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Abstract
Coherent population trapping (CPT) resonance signals have promise in a wide range of applications involving precision sensing. Generally, the CPT phenomenon occurs in a three-level Λ system with a bichromatic phase-coherent light fields. We theoretically and experimentally studied an Rb vapor-cell-based atomic system involving bichromatic CPT optical fields and an external microwave (MW) field simultaneously. In such a mixing scheme, the coherence of the ground states could be controlled either by the Rabi frequency of the microwave field or by the relative phase between the optical fields and the MW field. Moreover, we investigated the Rabi resonance in this mixing scheme. The Rabi frequency of the MW field can be measured SI (International System of Units)-traceably based on the Rabi resonance lineshape, and thus holds the potential to realize intensity stabilization of the optical field in this system. Simple theoretical models and numerical calculations are also presented to explain the experimental results. There is scope to use the proposed technique in future development of SI-traceable optical field strength standards.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Coherent population trapping (CPT) is a type of quantum interference process that can usually be described by a simple three-level $\Lambda$ system, which can easily be found in the D lines of alkali atoms (Cs, Rb, K, etc.) [1–4]. It occurs when two phase-coherent optical lines couple two ground states of an atomic species to a common excited state. CPT resonance signals have been widely investigated for precision sensing with potential applications in microwave atomic clocks based on vapor cells and cold atoms, high-resolution spectroscopy, magnetometers, terahertz references, atomic interferometers, phase-sensitive amplification, light storage, and slow light [5–18]. In these CPT-based applications, the microwave (MW) interrogation that probes the atoms is usually optically carried, and the MW field is not involved. There are also various experiments have investigated mixing of optical fields and an MW field simultaneously in a $\Lambda$ system, such as four-wave mixing (FWM) realized in a vapor cell [19,20], an electromagnetically induced transparency (EIT) transmission signal controlled by an MW field [21], and a solid system driven by an MW field [22].
The CPT optical fields are highly phase-coherent and have equal Rabi frequencies. Herein, we theoretically and experimentally investigate an atomic system simultaneously involving CPT light fields and an MW field coupled to the hyperfine levels of the alkali atoms. The coherence of the ground state can be controlled by both the strength and relative phase of the optical fields and MW field. Thus, the CPT absorption/transmission field in an atomic vapor cell can be adjusted by an external MW field. With this feature, we also studied Rabi resonance [23–25] in the CPT-MW mixing scheme. In our previous works [26,27], an SI-traceable MW field detection technique is reported using Rabi resonance with two hyperfine levels of the ground states of alkali atoms ($^{87}$Rb and $^{133}$Cs). A notable advantage of our technique is that the field strength can be translated into a Rabi frequency via atomic constants, making this technique a candidate method to link MW quantities with SI units [28]. This Rabi resonance-based technique can also be used to stabilize the field strength in a manner similar to a frequency stabilization to a resonant transition of atoms [23], which is the so-called “atomic candle".
Although we have demonstrated a Rabi resonance-based experimental system with an Rb-Cs hybrid vapor cell in a bandwidth of 4.8 GHz around 8.1 GHz [27], the frequency range is still limited in the MW band. A possible method of extending the Rabi resonance-based technique to the optical band is to directly interrogate the ground state and excited state of alkali atoms with a phase-modulated resonant optical field. However, the laser intensity must be greater than 10 W/cm$^{2}$ to obtain a sufficient signal-to-noise ratio [23]. It is not common to have such high-power laser sources in research labs and application scenarios. The Rabi resonance in the CPT-MW mixing scheme presents an alternative approach for implementing the Rabi resonance technique in optical field applications, and shows potential for stabilizing the optical field strength by controlling and measuring the Rabi frequency of the external MW field.
Here, we investigate the Rabi resonance behavior in the CPT-MW mixing scheme. The MW field is phase-modulated at $\omega _m$ and coupled with the hyperfine splitting of the ground state, whereas the phase-coherent optical fields are coupled with both the ground state and excited state. The probability of the population in the excited state oscillates at 2$\omega _{m}$ and is determined by the MW field strength interaction with the atoms [23–25].
