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Abstract
A dual-frequency light field scheme, composed of counterpropagating pump and probe light waves with equal circular polarizations and different intensities, is proposed for the detection of subnatural-linewidth electromagnetically induced absorption (EIA) resonances. In this scheme, the bright-type EIA resonance is obtained at fixed static magnetic field by tuning the frequency difference between both optical fields and can be used as a frequency reference in an atomic clock. Using a 5-mm long buffer-gas-filled Cs vapor cell, an EIA-based atomic clock with a short-term fractional frequency stability of 5.8 × 10−12 τ−1/2 until 20 s integration time is reported. These performances are found to be in correct agreement with the signal-to-noise/linewidth ratio of the resonance. The proposed EIA scheme can be considered as an alternative approach to the coherent population trapping (CPT) technique for the development of compact atomic clocks and sensors.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum coherent superposition of atomic states created by the simultaneous interaction of two coherent light fields are extensively used in nonlinear spectroscopy [1] and find many exciting applications in quantum sensing, quantum information, laser cooling [2] and atomic devices [3]. These phenomena, including mainly electromagnetically induced transparency (EIT) [4], coherent population trapping (CPT) [5,6], electromagnetically induced absorption (EIA) [7] and other atomic three-level configurations [8], imply the creation of a specific non-absorbing (non-transmitting) quantum "dark" ("bright") state that yields the detection of narrow subnatural-linewidth dark (bright) resonances in the absorption profiles.
Among these different approaches, the coherent population trapping phenomenon has known a great success for the development of atomic clocks. In CPT physics [5], atoms are driven by the combined action of two resonant optical fields into a non-interacting dark state uncoupled from the excited state when the null Raman detuning condition is satisfied. The microwave CPT resonance linewidth is limited by the relaxation rate of the atomic ground state coherence and can be reduced with buffer-gas-filled cells [9,10], wall-coated cells [11–13], or through the use of cold atom samples [14]. The CPT resonance contrast, defined as the ratio between the resonance signal amplitude and the dc background level, depends strongly on the light-field polarization scheme.
In most compact CPT clocks, atoms interact with a circularly polarized field ($\sigma -\sigma$ configuration) [15]. However, this scheme tends to accumulate many atoms into extreme Zeeman sub-levels and yields the detection of low-contrast CPT resonances onto the magnetic-field insensitive "0-0" clock transition. To circumvent this issue, different optimized CPT pumping schemes, generally based on the alkali D$_1$ line [16], have been proposed [17–24]. Such polarization schemes, combined with pulsed Ramsey-like interrogation protocols [21,25], have led to the demonstration of high-stability CPT atomic clocks [26–29].
At the contrary of the CPT or EIT [30] approaches involving the detection of dark resonances, the exploitation of coherently driven bright resonances has been rarely explored for the development of atomic clocks. In [31], a bright three-photon N-type resonance has been used on the Rb D$_1$ line for the demonstration of an atomic clock. The electromagnetically induced absorption (EIA) two-photon resonance, discovered in [7] and explained initially as the result of a spontaneous transfer of Zeeman coherences or magnetic sub-level populations from the excited state to the ground state [32–34], is an alternative option for the detection of narrow "bright" resonances.
A relevant number of EIA-based schemes and experiments using single-frequency magneto-optical (Hanle-like) configurations or dual-frequency light-field configurations have been reported in the literature [35–54]. It should be noted that in many of these works, the physical origin of the EIA effect differs from the process of spontaneous transfer of Zeeman coherences or populations identified in pioneering EIA studies [7,36]. In any case, none of the previously proposed EIA schemes was suitable for atomic clock operation since the high-quality EIA resonance was observed in the magneto-optical configuration (mostly single-frequency Hanle-like resonances). To demonstrate an EIA-based atomic clock, the EIA scheme should involve a light field composed of at least two optical frequencies $\nu _1$ and $\nu _2$ separated by the microwave frequency $\nu _{\mu } = \nu _{1} - \nu _2$. To the best of our knowledge, such an EIA-scheme compliant with the development of an EIA-based clock has never been proposed.
