Patent Publication Number: US-2022236060-A1

Title: Shaken-lattice matter-wave gyro

Description:
BACKGROUND 
     Inertial navigation systems, e.g., those used in cars, ships, submarines, aircraft, missiles, and spaceships, help track linear and angular motion using, respectively, accelerometers and gyroscopes, “gyros”. Herein, a “gyro” is a device used for measuring angular velocity. A classical mechanical gyroscope includes a spinning wheel or disc in which the axis of rotation is free to assume any orientation by itself. When rotating, the orientation of this axis is unaffected by tilting or rotation of the mounting, due to the conservation of angular momentum. 
     Laser gyros, e.g., ring-laser gyros and fiber-optic gyros, measure angular velocity as a function of shifts in interference patterns between two counter-propagating laser beams. While they can achieve greater precision than mechanical gyros, the precision of a laser gyro is limited by the wavelength of the laser light used to create the interference pattern. In principle, smaller wavelengths can be achieved using matter waves, i.e., de Broglie waves associated with atoms, to enable low cost, robust, and highly-accurate inertial sensors. Development work on atom-based gyros is ongoing. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram of a matter-wave, e.g., atom, gyro with counter-propagating traps. 
         FIG. 2  is a schematic diagram of a cross-section of a 3-D matter-particle, e.g., ultra-cold atom, trap, e.g., optical lattice. 
         FIG. 3  is a flow chart of a matter-wave gyro process. 
         FIG. 4  is a schematic diagram of a laser system of the gyro of  FIG. 1 . 
     
    
    
