Method and sensor for determining an angular acceleration of an external rotation

Abstract

The invention relates to a method and a sensor (100) for determining an angular acceleration of an external rotation, comprising preparing a plurality of atoms (100) in a first internal electron state (12) in a trap (200), irradiating a first pulse of a first analysis laser beam (26) and a second analysis laser beam (28) onto the atoms (100) trapped in the trap (200) to generate a uniformly distributed superposition between a first and second angular momentum state (13, 15) of a center of gravity movement of the atoms (100); irradiating a second pulse of the first and second analysis laser beams (26, 28) onto the atoms (100) to exchange a population (50) of the atoms (100) between the angular momentum states (13, 15); irradiating a third pulse of the analysis laser beams (26, 28) onto the atoms (100); Determine the population (50) of atoms (100) in the first internal electron state (12).

Description

State of the art

[0001] The invention relates to a method for determining an angular acceleration of an external rotation and to a sensor for carrying out a method for determining an angular acceleration of an external rotation.

[0002] An external rotation generally leads to three non-inertial forces, the Coriolis force (F C ), the centrifugal force (F Z ) and the Euler force (F E ). The first two are determined by the angular velocity Ω, F C ∝ Ω and F Z ∝ Ω 2 However, if this changes over time, there is also an angular acceleration Ω̇ and the Euler force, F E ∝ Ω̇, comes into play.

[0003] In the case of external rotations with constant angular velocity Ω and low amplitude, the Coriolis force always dominates the influence on the system under investigation. Such external rotations can be precisely measured using classical and quantum gyroscopes based on so-called Sagnac interferometers with electromagnetic waves or matter waves. As described, for example, in Ramanathan, AK (2011), "A ring with a spin: Superfluidity in a toroidal Bose-Einstein condensate", University of Maryland, College Park, torus geometries exist for matter waves in which the interferometer arms enclose a circular area (multiple times), thereby measuring the Sagnac phase.

[0004] Furthermore, the publication by Garrido Alzar, CL: Compact chipscale guided cold atom gyrometers for inertial navigation: Enabling technologies and design study. In: AVS Quantum Sci., 1, December 2019, 1, 014702-1 - 014702-18. DOI: 10.1116 / 1.5120348 provides an overview of Sagnac interferometers.

[0005] In addition, the influence of the Coriolis force on freely falling atoms can be measured in unguided atom interferometers and the angular velocity Ω can be determined.

[0006] In the case of time-varying rotational movements with non-vanishing angular velocity Ω, the procedure may differ depending on the amplitude of this value.

[0007] For changes in angular velocity much slower than the repetition rate of the experiment, the direct influence of the Euler force can be neglected, and the angular velocity Ω can be measured using the techniques mentioned above. As shown in Schreiber, KU, Kodet, J., Hugentobler, U., Klügel, T., & Wells, JPR (2023). Variations in the Earth's rotation rate measured with a ring laser interferometer. Nature Photonics, 17(12), 1054-1058, by making a sufficient number of measurements, Ω can be plotted completely as a function of time.

[0008] However, for rapid changes, such as in the field of seismology or to improve navigation technology, a direct measurement of the angular acceleration Ω is of interest.

[0009] For this purpose, conventional devices exist, such as those described in Nusbaum, U., Rusnak, I., & Klein, I. (2019). Angular accelerometer-based inertial navigation system. Navigation, 66(4), 681-693, based on various technologies. For example, angular acceleration sensors based on liquid rotors, microfluidic channels, amorphous wires, piezoelectric elements, or micro-electromechanical systems (MEMS) exist.

[0010] From WO2020141536A1, a measuring device is known for the accurate and precise measurement of a property selected from the group consisting of external linear acceleration or effective accelerations due to magnetic field strength, external rotation with constant angular velocity, and gravitational field strength, comprising an optical grating generated by coherent electromagnetic waves generated by 3, 4, or 6 coherent lasers arranged in a ring at 120-degree, 90-degree, or 60-degree intervals, and individual atoms trapped in the optical grating, and a system comprising an absorption line detector for monitoring shifts in absorption lines of atoms in the optical grating.

[0011] US20170299389A1 discloses an inertial measurement unit based on atom interferometry. The inertial measurement unit comprises a vacuum chamber, first and second atom trapping sites housed in the vacuum chamber, each of the first and second atom trapping sites being selectively configured to trap and cool first and second atom samples of different atomic species, an atom interferometry region disposed between the first and second atom trapping sites, and first and second atom interferometers operating in the atom interferometry region.

[0012] The first atom interferometer is configured to produce a first measurement corresponding to the common inertial parameters (linear acceleration and angular velocity) based on the first atom sample, and the second atom interferometer is configured to produce a second measurement corresponding to the same common inertial parameters based on the second atom sample. Disclosure of the invention

[0013] The object of the invention is to provide an improved method for determining an angular acceleration of an external rotation.

[0014] A further object is to provide a sensor for carrying out such a method for determining an angular acceleration of an external rotation.

[0015] The objects are achieved by the features of the independent claims. Advantageous embodiments and advantages of the invention emerge from the further claims, the description, and the drawings.

