The accelerometer plays a crucial role in inertial navigation. The performance of conventional accelerometers such as lasers is usually limited by the sensing elements and shot noise limitation (SNL). Here, we propose an advanced development of an accelerometer based on atom–light quantum correlation, which is composed of a cold atomic ensemble, light beams, and an atomic vapor cell. The cold atomic ensemble, prepared in a magneto-optical trap and free-falling in a vacuum chamber, interacts with light beams to generate atom–light quantum correlation. The atomic vapor cell is used as both a memory element storing the correlated photons emitted from cold atoms and a bandwidth controller through the control of free evolution time. Instead of using a conventional sensing element, the proposed accelerometer employs interference between quantum-correlated atoms and light to measure acceleration. Sensitivity below SNL can be achieved due to atom–light quantum correlation, even in the presence of optical loss and atomic decoherence. Sensitivity can be achieved at the level, based on evaluation via practical experimental conditions. The present design has a number of significant advantages over conventional accelerometers such as SNL-broken sensitivity, broad bandwidth from a few hundred Hz to near MHz, and avoidance of the technical restrictions of conventional sensing elements.
1. INTRODUCTION
Accelerometers have extensively been applied as weak force probes, especially in situations that involve inertial navigation, land-based resource exploration, and seismic monitoring [1–4]. There are many types of accelerometers, including capacitive [5,6], piezoelectric [7], tunnel-current [8], thermal [9], and optical [10–16]. Almost all utilize mass-spring-damper systems as displacement sensors [17]. The performance of such accelerometers highly relies on fabrication of the displacement sensors, which directly results in technical limitations to the bandwidth, quality factor , and noise level [18]. Among numerous accelerometers, optical ones have attracted much attention due to their high sensitivity. However, the sensitivity of such accelerometers is fundamentally subject to shot noise limitation (SNL) [19,20]. Developing methods and technologies to break the SNL and remove the technical limitations of conventional displacement sensors is desired for the innovation of accelerometers.
In this paper, we present a memory-assisted quantum accelerometer (MQA), which consists of atomic ensembles, light beams, and optical elements. The MQA has three advantages over conventional accelerometers. First, instead of a mass-spring-damper system, a cold atomic ensemble [21] acts as the displacement sensor, which can avoid the technical restrictions of a mass-spring-damper system. Second, the SNL-broken sensitivity in measurement of acceleration can be achieved by the quantum interference of correlated atoms and light [22–24]. The third merit is that the accelerometer allows a tunable bandwidth, which is determined by the controllable memory time of the correlated photons stored in a memory element [25–27]. We calculate and analyze the sensitivity and bandwidth of the MQA. An inverse relation exists between sensitivity and bandwidth. Sensitivity can reach below the SNL in the range of bandwidth from a few hundred Hz to near MHz due to atom–light quantum correlation, even in the presence of the optical losses and atomic decoherence. An optimal -level sensitivity without loss is anticipated at frequencies for a feasible average atomic number and an available memory time .
2. RESULTS
A. Principle of MQA
A schematic diagram of the accelerometer is shown in Fig. 1. The cores are composed of two atomic systems: cold ensemble and atomic vapor cell . is prepared in a magneto-optical trap (MOT), free-falling in the vacuum chamber to generate atom ()–light () quantum correlation and realize atom–light interference via two Raman scattering processes. Atomic vapor cell , containing thermal atoms and buffer gas, is a memory element to store the correlated photonic signal from , which acts as a bandwidth modulator. In principle, the memory element can be any quantum memorizer for photons with long coherence times, such as rare-earth-doped crystals [28,29]. All optical devices including the vacuum chamber and memory element are fixed onto a mobile platform. When MOT is turned on, and move together with the platform. The distance between two atomic systems is fixed as at any velocity or acceleration of the platform. When MOT is turned off, is free-falling in vacuum, acting as the static frame reference, and remains moving with the platform at acceleration . changes with acceleration, where is the displacement due to acceleration . , where is the free-evolution time duration after the first Raman scattering process and before the second one, that is, the atom–light “wave-splitting” and “wave-recombining” processes, respectively. is independent of acceleration and the velocity of the platform. can be achieved via atom–light quantum interference. Acceleration can be achieved by the variation of with time . The acceleration sensitivity is no longer limited by the performance of the mass-spring-damper system. This is one of the advantages of our scheme.
