Noise suppression by differential detection of simultaneous electromagnetically induced transparency and absorption in counterpropagating bichromatic light

Well-established electromagnetically induced transparency (EIT) and absorption (EIA) have been applied to various applications including quantum computing, light storage, and precision measurement. Here, we propose and implement a differential detection scheme based on coexisting EIT and EIA signals in a double-Λ system with counterpropagating bichromatic laser fields, in which a differential coherent population trapping (diff-CPT) signal is extracted with a desired enhanced amplitude and highly suppressed common-mode noise. Compared to that of either EIT or EIA, the observed signal-to-noise ratio of the proposed method’s diff-CPT signal improved by one order of magnitude, which would benefit the implementation of high-performance CPT clocks. This technique may also be applied to magnetometers and precision spectroscopy.


Introduction
Coherent population trapping (CPT) is a fundamental and widely used quantum interference [1,2], in which coherent bichromatic light couples two ground states to a common excited state in a three-level Λ atomic system.When the frequency difference of the bichromatic light is equal to the splitting of the two ground states, an atomic ensemble is prepared into a coherent dark state.A narrow transmission peak appears when transmitted light is detected, which is also called electromagnetically induced transparency (EIT) [3].This effect has various applications in high-resolution spectroscopy [4], laser cooling [5,6], superluminal and slow light propagation [7,8], light storage [9], coherent population transfer [10], atomic clocks [11], and magnetometers [12].
Electromagnetically induced absorption (EIA), the opposite of EIT, was discovered in the 1990s [13,14], and its application to atomic clocks was recently demonstrated by Denis Brazhnikov in 2019 [15], where an EIA signal was produced by two counterpropagating dual-frequency beams with the same circular polarization.The obtained EIA signal leads to promising short-term frequency stability at a level of 5.8 × 10 −12 τ −1/2 (τ ⩽ 20 s).In this EIA clock, the contribution of the laser frequency modulation (FM) and amplitude modulation (AM) noise is at a level of 4.6 × 10 −12 (@ 1 s), which dominates the short-term performance [16].
The common-mode noise (such as the laser noise, detector noise, etc.) of CPT atomic clocks can be largely suppressed by differential detection [17,18]; this suppression leads to CPT resonance with an enhanced signal-to-noise ratio (SNR), which would help improve the short-term frequency stability of atomic clocks.However, the existing differential detection schemes exhibit weak signal amplitude and contrast.
Based on our previous configurations involving the differential detection of CPT resonance using copropagating bichromatic light [19], we developed and implemented a simpler differential detection scheme, in which a counterpropagating bichromatic pump and probe beams were used to interact with the atomic ensemble; EIT and EIA signals were observed simultaneously, and their differential CPT (diff-CPT) signals were extracted continuously in the time domain rather than in the pulse mode, as in our previous scheme.Using this method, we benefited from the counter-propagating configuration and differential detection by obtaining an improved CPT signal amplitude and suppressed common-mode noise.The relatively simple structure and high-SNR clock signal observed in this scheme make it attractive for implementation of compact and high-performance CPT atomic clocks.
A quantitative theoretical analysis of coexisting EIT and EIA is presented in section 2. The experimental setup is described in section 3. The experimental results are presented and discussed in section 4. Typical EIT, EIA, and diff-CPT are observed, critical parameters such as the signal amplitude and power spectral density of noise under different incident laser powers are investigated, and the improvement in the CPT clock performance is estimated.Summaries are given in section 5.