     DETAILED DESCRIPTION 
     In a matter-wave gyro, particle-bearing traps counter-propagate plural times about a closed path. Upon detection, the particles form an interference pattern, from which an angular momentum or related parameter can be evaluated. In accordance with their wave functions, each particle propagates with each trap. The wave functions collapse when the traps are probed by lasers, resulting in an interference pattern, the position of which is a function of angular velocity. In an embodiment, in this case, a shaken lattice atomic gyro (aka, “SLAG”) the particles are ultra-cold atoms from Bose-Einstein Condensates (BECs) and the traps are three-dimensional optical lattices formed by interfering laser beams. The optical lattices are moved about the closed path (e.g., a Sagnac loop) by “shaking”, i.e., phase and frequency modulating, the laser beams. Frequency and phase modulation can also be used to cool and load the atoms into the lattices and to recombine them for the angular velocity measurement. Herein, the term “particle” encompasses atoms and molecules. “Atom” encompasses neutral atoms and ions. 
     The particles are trapped so that their positions relative to each are substantially fixed as they go around the closed path. This helps limit inter-particle interactions that might otherwise contribute noise to the desired angular momentum measurements. The speed of counter-propagation can be fast enough so the particles are not significantly disturbed by the passing of the other trap and its contents. This allows the counter-propagation to proceed two or more times around the loop before or between measurements, increasing the effective area circumnavigated, and thus increasing the spatial resolution of the measurements. In fact, one can trade off spatial resolution and temporal resolution by setting a number of revolutions between measurements. This is in contrast to some other Sagnac-type gyros for which the effective path length is fixed to a single traversal of the path. 
     As shown in  FIG. 1 , a SLAG  100  includes a laser system  102 , a vacuum chamber  104 , an imager  106 , an image analyzer  108 , and a machine learning engine  110 . Laser system  102  can include one or more lasers, e.g., a cooling laser  112 , a lattice laser  114 , and a probe laser  116 . Cooling laser  112  is used to cool atoms as they are introduced; lattice laser  114  is used to establish the lattices to be counter-propagated, as well as other lasers to transfer particles. The probe laser  116  is designed to interrogate the trapped ultra-cold atoms so that they can be imaged. Other lasers can include a laser for forming a potential well for evaporating cold atoms to yield ultra-cold atoms, and a laser for producing lattices that can be chirped to transfer cold and ultra-cold atoms. However, depending on the embodiment, some lasers can perform multiple functions. 
     Laser system  102  includes optical elements  118  for guiding, splitting, and recombining laser beams. Most of these optical elements reside in vacuum device  104 , the interior of which can be accessed by lasers through transparent walls and/or windows of vacuum device  104 . Laser system  102  includes a modulator  120 , which serves to modulate laser beams produced by laser system  102 , e.g., to cause the fringes of interference patterns produced by the laser system to shift, e.g., so that entrained atoms can be transported. 
     Vacuum device  104  includes an atom source  130 , e.g., a Rubidium, Cesium, or Strontium source. In other embodiments, a source of other particles, e.g., other neutral atoms, ion (charged atoms), and molecules, can be used. In the illustrated embodiment, source  130  is located within the vacuum device  104 . However, in other embodiments, the source can be external and the particles are injected into the vacuum device. 
     Atoms introduced from atom source  130  are cooled by cooling laser  112  as they enter a cooling region  132 , which may be the site of a trap such as a magneto-optical trap (MOT) or an optical trap. The resulting cold atoms are further cooled at evaporating region  134  to produce a Bose-Einstein Condensate of ultra-cold atoms. Evaporation region  134  may be the site of a MOT or optical trap. An optical trap is preferred as the laser light used to form a potential well for the evaporative cooling is much less prone to affect nearby regions than are the magnetic fields that would be used in a MOT. 
     Lattice laser  114  is used to form multi-dimensional lattices  140 , including lattices  141  and  142 , within vacuum device  104 . As shown in  FIG. 2 , three-dimensional lattice  141  is formed using three pairs of interfering beams. Laser beams  201  and  202  propagate in the −X and +X directions, respectively, and so interfere to produce a one-dimensional lattice of interference fringes that extend vertically in  FIG. 2 . Laser beams  203  and  204  propagate in the +Y and −Y directions respectively and, so, produce a one-dimensional lattice of interference fringes that extend horizontally in  FIG. 2 . Laser beams  205  and  206  extend in the +Z and −Z directions respectively, forming a one-dimensional lattice of interference fringes (not shown) that extend parallel to the sheet of  FIG. 2 . Lattice  142  is structured analogously. 
     The one-dimensional lattices formed by laser beams  201 - 204  intersect to define a two-dimensional lattice with intersecting fringes. Atoms  210  tend toward the intersection of bright fringes, as indicated in the detail of  FIG. 2 . Note that lattice  141  may be sparsely populated to minimize inter-atom interactions. The one-dimensional lattice formed by laser beams  205  and  206  serves to prevent atoms from escaping from the two-dimensional lattice. 
     Ultra-cold atoms have negligible kinetic energy. A loader  144  transfers the ultra-cold atoms from evaporation region  134  to lattices  140 . Loader  144  includes elements of laser system  102  that form a one-dimensional lattice, which is chirped (the laser frequency is ramped up or down) so the interference fringes of the loader lattice move toward lattices  140 . Once the ultra-cold atoms are in place, the power for lattices  140  is ramped up, while the power to the loader lattice is extinguished, thus trapping the ultra-cold atoms in lattices  140 . 
     Once lattices  140  are loaded, they can be counter-propagated about a closed path, in this case a circular path  146 . As shown in  FIG. 1 , lattice  141  is translated (without changing its orientation) about path  146  in a counter-clockwise direction, while lattice  142  is translated in a clockwise direction about path  146 . Typically, each lattice traverses path  146  more than once before a measurement or between measurements. 
     Vacuum device  104 , includes an atom chip  148 , which, advantageously, constitutes a wall of vacuum device  104  and provides for electrical access to the interior of vacuum device  104 . One function of atom chip  148  is to provide currents to form magnetic fields, used, for example, in some embodiments to form magneto-optical traps for cooling and for evaporation. In other embodiments, the traps are fully optical. 
     Note that each ultra-cold atom can be characterized by a distribution of possible locations, some of which “belong” to lattice  141  and some of which belong to lattice  142 . The positions may be interrogated at some point while they lattices are counter-propagating using probe laser  116 . Probe laser  116  may dislodge some or all of the ultra-cold atoms so that they impact imager  106  to capture an interference pattern. The image can be analyzed by image analyzer  108  to evaluate any shift relative to a position associated with zero angular momentum. Any such shift is then converted to an angular momentum or another parameter associated with angular momentum. 
     Machine learning engine  110  is used in a calibration mode to set the phase and frequency functions according to which modulator  120  controls lattice laser  114  and other lasers. To this end, during calibration mode, training angular velocity data  150  can be used. Machine learning engine  110  can be set to minimize errors, that is, differences between training angular velocity values and angular velocity values  152  output from image analyzer  108 . 
     A matter-wave-gyro process, flow charted in  FIG. 3 , includes a calibration phase  310 , a state-preparation phase  320 , and a measurement phase  330 . During calibration phase  310 , at  311 , a loop number N is selected representing the number of times the multi-dimensional lattices are to move around the closed path before a measurement or between measurements. Selecting a higher N, e.g.,  32 , results in greater sensitivity (spatial resolution), while a lower N, e.g.,  3 , can reduce the time between measurements to achieve greater temporal resolution. 
     At  312 , machine learning engine  110  trains, e.g., programs, modulator  120  with optimal modulation phase and frequency functions of time, and trains image analyzer with functions for converting interference images to angular momentum based on training data with known angular momenta. 
     At  321  of state preparation phase  320 , particles (e.g., neutral atoms, ions, molecules) are introduced in a vacuum device. At  322 , the particles are cooled to yield a population of cold particles. At  323 , the cold-particle population is evaporated, i.e., the higher energy cold atoms are allowed to escape, leaving a Bose-Einstein Condensate (BEC) of ultra-cold particles. At  324 , the ultra-cold particles are transferred to the multi-dimensional lattices. At  325 , sideband cooling is performed. Sideband cooling is a laser cooling technique allowing cooling of tightly bound atoms and ions beyond the Doppler cooling limit, potentially to their motional ground state. 
     At  331  of measurement phase  330 , the multi-dimensional lattices are counter propagated plural times about a closed path. Since the lattices are at least partially populated with ultra-cold particles, the portions of the probability distribution for each ultra-cold particle are also counter-propagated about the closed path. In the illustrated embodiment, the closed path is circular with a radius of 1 mm and an area of pi mm 2 . In other embodiments, other areas are enclosed, e.g., areas between 1 mm 2  and 10 mm 2 . Other areas may be appropriate for other atom moieties and other particle types. 
     At  332 , the circulating ultra-cold particles are imaged. To this end, they may be probed by a laser, dislodged from the lattice so that they fall on a imager to form an interference pattern. Note that the z-dimension lattices may be shut off during imaging to allow ultra-cold particles to exit the multi-dimensional lattices. At  333 , the interference-pattern image is analyzed to determine an offset of the interference pattern and to determine a value for angular momentum or related parameter based on the offset. 
     Of course, process  300  can be repeated to provide a series of angular momentum measurements. For example, the successive iterations can be discrete in that process  300  particle introduction  321  for a second iteration follows imaging  332  for a first iteration. (Calibration  311  need not be repeated for every iteration.) On the other hand, iterations of process  300  can be pipelined. For example, during image analysis  333  for a fourth iteration, counter-propagation  331  can be performed for a third iteration, while BEC production  323  is performed for a second iteration, and laser cooling  322  occurs for a first iteration. Pipelining iterations of process  330  increases the repetition rate for measurements so as to achieve higher temporal resolution for a series of measurements. 
