Optical atomic telecommunications-band clock

Abstract

An optical clock including a magneto optical trap configured to cool Titanium atoms using a first set of laser beams, an optical trap configured to trap the cooled Titanium atoms in a vacuum chamber using a second set of laser beams, and a clock laser. Each laser beam of the first set of laser beam has a respective cooling wavelength correspond to a first state transition between an excited state and a first state. Each laser beam of the second set of laser beams has a respective magic trapping wavelength. The clock laser emits a clock laser beam having a clock wavelength to cause trapped Titanium atoms in the first state to transition to a second state. The clock wavelength correspond to a second state transition between the first state and the second state. The clock wavelength is in the telecom band.

Description

Attorney Docket # 00495-0014 (B2023-88) ` OPTICAL ATOMIC TELECOMMUNICATIONS-BAND CLOCK STATEMENT OF GOVERNMENT SUPPORT

[0001] This invention was made with government support under Grant Numbers W911NF- 19-1-0017 and W911NF-20-1-0266 awarded by the Army Research Office, Grant Numbers N00014-20-1-2513 and N00014-22-1-2280 awarded by the U.S. Office of Naval Research, Grant Numbers PHY-2012068, OMA-2016245 and OAC-1919839 awarded by the US National Science Foundation, and Abstract Number 1919839 awarded by the US National Science Foundation. The government has certain rights in the invention. REFERENCE TO RELATED APPLICATIONS

[0002] The present application claims priority to and the benefit of U.S. Provisional Patent Application Serial No. 63 / 532,578, filed August 14, 2023, and entitled “OPTICAL ATOMIC TELECOMMUNICATIONS-BAND CLOCK” the entire contents of which are hereby incorporated by reference herein. FIELD

[0003] The present invention generally relates to optical atomic clocks operating in the telecommunication wavelength. BACKGROUND

[0004] Optical atomic clocks have taken a giant leap in recent years, with several experiments reaching uncertainties at the 10−18level [1-3]. The comparison of clocks based on different atomic standards [4] or placed in separate locations [5] enables important applications such as relativistic geodesy [6], tests of fundamental physics [7], and dark matter searches [8]. These applications motivate the development of synchronized clock networks and transportable clocks that operate in extreme and distant environments [9].

[0005] The leading neutral-atom optical clocks operate on wavelengths of 698 nm (Sr) and 578 nm (Yb). Light at these wavelengths is strongly attenuated in optical fibers, posing a challenge to long-distance time transfer. These wavelengths are also inconvenient for constructing the ultrastable lasers that are an essential component of optical clocks.

[0006] By comparison, an optical atomic clock operating in the telecommunication wavelength band would have clear advantages. The S-, C- and L-bands, ranging altogether between about 1460 and 1625 nm, feature low losses in standard optical fibers. Stable light sources and robust optical amplifiers are also available across these ranges. These features 1 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) would support the development of fiber-linked terrestrial clock networks over continental distances.

[0007] What is needed is a way to solve inaccuracies in precision measurements introduced by magnetic dipole-dipole interactions in atomic optical clock systems. SUMMARY

[0008] In view of the above, it is an object of the present invention to provide a technological solution to address the long felt need and technological challenges faced in conventional atomic optical clock systems in which inaccuracies in precision measurements are introduced by magnetic dipole-dipole interactions by providing an optical atomic clock operating in the telecommunication wavelength, employing a cooling laser, plurality of lattice lasers, and clock lasers against Titanium atoms.

[0009] In exemplary embodiments, an optical clock includes (A) a magneto optical trap configured to cool Titanium atoms using a first set of laser beams, each having a respective cooling wavelength corresponding to a first state transition between an excited state and a first state; (B) an optical trap configured to trap the cooled Titanium atoms in a vacuum chamber using a second set of laser beams, each having a respective magic trapping wavelength; and a clock laser configured to cause trapped Titanium atoms in the first state to transition to a second state using a clock laser beam having a clock wavelength corresponding to a second state transition between the first state and the second state.

[0010] Other features and advantages of the present invention will become readily apparent from the following detailed description and the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] The above and related objects, features and advantages of the present invention will be more fully understood by reference to the following, detailed description of the preferred, albeit illustrative, embodiment of the present invention when taken in conjunction with the accompanying figures, wherein:

[0012] FIG. 1A illustrates the relevant atomic structure in Ti for an optical clock in accordance with embodiments of the present invention;

[0013] FIG 1B is a schematic diagram of a set of lattice laser beams and a clock beam in accordance with exemplary embodiments of the present invention;

[0014] FIG. 2 illustrates scalar dynamic polarizability of the mJ = 0 sublevels of the a3F4 and a5F5states in Ti from 1100 nm to 750 nm as calculated by the sum-over-states method in accordance with embodiments of the present invention; 2 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88)

[0015] FIG.3 illustrates secular dipole-dipole interaction of two atoms in the a3F4 and a5F5 clock states in accordance with embodiments of the present invention;

[0016] FIG. 4 illustrates Ramsey contrast of a clock transition as a function of the free evolution time T in accordance with embodiments of the present invention;

[0017] FIG. 5 illustrates two-atom level structure with and without ac Stark shifts applied in accordance with embodiments of the present invention;

[0018] FIGS. 6A and 6B are diagrams illustrating the beam configuration of magneto optical traps used to cool Titanium atoms in accordance with exemplary embodiments of the present invention; and

[0019] FIGs.7A-7C are diagrams illustrating the beam configuration of a set of laser beams forming an optical trap designed to trap cooled Titanium atoms in accordance with exemplary embodiments of the present invention. DETAILED DESCRIPTION

[0020] The present invention generally relates to optical atomic clocks operating in the telecommunication wavelength.

[0021] In accordance with exemplary embodiments, an optical clock is configured to cool Titanium atoms in an excited state to a first state using a first set of laser beams (e.g., to form a cloud of Titanium atoms), trap the cooled Titanium atoms in a vacuum chamber using a second set of laser beams, and cause trapped Titanium atoms in the first state to transition to a second state using a clock laser beam. In embodiments, as discussed further below, each of the first set of laser beams has a respective cooling wavelength corresponding to a first state transition between an excited state and a first state. In embodiments, as discussed further below, each of the second set of laser beams has a respective magic trapping wavelength for Titanium atoms. In embodiments, as discussed further below, the clock laser beam has a clock wavelength corresponding to the second state transition between the first state and the second state (e.g., in the telecom band).

[0022] In embodiments, the first set of beams, second set of beams, and the clock laser beam are generated using one or more cooling lasers of an magneto optical trap, one or more lasers of an optical trap, and a clock laser, respectively. In embodiments, the number, position, and orientation of the lasers may be selected based on application and other considerations, such as space and cost constraints. For example, a magneto optical trap using grating may be selected when the optical clock forms part of an integrated circuit, and a magneto optical trap 3 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) using mirrors paired with lasers may be used in a setting with fewer space constraints. For purposes of illustration, various embodiments, not intended to limit the scope of the claimed invention, are disclosed herein.

[0023] FIGS. 6A and 6B are diagrams illustrating the beam configuration of magneto optical traps used to cool Titanium atoms in accordance with exemplary embodiments of the present invention. The dotted lines in FIGs. 6A and 6B are used to illustrate the geometry of the beam arrangement.

[0024] In embodiments, the optical clock includes a magneto optical trap configured to cool Titanium atoms using a first set of laser beams. In embodiments, each laser beam of the first set of laser beams has a respective cooling wavelength corresponding to a first state transition between an excited state and a first state. In embodiments, the respective cooling wavelength corresponds to a first state transition between an excited state and a first state (e.g., a5F5). In embodiments, the cooling wavelength of a given laser beam may be between 497.9 nm and 498 and / or between 1039.9 nm and 1040.1 nm. For example, a first beam of the first set of beams may have a wavelength of 498 nm and a second beam of the first set of beams may have a wavelength of 1040 nm. In embodiments, the respective wavelength of a given laser beam is selected so that it is offset from the wavelength of another laser beam of the first set of laser beams (E.g., for laser beams oriented in an orthogonal direction, such as beams 608 and 610). In embodiments, the respective wavelength of a given laser beam is selected so that it is not offset from the wavelength of a given laser beam (e.g., for laser beams oriented along the same direction, such as laser beams 606 and 606a).

[0025] FIG. 6A is a diagram illustrating the beam configuration of a magneto optical trap with four beams (laser beam 602 and diffracted beams 604a, 604b, 604c) used to cool Titanium atoms 600 in accordance with exemplary embodiments of the present invention. A magneto optical trap (MOT) is an apparatus which uses laser cooling and a spatially-varying magnetic field (a magnetic gradient) to create a trap which can produce samples of cold, neutral atoms. In embodiments, the samples are formed in a vacuum chamber. As an example, in embodiments, the magneto optical trap includes a first cooling laser (e.g., cooling laser 610) configured to emit a first laser beam (e.g., laser beam 602) of the first set of laser beams. Continuing the example, in embodiments, the magneto optical trap includes grating (not shown) configured to diffract the first laser beam to generate a first plurality of laser beams (e.g., diffracted beams 604a, 604b, 604c, to give an example) of the first set of laser beams. As 4 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) another example, in embodiments, the first plurality of laser beams are generated using a pyramid mirror system.

[0026] FIG.6B is a diagram illustrating beam configuration of a magneto optical trap with six beams (laser beams 606, 606a, 608, 608a, 610, 610a) used to cool Titanium atoms 600 in accordance with exemplary embodiments of the present invention. In embodiments, magneto optical trap includes a plurality of cooling lasers (e.g., cooling laser 620, 622, 624), wherein each respective cooling laser of the plurality of cooling laser is configured to emit a respective laser beam (e.g., laser beam 606, 608, 610) of a set of laser beams. In embodiments, the magneto optical trap includes a plurality of mirrors, each configured to reflect a respective laser beam of the first set of laser beams emitted from a respective cooling laser beam of the plurality of cooling lasers. For example, as shown in FIG. 6B, mirror 630 is configured to form beam 606a by reflecting beam 606, mirror 634 is configured to reflect beam 610a by reflecting beam 610, and mirror 632 is configured to form beam 608a by reflecting beam 608. In embodiments, the reflected beams may be along (e.g., follow) a path of the respective laser beam which is reflected by each of the mirrors. In embodiments, as shown in FIG.6B, the plurality of cooling lasers comprises three cooling lasers and the plurality of mirrors comprises three mirrors.

