# Atomic superheterodyne receiver Sensitivity estimation based on homodyne
readout

Shanchi Wu, Chen Gong, Rui Ni Shanchi Wu, Chen Gong are with University of
Science and Technology of China, Email address<EMAIL_ADDRESS><EMAIL_ADDRESS>Ni is with Huawei Technology, Email address:
<EMAIL_ADDRESS>
###### Abstract

The electric field measurement sensitivity based on the Rydberg atomic vapor
cell has great theoretical advantages over traditional dipole antennas. We
combine the Rydberg atomic heterodyne receiver and the Mach-Zehnder
interferometer (MZI) with high phase detection sensitivity to evaluate the
system reception sensitivity based on the transmitted laser phase shift. We
conduct a theoretical investigation into the impacts of local microwave
electric field frequency detuning, and laser frequency detuning on enhancing
the sensitivity of heterodyne Rydberg atomic receiver based on MZI. To
optimize the output signal’s amplitude given the input microwave signal, we
derive the steady-state solutions of the atomic density matrix. Numerical
results show that laser frequency detuning and local microwave electric field
frequency detuning can improve the system detection sensitivity, which can
help the system achieve extra sensitivity gain. It also shows that the phase-
based readout scheme of heterodyne Rydberg atomic receiver based on MZI can
achieve better sensitivity than the intensity-based readout scheme of
heterodyne Rydberg atomic receiver.

###### Index Terms:

Rydberg atom, phase, homodyne readout, frequency detuning, sensitivity
optimization.

##  I Introduction

Rydberg atoms show extremely strong microwave transition electric dipole
moments, which are sensitive to external electromagnetic fields. At room
temperature, electromagnetic fields can be measured with high sensitivity and
precision using atomic quantum coherence effects. Due to their small size,
high sensitivity, and wide operating frequency band, Rydberg atomic electric
field sensors offer great potential for the detection of radio frequency
signals.

Utilizing EIT (electromagnetically induced transparency) spectroscopy, a
Rydberg atomic sensor with sensitivity 30 uV$\cdot$cm-1Hz-1/2 and minimum
detectable electric field intensity of $8$ uV/cm has been demonstrated [1].
Rydberg atomic sensors’ sensitivity can be raised to $2$ uV$\cdot$cm-1Hz-1/2
using balanced homodyne detection and frequency modulation methods based on
optical interferometers [2, 3]. Combining the Rydberg atomic sensor with the
traditional superheterodyne approach results in a significant increase in
sensitivity. Such method introduces a local microwave signal and the phase and
frequency measurement of microwaves can be realized by the Rydberg atomic
sensor, improving the sensitivity of microwave electric field detection to
$55$ nV$\cdot$cm-1Hz-1/2 [4]. The setting of laser parameters also affects the
electric field detection sensitivity of the Rydberg atomic sensor. Numerical
simulations and experimental results show that the sensitivity of the Rydberg
atomic sensor is related to the amplitude intensity of the two lasers and
microwave electric. The microwave electric detection sensitivity can be
improved to $12.5$ nV$\cdot$cm-1Hz-1/2 by detuning the coupling laser
frequency, which is the highest sensitivity achieved in experiments so far
[5]. For applications in communication systems, improving the detection
sensitivity and predicting the Rydberg atomic sensor performance under
different parameter settings through theoretical analysis and numerical
simulation are significant.

Rydberg atomic sensors’ detection sensitivity is fundamentally constrained by
quantum noise, which is orders of magnitude less prevalent than thermal noise.
The coherence time of the atoms can be decreased in real systems by a variety
of variables, which lowers the system’s sensitivity to detection. The main
relaxation mechanisms that affect Rydberg atomic sensor’s sensitivity include
collision broadening, transition time broadening [6], power broadening [7] and
laser linewidth [8, 9]. Atomic collisions with cavity walls as well as atomic
collisions with one another cause the collision broadening effect. The
collision broadening effect can be efficiently decreased by low atomic
density. The atoms’ mobility causes the transition time broadening effect,
which depends on the temperature, atomic mass, and laser beam size. The power
broadening originates from the instability of laser power, which can be
squeezed by the power stabilization module in experiment. The laser linewidth
is an inherent property of lasers, and narrow linewidth laser can improve the
performance of Rydberg atomic sensors. Finding technical methods to eliminate
or reduce the influence of these factors is crucial for future applications in
communication system.