     Some embodiments provide for transfers  324  while the lattices are stationary and superimposed. However, some embodiments provide for an alternative mode in which the lattices are loaded while they are moving. For example, the ultra-cold atoms can be split during transfer and the resulting branches can be directed so that they match the tangential velocities of the lattices. Once the branches are synchronized with respective lattices, the power to the lattices can be increased while the transfer lattices are extinguished, In some embodiments, imaging  332  only partially depletes the lattices and the synchronized loading is used to replenish the partially depleted lattices. 
     A portion of laser system  102  is shown in  FIG. 4 , illustrating how one (e.g., the lattice laser) can produce the laser beams used for counter-propagation and measurement. An 852 nanometer (nm) laser master  402  emits light of a desired wavelength. 852 nm is chosen in the illustrated embodiment as it is readily tuned using a Cesium-based spectroscopy lock  404 . As is understood by those skilled in the art, the laser light can be redirected using fully-reflective mirrors and can be split using mirrors that are partially reflective and partially transmissive. 
     852 nm light emitted by master  402  is tapped as an input to spectroscopy lock  404  and the remainder is split between intensity controllers  406  and  408 . The outputs of intensity controllers  406  and  408  are input respectively to optical amplifiers  410  and  412 . The outputs of optical amplifiers  410  and  412  are respectively tapped to provide respective inputs to detectors  414  and  416 . The outputs of detectors  414  and  416  are fed back as control signals to intensity controllers  406  and  408  so that the outputs of optical amplifiers  410  and  412  can be regulated independently. 
     The untapped portion of the output from optical amplifier  410  is directed to a switch  420 . Switch  420 , when in a first switch position, directs light from optical amplifier  410  for use as a Y dipole  422  during measurement. Switch  420 , when in a second switch position, directs light from optical amplifier for use in forming lattice  140 . More specifically, the light output from switch  420 , when in its second position, is divided into three beams. One of the three beams is directed to a shift (modulator)  424  that outputs a beam  425  used for form a −y one-dimensional lattice, another of the three beams is directed to a shift  426  that outputs a beam  427  used to form the +y one-dimensional lattice that forms a +Y lattice. The third beam  428  is used to form the +Z lattice. 
     The untapped portion of the output from optical amplifier  412  is directed to a switch  430 . Switch  430 , when in a first switch position, directs light from optical amplifier  412  for use as an X dipole  432  during measurement. Switch  420 , when in a second switch position, directs light from optical amplifier for use in forming lattice  140 . More specifically, the light output from switch  430 , when in its second position, is divided into three beams. One of the three beams is directed to a shift (modulator)  434  that outputs a beam  435  used in forming a −x one-dimensional lattice, another of the three beams is directed to a shift  436  that outputs a beam  437  used to form the +x one-dimensional lattice that forms a +x lattice. The third beam  428  is used to form the −Z lattice. 
     Gyro  100  ( FIG. 1 ) is more specifically characterized as a Shaken-lattice atomic gyroscope (SLAG). This type of gyro is based on shaken-lattice interferometry (SLI), a technique first proposed and subsequently demonstrated by the Anderson group at the JILA Institute of the University Of Colorado (UCB). C. A. Weidner, H. Yu, R. Kosloff, and D. Z. Anderson, “Atom interferometry using a shaken optical lattice,”  Phys Rev A , vol. 95, no. 4, p. 043624, April 2017. C. A. Weidner and D. Z. Anderson, “Experimental Demonstration of Shaken-Lattice Interferometry,”  Phys. Rev. Lett ., vol. 120, no. 26, 2018. 
     In its simplest rendition, SLAG utilizes atoms confined to a two-dimensional optical lattice produced by a pair of intersecting, mutually incoherent, optical lattice fields, as illustrated in  FIG. 2 . The two light fields are phase and frequency modulated (“shaken”) in such a way that atoms propagate in a loop that encloses finite and closed area. Physical operations analogous to those used in a fiber optic gyroscope (FOG) are produced by appropriately modulating the optical lattice. The path can be a nominally circular, 1 mm radius, Sagnac loop. “Sagnac” is named after Georges Sagnac who, in 1913, noted the effect of a rotation on a fringe phase shift. See P. Boyer, “The Centenary of Sagnac Effect and Its Applications: from Electromagnetic to Matter Waves,  ISSN  2075_1087 , Gyroscopy and Navigation,  2014, Vol. 5, No. 1, pp. 20-26. Pleiades Publishing, Ltd., 2014. 
     One advantage of this configuration is that various tradeoffs between response time (temporal resolution) and sensitivity (spatial resolution) can be selected by choosing the number of round trips over the circular path taken prior to a measurement or between measurements. For example, a 100 mm 2  enclosed area can be achieved with 32 round trips traversed by ultra-cold rubidium ( 87 Rb) atoms in 1 s. System demonstration under this effort requires approximately 6 W of optical power at λ L =852 nm. This system meets the challenging demands of high-performance rotation sensing in a real-world dynamic environment. 
     SLAG utilizes trapping forces that can be as high as several tens of g&#39;s. (where g is the acceleration due to the Earth&#39;s gravity) thereby making the interferometer robust against dynamic forces of a real-world environment and virtually insensitive to orientation. The straightforward ability to increase effective enclosed area with multiple circuits around the Sagnac loop means that the gyro sensitivity can scale largely independently of system size (in contrast to free-space interferometers). The large trapping forces enable high-sensor-bandwidth operation by enabling atoms to be accelerated to high velocity. In the shaken-lattice context, they also provide a straight-forward way to dynamically adjust the tradeoff between rotation sensor bandwidth and sensitivity. 
     Sagnac loop geometry, and therefore scale factor, are primarily set by digitally controlled, highly reproducible phase/frequency modulation signals applied to light beams and thus is relatively immune to drifts due to thermal effects, vibration, and so forth. While strong trapping in an atom waveguide approach to interferometry tends to exacerbate the deleterious effects of atom interactions, a lattice can be made sparsely populated to minimize atom interactions while still accommodating a sufficient number of atoms to attain good atom shot-noise performance. 
     Moreover, in an embodiment, the shaken-lattice approach is implemented in PIC (Photonic Integrated Circuit) integration and integration with all optical approaches to BEC (Bose-Einstein Condensate). In particular, the illustrated embodiment is based on standard 780 nm and 852 nm diode laser wavelengths, and operation involves intensity and phase/frequency control that can all be accomplished with integrated electro-optic modulators and linear optical elements. See C. J. E. Straatsma, M. K. Ivory, J. Duggan, J. Ramirez-Serrano, D. Z. Anderson, and E. A. Salim, “On-chip optical lattice for cold atom experiments,”  Opt Lett , vol. 40, no. 14, pp. 3368-3371, 2015. 
     In summary, the following advantages have been achieved. Atoms are confined to an optical lattice with high trapping forces thus allowing a Sagnac device to operate in relatively harsh dynamic environments. For the purposes here, they enable orientation invariance of system operation. Among the various approaches to trapping atoms (optical fields, magnetic fields, radio-frequency fields) optical lattices have been the most successfully used to provide such large trapping forces without introduction of substantial noise that would cause decoherence in an interferometer. 
     The entire laser system, including phase and frequency modulation, is amenable to PIC integration. Scale factor is real-time programmable and knowable to high precision as it is precisely determined by lattice wavelength and digitally reproducible signals driving phase and frequency shifters. For similar reasons, gyro scale factor is robust against thermal changes and other environmental factors. The real-time programmable scale factor allows tradeoff between gyro sensitivity and bandwidth. 
     A high-repetition rate Bose-Einstein Condensate (BEC) provides the low temperature, quantum state atoms needed to load the Sagnac interferometer, as well as the shaken-lattice method and atom transport, as described above. See D. M. Farkas, K. M. Hudek, E. A. Salim, S. R. Segal, M. B. Squires, and D. Z. Anderson, “A compact, transportable, microchip-based system for high repetition rate production of Bose-Einstein condensates,”  Appl Phys Lett,  vol. 96, no. 9, p. 093102, 2010. 
     This approach leverages the considerable community knowhow regarding ultra-cold atoms trapped by optical lattices, which, in fact, is the basis of the world&#39;s most accurate clocks. A. D. Ludlow and J. Ye, “Progress on the optical lattice clock,”  Comptes Rendus Physique , vol. 16, no. 5, pp. 499-505, June 2015. This is evidence that lattice light technical noise that might otherwise lead to gyroscope performance degradation can be sufficiently mitigated. Performance metrics such as Allan deviation, which impose constraints on experimental parameters such as atom number, are by themselves reasonably modest (i.e. atom numbers in the range of a few thousand are routine. Likewise, metric-derived specifications such as atom lifetime (which in turn constrains vacuum performance), are achieved. Algorithms are used for the control and manipulation of the ultra-cold atoms trapped in the lattice. 
     Herein, the “Sagnac lattice potential” is a superposition of moving lattices and provides the basis of rotation sensing and serves as the context for sensor design and gyro system design. A classical interpretation of lattice forces is presented below to explain orientation invariance and robustness in dynamic environments. 
     An objective of atom-based Sagnac interferometry is to cause matter waves to be split into two counter-propagating waves propagating around a closed path, e.g., a loop, and then be recombined to produce a phase signal indicative of rotation rate in the plane of the loop. In the shaken-lattice approach, atom trapping forces are provided by an optical lattice, and interferometry, that is, splitting, propagation, and recombination, is carried out by appropriately time-varying the position of the nodes and antinodes of the lattice. While the Sagnac gyroscope requires, in principle, a two-dimensional lattice, a one-dimensional lattice is considered initially for the sake of gaining physical insight. The atomic potential generated by a standing light field (one dimensional lattice) of wave number k L  is given by equation 1: 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         V 
                         0 
                       