[0027] In embodiments, the plurality of cooling lasers includes six cooling lasers. For example, in embodiments the three mirrors shown in FIG.6B may be replaced by three cooling lasers. In general, in embodiments, additional cooling laser and mirrors may be added to direct beams the Titanium atom 600 in order to cool it.

[0028] In embodiments, at least one of the first set of laser beams is split to form a plurality of split cooling laser beams.

[0029] In embodiments, the magneto optical trap further includes a magnetic field generator configured to generate a magnetic field gradient.

[0030] In embodiments, the optical clock does not include an magneto optical trap but instead is configured to cool Titanium atoms using one or more cooling lasers in accordance with the present invention (e.g., using what is known as optical molasses, to give an example).

[0031] FIGs.7A-7C are diagrams illustrating the beam configuration of a set of laser beams forming an optical trap designed to trap cooled Titanium atoms in accordance with exemplary embodiments of the present invention.

[0032] In exemplary embodiments, the optical clock includes an optical trap (e.g., optical tweezers 702, 1D lattice 712, 3D lattice 722) configured to trap the cooled Titanium atoms in a vacuum chamber using a second set of laser beams. In embodiments, each laser beam of the 5 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) second set of laser beams has a respective magic trapping wavelength. In embodiments, the magic trapping wavelength of a given laser beam may be between 1036 nm and 1037 nm, between 883 nm and 891 nm, between 786.8 mm and 794 nm, and / or between 774 nm and 784 nm. For example, a first beam of the second set of beams may have a wavelength of 774 nm and a second beam of the first set of beams may have a wavelength of 787 nm. In embodiments, the respective wavelength of a given laser beam may be selected so that it is offset from the wavelength of another laser beam of the second set of laser beams (e.g., laser beams oriented in orthogonally to each other, such as laser beams 720 and 722 depicted in FIG.7C, may both be near, but offset from, one another and a magical trapping wavelength). In embodiments, the respective wavelength of a given laser beam is selected so that it is not offset from the wavelength of a given laser beam (e.g., laser beams which are counter-propagating to one another, such as laser beams 714 and 716 in FIG. 7B, may be selected to have the same wavelength).

[0033] In embodiments, the second set of laser beams form optical tweezers.

[0034] FIG. 7A is a diagram illustrating a beam configuration of a set of laser beams forming optical tweezers 702 configured to trap cooled Titanium atoms in accordance with exemplary embodiments of the present invention. As shown in FIG. 7A, in embodiments, the optical clock includes focusing optics (e.g., focusing optics 704) configured to form optical tweezers from the second set of beams (e.g., beams 706). The optical tweezers operate to trap cooled Titanium atoms at one or more positions 700, allowing them to be probed by a clock laser beam 710. In embodiments, multiple wavelengths of light may be sent through the focusing optics. For example, in embodiments, a first magical trapping wavelength may be transmitted through the focusing optics for initial tweezer trapping, and then another magical trapping wavelength may be used during the actual clock operation. In embodiments, the focusing optics may also be used as a part of a detection method by collecting light emitted from the atoms at another wavelength during a transition (e.g., at one of the laser cooling wavelengths).

[0035] In embodiments, the second set of laser beams form an optical lattice. In embodiments, the optical lattice comprises a one-dimensional lattice of potential wells configured to trap Titanium atoms. In embodiments, the optical lattice comprises a two- dimensional lattice of potential wells configured to trap Titanium atoms. In embodiments, the optical lattice comprises a three-dimensional lattice of potential wells configured to trap Titanium atoms. 6 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88)

[0036] FIG. 7B is a diagram illustrating a beam configuration of a set of laser beams forming a one-dimensional lattice in accordance with exemplary embodiments of the present invention. As shown in FIG.7B, in embodiments, a one-dimensional optical lattice consists of potential wells 712 formed by the interference pattern of two counter-propagating laser beams 714 and 716 configured to trap cooled Titanium atoms. In embodiments, a clock beam 710 may be used to probe the trapped Titanium atoms. In embodiments, the first optical clock includes a first laser configured to emit a first laser beam (e.g., laser beam 714) and a second laser configured to emit a second laser beam (e.g., laser beam 716). In embodiments, the first optical clock includes a first laser configured to emit a first laser beam (e.g., laser beam 714) and a mirror configured to reflect the first laser beam to form a second laser beam (e.g., laser beam 716).

[0037] FIG. 7C is a diagram illustrating a beam configuration of a set of laser beams forming a three-dimensional lattice in accordance with exemplary embodiments of the present invention. As shown in FIG. 7C, in embodiments, a three-dimensional optical lattice consists of potential wells 718 formed by optical interference from the second set of laser beams (e.g., beams 720, 722, 724, 730, 732, and 734) configured to trap cooled Titanium atoms. In embodiments, a clock beam 710 may be used to probe the trapped Titanium atoms.

[0038] In embodiments, the optical clock includes a first optical trap laser configured to emit a first laser beam of the second set of laser beams (e.g., beam 720, to give an example), and a plurality of mirrors configured to reflect one or more of the second set of laser beams (e.g., as beams 722, 724, 730, 732, 734, to give an example). In embodiments, the first laser beam and the reflected one or more of the second set of laser beams form an optical lattice.

[0039] In embodiments, the optical clock includes a first optical trap laser, configured to emit a first laser beam (e.g., laser beam 720) of the second set of laser beams, a first mirror configured to reflect the first laser beam to form a second laser beam (e.g., laser beam 730) of the second set of laser beams, a second optical trap laser, configured to emit a third laser beam (e.g., laser beam 722) of the second set of laser beams, a second mirror configured to reflect the third laser beam to form a fourth laser beam (e.g., laser beam 732) of the second set of laser beams, a third optical trap laser, configured to emit a fifth laser beam (e.g., laser beam 724) of the second set of laser beams; and a third mirror configured to reflect the fifth laser beam to form a sixth laser beam (e.g., laser beam 734) of the second set of laser beams. In embodiments, the second set of laser beams form a three-dimensional optical lattice configured to trap Titanium atoms. In embodiments, first mirror is configured to reflect the first laser beam back 7 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) along a path of the first laser beam. In embodiments, first mirror is configured to reflect the third laser beam back along a path of the third laser beam. In embodiments, the third mirror is configured to reflect the fifth laser beam back along a path of the fifth laser beam.

[0040] In embodiments, the optical clock includes six optical trap lasers (e.g., a first, second, third, fourth, fifth, and six optical trap laser), wherein each of the plurality of lattice lasers are configured to emit a respect laser beam of the second set of laser beams (e.g., a first laser beam 720, a second laser beam 722, a third laser beam 724, a fourth laser beam 730, a fifth laser beam 732 and sixth laser beam 724). In embodiments, the first and fourth laser beam are configured to have the same magic wavelength. In embodiments, the second and fifth laser beam are configured to have the same magic wavelength. In embodiments, the third and sixth laser beam are configured to have the same magic wavelength. In embodiments, the second set of laser beams form a three-dimensional optical lattice configured to trap Titanium atoms. In embodiments, each of the optical trap lasers operate at 3.5 Watts and contain a beam waist of less .1 mm or less.

[0041] In embodiments, the second set of laser beams is configured to reduce and / or suppress magnetic interactions between the Titanium atoms.

[0042] In exemplary embodiments, the optical clock includes at least one clock laser. In embodiments, the clock laser is configured to cause trapped Titanium atoms in the first state to transition to a second state using a clock laser beam having a clock wavelength. In embodiments, the clock wavelength corresponds to a state transition between the first state and the second state. In embodiments, the clock wavelength (e.g., 1548.926nm or 1573.346nm, for example) is in the telecom band. In embodiments, the telecom band is in a short band between 1460 and 1530 nm. In embodiments, the telecom band is in a conventional band, between 1530 and 1565 nm. In embodiments, the telecom band is in the long band between 1625 nm. In embodiments, the clock wavelength is between 1260 and 1625 nm. n embodiments, the second state is a3F4. In embodiments, the third wavelength is adjusted based on measurements regarding the frequency of the second wavelength. In embodiments, the frequency of the third wavelength is configured to be substantially independent from the intensity of magic trapping light.

[0043] In embodiments, the clock frequency of the optical clock does not substantially shift as a result of black body radiation. In embodiments, the linewidth of the telecom band transitions of the telecom band is less than 60 microhertz. In embodiments, the linewidth of the 8 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) telecom band transitions of the telecom band is less than 250 microhertz. In embodiments, the linewidth of the telecom band transitions of the telecom band is less than 1 millihertz.

[0044] In embodiments, the optical clock further includes an optical detector. In embodiments, the optical detector is configured to illuminate titanium atoms. In embodiments, the optical detector is configured to measure state-dependent light scattering. In embodiments, the optical clock is configured to make adjustments (e.g., to the clock wavelength) based on measurements of the clock wavelength. In embodiments, the optical detector may include one or more photosensors, such as photodetectors, photodiodes, avalanche photodiodes, phototransistors, charge-coupled devices (CCDs), CMOS Image sensors (CIS), and photomultiplier tubes (PMTs), to give a few examples.

[0045] In embodiments, the optical clock is used in fiber-linked clock networks (e.g., terrestrial clock-networks linked by fiber cables).

[0046] In embodiments, the optical clock includes an optical frequency comb. In embodiments, the optical clock includes a second clock laser. In embodiments, the second clock laser has a second clock wavelength.

[0047] Embodiments of the present invention are directed to an optical clock based on narrow, spin-forbidden M1 and E2 transitions in laser-cooled neutral titanium. These transitions exhibit much smaller black body radiation shifts than those in alkaline earth atoms, small quadratic Zeeman shifts, and have wavelengths in the S-, C-, and L-bands of fiber-optic telecommunication standards, allowing for integration with robust laser technology. In embodiments of the present invention, lifetimes; transition matrix elements; dynamic scalar, vector, and tensor polarizabilities; and black body radiation shifts of the clock transitions are calculated using a high-precision relativistic hybrid method that combines a configuration interaction and coupled cluster approaches. Line strengths and branching ratios of the transitions used for laser cooling are also calculated. The present invention addresses challenges that arise in inaccuracies of precision measurements due to magnetic dipole-dipole interactions and describe an approach to overcome them. The technical solution described herein enables direct access to a telecommunications-band atomic frequency standard which will aid the deployment of optical clock networks and clock comparisons over long distances.

