"robust atom interferometer"

Abstract

This disclosure relates to an inertial sensor for measuring an inertial quantity along a sensing axis. The sensor comprises an atom interferometer comprising a pulse generator to generate one or more pulsed light beams, defined by a respective pulse duration to place atoms into a superposition, and re-combine the atoms to measure an interference of the atoms and calculate the inertial quantity based on the measured interference. The sensor further comprises an auxiliary acceleration sensor configured to measure acceleration transverse to one or more of the pulsed light beams; and a control system configured to calculate a transversal offset of the atoms relative to the pulsed light beams caused by the measured acceleration transverse to the one or more of the light beams, and increase the pulse duration to compensate for a reduced beam intensity applied to the atoms as a result of the transversal offset.

Description

"Robust interferometer" Cross-Reference to Related Applications

[0001] The present application claims priority from Australian Provisional Patent Application No 2022902177 filed on 3 August 2022, the contents of which are incorporated herein by reference in their entirety. Technical Field

[0002] This disclosure relates to measuring an inertial quantity, such as acceleration and rotation, using an atom interferometer. Background

[0003] Quantum accelerometers and gyroscopes based on atom interferometry possess potentially superior sensitivity and stability when compared with state-of-the- art classical sensors. In many atom interferometric sensors, pulsed interactions between a laser beam and an expanding, free-falling cloud of atoms are used to form equivalents of mirrors and beam splitters for atomic matter-waves. These pulses are then applied in an interferometer sequence to divide, reflect, and interfere atomic matter-waves. Although these devices possess a large intrinsic sensitivity to accelerations and rotations, they currently have limited use outside controlled lab-based environments.

[0004] Atom interferometers exploit the quantum-mechanical wave nature of atoms by measuring the interference between atomic matter-waves that have been divided to follow separate paths by the matter-wave equivalent of a beam splitter. Atoms make excellent identical test masses due to their multiple internal and extrinsic degrees of freedom - which may be exploited for precise control using optical fields - and for their sensitivity to inertial, electromagnetic, and gravitational effects. Applications of atom interferometers include state-of-the-art measurements of the fine-structure andgravitational constants, tests of the principle, searches for exotic physics, geophysics, civil engineering, and inertial navigation.

[0005] Any discussion of documents, acts, materials, devices, articles or the like which has been included in the present specification is not to be taken as an admission that any or all of these matters form part of the prior art base or were common general knowledge in the field relevant to the present disclosure as it existed before the priority date of each of the appended claims.

[0006] Throughout this specification the word "comprise", or variations such as "comprises" or "comprising", will be understood to imply the inclusion of a stated element, integer or step, or group of elements, integers or steps, but not the exclusion of any other element, integer or step, or group of elements, integers or steps. Summary

[0007] An inertial sensor for measuring an inertial quantity along a sensing axis comprises: an atom interferometer comprising a pulse generator to generate one or more pulsed light beams, defined by a respective pulse duration to place atoms into a superposition, and re-combine the atoms to measure an interference of the atoms and calculate the inertial quantity based on the measured interference; an auxiliary acceleration sensor configured to measure acceleration transverse to one or more of the pulsed light beams; and a control system configured to: calculate a transversal offset of the atoms relative to the pulsed light beams caused by the measured acceleration transverse to the one or more of the light beams, and increase the pulse duration to compensate for a reduced beam intensity applied to the atoms as a result of the transversal offset.

[0008] In some embodiments, the atom interferometer comprises:an atomic source to provide the moving relative to the sensing axis; the pulsed light beams being configured to: place the atoms into a superposition of a first state and a second state, where the first state relates to a first path and the second state relates to a second path, and re-combine the first state and the second state to create interference between the first state and the second state; a measurement system to measure a physical effect indicative of the interference; and a processor to calculate the inertial quantity based on the measured physical effect.

[0009] In some embodiments, the pulsed light beams are further configured to apply a mirror operation to the atoms between the placing and the re-combination.

[0010] In some embodiments, the inertial quantity is a linear acceleration or a rotation.

[0011] In some embodiments, measuring the physical effect comprises measuring a population of atoms in different superposition states.

[0012] In some embodiments, increasing the pulse duration comprises applying an upper limit to the pulse duration.

[0013] In some embodiments, the atoms have a velocity distribution and the upper limit relates to a pulse duration at which the pulse still interacts with a threshold quantity of the atoms.

[0014] In some embodiments, the upper limit is set by a peak two-photon Rabi frequency of the sensor and the longitudinal velocity width of the atoms.

[0015] In some embodiments, increasing the pulse duration comprises:calculating a degradation as one more pulse adjustment factors that relate an ideal interaction of the atoms with the pulsed light beams to a predicted interaction of the atoms with the pulsed light beams under the transversal offset; and increasing the pulse duration by the one or more adjustment factors.

[0016] In some embodiments, the ideal interaction and the predicted interaction are represented by corresponding two-photon Rabi frequencies.

[0017] In some embodiments, the predicted interaction is predicted taking into account a spatial shape of the pulsed light beams.

[0018] In some embodiments, the pulse duration is increased differently for each of the pulsed light beams by calculating the degradation as a function of time.