                       2 
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               k 
                               L 
                             
                             ⁢ 
                             x 
                           
                           + 
                           ϕ 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where the amplitude V 0  is determined by the optical intensity and φ is, for now, an arbitrary phase reference. 
     It is conventional to express the lattice amplitude in units of the atomic recoil energy, 
     
       
         
           
             
               
                 
                   
                     E 
                     R 
                   
                   = 
                   
                     
                       
                         ℏ 
                         2 
                       
                       ⁢ 
                       
                         k 
                         L 
                         2 
                       
                     
                     
                       2 
                       ⁢ 
                       m 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where ℏ is the reduced Planck&#39;s constant and m is the atomic mass ( 87 Rb in our case, and for λ L =852 nm, E R /ℏ≅2π×3.2 kHz. The possibility to implement interferometry in a lattice, Sagnac or otherwise, is based on the pioneering work by Pötting et al. See S. Pötting, M. Cramer, and P. Meystre, “Momentum-state engineering and control in Bose-Einstein condensates,”  Phys. Rev. A , vol. 64, no. 6, p. 063613, November 2001. These authors showed that the momentum state of an ensemble of atoms can be transformed from an initial state to some desired final state by appropriately shaking the lattice, that is, by causing the phase of the lattice to become time-dependent, ϕ→ϕ(t), in a specific way. 
     Some insight into the realization of Potting et al can be gleaned by considering Bloch state diagrams that present the energy versus quasi-momentum for lattices of two different depths. As Bloch states form a complete set, any state within the lattice can be expressed as a superposition of Bloch states. Transforming from one state to another involves transitions among the various Bloch states. Thus, the phase modulation can be expected to be comprised of combinations of transition frequencies. See C. A. Weidner and D. Z. Anderson, “Simplified landscapes for optimization of shaken lattice interferometry,”  New J. Phys ., vol. 20, no. 7, 2018. In general, it is not possible to discover an appropriate modulation analytically. Pötting et al. thus utilized a genetic algorithm to “learn” a ϕ(t) that served to accomplish the momentum transformation of interest. 
     In one-dimension, interferometry is accomplished by a sequence (or sequences) of four physical operations: beam splitting, propagation (or transport), reflection, and recombination. Splitting and recombination are typically the first and last steps, while sequences of propagation and reflection determine interferometer geometry. 
     In recent work, Weidner and Anderson showed that the basic shaken-lattice concept could be applied to interferometry by learning a set of phase modulation functions {ϕ(t), i=1, 2, 3 . . . } that would accomplish the appropriate operations. Consider the beam splitting operation: the initial momentum state is that of the atomic ground state of the lattice. One would like to transform that initial state into one that consists of a pair of momenta having equal magnitudes but opposite direction. The beam-splitting operation can be implemented by learning an appropriate ϕ s (t) that accomplishes the task. 
     Taking just this first operation as an example, it is not intuitively obvious, yet it is nevertheless the case, that shaking the lattice can be used to transform the ground state into a pair of oppositely directed matter waves as suggested by Pötting et al. Weidner and Anderson (ibid) specifically demonstrated an accelerometer based on a matter wave Michelson interferometer by learning a sequence of five phase modulation functions, one each for splitting, propagating, reflecting, reverse propagating, and recombining. 
     Ideally, an ensemble of atoms in a lattice is characterized by a single wavefunction ψ({right arrow over (k)}), in which the momentum space is one, two, or three dimensional corresponding to the dimensionality of the problem. In the case of a lattice, the momentum states are discrete, having values k n =±2nk L , n=1, 2, 3 . . . in one dimension. Experimental measurement can provide only populations indicative of the wave function and not, directly, phase information. Thus, one can measure a set of momentum state populations {pn,n=0, ±1, +2 . . . }. This is normally done by time-of-flight (TOF) imaging in which atoms are released from the lattice by extinguishing the laser light, allowed to fall for a fixed time under the influence of gravity, and then imaged. 
     One can treat momentum populations as a vector. During the learning process one has a target, or desired momentum population vector {right arrow over (p)} (d) . In the case of +2ℏk L , beam splitting, for example, the desired momentum vector is {right arrow over (p)} (d) =( . . . 0,0,1,0,1,0,0, . . . )/√{square root over (2)}. One can begin with a guess of a shaking function ϕ (0) (t) and measure a resulting set of trial populations {right arrow over (p)} (0)  and form an error parameter 
     
       
         
           
             
               
                 
                   ℰ 
                   = 
                   
                     1 
                     - 
                     
                       
                         
                           
                             p 
                             → 
                           
                           
                             ( 
                             d 
                             ) 
                           
                         
                         · 
                         
                           
                             p 
                             → 
                           
                           
                             ( 
                             0 
                             ) 
                           
                         
                       
                       
                         
                            
                           
                             
                               p 
                               → 
                             
                             
                               ( 
                               d 
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                            
                         
                         ⁢ 
                         
                            
                           
                             
                               p 
                               → 
                             
                             
                               ( 
                               0 
                               ) 
                             
                           
                            
