• Photonics Research
  • Vol. 10, Issue 4, 1022 (2022)
Zhifei Yu1, Bo Fang1, Liqing Chen1、2、*, Keye Zhang1, Chun-Hua Yuan1、2、5, and Weiping Zhang2、3、4、6
Author Affiliations
  • 1State Key Laboratory of Precision Spectroscopy, School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
  • 2Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
  • 3School of Physics and Astronomy, Shanghai Jiao Tong University, and Tsung-Dao Lee Institute, Shanghai 200240, China
  • 4Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
  • 5e-mail: chyuan@phy.ecnu.edu.cn
  • 6e-mail: wpzhang@phy.ecnu.edu.cn
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    DOI: 10.1364/PRJ.453940 Cite this Article Set citation alerts
    Zhifei Yu, Bo Fang, Liqing Chen, Keye Zhang, Chun-Hua Yuan, Weiping Zhang, "Memory-assisted quantum accelerometer with multi-bandwidth," Photonics Res. 10, 1022 (2022) Copy Citation Text show less

    Abstract

    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 ng/Hz 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 [14]. There are many types of accelerometers, including capacitive [5,6], piezoelectric [7], tunnel-current [8], thermal [9], and optical [1016]. 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 Qf, 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 [2224]. The third merit is that the accelerometer allows a tunable bandwidth, which is determined by the controllable memory time tM of the correlated photons stored in a memory element [2527]. 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 ng/Hz-level sensitivity without loss is anticipated at frequencies 100  Hz for a feasible average atomic number and an available memory time tM=10  ms.

    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 A1 and atomic vapor cell A2. A1 is prepared in a magneto-optical trap (MOT), free-falling in the vacuum chamber to generate atom (S^a1(1))–light (a^1) quantum correlation and realize atom–light interference via two Raman scattering processes. Atomic vapor cell A2, containing thermal atoms and buffer gas, is a memory element to store the correlated photonic signal a^1 from A1, 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, A1 and A2 move together with the platform. The distance L between two atomic systems is fixed as L0 at any velocity or acceleration of the platform. When MOT is turned off, A1 is free-falling in vacuum, acting as the static frame reference, and A2 remains moving with the platform at acceleration a. L=L0+ΔL changes with acceleration, where ΔL is the displacement due to acceleration a. ΔL=12aT2, where T 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. T is independent of acceleration a and the velocity of the platform. ΔL can be achieved via atom–light quantum interference. Acceleration a can be achieved by the variation of ΔL with time T. 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.

    Schematic diagram of the MQA. The cold atomic ensemble A1 is free-falling in the vacuum chamber. Atomic vapor A2, the vacuum chamber, and all optical elements are fixed on and move with the platform at acceleration a. The distance between A1 and A2 is changed from L0 to L0+ΔL. ΔL is the acceleration-dependent displacement achieved via atom–light quantum interference, which is realized in three steps. Step 1: a^1 and S^a1(1) are generated by the first stimulated Raman scattering (SRS) in A1 via input seed a^0 and Raman pump P1. The atomic spin wave is S^aj(i), where superscript i (i=1, 2) indicates different atomic ensemble A1 or A2, and subscript j (j=1, 2, 3) represents the evolution state at different times of atomic ensemble Ai. Step 2: a^1 is stored in A2 as S^a1(2) is driven by the strong write pulse W. After memory time tM, S^a2(2), evolved from S^a1(2) due to atomic decay, is retrieved back to a^2 by the read pulse R. Step 3: S^a2(1), evolved from S^a1(1) during the memory time, and a^2 interfere by the second SRS via Raman pump P2.

    Figure 1.Schematic diagram of the MQA. The cold atomic ensemble A1 is free-falling in the vacuum chamber. Atomic vapor A2, the vacuum chamber, and all optical elements are fixed on and move with the platform at acceleration a. The distance between A1 and A2 is changed from L0 to L0+ΔL. ΔL is the acceleration-dependent displacement achieved via atom–light quantum interference, which is realized in three steps. Step 1: a^1 and S^a1(1) are generated by the first stimulated Raman scattering (SRS) in A1 via input seed a^0 and Raman pump P1. The atomic spin wave is S^aj(i), where superscript i (i=1, 2) indicates different atomic ensemble A1 or A2, and subscript j (j=1, 2, 3) represents the evolution state at different times of atomic ensemble Ai. Step 2: a^1 is stored in A2 as S^a1(2) is driven by the strong write pulse W. After memory time tM, S^a2(2), evolved from S^a1(2) due to atomic decay, is retrieved back to a^2 by the read pulse R. Step 3: S^a2(1), evolved from S^a1(1) during the memory time, and a^2 interfere by the second SRS via Raman pump P2.