[0019] In some embodiments, the degradation is a function of an interrogation time.

[0020] In some embodiments, the auxiliary acceleration sensor is configured to measure the acceleration transverse to one or more of the pulsed light beams multiple times during the travel of the atoms through the atom interferometer; and the control system is configured to compensate for degradation at each of the pulsed light beam using the corresponding measured acceleration.

[0021] In some embodiments, the pulsed beams place the atoms into the superposition and re-combine the atoms by stimulating a Bragg transition or a Raman transition.

[0022] In some embodiments, the superposition is a superposition of two states with different momenta that are separated by an integer number of two-photon recoils.

[0023] In some embodiments, the control system is further configured to adjusting a pulse property other than duration to perform arbitrary further manipulations of a motional state of the atoms.

[0024] In some embodiments, the system is further configured to adjust a pulse timing to compensate for the reduced beam intensity applied to the atoms as a result of the transversal offset.

[0025] In some embodiments, the control system is further configured to adjust an intensity of the one or more light beams to compensate for the reduced beam intensity applied to the atoms as a result of the transversal offset.

[0026] A method for measuring an inertial quantity along a sensing axis comprises: generating a cloud of atoms; generating one or more pulsed light beams, defined by a respective pulse duration to place the atoms into a superposition, and re-combine the atoms; measuring acceleration transverse to one or more of the pulsed light beams using an auxiliary acceleration sensor; calculating a transversal offset of the atoms relative to the pulsed light beams caused by the measured acceleration transverse to the one or more of the light beams, and increasing the pulse duration of the pulsed light beams to compensate for a reduced beam intensity applied to the atoms as a result of the transversal offset. measuring an interference of the atoms; and calculating the inertial quantity based on the measured interference.

[0027] A computer-implemented method for controlling an atom interferometer comprises: receiving, from an auxiliary acceleration sensor, acceleration sensor data indicative of an acceleration transverse to one or more of multiple pulsed light beams, defined by a respective pulse duration to place atoms into a superposition, and re- combine the atoms to measure an interference of the atoms; calculating a transversal offset of the atoms relative to the pulsed light beams caused by the measured acceleration transverse to the one or more of the light beams, increasing the pulse duration of the pulsed light beams to compensate for a reduced beam intensity applied to the atoms as a result of the transversal offset.Brief Description of Drawings

[0028] Fig.1 illustrates the working principle of an atom interferometer under an arbitrary acceleration.

[0029] Fig.2 is a depiction of two possible free-falling atomic cloud trajectories under a) no transverse platform acceleration and b) a constant transverse platform acceleration. The Gaussian intensity profile of the interferometry beams is also shown. If atoms are accelerated to a position in the beam with lower intensity during the interferometer, the pulse area conditions disclosed herein are no longer met and the contrast of the fringes is reduced.

[0030] Fig.3: a) Simulation of the expected change in two-photon Rabi frequency multiplier ^(t ) = I ( r ( t )) / I for an expanding ens50 emble of 10 atoms subject to aax= 30 ms^ 2constant transverse acceleration in the x direction. HereIis the spatially-dependent beam intensity profile,I0the peak intensity, and r ( t ) is the classical trajectory of an atom over time. Different atoms in the cloud have different initial positions and velocities, and therefore different trajectories r ( t ) ; the shaded region shows the full-range of variation due to different atomic trajectories throughout an interferometer with a 15 ms interrogation time. b) depicts the distribution of two- photon Rabi frequency multipliers during each pulse. Solid black lines indicate kinematic estimates of the average two-photon Rabi frequency multiplier computed using rkin(t) =^1axt2. The transverse temperature of the atoms was 5 ^ K , and the initial atoma Gaussian spatial distribution with a standard deviation of 1 mm and was centred on the beam axis. The expected change in ^ (t ) was calculated by solving the equations of motion for each atom and assuming a Gaussian beam with a1 / e2radius of 20 mm.

[0031] Fig.4a illustrates a simulated interferometer fringe contrast and Fig.4b illustrates longitudinal sensitivity with software-gimballing (gimbal control) andwithout (conventional) for a range of transverse accelerations. We have assumed that the peak two-photon Rabi frequency of the Raman pulses is 100 kHz, and that the standard deviation of the atom cloud’s longitudinal momentum distribution is 0.125 ^keff. To determine the fringe contrast for each acceleration, we calculate the population difference between the states |1 ^ and | 2 ^ after simulating the entire pulse sequence while varying the phase of the final pulse. The contrast is taken to be the amplitude of a sinusoidal fit to the fringes after averaging over 104atoms with random initial positions and velocities drawn from their respective distributions. The interrogation time was 25 ms. The standard deviation of the initial (Gaussian) cloud was 1 mm, the (Gaussian) beam 1 / e2radius was 20 mm, and the transverse temperature was 5^K. The maximum duration scaling factor was 20 ^ . Our model includes the effects of ballistic expansion, which are used to calculate the two-photon Rabi frequency during each time-slice.

[0032] Fig.5 illustrates an inertial sensor for measuring an inertial quantity, such as linear acceleration or rotation.

[0033] Fig.6 illustrates a method for measuring an inertial quantity.

