2. Enhancing light-dragging using EIT

EIT can be implemented by coupling two ground states, |1⟩ and |2⟩ to a mutual excited state,|3⟩, in a Λ-scheme (Fleischhauer et al., Reference Fleischhauer, Imamoglu and Marangos2005).

Here, Ω _p and Ω _c are the Rabi frequencies of the probe and control field, coupling the ground states |1⟩ and |2⟩ to a common excited state |3⟩, with frequencies ω _p and ω _c , respectively. The probe field has a detuning δ, which is used to scan the frequency of the probe field and observe the EIT phenomenon, Figure 1(a). The EIT condition is satisfied by having Ω _p ≪ Ω _c , where sweeping ω _p around the resonance reveals a narrow frequency window where the medium exhibits full transparency. The transparency window has a linewidth γ _EIT = γ ₁₂ + |Ω _C |²/γ ₁₃, where γ _12/13 is the decay rate between states |1⟩−|2⟩ and |1⟩−|3⟩. In this condition, the medium simultaneously exhibits high dispersion and low absorption, Figure 1(b), leading to a large dragging coefficient, that is, F _d > 1. This high dispersion also leads to a reduced group velocity of the probe field (Fleischhauer and Lukin, Reference Fleischhauer and Lukin2000):

(1) $${v_g} = {c \over {1 + {{{g^2}N} \over {{\Omega _c}^2}}}}$$

where g is the coupling strength and N is the number of atoms. Fine-tuning the ratio ${\Omega _{p} \over \Omega _{c}}$ (Kash et al., Reference Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully1999), as well as increasing the number of atoms (Strekalov et al., Reference Strekalov, Matsko, Yu and Maleki2004), that is, optical depth, can result in extremely slow group velocities in the probe field, and a slow group velocity means the possibility of compressing a spatially extended optical pulse within a few-centimeter-long vapor cell.

Figure 1. Electromagnetically induced transparency. a) The relevant Λ-scheme and b) dispersion (red) and absorption (blue) characteristics, which are related to the real and imaginary parts of the susceptibility, respectively. See text for details.

In this regime, the accumulated phase caused by light drag can be calculated from the formula (Davuluri and Rostovtsev, Reference Davuluri and Rostovtsev2012; Kuan et al., Reference Kuan, Huang, Chan, Kosen and Lan2016; Chen et al., Reference Chen, Lim, Huang, Dumke and Lan2020):

(2) $${\rm{\phi}} =-{k_{0}VL \over v_{g}}=-k_{0}v\tau _{d}$$

where k ₀ is the wave vector of light in the lab frame, L is the interaction length of the medium and τ _d ≡ L/v _g is the time light spends in the medium. However, since the time delay τ _delay ∼ 100μs (Strekalov et al., Reference Strekalov, Matsko, Yu and Maleki2004; Kuan et al., Reference Kuan, Huang, Chan, Kosen and Lan2016; Safari et al., Reference Safari, De Leon, Mirhosseini, Magaña-Loaiza and Boyd2016; Chen et al., Reference Chen, Lim, Huang, Dumke and Lan2020), detecting the phase shift from the dragging effect when dealing with slow medium velocities, V ≪ 1, becomes extremely challenging.

3. Velocimetry using light storage

In velocimetry using EIT (Kuan et al., Reference Kuan, Huang, Chan, Kosen and Lan2016; Chen et al., Reference Chen, Lim, Huang, Dumke and Lan2020], the Fizeau’s light-dragging effect is at the core of the process, and the phase shift results from the change in the phase velocity of light as the medium moves; EIT enhances this effect by increasing the dispersion of the medium and reducing the group velocity of the probe field; therefore, the delay time τ _d is significantly increased. In addition to reducing the speed of light, EIT also allows the light pulses to be completely halted and stored in the medium for some storage time τ _s . In our proposed method, the measurement of the phase shift in the beat note is similar to the previous works; however, by leveraging light storage in atomic media, the phase shift is rooted at a fundamentally different physical phenomenon. The setup proposed here measures the phase shift of the beating signal by comparing the phase of the signal at two separate spatial points along a reference field. The process may be thought of as a delayed imbalanced Mach–Zehnder interferometer, such that the length of the delay arm is set by the storage time and the phase evolution of the probe field in the delay arm is set by the phase coherence of the control field and the storage process. In spite of this difference, it will be seen that the mathematical description for measuring the velocity of the moving medium remains the same, giving us a framework for comparing the sensitivity of the two methods.