    The accelerometer is operated in three steps: generation of atom–light quantum correlation via the first Raman scattering in A1, atomic memory in A2, 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 A1, which is effectively an atom–light wave-splitting process. A Stokes seed a^0 and a strong Raman beam P1 interact with the atoms in A1 to generate Stokes light a^1 and atomic spin wave S^a1(1) [30]. The input–output relation of SRS can be written as a^1=G1a^0+g1S^a0(1)eiθP1, S^a1(1)=G1S^a0(1)+g1a^0eiθP1. S^a0(1) describes the initial spin wave, which starts from the ground state of the atomic ensemble. G1 and g1 are Raman gains that satisfy G12=g12+1, and θP1 is the phase of beam P1.

    After SRS, the generated Stokes signal a^1, transmitting out of A1 and entering A2, quantum-mechanically correlates with the induced atomic spin wave S^a1(1) that remains in A1. The intensity fluctuations of both a^1 and S^a1(1) are amplified to the level above SNL as a result of Raman amplification. But the relative intensity fluctuation S^a1(1)S^a1(1)a^1a^1 is squeezed below SNL by 2G121 times, due to the quantum correlation between a^1 and S^a1(1) (see Appendix A).

    C. Atomic Memory

    Stokes signal a^1 propagates into atomic vapor cell A2, and then is stored as atomic spin wave S^a1(2) in the writing process driven by the write beam W. The evolution of the spin wave obeys the Heisenberg propagation equation tS^a1(2)=i(|ΩW|2Δδ+ΔkW·v)S^a1(2)iχa^1,where ΩW is the Rabi frequency of W, Δ is single-photon detuning, δ is two-photon detuning, ΔkW=kWks is the wave vector difference of W and a^1, χ is the Raman coupling coefficient, v is the center of mass velocity of the atoms, and ΔkW·v is the Doppler frequency shift. The solution to Eq. (1) is (see Appendix B) S^a1(2)=(iηWeiθWa^1eiφ+eiθW01ηWS^a0(2))×ei(|ΩW|2Δδ+ΔkW·v)tW/2,where ηW is the writing efficiency determined by the coupling coefficient χ; φ=ksL0, and ks=2π/λs, with λs the Stokes wavelength. θW is the phase of the write field W; tW is the writing time. S^a0(2) is the initial spin wave in A2, which is in the vacuum state. The term |ΩW|2Δ corresponds to the Stark effect. |ΩW|2Δ, δ, and θW0 are acceleration-independent parameters, which can be considered as fixed values.

    After memory time tM, S^a2(2), evolving from S^a1(2) with decay rate Γ2 due to atomic collisions, can be read out as Stokes a^2 with efficiency ηR by the read beam R. Assuming a reading time tRtW and defining η=ηWηR, Stokes a^2 can be simplified as (see Appendix C) a^2=ηei(Δφa+Δφvφ0)eΓ2tMa^1+1ηe2Γ2tMV^,where V^ is the effective vacuum field. The phase offset φ0 can be safely set to zero since it is independent of the measured acceleration a (see Appendix C). The velocity-dependent phase shift Δφv=ΔkW·vtM is induced by the Doppler effect due to the center of mass motion of atoms. The acceleration-dependent phase shift ΔφaΔkatM(tMtW)/2ks(ΔL), where Δk is the projection of ΔkW along the acceleration. ΔL=12aT2=12a(t1+t2+tM)2, where t1 and t2 are the flying times of the Stokes light between cold ensemble A1 and vapor cell A2 forth and back, respectively. Normally, ks|Δk| and tMtW,Rt1,2, i.e., ΔφaksatM2/2. The phase shift of the Stokes field can be measured through atom–light interference between the readout field a^2 returning to A1 and atomic spin wave S^a1(1) remaining in A1.

