Atomic superheterodyne receiver Sensitivity estimation based on homodyne readout

A schematic diagram of the Rydberg atomic superheterodyne receiver based on homodyne readout is shown in Fig. 1. It can be seen that compared with the Rydberg atomic heterodyne receiver based on intensity readout, the Rydberg atomic heterodyne receiver based on homodyne readout studied in this paper uses the MZI in the transmitted laser readout. The probing laser and coupling laser are locked on the desired wavelength and simultaneously excite the atoms in the atomic vapor cell. When the microwave signal radiates cross the atomic vapor cell, the atomic vapor cell changes the phase of transmitted probing laser, which is superimposed with the reference light through a beam splitter. Finally, the photodetector receives the mixed light for subsequent signal processing. The system response is sensitive to the direction of polarization. It has been demonstrated that the microwave electric field polarization can also be obtained by atomic system[15]. In this work, we assume that the lasers and microwave signal have the same polarization.

In order to obtain a narrower laser linewidth and lower power fluctuations, the actual system is more complex. Fig. 2 shows the proposed system setup. The 852nm laser uses a frequency stabilization scheme based on atomic absorption spectroscopy, while the 509nm laser uses a frequency stabilization scheme based on EIT effect of atomic absorption spectra. The two lasers pass through the power stabilization module and then enter the detection atomic vapor cell to excite the atoms to the Rydberg state. Under the radiation of electric field, the atomic vapor cell changes the phase of the transmitted laser, which is received by the photodetector through a MZI. The frequency stabilization scheme can be replaced by other higher performance schemes, such as ultra-low expansion cavity scheme.

The diagram for a typical atomic four-level structure is shown in Fig. 3. The probing laser and coupling laser are tuned with detuning $\Delta_{p}$ and $\Delta_{c}$, respectively; and the local microwave electric field is tuned with detuning $\Delta_{L}$. Assume that the frequency and phase difference between local microwave electric field and signal microwave electric field is $\delta_{s}$ and $\phi_{s}$, respectively. The corresponding Hamiltonian in the rotating frame is given by

where $\Omega_{p}$ and $\Omega_{c}$ are the corresponding Rabi frequencies for transitions $\left.|1\right>\rightarrow\left.|2\right>$ and $\left.|2\right>\rightarrow\left.|3\right>$, respectively. The local microwave electric field and signal microwave electric field couple with Rydberg transition between state $\left.|3\right>$ and state $\left.|4\right>$, with Rabi frequencies $\Omega_{L}$ and $\Omega_{s}$, respectively. Let $\Omega=|\Omega_{L}+\Omega_{s}e^{-iS\left(t\right)}|$, where $S\left(t\right)=2\pi\delta_{s}t+\phi_{s}$ is the cumulative phase difference of the signal microwave relative to the local microwave.

Due to the relaxation effect caused by the spontaneous radiation, collision of atoms and transit relaxation effect, as well as the atoms repopulation, the complete Liouville equation for the rotating-frame density matrix of the system can be written as [16]

where $\rho$ is the density matrix. Relaxation matrix $\Gamma$, which represents the relaxation rate of each state, is given by

where $\gamma_{2}$, $\gamma_{3}$ and $\gamma_{4}$ represent the spontaneous decay rates of atoms on the three high levels; $\gamma_{c}$ and $\gamma$ are the relaxation rates of the atoms collision and transit effect, respectively. We assume that each level undergoes the same relaxation rate $\gamma$ due to the exit of atoms from the laser beam.

In addition, the atoms that spontaneously decay from the upper states also repopulate the lower sates. In this work, we only consider the decay paths shown in Fig. 3. Repopulation matrix $\Lambda$ is given by

The dynamics equation of the atomic system can be expressed as

where $L=-\frac{1}{2}\left(\Gamma\rho+\rho\Gamma\right)+\Lambda$ represents the Lindbladian operator that defines the relaxation progress in the system, which matrix form is given by

We begin with analytical derivations based on cold atoms. Within the adiabatic approximation, the probe laser transmission associated with the instantaneous steady state is written in terms of the imaginary component of the susceptibility as $P\left(t\right)=P_{i}e^{-k_{p}L\Im\left(\chi\right)}$. Here, $P_{i}$ is the incident light power, $L$ is the length of atomic vapor cell and $k_{p}=2\pi/\lambda_{p}$ is the probing laser wavevector [4]. The linear susceptibility can be given by

where $N_{0}$ is the total density of atoms, $\mu_{21}$ is the dipole moment of transition $|1\rangle\rightarrow|2\rangle$ and $\epsilon_{0}$ is the permittivity in vacuum, and $\rho_{21}$ is the density matrix element that describes the atomic coherence between the ground and first excited state and is found by solving Eq. 5 in steady state.