In order to store an optical pulse using EIT, once the probe pulse is compressed within the medium, the control field is adiabatically switched off. This process converts the input photonic excitation to a coherence between the ground states |1⟩ and |2⟩, also called a dark-state polariton (Fleischhauer and Lukin, Reference Fleischhauer and Lukin2000). The retrieval is the time reversal of this process: control field is adiabatically switched back on, and the stored excitation is converted back to an optical field, which then leaves the medium along its initial direction (Phillips et al., Reference Phillips, Fleischhauer, Mair, Walsworth and Lukin2001; Liu et al., Reference Liu, Dutton, Behroozi and Hau2001]. The storage time in this scheme depends on several factors, such as ground state decoherence rate, electric and magnetic field noise and optical beam diameters in warm gases (Finkelstein et al., Reference Finkelstein, Bali, Firstenberg and Novikova2023]. The phase information has been shown to be preserved and coherently retrieved during writing and retrieving the optical pulse (Mair et al., Reference Mair, Hager, Phillips, Walsworth and Lukin2002]. Furthermore, precise measurement and control of the phase of the retrieved pulse have been demonstrated (Park et al., Reference Park, Zhao, Lee, Chough and Kim2016; Jeong et al., Reference Jeong, Park and Moon2017). These features make EIT-based optical memories suitable tools for utilizing optical interferometry in developing displacement sensors.

Spatial transport of stored optical pulses (Li et al., Reference Li, Islam and Windpassinger2020) has led to new proposals for quantum communications (Gündoǧan et al., Reference Gündoǧan, Sidhu, Krutzik and Oi2024). In our proposed method, displacement is detected by storing the probe field in a moving alkali vapor cell and monitoring the phase of an optical beating signal between the probe field and a reference field, which is detuned far enough not to be affected by the EIT condition. Before entering the medium, the phase of the beating signal is initially detected by the photodetector D _I . The control field is launched simultaneously, creating the conditions for light storage. At this moment, the probe field is stored for a storage time τ _s ; meanwhile, the medium moves at some average speed V, resulting in a displacement of Δx = Vτ _s . The probe field is retrieved after τ _s , and the phase of the beat signal is then detected by the second detector D _II . During this process, the phase of the beating signal is influenced by both the storage time and the medium’s displacement, which can be measured by comparing the phase of the detected signals. In order to have a high efficiency in storing the probe pulse and reading the phase of the beating signal, two factors should be considered: first, the bandwidth of the probe field pulse should be narrow enough to fall into the EIT linewidth γ _EIT and, simultaneously, much higher than the frequency of the beating signal, to include sufficient number of oscillations for phase measurements. The schematic of the proposed setup is presented in Figure 2(a).

Figure 2. Setup and the theory scheme. a) Schematic of the optical setup. The inset shows the pulse sequence matched with the position of the vapor cell. b) A time-space visualization of the velocimetry protocol: detector D_I records the phase of the beating signal at times t₀ or t′₀; depending on the movement of the vapor cell, the time of the recorded phase of by detector D_II is labeled as t₁ or t′₁; comparing phase shifts in the time window of t₀ → t₁ and t′₀ → t′₁ will reveal the displacement of the cell and hence its velocity. c) In each step of the protocol in “b,” the phase of the beating signal is measured with respect to the symmetry point of the pulse profile. The velocity of the moving medium is then calculated by comparing these different measured phases.

Figure 2(b) shows the principle underlying the stopped light velocimetry. The process can be split into two steps. In the first step, at time t ₀, the beating signal between the probe field and the reference field is detected by D _I , and the phase information is recorded as Φ _I (t ₀). This phase information sets a reference point, relative to which, all the changes in the phase of the beating signal can be measured. Simultaneous with this initial phase measurement, the light is stored in the vapor cell; after a storage time of τ, without moving the cell, the phase information of the beating between the retrieved pulse and the reference field will be detected by D _II , and recorded as Φ _II (t ₁,Δx = 0), where t ₁ = t ₀ + τ _s . Any change in the phase, as compared with the reference phase Φ _I (t ₀), will reveal the phase accumulation during the light storage coming from the beating frequency and the inner dynamics of the storage process (Jeong et al., Reference Jeong, Park and Moon2017; Katz and Firstenberg, Reference Katz and Firstenberg2018). We call this step the calibration step:

(3) $$\Delta \Phi _{Cal.}\equiv \Phi _{II}(t_{1},x_{0}\Delta x=0)-\Phi _{I}(t_{0})$$