[0034] Fig.7a illustrates the simulated interferometer fringe contrast with software- gimballing (gimbal control) and without (conventional) for a range of constant transverse accelerations. Gimbal control is depicted with and without pulse timing adjustments (gimbal control: duration and gimbal control: duration and timing, respectively). Pulse timing adjustments are calculated for each value of transverse acceleration to minimise the dependence of the interferometer phase on longitudinal atomic momenta. We have assumed the peak two-photon Rabi frequency is 50 kHz, and that the standard deviation of the atom cloud’s longitudinal momentum distribution is 0.25 ħkeff. To determine the fringe contrast for each acceleration, we calculate the population difference between the states |1> and |2> after simulating the entire pulse sequence while varying the phase of the final pulse. The contrast is taken to be the amplitude of a sinusoidal fit to the fringes after averaging over 5ൈ105atoms withrandom initial positions and velocities from their respective distributions. The interrogation time was 30 ms. The standard deviation of the initial (Gaussian) cloud was 1 mm, the (Gaussian) beam 1 / e2radius was 10 mm, and the transverse temperature was 1 μK. The maximum duration scaling factor was 10. Our model includes the effects of ballistic expansion, which are used to calculate the two-photon Rabi frequency during each time-slice.

[0035] Fig.7b illustrates the simulated interferometer fringe phase shift with software-gimballing (gimbal control) and without (conventional) for a range of constant offsets to the longitudinal momentum distribution (longitudinal momentum distribution asymmetry). A constant transverse acceleration of magnitude 0.5g was assumed in the model. Gimbal control is depicted with and without pulse timing adjustment (gimbal control: duration and gimbal control: duration and timing, respectively). The pulse timing adjustment is calculated to minimise the dependence of the interferometer phase on longitudinal atomic momenta caused by the duration adjustment made in response to the transverse acceleration. We have assumed the peak two-photon Rabi frequency is 50 kHz, and that the standard deviation of the atom cloud’s longitudinal momentum distribution is 0.25 ħkeff. To determine the fringe phase for each acceleration, we calculate the population difference between the states |1> and |2> after simulating the entire pulse sequence while varying the phase of the final pulse. The interferometer phase shift is taken to be the phase of a sinusoidal fit to the fringes after averaging over 5ൈ104atoms with random initial positions and velocities drawn from their respective distributions. The interrogation time was 30 ms. The standard deviation of the initial (Gaussian) cloud was 1 mm, the (Gaussian) beam 1 / e2radius was 10 mm, and the transverse temperature was 1 μK. The maximum duration scaling factor was 10. Our model includes the effects of ballistic expansion, which are used to calculate the two-photon Rabi frequency during each time-slice. Description of Embodiments

[0036] Fig.1 illustrates a light-pulse atom interferometric sensor 100 (also referred to as simply “atom interferometer”) comprising an atomic source 101, often cooled and / orvelocity-selected to temperatures of a or less and prepared in a single ground electronic atomic state. An electronic state is defined by the electron configuration of the system, and by the quantum numbers of each electron contributing to that configuration. Each electronic state corresponds to one of the energy levels of the atom.

[0037] In some examples, alkali-metal atoms are used (e.g.87Rb, but others could equally be used), in which case trapping and laser cooling may be achieved by using a magneto-optical-trap (MOT) and optical molasses. In other examples, an ultracold atomic sample such as a Bose-Einstein Condensate is used as a source of atoms. Atoms may also be subsequently velocity-selected along the interferometry beam axis and prepared in a single magnetically-insensitive ground hyperfine state using a series of optical and / or microwave pulses.

[0038] After the atomic source is appropriately prepared as described above (or otherwise), the atomic source is subjected to a sequence of optical interferometry pulses. More particularly, there is a first pulse 102 that acts as a beam splitter (BS1) in the sense that the pulse places the atoms from source 101 into a superposition of a first state and a second state. These two states have different momenta and therefore, they separate spatially along the interferometry beam axis. A second pulse 103 swaps the states of each part of the atomic superposition created after the first pulse 102, which is why the second pulse 103 is also referred to as a mirror (M). Finally, the third pulse 104 re-combines the atoms in the superposition to create an interference, which is also referred to as a second beamsplitter (BS2). A counter counts the population of atoms in each state after the third pulse 104. In the absence of any acceleration, there should be no relative phase shift between the states in a perfectly operating atom interferometer, which means after the interference all atoms should be in the same state.

[0039] However, in the presence of an acceleration with a non-zero component in the z-direction, there will be a different phase shift accumulated by atoms along each path, which means after third pulse 104, a proportion of atoms will be in a different state than what would have been the case without acceleration. This population difference may be“read out” using basic principles of If the states are the same the matter waves add constructively and contribute maximally to an optical probe signal. On the other hand, if the states differ by a phase shift of ^^ / 2, they add destructively and minimize an optical signal. Arbitrary phase shifts between these extrema represent a continuously variable measurement outcome.