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Figure 1.Schematic diagram of the MQA. The cold atomic ensemble is free-falling in the vacuum chamber. Atomic vapor , the vacuum chamber, and all optical elements are fixed on and move with the platform at acceleration . The distance between and is changed from to . is the acceleration-dependent displacement achieved via atom–light quantum interference, which is realized in three steps. Step 1: and are generated by the first stimulated Raman scattering (SRS) in via input seed and Raman pump . The atomic spin wave is , where superscript (, 2) indicates different atomic ensemble or , and subscript (, 2, 3) represents the evolution state at different times of atomic ensemble . Step 2: is stored in as is driven by the strong write pulse . After memory time , , evolved from due to atomic decay, is retrieved back to by the read pulse . Step 3: , evolved from during the memory time, and interfere by the second SRS via Raman pump .
The accelerometer is operated in three steps: generation of atom–light quantum correlation via the first Raman scattering in , atomic memory in , and acceleration acquisition via atom–light interference. Below, we describe the calculation and analysis in detail.
B. Generation of Quantum Correlation
Atom–light quantum correlation is generated via stimulated Raman scattering (SRS) in , which is effectively an atom–light wave-splitting process. A Stokes seed and a strong Raman beam interact with the atoms in to generate Stokes light and atomic spin wave [30]. The input–output relation of SRS can be written as , . describes the initial spin wave, which starts from the ground state of the atomic ensemble. and are Raman gains that satisfy , and is the phase of beam .
After SRS, the generated Stokes signal , transmitting out of and entering , quantum-mechanically correlates with the induced atomic spin wave that remains in . The intensity fluctuations of both and are amplified to the level above SNL as a result of Raman amplification. But the relative intensity fluctuation is squeezed below SNL by times, due to the quantum correlation between and (see Appendix A).
C. Atomic Memory
Stokes signal propagates into atomic vapor cell , and then is stored as atomic spin wave in the writing process driven by the write beam . The evolution of the spin wave obeys the Heisenberg propagation equation where is the Rabi frequency of , is single-photon detuning, is two-photon detuning, is the wave vector difference of and , is the Raman coupling coefficient, is the center of mass velocity of the atoms, and is the Doppler frequency shift. The solution to Eq. (1) is (see Appendix B) where is the writing efficiency determined by the coupling coefficient ; , and , with the Stokes wavelength. is the phase of the write field ; is the writing time. is the initial spin wave in , which is in the vacuum state. The term corresponds to the Stark effect. , , and are acceleration-independent parameters, which can be considered as fixed values.
After memory time , , evolving from with decay rate due to atomic collisions, can be read out as Stokes with efficiency by the read beam . Assuming a reading time and defining , Stokes can be simplified as (see Appendix C) where is the effective vacuum field. The phase offset can be safely set to zero since it is independent of the measured acceleration (see Appendix C). The velocity-dependent phase shift is induced by the Doppler effect due to the center of mass motion of atoms. The acceleration-dependent phase shift , where is the projection of along the acceleration. , where and are the flying times of the Stokes light between cold ensemble and vapor cell forth and back, respectively. Normally, and , i.e., . The phase shift of the Stokes field can be measured through atom–light interference between the readout field returning to and atomic spin wave remaining in .
D. Atom–Light Interference
When returns to , the atomic spin wave has experienced free evolution for the duration of memory time . As a result, the spin wave has the form , with Doppler dephasing being negligible in the cold ensemble (see Appendix D), where is the quantum Langevin operator, reflecting the collision-induced fluctuation and satisfying [24]. The decay rate represents the decoherence effect due to atomic collisions and flying off the laser beam. For a general cold atomic ensemble, the root mean square velocity is ∼ several cm/s, and the diffusion of cold atoms is after a memory time of 10 ms. The whole atomic ensemble will drop 0.5 mm in gravity direction after 10 ms. The mismatch of the spot expansion and dropping of the cold atomic ensemble plays an opposite role in quantum enhancement by destroying atom–light quantum correlation. However, these two effects can be solved using laser beam expansion and light path adjustment in the experiment.
and interact with each other in , which is driven by with , and generate Stokes with atomic spin wave by a second Raman scattering. Setting and , we have the final output (see Appendix D), where is the normalized noise operator, , , , , and .