Φ _Cal. is only dependent on τ _s and sets our next reference point for measuring the influence of the motion of the stage. In the next step, the phase of the beating signal is again detected at detector D _I at time t′₀ and simultaneously stored in the medium. However, this time, during the storage time τ _s , the vapor cell is moved at a constant speed V, ending up in the position x ₀ + Vτ _s . After retrieval of the stored light, the phase of the beating signal between the probe field and the reference field is measured at the second detector and the phase information Φ _II (t′₁, Δx = Vτ _s ), where t′₁ = t′₀ + τ _s is recorded. We name this step the measurement step:

(4) $$\Delta \Phi _{Meas.}\equiv \Phi _{II}(t{\rm '}_{1},\Delta x=V\tau _{s})-\Phi _{I}(t{\rm '}_{0})$$

As indicated in Figure 2(b), the difference between ΔΦ _Meas. and ΔΦ _Cal. should only be originating from the movement of the vapor cell:

(5) $$\Delta \Phi _{Meas.}-\Delta \Phi _{Cal.}=kV\tau _{s}$$

with k being the wave vector of the reference field. Figure 2(c) shows a hypothetical phase measurement sequence. The “reference” curve (green) shows the beating signal as it is recorded by D _I at times t ₀ or t′₀. For simplicity, the curves for these two instances are plotted with the same phase. The “calibration” curve (red) is the recorded beating signal by D _II at time t ₁. The “measurement” curve (orange) is the recorded beating signal by D _II at time t′₁. By knowing the phase shift in the calibration step, one can calculate the phase shift purely resulted from the movement of the cell. Comparing Equations 2 and 5, it is clear that, in both cases, the sensitivity in measuring the velocity is inversely proportional to the storage/delay time, that is, δV ∝ τ _s/d . The EIT delay times reported in the literature are usually in the range of ∼ 10μs − ∼ 100μs (Chen et al., Reference Chen, Lee, Wang, Du, Chen, Chen and Yu2017,Reference Chen, Lim, Huang, Dumke and Lan2020; Kash et al., Reference Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully1999; Xiao et al., Reference Xiao, Klein, Hohensee, Jiang, Phillips, Lukin and Walsworth2008), whereas the range for storage time of optical pulses in these media can lie in the range of ∼ 1ms − 1 s (Katz and Firstenberg, Reference Katz and Firstenberg2018; Wang et al., Reference Wang, Craddock, Sekelsky, Flament and Namazi2022). Comparing these two time ranges indicates that by using the light storage protocol instead of only delayed light, in principle, this sensitivity can be enhanced by several orders of magnitude.

4. Potential implementation

In this section, we demonstrate in detail the features of a suitable experimental setup for testing the proposed velocimetry scheme using alkali warm vapor. At the heart of the setup is a warm vapor cell of an alkali metal, and, among different options, the D1 transition lines of ¹³³Cs and ⁸⁷Rb offer suitable conditions with sufficient energy splitting between the excited levels so that the competing EIT paths of the overlapping excited states do not affect the EIT visibility (Mishina et al., Reference Mishina, Scherman, Lombardi, Ortalo, Felinto, Sheremet, Bramati, Kupriyanov, Laurat and Giacobino2011; Scherman et al., Reference Scherman, Mishina, Lombardi, Giacobino and Laurat2012). Besides the choice of the alkali metal, the buffer gas filling the vapor cell can play an important role in order to reduce the diffusion of the atoms outside of the optical fields, as well as decoherence due to inelastic collisions between the alkali atoms (McGillis and Krause, Reference McGillis and Krause1967; Reim et al., Reference Reim, Michelberger, Lee, Nunn, Langford and Walmsley2011). Several studies show that noble gasses or molecular nitrogen with pressures ranging from 10 to 30 Torr improve the performance of such optical memories (Thomas et al., Reference Thomas, Munns, Kaczmarek, Qiu, Brecht, Feizpour, Ledingham, Walmsley, Nunn and Saunders2017; Wang et al., Reference Wang, Craddock, Sekelsky, Flament and Namazi2022). For containing the alkali metal vapor and the buffer gas, usually, quartz cells with wedged walls attached at an angle are used. The walls are wedged in order to minimize the interference effects due to surface reflections; however, such a wedge deflects the path of the beam as it passes through the cell walls, leading to off-axis displacement of the retrieved light as the cell moves forward. Therefore, for the suggested setup, having a vapor cell with wedged walls should be avoided. The cell is placed inside a holder designed to hold heaters and thermostats to control and monitor the temperature of the cell and achieve the desired optical depth. Shielding the medium from the noisy electromagnetic fields in the surrounding is another key factor in obtaining an efficient optical memory. Enclosing the vapor cell in two layers of concentric cylindrical μ−metals, with dimension carefully chosen to minimize surrounding electromagnetic fields in the transverse direction (Dubbers, Reference Dubbers1986), could create the sufficient conditions for achieving storage times close to 1 ms in a compact and portable setup (Wang et al., Reference Wang, Craddock, Sekelsky, Flament and Namazi2022). In order to mimic the movement of the medium, automated translation stages with minimum incremental motion as low as Δx ∼ 1nm is available in the market and can test the precision of the proposed scheme down to nm/s scale.