[0048] In embodiments, a stable reference for narrow linewidth optical radiation in the fiber telecom band is realized using titanium and / or ruthenium atoms. This optical atomic clock makes use of one or more of the following innovations: (1) laser cooling of titanium atoms in the metastable a5F5 atomic state on cycling transitions at the wavelengths of 498 and / or 1040 9 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) nm, (2) laser cooling of ruthenium atoms in the ground a5F5atomic state on a cycling transition at the wavelength of 350 nm, (3) state preparation in the fine-structure manifold of the a5F or a3F terms of titanium and / or ruthenium, (4) establishment of internal-state atomic coherence at the telecom frequency range based on superpositions of the a5F and a3F transitions within titanium and / or ruthenium, including (5) use of a two-photon transition (one telecon-frequency photon (E2 or M1) + 1 radio or microwave transition photon) to drive transitions between levels that both have magnetic quantum numbers mJ of zero, (6) use of aforementioned cycling transitions for readout detection of the atomic clock state, (7) use of magic-wavelength optical potentials that trap titanium atoms while providing zero light shift on the clock transition (mJ=0 to m'J=0), and (8) use of magic trapping wavelength optical potentials that also provide a tensor light shift that cancels out effects of magnetic-dipole interactions in titanium or ruthenium. Here, telecom frequency bands are defined as infrared optical radiation spanning the wavelength range between 1260 nm and 1625 nm, comprising the O, E, S, C, and L bands. Several transitions between fine structure states of the a5F and of the a3F terms fall in this frequency range, and are connected by E2 and M1 optical transitions. Techniques disclosed in connection with the present invention are applicable to atomic optical clocks based on any of these transitions using any of the innovations listed above and as described herein.

[0049] FIG.1A illustrates the relevant atomic structure in Ti for an optical clock. The a3F (state 104) and a5F (state 106) terms serve as the basis for the optical clock, while the excited y5Go6 level (e.g., state 102) serves as the excited state for laser cooling of Ti. The two optical clock transitions highlighted in the text are shown as arrows 108 and 110; the laser cooling transition is shown as 112. FIG.1B is a schematic diagram of a set of lattice laser beams and a clock beam in accordance with exemplary embodiments of the present invention. As shown in FIG. 1B, in embodiments, an optical clock includes a titanium atom, or cloud of atoms, 120, along with six lattice beams 122, and clock beam 124. Polarizations are indicated on a given beam by a small arrow on the diagram of the beam itself.

[0050] Numerous transition-metal elements, including Ti, can be laser-cooled on near- cycling optical transitions

[0013] , allowing for the adoption of optical lattice or tweezer trapping techniques

[0014] used in leading neutral-atom clocks. Several transitions between the 3d24s2a3F and 3d3(4F)4s a5F fine structure manifolds in Ti with transition wavelengths between 1483 and 1610 nm can serve as optical clock references for ultra-stable telecommunication-band light sources. 10 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) J J’ λ (nm) Tele. DM1 DE2 Γ Band (10-3µB) (a.u.) (10-6s-1) 4 5 1548.926 C 1.0(5) 0.140(4) 242(5) 4 4 1573.346 L 0.36(18) 0.134(8) 239(5) ) ) ) ) ) ) )

[0051] Tus ra es a s o op ca coc rans ons n . e wo cock transitions focused on herein are in bold. All transitions are between the lower (upper) a3F (a5F) terms. The lower (upper) states are indexed by J (J')

[0019] . The telecom band is indicated, with S (short), C (conventional) and L (long) bands noted. M1, E2 reduced matrix elements DM1, DE2 and transition linewidths are calculated. Transitions between the a3F and a5F manifolds occur via spin forbidden electric quadrupole (E2) and magnetic dipole (M1) transitions. Calculated reduced matrix elements for these transitions are tabulated. The calculated natural linewidths account for both the decay of the upper state to the lower manifold on the listed E2 and M1 transitions and the M1 decays within each fine-structure manifold.

[0052] From a numerical calculation of the Ti level structure, several key features that make Ti an attractive atom for clock applications can be identified. By way of example, the extreme narrowness of the candidate clock transitions (allowing for optical atomic clocks with long coherence times, a weak clock sensitivity to blackbody radiation shifts, and the existence of several magic trapping wavelengths for optical trapping, all make Ti an attractive choice. In view of the identified challenges posed by the nonzero angular momentum of the clock states in Ti, a proper magic-wavelength condition for optical trapping is disclosed herein, which imposes a significant differential tensor ac Stark shift, mitigates their effects.

[0053] High precision atomic structure calculations were used to characterize 85 levels of neutral Ti. For this, a hybrid method is employed that combines the configuration interaction (CI) and linearized coupled-cluster (CC) approaches (referred to as CI + all order method [15, 16]). In this method, the correlations between four valence electrons are included via a 11 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) largescale CI computation using a highly parallel message passing interface (MPI) CI code [16, 17]. Several computations with increased number of configurations were carried out to ensure convergence. The core-core and core-valence correlations are included using an effective Hamiltonian formalism

[0015] . The effective Hamiltonian is constructed using second-order many-body perturbation theory (MBPT) and more accurate CC methods. The difference between these results gives the size of the higher-order corrections, which is used to estimate uncertainties on all theory values

[0016] . The results are used to calculate transition rates, dynamical polarizabilities, and systematic shifts in the clock transitions. Further details of the computational methods are provided, as discussed below.

[0054] Embodiments of the present invention include, but are not limited to, the a3F4→ a5F5 (e.g., from state 106 to state 104 of FIG.1A) transition at 1549 nm (e.g., transition 108 of FIG.1A), as shown in the first row of Table 1. An advantage of this transition is that the a5F5 state is the lower level of the near-cycling 498 nm transition (e.g., transition 112 of FIG. 1A), which is suited for laser cooling. In embodiments, the cooling laser also includes an optical detector. The calculations predict that the cooling transition has low branching ratios to other even parity states (<10−6), enabling single-laser state preparation and readout for atoms in the upper clock state. An additional benefit is that light at the 1549 nm clock wavelength can be generated by narrow-linewidth, high-power Er-doped fiber lasers, simplifying the required optical setup.

[0055] Three titanium isotopes which have zero nuclear spin, and therefore have no hyperfine structure (46,48,50Ti), are considered. To make the clock insensitive to first-order differential Zeeman shifts from stray magnetic fields, the ^^^= 0 ^ → ^^^= 0^ transition is driven, with mJ being the magnetic quantum number and the primed symbols and numbers referring to the upper a5F state. Because the E2 matrix element for this transition is zero, only the M1 matrix element contributes to a direct one-photon drive of the clock transition. Choosing quantization, clock-laser polarization, and clock-laser propagation axes as shown in FIG. 1A and 1B, a driving intensity of 0.1 W / mm2is calculated, and clock Rabi frequency of 91(46) Hz is achieved.

[0056] To compare the strength of this M1 transition to that of an E2 transition in the same set of transitions, driving the ^^^^^, ^^= 0^ → ^^^^^, ^^^= 0^ transition at a wavelength of 1573 nm is considered. Forelement vanishes while the E2 matrix element does not. With the same intensity and polarization as in FIG.1, but propagating along 12 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) the z axis, the Rabi frequency for such an E2 transition is 214(13) Hz. For a detailed derivation of these Rabi frequencies, as discussed below. λmagic α α^^^^^ ^ α ^^^^α^^ ^^^^α ^^^^α^^ ^^^^α^^^^^^.^ )))5)5F5clock transition. Wavelengths are given in units of nm, and polarizabilities are given in atomic units.

[0058] Neutral-atom optical clocks often use optical lattice potentials to confine atoms, allowing for a long interrogation time. In order to avoid imposing large differential ac Stark shifts between the upper and lower states of the clock transition, it is necessary to use lattice light which is at a “magic trapping wavelength”, at which the dynamic polarizabilities of the lower and upper clock states are identical

[0020] .

[0059] FIG. 2 illustrates the scalar dynamic polarizability of the mJ= 0 sublevels of the a3F4 (line 202) and a5F5 (curve 204) states in Ti from 1100 nm to 750 nm as calculated by the sum-over-states method. The angle between the polarization and the B field direction is set to 90°. The locations of magic trapping wavelengths (e.g., wavelengths 206a, 206b, 206c, and 206d) considered herein are circled.

[0060] In FIG. 2 and Table II, several magic trapping wavelengths for the ^^^^^, ^^ =0^ → ^^^^^, ^^^ = 0^ clock transition are reported. As with most states in Ti, the clock statesexperience significant vector and tensor ac Stark shifts

[0013] , owing to their non-zero angular momentum and Ti’s complex spectrum. To account for these shifts, the specific lattice configuration shown in FIG.1 is considered. Here, a magnetic field (not shown) applied in the z direction imposes a linear Zeeman shift and defines the quantization direction. All lattice light is linearly polarized in the transverse x – y plane. In this configuration, the clock transition is shifted only by the differential scalar and tensor ac Stark effects (the vector shift is zero on the mJ = m'J = 0 sublevels). The sum of the scalar and tensor dynamic polarizabilities (S and T respectively) on the transition is then given by: ∆^ = α^^^^^− α^ ^^^^+^^ ^ α^^ ^^^^− α^^^^^^4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88)

[0061] At the identified magic trapping wavelengths, the net transition ac Stark shift is zero.

[0062] Calculations of the polarizabilities were performed by two methods. First, the sum- over-states method was used to roughly calculate polarizabilities over a wide range of frequencies. The 76 transitions with the largest contributions to dc polarizability were used in the case of the a3F4states, while 51 transitions were used in the case of the a5F5states. Once promising candidates for magic trapping wavelengths were found, direct dynamical polarizability calculations was performed to identify the location of the magic trapping wavelengths more precisely. Direct computations for two of the magic trapping wavelengths allow for predicting the remaining values accurately. Previously, the direct computation method was only used for divalent systems such as Sr [21, 22], Mg

[0023] , Yb [24, 25], Cd

[0026] , or Tm

[0027] .

[0063] For more complex atoms, the rapidly increasing number of relevant configurations makes such a direct computation intractable. Here, a truncation approximation was applied, where the configurations are ordered by weight to select the most important ones and then start removing configurations while checking the accuracy of the energies and relevant matrix elements. This procedure drastically reduces the number of Slater determinants required to maintain numerical accuracy. Further details on this method are discussed below.

[0064] This approach is not specific to Ti, and allows for the computation of polarizabilities, magic trapping wavelengths, and other atomic properties for other atoms with a complex electronic structure.