Recent progress in experiments show that reducing the laser linewidth and
power fluctuation are feasible improvement methods. Experimental techniques
with ultrastable cavities can achieve extremely narrow linewidth laser beam
and high power stabilization, which can increase the system sensitivity to the
level higher than $100$ nV$\cdot$cm-1Hz-1/2. The optical readout noise is the
dominant factor limiting the detection sensitivity [4, 5]. Except for
apparatus performance improvement, new detection and readout techniques are
the options left. In the perspective of readout techniques, the experiment
based on optical interferometer provides ideas for reducing readout noise.
Methods that combine circulating cavity with compressed state light have the
potential to further reduce the readout noise limit of the system. Existing
experimental schemes adopt different detection techniques depending on the
atomic response regime, such as AT splitting effect of atomic resonance [1],
AC Stark effect under off-resonant [10], and the heterodyne readout technique
with tuning capability [11]. A continuously tunable electric field measurement
based on the far off-resonant AC stark effect in a Rydberg atomic vapor cell
shows comparable detection sensitivity with a resonant microwave-dressed
Rydberg heterodyne receiver using the same system [12]. Additional research
about electric field detuning can make smooth transitions among these
operating regimes, especially for modulated signals. The frequency modulation
spectroscopy [3], lower noise photodetector, and the setting of operating
state in the experiment are all sensitivity optimization methods worth trying.

In previous work, the intensity-based readout scheme has been widely studied
[4, 5], and the most sensitive Rydberg atomic heterodyne receiver receives the
intensity signal of the transmitted laser through a photodetector[5]. Although
phase-based homodyne readout scheme also has been demonstrated[2, 13, 14], the
relationship between the transmitted laser phase shift and electric field
strength remains to be studied. Considering that atomic media changes the
phase of transmitted laser, and optical interferometer is highly sensitive to
the phase shift, we construct the phase-based Rydberg atomic heterodyne
receiver embedded in Mach-Zehnder interferometer. This scheme is expected to
further improve the performance of atomic-based electric field sensing
sensitivity. In this work, we further characterize the effects of local
microwave, probing laser and coupling laser frequency detuning on the
sensitivity of Rydberg atomic superheterodyne receiver based on homodyne
readout.

The reminder of this paper is organized as follows. In Section II, we
introduce the system structure of proposed scheme, and outline the basic
detection principles based on homodyne readout. In Section III, we optimize
the frequency parameters based on atomic four-level model. We theoretically
investigate the detection response to the local microwave, coupling laser and
probing laser frequency detuning. In Section IV, we numerically characterize
the system sensitivity variation. In Section V, we evaluate the electric field
detection sensitivity based on different noise baselines. Finally in Section
VI, we conclude this work.

##  II System Model

###  II-A System Structure

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.

Figure 1: The diagram of the atomic heterodyne detection system based on
homodyne readout.

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.

Figure 2: Overview of the system setup. We have used the following notations:
(1) ISO: isolator, (2) HWP: half wave plate, (3) PBS: polarized beam splitter,
(4) AOM: The acousto-optic modulators which shift the frequencies of the
probing and coupling lights to atomic resonances, (5) PSM: power stabilization
module, (6) PD: photodetector, (7) DM: dichroic mirror, (8) RM: reflective
mirror, (9) LFSM: laser frequency stabilization module, (10) MZI: Mach-Zehnder
interferometer.

###  II-B Theoretical Model

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

$H=\frac{{\hbar}}{2}\left(\begin{matrix}0&\Omega_{p}&0&0\\\
\Omega_{p}&2\Delta_{p}&\Omega_{c}&0\\\
0&\Omega_{c}&2\left(\Delta_{p}+\Delta_{c}\right)&\Omega_{L}+\Omega_{s}e^{-iS\left(t\right)}\\\
0&0&\Omega_{L}+\Omega_{s}e^{iS\left(t\right)}&2\left(\Delta_{p}+\Delta_{c}+\Delta_{L}\right)\\\
\end{matrix}\right),$ (1)

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.

Figure 3: The diagram of four-level configuration.