The presence of the atomic ensemble causes absorption and phase shifts on the probe light that depend on the density matrix via the differential equations [17]

and

We analyze the thin optical depth case and ignore the space dependency of $\Omega_{p}$. This yields the phase difference through the atomic vapor cell,

Then, the output electric field through the atomic medium is given by

In the framework of MZI, the electric field of signal arm (through the atomic vapor cell) and reference arm received by the photodetector are given by

where $E_{1}$ and $L_{\text{SIG}}$ are the incident electric field and optical path length of signal arm, and $E_{2}$ and $L_{\text{REF}}$ are the incident electric field and optical path length of reference arm; $n_{0}$ is the refractive index. The differential signal of the balanced homodyne detectors (not shown in Fig. 2) is given by [14]

According to the signal of photodetector, the small phase shift $\Delta{\phi}$ can be inferred. The strength of signal electric field can be extracted based on Eq. (10) and the relationship between $\Re({\chi})$ and $\Omega_{s}$, which is investigated in this work. Based on the steady-state solution of the system Liouville equation, we derive the expression of the linear susceptibility and phase shift, and then find the optimal values.

In experimental systems, in order to reduce the interaction and collision rate between atoms, the atomic density and laser power are usually limited, although increasing the laser power can theoretically increase the signal-to-noise ratio of the output signal and improve the sensitivity. Therefore, in the current experimental setups, the detection laser power is usually at a weak level, and the coupled laser power is also limited. For Rydberg atom sensors using EIT readout, the optimal Rabi frequencies for transitions $|1\rangle\rightarrow|2\rangle$ and $|2\rangle\rightarrow|3\rangle$ depend on the decay rate of each state and relaxation rate caused by atom collision [17]. We analyze the influence of laser frequencies and local microwave electric field frequency detuning on the system sensitivity.

We set a detuning parameter $\Delta$ (which can be $\Delta_{p}$, $\Delta_{c}$, or $\Delta_{L}$). Assuming $\gamma_{3}=\gamma_{4}=\gamma_{c}=\gamma=0$, the steady-state solution $\rho_{21}\left(\Delta\right)$ can be obtained based on Eq. (5). Assuming $\Omega_{s}\ll\Omega_{L}$, the output optical power has the formalism as follows,

where $\kappa$ is a conversion coefficient [4]. Similarly, the corresponding linear susceptibility real component of the atomic medium can be written as

which has a time-invariant part $\chi_{0}\left(\Delta\right)$ and the coefficient of time variant part $\chi_{1}\left(\Delta\right)$. The first part contributes a constant phase shift, and the second part represents a phase variation that depends on the electric field strength.

The probing laser transmission phase shift is associated with the real component of the susceptibility as

Fig. 4 shows the transmitted laser phase shift evolution over time with different signal electric field strengths.

Since the amplitude of microwave signal is linearly dependent on its corresponding Rabi frequency $\Omega_{s}$, we can optimize coefficient $\chi_{1}\left(\Delta\right)$ in Eq. (16) to maximize the system sensitivity. It should be noted that the case of weak signal corresponds to small variation of probing laser transmission power. We analyze phase shift individually and do not consider laser power variation with time.

Assuming that $\Delta_{c}=\Delta_{p}=\gamma_{3}=\gamma_{4}=0$, the steady-state solution $\rho_{21}(\Delta_{L})$ can be obtained according to Eq. (5) as follows,

The real component is given by

In the weak signal regime, $\Omega_{s}\ll\Omega_{L}$, the first order approximation of $\Omega_{s}/\Omega_{L}$ is given by,

where $C_{L-DC}$ and $C_{L-AC}$ are as follows,

and

The corresponding linear susceptibility of the atomic system can be written as

which has a time-invariant part,

and a time variant part coefficient,

(P1): Sensitivity Maximization in General Case via Local Microwave Detuning

Assuming that $\Delta_{c}=\Delta_{L}=\gamma_{3}=\gamma_{4}=\gamma_{c}=\gamma=0$, the steady-state solution $\rho_{21}(\Delta_{p})$ can be obtained according to Eq. (5) as follows,

The real component is given by

In weak signal regime, $\Omega_{s}\ll\Omega_{L}$, the first order approximation of $\Omega_{s}/\Omega_{L}$ is given by,

where $C_{P-DC}$ and $C_{P-AC}$ are as follows,

and

The corresponding real component of linear susceptibility of the atomic system can be written as

which has a time-invariant part,

and a time variant part coefficient,

(P2): Sensitivity Maximization in General Case via Probing Laser Frequency Detuning

Assuming that $\Delta_{p}=\Delta_{L}=\gamma_{3}=\gamma_{4}=\gamma_{c}=\gamma=0$, the steady-state solution $\rho_{21}(\Delta_{c})$ can be obtained according to Eq. (5) as follows,

The real component is given by

In weak signal regime, $\Omega_{s}\ll\Omega_{L}$, the first order approximation of $\Omega_{s}/\Omega_{L}$ is given by,

where $C_{C-DC}$ and $C_{C-AC}$ are as follows,

and

The corresponding real component of linear susceptibility $\chi\left(t,\Delta_{c}\right)$ of the atomic system can be written as

which has a time-invariant part,

and a time variant part coefficient,

(P3): Sensitivity Maximization in General Case via Coupling Laser Detuning

We numerically characterize the frequency detuning effect on the weak signal response in Section IV.