[0065] Using the lattice configuration and magic trapping wavelength described above not only eliminates the differential light shift, but also protects against the effects of dipole-dipole interactions between Ti atoms. These effects are not present in lattice clocks of Sr, Yb, or Hg as those clocks operate on transitions between non-magnetic J = 0 states. In contrast, themagnetic moments of the proposed Ti clock states are both large, with µ^^^^ = 5.00 µ^ andµ^^^^ = 7.05 µ^ .

[0066] There are three processes associated with the dipole-dipole interaction that are considered: dipolar relaxation, elastic spin-spin energy shifts, and inelastic spin-spin mixing

[0028] . Dipolar relaxation is the process by which Zeeman energy is converted to kinetic energy, depleting atoms from the clock states. Such relaxation can be suppressed for atoms trapped in a deep 3D optical lattice by ensuring the bandgap far exceeds the Zeeman energy

[0029] . The band energy scale in a lattice is set by the lattice recoil energy Er= h2 / (8ma2), where a is the 14 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) lattice spacing. For a 48Ti atom in the magic-wavelength lattice described above, the recoil energy is Er= h × 6.8 kHz. 3D optical lattice clocks typically use deep lattices to suppress tunneling and atom-atom contact interactions. As of 2019, the fermionic Sr 3D lattice clock at JILA operated at a lattice depth of V = 80Er

[0030] . In deep lattices, the gap above the ground 0 band is ^^≈ 2^^^^^). In the case of Ti, a comparable lattice operating at the magic trapping near 781 nm could be achieved by intersecting six 3.5 W beams with waists of 0.1 mm. This would give a lattice depth of V0 = 79, Er = 5 x 540 kHz and gap of Eg = 18Er = h x 120 kHz. Setting the Zeeman energy below this band gap requires the ambient magnetic field be well below B~Eg / uB=60mG.

[0067] The second two processes associated with the dipole-dipole interaction are captured in the so-called secular Hamiltonian, which is obtained by time-averaging the dipole-dipole interaction over Larmor precession: ^^^^ ^^ ^^^ ^ ^ ^^^ = ^(2)

[0068] Here, i and jby a distance vector of length rij and polar angle θij with respect to the quantization axis. gJi is the Lande g-factor of the atom at lattice site i.

[0069] The elastic spin-spin energy shift corresponds to the JziJzj term in the secular Hamiltonian. In theory, this term generates shifts to the transition frequency between atomic states with nonzero angular momenta. However, for a clock transition between mJ = m'J = 0 magnetic sublevels, the shift is zero and can be ignored.

[0070] The final process is the spin-mixing interaction, which corresponds to the J+iJ-j+ J- iJ+j term in the Hamiltonian. This term couples atoms in an initial two-body state ^m(1)J =0,m(2)J =0^ to final states ^m(1)J =±n,m(2)J =∓n^, n∈{1,...,J}. If not controlled, this would lead to rapid loss of population from the mJ=0 clock states. The maximal strength of the coupling is ℏΩSM=μ0μ2B g2JJ(J+1)√2 / 16π(λ / 3)3. In aλ=781 nm optical lattice, this gives spin mixinglower (upper) clock state manifold. Spin mixing between atoms in the upper and lower clock states is energetically suppressed because of the 15 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) significant differential Zeeman splitting. For a 30 mG magnetic field, the splitting between the ^m(1)J =0,m'(2)J =0^ and ^m(1)J =±1,m'(2)J =∓1^ states is h × 6.7 kHz.

[0071] In the case where both atoms occupy either the upper or lower clock state, spin mixing is suppressed by the tensor ac Stark shift imparted by the optical lattice light. The tensor light shift creates an energy splitting between the ^m(J1)=0,m'J(2)=0^ and ^m(J1)=±n,m'J(2)=∓n^ two- atom states. Using the same optical lattice the m(1^)J =0,m'(2)J =0^and^m(1)J =±1,m'(2)J =∓1^is ΔEtens=hh×4(2) kHz (h×4.8(6) kHz) within the are much,

[0072] In this regime, spin mixing enters as a second-order perturbative effect. The ^m(J1)=0,m'J(2)=0^ two-atom states in both the lower and upper clock manifolds are weaklycorresponding ^m(J1)=±1,m'J(2)=∓1^ sates by ΩSM. Both clock states experience an energy shift on the order of ~Ω^^^ / Δ^^^^^. The difference between the shifts leads to a shift of the clock frequency, while the sum of the shifts leads to decoherence between the clock states. For two atoms, the shift is ~3 mHz and the rate of decoherence is ~ 6 mHz. For more discussion of the dipole-dipole interaction, see below.

[0073] Another significant systematic uncertainty in optical clocks is the blackbody radiation (BBR) shift, which has been the subject of significant past investigation [21, 35]. The BBR shift for the Ti clock line as: ^^^^∆^^^= − ^ ^^^^^^− ^^^^^^ ^ ^^^^ (1 + η) (3) where ^ =^^proportionality, ^^^is the dc scalar polarizability of the i state of Ti, T is the thermal background temperature in K, and η is a small dynamical correction omitted in the present work. The same CI+all-order approach is used to compute dc and dynamic polarizabilities. In this case, it is determined that ^^^^^^=128.53 a.u. and ^^^^^^=100.39 a.u., which leads to ΔBBR = −0.24 Hz at T = 300 K.lower than that in Sr, where the BBR shift is known to be -2.2789 Hz

[0036] . 16 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88)

[0074] The final systematic uncertainty that is considered is the quadratic Zeeman shift (QZS). For the46,48,50Ti isotopes, the effect will be small since it will arise only from the mixing of neighboring fine structure states, whereas in atoms with nonzero nuclear spin, a stronger QZS arises from mixing of hyperfine states. For the states in the Ti clock, the QZS of the mJ=0 sublevels are Δ^a3FQZS4^=0.129[Hz / G2]^^and Δ^a5FQZS5^=0.434[Hz / G2]^^, and the QZS on the transition is thus Δ^^^=0.305[Hz / G2]^^. Given that a Ti clock may operate at a magnetic field well below 60 mG to suppress dipolar relaxation, the QZS of the clock transition will be below 1 mHz. This is approximately an order of magnitude lower than the QZS that is present in Sr optical lattice clocks, of almost 10 mHz [30, 31].

[0075] In view of the above, laser-cooled Ti is an attractive choice for realizing a telecommunications-band optical atomic clock. Operating Ti clocks on several of the available telecommunications-band optical transitions allow for clock comparisons as a powerful method for identifying and reducing systematic corrections. The present invention advances atomic structure calculations to determine critical properties of such clocks, including identifying magic trapping wavelengths for optical trapping, estimating clock transition widths and line strengths, and determining that the BBR shift for Ti clock transitions is an order of magnitude smaller than the shift that dominates current Sr-based clock systematics [30, 31]. The present invention also describes potential effects of, and mitigation measures against, magnetic dipole- dipole interactions. These measures are relevant to other potential applications of dipole- interacting atoms and molecules for precision measurement.

[0076] FIGs. 3-5 illustrate a CI+all-order method [S1] that combines linearized coupled cluster and configuration interaction (CI) approaches. In this method, the electrons are separated into the 1s22s22p63s23p6core and four remaining valence electrons. First, the coupled cluster method is used to construct the effective Hamiltonian Heffthat accounts for core and core-valence correlations and can be constructed using second-order many-body perturbation theory in the CI + MBPT method or the coupled cluster approach (CI + all-order method). The CI method is used to correlate the remaining four valence electrons using this effective Hamiltonian rather than the usual bare Hamiltonian. This procedure effectively includes all types of correlation effects in the core and valence spaces.

[0077] The CI wave function is constructed as a linear com-bination of all distinct states of a specified angular momentum J and parity, 17 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) ^^= ^^^^^,(S1) ^ where {Φi} is the set ofelectrons from the reference configuration to higher orbitals. The many-electron Schrödinger equation is expressed as ^^^^^ = ^^, (S2) where the effective Hamiltonian has the form of ^^^^ = ^^^ + Σ.(S3)

[0078] Here, HCI is the CI Hamiltonian described by the equation ^^^ ^^^^^^+ ^ ℎ^,^^ + ^ ^^^,(S4)where Ecoreis energy of the frozen core, Ncoreis the number of core electrons, HCIrepresents the kinetic energy of the valence electrons and their interaction the central field and Vij accounts for the valence–valence correlations.

[0079] The core-valence correlation potential, ^ = ^ + ^ , is obtained from the all-orderone- and two-electron parts of the core–valence correlation potential, respectively. After Eq. (S2) is solved using the CI technique and the wave functions are obtained, they are used to calculate matrix elements of the electric-dipole, magnetic-dipole, electric-quadrupole, and other one-electron operators.

[0080] When applying the CI+all-order method to atomic Ti, a VN−4potential of the 1s22s22p63s23p6frozen core is used. The Dirac-Hartree-Fock equations are solved in this potential to generate 3d, 4s, 4p, 5s, 4d, 5p, and 4f orbitals. All other orbitals are constructed in a spherical cavity of 40 a.u. using B-splines.

[0081] The set of CI configurations has to be constructed separately for even and odd states. Several calculations are carried out with increasing number of configurations to ensure 18 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) convergence of the CI with the number of included configurations. For even states, it is sufficient to make all possible single and double excitations to a 20spd18f16g basis starting from 4s23d2, 4s3d25s, 4s23d4d, 3d24p2, 3d25s2, 4s3d3, 4s3d24d, 3d35s, and 3d25s4d configurations. It is verified that a subset of triple excitations give a negligible contribution. For odd states, the CI configuration space is sufficiently saturated by the single and double excitations to the same large basis from the 4s3d24p, 3d34p, 4s3d25p, 3d24p5s, 4s3d4p4d, 3d25s5p, and 3d24p4d configurations.

[0082] The calculation of energies were carried out for 85 levels and are compared against experimental data. Most of the theoretical energies differ from experimental values by only 0.1-2.5 Results presented herein are only for the lines and levels that can be used to pump titanium optically into the metastable state to perform laser cooling and to drive the relevant clock transitions. The selected energy levels in cm−1are listed in Tables S1 and S2 provided below.