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Note that this error parameter is zero if the trial is identical to the desired momentum population vector. Based on the error, a learning algorithm updates the guess ϕ (0) (t) and iterates. Algorithms vary in their details, but the objective is, roughly, to sample the landscape of possible modulation functions and converge on one that minimizes the error. The application of learning techniques in the quantum realm is generally the domain of what is referred to as the field of “quantum control” and more broadly as “optimal control”. There is considerable science and art the choice of: 1) parameters that are used to vary the modulation function; and 2) the algorithm used to perform learning using real data. The learning of the set of modulation functions {ϕ 0 (t)} occurs exactly one time as a calibration operation: once the modulation functions are learned, the interferometer can then be used as a sensor. 
     In standard atom interferometry the rotation or acceleration signal is a phase which is inferred by measuring a pair of momentum populations in analogy with the two ports of an optical interferometer in which one determines an optical phase from measured intensities. In the lattice case, the measured state of atoms is represented by a momentum population signal vector {right arrow over (p)} (s)  having several rather than two values (for a typical lattice depth of V 0 ≅10E R -20E R  between 5 and 7 momentum values have significant population). 
     The signal in the shaken lattice interferometer thus gives generally more information than the conventional interferometer. For example, there is ambiguity in the direction of rotation or acceleration in standard atom interferometry that must be resolved using one of a few available methods. Directional information is generally present, however, in the larger momentum population vector of the shaken lattice. The phase information is extracted using the Fisher information to calibrate the output momentum population vector, that is, given the interest in rotation sensing, {right arrow over (p)} (s) (Ω). The Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. 
     Weidner and Anderson (ibid) showed both numerically and experimentally that the sensitivity of their atom Michelson interferometer to acceleration has the same τ 2  dependence of acceleration sensitivity on interrogation time as the free-space interferometer. One should expect this, since Schrödinger&#39;s equation is linear in the atomic potential. Likewise, the shaken lattice Sagnac gyroscope sensitivity scales with enclosed area in the same way as does the free-space gyroscope. 
     Interferometer sensitivity, whether for acceleration or rotation, scales with dimensions of the interferometer. Moving atoms over a characteristic distance of 1 mm, as in the illustrated embodiment, can be considered a large distance, most meaningfully measured in units of the lattice spacing. Perhaps in contrast to the beam-splitting and propagation operations described above, the possibility of large-distance transport of trapped atoms is easy to understand. A moving lattice can be produced by imposing a frequency difference δω between a pair of counter-propagating laser beams that is small relative to their optical frequency. In the one-dimensional case the atomic potential: 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         V 
                         0 
                       
                       2 
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               k 
                               L 
                             
                             ⁢ 
                             x 
                           
                           + 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     describes a lattice that moves with phase velocity 
     
       
         
           
             
               
                 
                   
                     
                       v 
                       p 
                     
                     = 
                     
                       
                         λ 
                         L 
                       
                       · 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                         
                         
                           4 
                           ⁢ 
                           π 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where λ L  is the lattice laser wavelength. Clearly, such a lattice has identical Bloch states as a stationary one, except translated to account for the energy and momentum shifts relative to a stationary frame. 
     By chirping the frequency difference, one can also accelerate or decelerate atoms. For example, the moving lattice potential: 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         V 
                         0 
                       
                       2 
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               k 
                               L 
                             
                             ⁢ 
                             x 
                           
                           + 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 vt 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     accelerates from a standstill, then decelerates the lattice to a standstill, translating it (and trapped atoms) a distance Δx=λ L δω/v in a time 2π. In a time the lattice returns to its original position. The Anderson group, for example, used a frequency chirped standing light field to transport cold cesium atoms a distance of over 0.5 mm and back again in a time of 20 ms, achieving a peak velocity of over 200 mm/s. See B. A. Dinardo and D. Z. Anderson, “A technique for individual atom delivery into a crossed vortex bottle beam trap using a dynamic 1D optical lattice,”  Review of Scientific Instruments,  vol. 87, no. 12, p. 123108, December 2016. Note that the latter corresponds to imposing a mean acceleration of about 4 g onto the atoms. 
     The potential 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         V 
                         0 
                       
                       4 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   k 
                                   L 
                                 
                                 ⁢ 
                                 x 
                               
                               - 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   k 
                                   L 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     evidently describes a pair of moving standing light fields propagating with equal speeds moving in opposite directions. It proves to be the case that for sufficiently large lattice velocities v p &gt;&gt;ℏk L /m, atoms that are trapped in one moving lattice “see” only the time-averaged potential of the other moving lattice rather than its bumps and valleys. The SLAG approach utilizes this property to establish high-speed large-distance counter-propagating transport of atoms. 
     Potentials such as the one of Eq. (7) that involve frequency shifting can be well-approximated by phase modulation such as: 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         V 
                         0 
                       
                       2 
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               k 
                               L 
                             
                             ⁢ 
                             x 
                           
                           - 
                           
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ω 
                                     m 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     A Bessel function expansion of this potential shows that setting α=2.408, i.e. to the root of the zeroth order Bessel function, yields the counter-propagating potential with δω=ω m  plus higher order sidebands that can be ignored for sufficiently large ω m . 
     Sagnac interferometry requires the transport of atoms around a loop in two dimensions. To this end, the SLAG can use a two-dimensional lattice produced by a pair of lattices, one each oriented in the x- and y-direction, formed from mutually incoherent laser beams (though with nearly the same wavenumber). While nearly any two-dimensional path can be achieved, a two-dimensional moving lattice executing circular transport is remarkably easy to produce with frequency shifting. 
     The potential: 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         V 
                         0 
                       
                       2 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 
                                   k 
                                   L 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           vt 
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 
                                   k 
                                   L 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     y 
                                     - 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           vt 
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     circular motion with radius R and period τ=2π/v. One can think of such a lattice as an egg crate that is held fixed in orientation but is moved in a circle. The potential can be created by frequency shifting: 
       δω x ( t )=− Rv  sin( vt )
 
       δω y ( t )= Rv  cos( vt )  (10)
 
     A pair of counter-propagating egg crates is produced by imposing counter-propagating lattices in the x-direction as in Eq. (7): 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       s 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         V 
                         0 
                       
                       4 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 
                                   k 
                                   L 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           vt 
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           cos 
                           [ 
                           
                             
                               2 
                               ⁢ 
                               
                                 
                                   k 
                                   L 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     + 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           vt 
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     2 
                                     ⁢ 
                                     
                                       
                                         k 
                                         L 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           y 
                                           - 
                                           
                                             R 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 vt 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Herein, this is referred to as the “Sagnac Lattice Potential” or “SLP”. The SLP plays a primary role in the Sagnac gyroscope, in particular, in propagating atoms one or multiple times around the Sagnac loop. For a visual representation of an “egg crate”, see FIG. 14 of H. H. Metcalf “Laser Cooling and Trapping of Neutral Atoms”, Journal of the Optical Society of America 8—May 2003, pp. 975-1014. 
     In a classical interpretation, an atom that is trapped in the SLP and undergoing circular motion is subject to a centripetal force that is constant in magnitude. For a rubidium atom executing circular motion having radius R=1 mm and period T s =30 ms, the centripetal acceleration is ac=v2/R=4π2R/τ S≅4.5 g   2 , i.e. larger than but comparable to the force of gravity. The centripetal force is supplied by the lattice potential; viewed classically, it is given by the negative gradient of the Sagnac potential. Estimate of the maximum is simply F max =V 0 k L . A typical potential amplitude is V 0 ˜10E R , corresponding to a maximum force of 110 g, i.e., adequate to accommodate substantial dynamic environments. 
     Sagnac sensitivity to input rotation scales as the enclosed area and, therefore, as the square of the circular loop radius. The interferometer phase difference ΔΦ is determined by the input rotation rate Ω according to Equation 12: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Φ 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         m 
                         ℏ 
                       
                       ⁢ 
                       A 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Ω 
                     
                     = 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                         m 
                         ℏ 
                       
