both relative frequency stability and precision. Ion clocks have introduced single-particle detection and control, but they typically operate with only a single ion. Optical lattice clocks (OLCs) interrogate thousands of atoms to improve short-term stability, but single-atom detection remains an outstanding challenge.

[0841] Embodiments of the present invention create a third optical clock platform, the atomic array optical clock (AOC), which incorporates many of the advantages of ion and lattice clocks; that is, large arrays of isolated atoms that can be individually read out and controlled. The exemplary platform described herein combines a set of highly desirable features: essentially no interaction displacement, short dead time, multi-atom operation with single-atom resolution, and relative experimental simplicity.

[0842] In particular, embodiments of the invention apply single-atom trapping and single-atom readout in tweezer arrays to ultra-narrow spectroscopy problems that stabilize local oscillators, e.g., stabilizing lasers to atomic transitions of an ensemble of alkaline earth metal atoms trapped in an optical tweezer array. It merges ultra-high precision interrogation of the clock state of neutral alkaline earth metal-like atoms with detection and control techniques for individual atoms.

[0843] In addition to describing the operation of an AOC employing single-atom readout in arrays of variable size, embodiments of the present invention further describe a novel Monte Carlo computer simulation that fully incorporates feedback dynamics, readout noise, finite temperature, and laser noise, thereby providing a more modern and predictive framework for clock modeling and analysis than previously used analytical methods.

[0844] Embodiments of the present invention will have lasting impact in at least four independent directions: 1) advance fixed and portable optical clock systems by achieving the highest levels of stability and precision while maintaining robustness and experimental simplicity; 2) provide a much more advanced platform for controlling and detecting metrology quantum enhancement through entanglement; 3) lay the experimental foundation for realizing a large number of quantum computing, simulation and networking proposals based on controlling individual alkaline earth metal atoms; 4) initiate the widespread use of the Monte Carlo modeling methods developed here, not only for AOCs, but also for other clock platforms. Each of these advances can be implemented in commercial products such as portable optical clocks with quantum enhanced performance for geodetic measurements, and benchmarking simulation packages for characterization and optimal control of optical clocks.

[0845] Processing environment

[0846] Fig. 27 An exemplary system 2700 is shown that includes a computer 2702 for implementing the processing elements required to perform the computing and control functions described herein. The computer 2702 may be a user / client computer, a server computer, or may be a database computer and may include peripheral devices.

[0847] The computer 2702 includes a hardware processor 2704A and / or a dedicated (hardware) processor 2704B (hereinafter collectively referred to as a processor, interchangeably) and a memory 2706 such as a random access memory (RAM). Typically, the computer 2702 operates under the control of an operating system 2708 stored in the memory 2706, and interacts with a user / other computer to accept input and commands (e.g., analog or digital signals), and presents results through an input / output (I / O) module 2780 or device. In one or more instances, the I / O module includes a display, a graphical user interface (GUI), a keyboard, a printer 2728, and / or a pointer / cursor control device (e.g., a mouse). The output / results may be presented on a display or provided to another device for presentation or further processing or action. For example, an image may be provided by a GUI module 2718. Although the GUI module 2718 is depicted as a separate module, the instructions for performing GUI functions may reside or be distributed in the operating system 2708, the computer program 2710, or implemented with a dedicated memory and processor.

[0848] In one or more embodiments, the computer 2702 may be connected to or may include a portable device 2732 (e.g., a cellular / mobile device, smartphone or laptop computer, multi-touch, tablet device, or other Internet-enabled device) executing on a variety of platforms and operating systems.

[0849] In one embodiment, the computer 2702 operates through a hardware processor 2704A executing instructions defined by a computer program 2712 under the control of an operating system 2708. The computer program application 2712 accesses and manipulates data stored in the memory 2706 of the computer 2702. The computer program 2712 and / or the operating system 2708 may be stored in the memory 2706 and may interact with users and / or other devices to accept input and commands, and provide output and results based on such input and commands and the instructions defined by the computer program 2712 and the operating system 2708.

[0850] Some or all of the operations performed by computer 2702 according to computer program 2712 instructions can be implemented in special purpose processor 2704B. In this embodiment, some or all of the computer program 2712 instructions can be implemented via firmware instructions stored in read-only memory, programmable read-only memory or flash memory in memory 2706 or in special purpose processor 2704B. Special purpose processor 2704B can also be hardwired by circuit design to perform some or all of the operations for implementing the present invention. In addition, special purpose processor 2704B can be a hybrid processor, which includes a dedicated circuit for performing a subset of functions, and other circuits for performing more general functions (e.g., responding to computer program 2712 instructions). In one embodiment, special purpose processor 2704B is an application specific integrated circuit (ASIC).

[0851] The computer 2702 may also implement a compiler 2714 that allows applications or computer programs 2712 written in programming languages ​​such as C, C++, Assembly, SQL, PYTHON, PROLOG, MATLAB, RUBY, RAILS, HASKELL, or other languages ​​to be translated into processor 2704 readable code. Alternatively, the compiler 2714 may be an interpreter that directly executes instructions / source code, translates source code into an intermediate representation that is executed, or executes stored precompiled code. Such source code may be written in various programming languages ​​such as JAVA, JAVASCRIPT, PERL, BASIC, etc. Upon completion, the application or computer program 2712 accesses and manipulates data received from I / O devices and stored in the memory 2706 of the computer 2702 using the relationships and logic generated using the compiler 2714.

[0852] Computer 2702 also optionally includes external communications equipment such as a modem, satellite link, Ethernet card, or other device for accepting input from other computers 2702 and providing output to other computers 2702.