There are additional interesting potential applications for the Rabi resonance signal in the CPT-MW mixing scheme. The optical field in this mixing scheme can be considered as a dispersion-controllable slow light. Absorption coefficients and indices of refraction can be precisely measured based on the measured Rabi frequency of the MW field. The Rabi resonance in the CPT-MW mixing scheme can be also used to measure and control the effective MW strength in the MW cavity, which can play an important role in calibrating the MW field coupling strength. MW–optical frequency conversions were studied in the similar $\Lambda$ levels based on an atomic vapor cell [19], which demonstrates the possibility of an MW–optical power/frequency conversion within the same atomic system . If we applied phase modulation to the MW field in this FWM scheme, we could also code the information of the MW band to the optical band. It also holds the potential to explore applications for precision metrology based on optical/MW interrogation and MW–optical transduction of qubits [29–32].
Section 2 of this paper presents the theoretical analysis of CPT controlled by an MW field and Rabi resonance in the CPT-MW mixing scheme. Section 3 describes our experimental apparatus. In Sec. 4, we report the experimental results that demonstrate CPT transmission signal variation by tuning the phase and strength of the MW field, and the Rabi resonance lineshapes of the MW-controlled optical transmission signals are plotted. The results are characterized and show interesting potential in developing atomic applications. The paper is concluded by Sec. 5, with a discussion and a summary of the study.
2. Theory
2.1 CPT controlled by MW field
First, we consider two optical fields and a non-phase-modulated MW field coupled with three-level $^{87}$Rb atoms, as sketched in Fig. 1. Two circularly polarized optical fields, $\Omega _{1}$ and $\Omega _{2}$, are propagated along the $z$-axis of a vapor cell. The vapor cell filled with $^{87}$Rb is placed in an MW cavity with an MW field $\Omega _{\mu }$ applied. The Hamiltonian of the light-atom interaction can be written as $(\hbar =1)$:
Fig. 1. Energy levels of the $\Lambda$ system. $\Omega _{1}$ and $\Omega _{2}$ are the Rabi frequencies of the bichromatic light fields.
In the rotating frame, the Hamiltonian is simplified as
The Hamiltonian shows the interaction between the three fields and atoms. Atoms can be trapped in the dark state undergoing the interaction with two optical fields, and can also be pulled out of the dark state in the presence of an MW field. Thus, to simplify the calculation, a new set of basic vectors is used to acquire the density matrix elements, which consist of the dark state $\left |a\right \rangle =\frac {\sqrt {2}}{2}(\left |1\right \rangle -\left |2\right \rangle )$, bright state $\left |b\right \rangle =\frac {\sqrt {2}}{2}(\left |1\right \rangle +\left |2\right \rangle )$, and excited state $\left |c\right \rangle =\left |3\right \rangle$. Then, to solve the absorption and dispersion of the atomic media, the steady-state solution of the atomic density matrix elements is calculated by the master equation
To obtain the transmission of optical fields based on the results calculated above, the Maxwell–Schrodinger equation is used to describe the propagation of optical fields in the medium as [33]:
From Eq. (7), it is obvious that optical field transmission is influenced by the MW coupling, as $\langle \rho _{\mathrm {b}c}\rangle$ is a function of $\Omega _{\mu }$ and $\phi _{\mu }$. Thus, we can adjust the amplitude and phase of the MW field to control the optical field transmission; the numerical results of the optical field transmission with different relative phases $\phi _{\mu }$ are shown in Fig. 2. The optical field transmission demonstrated a periodic variation with different relative phases $\phi _{\mu }$. The result can be explained by the fact that atoms are trapped in the dark state when the relative phase is ${n}\pi$, and the coherence is destroyed completely when the relative phase is $\frac {1}{2}{n} \pi$, where $n\in \mathbb {Z}$ is an integer.