                       ⁢ 
                       
                         R 
                         2 
                       
                       ⁢ 
                       Ω 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Were it not for other technical limitations, one would choose the single-turn loop radius as large as possible. In addition to the direct sensitivity benefit, a single large-area loop also indirectly ameliorates the effects of atomic collisions, since the average distance between atoms, keeping the total number constant, can be larger than with a smaller loop. The primary technical impediment to a larger ring is lattice laser power, since for fixed lattice depth the required power scales at least linearly with radius yet, in reality, more quickly because of beam diffraction. The other technical challenge limiting radius is maintaining uniform lattice beam intensity over a larger cross-section. 
     R=1 mm is a reasonable compromise between loop size and required laser power and beam intensity uniformity. A single turn of the loop thus encloses A=πR≅3.14 mm 2  area. Satisfying the metrics requires about 3 turns around the loop for 10 mm 2  and 32 turns around the loop for 100 mm 2 . Atoms thus propagate around the loop in about 0.031 milliseconds (ms) in the former case and in about 0.2 ms in the latter case. 
     The lattice laser provides the trapping forces and the ability to transport atoms in the Sagnac interferometer. The choice of laser wavelength involves a direct tradeoff among required power, lattice potential, and atom lifetime due to spontaneous emission. For fixed trapping potential, the required power increases linearly with the detuning of the wavelength away from atomic resonance, while the scattering rate falls off with the square of detuning. It is common in optical lattice experiments to use trapping light of wavelength λ=1.04 μm. 
     Some embodiments take advantage of the lower power requirements with the choice of λ L =852 nm. Moreover, this wavelength is accessible with diode laser technology and also with PICs. 852 nm is selected instead of, for example, 850 nm, because cesium can then be used as a reference to stabilize the frequency if need be. As gyro scale factor is directly tied to the lattice laser wavelength, one can set the scale factor with a precision to well better than a part per billion. There is no extra cost or burden in choosing the 852 nm wavelength. 
     The Sagnac loop radius of 1 mm requires a lattice beam of somewhat over 2 mm in width. Assuming a target beam thickness of 40 μm as used in the shaken lattice work of Weidner and Anderson, about 0.4 W of laser power at 852 nm is sufficient to provide a lattice depth of V 0 ≅10E R . Effective depth is decreased by a factor of 2 in the case of superposed moving standing light fields. At 0.4 W power, atom lifetime due to spontaneous emission is about 20 s, i.e. sufficient for the is measurement time targeted for the SLAG. 1 W of power per lattice beam can suffice. 
     Interrogation time is defined as the length of time given to the atoms to enclose 100 mm 2 . The Sagnac atom interrogation time is 1 s, which is a reasonable tradeoff among a number of technical constraints, keeping in mind the performance metrics. Indeed, taken together, interrogation time and duty cycle determine the time-averaged number of atoms that are participating in a rotation measurement. A large mean number of atoms in principle improves Allan deviation. On the other hand, a low atom number reduces the deleterious impact of atom interactions on system performance. A limiting factor is also the time it takes to produce ultra-cold atoms; production times significantly less than 1 second (s) are a challenge. Therefore the 50% duty cycle in which half the time is spent producing ultra-cold atoms and the other half of the time is spent on the measurement may be about optimal. 
     The atom shot noise limited Allan deviation of the interferometer phase corresponding to a measurement time τ is given by the number of atoms N(τ) participating in the measurement by equation 13: 
     
       
         
           
             
               
                 
                   
                     
                       σ 
                       Φ 
                     
                     ⁡ 
                     
                       ( 
                       τ 
                       ) 
                     