Quantitative theoretical analysis of simultaneous EIT and EIA in a double-Λ system
As shown in figure 1, our model is based on a four-level double-Λ system of the D 1 line of 87 Rb, which interacts with a counterpropagating bichromatic pump and probe beams.The circularly polarized bichromatic pump beam prepares certain atoms to a CPT dark state.Then, the counterpropagating linearly polarized probe beam interacts with the dark state atoms.Thus, constructive and destructive interferences are induced by the two counter-rotating circularly polarized components of the probe beam.
The pump beam propagates along the z direction, and the electric field of the pump and probe beams can be written as where In the rotating frame and at Raman resonance, i.e. ∆ = (ω 1 − ω 2 ) − ω hf = 0, where ω hf is the ground hyperfine transition frequency, the dark state prepared by the pump beam is [20,21] with Similarly, the dark states generated by the two counter-rotating circular polarization components of the probe beams are The minus sign before E 3 in (3) is due to the opposite sign of its electric dipole moment compared to the other three in the four-level double-Λ atomic system of 87 Rb [22] When E 3 = E 4 , leading to |⟨d l |d r ⟩| = 0, the two dark states (| ⟨d l , | ⟨d r ) are out of phase, i.e. the dark state prepared by one circular polarization component is a bright state for its counter-rotating state.In this case, when the transmitted light is directly detected without polarization separation, CPT signals are not observed owing to the destructive interference of these two dark states [23].This result was confirmed in the lin//lin scheme [24], where there is no CPT signal when coherent bichromatic light couples 'm F =0 ↔ m F = 0.' Here, m F is the magnetic quantum number.However, with the polarization separation of the probe beam, CPT resonance signals can be observed by appropriately adjusting the Raman phase difference between the pump and probe beams.Here, the Raman phase refers to the phase difference between the two frequency components of the bichromatic light.According to [15], assuming E 1 = E 2 , the product of the dark states of the pump and probe beams can be expressed as When we fine-tune the mirror position along the beam propagation direction to meet where n is an integer, an EIT signal is observed in one detection channel (e.g.CH1) of the balanced photodetectors; the corresponding position is denoted as z CH1-EIT .On the one hand, |⟨d + |d l ⟩| = 1, the two dark state | d + ⟩and | ⟨d l are decoupled from both light fields (E p (z, t) and E l (z, t)), and an EIT signal is observed on CH1 with detection of the left circularly polarized component of the probe field [20,25].In this case, the two dark states (|⟨d + ,| ⟨d l ) are parallel or in-phase, which we refer to as the common dark state.On the other hand, |⟨d )); thus, the right circularly polarized component of the probe field is strongly absorbed [26,27], appearing as an EIA signal on the other detection channel (CH2) of the balanced photodetectors.In this case, they are orthogonal or out of phase, and there is no common dark state.Thus, with the polarization separation of the transmitted probe beam, the EIT and EIA signals can be observed simultaneously.
Similarly, when the mirror positions move to the role of the two dark states of the probe beam is switched, i.e. an EIT on CH1 converts to an EIA signal, and vice versa.The signal amplitude oscillates periodically [28][29][30], with a period (z p ) half of the ground-state hyperfine transition wavelength (λ hf ), i.e. z p = λ hf /2, which is approximately 22 mm for 87 Rb atoms.
In addition to the distance between the vapor cell and mirror, the signal amplitude is also affected by the length of the vapor cell.When the length of the vapor cell L ≥ π /k 12 , the signal amplitude does not increase significantly because the periodic oscillation fluctuations in space are averaged [15].When L << π /k 12 , its amplitude exhibits a spatially substantial periodic oscillatory variation and is significantly enhanced at the z CH1-EIT or z CH1-EIA position; thus, our method could also apply to chip scale atomic clocks (CSACs), which use a millimeter-scale atomic cell.