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]

$i{\hbar}\frac{d}{dt}\rho=\left[H,\
\rho\right]-i{\hbar}\frac{1}{2}\left(\Gamma\rho+\rho\Gamma\right)+i{\hbar}\Lambda,$
(2)

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

$\Gamma=\left(\begin{matrix}\gamma&0&0&0\\\ 0&\gamma+\gamma_{2}&0&0\\\
0&0&\gamma+\gamma_{3}+\gamma_{c}&0\\\ 0&0&0&\gamma+\gamma_{4}\\\
\end{matrix}\right),$ (3)

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

$\Lambda=\left(\begin{matrix}\gamma+\gamma_{2}\rho_{22}+\gamma_{4}\rho_{44}&0&0&0\\\
0&\gamma_{3}\rho_{33}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{matrix}\right).$ (4)

The dynamics equation of the atomic system can be expressed as

$\frac{d}{dt}\rho=-\frac{i}{{\hbar}}\left[H,\rho\right]+L,$ (5)

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

$L=\left(\begin{matrix}\gamma+\gamma_{2}\rho_{22}+\gamma_{4}\rho_{44}&-\frac{\gamma_{2}+2\gamma}{2}\rho_{12}&-\frac{\gamma_{3}+\gamma_{c}+2\gamma}{2}\rho_{13}&-\frac{\gamma_{4}+2\gamma}{2}\rho_{14}\\\
-\frac{\gamma_{2}+2\gamma}{2}\rho_{21}&\left(\gamma_{3}+\gamma_{c}+\gamma\right)\rho_{33}-\left(\gamma_{2}+\gamma\right)\rho_{22}&-\left(\frac{\gamma_{23}+\gamma_{c}+2\gamma}{2}\right)\rho_{23}&-\left(\frac{\gamma_{24}+2\gamma}{2}\right)\rho_{24}\\\
-\frac{\gamma_{3}+\gamma_{c}+2\gamma}{2}\rho_{31}&-\left(\frac{\gamma_{23}+\gamma_{c}+2\gamma}{2}\right)\rho_{32}&-\left(\gamma_{3}+\gamma_{c}+\gamma\right)\rho_{33}&-\left(\frac{\gamma_{34}+\gamma_{c}+2\gamma}{2}\right)\rho_{34}\\\
-\frac{\gamma_{4}+2\gamma}{2}\rho_{41}&-\left(\frac{\gamma_{24}+2\gamma}{2}\right)\rho_{42}&-\left(\frac{\gamma_{34}+\gamma_{c}+2\gamma}{2}\right)\rho_{43}&-\left(\gamma_{4}+\gamma\right)\rho_{44}\\\
\end{matrix}\right).$ (6)

###  II-C Principle

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

$\chi=-\frac{2N_{0}\mu_{21}^{2}}{{\hbar}\varepsilon_{0}\Omega_{p}}\rho_{21},$
(7)

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]

$\frac{d\Omega_{p}}{dz}=\frac{k_{p}N_{0}\mu_{21}^{2}}{2{\hbar}\varepsilon_{0}}\Im\left(\rho_{21}\right),$
(8)

and

$\frac{d\phi}{dz}=\frac{k_{p}N_{0}\mu_{21}^{2}}{2{\hbar}\varepsilon_{0}\Omega_{p}\left(z\right)}\Re\left(\rho_{21}\right).$
(9)

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,

$\Delta\phi=k_{p}L\Re\left(\chi\right)/4.$ (10)

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

$E_{out}=E_{i}\cdot e^{-k_{p}L\Im\left(\chi\right)/2}\cdot e^{i\Delta\phi}.$
(11)

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

$\displaystyle E_{\text{SIG}}$ $\displaystyle=E_{1}\cdot
e^{ikn_{0}L_{\text{SIG}}}\cdot e^{-k_{p}L\Im\left(\chi\right)/2}\cdot
e^{i\Delta\phi},$ (12) $\displaystyle E_{\text{REF}}$
$\displaystyle=E_{2}\cdot e^{ikn_{0}L_{\text{REF}}},$

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]

$\Delta I_{d}\propto
2|E_{\text{REF}}||E_{\text{SIG}}|e^{-k_{p}L\Im\left(\chi\right)/2}\cos\left(\phi_{\text{REF}}+\Delta\phi\right).$
(13)

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.

##  III Parameters Optimization

###  III-A Methodology

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,

$P\left(t\right)\approx\bar{P}_{0}+\kappa\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right),$
(14)

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

$\Re\left(\chi\left(t,\
\Delta\right)\right)\approx\chi_{0}\left(\Delta\right)+\chi_{1}\left(\Delta\right)\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right),$
(15)

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

$\displaystyle\Delta\phi=k_{p}L\Re(\chi)/4=k_{p}L\chi_{0}(\Delta)+k_{p}L\chi_{1}(\Delta)\Omega_{s}\cos(2\pi\delta_{s}+\phi_{s}).$
(16)

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

Figure 4: Transmitted laser phase shift $\Delta\phi$ evolution over time.