    D. Atom–Light Interference

    When a2 returns to A1, the atomic spin wave S^a1(1) has experienced free evolution for the duration of memory time tM. As a result, the spin wave has the form S^a2(1)=S^a1(1)eΓ1tM+F^(1), with Doppler dephasing being negligible in the cold ensemble (see Appendix D), where F^(1) is the quantum Langevin operator, reflecting the collision-induced fluctuation and satisfying [F^(1),F^(1)]=1e2Γ1tM [24]. The decay rate Γ1 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 1.0mm 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.

    a^2 and S^a2(1) interact with each other in A1, which is driven by P2 with θP2=θP1, and generate Stokes a^out with atomic spin wave S^aout(1) by a second Raman scattering. Setting T1eΓ1tM and T2eΓ2tM, we have the final output a^out=ς1a^0+ς2S^a0(1)+ς3V^+ς4f^ (see Appendix D), where f^ is the normalized noise operator, f^F^(1)/1T1, ς1g1g2T1G1G2ηT2ei(Δφa+Δφv), ς2eiθP2[G1g2T1g1G2ηT2ei(Δφa+Δφv)], ς3G21ηT2, and ς4g2eiθP21T1.

    Here, we emphasize that atomic vapor cell A2 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 μm, with vrms300  m/s. 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 108 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 Γ2 is small [32]. Doppler dephasing (Δφv) can be avoided by adopting near-degenerate Zeeman two-photon transitions, as shown in Appendix E. Now the Stokes field a^out, containing only phase Δφa, 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 A1 and atomic vapor cell A2 move together with the platform. The distance L between two atomic systems is fixed as L0 at any velocity v0 or acceleration of platform a. 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 v0. 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 L0 to L0+ΔL, where ΔL=Δv0T+12aT2 is the displacement due to acceleration, with Δv0 the initial relative velocity of A1 and A2, and T is independent of the acceleration a and velocity of the platform. ΔL causes a phase shift. Here, Δv0 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 Δφa as follows: a=λsΔφaπtM2.

    Using the HD of the quadrature of Stokes field a^out, acceleration is measured, and sensitivity is given by (see Appendix E) Δa=1QeΔaSNL,where Qe is the quantum enhancement factor. Sensitivity breaks the SNL when Qe>1. In general, Qe=2G1G2G12+g12ηeΓ2tMn=14|ςn0|2,where ςn0 are the values of ςn(n=1,,4) at the dark fringe. Defining N0a^0a^0 for the input mean photon number, we work out the SNL for acceleration measurement: ΔaSNL=λsπ(G12N0+g12N0)tM2,where G12N0 and g12N0 are the particle numbers of Stokes signal a^1 and atomic spin wave S^a1(1), respectively. Acceleration sensitivity depends on the phase sensitivity of atom–light interference, whose SNL is determined by the total phase-sensitive particle number (G12+g12)N0 of two interference beams, signal a^1 and atomic spin wave S^a1(1). Here, quantum correlation between S^a1(1) and a^1 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 ΔaSNL itself and increase quantum enhancement factor Qe. In general, ΔaSNL can be reduced by increasing the input particle number N0 and prolonging memory time tM. 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 Qe is complicated, and we numerically analyze the behavior of Qe in Fig. 2(a) as a function of T1 and T2 (T2ηT2) for a different Raman gain G2 under a given G1. The larger T1, T2, and G2, the larger Qe. The red solid curves with Qe=1 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 Qe>1, representing quantum enhancement.

    (a) Quantum enhancement factor Qe versus T1 and T2′ when G2=2 and 8, respectively. (b) Sensitivity versus s and G2 when T1=0.5 and 1, respectively. s=T2′/T1 is the ratio of two beams’ losses. The red curves mark Qe=1 for SNL. Sensitivities within the red curves can beat the SNL. Parameter settings: N0=106, G1=8, λs=795 nm, and tM=1 ms.

    Figure 2.(a) Quantum enhancement factor Qe versus T1 and T2 when G2=2 and 8, respectively. (b) Sensitivity versus s and G2 when T1=0.5 and 1, respectively. s=T2/T1 is the ratio of two beams’ losses. The red curves mark Qe=1 for SNL. Sensitivities within the red curves can beat the SNL. Parameter settings: N0=106, G1=8, λs=795  nm, and tM=1  ms.