In this article, we report a novel excitation scheme for the detection of subnatural-linewidth EIA resonances on the cesium D$_1$ line, adapted for atomic clock operation. A buffer-gas-filled vapor cell is irradiated by dual-frequency counterpropagating control (pump) and probe beams with equal circular polarizations and different intensities. The bright-type EIA resonance is detected by tuning the laser frequency difference between both optical fields, at fixed static magnetic field. The proposed scheme is applied for the demonstration of an all-optical EIA-based atomic clock using a 5-mm long cell. The latter demonstrates a preliminary short-term fractional frequency stability of 5.8 $\times$ 10$^{-12}$ $\tau ^{-1/2}$ until 20 s integration time, found to be in correct agreement with the signal-to-noise/linewidth limit. This EIA scheme might be viewed as an alternative to CPT for the development of compact atomic clocks and quantum sensors.
2. EIA scheme
2.1 Experimental setup
Figure 1(a) depicts the experimental setup used for EIA resonance spectroscopy and clock. The laser field is produced by a distributed-Bragg resonator (DBR) laser tuned on the Cs D$_1$ line ($\lambda =894.6$ nm), with a spectral linewidth narrower than $1$ MHz. A fiber-coupled intensity Mach-Zehnder electro-optic modulator (EOM, iXblue NIR-MX950-LN-20) driven by a microwave frequency synthesizer is used to produce an optical field composed of two resonant angular optical frequencies $\omega _1$ and $\omega _2$ separated by $\Delta _{g} \approx 2\pi \times 9.192$ GHz. At the output of the EOM, a neutral-density filter (NF) controls the total light power sent into the cell. A polarizing beam splitter PBS1 serves as a polarizer. The combination of PBS1 and PBS2 allows to separate the laser beam into a probe beam and a control beam, propagating ultimately into the Cs vapor cell in opposite directions with equal circular polarizations (obtained thanks to quarter-wave plates QWP1 and QWP2) and different intensities. The PBS2 is also used to send the probe beam at the output of the cell into the photodiode PD. The PBS3 is used to filter the backward control beam. The Gaussian diameter of the control beam is about $5$ mm. The probe beam diameter is controlled by an iris diaphragm and equals $5$ mm.
Fig. 1. (a) Experimental setup. DBR: DBR laser, EOM: intensity electro-optic modulator, $\mu$-wave: microwave frequency synthesizer, PBS: polarizing beam splitter, PD: photodetector, QWP: quarter-wave plate, M1 and M2: two mirrors on a translation stage, D: iris diaphragm. (b) Simplified scheme of the Cs D$_1$ line energy levels involved in the EIA experiment. Atoms are assumed to be at rest. The vertical lines denote $\sigma ^+$ light-induced transitions which are in two-photon resonance with the atoms. Blue and red lines stand for $\omega _1$ and $\omega _2$ angular optical frequencies, respectively. The other transitions are not in resonance and are not shown. $\Delta _g$ stands for the microwave frequency of the ground-state hyperfine splitting. A static magnetic field is applied parallel to the wave vectors to frequency-split the ground-state levels. The splitting of the excited-state levels is not relevant in our study. (c) $\Lambda$-scheme of atomic energy levels under the two-photon resonance (left) and not in the resonance (right). Circles reflect qualitatively the energy-level populations. The pink color is for the atoms in the CPT dark state, while the yellow color is for other atoms. Wavy arrows denote spontaneous decay.
The output signal from the photodiode is used both for the stabilization of the microwave synthesizer local oscillator (LO) frequency onto the EIA resonance, and for the stabilization of the laser frequency onto the Doppler absorption profile. The cell is a borosilicate glass-blown Cs vapor cell with length 5 mm and diameter 25 mm. The cell is filled with Ne-Ar buffer-gas mixture (Ne:Ar = 20:2 Torr). The cell is heated by a thermochamber made of copper. A longitudinal static magnetic field of $450$ mG is produced by a solenoid in order to eliminate the degeneracy of the cesium ground states and isolate the "0-0" two-photon clock transition. A three-layer $\mu$-metal magnetic shield is used to prevent perturbations from the ambient magnetic field on the clock transition.