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.

###  III-B Local Microwave Detuning

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,

$\rho_{21}(\Delta_{L})=-\frac{i\gamma_{2}\Omega_{p}\Omega^{4}-2\Delta_{L}\Omega_{p}\Omega^{2}\Omega_{c}^{2}}{\gamma_{2}^{2}\Omega^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega^{2}\Omega_{p}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}+\Omega^{2}\right)}.\\\
$ (17)

The real component is given by

$\Re\left(\rho_{21}\left(\Delta_{L}\right)\right)=\frac{2\Delta_{L}\Omega_{p}\Omega^{2}\Omega_{c}^{2}}{\gamma_{2}^{2}\Omega^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega^{2}\Omega_{p}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}+\Omega^{2}\right)}.$
(18)

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

$\Re\left(\rho_{21}\left(\Delta_{L}\right)\right)\approx\frac{C_{L-DC}+C_{L-AC}\Omega_{s}\cos\left(S\left(t\right)\right)}{\left(\gamma_{2}^{2}\Omega_{L}^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}+\Omega_{L}^{2}\right)\right)^{2}},$
(19)

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

$C_{L-DC}=2\Delta_{L}\Omega_{p}\Omega_{c}^{2}\Omega_{L}^{2}\left(\gamma_{2}^{2}\Omega_{L}^{4}+4\Delta_{L}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)^{2}+2\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}+\Omega_{L}^{2}\right)\right),$
(20)

and

$C_{L-AC}=4\Delta_{L}\Omega_{c}^{2}\Omega_{p}\Omega_{L}\left(4\Delta_{L}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)^{2}-\Omega_{L}^{4}\left(\gamma_{2}^{2}+2\Omega_{p}^{2}\right)\right).$
(21)

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

$\Re\left(\chi\left(t,\
\Delta_{L}\right)\right)=\chi_{0}\left(\Delta_{L}\right)+\chi_{1}\left(\Delta_{L}\right)\Omega_{s}\cos
S\left(t\right),$ (22)

which has a time-invariant part,

$\chi_{0}\left(\Delta_{L}\right)=\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\varepsilon_{0}}\frac{C_{L-DC}}{[\gamma_{2}^{2}\Omega_{L}^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}+\Omega_{L}^{2}\right)]^{2}},$
(23)

and a time variant part coefficient,

$\chi_{1}\left(\Delta_{L}\right)=\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\varepsilon_{0}}\frac{C_{L-AC}}{\left[\gamma_{2}^{2}\Omega_{L}^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}+\Omega_{L}^{2}\right)\right]^{2}}.$
(24)

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

$\Delta_{L}^{*}=\underset{\Delta_{L}}{\text{argmax }}|\chi_{1}(\Delta_{L})|.$
(25)

###  III-C Probing Laser Frequency 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,

$\displaystyle\rho_{21}(\Delta_{p})$
$\displaystyle=\frac{\Omega_{p}\left(4\Delta_{p}^{2}-\Omega^{2}\right)\left(i\gamma_{2}\left(4\Delta_{p}^{2}-\Omega^{2}\right)+2\Delta_{p}\left(4\Delta_{p}^{2}-\Omega_{c}^{2}-\Omega^{2}\right)\right)}{C_{P}(\Omega)},$
(26) $\displaystyle C_{P}(\Omega)$
$\displaystyle=64\Delta_{p}^{6}+\gamma_{2}^{2}\left(\Omega^{2}-4\Delta_{p}^{2}\right)^{2}+4\Delta_{p}^{2}\left(\left(\Omega^{2}+\Omega_{c}^{2}\right)^{2}+2\Omega_{p}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}-2\Omega^{2}\right)\right)$
$\displaystyle-32\Delta_{p}^{4}\left(\Omega^{2}+\Omega_{c}^{2}-\Omega_{p}^{2}\right)+2\Omega_{p}^{2}\Omega^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}+\Omega^{2}\right).$

The real component is given by

$\Re\left(\rho_{21}\left(\Delta_{p}\right)\right)=\frac{2\Delta_{p}\Omega_{p}\left(4\Delta_{p}^{2}-\Omega^{2}\right)\left(4\Delta_{p}^{2}-\Omega^{2}-\Omega_{c}^{2}\right)}{C_{P}\left(\Omega\right)}.$
(27)