Experimental setup
The experimental setup is shown in figure 2. Similar to our previous demonstration [31], a distributed Bragg reflector (DBR) laser is adopted, which emits a 795 nm laser beam corresponding to the D 1 line of 87 Rb.Owing to the high modulation bandwidth and the narrow linewidth (∼1 MHz) of the DBR laser, we can directly modulate the beam with half of the ground hyperfine transition frequency of 87 Rb (∼3.417GHz); thus, highly coherent bichromatic light is generated with the ±1st sidebands of the laser.The bichromatic light is split into two beams, one of which is sent to a reference cell for laser frequency locking.The reference cell is filled with pure 87 Rb atoms, an error signal is obtained through dual-frequency sub-Doppler spectroscopy [26,32], and the laser carrier frequency is then locked to the midpoint of the two transitions, i.e. 5 2 S 1/2 , F = 1&2 → 5 2 P 1/2 , F ′ = 2 .The other beam, with an expanded diameter of 7 mm, is sent to a clock cell for CPT experiments.The cylindrical clock cell, with a length of 10 mm and diameter of 20 mm, is filled with 87 Rb and 25 Torr buffer gas mixture of Ar and N 2 (with a pressure ratio of 1.5), and its temperature is stabilized at approximately 62 • C. The introduction of a buffer gas is aimed at achieving a narrow linewidth CPT resonance spectrum through the Dicke narrowing effect [33].A static magnetic field of 1.1 µT is applied along the axial direction of the vapor cell to provide a quantized axis and remove the Zeeman degeneracy.Two layers of permalloy surround the outside of the vapor cell as a magnetic shield to prevent any disturbance from the ambient magnetic field.
After passing through a quarter-wave plate and non-polarizing beam splitter (BS), the circularly polarized light beam (pump beam) interacts with the 87 Rb atoms for the first time.The transmitted light is linearly polarized by a polarizer, reflected by a movable mirror, and utilized as a probe beam, which enters the vapor cell and interacts with the atomic ensemble for a second time.The BS separates the counterpropagating probe beam from the pump beam, whose reflectance-to-transmittance ratio is 90:10, to obtain a large fractional ratio for the probe beam.Using a half-wave plate and polarizing BS, the probe beam is polarization-separated and finally detected by balanced detectors.

Simultaneously observed EIT, EIA, and diff-CPT signals
A typical Zeeman spectrum of EIT, EIA, and diff-CPT is shown in figure 3(a), and the central peak, which is insensitive to the magnetic field-induced first-order Zeeman shift known as '0-0' clock transition, is zoomed in figure 3(b).Several features can be observed from this figure.First, a mirror reflection of the EIT and EIA spectra is clearly observed.Here, the amplitudes (2.1 mV) and contrast (1.9%) of the central CPT clock transitions for the EIT and EIA are essentially similar, where the contrast (C) is defined as the ratio of the signal amplitude to the background level.However, their linewidths (full-width at half-maximum (FWHM)) are 1232 and 1414 Hz for the EIT and EIA, respectively; the reason for this difference is under investigation.In addition, a diff-CPT signal is obtained by subtracting the EIT and EIA signals, with an amplitude doubled to 4.26 mV and linewidth of 1330 Hz.
Second, the three Zeeman CPT peaks of each signal channel, which correspond to 'm F ↔ m F ' CPT transitions from left to right with a magnetic quantum number m F of −1, 0, and 1, increase in amplitude with Raman detuning (∆).This is due to the optical pumping effect induced by the left-hand circularly polarized pump beam and the power of the linearly polarized probe beam, which is considerably weaker than that of the pump beam.
Third, a Lorentzian line shape fits well with the '0 ↔ 0' clock transition, whereas a slight dispersive Lorentzian component appears in the non-clock transitions ('−1 ↔ −1' and '1 ↔ 1').One explanation is that the relationship of the involved electric dipole moments is neither symmetric nor antisymmetric, that is, |d 23 /d 13 | ̸ = |d 24 /d 14 |.Another reason could be the inhomogeneity of the applied magnetic field; non-clock transitions are more sensitive to magnetic fields than clock transitions.

Mirror position-dependent Raman phase difference
The EIT or EIA signals can be observed by each detector, which is dependent on the Raman phase difference between the circularly polarized components of the probe and pump beams.This can be adjusted by the position of the mirror, polarization of the probe beam, and relative Raman phase of the bichromatic light.Here, the amplitudes of the EIT, EIA, and diff-CPT signals versus the position of the mirror are shown in figure 4, with a spatial modulation period of approximately 22 mm, which corresponds to the half wavelength of the ground hyperfine splitting of the 87 Rb atom and is in good agreement with the above theoretical analysis.We will mainly study the effect of the movable mirror position and the laser power on the EIT, EIA, and diff-CPT, which are the critical parameters for each resonance, as well as the CPT atomic clock applications.
In this plot, for the EIT channel, the Lorentzian line shape of the clock CPT resonance at a relative mirror position z 0 of 9 mm evolves to a dispersion line shape at z ≈ z 0 + 5.5 mm and again to a Lorentzian line shape at z ≈ z 0 + 11 mm, where the line shape is a mixture of Lorentzian and dispersion line shapes, similar to the EIA and diff-CPT signal channels.Thus, we define the signal amplitude as (A-B)-(B-C) [28] in this plot, where A, B, and C denote the maximum, background level, and minimum of the CPT clock signal, respectively.
We also observed some features that deviated from the theoretical expectations.When the position of the mirror is moved, the EIA signal is transformed into an EIT signal for the EIA channel; however, the maximum EIT amplitude in this channel is less than that in the EIT channel.For the EIT channel, the EIT signal is not transformed into an EIA signal but only changes its signal amplitude when we move the mirror position.This deviation may be caused by the deterioration of the ideal polarization of the pump and probe beams, e.g. the beam expander and the imperfect parallelism and flatness of the cell windows.The overlapping degradation of the pump and probe beams may also induce this effect when we move the mirror position.