[0016] According to one aspect of the invention, a method for determining an angular acceleration of an external rotation is proposed, comprising preparing a plurality of atoms in a first internal electron state in a trap which restricts movement of the atoms to a region within a torus and cooling the plurality of atoms in the trap to a ground state, wherein the atoms move with quantized angular momentum in the same direction within the torus; irradiating a first pulse of a first analysis laser beam, in particular a Laguerre-Gaussian laser beam, and a second analysis laser beam, in particular a Gaussian laser beam, onto the atoms trapped in the trap to generate a uniformly distributed superposition between a first and a second quantum mechanical angular momentum state of a center-of-mass movement of the atoms; developing the atoms in the trap for an interrogation time;Irradiating a second pulse of the first and second analysis laser beams onto the atoms to completely exchange a population of the atoms between the two angular momentum states; developing the atoms in the trap for an interrogation time; irradiating a third pulse of the first and second analysis laser beams onto the atoms to superpose the two quantum mechanical angular momentum states of the center-of-mass motion of the atoms and to exchange at least a portion of the population of the atoms between the two angular momentum states; determining the population of the atoms in the first internal electron state; determining the angular acceleration from the population in the first internal electron state.

[0017] According to the proposed method, the angular acceleration Ω̇ of an external rotation is directly measured using interferometric means, such as the interference of matter waves. The method is based on a quantum mechanical sensor. A torus geometry compensates for the Coriolis force and centrifugal force, thereby directly measuring the Euler force. No spatial area is enclosed in the interferometer, thus eliminating the Sagnac phase.

[0018] A major radius of the torus, which describes the distance from the center of the torus to the center of the torus tube, can advantageously be chosen significantly larger than a minor radius, which describes the mean radius of the tube. This creates a narrow torus, so that the movements of the atoms in the torus are subject to effective one-dimensional dynamics along the azimuthal angle, and no relevant movements of the atoms occur in the radial direction.

[0019] In contrast to existing classical, mostly mechanical sensors for measuring angular acceleration, the proposed method utilizes the behavior of ultracold quantum gases. The Euler force acting on the atoms manifests itself in the phase of the wave function, and this effect can then be read interferometrically through the interference properties of the matter waves.

[0020] Unlike existing atom interferometers with freely falling atoms, which are used as sensors for the angular velocity Ω, a toroidal dipole trap can be used, so that the influence of Coriolis force and centrifugal force is compensated.

[0021] In contrast to existing concepts and setups of torus matter-wave interferometers, in which atoms are guided around the torus in both directions and thus enclose a surface upon joining, making these Sagnac interferometers sensitive to the angular velocity Ω, the proposed method involves atoms moving in the same direction around the torus. They traverse the same path in the two possible interferometer arms with a time offset. Accordingly, no spatial surface is enclosed, and the Sagnac phase disappears. The time offset of the two movements allows the angular acceleration Ω̇ to be measured.

[0022] In addition to its high sensitivity, one advantage of this method is its drift-free nature. This means it is independent of calibration with other sensors and is therefore of interest in the field of inertial / autonomous navigation, for example.

[0023] In cross-section, a Gaussian beam exhibits a profile according to a Gaussian curve with a width that varies along the propagation axis. The beam tapers approximately linearly until it reaches its narrowest point, known as the focus or waist, and then grows again in the same way. Along the propagation axis, the spatial intensity of the beam exhibits a Lorentzian profile, with the maximum occurring at the waist.

[0024] The electromagnetic field of a Gaussian beam is derived from Maxwell's equations for a constant frequency ω, i.e., from the Helmholtz equation, using a paraxial approximation. For a given propagation direction and wavelength, the Gaussian beam is completely determined by specifying the location and beam diameter of the waist.

[0025] Laguerre-Gaussian beams are typical vortex beams with orbital angular momentum, spiral wavefront phase, and toroidal intensity distribution. Laguerre-Gaussian beam generation methods include passive and active methods. In passive methods, Gaussian beams are modulated outside a resonator by the phase elements in Laguerre-Gaussian beams; in active methods, the high-order transverse modes are directly excited in the resonant cavity to obtain Laguerre-Gaussian beams.

[0026] According to the proposed method, a timeline for interferometric measurement of angular acceleration can be created.

[0027] First, the atoms are prepared in a first internal electron state |1〉 and cooled to the motional ground state of the trap with a vanishing angular momentum.

[0028] During the two-photon transition with the two analysis laser beams, angular momentum ℏl is transferred through the absorption of the Laguerre-Gaussian laser beam. As a result, after the transition, the atoms are in the second internal electron state |2> and have the angular momentum state ℏl in the center-of-mass motion.

[0029] In this way, the angular momentum states of the external degree of freedom of the center of mass motion of the atoms are linked to the internal electron states in the sequence.

[0030] The first pulse, a so-called π / 2 pulse, creates a uniformly distributed superposition of the first angular momentum state in the first electron state |1〉 and the second angular momentum state in the second electron state |2〉. The portion of the atoms in the second angular momentum state, and thus in the second electron state, rotates with constant angular momentum ℏl in the torus of the trap. The atoms in the first electron state |1〉 have no angular momentum.

[0031] Advantageously, the populations of the two angular momentum states can be varied sinusoidally as a function of the combined effective pulse area. If the area is chosen such that the population is completely swapped, it is called a π pulse. In this case, the area is equal to π. If the area is half as large, it is a π / 2 pulse.

[0032] After this pulse (at a time t = 0), the atoms in the trap are developed for the interrogation time T.

[0033] At time t = T, a second pulse, a π-pulse, is applied. This swaps the internal electron states and angular momentum. The atoms in the first electron state |1〉, with no angular momentum, are transferred to the second electron state |2〉, with angular momentum ℏl. Accordingly, the fraction that was previously in the second electron state |2〉, with angular momentum ℏl, is transferred to the first internal electron state |1〉, with no angular momentum.