[0040] This principle can also be explained using a single atom. The single atom is placed into a superposition of two states by first pulse 102. In the case of zero acceleration in the longitudinal direction, the third pulse 103 places the atomic superposition into a single state. Under the influence of an acceleration, however, the states in the superposition accumulate a non-zero relative phase before the third pulse is applied. As a result, the third pulse 104 places the atom not exactly in the ground state. In other words, there is a non-zero probability that the atom is measured in a different state. Therefore, if the experiment is repeated with many atoms, some of them will be observed to be in a different state, which is a physical phenomenon that is indicative of the experienced acceleration.

[0041] The interferometer is sensitive to accelerations along the interferometer beam axis, which is referred to as the longitudinal direction z. The perpendicular ( xy ) plane is referred to as the transverse plane, with accelerations in this plane referred to as transverse accelerations.

[0042] In some examples, the atoms have freedom of movement along the interferometer beam axis. In some examples, the atoms also have freedom of movement in the transverse plane (called “unguided” atom interferometers), such that the atomic source ballistically expands in the absence of external forces. In other examples, the dynamics of the atoms in the transverse plane are modified through optical fields, for instance; these are called “guided” atom interferometers. In yet another example, the atoms are held against gravity using e.g. Bloch Oscillations or extended sequences of pulses.

[0043] Atom interferometry pulses 104 typically effect either two-photon Bragg or Raman transitions. These atom-light interactions are induced by twointerferometry beams with frequencies^i= c | ki|(i = 1, 2 indexing the two beams,and c is the speed of light in vacuum) and couple two motional states separated in momentum by an integer number of two-photon recoils ^keff=n ( k1^ k2) . In other words, ^^^and ^^ଶare the wave numbers of the two beams that also define the direction of propagation. For simplicity of presentation, this disclosure provides examples of atom interferometry with two-photon Raman transitions, whereby atoms in electronic state |1 ^ and momentum p are coupled to a different electronic state | 2 ^ with momentum p ^ ^ keffvia two counter-propagating beams of wavenumbers k1^ kLzˆand k2^ ^ kLzˆ, such that the effective momentum transferred by the pulses is ^keff^ 2 ^ kLHowever, the technique disclosed here is straightforwardly adapted to atom interferometry with other atom-light interactions, such as multi-photon Bragg transitions.

[0044] Resonant two-photon Raman transitions are described by the unitary matrix ^^^^i^^^ ^^cos^ ^ ^ie ^ sin^ ^ ^where^= ^^0 dt |^R( t ) |is the pulse area, ^ is the atom-light interaction time, ^R ( t ) isthe two-which gives the strength of the atom-light coupling at a given time t and is proportional to the intensity of the interferometry beams, and ^ is the relative phase of the optical beams. The atom-optical equivalent of a 50 / 50 beamsplitter is realized by selecting the pulse area ^ = ^ / 2 : U^ / ^ ^ = 1^^ ^ ie i ^ ^ ^ ^^whereas a U( ^ , ^ ) |1,p ^ = ^ iei ^| 2, p ^ ^ keff^ , (3a) U( ^ , ^ ) | 2,p ^ ^ keff^ = ^ ie^ i ^|1, p ^ . (3b)

[0045] For rectangular pulses with duration ^ and constant intensity, phase, and frequency, the pulse area is ^ = ^R^ . Thus, the conditions for a beamsplitter (BS) and mirror (M) pulse are then:^ BS= ^2^(4)R (5)

[0046] A possible measurement of acceleration with a cold-atom interferometer in the three-pulse Mach-Zehnder configuration (BS1-M-BS2) proceeds as follows. Acting the first beamsplitting pulse on an initial cloud of atoms in state |1,p ^ places each atom in a 50 / 50 superposition of atoms in |1,p ^ and | 2,p ^ ^ keff^ . A 50 / 50 superposition means that if the state of any given atom was measured, it would be observed in either state|1,p ^ or| 2,p ^ ^ keff^with 50% probability.

[0047] After some interrogation time T (during which the light is switched off), the mirror pulse redirects the two parts of the superposition such that, after a second period of interrogation time T , they spatially overlap. The second beamsplitter pulse then recombines the two states, after which the number of atoms in internal states |1 ^ and | 2 ^ are measured (P1and P2, respectively). The component of acceleration parallel tothe counter-propagating beams (which the interferometry beam axis) causes a relative phase shift ^ between the two internal states immediately prior to the second beamsplitter, which affects the measured population difference via: P1^ P2= N ^ cos( ^ ) (6) where N is the total number of atoms and ^ is the fringe contrast. In the limit where the pulse durations are much shorter than the interrogation time and the acceleration is constant, the phase shift relates to the acceleration a via ^=keff^ aT2. (7)

[0048] Without loss of generality, here the beams align with the acceleration suchthatkeff^ a = keffawithkeff=| keff|and a =| a | .