Here, we emphasize that atomic vapor cell moves with the platform. To achieve acceleration, this requires that the vapor atoms and cell wall move as a whole, which implies that a robust thermal equilibrium between the atoms and the cell wall is essential during memory time. This is realized with the assistance of buffer gas in the vapor cell. In a room-temperature cell with buffer gas at several-Torr pressure, the mean free path of atoms is , with . This allows the vapor atoms to remain in thermal equilibrium with the cell wall under acceleration (see Appendix D). The ground-state coherence can be preserved for up to collisions between the buffer gas and atoms [31]. The buffer gas can separate atoms and decrease the collision between them. Therefore, the value of the decay rate is small [32]. Doppler dephasing () can be avoided by adopting near-degenerate Zeeman two-photon transitions, as shown in Appendix E. Now the Stokes field , containing only phase , can be measured through homodyne detection (HD).
E. Acquisition of Acceleration
All optical devices including the vacuum chamber and memory element are fixed onto a mobile platform. When MOT is turned on, cold ensemble and atomic vapor cell move together with the platform. The distance between two atomic systems is fixed as at any velocity or acceleration of platform . When MOT is turned off, the movement of the cold atomic ensemble is independent of the platform. The cold atomic ensemble moves in uniform motion with constant velocity . The atomic vapor still moves with the platform under acceleration. Its velocity changes with acceleration, and the distance between two atomic systems also changes from to , where is the displacement due to acceleration, with the initial relative velocity of and , and is independent of the acceleration and velocity of the platform. causes a phase shift. Here, is zero because the velocities of two atomic systems are the same before MOT is turned off. Acceleration can be measured in the direction of the seed and pump fields passing the cold ensemble, and it can be expressed in terms of the phase as follows:
Using the HD of the quadrature of Stokes field , acceleration is measured, and sensitivity is given by (see Appendix E) where is the quantum enhancement factor. Sensitivity breaks the SNL when . In general, where are the values of at the dark fringe. Defining for the input mean photon number, we work out the SNL for acceleration measurement: where and are the particle numbers of Stokes signal and atomic spin wave , respectively. Acceleration sensitivity depends on the phase sensitivity of atom–light interference, whose SNL is determined by the total phase-sensitive particle number of two interference beams, signal and atomic spin wave . Here, quantum correlation between and is key to breaking the SNL in acceleration sensitivity.
F. Accelerometer Sensitivity
To obtain the best sensitivity in the acceleration measurement, it is essential to decrease itself and increase quantum enhancement factor . In general, can be reduced by increasing the input particle number and prolonging memory time . However, this is usually constrained in realistic experiments. In this sense, quantum enhancement provides just an alternative way to further improve sensitivity through SNL breaking.
Factor is complicated, and we numerically analyze the behavior of in Fig. 2(a) as a function of and () for a different Raman gain under a given . The larger , , and , the larger . The red solid curves with in Fig. 2(a) set a critical boundary of the SNL, which reflects a balance of competition between losses and quantum correlation. The region within the curves marks , representing quantum enhancement.
Figure 2.(a) Quantum enhancement factor versus and when and 8, respectively. (b) Sensitivity versus and when and , respectively. is the ratio of two beams’ losses. The red curves mark for SNL. Sensitivities within the red curves can beat the SNL. Parameter settings: , , , and .