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

$\Re\left(\rho_{21}\left(\Delta_{p}\right)\right)\approx-\frac{C_{P-DC}+C_{P-AC}\Omega_{s}\cos\left(S\left(t\right)\right)}{C_{P}\left(\Omega_{L}\right)^{2}},$
(28)

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

$C_{P-DC}=2\Delta_{p}\Omega_{p}\left(4\Delta_{p}^{2}-\Omega_{L}^{2}\right)\left(4\Delta_{p}^{2}-\Omega_{c}^{2}-\Omega_{L}^{2}\right)C_{P}\left(\Omega_{L}\right),$
(29)

and

$\displaystyle C_{P-AC}$
$\displaystyle=4\Delta_{p}\Omega_{p}\Omega_{L}\left(64\Delta_{p}^{6}\Omega_{c}^{2}-\gamma_{2}^{2}\left(4\Delta_{p}^{2}-\Omega_{L}^{2}\right)^{2}\Omega_{c}^{2}+2\Omega_{L}^{4}\Omega_{p}^{4}-32\Delta_{p}^{4}\left(\Omega_{c}^{4}+3\Omega_{p}^{4}+\Omega_{c}^{2}\left(\Omega_{L}^{2}+4\Omega_{p}^{2}\right)\right)\right)-$
(30) $\displaystyle
16\Delta_{p}^{3}\Omega_{p}\Omega_{L}\left(\Omega_{c}^{6}+4\Omega_{L}^{2}\Omega_{p}^{4}+2\Omega_{c}^{4}\left(\Omega_{L}^{2}+2\Omega_{p}^{2}\right)+\Omega_{c}^{2}\left(\Omega_{L}^{4}+8\Omega_{L}^{2}\Omega_{p}^{2}+4\Omega_{p}^{4}\right)\right).$

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

$\Re\left(\chi\left(t,\
\Delta_{p}\right)\right)=\chi_{0}\left(\Delta_{p}\right)+\chi_{1}\left(\Delta_{p}\right)\Omega_{s}\cos
S\left(t\right),$ (31)

which has a time-invariant part,

$\chi_{0}\left(\Delta_{p}\right)=\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\varepsilon_{0}}\frac{C_{P-DC}}{C_{P}(\Omega_{L})^{2}},$
(32)

and a time variant part coefficient,

$\chi_{1}\left(\Delta_{p}\right)=\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\varepsilon_{0}}\frac{C_{P-AC}}{C_{P}(\Omega_{L})^{2}}.$
(33)

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

$\Delta_{p}^{*}=\underset{\Delta_{p}}{\text{argmax }}|\chi_{1}(\Delta_{p})|.$
(34)

###  III-D Coupling 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,

$\displaystyle\rho_{21}(\Delta_{c})$
$\displaystyle=\frac{i\gamma_{2}\Omega_{p}\left(4\Delta_{c}^{2}-\Omega^{2}\right)^{2}-2\Omega_{p}\Omega_{c}^{2}\Delta_{c}\left(4\Delta_{c}^{2}-\Omega^{2}\right)}{C_{C}(\Omega)},$
(35) $\displaystyle C_{C}(\Omega)$
$\displaystyle=32\Delta_{c}^{4}\Omega_{p}^{2}+\gamma_{2}^{2}\left(\Omega^{2}-4\Delta_{c}^{2}\right)^{2}+2\Omega_{p}^{2}\Omega^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}+\Omega^{2}\right)$
$\displaystyle+4\Delta_{c}^{2}\left(\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)^{2}+\Omega_{p}^{2}\left(\Omega_{p}^{2}-4\Omega^{2}\right)\right).$

The real component is given by

$\Re\left(\rho_{21}\left(\Delta_{c}\right)\right)=-\frac{2\Delta_{c}\Omega_{p}\Omega_{c}^{2}\left(4\Delta_{c}^{2}-\Omega^{2}\right)}{C_{C}\left(\Omega\right)}.$
(36)

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

$\Re\left(\rho_{21}\left(\Delta_{c}\right)\right)=-\frac{C_{C-DC}+C_{C-AC}\Omega_{s}\cos\left(S\left(t\right)\right)}{C_{C}\left(\Omega_{L}\right)^{2}},$
(37)