[0049] We define the sensitivity as the smallest measurable constant acceleration, ^ a , which for a single interferometer run at the shot-noise limit is: ^a =1^ 2.(8)

[0050] The sensitivity depends upon the interrogation time T , the fringe contrast ^ , the momentum transferred to the atoms by the light ^keff, and the number of atoms N . If the pulse area conditions (Equations (4) and (5)) are not met for each pulse - resulting in imperfect beamsplitting and / or reflection - the contrast and hence sensitivity is degraded. Challenge to be addressed

[0051] One challenge to realizing high precision cold-atom accelerometry onboard mobile platforms is the substantial reduction in beamsplitter and mirror efficiencies due to platform motion. A non-zero platform acceleration in the plane transverse to theinterferometry beams will in general trajectories of the atoms relative to the platform, as shown in Fig.2. It is noted that the unguided atoms would be considered “freely falling” for certain types of measurements, such as gravitational measurements. More generally, “unguided” means that the atoms are not subjected to a force transverse to the interferometer’s primary sensitivity direction which is generated by the sensor system. Note that this is the case for unguided atom interferometers and also for some guided atom interferometers where transverse accelerations are sufficiently large to overcome any guiding forces.

[0052] Since the interferometry beams have a spatially-dependent profile, the atom cloud experiences a different peak intensity (and therefore a different peak two-photon Rabi frequency) during the initial beamsplitter, central mirror, and final beamsplitter pulses if the atoms’ position within the beam varies due to transverse acceleration. For the case of a Gaussian beam profile (depicted in Fig.2), the further an atom is from the beam axis, the lower the intensity and hence the lower the two-photon Rabi frequency. This lower-than-expected two-photon Rabi frequency changes the pulse area from the ideal ^ / 2 and ^ needed to realize perfect beamsplitters and mirrors, respectively. This can substantially degrade the fringe contrast, and therefore the sensitivity to accelerations along the Raman beam axis (Equation (8)).

[0053] It is possible to demonstrate how imperfect beamsplitting and reflection degrades contrast using the resonant Raman model provided by Eq. (1); assuming the atoms are initially on axis (x , y = 0 ) such that a non-zero transverse acceleration only affects the pulse area of the mirror and second beamsplitter, the output state is | ^out^ =U ( ^BS2, ^ ) U ( ^M,0) U ( ^ / 2,0) |1,p ^ . (9)

[0054] Computing populations P2 21 = N | ^ 1,p | ^out^ | and P2= N | ^ 2,p ^ ^ keff| ^out^ | , we obtain

[0055] Let us and a lateral acceleration that causes the peak intensity to drop by 60% and 90% for the mirror and second beamsplitter, respectively. Then ^M= 0.6 ^R^ ^ / ^R= 0.6 ^ and ^BS2= 0.1 ^R^ ^ / (2 ^R) = 0.1( ^ / 2) , yielding ^ ^ 0.05 - a factor of 20 reduction in contrast compared to ideal operation.

[0056] Given the interference fringe contrast of an atom interferometer is directly proportional to the signal-to-noise ratio of the measurement and inversely proportional to the smallest detectable change in acceleration, maintaining high-contrast fringes is desirable for all interferometric sensors but particularly for performant operation in field-based dynamic environments. Mitigating contrast loss

[0057] In this disclosure, there is provided an adaptive software-gimballing technique for mitigating contrast loss due to transverse platform accelerations. This disclosure proposes to combine a quantum measurement of longitudinal acceleration with at least one classical measurement of platform acceleration in the transverse plane (perpendicular to the Raman beam axis) by an auxiliary acceleration sensor. This measurement may be performed using a classical accelerometer (e.g. a Micro Electro- Mechanical System (MEMS) device). In addition, a classical gyroscope may be used to determine the change in orientation of the interferometric sensitivity axis with respect to a particular co-ordinate system, for example that defined by local gravity and the surface of the Earth. In this case, the rotation measurement can be used to determine the component of gravitational acceleration acting on atoms in the transverse plane. More generally, the disclosed method enables the use of gyroscopes to relate the body frame to some fixed inertial frame (or the navigation frame if applicable, such as where cold- atom sensors are used in a navigation application).

[0058] By combining this classical with knowledge of the initial position of the atomic cloud in the beam and the shape of the optical beam intensity profile, a control system can calculate the expected change in average two-photon Rabi frequency between the pulses, as depicted in Fig.3. This enables the system to make a feed-forward adjustment to the pulse shape (for example, the duration and / or amplitude) of the three pulses and hence maintain the pulse area requirement for each of the three interferometry pulses. In commonly encountered scenarios where most transverse accelerations will move atoms to positions in the beam with lower intensity, this means increasing the duration or amplitude of the pulses.