2.2 Qualitative description
The present EIA resonance detection scheme presents two key differences in comparison with the $\sigma ^+-\sigma ^-$ scheme proposed in [17] for the detection of a dark resonance. The first point is that counterpropagating beams with equal circular polarizations are used. The second point is that the condition $I_\textrm {p} \ll I_\textrm {c}$, with $I_\textrm {p}$ and $I_\textrm {c}$ the intensities of the probe and control beams respectively, must be satisfied. The observation of bright resonances in the proposed scheme can be explained by a simplified $\Lambda$-model (Fig. 1(c)). This model, commonly used to study CPT and related effects [6], implies two ground-state levels $|1\rangle$ and $|2\rangle$, an excited-state level $|3\rangle$ and an additional level $|0\rangle$ (optionally). The latter level does not interact with the light field, accumulates the atoms owing to optical pumping [20] and acts as a Zeeman trap state.
The laser field is composed of counterpropagating light beams traveling along the $z$-axis and can be described as:
In the case where only the two-frequency control field $\textbf {E}_1 + \textbf {E}_2$ is present in the cell and the two-photon resonance condition is satisfied (with $\Delta _\textrm {R} = \omega _1-\omega _2-\Delta _\textrm {g} = 0$ the Raman frequency detuning, see Figs. 1(b,c)), a non-coupled (NC) coherent superposition of levels $|1\rangle$ and $|2\rangle$ is created [6]. This state does not interact with the resonant light field and is responsible in the control-field vapor-cell transmission for the creation of a dark resonance. To derive an explicit form of the NC state, consider the operator of light-atom interaction in the dipole and rotating wave approximations (i.e., the time phases $i\omega _{1,2} t$ are excluded):
It can be easily shown that the following state in the non-coupled state for the operator $\hat {V}$ (also see [17,55]):
with $k_{12}=k_1-k_2$. This state is an eigenvector with a zero eigenvalue, i.e. On the other hand, if only the probe field crosses the cell, another non-coupled superposition state can be created:The key point is then to understand that $|\textrm {NC}_1\rangle$ and $|\textrm {NC}_2\rangle$ states can be similar or essentially different, depending on the product ${\bigl |}\langle \textrm {NC}_1|\textrm {NC}_2\rangle {\bigr |}$ such that:
In a short cell, the absorption of the medium remains about constant, so that the oscillations (6) over the cell length can be neglected, i.e. $z \equiv z_\textrm {c}$ can be considered in (6) with $z_\textrm {c}$ being the position of the cell. Analogously, the optimization of the probe-wave absorption can be achieved by adjusting the phase $\phi _{12}$. In our experiments, this adjustment is done by shifting the mirrors M1 and M2 shown in Fig. 1(a). Since shifting the mirrors by $\Delta z$ is identical to shifting the cell by $2\Delta z$, bright resonances alternate with dark resonances with a period $\Delta z/2 \approx 0.8$ cm during the mirrors scan.
To obtain an EIA resonance with the highest contrast, the probe-field absorption must be high at $\Delta _\textrm {R}=0$ and low at $\Delta _\textrm {R} \gg \Delta _\textrm {res}$, where $\Delta _\textrm {res}$ is the full-width at half-maximum (FWHM) of the resonance. At $\Delta _\textrm {R}=0$, the high absorption occurs when $|\textrm {NC}_1\rangle$ and $|\textrm {NC}_2\rangle$ are orthogonal. Figure 1(c) (right) shows that at $\Delta _\textrm {R} \gg \Delta _\textrm {res}$, a low probe field absorption can be achieved by pumping optically most of the atoms into the trap level $|0\rangle$. Indeed, if $\Delta _\textrm {R} \gg \Delta _\textrm {res}$, the non-coupled state $|\textrm {NC}_1\rangle$ is not created and the control field pumps atoms out of $|1\rangle$ and $|2\rangle$ levels.