    On the other hand, in terms of Eq. (7), the gains G1, g1, and N0 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 G11 for numeric analysis. Sensitivity Δa is shown as a function of s=T2/T1 and G2 under given G1 and N0 in Fig. 2(b). The quantum enhancing region associated with Qe>1 is marked as the same within the red solid curves. Obviously, high gain G2 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 s. The best sensitivity appears near s=1 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 T1T20.5.

    G. Measurement Bandwidth

    Sensitivity Δa is the result determined by one single-shot measurement approximately completed in memory time tM, which gives the upper limit of the MQA’s frequency fmax=1/tM. In realistic experiments, if the setup of the system takes time ts, and stable operation for M-times repeating measurements takes time to=MtM, then the overall sensitivity of the accelerometer is given by ΔaoΔa/feff(fmax)3/2, where feff=fmaxto/(ts+to).

    There is a trade-off between sensitivity Δao and bandwidth, as shown in Fig. 3. In general, sensitivity Δao 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.

    Sensitivity as a function of bandwidth under the conditions G=8, N0=106, λs=795 nm, and η=1. 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].

    Figure 3.Sensitivity as a function of bandwidth under the conditions G=8, N0=106, λs=795  nm, and η=1. 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., N0=106 per pulse, Raman gain G1=8, G2=2, MOT cycle ts=30  ms [33,34], memory time tM10  ms [35,36], we can achieve single-shot sensitivity 23 ng at frequency 100 Hz. Finally, the sensitivity is 4.6  ng/Hz.

    For the traditional interferometric accelerometer, the only method to improve sensitivity at a high bandwidth is to increase the quality factor Qf 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 2π. With λs of 795 nm, the dynamic range of acceleration of a single shot is from 23 ng to 1.59 mg with tM=10  ms, and from 2.3g to 1.59×105g with tM=1.0μs. It can be seen that the dynamic range is 48  dB with a fixed tM. Furthermore, the measurable range of acceleration is from 23 ng to 1.59×105g, 128  dB, only by adjusting tM to suitable values. In future applications of the MQA, a long atomic coherence time of A1,2 and high memory efficiency η are crucial to achieve high sensitivity. Large optical depths for A1,2 are required to ensure enough atomic numbers to achieve a large input particle number N0 and high gains, so as to lower the level of ΔaSNL and raise the quantum enhancement factor Qe.

    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 P1 (or P2) and the Stokes field a^1 (or a^3) 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 S^a. The coupling Hamiltonian is given by [37,38]H=iξa^S^a+h.c.,where ξ is the coupling coefficient. The time evolution for a^ and S^a is given by a^=Ga^0+gS^a0eiθP,S^a=GS^a0+ga^0eiθP.In the case of Raman amplification, G, being the Raman gain, is larger than one, and gG21 with θP the phase of Raman pump beam. In the following, we use Gj(j=1,2) to describe the gain of the jth SRS process.

    After the first SRS process in a cold ensemble A1 (see step 1 in Fig. 1), the Stokes light a^1 and atomic spin wave S^a1(1) are generated and written as a^1=G1a^0+g1S^a0(1)eiθP1,S^a1(1)=G1S^a0(1)+g1a^0eiθP1,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 a^1a^1=G12a^0a^0+g12G12a^0a^0,S^a1(1)S^a1(1)=g12a^0a^0+g12(G121)a^0a^0.The number difference operator a^1a^1S^a1(1)S^a1(1) describes the relative intensity fluctuations. After the first SRS process, the relative intensity fluctuation is given by Var(a^1a^1S^a1(1)S^a1(1))=Var(a^0a^0)=a^0a^0,where we consider an SNL input light Var(a^0a^0)=a^0a^0. 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 a^1a^1+S^a1(1)S^a1(1)=(2G121)a^0a^0.

    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, DOS=a^0a^0(2G121)a^0a^0=12G121.The variances in the number operator of one beam alone under SRS are Var(a^1a^1)=G14Var(a^0a^0)+G12(G121)a^0a^0=G12(2G121)a^0a^0,and Var(S^a1(1)S^a1(1))=(G121)2Var(a^0a^0)+G12(G121)a^0a^0=(G121)(2G121)a^0a^0.Using Eq. (A3), the DOS is DOS=2G121. This corresponds to a linear increase in the noise on the two beams as gains are increased.