In order to realize the Λ−scheme EIT in the D1 transition a strong control field, P ∼ mW, on the F = 4 ↔ F′ = 3 transition in Cs or on the F = 2 ↔ F′ = 1 in Rb, and a weak probe field, P ∼ μW, on the F = 3 ↔ F′ = 3 or F = 1 ↔ F′ = 1, in Cs and Rb, respectively, can be tuned. Two separate lasers can be used to achieve better tunability of the control and probe field frequency and energies. Using frequency modulation spectroscopy (Bjorklund et al., Reference Bjorklund1983), one low-power “primary” laser can be locked to the transition aimed for the probe field; then, a second high-power “secondary” laser will be offset-locked (Ivanov et al., Reference Ivanov, Esnault and Donley2011) to the primary laser, with an offset frequency matching the hyperfine splitting of the ground state in the D1 transition, 6.8GHz & 9.2GHz for Rb and Cs, respectively. This way, the secondary laser can act as a strong control field. The probe field can be obtained by using an electro-optical modulator (EOM), with the modulation frequency equal to the splitting of the ground states. The secondary laser passes through the EOM and then a high finesse cavity to get a clean side band with the energy matching to that of the probe field. This way, the phase coherence between the two optical fields is guaranteed, and any fluctuation in the wavelength of the laser does not alter the EIT conditions, as both control and probe field are tied together (Li et al., Reference Li, Islam and Windpassinger2020). On the other hand, the proposed scheme relies on having a strong coherence between the phase of the stored and retrieved pulse. As it has been shown (Park et al., Reference Park, Zhao, Lee, Chough and Kim2016; Jeong et al., Reference Jeong, Park and Moon2017; Katz and Firstenberg, Reference Katz and Firstenberg2018), the phase of the retrieved pulse is directly controlled by the phase of the control field at the moment of retrieving the stored pulse from the memory. Therefore, the maximum limit of storage time for extending the sensitivity of the velocimeter is inversely proportional to the linewidth of the laser used for the control and probe fields. External cavity diode lasers usually have a linewidth of ∼ 100 kHz, within ∼ 10 ms. Such a linewidth translates into a time window of ∼ 10μs. At this stage, the main limiting factor of the motion sensor is not the storage time of the memory, but rather the linewidth of the laser used. Nevertheless, different methods have managed to reduce the linewidth down to sub-kilohertz (Ludlow et al., Reference Ludlow, Huang, Notcutt, Zanon-Willette, Foreman, Boyd, Blatt and Ye2007; Saliba and Scholten, Reference Saliba and Scholten2009; Torrance et al., Reference Torrance, Sparkes, Turner and Scholten2016), which pushes the limit on the storage time beyond 1 ms, which is more or less the range achievable with a warm vapor optical memory based on resonant EIT (Wang et al., Reference Wang, Craddock, Sekelsky, Flament and Namazi2022).

Lastly, similar to the setup in Chen et al. (Reference Chen, Lim, Huang, Dumke and Lan2020), the vapor cell can be placed on a motorized moving stage with motion precision on a nanometer scale, and in order to measure the phase shift due to the movement of the cell, two high-dynamic range detectors, before and after the moving medium, can be used to detect the weak beat note of the probe+reference field. By comparing the RF signal from the detectors on an oscilloscope, the phase shift can be measured according to the protocol described in section “Velocimetry using light storage.”