Effect of the laser power on the CPT resonances
The incident laser power (P L ) of the pump beam is a critical parameter affecting the CPT signal quality.Here, P L represents the laser power of pump light, i.e. the light power incident on the vapor cell for the first time.Hence, we measured its effect on the amplitude, linewidth, contrast, and q-value of the CPT clock transition, and plotted them in figure 5, where the quality figure q is defined as the ratio of the contrast to FWHM, i.e. q = C/FWHM.
As shown in figures 5(a) and (b), the amplitudes and linewidths of the EIT, EIA, and diff-CPT signals are all approximately proportional to the laser power within the experimentally investigated laser power range.Furthermore, as shown in figure 5(c), the contrast of the EIA exhibits a typical behavior, i.e. it first increases with the laser power and then approaches a saturation value.However, the contrasts of the EIT show some unusual behavior; they first increase with the laser power but then decrease slightly with increasing laser power.This induces a difference in the q-value dependence on the laser power between the EIT and EIA signals, as shown in figure 5(d).The line shape and center frequency of the diff-CPT signal spectrum may undergo changes relative to EIT (or EIA), or there could be differences in the dependence of EIT and EIA resonances on laser power.These factors could potentially impact the medium-to long-term frequency stability of the CPT clock based on our differential detection method, and it will be investigated in our future studies.For the amplitude balance of the EIT and EIA signals, a laser power P L of 198 µW was chosen for the following noise rejection experiment.