[0083] TABLE III illustrates a comparison of theoretical even energy levels (in cm−1) with experimental values [S2]. 19 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) Configuration Term Expt Theory Diff Diff% 3d4s2a3F20 0 0 %

[0084] TABLE IV illustrates a comparison of theoretical odd energy levels (in cm−1) with experimental values [S2]. Configuration Term Expt Theory Diff Diff% 3d2(3F)4s4p(3Po)^5^^6 16459 16454 -5 0% 3d2^1^^0 24174 5^04 25927 26081 154 0.6% ^5^^6 26911 26982 71 0.3%

[0085] and (S2) are computed, where L and S are the total electron orbital and spin angular momentum operators, to obtain approximate quantum numbers L and S, where (L2) = L(L+1) and (S2) = S(S+1), which unambiguously identifies all terms in Tables III and IV. As a result, a level, 3d24s4p1So0, not listed in the NIST database is identified. This level is included in Table IV.

[0086] The forbidden transitions between the a3F and a5F terms are studied to identify the most suitable clock transition. The total clock transition linewidth accounts not only for spontaneous decay on the clock transition itself, but also for decay of the upper and lower levels of the clock transition to other states, leading to an overall clock transition linewidth that is larger than the spontaneous decay rate on the clock transition alone. In Table V, the contributions that determine the linewidth of the a3F4 → a5F5 clock transition are listed. It is noted that the clock transition linewidth is dominated by M1 decays of both the lower and upper states to other fine structure states within their respective manifolds. The same calculation was performed to calculate the linewidth of all other clock transitions. The random- phase approximation (RPA) corrections are included to the effective electric-dipole and 20 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) magnetic dipole operators, see, for example Ref. [S3]. Such effective operators account for the core-valence correlations in analogy with the effective Hamiltonian Heffas previously set forth.

[0087] The uncertainties in the values of matrix elements were estimated as difference between values obtained using CI+all-order and CI+MBPT methods.

[0088] To support ongoing experimental efforts to realize laser cooling of Ti, these two transitions are characterized theoretically using our atomic-structure calculations described herein. Specifically, the strengths of the two electric-dipole laser cooling transitions are calculated, and also, critically to experimental efforts, determined the small leakage rate (branching ratio) out of the laser cooling transitions. Results for two of the cooling transitions, 3d3(4F)4p y5Go6 - 3d3(4F)4s a5F5 at λ=498 nm and 2d3(3F)4s4p(3Po) z5Go6 - 3d3(4F)4s a5F5 at λ=1040 nm are listed in Table VI).

[0089] It was noted that while there do exist other even parity states to which a Ti atom in the excited state of the laser cooling transition can decay, such scattering would be strongly suppressed because the transitions are spin forbidden. The calculations provided herein confirm this expectation. Indeed, the branching to those states is exceptionally low, at the 10−6level for the 498 nm transition and at the 3 × 10−7level for the 1040 nm transition. These are low enough branching ratios to enable the typical tools of modern ultracold atomic physics, including both Doppler and sub-Doppler cooling techniques and single-atom fluorescence detection in quantum-gas microscopes or optical tweezers, without the need for additional repumping lasers.

[0090] To obtain transition rates and branching ratios for the cooling transitions (as shown in Table S4) the electric-dipole reduced matrix elements are calculated with the effective electric-dipole operator in the random-phase approximation. The other correction to the E1 operator beyond RPA is also considered: the core-Brueckner (σ), structural radiation (SR), two- particle (2P), and normalization (Norm) corrections [S5–S7]. As has been noted for the case of Sr, these corrections cannot be omitted at the 1% level of accuracy.

[0091] TABLE V. Results of calculations for the a3F4– a5F5clock transitions of Ti. Wavelength λ (in units of nm), E2 and M1 reduced matrix elements (ME) (in a.u. for E2 transitions, in μBfor M1 transitions), transition rates (in ×10−7s−1), and branch ratio. 21 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) Branching Upper Term Level Lower Term Level λ ME Tr. Rate Ratio 3d3(4F)4s a5F5 6843 3d24s2 a3F3 170 1498.6 E2 0.0472(7) 3.00(9) 0.00838(25) a3F4 387 1548.9 M1 0.0010(5) 7(6) 1.020(17) E2 0.140(4) 22.4(1.3) 0.063(4) 3d3(F)4s a5F4 6743 99794.4 M,1 3.632(10) 325.43(18) 0.909(16) E2 1.748(18)3.14(8) x8.78(22) x 10- 10-69 3d24s a3F4 387 3d24s2a3F3 170 46138. M1 2.597(0.026) 2059(53) 1

[0092] TABLE VII illustrates wavelengths λ (in nm), calculated and observed transition rates (in s−1×106), and branching ratios for optical pumping of Ti. The upper state for all transitions is the 3d2(3P)4s4p(3Po)5Do4level, at 25967 cm−1. Lower Term Level λ Tr. Rate, theory Tr. Rate, lit Branching Ratio 3d24s2a3F3 170 388.2 0.13(6) 0.28(6) 0.049 a3F4 387 391.5 1.8(8) 2.11(28) 0.68, Dtot (in a.u.), transition rates (in s−1), and branching ratios for cooling transitions of Ti I. The Dtotvalues (in a.u.) are calculated with CI+all-order method and include the random-phase approximation (RPA), the core-Brueckner (σ), structural radiation (SR), two-particle (2P), and normalization (Norm) corrections. For approximate values (indicated by a ~ symbol), the precise value of the matrix element is highly uncertain, and the reported value should be interpreted as correct only within an order of magnitude.

[0094] In Table VI, transition matrix elements, Dtot are included, obtained taking into account all the corrections mentioned above. The uncertainties were estimated by taking the difference between the values obtained using CI+all-order and CI+MBPT methods. For very small matrix elements (marked by the tilde symbol), approximate values are provided without explicit evaluation of errors. For these, the error should be assumed to be at the same order of magnitude as the quantity itself.

[0095] Gas phase Ti atoms may be produced with a thermal source that operates between 1200 C to 1800 C. At these temperatures, a low population of the Ti atoms produced would be in the a5F5laser-coolable state. It would therefore be necessary to transfer atoms to this state 22 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) from the a3F ground state manifold via optical pumping. Through our calculations, multiple potential excited states that could be used to transfer population efficiently to the a5F5state are examined. It is determined that the 3d2(3P)4s4p(3Po) y5Do4 state has a significant branching ratio to both the a5F5state and a3F states, allowing for optical pumping of atoms from the ground states to the laser cooling state. The transition wavelengths are also amenable to current laser technology.

[0096] The optical clock transitions of Ti can be driven by either the M1 or E2 multipoles of the electromagnetic field, both of which must be accounted for when calculating the overall Rabi frequency. In (mJ=0→m’J=0, J=4→J’=4,5) only one of the two multipoles is allowed.

[0097] Given a reduced M1 matrix element, DM1, for a transitions between lower and upper levels J and J', and a magnetic field B of the clock laser, the M1 Rabi frequency between a lower and upper sublevel mJand m'Jis given by the equation ^^1^^^^^^ê∗^^1 = − (−1)^^^^^^⋅ B ℎ

[0098] For an electric quadrupole transition with reduced matrix element DE2 driven by a plane-wave clock laser field E with polarization ε and propagation wavevector k, the Rabi frequency is Ω^^ =^^(−1)^^ ^^^ ^^^ ^^ ^ 2 ^^^ ^^ ^

[0099] Here, Mij(q) is a geometric factor given by ^^^^=√5^ ^ê^ ∙ê∗^^ ^ ^ê^ ∙ ê∗^^ ^ ^1 1 2^ (S7)where ^̂^and ^̂^are the cartesian and spherical basis vectors respectively.

[0100] In order to find the magic trapping wavelengths of the clock transitions, it is necessary to calculate the polarizability of Ti in both the a3F4and a5F5states over a wide range of frequencies. Performing this calculation directly remains a computational challenge for complex atoms like Ti. Direct calculation requires the inversion of huge matrices — 350,000 23 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) × 350,000 in the present case — which is computationally intractable. Instead, an iterative approach is included in the pCI Code Package implementing the CI+all-order technique to calculate the polarizability of the considered states of Ti. This approach allows for accurate results using the inversion of smaller matrices (15,000 × 15,000), making the task feasible. Static polarizabilities for both 3d24s2a3F4 and 3d3(4F)4s a5F5 levels are successfully obtained, but for dynamic polarizabilities the calculation diverged for wavelengths shorter than 750 nm (900 nm) for the a3F4 (a5F5) level. To overcome this, a combination of the CI+all-order technique and the sum-over-states method are used to calculate the scalar, vector and tensor polarizability from 1100 nm to 400 nm.

[0101] The sum-over-states method involves using only bound states of an atom, and there is always some inaccuracy due to missing contributions from continuum states and bound states not included in the calculation. For the best accuracy, it is advisable to use as many states as possible in the sum. However, there are always limits on the accuracy of calculations of highly excited states. To balance these trade-offs, in the sum-over-state method for the lower lying states are used that contribute the majority of the dc polarizability but still can be properly calculated. For this purpose, 73 states are used to obtain the polarizability of the 3d24s2a3F4 level and 49 states for the polarizability of the 3d3(4F)4s a5F5level. The most important contributions to both polarizabilities are shown in Tables VIII, IX. The polarizability is generally divided into three terms: S - scalar, V -vector, and T - tensor polarizabilities. They are represented for an arbitrary state is as follows [S9, S10]: ^^ =2^(^ − ^ ) | | ^^(^)^ ^ |^^^| ^ |^24 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) Configuration Term Exp. Level Theor. Level Diff % D Contrib. to α0 3d2(3F)484p(3Po) z5^^^16459 16454 0.0% 0.86(3) 1.00(8) z5^^^17215 17185 -0.2% 2.48(6) 7.7(4) z5^^^18695 18769 0.4% 2.08(3) 4.77(16)Configuration Term Exp. Level Level Diff % D Contrib. to α0 3d2(3F)484p(3Po) z3^^^19575 19632 0.3% 1.682(13) 2.39(4) z3^^^20126 20115 -0.1% 1.467(9) 1.774(22)25 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) ^^^^^ −1^ ^^11 2 ^ (^) = ( )^ ^^ ^^^^^^^ ^ where^^= −2^6^^and5^^ (2^ − 1)^ = ^ ^^ 4 .

[0102] The index n refers to the states in the sum-over-states that contribute to the polarizability of the state i; En,i is the energy and Jn,i is the total angular momentum of the state; ^n||D||i^ is the reduced matrix element between the two states, and ω is the frequency of the external electric field. The total dynamic polarizability can be expressed as follows: ^^ ^^^ ^^(^) = ^^ (^) + ^ cos(^^)^^^^^ (^) +(S11)

[0103] where ε is the ellipticity of the polarization, θkis the angle between the direction of propagation of the light and the quantization axis, θp is the angle between the polarization of the light and the quantization axis, and mJiis a magnetic quantum number.