[0034] After this pulse (at t = T), the atoms are again developed in the trap for time T.

[0035] At time t = 2T, the interferometer is closed with a third pulse, a π / 2 pulse.

[0036] After this pulse, the population in the first internal electron state |1〉 is determined, for example, by microwave excitation followed by absorption imaging.

[0037] The angular acceleration can then be determined from the population.

[0038] With a favorable refinement of the method, the population of atoms in the first internal electron state can be determined by microwave excitation followed by absorption imaging. This advantageously allows the population to be determined using a proven method.

[0039] According to a favorable design of the process, the pulse area acting on the atoms of the first and third pulses can be half the size of the pulse area acting on the atoms of the second pulse. Advantageously, the populations of the two angular momentum states can be changed sinusoidally as a function of the combined effective pulse area. If the area is chosen such that the population is completely swapped, it is called a π pulse. In this case, the area is equal to π. If the area is half as large, it is a π / 2 pulse.

[0040] In a favorable embodiment of the method, the first analysis laser beam and the second analysis laser beam can be phase-locked. This allows the populations to be advantageously swapped.

[0041] According to a favorable embodiment of the method, the angular acceleration can be determined from a period of the population in the first internal electron state for different interrogation times. In particular, the angular acceleration can be determined from the period of the population in the first internal electron state as a function of the square of the interrogation time.

[0042] Such a relationship can be advantageously used to determine the population.

[0043] According to a favorable embodiment of the method, a Raman transition between at least two internal electron states can be triggered by the first and second analysis laser beams, whereby a quantized angular momentum is transferred to the center-of-mass motion of the atoms. In particular, the Raman transition can induce Rabi oscillations between the at least two internal electron states. The populations in these states can be changed sinusoidally as a function of the combined effective pulse area.

[0044] According to a favorable embodiment of the method, irradiation of the first and second analysis laser beams can cause a transition between the at least two angular momentum states of the center-of-gravity motion of the atoms trapped in the trap. This advantageously allows for population exchange.

[0045] According to a favorable embodiment of the method, the atoms can be localized in the torus, and a transition between the at least two angular momentum states of the atoms' center-of-mass motion can be achieved by transferring a pulse to the trapped atoms via a Bragg pulse. Alternatively, switching between two angular momentum states can be achieved by dispersing the atoms not throughout the torus, but rather by localizing them, and applying a so-called Bragg pulse to transfer a pulse to the atoms at this point without changing their internal electronic state.

[0046] According to a favorable embodiment of the method, the trap can be an optical dipole trap, which is formed by superimposing a light sheet and a laser beam. The light sheet confines the atoms along a spatial axis to a spatial segment that is bounded in both directions, particularly a planar one. The laser beam is oriented perpendicular to the spatial segment and defines a ring-shaped intensity profile in the spatial segment and a torus. The atoms are trapped in the torus, and their movement is limited to an azimuthal angle. This advantageously confines the movement of the atoms to the torus of the optical dipole trap.

[0047] The light sheet represents a disc-shaped illuminated area that is quasi-one-dimensional in a direction perpendicular to the propagation direction. The light sheet can be generated, for example, by scanning a laser beam perpendicular to the propagation direction.

[0048] According to a further aspect of the invention, a sensor for carrying out a method for determining an angular acceleration of an external rotation is proposed, comprising a plurality of, in particular ultra-cold, atoms which have a transition from an energetic ground state to at least one energetically excited state of their electron system, a radiation-induced trap which restricts a movement of the atoms to a torus, wherein the atoms have at least a first and a second internal electron state with which the atoms are trapped in the trap, and a means for coherently switching between the at least two internal electron states with an angular momentum transfer to the center of mass movement of the atoms.

[0049] To implement the proposed method, a sensor can advantageously be used that contains ultra-cold atoms with a suitable transition between electronic states. Furthermore, the sensor comprises a trap for the atoms, which restricts their motion to a torus. The atoms trapped in the trap exhibit two specific angular momentum states. Furthermore, the sensor has the ability to coherently switch between the two internal electron states with an angular momentum transfer to the center-of-mass motion of the atoms. This effectively allows a system of two angular momentum states to be realized and applied.

[0050] Such sensors function in modified forms, for example, with different atoms, different transitions, different modes of operation of the torus-shaped trap, or different two-photon transitions, as long as they transfer an effective angular momentum. Furthermore, switching between two angular momentum states can be achieved in an alternative way by dispersing the atoms not across the entire torus but rather localizing them, and applying a so-called Bragg pulse to the atoms at this point without changing their internal electronic state.

[0051] According to a favorable design of the sensor, the angular acceleration can be determined by determining a respective population of atoms in at least two angular momentum states of the center-of-gravity motion of the atoms. A relationship between angular acceleration and time-dependent population can be advantageously exploited.

[0052] According to a favorable sensor design, the angular acceleration from a period of the population in the first angular momentum state can be determined for different interrogation times. In particular, the angular acceleration from the period of the population in the first angular momentum state can be determined as a function of the square of the interrogation time. Such a relationship can be advantageously exploited for population determination.

[0053] With a favorable sensor design, an imaging device can be provided that allows the population to be determined by microwave excitation followed by absorption imaging. This advantageously allows the population to be determined using a proven method.