On the other hand, in terms of Eq. (7), the gains , , and determine the SNL for acceleration sensitivity. To set an SNL as low as possible in the measurement, it is essential to achieve gains as high as possible. Hence, we assume for numeric analysis. Sensitivity is shown as a function of and under given and in Fig. 2(b). The quantum enhancing region associated with is marked as the same within the red solid curves. Obviously, high gain and low losses can greatly enhance sensitivity due to the sufficient exploitation of quantum correlation. In addition, quantum enhancement of sensitivity well exhibits a loss tolerance, and highly depends on the balance of losses defined by the ratio . The best sensitivity appears near for high gains, where the atomic and optical losses are well balanced. In particular, sensitivity can keep beating the SNL through quantum enhancement until losses approach the limit .
G. Measurement Bandwidth
Sensitivity is the result determined by one single-shot measurement approximately completed in memory time , which gives the upper limit of the MQA’s frequency . In realistic experiments, if the setup of the system takes time , and stable operation for -times repeating measurements takes time , then the overall sensitivity of the accelerometer is given by , where .
There is a trade-off between sensitivity and bandwidth, as shown in Fig. 3. In general, sensitivity has a 3/2 power-law dependence of bandwidth, indicating that sensitivity degrades in the high-frequency region. In Fig. 3, we see that when the losses of the system remain low enough, the accelerometer exhibits well the quantum advantage to achieve an SNL below sensitivity with a broad range of frequencies. Sensitivity rises above the SNL at low frequencies with losses. As a comparison, some available data of sensitivities for previously reported accelerometers are labeled in Fig. 3 as well. Overall, the MQA has great potential in highly-sensitive measurements of acceleration.
Figure 3.Sensitivity as a function of bandwidth under the conditions , , , and . The data of other reported accelerometers are given as comparison. Dark green diamond: microchip optomechanical accelerometer [14]. Orange pentagram: micromechanical capacitive accelerometer [18]. Purple hexagram: MEMS accelerometer [6]. Gray circle: optical accelerometer [15].
With the available experimental conditions, e.g., per pulse, Raman gain , , MOT cycle [33,34], memory time [35,36], we can achieve single-shot sensitivity 23 ng at frequency 100 Hz. Finally, the sensitivity is .
For the traditional interferometric accelerometer, the only method to improve sensitivity at a high bandwidth is to increase the quality factor of the damper, which is technically hard to implement. In our scheme, sensitivity at a high bandwidth can be improved by increasing the Raman gain or the number of initial photons and trapping cold atoms, making it much easier to operate.
3. DISCUSSION AND CONCLUSION
We present an innovative principle for a quantum enhanced accelerometer. There are several significant advantages over other interferometer-based accelerometers: broad bandwidth, SNL below sensitivity, and mechanics-free sensing flexibility compared to mass-spring-damper systems. The dynamic range of acceleration depends on that of phase measurement, which is from the phase sensitivity to . With of 795 nm, the dynamic range of acceleration of a single shot is from 23 ng to 1.59 mg with , and from to with . It can be seen that the dynamic range is with a fixed . Furthermore, the measurable range of acceleration is from 23 ng to , , only by adjusting to suitable values. In future applications of the MQA, a long atomic coherence time of and high memory efficiency are crucial to achieve high sensitivity. Large optical depths for are required to ensure enough atomic numbers to achieve a large input particle number and high gains, so as to lower the level of and raise the quantum enhancement factor .
We emphasize that this work building the MQA is based on quantum correlation. The physics behind this is universal. Hence, the principle presented in this paper is not limited only to the atom–light coupling system, but can be extended to other systems that can generate and preserve quantum correlation, such as rare-earth-doped crystals [28,29]. Specifically, one crystal can be trapped and released to generate light–crystal correlation. The other crystal, fixed on platform, acts as the quantum memorizer. Acceleration sensitivity also depends on the phase-sensitive particle number and memory time.
APPENDIX A: GENERATION OF QUANTUM CORRELATION
Step 1 and step 3 in the accelerometer scheme are SRS processes. The atomic levels and optical frequencies are shown in Fig. 1. In the SRS process, a pair of lower-level meta-stable states is coupled to the Raman pump (or ) and the Stokes field (or ) via an upper excited level. After adiabatically eliminating the upper excited level, this is a three-wave mixing process involving the Raman pump field, Stokes field, and a collective atomic pseudo-spin field . The coupling Hamiltonian is given by [37,38]where is the coupling coefficient. The time evolution for and is given by In the case of Raman amplification, , being the Raman gain, is larger than one, and with the phase of Raman pump beam. In the following, we use to describe the gain of the th SRS process.