Fig. 2. Numerical results of the optical transmission field dependence on the relative phase between the CPT optical fields and the MW field $\phi _{\mu }$, with $\Omega _{\mu }=0.01\,\mathrm {MHz}$, $\Omega (0)=1\,\mathrm {MHz}$, $\Gamma =2\pi \times 6\,\mathrm {MHz}$, $\gamma _{12}=10^{-3}\Gamma$, $\gamma _{13}=\gamma _{23}=\Gamma /2$, $\Delta _{3}=2\,\mathrm {MHz}$ , $T=300\,\mathrm {K}$.
2.2 Rabi resonance in the CPT-MW mixing scheme
Based on the above analysis, we can determine that optical transmission varies periodically with MW phase. Thus, if the MW phase is modulated with modulation frequency $\omega _{m}$, we speculate that the optical transmission will also oscillate. Then, inspired by the study of Rabi resonance, the optical transmission amplitude with oscillation frequency $2\omega _{m}$ is investigated. The oscillation of the optical transmission results from the oscillation of the atomic density matrix elements, which is caused by the modulated MW field. The Hamiltonian with a modulated MW field should be same as in Eq. (3), where the phase is replaced with
Here, the initial phase of the MW field is $\phi _{\mu }$ and the modulated phase is $\xi \sin (\omega _{m}{t})$, where $\xi$ represents the modulation index and $\omega _{m}$ is the modulation frequency. According to the Jacobi-Anger expansion $e^{\pm i\xi \mathrm {sin}\theta }=J_{0}(\xi )+2\sum _{n=1}^{\infty }J_{2n}(\xi )\mathrm {cos}(2n\theta )\pm 2i\sum _{n=0}^{\infty }J_{2n+1}(\xi )\mathrm {sin}((2n+1)\theta )$, where $J_{n}$ is the $n$-th Bessel function, the system Hamilton can be rewritten as:Fig. 3. The simulation result of the Rabi resonance lineshape in the CPT-MW mixing scheme, which is the ensemble average result considering the Doppler effect. In the calculation, we take $\gamma _{13}=\gamma _{23}=\pi \times 5.7MHz,\gamma _{12}=10^{-3}\gamma _{13},\Omega _{1}=\Omega _{2}=2MHz,\Delta _{3}=1MHz,\phi _{\mu }=\pi /4,\xi =0.001$.
3. Experimental setup
Figure 4 shows a schematic diagram of the experimental apparatus. Most CPT clock experiments were performed with bichromatic light fields consisting of two phase-locking lasers. The phase coherence of the CPT light fields eliminates the population trapped in the dark state [10,34]. In our configuration, the phase coherence between the optical fields and the MW field also enhances or destroys the ground-state coherence. We develop a simple architecture based on a single laser source and a single MW source to generate high phase coherence between the fields.
Fig. 4. Experimental apparatus. ECDL: external cavity diode laser; Sat. Abs.: saturated absorption laser stabilization; EOM: electro-optical modulator; QWP: quarter waveplate; PD: photodiode; DDS: direct digital synthesizer; FFT: fast Fourier transform.
The laser source is a 100 kHz linewidth external cavity diode laser (ECDL) tuned to the $^{87}$Rb D1 transition at 795 nm. The laser frequency is stabilized at the F=2 to F${'}$=2 optical transition by saturated absorption configuration with an auxiliary Rb-filled vapor cell. The bichromatic CPT light fields are generated by a 20-GHz-bandwidth polarization-maintaining pigtailed electro-optical modulator (EOM) modulated at 6.835 GHz. The single laser and EOM setup ensure high phase coherence between the sideband and carrier signal [6,11,12]. The carrier signal and negative first-order sideband form the CPT $\Lambda$-system of $^{87}$Rb atoms. The EOM drive signal is divided from a low-noise commercial synthesizer. The power of the first-order sidebands is equal to the carrier signal, so as to maximize the CPT transmission signal [6].