                   
                   = 
                   
                     1 
                     
                       
                         N 
                         ⁡ 
                         
                           ( 
                           τ 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     A large atom number per measurement thus gives small uncertainty, yet it also gives rise to unwanted atom interactions. 3000 atoms participating in a measurement along with a 50% duty cycle indicates a shot noise performance of σΦ=25 mrad/√{square root over (s)}. In reality Allan deviation is typically dominated by other-than shot noise sources. 
     Shaken lattice Sagnac interferometry requires an at least two-dimensional lattice. However, the illustrated embodiment also confines atoms in the axis transverse to the plane of the Sagnac loop with an additional z-axis lattice having modest power. In the one-dimensional shaken lattice work of Weidner and Anderson, interferometer performance was limited by heating of atoms, in particular, excitation of the transverse modes of the atoms. Wholly confining atoms in the transverse plane substantially mitigates transverse excitation. Moreover, the z-axis lattice will address acceleration/tilt invariance metrics. The configuration of the three-dimensional lattice has a high aspect ratio: approximately 2.5 mm×2.5 mm×0.04 mm. 
     In an embodiment, the SLAG utilizes forced evaporation in an atom-chip-based magnetic potential. However, another embodiment utilizes an optical BEC for two reasons: 1) the needed laser power is already available from the lasers that also supply the lattice light (since BEC and lattices are done sequentially) and, therefore, optical BEC lowers the total system power requirement; and 2) it simplifies the design and implementation of the 3D lattice since the need for magnetic fields from the atom chip is eliminated, which otherwise interferes with easy optical access of the third dimension of the lattice. Furthermore, higher power metrics are more accessible with the implementation of optical BEC. 
     Operationally a rotation measurement takes place as a sequence of steps. They are: atom cooling, lattice loading, ground-state cooling, interferometer operation (matter wave beam splitting, Sagnac loading and transport, matter-wave beam combining), then imaging. In an embodiment, the gyro system includes two laser systems each connected via optical fibers to a physics package that serves as the interface between the laser systems and the vacuum system in which the manipulation and control of atoms takes place. The laser system utilized for laser cooling of the rubidium atoms is now standard, as is the use of a double-MOT scheme to enable fast (˜1 s) BEC production times. 
     An 852 nm laser system can be used for producing a 3D lattice as well as optical BEC. In this design, independent frequency control is provided to counter-propagating light. This allows complete flexibility in the ability to apply both frequency and phase modulation to the lattice beams, e.g., phase modulation for carrying out beam splitting, and frequency modulation to establish the Sagnac lattice potential, Eq. (11). 
     Output from the laser systems can be coupled via optical fibers to the physics package. The physics packages are variations on ColdQuanta commercial “Physics Station” products. 2-D and 3-D MOT beam distribution as well as imaging are already built into the Physics Station. Moreover, the system incorporates magnetic coils and an atom chip for the production of BEC using RF forced evaporation. The Physics Station has been modified to accommodate the lattice and optical BEC beams, as well as intensity monitoring detectors for the lattice beams. 
     A physics package used by Weidner and Anderson for the shaken lattice experiments has been modified to accommodate a two-dimensional lattice. The system has been upgraded to accommodate a new vacuum cell specifically designed to accommodate a 3D lattice. The upgraded system is capable of tilt. A more advanced system incorporates a further advanced vacuum cell and is capable of essentially arbitrary orientation in order to carry out acceleration invariance measurements. 
     The core of the physics package is the vacuum cell that accommodates a 2D MOT, a 3D MOT and BEC. The cell system utilized here is based on ColdQuanta&#39;s commercial RuBECi system (which in turn is based on technology developed at The University of Colorado at Boulder under the DARPA g-BECi program). The majority of beams have standard Gaussian beam cross-sections. Of particular concern, though, is the lattice beam shape, which ideally has uniform intensity across a high-aspect ratio rectangular beam shape and a planar wave front. While classical optical line generators are used in some embodiments to approximate the ideal, other embodiments achieve greater tolerance limits on intensity and wave-front uniformity to achieve higher performance. 
     The various operations are carried out using a control system based on Lab View software. (Laboratory Virtual Instrument Engineering Workbench is a system-design platform and development environment for a visual programming language from National Instruments.) The system has a user interface tailored to the kind of event timing need for the production and utilization of cold atoms. The present work has additional requirements for control of the lattice beams in particular and therefore the current control system will need upgrading to handle the larger number of control parameters. In addition, the current control systems are not designed to update their control parameters on the basis of measurement feedback from the experimental system. Such capability can greatly speed experimentation and development. 
     An embodiment uses an integrated vacuum package compatible with the atomic lattice gyro (ALG) based on ColdQuanta&#39;s ultrahigh vacuum (UHV) “channel cell” technology. Channel cell technology utilizes a silicon-and-glass construction which lends itself to direct integration with planar waveguide structures and minimizes the complexity of the optical access into the vacuum system. Channel cells can be made to include precision optical components, such as mirrors, gratings, and light collection optics; as well as all necessary UHV system components, including rubidium sources, and passive and active vacuum pumps. Being fabricated from glass and silicon, an incorporating system is able to produce a vacuum system that would normally occupy several liters in a package that is less than 100 cc while maintaining a vacuum in the nano-torr regime. Also, because of the monolithic construction, channel cells provide a high-performance package that is sufficiently robust for demanding environments. In June of 2017, ColdQuanta demonstrated an airborne channel cell-based laser cooling system operating aboard an unmodified Cirrus SR-22—a four seat, single piston engine, general aviation aircraft. 
     Chip-format Liquid Rb Dispensers (LRD) are used in an embodiment so that rubidium can be delivered with only 30 mW of electrical power. An alternative LRD design incorporates a reservoir with channel features for liquid Rb source material, membrane orifice for puncture/flow, and temperature control features. The LRD takes advantage of micromachining. For the purposes of the channel cell MOT, a relatively simple optics package uses conventional OTS optics. However a more advanced design uses PIC beam distribution technology that utilizes a channel cell vacuum package in conjunction with a PIC distribution system to make a compact, robust, and manufacturable vacuum package, where the precision in the optical alignment has been primarily leveraged onto the PIC. 
     An embodiment incorporates circularly-polarized grating couplers (CPGCs) which allow any desired polarization of light to be delivered to trapped atoms via on-chip waveguides. These are based upon on-chip single-mode waveguides at the variety of wavelengths useful for atomic physics applications. On-chip waveguides guide a single optical mode below the surface of the trap chip by utilizing the index contrast between the waveguide material and the cladding. Embodiments herein use optimized materials and processes to achieve low optical losses in waveguides over the visible and near infra-red (NIR) spectrum; losses as low as 3 dB/cm have been achieved for wavelengths as low as 370 nm. The waveguides support two orthogonal polarizations (TE and TM) which are transformed by grating coupler into two orthogonal linear polarizations of free-space beams. 
     The grating couplers direct light out of the chip plane and focus the beam to a desired spot above the chip surface. These gratings are formed by etching the waveguide layer to create a periodic change in refractive index. The grating couplers deliver light with a defined linear polarization, focused spot size as low as 4 μm, and with the angle of grating coupler emission accurately predicted by simulations. Elliptical and circular polarizations of light can be achieved by interfering the linear light polarizations launched from a grating coupler with input from two orthogonal waveguides with a well-defined phase relationship between the two. 
     The phase relationship between the two beams depends on the waveguide path length difference between the beams, which cannot be defined in fabrication to the sub-wavelength tolerance needed to reproducibly achieve a particular circular polarization of light. Instead, a small heater fabricated into the chip (made of TiN or similar material) can locally heat one of the two waveguide branches to achieve the necessary phase relation between the two beams and achieve circular polarization. Initial calibrations of each CPGC is used to determine the necessary heater current to achieve a desired polarization. The waveguides are sufficiently stable that this calibration step need only be performed once. Embodiments use CPGCs at 780 nm, Other embodiments use CPGCs at other wavelengths. The CPGCs are used for laser cooling. These CPGCs are used in embodiments in which the cold-atom gyroscope is packaged in PIC form. 