5. Gravimetry using light storage in ultra-cold atomic clouds

Here, we propose to use the same protocol presented in section “Velocimetry using light storage” for measuring local gravitational acceleration. The principle behind the measurement and the experimental steps will remain basically the same; however, the vapor cell will be replaced by a cold atom cloud, similar to atom interferometry gravimeters. The atom cloud would be held fixed at a position h ₀, and the calibration step is conducted by measuring the phase of the beating signal before the signal reaches the cloud and then after storage and retrieval of the probe field in the atoms. In the next step, the phase shift of the beat signal during storage is measured, while the atom cloud is released at the moment of storage. Similar to the analysis given in section “Velocimetry using light storage,” the change in the phase between the two steps in related to the vertical displacement of the atomic cloud:

(6) $$\Delta \Phi _{Meas.}-\Delta \Phi _{Cal.}=k\Delta h={1 \over 2}kg\tau _{s}^{2}.$$

It is worthwhile to draw some comparisons between this proposed scheme and the well-established Mach–Zehnder type atom interferometry used for gravimetry (Kasevich and Chu, Reference Kasevich and Chu1991, Reference Kasevich and Chu1992; Peters et al., Reference Phillips, Fleischhauer, Mair, Walsworth and Lukin1999, Reference Rastogi, Saglamyurek, Hrushevskyi, Hubele and LeBlanc2001), in which the phase shift due to the fall of the atom cloud amounts to ΔΦ = kg(T/2)², with T being the total fall time. In comparison, in our proposal, the storage time τ _s ≡ T; therefore, for an equal fall duration, the accumulated phase in our proposed setup is twice as big as the usual atom interferometry method. Moreover, the physical principle behind this concept is similar to the usual classical approach to gravimetry, that is, the free-falling corner-cube gravimeters (Alasia et al., Reference Alasia, Cannizzo, Cerutti and Marson1982; Faller and Marson, Reference Faller and Marson1988), which relies on a Michelson-type interferometry and tracks the vertical location of a free-falling object, and hence its gravitational acceleration, by monitoring the changes in the phase of the fringe pattern. However, due to the negligible mass of the free-falling atom cloud, it can be immune to some systematic errors faced by the free-falling corner cubes, such as self-attraction or recoil, which can limit the sensitivities of such gravimeters to reach the ultimate shot noise limit (Niebauer et al., Reference Torrance, Sparkes, Turner and Scholten1995, Reference Wang, Craddock, Sekelsky, Flament and Namazi2012). Free-falling Bose–Einstein condensates could be suitable media for this purpose, as it has been proposed that they can be manipulated in a way to achieve long-lived optical memories (Ros et al., Reference Ros, Kanthak, Sağlamyürek, Gündoğan and Krutzik2023).

On the other hand, in atom interferometry, all the atoms contribute to the measurement signal, and the quantum shot noise (QSN) limit scales with the square root of the number of atoms. Therefore, in order to approach the QSN limit observed in atom interferometry with stored light interferometry, one should reach the regime where the number of stored photons approaches the number of atoms in the cloud. So far, most of the analysis of EIT and EIT-based QMs relies on the so-called “weak probe” regime, where Ω _P /Ω _C ≪ 1 and the number of photons in the probe field is much smaller than the number atoms interacting with the optical fields, that is, n _photons ≪ n _atoms (Fleischhauer and Lukin, Reference Fleischhauer and Lukin2000; Fleischhauer et al., Reference Fleischhauer, Imamoglu and Marangos2005; Rastogi et al., Reference Rastogi, Saglamyurek, Hrushevskyi, Hubele and LeBlanc2019). In these conditions, classical optical pulses can be coherently stored in ultra-cold atoms for seconds (Zhang et al., Reference Zhang, Garner and Hau2009; Dudin et al., Reference Dudin, Li and Kuzmich2013). Some works have investigated EIT in the strong probe field regime, in a ladder scheme (Wielandy and Gaeta, Reference Wielandy and Gaeta1998; Dutton and Hau, Reference Dutton and Hau2004) and Λ-scheme in Bose–Einstein condensates (Pandey and Natarajan, Reference Pandey and Natarajan2008); based on these studies, one can claim that EIT can indeed be observed in this regime, and in specific conditions, light storage will be coherent. Nonetheless, a complete analysis of EIT in cold atom clouds with n _photons ∼ n _atoms , which could address all the concerns related to our proposal, is yet to be presented. More specifically, such an analysis should answer how the efficiency of the memory would be affected in this regime or whether the phase information would be also stored and retrieved coherently. The latter is of utmost importance since the success of motion sensing using QMs relies on full coherent control over the phase of the probe field, a feature which has been shown in the weak field regime.