In EIT or CPT resonances, the configuration $\sigma ^+-\sigma ^-$ is often used in order to eliminate the trap states and to increase the contrast of the dark resonances [18]. At the opposite, Zeeman trap states can play a positive role in the case of bright EIA resonances by inducing a reduced absorption at $\Delta _\textrm {R} \gg \Delta _\textrm {res}$. However, since the probe-wave absorption is also high at $\Delta _\textrm {R}=0$ due to the creation of the $|\textrm {NC}_1\rangle$ state, the final contrast of the EIA resonance could be quite low. To circumvent this issue, counterpropagating beams with identical circular polarizations are used in the present work to let the trap states survive and to observe the EIA effect.
This analysis explains also the reason why the probe field should be much weaker than the control field. With two counterpropagating beams of equal intensities, any non-coupled state cannot be created due to the "competition" between orthogonal $|\textrm {NC}_1\rangle$ and $|\textrm {NC}_2\rangle$ states. In this case, most of the atoms are transferred to a trap level $|0\rangle$ by optical pumping, leading to a significant decrease of the probe-field absorption at the center of the resonance ($\Delta _\textrm {R}=0$).
2.3 EIA spectroscopy
Figure 2 shows some examples of EIA resonances observed in the probe wave transmission by sweeping the Raman frequency detuning $\Delta _\textrm {R}$. In the first case, the resonance line, shown in green, is a typical EIA clock resonance. Obtained here with a control beam power $P_\textrm {c}$ and probe beam power $P_\textrm {p}$ of 40 and 4.5 $\mu$W respectively, the EIA resonance exhibits a FWHM of 850 Hz and a contrast of 2.5$\%$ with respect to the background light transmission. The second case is an illustration of the quasi-disappearance of the EIA resonance when required conditions are not satisfied. As mentioned previously, the sign of the subnatural-linewidth resonance (enhanced absorption or transparency) depends on the relative phase between both counterpropagating beams. In addition, the EIA resonance disappears when both probe and control beams have similar intensities. This situation is illustrated by the blue line on Fig. 2 where the resonance contrast is degraded to 0.1$\%$. Note that the experiment is performed in a large optical density regime in which both the probe wave and the pump wave experience significant absorption in the cell (about 50$\%$). Therefore, the intensity of both waves in the cell is not constant and the probe wave can be weaker than the control one in some regions of the cell, explaining the residual EIA dip in the center of the blue line profile. For illustration, let us consider a sketch shown in Fig. 3. When light wave intensities are equal at the cell entrance, only the central part of the cell is exposed to light waves of equal strengths (region "II"). In this region, there are no $|\textrm {NC}_1\rangle$ as well as $|\textrm {NC}_2\rangle$ states because of their "competition". At the same time, in the regions "I" and "III", the light wave intensities are different, leading to the EIA effect in the former case and to the EIT effect in the latter case. This basic treatment helps to explain the non-vanishing EIA resonance in Fig. 2 (blue curve) superimposed with a broad EIT background profile.
Fig. 2. Detection of EIA resonances. In the first case with light beams of different intensities (green steep dip, left "y" axis), the typical clock EIA resonance is observed. The resonance linewidth is 850 Hz and the resonance contrast is 2.5%. Experimental data are fitted by a Lorentzian function. The cell temperature is 60$^{\circ }$C. The control and probe beam powers are $40$ $\mu$W and $4.5$ $\mu$W respectively. In the second case (blue line and right "y" axis), the EIA resonance is almost absent, having extremely low contrast of about 0.13%. This occurs when the beams have close powers: $P_\textrm {c}\approx P_\textrm {p}=40$ $\mu$W.
Fig. 3. Sketch of light field absorption in a vapor cell under the regime of equal intensities of counterpropagating waves. EIA and EIT denote the regions which contribute to the creation of subnatural-linewidth absorption and transparency resonances in the probe wave transmission signal (see blue curve in Fig. 2). The effect of spatial oscillations of the light field absorption within the cell is neglected since the cell is sufficiently short.