The bichromatic beam is circularly polarized by a quarter-wave plate and the beam diameter is 6 mm. The laser beam is then directly sent to a physical package. The physical package contains a quasi-TE011 MW cavity and a 3-cm-long and 2-cm-diameter vapor cell filled with Rb-$40$ Torr N$_{2}$ buffer gas. The buffer gas is often used to eliminate the relaxation rate of alkali atoms. The vapor cell is temperature-stabilized at 70 $^{\circ }$C and surrounded by a static magnetic field of 120 mG parallel to the beam direction to lift the Zeeman degeneracy. The MW signal applied on the MW cavity is generated from the same synthesizer that drives the EOM, making the MW signal phase-coherent with the CPT bichromatic light fields. The cavity is isolated from external electromagnetic perturbations by a $\mu$-metal shield.
As presented above, two methods are used to adjust the relative phase between the optical fields and MW field. One way is to fix the physical package at a translated plate, and change the relative phase by adjusting the position of the physical package along the optical axis. Alternatively, the phase of the MW signal sent to the cavity can be changed using a voltage-controlled phase shifter. The powers of the MW signals applied on the EOM and cavity are also both controlled by the MW power attenuators connected to the MW splitter. The Rabi resonance is realized by phase-modulating the MW field injected into the cavity. A direct digital synthesizer (DDS) generates a modulation signal of $\omega _{m}$ and sends it to the phase shifter. By scanning $\omega _{m}$, we can obtain the Rabi resonance signal and measure the Rabi frequency of the field.
In the absence of the MW field, the atoms are trapped in the dark state. The light is transmitted through the vapor, and the transmission signal is received by a photodetector. When the resonance MW field is presented, atoms are pumped out from the dark state, and the transmission signal is decreased. Any MW-induced oscillation of the population will be observed as oscillation in the optical transmission signal and can be extracted by a FFT spectrum analyzer.
4. Results and discussion
4.1 CPT transmission signal controlled by MW field
Figure 5(a) shows a conventional CPT transmission resonance signal without an MW field. The amplitude of transmission reflects the completeness of dark state formation and dominance by the phase coherence of hyperfine levels [12]. In the CPT-MW mixing scheme, the hyperfine level coherence can be controlled by the MW field. A plot of optical transmission signal versus detuning with various MW field amplitudes is shown in Fig. 5(b). The presence of the MW field affects the optical transmission as predicted by the theoretical analysis (Eq. (7)). The peak amplitude of the lineshape decreases with increasing MW power as expected. The depth of the lineshape dip becomes deeper with increasing MW power, indicating that the atoms trapped in the dark state by the CPT light fields are pumped out in the presence of the MW field. Fig. 4(b) also indicates that the linewidth of CPT resonance is narrower than that of the resonance with the MW field. The asymmetry of the resonance signals is partially due to the detuning and disequilibrium of CPT transitions [11].
Fig. 5. Optical transmission signal as a function of detuning. (a) Conventional CPT transmission signal. (b) Optical transmission signal in the CPT-MW mixing scheme for MW powers of $-31.15$ dBm (black), $-22.74$ dBm (red), and $-19.79$ dBm (blue).
The relation between the optical transmission signal and relative phase between the optical fields and MW field is plotted in Fig. 6. The relative phase is modified by moving the physical package (Fig. 6(a)) or adjusting the MW phase shifter (Fig. 6(b)). The period of variation in Fig. 6(a) is measured to be 2.2 cm, which is half the wavelength of the hyperfine-level MW transition. In Fig. 6(b), the MW field phase is tuned by a voltage-control phase shifter connected to the MW cavity, and the phase of the optical field is fixed in this case. Both methods demonstrate that the transmission signal can be enhanced or suppressed by controlling the relative phase of the fields. The experimental results show similar behavior to the numerical results in Fig. 2.
Fig. 6. Amplitude of optical transmission peaks as a function of (a) position of the physical package and (b) phase shift of the MW field.