[0853] In one embodiment, the instructions implementing the operating system 2708, computer program 2712, and compiler 2714 are tangibly embodied in a non-transitory computer-readable medium, such as a data storage device 2721, which may include one or more fixed or removable data storage devices, such as a zip drive, a floppy disk drive, a hard drive, a CD-ROM drive, a tape drive, etc. In addition, the operating system 2708 and the computer program 2712 consist of the following computer program 2712 instructions: when accessed, read, and executed by the computer 2702, it causes the computer 2702 to perform the steps necessary to implement and / or use the present invention, or loads the instruction program into the memory 2706, thereby creating a special data structure, which causes the computer 2702 to operate as a specially programmed computer that performs the method steps described herein. The computer program 2712 and / or operating instructions may also be tangibly embodied in the memory 2706 and / or the data communication device 2730, thereby manufacturing a computer program product or article of manufacture according to the present invention. Thus, the terms "article of manufacture," "program storage device," and "computer program product" as used herein are intended to encompass a computer program accessible from any computer-readable device or medium. In one embodiment, the special-purpose processor 2704B is an application-specific integrated circuit (ASIC). Further examples include, but are not limited to, a computer 2702 connected to or including a personal computer (e.g., a desktop computer (e.g., an HP Compaq TM ), portable or media viewing / listening devices (e.g., cellular / mobile device / phone, laptop, tablet, personal digital assistant, etc.), or integrated circuits, chips, or field programmable gate arrays (FPGAs). In yet another embodiment, computer 2702 may include a multi-touch device, gaming system, or other Internet-enabled device that executes on a variety of platforms and operating systems. In one or more instances, computer 2702 or processor 2704B includes a controller or control platform.

[0854] Those skilled in the art will recognize that many modifications may be made to this configuration without departing from the scope of the present disclosure. For example, those skilled in the art will recognize that any combination of the above components, or any number of different components, peripherals and other devices may be used.

[0855] in conclusion

[0856] This concludes the description of the preferred embodiments of the present invention. The foregoing description of one or more embodiments of the present invention has been presented for the purpose of illustration and description. It is not intended to be exhaustive or to limit the invention to the exact form disclosed. Many modifications and variations are possible in light of the above teachings. It is intended that the scope of the present invention is not limited by this detailed description, but rather by the appended claims.

Claims

1. A device comprising: A plurality of alkaline earth metal atoms or alkaline earth metal-like atoms, each of the atoms having four energy levels including: First energy level 1 S0; A second energy level higher than the first 1 P1; The third energy level 3 P J ;and The fourth energy level is higher than the first energy level and lower than the third energy level 3 P0; One or more lasers emitting one or more first laser beams, one or more second laser beams, one or more third laser beams, and one or more fourth laser beams, wherein: the one or more first laser beams generating an array of wells comprising trapping potentials, the trapping potentials each trapping only one of the atoms; The one or more first laser beams have a wavelength that is magic for the first energy level and the fourth energy level, but not magic for the third energy level; and the one or more fourth laser beams excite a clock transition between the first energy level and the fourth energy level, an imaging system including a detector that detects fluorescence generated by irradiating the atoms with the one or more second laser beams to image each of the atoms individually; a cooling system that individually cools each of the atoms using the one or more third laser beams, wherein each of the atoms is cooled to prevent loss of each of the atoms from their respective trapping potentials caused by the imaging, wherein: The one or more second laser beams have a frequency and polarization that are tuned to excite a transition between a first energy level and a second energy level such that the fluorescence includes spontaneous emission from the second energy level back to the first energy level; the detector detecting the presence or absence of the fluorescence to generate a signal representing the presence or absence of each of the atoms in a ground state in which the electrons are in a first energy level or in a clock state in which at least one of the electrons is in a fourth energy level, the detector imaging each of the atoms individually; The detector detects the signal multiple times as follows: (1) after preparing the atom to a ground state, such that the absence of the fluorescence indicates that the atom does not occupy the well; (2) after being excited from the first energy level to the second energy level using one or more fourth laser beams detuned from the clock transition red, such that the presence of the fluorescence indicates that the atom is in an excited state in which at least one of the electrons is in the first energy level; (3) after preparing the atom to a ground state after step (2), such that the absence of the fluorescence indicates that the atom is not occupying its corresponding well; (4) after excitation from the first energy level to the second energy level using one or more fourth laser beams detuned from clock transition blue, such that the presence of the fluorescence indicates that the atom is in an excited state in which at least one of the electrons is in the first energy level; and A computer / processor that uses the signal to generate an error signal, comprising: determining which of the wells are occupied; and For each of the occupied wells: determining a first occupancy number of atoms in an excited state in which electrons are in a first energy level after excitation with one or more fourth laser beams having a red detuning, and determining a second occupancy number of atoms in an excited state in which the electrons are in the first energy level after excitation with one or more fourth laser beams having a blue detuning, determining an error signal for each of the wells comprising a difference between a first occupancy number and a second occupancy number in each of the wells; converting the error signal to one or more error-corrected frequencies; and One or more modulators modulate the one or more fourth laser beams with the error-corrected frequencies to stimulate a clock transition between a first energy level and a fourth energy level using the one or more fourth laser beams having the one or more error-corrected frequencies.

2. The apparatus of claim 1, wherein: the computer averages the error signal over time and / or for each of the wells to obtain an average error signal for use in generating an error-corrected frequency comprising an average frequency, or generating one of said error signals for each atom in its respective trap using respective imaging of said atom so as to correct the frequency of the fourth laser beam exciting said clock transition for each atom, or An average error signal is generated that includes an average of the error signals of the atoms, so that a frequency of a fourth laser beam that excites the clock transition is generated from the average error signal.

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