[0054] According to a favorable design of the sensor, the atoms can be sodium atoms, which allow a transition of a valence electron from the 3 2 S 1 / 2 Ground state in the excited 3 2P 3 / 2 Such a system can be used to advantageously implement a sensor for the proposed method.

[0055] According to a favorable embodiment of the sensor, the trap can be designed as an optical dipole trap formed by superimposing a light sheet and a laser beam. The light sheet restricts the movement of the atoms along a spatial axis to a spatial segment that is bounded in both directions, particularly a planar one. The laser beam is oriented transversely, particularly perpendicularly, to the spatial segment and has an annular intensity profile in the spatial segment, defining a torus. The atoms are trapped in the torus, and their movement is restricted to an azimuthal angle. This advantageously restricts the movement of the atoms to the torus of the optical dipole trap.

[0056] According to a favorable design of the sensor, a magnetic field can be applied parallel to the light sheet, which creates a Zeeman splitting of the total angular momentum of the atoms into the at least two internal electron states. The magnetic field advantageously allows for the exploitation of a quadratic Zeeman effect, so that the transition frequencies between different transitions are not the same.

[0057] According to a favorable embodiment of the sensor, a first analysis laser beam, in particular a Laguerre-Gaussian laser beam, and a second analysis laser beam, in particular a Gaussian laser beam, can be formed parallel to the first analysis laser beam. The first and second analysis laser beams have a frequency difference that corresponds to an energy difference between the at least two internal electron states. In this way, populations can be advantageously exchanged between the different angular momentum states.

[0058] With a favorable sensor design, the first and second analysis laser beams can be phase-locked to each other. This allows for advantageous population swapping.

[0059] According to a favorable embodiment of the sensor, the first and second analysis laser beams can be configured to trigger a Raman transition between the at least two internal electron states, whereby a quantized angular momentum can be transferred to the center-of-mass motion of the atoms. In particular, the Raman transition can result in Rabi oscillations between the at least two internal electron states of the atoms. The populations in these states can be changed sinusoidally as a function of the combined effective pulse area.

[0060] According to a favorable design of the sensor, a transition between the at least two angular momentum states of the atoms' center-of-gravity motion can be effected by irradiating the first and second analysis laser beams onto the atoms trapped in the trap. In this way, the populations of the at least two angular momentum states can be advantageously swapped.

[0061] According to a favorable design of the sensor, the atoms can be localized in the torus, and a transition between the at least two angular momentum states of the atoms' center-of-gravity motion can be achieved by transferring a pulse to the atoms trapped in the trap via a Bragg pulse. Alternatively, switching between two angular momentum states can be achieved by dispersing the atoms not throughout the torus, but rather by localizing them, and applying a so-called Bragg pulse to transfer a pulse to the atoms at this point without changing their internal electronic state. drawing

[0062] Further advantages will become apparent from the following description of the drawings. The figures illustrate exemplary embodiments of the invention. The figures, the description, and the claims contain numerous features in combination. Those skilled in the art will also expediently consider the features individually and combine them into useful further combinations.

[0063] Examples include: Fig. 1 shows an energy diagram of an interferometer of a sensor for determining an angular acceleration of an external rotation using a method according to an embodiment of the invention; Fig. 2 an optical dipole trap of the sensor in a schematic representation; Fig. 3 a population of atoms in a first angular momentum state as a function of a square of an interrogation time; and Fig. 4 a block diagram of a sensor for determining an angular acceleration of an external rotation according to an embodiment of the invention. Embodiments of the invention

[0064] In the figures, components of the same type or function similarly are designated by the same reference numerals. The figures are merely examples and are not to be construed as limiting.

[0065] The directional terminology used below, including terms such as "left," "right," "top," "bottom," "before," "behind," "after," and the like, is intended solely to enhance understanding of the figures and is in no way intended to limit the scope of the invention. The components and elements depicted, as well as their design and use, may vary according to the considerations of a person skilled in the art and may be adapted to specific applications.

[0066] Fig. 1 shows an energy diagram of an interferometer of a sensor 1000 for determining an angular acceleration Ω̇ of an external rotation using a method according to an embodiment of the invention. In Fig. 2, an optical dipole trap 200 of the sensor 1000 is shown schematically.

[0067] The sensor 1000 comprises a plurality of, in particular ultra-cold, atoms 100, which exhibit a transition from an energetic ground state 10 to at least one energetically excited state 16 of their electron system, a radiation-induced trap 200, which restricts a movement of the atoms 100 to a torus 22, wherein the atoms 100 have at least a first and a second internal electron state 12, 14, with which the atoms 100 are trapped in the trap 200, and a means 25 for coherently switching between the at least two internal electron states 12, 14 with an angular momentum transfer to the center of mass movement of the atoms 100.

[0068] A major radius of the torus 22, which describes a distance from the center of the torus 22 to the center of the tube of the torus 22, can advantageously be chosen to be significantly larger than a minor radius, which describes a mean radius of the tube. This creates a narrow torus 22, so that movements of the atoms 100 in the torus 22 are subject to effective one-dimensional dynamics along the azimuthal angle, and no relevant movements of the atoms 100 occur in the radial direction.

[0069] The angular acceleration Ω̇ can be determined by determining a respective population 50 of the atoms 100 in the at least two internal electron states 12, 14 of the center of mass motion of the atoms 100.

[0070] As means 25, a first analysis laser beam 26, in particular a Laguerre-Gaussian laser beam, and a second analysis laser beam 28, in particular a Gaussian laser beam, are formed parallel to the first analysis laser beam 26. The first and second analysis laser beams 26, 28 have a frequency difference that corresponds to an energy difference between the at least two internal electron states 12, 14. The first and second analysis laser beams 26, 28 can advantageously be phase-locked to one another.