    After SRS, the intensity fluctuation for a^1 or S^a1(1) alone is larger than SNL with G1>1. But relative intensity fluctuation a^1a^1S^a1(1)S^a1(1) is smaller than SNL by 2G121 times. That is, a^1 or S^a1(1) is quantum correlated. The second SRS process (see step 3 in Fig. 1) for correlation generation between a^3 and S^a2(1) is similar to the above result.

    APPENDIX B: MEMORY PROCESS

    The Hamiltonian in the writing process can be written as follows: H=χa^S^a+h.c.,where χ is the coupling coefficient. Since the pulse time is relatively short, the writing process can be considered approximately lossless. The evolution equations for a^ and S^a are ta^=iχ*S^a,tS^a=iχa^,where χ is the coefficient associated with the write beam. Considering the detuning and the Doppler effect due to atomic center of mass motion with velocity v, the evolution of S^a, coupling to field a^, can be described by the matrix equation t(a^S^a)=i(0χ*χαW)(a^S^a),where αW=|ΩW|2/Δδ+ΔkW·v, ΩW is the Rabi frequency of the write field W, Δ is single-photon detuning, δ is two-photon detuning, and ΔkW=kWkS is the wave vector difference of W and a^1. By substituting the following initial conditions into coupling Eq. (B3): a^(t=0)=a^1eiφ,φ=ksL0,S^a(t=0)=S^a0(2),we obtain the following solutions: S^a1(2)=eiαWtW/2{[i2|χ|βWeiθWsin(βW2tW)]a^1eiφ+[iαWβWsin(βW2tW)+cos(βW2tW)]S^a0(2)},a^L=eiαWtW/2{[i2|χ|βWeiθWsin(βW2tW)]S^a0(2)+[iαWβWsin(βW2tW)+cos(βW2tW)]a^1eiφ},where θW=arg(AW) is the write beam phase, βW=αW2+4|χ|2, and a^L is the Stokes signal leaked out from atomic vapor cell A2 during imperfect storage. It is easy to find that |i2|χ|βWeiθWsin(βW2tW)|2+|iαWβWsin(βW2tW)+cos(βW2tW)|2=1.We can set ηW=2|χ|βWsin(βW2tW), 1ηWeiθW0=iαWβWsin(βW2tW)+cos(βW2tW), and ηW represents the write efficiency. θW0 is the phase induced by the writing process, which can be considered as a fixed value. Then, S^a1(2)=eiαWtW/2[iηWeiθWa^1eiφ+eiθW01ηWS^a0(2)],a^L=eiαWtW/2[iηWeiθWS^a0(2)+eiθW01ηWa^1eiφ].During the storage period tM, atomic vapor is subject to acceleration a, which causes a change of atomic center of mass velocity vv+at, and to the decoherence with decay rate Γ2 due to atomic collisions. These result in the evolution of the spin wave S^a1(2) into S^a2(2)=S^a1(2)eΓ2tMeiϕ(tM)+F^(2), where ϕ(tM)=0tMΔkW·vdt=ΔkW·vtM+ΔkW·atM2/2, and F^(2) is the Langevin operator describing the noise, and satisfies [F^(2),F^(2)]=1e2Γ2tM [24].

    APPENDIX C: READOUT PROCESS

    After storage, the spin wave is read out by a read beam with Rabi frequency ΩR, and the readout field propagating into cold ensemble A1 has the form a^2=eiαRtR/2eiks(L0+ΔL)(iηReiθRS^a2(2)+eiθR01ηRb^),where L0 is the distance between cold ensemble A1 and atomic vapor cell A2 before MOT is turned off, ΔL is the acceleration-induced distance change, αR=|ΩR|2Δδ+ΔkR·v, βR=αR2+4|χ|2, θR is the read beam phase, tR is the reading time, and ΔkR=kRkS is the wave vector difference of read field R and a^2. We set ηR=2|χ|βRsin(βR2tR), 1ηReiθR0=iαRβRsin(βR2tR)+cos(βR2tR), with ηR the read efficiency. θR0 is the phase induced by the readout process, which can be considered as a fixed value.