Noise rejection and SNR improvement of diff-CPT signals
Laser sources contribute to the detection noise through two mechanisms: laser frequency noise and amplitude noise, which are converted to amplitude noise after passing the vapor cell [16].The benefits of differential techniques in suppressing common-mode noise from laser sources have been investigated, particularly the FM to AM (FM-AM) noise, AM to AM (AM-AM) noise caused by frequency-and power-dependent optical absorption processes.To demonstrate the noise suppression in our method, we first set the microwave frequency at the waist of the CPT resonance signal and then measure the power spectral density (PSD) of the EIT, EIA, and diff-CPT signals using a dynamic signal analyzer (Stanford Research SR785).
As clearly shown in figure 6, the noise of the diff-CPT is highly rejected (75%) compared to that of the EIT and EIA over the entire analyzer measurement range (1 Hz-100 kHz).At a typical microwave modulation frequency f m = 125 Hz, the noise levels of the EIT and EIA signals are similar (∼2.1 × 10 −7 V/√Hz), whereas the noise of the diff-CPT is significantly reduced by four times to a level of 5.2 × 10 −8 V/√Hz.
The noise of the EIT, EIA, and diff-CPT signals at f m = 125 Hz versus the laser power are measured and plotted in figure 7(a).The noise dependence on the laser power can be modeled with a simple phenomenological noise term where i = 1, 2, representing the EIT and EIA signals, respectively.N AM-AM and N FM-AM represent the laser AM-AM and FM-AM noise, respectively; N PD1 , N PD2 , and N PD3 are the detector noises in the EIT, EIA, and differential channels without incident light, respectively.N SN is the shot noise of a single channel of a balanced detector, which can be evaluated as √ 2eIG R .e denotes the electron charge, I represents the photocurrent received by the detector (proportional to laser power), and the transimpedance gain of a single channel is G R = 1.4 × 10 4 V A −1 .Here, the solid black line in figure 7(a) represents the expected shot noise of the diff-CPT signal, and the expression is given by √ 2N SN .Within the low laser power range, the contribution of the expected shot noise is relatively significant, and the noise suppression of the diff-CPT signal is more apparent.However, as the laser power increases and without a laser power active stabilization, the laser AM-AM noise becomes the dominant contributor, revealing the noise can be only suppressed partially with our differential detection method.α is the noise suppression ratio, indicating the suppression effect of the laser AM-AM noise.The larger the value of α, the greater is the suppression effect, and vice versa.The laser FM-AM noise of the differential channel is negligible in equation (10) owing to the laser frequency locking, resulting in less noise and less power dependence; thus, we reasonably assume that it is fully suppressed.The appendix presents detailed investigations of the noise processing reported in this subsection.
In addition, we plot the SNR of the EIT, EIA, and diff-CPT signals versus laser power in figure 7(b), where the SNR is as the ratio of the signal amplitude to its noise level (PSD) at f m .All of them show similar behavior, i.e. they first increase with laser power, and then approximate a saturation value.The SNR of the EIT and EIA signals increase sharply in the small laser power range, which is attributed to the signal amplitude being quadratic with the laser power [11].However, this relationship changes from quadratic to linear over an extended range of laser power.
The common-mode rejection ratio (CMRR) is defined as the ratio of the average noise of the EIT and EIA to the diff-CPT The CMRR (blue squares) and SNR improvement factor (green circles) versus the laser power are shown in figure 8.The SNR improvement factor is defined as the SNR ratio between the diff-CPT and the average of the EIT and EIA signals.The red curve is the fitting curve of the CMRR obtained according to equation (11) with α = 0.38, and the orange curve represents its double.
The following features are revealed from the plots.First, the noise of the EIT, EIA, and diff-CPT increases with the laser power, as shown in figure 7(a).In particular, the noise of the diff-CPT is linearly dependent on the laser power.This can be explained by the laser AM-AM noise becoming dominant at moderate and high laser power, which is induced by the laser relative intensity noise (RIN) and depends linearly on the laser power within optical power range of the present study.Second, the laser FM-AM noise, detector noise, and common-mode part of the AM-AM noise are suppressed by differential techniques in this configuration.Specifically, the laser AM-AM noise is suppressed by approximately 4 dB (38%) around P L = 198 µW (figure 6).Third, at low laser power, the CMRR is dominated by the noise rejection ratio (r 0 ≈ 29) of the balanced detectors (Thorlabs, PDB210A).When we increase the laser power, the laser AM-AM noise increases linearly with the laser power and becomes dominant.In our system, the noise suppression ratio (α) of the differential detection is smaller than r 0 , which leads to a decrease in the CMRR of the diff-CPT signals as the laser power increases (figure 8).Finally, the SNR improvement factor is twice that of the CMRR, which is attributed to the double amplitude enhancement of the diff-CPT signals relative to the EIT or EIA case.When the linewidths are similar in the three cases, the relatively high CMRR value and SNR improvement factor of the diff-CPT signals are beneficial for implementing compact and high-performance CPT atomic clocks, as indicated in the following short-term frequency stability estimation.

Expected improvement in the short-term frequency stability
For the application of CPT atomic clock, we care only about the noise around microwave modulation frequency (f m ).That is due to the fact of a coherent demodulation technology is often-used to modulated and demodulated the CPT signal to get an error signal, and then lock the local oscillator to realize an atomic  frequency standard.In this technology, the noise around microwave modulation frequency is used to calculate the SNR [11,34], then the SNR-limited short-term frequency stability can be estimated by [35] where ∆ν is the FWHM of the signal and τ is the averaging time.As demonstrated in figure 9, the expected short-term frequency stability of the EIT-, EIA-, and diff-CPT-based atomic clocks as a function of the laser power is estimated.As we can see from the diff-CPT-based atomic clock, its frequency stability at 1 s shows a tendency to first decrease rapidly and then gradually deteriorate with the laser power.The expected short-term stability is improved in the small laser power range owing to the rapid increase in the SNR and the corresponding narrow linewidth.When the SNR tends to saturate (figure 7(b)), the linewidth increases linearly (figure 5(b)), leading to the σ y (1 s) gradually deteriorating in the large laser power range.The EITand EIA-based atomic clocks show similar behavior, but with one order of magnitude worse.