[0071] By irradiating the first and second analysis laser beams 26, 28 onto the atoms 100 trapped in the trap 200, a transition between at least two angular momentum states 13, 15 of the center of gravity motion of the atoms 100 can be effected.

[0072] Alternatively, the atoms 100 may be localized in the torus 22 and a transition between the at least two angular momentum states 13, 15 of the center of mass motion of the atoms 100 may be effected by transferring a pulse by a Bragg pulse to the atoms 100 trapped in the trap 200.

[0073] The atoms 100 can be, for example, sodium atoms ( 23 Na), which involves a transition of a valence electron from the 3 2 S 1 / 2 Ground state 10 in the excited 3 2 P 3 / 2 state 16.

[0074] The optical spectral line D2 (λ D2 ≈ 589 nm) to manipulate the atoms 100. The D2 line corresponds to the transition of the valence electron from the 3 2 S 1 / 2 Ground state 10 in the excited 3 2 P 3 / 2 State 16, which is caused by the spin-orbit coupling of the 3 2 P 1 / 2 state is degenerate.

[0075] In Fig. 1 shows the D2 line and the energy levels used for the interferometer of sensor 1000. The thick arrow corresponds to the Laguerre-Gaussian laser beam 20 of the dipole trap 200 and shows its frequency. This is far detuned from the transition and thus creates the torus-shaped trap 200 for all states of the 3 2 S 1 / 2 Manifold. The Raman beam pair of the Laguerre-Gaussian laser beam 26 in combination with the Gaussian laser beam 28 is detuned against the D2 line, but together resonant to the |1〉 ↔ |2〉 transition. The choice of the respective polarizations 27, 29 ( Fig. 2) and the quadratic Zeeman effect then leads to an effective two-level system.

[0076] To spatially trap the sodium atoms, an optical dipole trap 200 is used, as shown, for example, in Ramanathan, AK (2011). A ring with a spin: Superfluidity in a toroidal Bose-Einstein condensate. University of Maryland, College Park and in Ramanathan, A., Wright, KC, Muniz, SR, Zelan, M., Hill III, WT, Lobb, CJ, ... & Campbell, GK (2011). Superflow in a toroidal Bose-Einstein condensate: an atom circuit with a tunable weak link. Physical Review Letters, 106(13), 130401.

[0077] In this dipole trap 200, light with a redshift (ω Dipol - ω D2 < 0, where ω D2 = [E(3 2 P 3 / 2 ) - E(3 2 S 1 / 2 )] / ℏ, Fig. 1) used for D2 transition: λ Dipol ≈ 1030 nm), whereby the atoms feel a potential proportional to the light intensity and can be trapped in regions of high intensity. To create a torus-shaped trap 200 for the atoms 100, as in Fig. To illustrate the situation shown in Figure 2, two traps are combined. First, a so-called light sheet 24 in the form of a, particularly planar, spatial segment 32 is used, which effectively confines the atoms 100 in the z-direction to the z = 0 plane.

[0078] The light sheet 24 represents a disk-shaped illuminated area that is quasi-one-dimensional in a direction z perpendicular to the propagation direction y. The light sheet 24 can be generated, for example, by scanning a laser beam perpendicular to the propagation direction y.

[0079] In addition, a Laguerre-Gaussian laser beam 26 is used, which propagates along the z-axis. The spatial intensity profile orthogonal to the propagation direction of the Laguerre-Gaussian laser beam 26 is given by a radial parameter p and an azimuthal index l. By choosing p = 0 and l ≥ 1, a torus-shaped intensity profile is generated. Together, this results in an optical dipole trap 200, which 23 Na atoms in 3 2 S 1 / 2 Ground state 10 on a torus 22 and thus restricts the movement of atoms 100 to the azimuthal angle φ, as in Fig. 2 recognizable.

[0080] Fig. Figure 2 shows the dipole trap 200, consisting of the light sheet 24 in the form of the, in particular, planar, space segment 32, which traps the atoms 100 in the z-direction on the plane z = 0, and the Laguerre-Gaussian laser beam 20 (thick arrow). Together, this effectively creates a trap 200 in the form of a torus 22 for the 23Na atoms. The Raman transition is generated by the Laguerre-Gaussian laser beam as the first analysis laser beam 26 in combination with the Gaussian laser beam as the second analysis laser beam 28 (thinner arrows). The polarizations 27, 29 are shown relative to the magnetic field 40 (B) and were chosen accordingly to enable the transition.

[0081] The trap 200 is designed as an optical dipole trap formed by the superposition of the light sheet 24 and the laser beam 20. The movement of the atoms 100 along the spatial axis z is limited by the light sheet 24 to the spatial segment 32, which is bounded in both directions and is particularly planar. The laser beam 20 is oriented transversely, particularly perpendicularly, to the spatial segment 32 and has an annular intensity profile in the spatial segment 32, defining a torus 22. The atoms 100 are trapped in the torus 22. The movement of the atoms 100 is thus limited to an azimuthal angle.

[0082] The magnetic field 40 is applied parallel to the light sheet 24, by which a Zeeman splitting of the total angular momentum 11 of the atoms 100 into the at least two internal electron states 12, 14 is formed.