[0059] Provided below is a summary of how the adaptive software-gimballing approach works using rectangular Raman pulses within an example measurement cycle as follows (steps 2 to 5 comprise the software-gimballing procedure): 1. Cool, trap, release, and state-prepare atoms 2. Determine the transverse ( xy ) acceleration a ^ (t ) by using at least one classical accelerometer and / or gyroscope which is fixed to the device platform. The classical measurement of transverse acceleration may be performed once during the interferometer (in which case it is assumed to be constant for a single measurement) or it may be performed multiple times to compute the position of the atomic cloud in the beam with greater accuracy. 3. Compute the expected average two-photon Rabi frequency during each pulse by solving equations of motion for the atomic cloud center-of-mass in the ( xy ) planer (t) =^ ^a ^(t ) d t and using the measured intensity profile of the beams. For example,in the case of interferometry beams with a Gaussian intensity provide 2 ^(t ) =^ maxexp2 r ^RR (^( t )2).(11) wHere ^maxR is the peak two-photon Rabi (corresponding to maximum beam intensity I ), attained at2 20 the beam center on-axis (x = y = 0 ), r^( t ) = x ( t ) ^ y ( t ) is the radial displacement of the cloud center from the beam (z) axis andwis the 1 / e2beam radius, which is determined empirically. The intensity profile is not restricted to a Gaussian shape and the disclosed technique is applicable for any measurable and / or known beam profile. 4. Choose the duration of each pulse ^i^for i = {BS1,M,BS2} to give a pulse area close to their ideal values.: ^ max ^^= R i^ ^i(12)^twhere^iis the ideal pulse duration in the absence of lateral accelerations (e.g. Eq. (4) and Eq. (5) for rectangular pulses). By ideal it is meant the pulse area that maximises the contrast of the interferometer. The “ideal” pulse areas for non-rectangular (e.g. composite) beam splitter and mirror pulses are not limited to the values ^ / 2 and ^ , respectively. It is noted that the rectangular pulses are turned on (left hand edge) at times t=0 for BS1, t=T for M and t=2T for BS2. 5. Apply the interferometry sequence with corrected pulse durations ^i^6. Perform a measurement of population difference between two states

[0060] In one example, the system uses an upper limit on the duration scaling (i.e. ^i^^ ^ max i ) to avoid making the pulses prohibitively velocity selective in the longitudinal dimension (in general, longer pulse durations are resonant with a narrower class of atomic velocities). This limit is set by the peak two-photon Rabi frequency of the experiment and the longitudinal momentum width of the atomic source: themaximum scaling can be set such that a with maximum length scaling still interacts with all atoms in the velocity distribution with high fidelity.

[0061] Although rectangular pulse shapes are used in this disclosure, the software- gimballing technique works without modification for pulse shapes with variable ^R( t ) , and also for a more general definition of “pulse shape” that includes additional control parameters such as the laser beam frequency and phase. For example, the total duration or amplitude of Gaussian pulses, composite pulses, and pulses designed using robust control techniques may also be scaled to satisfy pulse area requirements. As another example, in the case of some composite pulses, both the pulse amplitude and the time- dependent sweeps of the laser frequency can be scaled to meet the needed resonant conditions and pulse area requirements for high efficiency operation.

[0062] It is noted that this procedure will be applicable to any interferometric sensor architecture that uses atoms free to move in a plane perpendicular to the interrogation beam (e.g., “unguided” atom interferometers), including: rotation sensors, multi-axis accelerometers, and gravity gradiometers. It is also applicable to interferometers using different atomic species and alternative atom-light interactions such as single-photon transitions and two-photon Bragg transitions. In the case of Bragg pulses, it is difficult to compensate changes in pulse area by scaling the duration using Equation (12). Instead, the disclosed adaptive software gimballing technique works by providing a look-up table that enables one to select in real-time a pre-optimised pulse shape. The selected Bragg pulse may be optimised for the two-photon Rabi frequency calculated in Eq. (11). Advantages

[0063] One advantage of the disclosed software-gimballing technique is a significant reduction in the fringe contrast decay caused by transverse accelerations of the device platform. Without such compensation, the decay of contrast caused by interrogating atoms in a region of the beam with lower intensity may degrade the device sensitivity. To quantitatively illustrate this, we have performed a Monte-Carlo simulation of anentire unguided interferometry rectangular Raman pulses with and without the disclosed software-gimballing technique. Our simulation includes the effects of cloud expansion, and uses a piecewise-constant approximation of the Raman Hamiltonian for quantum state propagation. The results are depicted in Figure 4.

[0064] In our simulation, we have assumed that the initially prepared atomic source has an initial Gaussian spatial distribution with standard deviation 1 mm and that it expands due to a transverse temperature of 5 ^ K during an unguided interferometry sequence with an interrogation time of 25 ms. We have also assumed the peak two- photon Rabi frequency is 100 kHz and that the standard deviation of the atom cloud’s (e.g., Gaussian) longitudinal momentum distribution is 0.125 ^keff. For our software- gimballing algorithm, we set an upper limit on the length scaling of our pulses to be ^maxi = 20 ^i. We find that the software-gimballing technique significantly reduces contrast decay caused by transverse acceleration, which translates to a 10 ^ improvement in sensitivity for transverse accelerations greater than 2 g . We stress that although the simulated range of transverse accelerations is large, smaller accelerations will cause significant contrast decay without applying our software-gimballing method if the interrogation time is increased, making the technique relevant for smaller transverse accelerations.

[0065] In summary, this disclosure provides an adaptive software-gimballing technique designed to reduce contrast loss in atom interferometric quantum sensors caused by platform accelerations perpendicular to the quantum measurement axis. The disclosed technique uses a simultaneous measurement of platform accelerations using at least one adjacent classical auxiliary sensor. Using this measurement, we compute the expected trajectory of the atomic cloud within the interferometry beams and hence estimate the change in laser intensity and two-photon Rabi frequency during the interferometer pulse sequence. The result is used to make a feed-forward adjustment to the pulse lengths or pulse shapes. This restores the beamsplitter and mirror pulse areas to that needed for highly efficient beam splitting and reflection, and hence maintainshigh-contrast interference fringes. This should therefore improve the stability and sensitivity of atom interferometers operating in dynamic sensing environments.