Figure 4 shows the evolution of the EIA resonance linewidth and height versus the control beam intensity $I_\textrm {c}$. Experimental linewidth data are correctly fitted by a linear function such that FWHM [kHz] = 3.2 $I_\textrm {c}$ + 0.58, with $I_\textrm {c}$ in mW/cm$^2$. This linear behavior is typical for subnatural-linewidth resonances in buffer-gas-filled vapor cells [9,10]. In our experiment, the main contribution to the zero-intensity linewidth is the time-of-flight relaxation process, due to the finite light beam size and then the limited atom-field interaction time. Using expressions reported in [57], this contribution is calculated to give a zero-intensity linewidth of 450 Hz, in correct agreement with the experimental result of 580 Hz.
Fig. 4. FWHM (squares) and signal (triangles) of the EIA resonance versus the control wave intensity. Experimental linewidth data are fitted by a linear function, yielding a linewidth extrapolated at null laser intensity of 580 Hz. The cell temperature is 60$^{\circ }$C. The probe beam power is about $1/9$ of the control beam power.
The height of the EIA resonance increases rapidly at small control wave intensities, while at higher intensities this growth is slower. The latter can be interpreted as the saturation of the differential probe-wave absorption coefficient $\Delta \alpha (I_c)=\alpha _0-\alpha _1$ with $\alpha _{0,1}$ the absorption coefficients in the center of the EIA resonance (0) and out of it (1). From linewidth and signal data, we were able to find that the signal/linewidth ratio was maximized for quite low control laser intensities, typically lower than 0.5 mW/cm$^2$.
3. EIA-clock short-term stability
Figure 5 reports the preliminary Allan deviation of the EIA-based atomic clock. This measurement was performed for laser powers ($P_\textrm {c}=40$ $\mu$W, $P_\textrm {p}=4.5$ $\mu$W) and cell temperature (60$^{\circ }$C) conditions optimizing the signal/linewidth ratio. In this experiment, the laser frequency is simply stabilized onto the bottom of the homogeneously-broadened absorption profile using a standard synchronous modulation-demodulation technique. No sub-Doppler spectroscopy technique for enhanced laser frequency stabilization is used. The microwave frequency is stabilized using the Pound-Drever-Hall (PDH) technique. Modulating the LO frequency, each optical sideband is accompanied by two additional first-order radiofrequency (RF) sidebands, each separated from the optical sideband by the LO modulation frequency (9.36 kHz in our case). In the case where the LO modulation frequency is higher than the linewidth of the EIA resonance, three separate EIA resonances are observed at the output of the detection photodiode. Phase-sensitive demodulation is then produced and the quadrature signal is used for stabilization of the microwave frequency. The phase of the modulation signal is here a crucial parameter and was optimized by maximizing the slope of the output error signal. Compared to conventional synchronous modulation-demodulation approaches, the main advantage of the PDH technique is to modulate the LO frequency at higher frequency, yielding a relevant reduction of the detection noise level. The LO modulation frequency (9.36 kHz) was chosen to avoid any spurious intermodulation effects with other frequency components existing in our electronics (50 Hz and harmonics for example) and any cross-talk issues with the laser frequency stabilization loop. The EIA resonance used for clock operation is shown in Fig. 2 (green line). The frequency stability of the stabilized LO is measured by comparison with a passive hydrogen maser reference standard.
The short-term frequency stability of an atomic clock can be well-approximated by $\sigma _y$ (1 s) $\approx$ (FWHM / $\nu _0$) $\times$ ($1$/SNR), with $\nu _0$ the clock frequency and SNR the signal-to-noise ratio of the resonance in a 1 Hz bandwidth. In our experimental setup, measurements of the total detection noise ($7.6 \times 10^{-6}$ V/$\sqrt {\textrm {Hz}}$ at $f_\textrm {m}=9.36$ kHz where $f_\textrm {m}$ is the LO modulation frequency) at the output of the photodiode led to a measured SNR of about 20330, yielding an expected short-term stability of $4.6 \times 10^{-12}$ at 1 s. This expected value is in correct agreement with the measured stability value shown on Fig. 5, found to be $5.8 \times 10^{-12}$ $\tau ^{-1/2}$ until $20$ s and at the level of $10^{-12}$ at $60$ s averaging time.