The power sensitivity of a conventional receiver is usually expressed by the minimum power received by the antenna connector to guarantee certain throughput or BER (bit-error-rate) requirements. Assuming that the length of the atomic vapor cell is $L$, the probing laser beam diameter is $d$, the effective signal acquisition area $A_{\text{eff}}$ equals to L times d. The equivalent signal power sensitivity is given by

where $\varepsilon_{0}$ and $c$ are the dielectric constant and speed of light in vacuum, $E_{min}$ is the minimum detectable electric field strength, and $\vec{S}$ is the Poynting vector of electromagnetic field.

Theoretically, the minimum detectable electric field strength corresponding to the shot noise limit is given by [18]

where $T$ is the integration time, $N$ is the atom involved in the measurement in $T$ time, and $h$ and $\mu_{E}$ represent Planck constant and transition dipole momentum coupled electric field, respectively. Considering the limit of the actual EIT (electromagnetic induced transparency) process on time $T$ ($T$ cannot be longer than the $T_{2}$ time of the EIT process), a more useful form can be obtained. The measurement that occurs within $T^{{}^{\prime}}$ can be understood as $N=N_{at}T^{{}^{\prime}}/T_{2}$, where $N_{at}$ is the number of atoms involved in the measurement. The minimum detectable electric field strength of the EIT process can be expressed as

Taking $T^{{}^{\prime}}=1$ s, we can get the time-normalized electric field detection sensitivity as

The Rydberg atomic heterodyne receiver has achieved great improvement in sensitivity, the system electric field measurement sensitivity obtained in the experiment reached $55$ nV$\cdot$cm^-1Hz^-1/2 [4]. Using the power sensitivity calculation method in Eq. (44), the power sensitivity of the system is approximately -144.7 dBm$\cdot$Hz^-1 based on the data provided in the literature. Based on the Rydberg atomic heterodyne receiver, the optimal electric field measurement sensitivity obtained by combining coupled laser frequency detuning is about $12.5$ nV$\cdot$cm^-1Hz^-1/2 [5], which corresponds to a power sensitivity about $-157.5$ dBm$\cdot$Hz^-1 (assuming that the vapor cell size and laser beam diameter parameters are consistent with the parameters in the work of Jing et al. [4]). The detection sensitivity of the Rydberg atomic heterodyne receiver is mainly limited by the thermal noise of electronic devices, and the technical noise caused by atomic transition broadening and laser linewidth. The atomic transition broadening can be reduced by amplifying the probing laser and coupling laser beam waist. Replacing balanced detection with homodyne or heterodyne detection can reduce photodetector noise. Using a more stable laser source can eliminate laser frequency noise. These factors can be combined to allow the system to approximate the quantum projection noise limit of atomic sensors, about $700$ pV$\cdot$cm^-1Hz^-1/2 [4]. The power detection sensitivity corresponding to the quantum projection noise limit is about $-182.6$ dBm$\cdot$Hz^-1.

As described in Section V.A, intensity-based readout methods have achieved great improvement in sensitivity, but are still far from the theoretical limit. In this paper, the performance of the zero-difference readout method is evaluated, and the relationship between the phase conversion coefficient and frequency detuning under different local oscillator electric field strengths is obtained. Referring to the results of the phase noise, the corresponding minimum detectable electric field strength can be given to calculate the power sensitivity.

We model an ideal measurement by assuming a photon-shot-noise limited homodyne measurement of the acquired phase $\phi$ on the transmitted probe light, where $\phi$ is calculated by Eq. (16). The noise density $phi_{N}$ in measuring this quantity for a coherent state is defined as

where detection efficiency assumed to be unity in this work.

The SNR (signal-to-noise ratio) density is given by

Assume that readout requirement is $0$ dB, the minimum detectable electric field is given by

Numerical results in Fig. 5, Fig. 6, and Fig. 7 show that the largest conversion coefficient is obtained in the probing laser frequency detuning case with $\Omega_{L}=2\pi\times 6$ MHz, $\chi_{1}(\Delta_{p})=1.430\times 10^{-7}$. According to Eq. (50), the electric field detection sensitivity is $0.185$ nV$\cdot$cm^-1Hz^-1/2, and the corresponding power detection sensitivity is $-194.1$ dBm$\cdot$Hz^-1. The phase-based readout method improves the sensitivity with $36.6$ dB gain. It is important to note that we consider the sensitivity improvement of three frequency components detuning independently for three arbitrarily selected local oscillator electric field strengths. The results obtained are not theoretically optimal sensitivity of the proposed method in this work.