Q Li et al
Because of the excellent noise rejection of our differential detection, frequency stability at a level of 1.6 × 10 −12 (τ =1 s) with a laser power of approximately 50 µW is expected, which is an order of magnitude better than the EIT-and EIA-based cases.Our configuration is of considerable interest for high-performance compact CPT atomic clocks, particularly CSACs, in which a lower light power emitted (∼1 mW) vertical cavity surface emitting laser is used.
It is worth noting that, besides the suppression of the detector intrinsic noise in our method, the noise rejection of the AM-AM noise and FM-AM noise is indeed equivalent to the improvement of laser performance.However, the mid-and long-term amplitude and frequency fluctuations of the laser source still need to be well stabilized, which are crucial for implementing a high-performance CPT atomic clock with better mid-and long-term frequency stability.Thus the mid-and long-term effect of noise suppression in our method is worth investigating in further research.

Conclusion
We proposed and implemented differential detection of coexisting EIT and EIA with counterpropagating bichromatic light, in which a diff-CPT signal is generated with an increased amplitude that highly rejected common-mode noise.With an enhanced SNR in this scheme, the improvement of the short-term frequency stability of CPT atomic clocks by one order of magnitude is expected.The structure of this scheme is relatively simple and suitable for compact CPT clocks using sub-centimeter vapor cells.In addition, it can be applied to atomic magnetometers with magnetic field-sensitive non-clock transition and high-resolution spectroscopy.
are the wave vector values, λ 1(2) and ω 1(2) are the corresponding wavelength and frequency, respectively, and c is the speed of light in vacuum.The first (last) two terms on the right side of equation (1) represent the two frequency components of the circularly (linearly) polarized pump (probe) beam.Here, e ±1 = ∓ e x ± ie y / √ 2 are the unit vectors of the left (σ + ) and right (σ − ) circular polarizations, respectively.
, i.e. −d 23 = d 13 = d 24 = d 14 , where d ij is the electric dipole moment related to states | i⟩ and | j⟩.

Figure 1 .
Figure 1.The four-level double-Λ atomic system interacted with counterpropagating bichromatic pump and probe beams (the 87 Rb atom is used in the experiment), the left (a) and right (b) circularly polarized component of the probe beam (dashed arrows) interacted with the CPT state produced by the pump beam (solid arrows).

Figure 3 .
Figure 3. Experimentally-observed Zeeman spectrum of EIT, EIA and diff-CPT signals (a) with incident laser power PL = 198 µW.The inset shows the involved atomic levels and coupled pump bichromate light, the atomic population distributions is just for an illustration.The center peaks are the '0 ↔ 0' clock transition, which is zoomed in (b).

Figure 4 .
Figure 4.The signal amplitude of CPT clock resonance as a function of relative mirror position for EIT, EIA, and diff-CPT.Same conditions as in figure 3.

Figure 5 .
Figure 5. CPT clock transition amplitude (a), linewidth (b), contrast (c), and q value (d) of EIT, EIA, and diff-CPT dependence on the laser power PL.

Figure 6 .
Figure 6.Noise spectra EIT (a), EIA (b), and diff-CPT signals (c), the noise floor (d) of the SR785 is also plotted for information.The same conditions as in figure 3.

Figure 7 .
Figure 7. Noises level at Fourier frequency of f m (a) and SNR (b) versus the laser power PL of EIT, EIA, and diff-CPT signals.Other parameters are the same as those in figure 5.

Figure 8 .
Figure 8. Common-mode rejection ratio (blue squares) and SNR improvement factor (green circles) versus the laser power PL, as well as the CMRR numerical calculation curve (red curve), and its twice (orange curve).Other parameters are the same as those in figure 5.

Figure 9 .
Figure 9.The expected short-term Allan deviation at 1 s of EIT, EIA, and diff-CPT based atomic clocks as a function of the laser power PL.Other parameters are the same as those in figure 5.