[0104] To obtain an accurate value of dynamic polarizability with the sum-over-states method, one has to estimate the residual contribution to the polarizability from states that are not included in summation. This estimate is made using the fact that |ω / (En−Ei)|"≪ ” 1 for all such residual states in the range of wavelengths that are considered. Using this ratio as a small parameter and expanding to the lowest non-zero order in, Eq. S8 can be simplified to ^^^(^) = ^^^^(^) + ^^^^^^(^) (S12)26 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) ^2 (^^ − ^^)|^^|^|^^|^^^^^(^) =3(2^ + 1) ^(^ − ^ ^ ^(S13) ^^ ^) − ^and where Nused in direct summation. Moreover,“^” _”^” ^(“^” _”^^^” ) “(^)” is expressed as:^^^^^^(^)= ^^+ ^^^^(S14)where A0 and B0 are polarizability. The same simplification works for αVi (ω) and αTi (ω), by writing αViN(ω) and αTiN(ω)) as a summation as in S13. This yields the expansion of the long wavelength residuals: ^^^^^^(^) = ^^^ where B1, A2, and B2 are

[0105] Using the results of the sum-over-states method, the four candidate magic trapping wavelengths discussed in the main text are identified. The atoms were assumed to have magnetic quantum numbers mJ = 0 and the optical field parameters were ε = 0 and θp = θk = 90° . However, the more accurate iterative approach used by the pCI code package was found to not converge at all of the candidate wavelengths. Using the fact that convergence in the iterative polarizability can be achieved at ω = 0.043989 a.u. (λ = 1035.8 nm) and for all smaller values of ω = 0, the values of αi(0.043989) and αi(0) obtained from the sum-over-states method from the corresponding results computed with the pCI code are subtracted. In this way, the residuals αSires(0), αTires(0) and αSires(0.043989), αVires(0.043989), αTires(0.043989) are found. This allowedEqs. S14, S15 and thus extend the accuracy of the direct iterative polarizability calculation to frequencies where the iterative calculation fails to converge.

[0106] The uncertainty on the polarizabilities were obtained by comparing the calculations of the more accurate CI+all-order technique with CI+MBPT at the points where the direct calculation converged. The difference between the values of the appropriate reduced matrix elements |^n||D||i^|CI+all-orderand |^n||D||i^|CI+MBPTis an additional possible inaccuracy of the method and thus is used to estimate the uncertainties of the polarizabilities. Furthermore, possible errors in the parameters A0,2and B0,1,2were included in the final uncertainties as the 27 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) difference between the correspondent parameters calculated with the CI+all-order and CI+MBPT methods.

[0107] As discussed in the main text, dipole-dipole interactions underlie several effects that impact the operation of a Ti atomic clock. One can distinguish between interactions that are inelastic or elastic in the motional degrees of freedom. Inelastic interactions can be described as spin relaxation, a dipole-mediated conversion of internal Zeeman energy into external motional energy. For example, in an optical lattice spin relaxation may couple atoms from lower to higher bands of the lattice. As previously set forth, inelastic motional interactions can be suppressed by trapping atoms in a lattice with a sufficiently large band gap.

[0108] Elastic motional interactions are purely in the spin sector using the secular Hamiltonian, which accounts for cycle-averaging over the Larmor precession of the atomic spins [S12]. As previously set forth, the secular Hamiltonian is given for two atoms as ^^^^^ ^^^^^^ ^^^ =^^^^ (1 − 3cos^ ^^^)with 1 and 2 labelling the two atoms, and r12 and θ12 describing their position difference vector and the angle it makes with the quantization axis.

[0109] Two effects of this secular Hamiltonian are considered separately. The first is a spin-elastic interaction, which is diagonal in the separable basis of magnetic sub-level states. This shift is zero on the mJ= 0 magnetic sublevels and therefore does not affect a clock based on an mJ = m'J = 0 transition.

[0110] The second effect is the spin mixing interaction, which is off-diagonal in the separable magnetic sublevel basis. This interaction conserves the total magnetic quantum number of the two atoms, m(tot) (1) (2)J =mJ+mJ, but changes the magnetic quantum numbers of the individual atoms. In the absence of dipolar interactions, the entire manifold of two-atom states with identical m(tot)J is degenerate for atoms with equal Zeeman splittings. Dipolar interactions can then generate significant mixing and energy shifts within this manifold, leading to imprecise measurement of the clock transition.

[0111] While several methods have been developed in NMR to control dipolar spin mixing, e.g. multiple pulse sequences [S13] and magic angle spinning [S14], these techniques are not needed to suppress spin mixing in a Ti clock system. As previously set forth, spin mixing within the clock states is mitigated by the tensor light shift imposed by the optical lattice beams. From 28 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) Eq. S11, the tensor light shift includes a part proportional to m2J, which splits the degeneracy of the aforementioned states and energetically suppresses the spin mixing process.

[0112] FIG.3 illustrates the secular dipole-dipole interaction of two atoms 302 and 304 in the a3F4(g) and a5F5(e) clock states. The g and e states (shown as states 104 and 106, respectively) have angular momentum quantum numbers J = 4 and J = 5, giving rise to 2J+1 magnetic sublevels each. FIG. 3 shows only three of these mJsublevels in each state, split by Zeeman shifts. The clock drive ΩM1couples the ^gmJ=0^ and ^emJ=0^ states, and Hddcouples the subspaces of the two atoms.

[0113] FIG. 4 illustrates Ramsey contrast as a function of the free evolution time T. Curve 402 represents the stark ac shift, Hac being on, Curve 404 represents Hac being off. The contrast dies quickly when the ac Stark shift, Hac, is excluded from the simulation, as shown by curve 404. The split in the T axis and change in scale at 0.25s is noted. The resulting Ramsey fringes for T = 0.2s are indicated by circles. The probability is plotted for an atom found in state|e0^as a function of the detuning δ from the clock transition.

[0114] To study the suppression of spin mixing, the secular dipole-dipole interaction of two nearest neighbor atoms were simulated in an optical lattice with and without accounting for the lattice-induced ac Stark shift, as shown by FIG.3. The single atom states are notated by ^ξ(i)mJ^ where ξ∈{g,e} refers to the upper or lower clock states, the superscripts denotes the ithand the mJsubscript is the magnetic sublevel. g is used to refer to the lower a3F4clock state, and e to refer to the upper a5F5 state. The two atom system consists of [(2Jg + 1) + (2Je + 1)]2, totaling 400 states.

[0115] A simple Ramsey interferometry sequence can be simulated which includes the dynamics of Hdd, the Zeeman Hamiltonian HZ, and, optionally, the ac Stark Hamiltonian Hac. Two atoms are initialized in the ^g(1)(2) 0 ;g0^, state, a π / 2 pulse is applied to each atom on the g →e0 clock transition, theis allowed to evolve for a time T under HZ , Hdd and (optionally) Hac; and a second π / 2 pulse is next applied to each atom; and then the probability of an atom being in the e0 state is determined. The detuning of the clock drive frequency is varied to generate a Ramsey interferometry fringe. From this fringe, the contrast C(T ) = (max Pe(T,δ)− min Pe(T,δ)) is obtained. FIG. 4 shows the results of the simulation. The left panel 400 demonstrates the decay of the Ramsey contrast with and without Hacincluded. The axis is broken at 0.25s to illustrate that the tensor ac Stark shift extends the decay time of the contrast significantly. Because unitary dynamics of two neighboring atoms are 29 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) simulated, revivals of the contrast are observed in the simulation beyond the times plotted in FIG.4, but these would not occur in a true many-body situation as the coherence would spread between many particles and be lost. The right panel 410 shows the resulting Ramsey fringes taken at the time indicated at points 406 on the left panel 400. When no optical lattice is applied, the coherence on the clock transition quickly vanishes, leading to a loss of the Ramsey signal. The fringe survives for ~85 s when the lattice beams are on.

[0116] FIG.5 illustrates the two-atom level structure with and with-out the ac Stark shifts applied (Hac). Only states that are symmetric under particle exchange are shown, as these are the only states involved in the dynamics. The states S ^^g(1)(2) mJ≠0;e-mJ^^ are omitted because they are split from the S ^^g(1)0 ;e(2) 0 ^^ state by theThe dotted lines show that the =0 sublevels is lifted when Hddis turned on. The five boxedlevels include the clock states and the states most strongly coupled to them by Hdd. The simplified picture includes only the levels boxed in the left panel. Hddcouples the states ^ξ(1)0 ;ξ(2)0 ^ and S^^ξ(1)0 ;ξ(2)0 ^^with rate ofΩξSM.

[0117] FIG.5 provides a simplified picture of the level structure at play. S(| i) is defined as the function that S(|ψ^)symmetrizes a multiparticle state by adding states with swapped mJand, if necessary, states in which the excited atom is switched. The initial state of the Ramsey sequence ^g(1)0 ;e(2) 0 ^ is symmetric under particle exchange and the Hamiltonian commutes with the exchange operator, so all states involved in the dynamics must remain symmetric. The left panel of FIG.5 illustrates that in the absence of a tensor light field or dipole-dipole shifts, the ^ξ(1)0 ;ξ(2)0 ^ states are degenerate with all the symmetrized states(2)-mJ^^=1 / √2 ^^ξ(1)mJ;ξ(2)-mJ^+ ^ξ(1)-mJ;ξ(2)mJ^^. On the other hand, the figure shows the singly with states of mixedstates have different Zeeman splittings.

[0118] The dotted lines show the lifting of the degeneracy of the spin-mixed states by the ac Stark shift (Hac). The splitting leads to the simplified level structure shown in the right panel of FIG. 5. After the splitting, only the S^^ξ(1)1 ;ξ(2)-1 ^^states remain energetically nearby the S^^ξ(1);ξ(2)^^ clock states.(1) (2) (1) (2)^ The splitting ^ ;ξ0^^ and S^^ξ1;ξ-1^^ states by the30 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) tensor ac light shift is denoted ΔEξtens. With the parameters previously used to describe the optical lattice, this splitting is h x 4(2) kHz for gg atoms, h×4.8(6) kHz for ee atoms. The spin mixing term in Hdd couples the nearby states with rates ΩξSM(h × 2.4 Hz for gg atoms, h × 4.6 Hz for ee atoms). The coupling produces first-order level mixing at the Ωξ SM / ΔEξtens~10-3level between these states – small enough to not directly impact the coherence of the clock.