    Finally, a^2 can be expressed as the following via the input light field a^1: a^2=eiαRtR/2eiks(L0+ΔL)(iηReiθRS^a2(2)+eiθR01ηRb^)=eiαRtR/2eiks(L0+ΔL)[iηReiθR(S^a1(2)eΓ2tMeiϕ(tM)+F^(2))+eiθR01ηRb^]=eiαRtR/2eiks(L0+ΔL){iηReiθR[eiαWtW/2(iηWa^1eiθWeiksL0+eiθ01ηWS^a0(2))eΓ2tMeiϕ(tM)+F^(2)]+eiθR01ηRb^}=ei[(αWtW+αRtR)/2+ϕ(tM)ks(2L0+ΔL)]ei(θWθR)ηWηReΓ2tMa^1+D^,where the total noise operator D^ei(αWtW+αRtR)/2·eiks(L0+ΔL)[iηR(1ηW)ei(θW0θR)eΓ2tMeiϕ(tM)S^a0(2)iηR·eiθReiαWtW/2F^(2)+eiθR01ηReiαWtW/2b^], S^a0(2) and b^ are vacuum inputs of the spin wave and light field, respectively, with operator b^ satisfying the bosonic commutation relation [b^,b^]=1. Here, the spin wave S^a0(2) can approximately be treated as a bosonic field [S^a0(2),S^a0(2)]1 because the number of atomic spin excitations is much smaller than the total atomic number. The commutation relation of operator D^ is [D^,D^]=1ηWηRe2Γ2tM,and for convenience, operator D^ can be normalized to D^=1ηWηRe2Γ2tMV^, with V^ being the effective vacuum field and satisfying [V^,V^]=1.

    In addition, without loss of generality, assuming that ΩW=ΩR, tW=tR, ΔkR=ΔkW, and considering vv+atM at the readout time, the phase term in Eq. (C2) is (αWtW+αRtR)/2+ϕ(tM)ks(2L0+ΔL)=(αW+αR)tW/2+ϕ(tM)ks(2L0+ΔL)=(|ΩW|2Δδ)tW+ΔkatM(tMtW)/2+ΔkW·vtMks(2L0+ΔL)Δφa+Δφvφ0,where Δk is the projection of ΔkW along a, and φ0(|ΩW|2Δδ)tW+2ksL0, ΔφvΔkW·vtM, ΔφaΔkatM(tMtW)/2ks(ΔL). φ0 is a determined overall phase offset accumulated by the Stark effect, two-photon detuning, and propagation distance L0. The velocity-dependent phase shift Δφv=ΔkW·vtM induces decoherence due to the dephasing process from atomic thermal motion. Δφa is the acceleration-dependent phase shift. The reading and writing beams originate from the same laser, and their phases satisfy θWθR=0. Finally, defining η=ηWηR Stokes a^2 can be simplified as a^2=ηei(Δφa+Δφvφ0)eΓ2tMa^1+1ηe2Γ2tMV^.φ0 can be set to zero by adjusting the initial setup of the system since it is independent of the measured acceleration a. The acceleration rate can be measured by reading out the phase shift Δφa through atom–light interference between the readout field a^2 returning to cold ensemble A1 and atomic spin wave S^a1(1) remaining in A1.

    APPENDIX D: ATOM–LIGHT INTERFERENCE

    First, we analyze the time evolution of the spin wave S^a1(1) remaining in cold ensemble A1 during the memory process. Spin wave S^a1(1) evolves to S^a2(1) with decay rate Γ1, which is given as S^a2(1)=S^a1(1)eΓ1tMeiδtMeiΔkP1·vtM+F^(1),where ΔkP1=kP1kS is the wave vector difference of pump field P1 and a^1, and F^(1) is the quantum statistical Langevin operator, which reflects the collision-induced fluctuation and satisfies [F^(1),F^(1)]=1e2Γ1tM. The decay rate Γ1 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 1.0  mm 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, tMms, dephasing can be safely ignored. Finally, considering the two-photon resonance in this work, S^a2(1)S^a1(1)eΓ1tM+F^(1).