[0083] For the interferometric design of the sensor 1000, two electron states 12, 14 are used, between which a coherent switch is possible. For this purpose, the 3 2 S 1 / 2 Ground state 10 can be used in the hyperfine structure, which is created by the interaction of the electrons with the nucleus. This splits the energy level into two states 11, 17 with the total angular momenta F = 1 and F = 2, see Fig. 1. An applied constant magnetic field 40 splits the F = 1 state 11 further into three energy levels (m F = 0, m F= ±1). The strength of the magnetic field 40 (B ≈ 5G) is chosen so that the quadratic Zeeman effect produces the transition frequencies |F = 1, m F = -1) ↔ |F = 1, m F = 0〉 and |F = 1,m F = 0) ↔ |F = 1, m F = 1) are not equal. For interferometry, the first internal electron state 12 |1〉 ≈ |F = 1, m F = -1) and the second internal electron state 14 |2〉 ≈ |F = 1, m F = 0).

[0084] To achieve a transition between these two internal electron states 12, |1〉 and 14 |2〉 of the 3 2 S 1 / 2 To produce levels, a combination of the Gaussian laser beam as the second analysis beam 28 and the Laguerre-Gaussian laser beam as the first analysis beam 26, with p = 0 and l ≥ 1, is used. These propagate in the z-direction ( Fig. 2), i.e., in the direction of the symmetry axis of the torus 22. They have a detuning 30 (Δ ≈ -2.3 GHz) relative to the D2 line, which prevents transitions to the excited state 16 with a single beam. The frequency of the Gaussian laser beam 28 is higher than that of the Laguerre-Gaussian laser beam 26 by the frequency difference |1〉 ↔ |2〉. Therefore, together, as shown in Fig. 1, resonant with the transition |1〉 ↔ |2〉. The linear polarization 27 (ê LG ) of the Laguerre-Gauss laser beam 26, which is parallel to the external magnetic field 40 (in Fig. 2 along the x-axis), leads to the fact that only transitions with Δm F = 0 are possible. In contrast, the linear polarization 29 (ê G ) of the Gaussian beam 28 perpendicular (in Figure 2 along the y-axis) to the external magnetic field 40 and leads to transitions with Δm F= ±1. Therefore, the transition |1〉 → |2〉 is possible and resonant through absorption of the Laguerre-Gaussian laser beam 26 and emission into the Gaussian mode. The angular momentum ℏl is transferred to the center-of-mass motion of the atoms 100 through the absorption of the Laguerre-Gaussian laser beam 26. The transition from |2〉 to the state |F = 1, m F = 1) is also detuned by the quadratic Zeeman effect, relative to the effective Rabi frequency, to such an extent that it is negligible. The result is an effective two-level system of internal electron states 12, |1〉, and 14, |2〉, with the second internal electron state 14 being associated with a higher angular momentum in the atomic motion by ℏl.

[0085] Thus, the internal electron state and external atomic motion are coupled and the product states of the interferometer are |a) = |1〉 ⊗ |L z = 0〉 (first internal electron state 12 and no angular momentum) and |b〉 = |2〉 ⊗ |L z= ℏl〉 (second internal electron state 14 and angular momentum ℏl). The two-photon Raman transition creates an effective Rabi oscillation between these two states. The populations 50 in these states can be changed sinusoidally as a function of the combined effective pulse area. If the area is chosen such that the population 50 is completely exchanged, it is called a π pulse. In this case, the area is equal to π. If the area is half as large, it is a π / 2 pulse.

[0086] The first and second analysis laser beams 26, 28 are configured to trigger a Raman transition between the at least two internal electronic states 12, 14, wherein a quantized angular momentum is transferred to the center-of-mass motion of the atoms 100. In particular, the Raman transition results in Rabi oscillations between the at least two internal electronic states 12, 14 of the atoms 100.

[0087] According to the proposed method, a timeline for the interferometric measurement of the angular acceleration Ω̇ can be created.

[0088] First, the atoms 100 are prepared in the first internal electron state 12, |1〉, and cooled to the motional ground state 10 of the trap 200 with a vanishing angular momentum.

[0089] During the two-photon transition with the two analysis laser beams 26, 28, an angular momentum ℏl is transferred through the absorption of the Laguerre-Gaussian laser beam 26. As a result, after the transition, the atoms 100 are in the second internal electron state 14 |2> and have the angular momentum state ℏl in the center-of-mass motion.

[0090] In this way, the angular momentum states 13 (L z = 0), 15 (L z = ℏl) of the external degrees of freedom of the center of mass motion of the atoms 100 are linked to the internal electron states 12, 14, as in Fig. 1 shown.

[0091] With a first pulse, a so-called π / 2 pulse, a uniformly distributed superposition of the first angular momentum state 13 in the internal electron state |1〉, and the second angular momentum state 15 in the internal electron state |2〉, the center of mass motion of the atoms 100, is generated. The part of the atoms 100 that is in the second internal electron state 14, |2〉, rotates with constant angular momentum L z = ℏl around the torus of the trap. The atoms 100 in the first internal electron state 12, |1〉, have no angular momentum (L z = 0).

[0092] After this pulse (at a time t = 0), the atoms 100 are developed in the trap 200 for the interrogation time T.

[0093] At time t = T, a second pulse, a π-pulse, is applied. This swaps the internal states and angular momenta. The atoms 100 in the first internal electron state 12, |1〉, with no angular momentum, are transferred to the second internal electron state 14, |2〉, with angular momentum ℏl. Accordingly, the fraction that was previously in the second internal electron state 14, |2〉, with angular momentum ℏl, is transferred to the first internal electron state 12, |1〉, with no angular momentum.