[0119] However at second-order, both of the clock states will experience an energy shift of ξ 2 ΔEξξ^ΩSM^ / ΔEξtensIt is clear that the difference between these two shifts leads to a shift of the from the simplified treatment is ~3 mHz for two atoms. The sum of thestates also leads to decoherence through entanglement generation. To understand this effect, the state of the two atom system is considered after it has evolved for atime t during the Ramsey sequence:|^(^)^ =1^^^∆^^^^ / ħ (^) (^)^^∆^^^ ^ / ħ(^) (^)2^ ^ ^^ ;^^ ^ + ^ ^^^ ; ^^ ^where only the non-trivial phases accumulated from the second-order dipole-dipole interaction are included. If the second atom state space is traced over, the single atom reduced density matrix is found as: ^(^)(^) = |^(^)^^ (^) (^) (^)^ ^^ | + |^^ ^^^^ |

[0120] From this, the coherence is observed to be fully lost when ^ΔEgg-ΔEee^t / ℏ=±π . Given the second-order dipole-dipole shifts calculated above, this corresponds to a full loss of coherence at t ~ 86 s. This is in agreement with the lifetime seen in the simulation from FIG. 4.

[0121] Now that embodiments of the present invention have been shown and described in detail, various modifications and improvements thereon can become readily apparent to those skilled in the art. Accordingly, the exemplary embodiments of the present invention, as set forth above, are intended to be illustrative, not limiting. The spirit and scope of the present invention is to be construed broadly. REFERENCES [1] S. Brewer, J.-S. Chen, A. Hankin, E. Clements, C. Chou, D. Wineland, D. Hume, 31 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) and D. Leibrandt, Al + 27 Quantum-Logic Clock with a Systematic Uncertainty below 10 - 18, Physical Review Letters 123, 033201 (2019). [2] C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, Optical clock comparison for Lorentz symmetry testing, Nature 567, 204 (2019). [3] T. Bothwell, D. Kedar, E. Oelker, J. M. Robinson, S. L. Bromley, W. L. Tew, J. Ye, and C. J. Kennedy, JILA SrI optical lattice clock with uncertainty of $2.0 \times 10ˆ{-18}$, Metrologia 56, 065004 (2019). [4] Boulder Atomic Clock Optical Network (BACON) Collaboration*, Frequency ratio measurements at 18-digit accuracy using an optical clock network, Nature 591, 564 (2021). [5] e. a. Barontini, G., Measuring the stability of fundamental constants with a network of clocks, EPJ Quantum Technology 9, 12 (2022). [6] W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Sch¨ a_er, K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley, G. Milani, M. Schioppo, T. H. Yoon, and A. D. Ludlow, Atomic clock performance enabling geodesy below the centimetre level, Nature 564, 87 (2018). [7] M. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Search for new physics with atoms and molecules, Reviews of Modern Physics 90, 025008 (2018). [8] e. a. S.. W. P. Antypas, D., New Horizons: Scalar and Vector Ultralight Dark Matter (2022), publisher: arXiv Version Number: 1. [9] O. Buchmueller, D. Carney, T. Cecil, J. Ellis, R. F. G. Ruiz, A. A. Geraci, D. Hanneke, J. Hogan, N. R. Hutzler, A. Jayich, S. Kolkowitz, G. W. Morley, H. Muller, Z. Pagel, C. Panda, and M. S. Safronova, Snowmass 2021: Quantum Sensors for HEP Science– Interferometers, Mechanics, Traps, and Clocks (2022), publisher: arXiv Version Number: 2.

[0010] M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, An optical lattice clock, Nature 435, 321 (2005).

[0011] N. Lemke, A. Ludlow, Z. Barber, T. Fortier, S. Diddams, Y. Jiang, S. Je_erts, T. Heavner, T. Parker, and C. Oates, Spin- 1 / 2 Optical Lattice Clock, Physical Review Letters 103, 063001 (2009).

[0012] P. J. Winzer, D. T. Neilson, and A. R. Chraplyvy, Fiberoptic transmission and networking: the previous 20 and the next 20 years [Invited], Optics Express 26, 24190 (2018).

[0013] S. Eustice, K. Cassella, and D. Stamper-Kurn, Laser cooling of transition-metal atoms, Physical Review A 102, 053327 (2020).

[0014] A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. 32 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) Schine, J. Ye, and A. M. Kaufman, Half-minute-scale atomic coherence and high relative stability in a tweezer clock, Nature 588, 408 (2020).

[0015] M. S. Safronova, M. G. Kozlov, W. R. Johnson, and D. Jiang, Development of a configuration-interaction plus all-order method for atomic calculations, Physical Review A 80, 012516 (2009).

[0016] E. B. Norrgard, D. S. Barker, S. P. Eckel, S. G. Porsev, C. Cheung, M. G. Kozlov, I. I. Tupitsyn, and M. S. Safronova, Laser spectroscopy of the y P J o 7 states of Cr i, Physical Review A 105, 032812 (2022).

[0017] C. Cheung, M. Safronova, and S. Porsev, Scalable Codes for Precision Calculations of Properties of Complex Atomic Systems, Symmetry 13, 621 (2021).

[0019] A. Kramida and Y. Ralchenko, NIST Atomic Spectra Database, NIST Standard Reference Database 78 (1999), type: dataset.

[0020] M. Takamoto and H. Katori, Spectroscopy of the S 01 - P 03 Clock Transition of S r 87 in an Optical Lattice, Physical Review Letters 91, 223001 (2003).

[0021] M. S. Safronova, S. G. Porsev, U. I. Safronova, M. G. Kozlov, and C. W. Clark, Blackbody-radiation shift in the Sr optical atomic clock, Physical Review A 87, 012509 (2013).

[0022] G. Kestler, K. Ton, D. Filin, M. S. Safronova, and J. T. Barreiro, Magic wavelengths of the Sr ( 5 s 2 S 01 – 5 s 5 p P 13 ) intercombination transition near the 5 s 5 p P 13 – 5 p 2 P 23 transition, Physical Review A 105,012821 (2022).

[0023] A. Kulosa, D. Fim, K. Zipfel, S. R¨ uhmann, S. Sauer, N. Jha, K. Gibble, W. Ertmer, E. Rasel, M. Safronova, U. Safronova, and S. Porsev, Towards a Mg Lattice Clock: Observation of the S 0 1 - P 0 3 Transition and Determination of the Magic Wavelength, Physical Review Letters 115, 240801 (2015).

[0024] M. S. Safronova, S. G. Porsev, and C. W. Clark, Ytterbium in Quantum Gases and Atomic Clocks: van der Waals Interactions and Blackbody Shifts, Physical Review Letters 109, 230802 (2012).

[0025] Z.-M. Tang, Y.-M. Yu, J. Jiang, and C.-Z. Dong, Magic wavelengths for the $6{s}ˆ{2}{}ˆ{1}{S} {0}\mbox{–}6s6p{}ˆ{3}{P} {1}ˆ{o}$ transition in ytterbium atom, Journal of Physics B: Atomic, Molecular and Optical Physics 51, 125002 (2018).

[0026] A. Yamaguchi, M. Safronova, K. Gibble, and H. Katori, Narrow-line Cooling and Determination of the Magic Wavelength of Cd, Physical Review Letters 123, 113201 (2019).

[0027] A. Golovizin, E. Fedorova, D. Tregubov, D. Sukachev, K. Khabarova, V. Sorokin, and N. Kolachevsky, Innershell clock transition in atomic thulium with a small blackbody radiation shift, Nature Communications 10,1724 (2019). 33 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88)

[0028] L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: A review of experiments with magnetic quantum gases (2022), number: arXiv:2201.02672 arXiv:2201.02672 [cond-mat,physics:physics, physics:quant-ph].

[0029] A. de Paz, A. Chotia, E. Mar´echal, P. Pedri, L. Vernac, O. Gorceix, and B. Laburthe-Tolra, Resonant demagnetization of a dipolar Bose-Einstein condensate in a three- dimensional optical lattice, Physical Review A 87, 051609 (2013).

[0030] S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Darkwah Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, and J. Ye, A Fermi- degenerate three-dimensional optical lattice clock, Science 358, 90 (2017).

[0031] T. Nicholson, S. Campbell, R. Hutson, G. Marti, B. Bloom, R. McNally, W. Zhang, M. Barrett, M. Safronova, G. Strouse, W. Tew, and J. Ye, Systematic evaluation of an atomic clock at 2 × 10-18 total uncertainty, Nature Communications 6, 6896 (2015).

[0032] S. Porsev, M. Safronova, U. Safronova, and M. Kozlov, Multipolar Polarizabilities and Hyperpolarizabilities in the Sr Optical Lattice Clock, Physical Review Letters 120, 063204 (2018).

[0033] I. Ushijima, M. Takamoto, and H. Katori, Operational Magic Intensity for Sr Optical Lattice Clocks, Physical Review Letters 121, 263202 (2018).

[0034] K. Ton, G. Kestler, D. Filin, C. Cheung, P. Schneeweiss, T. Hoinkes, J. Volz, M. S. Safronova, A. Rauschenbeutel, and J. T. Barreiro, State-Insensitive Trapping of Alkaline- Earth Atoms in a Nanofiber-Based Optical Dipole Trap (2022), publisher: arXiv Version Number: 1.

[0035] S. G. Porsev and A. Derevianko, Multipolar theory of blackbody radiation shift of atomic energy levels and its implications for optical lattice clocks, Physical Review A 74, 020502 (2006).

[0036] C. Lisdat, S. D¨ orscher, I. Nosske, and U. Sterr, Blackbody radiation shift in strontium lattice clocks revisited, Physical Review Research 3, L042036 (2021).