[0066] Fig.5 illustrates an inertial sensor 500 for measuring an inertial quantity along a sensing axis 501. Sensor 500 comprises an atom interferometer 502 comprising a pulse generator 503 to generate one or more pulsed light beams 504, 505, 506, defined by a respective pulse property to place atoms into a superposition, and re-combine the atoms to measure an interference of the atoms and calculate the inertial quantity based on the measured interference. Here, the interference is measured in the form of an atom count provided by counters 507 / 508.

[0067] There is also an auxiliary acceleration sensor 510, such as a MEMS or fiber optic sensor, configured to measure acceleration transverse to the pulsed light beams 504, 505, 506. A control system 511 is configured to increase the pulse duration of the pulsed light beams 504, 505, 506 to compensate for degradation of the placing and re- combination caused by the measured acceleration transverse to the one or more of the light beams.

[0068] Fig.6 illustrates a method 600 for measuring an inertial quantity along a sensing axis as described above. The method comprises generating 601 a cloud of unguided atoms and generating 602 one or more pulsed light beams, defined by a respective pulse property to place the atoms into a superposition, and re-combine the atoms. The method 600 then measures 603 an interference of the atoms and calculates 604 the inertial quantity based on the measured interference. In order to increase sensitivity, method 600 measures 605 acceleration transverse to one or more of the pulsed light beams using an auxiliary acceleration sensor and increases 606 the pulse duration of the pulsed light beams to compensate for degradation of the placing and re- combination caused by the measured acceleration transverse to the one or more of the light beams. It is noted that the steps in method 600 do not need to be performed in the given order. For example, measuring the acceleration in step 605 and then increasing the pulse duration 606 may be performed before generating the atoms in step 601.

[0069] It is noted that examples herein to a three-pulse BS1-M-BS2 Mach- Zehnder set-up, but many other atom interferometry schemes exist, which may use different numbers of pulses and non-standard splitting ratios. In those cases, it is also possible to measure acceleration transverse to the light beams and increase the pulse duration of the beams to compensate for degradation of the placing and re-combination caused by the measured acceleration. Adjusting timing and intensity

[0070] In addition to varying the pulse durations to compensate for a change in the magnitude of the two-photon Rabi frequency, the pulse timings may be adjusted to improve the contrast of the interferometer (Figure 7a) and reduce unwanted phase shifts caused by variations in the initial longitudinal velocity of the atomic cloud (Figure 7b). This may also be used to control the measurement scale factor of the cold-atom sensor where measurement scale factor is defined as the ratio between the measured interference fringe phase (Eq. (7)) and the longitudinal acceleration a (or the inertial quantity the atom interferometer is configured to measure).

[0071] For example, the time between the first and second interferometer pulses may be increased by an amount dt, where dt depends on the laser intensity profile and the transverse acceleration a^(t) measured by the classical co-sensor. This is calculated using the average two-photon Rabi frequency during each pulse (Eq. (11)). Specifically, ^^ ^^ ൌ ^^ఆಳೄమെ^ఆಳೄభ^, where ^^^ௌ^,^ௌଶare the magnitudes of the two-photon Rabi frequenciesand second beamsplitter pulses calculated using Equation (11), respectively. This pulse timing adjustment is chosen to minimise the variation of the interferometer phase as a function of longitudinal atomic momentum pz(Figure 7b). This improves the contrast of the interferometer (Figure 7a) because different longitudinal momenta atoms exit the interferometer with the same phase preventing the interference fringes “washing out”. Another advantage of this approach is that it provides robustness to variations in the initial longitudinal atomic momentum of the atoms and any asymmetry in their longitudinal velocity distribution.

[0072] The laser intensity may be during each pulse to compensate for the effect of a transverse motion of the atoms in the laser beam. As shown by Eq. (11), the magnitude of the two-photon Rabi frequency changes as a function of time due to a transverse acceleration. By calculating this decrease during each pulse, the peak laser intensity I0may be ramped upwards during each pulse such that the magnitude of the two-photon Rabi frequency remains constant. This will preserve the temporal symmetry and fidelity of the individual light pulses and minimise unwanted interferometer phase shifts arising due to the change in intensity during each pulse. Computer implementation

[0073] It is noted that the methods disclosed herein can be implemented by a computer system, which may comprise edge computing, cloud computing, local (desktop / laptop / tablet) computing, computing performed by a remote or local server, virtual machine, as well as hardware solutions like field programmable gate arrays, application specific circuits and other platforms.

[0074] In that sense, the computer system may comprise a processor (as stated above) that receives, from an auxiliary acceleration sensor, acceleration sensor data indicative of an acceleration transverse to one or more of multiple pulsed light beams. The pulsed light beams are defined by a respective pulse property to place atoms into a superposition, and re-combine the atoms to measure an interference of the atoms. The processor then increases the pulse duration of the pulsed light beams and / or adjusts timing and intensity to compensate for degradation of the placing and re-combination caused by the measured acceleration transverse to the one or more of the light beams.

[0075] It will be appreciated by persons skilled in the art that numerous variations and / or modifications may be made to the above-described embodiments, without departing from the broad general scope of the present disclosure. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.