[0094] After this pulse (at t = T), the atoms 100 are again developed in the trap 200 for the time T.

[0095] At time t = 2T, the interferometer is closed with a third pulse, a π / 2 pulse.

[0096] After this pulse, the population 50 in the first internal electron state 12, |1〉, is determined, for example, by microwave excitation followed by absorption imaging.

[0097] The resulting relative population 50, P 11 , in the first internal electron state 12, |1〉, after this sequence depends on the accumulated phase during the interferometer sequence. The analytical expression of this population 50 is given by: P11=12[1+cos(lΩ˙T2−δΦL)].

[0098] Here, l is the azimuthal index of the Laguerre-Gaussian laser beam 26, which corresponds to the transferred angular momentum ℏl during the state change. The contribution δΦL=ΦL(2T)−2ΦL(T)+ΦL(0) is caused by the phase Φ L (t) of the phase-locked analysis lasers 26, 28 for the π and π / 2 pulses.

[0099] In Fig. 3 shows a population 50 of atoms 100 in a first internal electron state 12 as a function of a square 60 of an interrogation time T.

[0100] For a fixed laser phase δΦ Lthe resulting relative population 50 in the first internal electron state 12, |1〉, is determined for different interrogation times T. If the population 50, P 11 , as a function of T 2 applied as in Fig. 3, an oscillating behavior with a period of 62.2π / (lΩ̇) is observed. Therefore, the angular acceleration Ω̇ can be determined directly from this series of measurements.

[0101] The angular acceleration Ω̇ can thus be determined from the period 62 of the population 50 in the first internal electron state 12 for different interrogation times. In particular, the angular acceleration Ω̇ can be determined from the period 62 of the population 50 in the first internal electron state 12 as a function of a square 60 of the interrogation time.

[0102] Advantageously, different atoms 100 can be used for the proposed method.

[0103] The result does not depend on the radius of the torus 22 of the trap 200 and the mass of the atoms 100.

[0104] The sensitivity is equivalent to the well-known Mach-Zehnder interferometer and therefore scales according to ~1 / (lNT2), where N describes the number of atoms 100 in the trap 200.

[0105] Fig. 4 shows a block diagram of a sensor 1000 for determining an angular acceleration Ω̇ of an external rotation according to an embodiment of the invention.

[0106] The sensor 1000 comprises the trap 200, the atoms 100 contained therein are cooled to a preferably ultra-cold temperature by means of a cooling device 400.

[0107] Furthermore, the sensor 1000 comprises a means 25 for coherently switching between the at least two internal electron states 12, 14 of the atoms 100 in the trap 200 and an imaging device 300 with which the population 50 can be determined by microwave excitation followed by absorption imaging.

[0108] Trap 200, means 25, device 300, and cooling device 400 can preferably be arranged in a housing 500. Optionally, for example, cooling device 400 can also be arranged outside of housing 500. Reference symbol 10 Basic state 11 Total angular momentum 12 first internal electron state 13 first angular momentum state 14 second internal electron state 15 second angular momentum state 16 excited state 17 Total angular momentum 20 laser beam 22 Torus 24 light sheets 25 funds 26 first analysis laser beam 27 Polarization of the first analysis laser beam 28 second analysis laser beam 29 Polarization of the second analysis laser beam 30 Detuning 32 space segment 40 Magnetic field 50 populations 60 square of interrogation time 62 period 100 atoms 200 traps 300 facilities 400 cooling device 500 housings 1000 sensors