Based on the theoretical model Eq. (7), the relative phase variation not only controls the transmission amplitude of the resonance signal, but also shifts the frequencies of resonance signals. Figure 7 shows the experimental and simulation results of CPT resonance signals for different relative phases. The relative phase in Fig. 7(a) is adjusted by the MW phase shifter, and the CPT amplitude data in Fig. 6(b) is extracted from Fig. 7(a).
Fig. 7. Experimental (a) and simulation (b) results of CPT resonance for various relative phases. The relative phase is adjusted by the MW phase shifter.
4.2 Rabi resonance in CPT-MW mixing scheme
The basic principle and measurement approach of Rabi resonance are similar to those of [26][27]. Here, we plot the Rabi resonance lineshapes in the CPT-MW mixing scheme with our experimental setup.
The Rabi resonance lineshapes for various MW field powers in the CPT-MW mixing scheme are plotted by scanning MW phase modulation frequency $\omega _{m}$, as shown in Fig. 8(a). The MW power is varied by tuning a voltage-controlled attenuator in the path connecting the cavity; the powers of the input MW signal to the cell are -31.15 dBm, -27.66 dBm, -22.74 dBm. -20.90 dBm, and -19.79 dBm, respectively. The values of the phase modulation frequency $\omega _{m}$ corresponding to the peak of the each lineshape are 7.5 kHz, 10 kHz, 14 kHz, 17 kHz, and 22.4 kHz, respectively. The relative phase between the optical fields and MW field is adjusted by $\frac {1}{4}\pi$ by moving the physical package.
Fig. 8. (a) Rabi resonance lineshapes of optical transmission signals in the CPT-MW mixing scheme for different cavity input MW powers. (b) Measured Rabi frequency as a function of input RF power. (c)Rabi Frequency of the transmission optical field as a function of the measured MW Rabi frequency.
The measured Rabi frequency of the MW field can be derived from the $\omega _{m}$ corresponding to the peak position of each resonance lineshape. In a relatively low-power regime, the peak amplitude of the Rabi resonance increases with increasing MW power, indicating that the CPT coherence is destroyed and that more atoms are involved in this Rabi resonance process. The lineshape broadens inhomogeneously compared to the calculated results because the atoms cannot sample the field’s spatial distribution on the time scale of a Rabi period. For a higher-MW-power regime, the increased power-broadening effect reduces the population imbalances of hyperfine levels and hence decreases the peak amplitude of resonance. Nevertheless, the MW Rabi frequency determined in the measurement results agrees with the calculated result from the theoretical model. Fig. 8(b) plots the measured Rabi frequencies (extracted from Fig. 8(a)) as a function of $\sqrt {{P}_{\mathrm in}}$ of the MW field, where ${P}_{\mathrm in}$ is the MW incident power. The measured Rabi frequencies are linearly dependent on the strength of the MW signal, which confirms that the MW field measurement based on the Rabi resonance is applicable for the CPT-MW mixing scheme.
Examining the relation between the optical transmission fields and the MW field using the measured Rabi frequency of the MW field shows a possible approach for stabilizing the strength of the optical field. The measured Rabi frequency of the MW field can be converted to the Rabi frequency of the optical transmission field based on our theoretical model. Fig. 8(c) shows the relation between the Rabi resonance-based measured MW Rabi frequency ($\Omega _{\mu }=2\omega _{m}$) and the Rabi frequency of the optical transmission field in the CPT-MW mixing scheme, assuming the Rabi frequency of the initial input optical field is 4 MHz with uncertainty of 0.5$\%$ and the relative phase is $\pi$/4. It indicates that the Rabi frequency of the optical transmission field can be expressed by the Rabi frequency of the MW field $\Omega _{\mu }$ and the phase modulation frequency of the MW field $\omega _{m}$. The strength of the optical transmission field decreases when the MW field increases, and a similar dependence is shown in Fig. 5(b). The “atomic candle" for stabilization of MW field strength is presented in [23], and the results here suggest the possibility of employing an “optical atomic candle" in the CPT-MW mixing scheme by detecting and adjusting the Rabi frequency of the MW field via the Rabi resonance technique. The phase modulation frequency can be used to generate an error signal indicating optical transmission field fluctuations, after which the MW strength can be adjusted to stabilize the optical field.