After the first SRS process in a cold ensemble (see step 1 in Fig. 1), the Stokes light and atomic spin wave are generated and written as and they are temporally correlated. This correlation can be demonstrated by calculating the relative intensity variance. After the first SRS has occurred, their intensities are The number difference operator describes the relative intensity fluctuations. After the first SRS process, the relative intensity fluctuation is given by where we consider an SNL input light . The SNL is then the shot noise that would be expected for a differential measurement made with equivalent total power. For the output fields, the SNL is .
Provided the input beams were originally SNL, this SRS process enables sub-shot noise measurements to be made. This is quantified by the “degree of squeezing” (DOS), which is the ratio of the variance of squeezed beams to the variance at the SNL, namely, The variances in the number operator of one beam alone under SRS are and Using Eq. (A3), the DOS is . This corresponds to a linear increase in the noise on the two beams as gains are increased.
After SRS, the intensity fluctuation for or alone is larger than SNL with . But relative intensity fluctuation is smaller than SNL by times. That is, or is quantum correlated. The second SRS process (see step 3 in Fig. 1) for correlation generation between and is similar to the above result.
APPENDIX B: MEMORY PROCESS
The Hamiltonian in the writing process can be written as follows: where is the coupling coefficient. Since the pulse time is relatively short, the writing process can be considered approximately lossless. The evolution equations for and are where is the coefficient associated with the write beam. Considering the detuning and the Doppler effect due to atomic center of mass motion with velocity , the evolution of , coupling to field , can be described by the matrix equation where , is the Rabi frequency of the write field , is single-photon detuning, is two-photon detuning, and is the wave vector difference of and . By substituting the following initial conditions into coupling Eq. (B3): we obtain the following solutions: where is the write beam phase, , and is the Stokes signal leaked out from atomic vapor cell during imperfect storage. It is easy to find that We can set , , and represents the write efficiency. is the phase induced by the writing process, which can be considered as a fixed value. Then, During the storage period , atomic vapor is subject to acceleration , which causes a change of atomic center of mass velocity , and to the decoherence with decay rate due to atomic collisions. These result in the evolution of the spin wave into , where , and is the Langevin operator describing the noise, and satisfies [24].
APPENDIX C: READOUT PROCESS
After storage, the spin wave is read out by a read beam with Rabi frequency , and the readout field propagating into cold ensemble has the form where is the distance between cold ensemble and atomic vapor cell before MOT is turned off, is the acceleration-induced distance change, , , is the read beam phase, is the reading time, and is the wave vector difference of read field and . We set , , with the read efficiency. is the phase induced by the readout process, which can be considered as a fixed value.
Finally, can be expressed as the following via the input light field : where the total noise operator , and are vacuum inputs of the spin wave and light field, respectively, with operator satisfying the bosonic commutation relation . Here, the spin wave can approximately be treated as a bosonic field because the number of atomic spin excitations is much smaller than the total atomic number. The commutation relation of operator is and for convenience, operator can be normalized to , with being the effective vacuum field and satisfying .
In addition, without loss of generality, assuming that , , , and considering at the readout time, the phase term in Eq. (C2) is where is the projection of along , and , , . is a determined overall phase offset accumulated by the Stark effect, two-photon detuning, and propagation distance . The velocity-dependent phase shift induces decoherence due to the dephasing process from atomic thermal motion. is the acceleration-dependent phase shift. The reading and writing beams originate from the same laser, and their phases satisfy . Finally, defining Stokes can be simplified as can be set to zero by adjusting the initial setup of the system since it is independent of the measured acceleration . The acceleration rate can be measured by reading out the phase shift through atom–light interference between the readout field returning to cold ensemble and atomic spin wave remaining in .