     System operation has two operational phases: 1) a learning/calibration phase in which the final lattice shaking algorithm is itself achieved through a learning algorithm; and 2) a sensing phase in which the now programmed lattice system is usable for rotation sensing. The learning process underlying calibration seeks to vary certain parameters, such as the Fourier components of a particular shaking function. The greater the number of Fourier components that are needed, in general, the slower the learning process. Thus, one would like to minimize the number of parameters, i.e., the dimensionality of the control space. The number of Fourier components can depend a great deal on the Bloch band structure of the lattice, and there is substantial freedom in the design since one can superpose, for example, moving and stationary lattices. Unlike many real-world learning scenarios one can manipulate the physical system to simplify the job of calibration. 
     Machine learning (ML) has become instrumental in advancing quantum device performance through its application to quantum control. Embodiments use machine learning techniques for state preparation and manipulation for shaken lattice interferometry and acceleration sensing demonstrated and apply similar ideas to rotating and counter-rotating atomic lattices used in a Sagnac interferometer gyroscope. Expertise at applying machine learning to quantum control of solid-state spin ensembles for magnetometry can be transferred to the design of the SLAG. 
     Various embodiments use stochastic, simplex, and gradient-based methods in determining optimal controls for state preparation and manipulation. The genetic algorithm, a stochastic multidimensional optimization technique, has enabled success in momentum-state engineering. See S. Pötting, M. Cramer, C. H. Schwalb, H. Pu, and P. Meystre, “Coherent acceleration of Bose-Einstein condensates,”  Phys. Rev. A , vol. 64, no. 2, August 2001 and in shaken-lattice interferometry. However, relatively slow algorithm convergence requires the use of the CRAB and dressed CRAB algorithms for quantum control optimization, which employ gradient-free minimization. See: 1) P. Doria, T. Calarco, and S. Montangero, “Optimal Control Technique for Many-Body Quantum Dynamics,”  Phys. Rev. Lett ., vol. 106, no. 19, 2011; T. Caneva, T. Calarco, and S. Montangero, “Chopped random-basis quantum optimization,”  Phys. Rev. A , vol. 84, no. 2, 2011; and N. Rach, M. M. Mueller, T. Calarco, and S. Montangero, “Dressing the chopped random-basis optimization: A bandwidth-limited access to the trap-free landscape,”  Phys. Rev. A , vol. 92, no. 6, 2015. 
     The underlying Nelder-Mead simplex optimization grows slow as the number of parameters approaches ten. While effective for one-dimensional shaken-lattice interferometry, increasingly powerful techniques are used ford for the shaken lattice Sagnac gyroscope. (See S. Machnes, E. Assémat, D. Tannor, and F. K. Wilhelm, “Tunable, Flexible, and Efficient Optimization of Control Pulses for Practical Qubits,”  Phys. Rev. Lett ., vol. 120, no. 15, p. 8, April 2018.) define quantum optimal control requirements i) flexibility, ii) numerical accuracy, iii) and speed, and propose the gradient optimization of analytic controls (GOAT) which meets these specifications. GOAT is particularly well-suited to shaken-lattice interferometry given the constraints set by band transition frequencies. 
     Focusing still on the calibration stage for example, how one treats the inevitable noise in the data has a substantial impact on the convergence of the learning process. Especially given the low atom number targeted for gyro experiments, image data will present noise to the learning processes. Reinforcement learning can be used for the control problem, in which complex system dynamics that are not completely known, produce noisy data. This applies to experiment, where both technical noise and atom counting (shot) noise cause significant deviation from simpler models in which conventional optimization succeeds. See S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, “Bose-Einstein condensation in a circular waveguide,”  Phys. Rev. Lett ., vol. 95, no. 14, 2005. 
     In the case of technical noise, deep reinforcement learning can directly learn control policies from high-dimensional sensory input, such as the pixelated screen of Atari games, by replacing the agent of a conventional reinforcement learning framework with a deep neural network. See V. Mnih, K. Kavukcuoglu, D. Silver, A. Graves, I. Antonoglou, D. Wierstra, and M. Riedmiller, “Playing Atari with Deep Reinforcement Learning,” arXiv.org, vol. cs.LG. 19 Dec. 2013; and V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,”  Nature, vol.  518, no. 7540, pp. 529-533, 2015. 
     For the shaken-lattice atomic gyroscope, the state is an image of the split atoms, the action a control protocol, and the reward a measure of interferometer or gyroscope performance, i.e. minimization of the Allan deviation of phase or rotation rate, respectively. This harnesses graphics processor unit (GPU) accelerated desktop and supercomputing systems for model training, validation, and testing. 
     For a review of atom Sagnac interferometry by one of the pioneers in free-space systems, see P. B. G. A. Navigation 2014, “The centenary of Sagnac effect and its applications: From electromagnetic to matter waves,” Springer.) The majority if not all of the work on trapped atom Sagnac matter-wave rotation sensors utilize trapping potentials that confine atoms relatively tightly in two dimensions and loosely, or not at all, in the third “waveguide” dimension. Trapping forces are produced by magnetic, optical fields or radio-frequency (RF) fields. Enclosed area is achieved by one of two concepts: either the potential itself is ring shaped, i.e., resembling a one-turn Sagnac fiber loop, or a one-dimensional waveguide is transported in the transverse direction, first one way and then returned, while the atoms are otherwise propagating along the guide. In the case of optical guides, ring structures can either be produced directly (e.g., an optical torus) or “painted” by scanning one or more laser beams to create the desired geometry at a rate that atoms do not “see” the time variations. See K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, “Experimental demonstration of painting arbitrary and dynamic potentials for Bose-Einstein condensates,”  New J Phys, vol.  11, no. 4, p. 043030, April 2009. Circular magnetic potentials can likewise be made in a few ways, such as sequentially excited circular segments (somewhat like a synchronous motor) or by an appropriate arrangement of coils. 
     All the approaches to date suffer from very similar sets of challenges that arise when the trapping forces are weak, of which there are two that are fundamental. First is the problem that a dynamic environment, even the modest vibration dynamics that occurs in a research laboratory, can cause unwanted excitations of the transverse modes. This will cause a reduction in contrast that becomes worse the longer the measurement time. The second is the deleterious effects of atom interactions. At best they cause phase-diffusion which also becomes worse as the interrogation time becomes longer, and at worst they cause heating and loss of atoms. J. A. Stickney, D. Z. Anderson, and A. A. Zozulya, “Increasing the coherence time of Bose-Einstein-condensate interferometers with optical control of dynamics,”  Phys Rev A , vol. 75, no. 6, June 2007. Unfortunately, these two issues are at odds: one can reduce the atom interactions by utilizing loose traps, but then the transverse excitations become exacerbated. There are other challenges facing trapped atom approaches to rotation sensing: magnetic trapping is prone to adding noise due to both technical (current noise in the power supplies) and fundamental (Johnson noise) sources. 
     Optical trapping also has its challenges, e.g. in a circular optical waveguide the intensity non-uniformities can cause heating of the atoms. The challenges of the weak-trapping approaches may indeed be overcome, but to date they have not provided high performance in even a modest dynamics environment. The shaken-lattice approach, by contrast, uses high trapping forces to minimize transverse mode excitations. Atom interactions are minimized in some embodiments by working at low atomic densities such that any given site has low probability of occupation. Importantly, the system can operate in dynamical environments as long as the corresponding forces are small compared with those holding the atoms. 
     In an embodiment, atom interference is observed, and the Allan deviation and phase repeatability is measured. Atom interference is at 10% contrast with trap and laser power less than 10 Watts (W) and a Sagnac area of better than 10 mm 2 . The instrument has a phase repeatability of 100 mrads over 2/24/2 hr on/off/on and an Allan deviation better than 100 mrad*s½ for 1 to 3600 seconds. The instrument can be subjected to a 10° tip over and yet continue operation with negligible change in performance. 
     In another embodiment, gyroscopic operation is observed and the Allan deviation and scale factor stability are measured. In addition, the invariance to acceleration, temperature stability and mechanical vibration tolerance of the atomic lattice gyro (ALG) are measured. The ALG is demonstrated with 80% atom interference contrast and a Sagnac area of at least 100 mm 2  with the volume of the sum of components less than 0.5 L and consuming less than 10 W power. In addition, the instrument exceeds an Allan deviation of 30 mrad*s½ for 1 to 3600 seconds and phase repeatability of 10 mrads over 2/24/2 hr on/off/on. Environmental apparatus are applied to the second ALG instrument to demonstrate phase stability to 1 mrad under +/−1 g acceleration with turn over tests at 0°, 90°, and 190°, 10 ppb/C temperature stability of the interferometer area, and stable ALG operation (gyroscopic measurement) under mechanical vibration of 0.01 g2/Hz from 10 Hz to 1000 Hz. 
     Herein, all art labeled “prior art”, if any, is admitted prior art; all art not labelled “prior art” is not admitted prior art. The illustrated embodiments as well as modifications thereto and variation thereon are provided for by the present invention, the scope of which is defined by the following claims.