The stability of such an “optical atomic candle" system usually can be predicted as:
where $S/N$ is the signal-to-noise ratio in a 1 Hz bandwidth and $\tau$ is the integration time.In the CPT-MW mixing scheme, the relation between the MW field and the optical transmission field is fixed with a certain relative phase, and the principle to stabilize the optical field is to adjust the MW field. Therefore, we consider the Q factor of the Rabi resonance lineshape of the MW field shown in Figure 8(a), and it is given as [23] $Q=\Delta \nu /\Omega _{\mu }=\frac {2\omega _{m}}{\gamma _{1}\sqrt {3}}$. Here, the Q factor is only related to the MW transition in the mixing scheme. The $S/N$ is related to the population involving the CPT and Rabi resonance process; here, we use the 0-0 sublevels and F’=2 to form the $\Lambda$-system. Thus, the stability of such an “optical atomic candle" could be improved by reducing the relaxation rate of the vapor cell and increasing the $S/N$. Optimized interrogation schemes (push–pull optical pumping, lin$\perp$lin, lin$\parallel$lin, $\sigma ^{+}$–$\sigma ^{-}$, etc.) [35–38] and laser-cooling atoms could significantly improve the $S/N$ of the resonance signal and reduce the relaxation introduced by the collision in the vapor cell [11,12].
It is also interesting to note that the Rabi resonance lineshape may lead to the development of other applications in the CPT-MW mixing scheme. The Rabi resonance peak response also depends on the relative phase between the optical fields and MW field $\phi$, and the peak amplitude shows an oscillation with the relative phase variation. This feature enables a conversion between the phase of the bichromatic optical field and the MW field with Rabi resonance, and could realize phase conversion. Moreover, the refractive index $n$ of the atomic vapor is determined by the density matrices $\rho _{13}$ and $\rho _{23}$, which can be controlled by the Rabi frequency of the MW field $\Omega _\mu$. Thus, this atomic system can be used as an atomic dispersion controller/compensator.
5. Conclusions
We have theoretically and experimentally studied optical CPT transmission signals controlled by an MW field in the CPT-MW mixing scheme. Moreover, the Rabi resonance in a $\Lambda$-system was investigated, with a phase-modulated MW field coupled to two hyperfine levels of $^{87}$Rb in this case. Based on the Rabi resonance response in the CPT-MW mixing scheme, the Rabi frequency of the MW field was measured. Combined with the relation between the optical fields and MW field, this feature can be used to realize phase conversions and controllable dispersion in this atomic system. We developed an experimental system combining a single laser, an EOM, and a physical package including a quasi-TE011 cavity and a Rb vapor cell filled with 40 Torr N$_{2}$. CPT optical transmission signal variations resulting from phase adjustment of the optical fields and MW field were reported. The Rabi resonance lineshape in the $\Lambda$-system was plotted. The theoretical simulation results show behavior similar to the experimental lineshape. The dark-state coherence and Rabi resonance lineshape strongly depend on the relaxation mechanism of the atoms. The population trapped in the dark state could also be improved by optimized interrogation schemes. Therefore, using cold atoms and a double-$\Lambda$-scheme, it may be possible to develop future SI-traceable optical field strength sensors/standards based on this Rabi resonance technique with the CPT-MW mixing scheme.
Funding
National Natural Science Foundation of China (61975194).
Acknowledgments
We are grateful to Dr. Fuyu Sun and Dr. Pengfei Wang for their many fruitful discussions and contributions to the design of the MW cavity.
Disclosures
The authors declare no conflicts of interest.
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