APPENDIX D: ATOM–LIGHT INTERFERENCE
First, we analyze the time evolution of the spin wave remaining in cold ensemble during the memory process. Spin wave evolves to with decay rate , which is given as where is the wave vector difference of pump field and , and is the quantum statistical Langevin operator, which reflects the collision-induced fluctuation and satisfies . The decay rate represents the decoherence effect due to atomic collisions and flying off a laser beam. For a general cold atomic ensemble, the root mean square velocity is ∼ several cm/s, and the diffusion of cold atoms is after a memory time of 10 ms. The whole atomic ensemble will drop 0.5 mm in gravity direction after 10 ms. The spot expansion and dropping of the cold atomic ensemble can result in a mismatch between returned light and atomic spin wave. However, these two effects can be solved using laser beam expansion and light path adjustment in the experiment. The dephasing induced by atomic motion is very slow. Thus, on the time scale, , dephasing can be safely ignored. Finally, considering the two-photon resonance in this work, .
Light fields and interact with each other in cold ensemble , and generate Stokes and atomic spin wave by a second Raman scattering . The input–output relation of the second SRS can be written as Raman beams and originate from the same laser, and their phases satisfy . The combination of and , output field from ensemble , is given by where the normalized noise operator , with , and The acceleration-dependent phase shift , with , where and are the flying times of Stokes light between and forth and back, respectively. Normally, and , i.e., . The phase shift can be measured through the readout field , which is affected by optical loss and atomic decoherence including , induced by collision, and induced by atomic thermal motion.
APPENDIX E: ACCELERATION SENSITIVITY USING HOMODYNE DETECTION
All optical devices including the vacuum chamber and memory element are fixed onto a mobile platform. When MOT is turned on, cold ensemble and atomic vapor cell move together with the platform. The distance between two atomic systems is fixed as at any velocity or acceleration of the platform. When MOT is turned off, is free-falling in vacuum, acting as a static frame reference, and remains moving with the platform at acceleration . changes with the acceleration, where is the displacement due to acceleration . , is initial relative velocity of and , and is the free-evolution time duration after the first Raman scattering process and before the second one. is zero because the velocities of two atomic systems are the same before MOT is turned off. is independent of acceleration and the velocity of the platform. causes the phase shift.
The phase shift induced by acceleration can be measured via atom–light quantum interference. It is noted that the scheme can measure acceleration only in the direction of the seed and pump fields passing the cold ensemble, and its sensitivity is [20,24] where is the measured operator related to the phase shift.
We use HD to measure the interference output field , where the measured operator is the quadrature operator , where is the phase of the local oscillator.
As is coherent light (), according to of Eq. (D3), the slope is given by The dephasing term comes from the velocity-dependent phase shift , where with the root mean square velocity of atoms [39,40]. When , the slope reaches its maximum.
The variance is where coefficients reach their minimum at , giving the minimal variance .
The decay terms , and the dephasing term degrade the sensitivity of acceleration measurement. The decoherence time depends on the and root mean square velocity of atomic thermal motion. To reduce the effect of Doppler dephasing, one can employ laser-cooled atomic gas with low or use near-degenerate sublevels to achieve very small . However, in our design, the second cell with atoms in it is assumed as a whole to be attached to the platform, sensing the acceleration. For this purpose, the atomic thermal motion in the cell must be rapid enough to remain in thermal equilibrium with the cell walls under acceleration. In this sense, a vapor cell with buffer gas is essential, ruling out the usage of laser-cooled atomic gas. The collision between buffer gas and atoms almost has no effect on hyperfine coherence in principle. In a reported paper, ground-state coherence can be preserved for up to collisions with buffer gas [31]. The atoms of buffer gas can separate the original atoms and decrease the collision between them. Therefore, the value of the decay rate in Fig. 3 is small [32]. For a negligible dephasing effect, e.g., , we work out . For a typical room-temperature vapor cell with and giving , . This can experimentally be realized by choosing Zeeman sublevels with frequency differences near MHz for two-photon transitions [36].
Based on the arguments above, the acceleration sensitivity is where is the SNL for the accelerometer, and is the quantum enhancement factor: where () are the values of when the MQA is operated at the dark fringe with a small phase shift.
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