Table 1 shows main contributions to the EIA-clock short-term stability budget. The main contribution to the EIA clock short-term stability is the total laser noise (laser FM noise through the FM-AM conversion process in the cell and laser AM noise). Note that the laser frequency noise contribution could be reduced by stabilizing the laser frequency onto an external reference cell using a Doppler-free spectroscopy technique. The measured short-term stability is 3.5 times higher than the calculated photon shot noise. On slightly higher time scales, we suspect that the clock frequency stability might be limited by the fluctuations of the optical beam path length. Indeed, a relevant sensitivity of the clock frequency to the mirrors translation at the level of $10$ mHz/$\mu$m, i.e. 1.1 $\times 10^{-12}$/$\mu$m in fractional value, has been measured. To date, no optical path length stabilization is implemented in our setup and many optical elements could experience acoustic and thermal fluctuations. We think that fractional length fluctuations of the optical path could be reduced at the level of 10$^{-9}$ in a properly-designed low-vibration and reduced-size system with high-precision temperature control at the mK level. Such a length stability level would reject this contribution on the clock frequency stability below the 10$^{-15}$ level. For improved length stabilization, interferometer-based techniques inspired from the domains of optical fiber links [58] or ultra-stable lasers [59] might be envisioned but at the expense of a relevant increase of the clock architecture complexity.
Table 1. Contributions to the EIA clock short-term frequency stability budget. Expressions similar to those reported in [27] combined with detection noise measurements were used for these estimations. FM and AM stand for laser frequency and amplitude noise, respectively.
The preliminary short-term frequency stability of the EIA-clock remains to date about a factor of 10–30 lower than those of best CPT-based vapor-cell atomic clocks [27,29]. Compared to these CPT clocks, an advantage of the present EIA scheme is to benefit from the use of short-length cells. The present EIA-clock uses a 5-mm long cell whereas CPT clocks described in [27,29] are based on 5-cm long cells. The current short-term stability performances of the EIA-clock remain encouraging since they compare favorably with those of a N-resonance atomic clock based on a 7.5-cm long vapor cell [31] or other CPT-based clock prototypes [18,60,61]. For further progress, the laser FM noise contribution to the short-term stability budget could be reduced by stabilizing the laser using sub-Doppler spectroscopy techniques [56,62]. In addition, we suppose that the EIA clock should benefit from the use of Ramsey-based interrogation protocols, both for reduction of the photon shot noise contribution since higher laser powers might be used and mitigation of light shift issues [28,29,63–65].
4. Conclusions
We have proposed a dual-frequency optical field configuration scheme that allows the detection of narrow-linewidth EIA resonances in a buffer-gas filled alkali vapor cell. At the contrary of previously-reported EIA schemes, the EIA resonance detection is achieved in the present setup by tuning the laser frequency difference between both optical fields at fixed static magnetic field. Microwave frequency stabilization onto a sub-kHz linewidth EIA resonance was performed in order to achieve an atomic clock. The EIA atomic clock exhibits a preliminary Allan deviation of $5.8 \times 10^{-12}\tau ^{-1/2}$ up to $20$ s, in correct agreement with the signal-to-noise ratio and linewidth of the detected resonance. Exploration paths to be studied in the future were suggested to optimize further the EIA clock fractional frequency stability performances.
Funding
Russian Science Foundation (17-72-20089); Agence Nationale de la Recherche (10-LABX-0048); Agence Nationale de la Recherche (11-EQPX-0033); Conseil régional de Bourgogne-Franche-Comté.
Acknowledgments
We thank Nikolay Kvashnin, Vadim Vasiliev and Alexey Lugovoy (Institute of Laser Physics SB RAS) for their help with electronics and software. R. Boudot thanks Jacques Millo, Enrico Rubiola and Vincent Giordano (FEMTO-ST) for fruitful discussions.
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