Claims

CLAIMS:

1. An inertial sensor for measuring an inertial quantity along a sensing axis, the sensor comprising: an atom interferometer comprising a pulse generator to generate one or more pulsed light beams, defined by a respective pulse duration to place atoms into a superposition, and re-combine the atoms to measure an interference of the atoms and calculate the inertial quantity based on the measured interference; an auxiliary acceleration sensor configured to measure acceleration transverse to one or more of the pulsed light beams; and a control system configured to: calculate a transversal offset of the atoms relative to the pulsed light beams caused by the measured acceleration transverse to the one or more of the light beams, and increase the pulse duration to compensate for a reduced beam intensity applied to the atoms as a result of the transversal offset.

2. The sensor of claim 1, wherein the atom interferometer comprises: an atomic source to provide the atoms moving relative to the sensing axis; the pulsed light beams being configured to: place the atoms into a superposition of a first state and a second state, where the first state relates to a first path and the second state relates to a second path, and re-combine the first state and the second state to create interference between the first state and the second state; a measurement system to measure a physical effect indicative of the interference; and a processor to calculate the inertial quantity based on the measured physical effect.

3. The sensor of claim 1 or 2, the pulsed light beams are further configured to apply a mirror operation to the atoms between the placing and the re- combination.

4. The sensor of any one of the preceding claims, wherein the inertial quantity is a linear acceleration or a rotation.

5. The sensor of any one of the preceding claims, wherein measuring the physical effect comprises measuring a population of atoms in different superposition states.

6. The sensor of any one of the preceding claims, wherein increasing the pulse duration comprises applying an upper limit to the pulse duration.

7. The sensor of claim 6, wherein the atoms have a velocity distribution and the upper limit relates to a pulse duration at which the pulse still interacts with a threshold quantity of the atoms.

8. The sensor of claim 6 or 7, wherein the upper limit is set by a peak two-photon Rabi frequency of the sensor and the longitudinal velocity width of the atoms.

9. The sensor of any one of the preceding claims, wherein increasing the pulse duration comprises: calculating a degradation as one or more pulse adjustment factors that relate an ideal interaction of the atoms with the pulsed light beams to a predicted interaction of the atoms with the pulsed light beams under the transversal offset; and increasing the pulse duration by the one or more adjustment factors.

10. The sensor of claim 9, wherein the ideal interaction and the predicted interaction are represented by corresponding two-photon Rabi frequencies.

11. The sensor of claim 9 or 10, wherein the predicted interaction is predicted taking into account a spatial shape of the pulsed light beams.

12. The sensor of any one of the claims, wherein the pulse duration is increased differently for each of the pulsed light beams by calculating the degradation as a function of time.

13. The sensor of any one of the preceding claims, wherein the degradation is a function of an interrogation time.

14. The sensor of any one of the preceding claims, wherein the auxiliary acceleration sensor is configured to measure the acceleration transverse to one or more of the pulsed light beans multiple times during the travel of the atoms through the atom interferometer; and the control system is configured to compensate for degradation at each of the pulsed light beam using the corresponding measured acceleration.

15. The sensor of any one of the preceding claims, wherein the pulsed beams place the atoms into the superposition and re-combine the atoms by stimulating a Bragg transition or a Raman transition.

16. The sensor of any one of the preceding claims, wherein the superposition is a superposition of two states with different momenta that are separated by an integer number of two-photon recoils.

17. The sensor of any one of the preceding claims, wherein the control system is further configured to adjusting a pulse property other than duration to perform arbitrary further manipulations of a motional state of the atoms.

18. The sensor of claim 17, wherein the control system is further configured to adjust a pulse timing to compensate for the reduced beam intensity applied to the atoms as a result of the transversal offset.

19. The sensor of any one of the claims, wherein the control system is further configured to adjust an intensity of the one or more light beams to compensate for the reduced beam intensity applied to the atoms as a result of the transversal offset.

20. A method for measuring an inertial quantity along a sensing axis, the method comprising: generating a cloud of atoms; generating one or more pulsed light beams, defined by a respective pulse duration to place the atoms into a superposition, and re-combine the atoms; measuring acceleration transverse to one or more of the pulsed light beams using an auxiliary acceleration sensor; calculating a transversal offset of the atoms relative to the pulsed light beams caused by the measured acceleration transverse to the one or more of the light beams, and increasing the pulse duration of the pulsed light beams to compensate for a reduced beam intensity applied to the atoms as a result of the transversal offset. measuring an interference of the atoms; and calculating the inertial quantity based on the measured interference.

21. A computer-implemented method for controlling an atom interferometer, the method comprising: receiving, from an auxiliary acceleration sensor, acceleration sensor data indicative of an acceleration transverse to one or more of multiple pulsed light beams, defined by a respective pulse duration to place atoms into a superposition, and re- combine the atoms to measure an interference of the atoms; calculating a transversal offset of the atoms relative to the pulsed light beams caused by the measured acceleration transverse to the one or more of the light beams, increasing the pulse duration of the pulsed light beams to compensate for a reduced beam intensity applied to the atoms as a result of the transversal offset.

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