[0037] https: / / udspace.udel.edu / handle / 19716 / 29071 (2021) [S1] M. S. Safronova, M. G. Kozlov, W. R. Johnson, and D. Jiang, Development of a configuration-interaction plus all-order method for atomic calculations, Physical Review A 80, 012516 (2009). [S2] A. Kramida and Y. Ralchenko, NIST Atomic Spectra Database, NIST Standard Reference Database 78 (1999), type: dataset. [S3] M. S. Safronova, S. G. Porsev, U. I. Safronova, M. G. Kozlov, and C. W. Clark, Blackbody-radiation shift in the Sr optical atomic clock, Physical Review A 87, 012509 (2013). 34 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) [S4] S. Eustice, K. Cassella, and D. Stamper-Kurn, Laser cooling of transition-metal atoms, Physical Review A 102, 053327 (2020). [S5] V. A. Dzuba, V. V. Flambaum, M. G. Kozlov, and S. G. Porsev, Using e_ective operators in calculating the hyperfine structure of atoms, Journal of Experimental and Theoretical Physics 87, 885 (1998). [S6] S. G. Porsev, Y. G. Rakhlina, and M. G. Kozlov, Electric-dipole amplitudes, lifetimes, and polarizabilities of the low-lying levels of atomic ytterbium, Physical Review A 60, 2781 (1999). [S7] S. G. Porsev, Y. G. Rakhlina, and M. G. Kozlov, Calculation of hyperfine structure constants for ytterbium, Journal of Physics B: Atomic, Molecular and Optical Physics 32, 1113 (1999). [S8] C. Cheung, M. Safronova, and S. Porsev, Scalable Codes for Precision Calculations of Properties of Complex Atomic Systems, Symmetry 13, 621 (2021). [S9] F. Le Kien, P. Schneeweiss, and A. Rauschenbeutel, Dynamical polarizability of atoms in arbitrary light fields: general theory and application to cesium, The European Physical Journal D 67, 10.1140 / epjd / e2013-30729-x (2013). [S10] J. Bai, X. Wang, X. Hou, W. Liu, and J. Wang, Angle- Dependent Magic Optical Trap for the 6S1 / 2-nP3 / 2 Rydberg Transition of Cesium Atoms, Photonics 9, 303 (2022). [S11] A. de Paz, A. Chotia, E. Mar´echal, P. Pedri, L. Vernac, O. Gorceix, and B. Laburthe-Tolra, Resonant demagnetization of a dipolar Bose-Einstein condensate in a three- dimensional optical lattice, Physical Review A 87, 051609 (2013). [S12] L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: A review of experiments with magnetic quantum gases (2022), number: arXiv:2201.02672 arXiv:2201.02672 [cond-mat, physics:physics, physics:quant-ph]. [S13] J. Choi, H. Zhou, H. S. Knowles, R. Landig, S. Choi, and M. D. Lukin, Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems, Physical Review X 10, 031002 (2020). [S14] T. Polenova, R. Gupta, and A. Goldbourt, Magic Angle Spinning NMR Spectroscopy: A Versatile Technique for Structural and Dynamic Analysis of Solid-Phase Systems, Analytical Chemistry 87, 5458 (2015) 35 4854-2293-8064v.3

Claims

Attorney Docket # 00495-0014 (B2023-88) CLAIMS: What is claimed is:

1. An optical clock comprising: (A)a magneto optical trap configured to cool Titanium atoms using a first set of laser beams, each having a respective cooling wavelength corresponding to a first state transition between an excited state and a first state; (B) an optical trap configured to trap the cooled Titanium atoms in a vacuum chamber using a second set of laser beams, each having a respective magic trapping wavelength; and (C) a clock laser configured to cause trapped Titanium atoms in the first state to transition to a second state using a clock laser beam having a clock wavelength corresponding to a second state transition between the first state and the second state.

2. The optical clock of claim 1, wherein the magneto optical trap comprises a magnetic field generator configured to generate a magnetic field gradient.

3. The optical clock of claim 1, wherein the magneto optical trap comprises a first cooling laser configured to emit a first laser beam of the first set of laser beams and grating configured to diffract the first laser beam to generate a first plurality of laser beams of the first set of laser beams.

4. The optical clock of claim 1, wherein the magneto optical trap comprises a plurality of cooling lasers, wherein each respective cooling laser of the plurality of cooling laser is configured to emit a respective laser beam of the first set of laser beams.

5. The optical clock of claim 4, wherein at least one of the first set of laser beams is split to form a plurality of split cooling laser beams.

6. The optical clock of claim 4, wherein the plurality of cooling lasers comprises six cooling lasers.

7. The optical clock of claim 4, wherein the magneto optical trap comprises a plurality of mirrors, each configured to reflect a respective laser beam of the first set of laser beams emitted from a respective cooling laser beam of the plurality of cooling lasers.

8. The optical clock of claim 7, wherein each mirror of the plurality of mirrors is configured to reflect the respective laser beam back along a path of the respective laser beam.

9. The optical clock of claim 7, wherein the plurality of cooling lasers comprises three cooling lasers and the plurality of mirrors comprises three mirrors. 36 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) 10. The optical clock of claim 1, wherein at least one laser beam of the first set of laser beams has a respective cooling wavelength between 497.9 nm and 498.1 nm.

11. The optical clock of claim 1, wherein at least one laser beam of the first set of laser beams has a respective cooling wavelength 1039.9 nm and 1040.1 nm.

12. The optical clock of claim 10, wherein at least one laser beam of the first set of laser beams has a respective cooling wavelength 1039.9 nm and 1040.1 nm.

13. The optical clock of claim 1, wherein the first state is a5F5.

14. The optical clock of claim 1, wherein the optical clock comprises focusing optics configured to form optical tweezers from the second set of laser beams.

15. The optical clock of claim 1, wherein the optical clock comprises a first optical trap laser configured to emit a first laser beam of the second set of laser beams, and a mirror configured to reflect the first laser beam to form a second laser beam of the second set of laser beams, wherein the first laser beam and the second laser beam form an optical lattice.

16. The optical clock of claim 1, wherein the optical clock comprises a first optical trap laser configured to emit a first laser beam of the second set of laser beams, and a second laser configured to emit a second laser beam of the second set of laser beams, wherein the first laser beam and the second laser beam form an optical lattice.

17. The optical clock of claim 1, wherein the optical clock comprises a first optical trap laser configured to emit a first laser beam of the second set of laser beams, and a plurality of mirrors configured to reflect one or more of the second set of laser beams, wherein the first laser beam and the reflected one or more of the second set of laser beams form an optical lattice.

18. The optical clock of claim 1, wherein the optical clock comprises: a. a first optical trap laser, configured to emit a first laser beam of the second set of laser beams; b. a first mirror configured to reflect the first laser beam to form a second laser beam of the second set of laser beams; c. a second optical trap laser, configured to emit a third laser beam of the second set of laser beams; d. a second mirror configured to reflect the third laser beam to form a fourth laser beam of the second set of laser beams; e. a third optical trap laser, configured to emit a fifth laser beam of the second set of laser beams; and 37 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) f. a third mirror configured to reflect the fifth laser beam to form a sixth laser beam of the second set of laser beams; wherein the second set of laser beams form a three-dimensional optical lattice configured to trap Titanium atoms.

19. The optical clock of claim 18, wherein the first mirror is configured to reflect the first laser beam back along a path of the first laser beam, the second mirror is configured to reflect the third laser beam back along a path of the third laser beam, and the third mirror is configured to reflect the fifth laser beam back along a path of the fifth laser beam.

20. The optical clock of claim 1, wherein the optical clock comprises: a. a first optical trap laser, configured to emit a first laser beam of the second set of laser beams; b. a second optical trap laser, configured to emit a second laser beam of the second set of laser beams; c. a third optical trap laser, configured to emit a third laser beam of the second set of laser beams; and d. a fourth optical trap laser, configured to emit a fourth laser beam of the second set of laser beams; e. a fifth optical trap laser, configured to emit a fifth laser beam of the second set of laser beams; f. a sixth optical trap laser, configured to emit a sixth laser beam of the second set of laser beams; and wherein the second set of laser beams form a three-dimensional optical lattice configured to trap Titanium atoms.

21. The optical clock of claim 1, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 1036 nm and 1037 nm.

22. The optical clock of claim 1, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 883 nm and 891 nm.

23. The optical clock of claim 21, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 883 nm and 891 nm.

24. The optical clock of claim 1, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 786.8 nm and 794 nm. 38 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) 25. The optical clock of claim 21, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 786.8 nm and 794 nm.

26. The optical clock of claim 22, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 786.8 nm and 794 nm.

27. The optical clock of claim 23, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 786.8 nm and 794 nm.

28. The optical clock of claim 1, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 774 nm and 784 nm.

29. The optical clock of claim 21, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 774 nm and 784 nm.

30. The optical clock of claim 22, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 774 nm and 784 nm.

31. The optical clock of claim 23, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 774 nm and 784 nm.

32. The optical clock of claim 24, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 774 nm and 784 nm.

33. The optical clock of claim 25, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 774 nm and 784 nm.

34. The optical clock of claim 26, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 774 nm and 784 nm.

35. The optical clock of claim 27, wherein at least one laser beam of the second set of laser beams has a respective magic trapping wavelength between 774 nm and 784 nm.

36. The optical clock of claim 1, wherein the second state is a3F4.

37. The optical clock of claim 1, further comprising an optical detector configured to measure state-dependent light scattering.

38. The optical clock of claim 1, wherein the frequency of the magic trapping wavelength is configured to be substantially independent from the intensity of the second set of laser beams.

39. The optical clock of claim 1, wherein the second set of laser beams is configured to reduce magnetic interactions between the Titanium atoms.

40. The optical clock of claim 1, wherein the clock wavelength is in the telecom band. 39 4854-2293-8064v.3Attorney Docket # 00495-0014 (B2023-88) 41. The optical clock of claim 40, wherein the telecom band is in a short band between 1460 nm and 1530 nm.

42. The optical clock of claim 40, wherein the telecom band is in a conventional band between 1530 and 1565 nm.

43. The optical clock of claim 40, wherein the telecom band is in a long band between 1565 nm and 1625 nm.

44. The optical clock of claim 40, wherein the clock wavelength is between 1260 and 1625 nm.

45. The optical clock of claim 40, wherein a telecom band transition of the telecom band has a linewidth of less than 60 microhertz.

46. The optical clock of claim 40, wherein a telecom band transition of the telecom band has a linewidth of less than 250 microhertz.

47. The optical clock of claim 40, wherein a telecom band transition of the telecom band has a linewidth of less than 1 millihertz.

48. The optical clock of claim 1, wherein the clock wavelength is 1548.926 nm.

49. The optical clock of claim 1, wherein the clock wavelength is 1573.346 nm.

50. The optical clock of claim 1, wherein the optical clock is configured to make adjustments the clock wavelength based on measurements of the clock wavelength.

51. The optical clock of claim 1, wherein the optical clock is used in fiber-linked clock networks.

52. The optical clock of claim 1, wherein the clock frequency of the optical clock does not substantially shift as a result of black body radiation. 40 4854-2293-8064v.3

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