Claims

[1] A method for determining an angular acceleration of an external rotation, comprising Preparing a plurality of atoms (100) in a first internal electron state (12) in a trap (200) which restricts movement of the atoms (100) to a region within a torus (22), and cooling the plurality of atoms (100) in the trap (200) to a ground state (10), wherein the atoms (100) move with quantized angular momentum in the same direction in the torus; Radiating a first pulse of a first analysis laser beam (26), in particular a Laguerre-Gaussian laser beam, and a second analysis laser beam (28), in particular a Gaussian laser beam, onto the atoms (100) trapped in the trap (200) to generate a uniformly distributed superposition between a first and a second quantum mechanical angular momentum state (13, 15) of a center of gravity movement of the atoms (100); Developing the atoms (100) in the trap (200) for an interrogation time; irradiating a second pulse of the first and second analysis laser beams (26, 28) onto the atoms (100) to completely exchange a population (50) of the atoms (100) between the two angular momentum states (13, 15); Developing the atoms (100) in the trap (200) for an interrogation time; irradiating a third pulse of the first and second analysis laser beams (26, 28) onto the atoms (100) to superpose the two angular momentum states (13, 15) of the center of gravity motion of the atoms (100) and to exchange at least a portion of the population (50) of the atoms (100) between the two angular momentum states (13, 15); Determining the population (50) of atoms (100) in the first internal electron state (12); Determine the angular acceleration from the population (50) in the first internal electron state (12). [2] The method of claim 1, wherein the population (50) of atoms (100) in the first internal electron state (12) is determined by microwave excitation followed by absorption imaging. [3] Method according to claim 1 or 2, wherein a pulse area of ​​the first and third pulse acting on the atoms (100) is half as large as a pulse area of ​​the second pulse acting on the atoms (100). [4] Method according to one of the preceding claims, wherein the first analysis laser beam (26) and the second analysis laser beam (28) are phase-locked. [5] Method according to one of the preceding claims, wherein the angular acceleration is determined from a period (62) of the population (50) in the first internal electron state (12) for different interrogation times, in particular wherein the angular acceleration is determined from the period (62) of the population (50) in the first internal electron state (12) as a function of a square (60) of the interrogation time. [6] Method according to one of the preceding claims, wherein a Raman transition between at least two internal electron states (12, 14) is triggered by the first and second analysis laser beam (26, 28), wherein a quantized angular momentum is transferred to the center of mass movement of the atoms (100), in particular wherein Rabi oscillations between the at least two internal electron states (12, 14) are caused by the Raman transition. [7] Method according to one of the preceding claims, wherein irradiation of the first and second analysis laser beams (26, 28) causes a transition between the at least two angular momentum states (13, 15) of the center of gravity movement of the atoms (100) to the atoms (100) trapped in the trap (200). [8] Method according to one of the preceding claims, wherein the atoms (100) are localized in the torus (22) and a transition between the at least two angular momentum states (13, 15) of the center of mass motion of the atoms (100) is effected by transferring a pulse by a Bragg pulse to the atoms (100) trapped in the trap (200). [9] Method according to one of the preceding claims, wherein the trap (200) forms an optical dipole trap formed by superposition of a light sheet (24) and a laser beam (20), wherein the light sheet (24) confines the atoms (100) along a spatial axis (z) to a spatial segment (32) which is limited in both directions and is in particular planar, and wherein the laser beam (20) is oriented perpendicular to the spatial segment (32) and has an annular intensity profile in the spatial segment (32) and defines a torus (22), wherein the atoms (100) are trapped in the torus (22) and a movement of the atoms (100) is limited to an azimuthal angle. [10] Sensor (100) for carrying out a method according to one of the preceding claims for determining an angular acceleration of an external rotation, comprising a plurality of, in particular ultra-cold, atoms (100) which exhibit a transition from an energetic ground state (10) to at least one energetically excited state (16) of their electron system, a radiation-induced trap (200) which restricts the movement of the atoms (100) to a torus (22), wherein the atoms (100) have at least a first and a second internal electron state (12, 14) with which the atoms (100) are trapped in the trap (200), a means (25) for coherently switching between the at least two internal electron states (12, 14) with an angular momentum transfer to the center of mass motion of the atoms (100). [11] Sensor according to claim 10, wherein the angular acceleration is determinable by determining a respective population (50) of the atoms (100) in at least two angular momentum states (13, 15) of the center of gravity movement of the atoms (100). [12] Sensor according to claim 11, wherein the angular acceleration from a period (62) of the population (50) in the first angular momentum state (13) is determinable for different interrogation times, in particular wherein the angular acceleration from the period (62) of the population (50) in the first angular momentum state (13) is determinable as a function of a square (60) of the interrogation time. [13] Sensor according to claim 12, wherein there is an imaging device (300) with which the population (50) can be determined by microwave excitation followed by absorption imaging. [14] Sensor according to one of claims 10 to 13, wherein the atoms (100) are sodium atoms which undergo a transition of a valence electron from the 3 2 S 1 / 2 Ground state in the excited 3 2 P 3 / 2 condition. [15] Sensor according to one of claims 10 to 14, wherein the trap (200) is designed as an optical dipole trap formed by superimposing a light sheet (24) and a laser beam (20), wherein a movement of the atoms (100) through the light sheet (24) along a spatial axis (z) is limited to a spatial segment (32) which is limited in both directions, in particular a planar one, wherein the laser beam (20) is oriented transversely, in particular perpendicularly, to the spatial segment (32) and has an annular intensity profile in the spatial segment (32) and defines a torus (22), wherein the atoms (100) are trapped in the torus (22) and a movement of the atoms (100) is limited to an azimuthal angle. [16] Sensor according to claim 15, wherein a magnetic field (40) is applied parallel to the light sheet (24), by which a Zeeman splitting of a total angular momentum (11) of the atoms (100) into the at least two internal electron states (12, 14) is formed. [17] Sensor according to claim 16, wherein a first analysis laser beam (26), in particular a Laguerre-Gaussian laser beam, and a second analysis laser beam (28), in particular a Gaussian laser beam, are formed as means (25) parallel to the first analysis laser beam (26), wherein the first and the second analysis laser beam (26, 28) have a frequency difference which corresponds to an energy difference between the at least two internal electron states (12, 14). [18] Sensor according to claim 17, wherein the first and second analysis laser beams (26, 28) are phase-locked to each other. [19] Sensor according to claim 17 or 18, wherein the first and second analysis laser beams (26, 28) are designed to trigger a Raman transition between the at least two internal electron states (12, 14), wherein a quantized angular momentum is transferable to the center of gravity movement of the atoms (100), in particular wherein Rabi oscillations are present between the at least two internal electron states (12, 14) of the atoms (100) due to the Raman transition. [20] Sensor according to one of claims 17 to 19, wherein a transition between the at least two angular momentum states (13, 15) of the center of gravity movement of the atoms (100) can be effected by irradiating the first and second analysis laser beams (26, 28) onto the atoms (100) trapped in the trap (200). [21] Sensor according to one of claims 11 to 20, wherein the atoms (100) are localized in the torus (22) and a transition between the at least two angular momentum states (13, 15) of the center of gravity motion of the atoms (100) can be effected by transmitting a pulse by a Bragg pulse to the atoms (100) trapped in the trap (200).

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