    Light fields a^2 and S^a2(1) interact with each other in cold ensemble A1, and generate Stokes a^out and atomic spin wave S^aout(1) by a second Raman scattering P2. The input–output relation of the second SRS can be written as a^out=G2a^2+g2S^a2(1)eiθP2.Raman beams P1 and P2 originate from the same laser, and their phases satisfy θP1θP2=0. The combination of a^2 and S^a2(1), output field a^out from ensemble A1, is given by a^out=ς1a^0+ς2S^a0(1)+ς3V^+ς4f^,where the normalized noise operator f^=F^(1)/1e2Γ1tM, with [f^,f^]=1, and ς1=g1g2eΓ1tMG1G2ηeΓ2tMei(Δφa+Δφv),ς2=eiθP2[G1g2eΓ1tMg1G2ηeΓ2tMei(Δφa+Δφv)],ς3=G21ηe2Γ2tM,ς4=g2eiθP21e2Γ1tM.The acceleration-dependent phase shift ΔφaΔkatM(tMtW)/2ks(ΔL), with ΔL=12a(t1+t2+tM)2, where t1 and t2 are the flying times of Stokes light between A1 and A2 forth and back, respectively. Normally, ks|Δk| and tMtW,Rt1,2, i.e., ΔφaksatM2/2. The phase shift Δφa can be measured through the readout field a^out, which is affected by optical loss η and atomic decoherence including e2Γ1tM, e2Γ2tM induced by collision, and eiΔφv 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 A1 and atomic vapor cell A2 move together with the platform. The distance L between two atomic systems is fixed as L0 at any velocity or acceleration of the platform. When MOT is turned off, A1 is free-falling in vacuum, acting as a static frame reference, and A2 remains moving with the platform at acceleration a. L=L0+ΔL changes with the acceleration, where ΔL is the displacement due to acceleration a. ΔL=Δv0T+12aT2, Δv0 is initial relative velocity of A1 and A2, and T is the free-evolution time duration after the first Raman scattering process and before the second one. Δv0 is zero because the velocities of two atomic systems are the same before MOT is turned off. T is independent of acceleration a and the velocity of the platform. ΔL 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] Δa=λsπtM2(ΔO^)21/2|O^/φ|,where O^ is the measured operator related to the phase shift.

    We use HD to measure the interference output field a^out, where the measured operator is the quadrature operator X=a^outeiθl+a^outeiθl, where θl is the phase of the local oscillator.

    As a^0 is coherent light |α (α=|α|eiθα), according to a^out of Eq. (D3), the slope is given by |X^(Δφa)|=2exp(tM2τ2)ηeΓ2tMG1G2N0sin(θl+θαΔφa).The dephasing term exp(tM2/τ2) comes from the velocity-dependent phase shift Δφv, where τ=2/(ΔkW)v¯rms with the root mean square velocity of atoms v¯rms=kBT/m [39,40]. When θl+θαΔφa=π/2, the slope |X^/(Δφa)| reaches its maximum.

    The variance is Δ2X^=|ς1|2+|ς2|2+|ς3|2+|ς4|2,where coefficients |ς1|=|eiΔφaexp(tM2τ2)eΓ2tMG1G2+eΓ1tMg1g2|,|ς2|=|G1g2eΓ1tMG2g1eiΔφaexp(tM2τ2)ηeΓ2tM|,|ς3|=G21ηe2Γ2tM,|ς4|=g21e2Γ1tM,reach their minimum at Δφa0, giving the minimal variance Δ2X^.

    The decay terms eΓ1tM, eΓ2tM and the dephasing term exp(tM2/τ2) degrade the sensitivity of acceleration measurement. The decoherence time τ=2/(ΔkWv¯rms) depends on the ΔkW and root mean square velocity v¯rms of atomic thermal motion. To reduce the effect of Doppler dephasing, one can employ laser-cooled atomic gas with low v¯rms or use near-degenerate sublevels to achieve very small ΔkW. However, in our design, the second cell A2 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 108 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 Γ2 in Fig. 3 is small [32]. For a negligible dephasing effect, e.g., tM/τ=tM(ΔkW)v¯rms/20.1, we work out ΔkW0.1×2/(v¯rmstM). For a typical room-temperature vapor cell with v¯rms300  m/s and giving tM10  ms, ΔkW2.3×2π  MHz/c. 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 Δa=λsπtM2Δ2X^|X^(Δφa)|=1QeΔaSNL,where ΔaSNL2λsG12+g12N0tM2 is the SNL for the accelerometer, and Qe is the quantum enhancement factor: Qe=2G1G2G12+g12ηeΓ2tMn=14|ςn0|2,where ςn0 (n=1,,4) are the values of ςn when the MQA is operated at the dark fringe with a small phase shift.

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    Zhifei Yu, Bo Fang, Liqing Chen, Keye Zhang, Chun-Hua Yuan, Weiping Zhang, "Memory-assisted quantum accelerometer with multi-bandwidth," Photonics Res. 10, 1022 (2022)
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