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

$C_{C-DC}=2\Delta_{c}\Omega_{p}\Omega_{c}^{2}\left(4\Delta_{c}^{2}-\Omega_{L}^{2}\right)C_{C}\left(\Omega_{L}\right),$
(38)

and

$C_{C-AC}=4\Delta_{c}\Omega_{p}\Omega_{c}^{2}\Omega_{L}\left(\gamma_{2}^{2}\left(-4\Delta_{c}^{2}+\Omega_{L}^{2}\right)^{2}+32\Delta_{c}^{4}\Omega_{p}^{2}+2\Omega_{L}^{4}\Omega_{p}^{2}-4\Delta_{c}^{2}\left(\Omega_{c}^{4}+4\Omega_{c}^{2}\Omega_{p}^{2}+4\Omega_{p}^{2}\left(\Omega_{L}^{2}+\Omega_{p}^{2}\right)\right)\right).$
(39)

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

$\Im\left(\chi\left(t,\Delta_{c}\right)\right)=\chi_{0}(\Delta_{c})+\chi_{1}(\Delta_{c})\Omega_{s}\cos
S(t),$ (40)

which has a time-invariant part,

$\chi_{0}\left(\Delta_{c}\right)=\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\varepsilon_{0}}\frac{C_{C-DC}}{C_{C}(\Omega_{L})^{2}},$
(41)

and a time variant part coefficient,

$\chi_{1}\left(\Delta_{c}\right)=\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\varepsilon_{0}}\frac{C_{C-AC}}{C_{C}(\Omega_{L})^{2}}.$
(42)

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

$\Delta_{c}^{*}=\underset{{\Delta_{c}}}{\text{argmax
}}|\chi_{1}(\Delta_{c})|.$ (43)

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

##  IV Numerical Results

We adopt some parameters from the experimental setups in [4]. Cs atoms were
filled in a vapour cell at room temperature. The cell contained ground-state
atoms at a total density of $N_{0}=4.89\times 10^{10}$ cm-3. For Rydberg atom
EIT, the residence time in the Rydberg state is small. Under typical
conditions, the population in Rydberg states is $~{}0.0001$. Only a small
distribution of atomic velocity classes in the vapor cell are selected by the
EIT lasers, roughly $~{}1/100$ for the scheme shown in Fig. 3 [18]. The
effective atoms density for atomic vapor is approximated as
$N_{0,\text{eff}}\approx 4.89\times 10^{8}$ cm-3. The four-level configuration
in Fig. 3 using four states in a Cs atom: $|1\rangle\rightarrow 6S_{1/2},F=4$;
$|2\rangle\rightarrow 6P_{3/2},F=5$; $|3\rangle\rightarrow 47D_{5/2}$; and
$|4\rangle\rightarrow 48P_{3/2}$. States $|2\rangle$, $|3\rangle$, and
$|4\rangle$ have the inverse lifetimes $\gamma_{2}=2\pi\times 5.2$ MHz,
$\gamma_{3}=2\pi\times 3.9$ kHz, and $\gamma_{4}=2\pi\times 1.7$ kHz. In this
section, we ignore the atoms collision effect, $\gamma_{c}=0$, the effective
Rabi frequencies for the transitions $|1\rangle\rightarrow|2\rangle$ and
$|2\rangle\rightarrow|3\rangle$ are $\Omega_{p}=2\pi\times 5.7$ MHz,
$\Omega_{c}=2\pi\times 0.97$ MHz, respectively. The frequency intervals
between the hyperfine states $6P_{3/2}$ are $151.2$ MHz ($F=2\rightarrow
F=3$), $201.2$ MHz ($F=3\rightarrow F=4$), and $251.1$ MHz ($F=4\rightarrow
F=5$) [19]. In the case of resonance, the system is locked to a specific
hyperfine state. We consider a detuning range from $-50$ MHz to $50$ MHz for
laser frequency and another range from $-100$ MHz to $100$ MHz for electric
field frequency in numerical results.

Local microwave frequency detuning case: The relationship between $\Delta_{L}$
and conversion coefficient $|\chi_{1}{\Delta_{L}}\Omega_{s}|$ is shown in Fig.
5. The maximum values of $|\chi_{1}{\Delta_{L}}\Omega_{s}|$ at
$\Omega_{L}/2\pi=2$ MHz, 4 MHz, 6 MHz are $4.125\times 10^{-9}$, $3.217\times
10^{-9}$, $2.491\times 10^{-9}$, respectively. It can be found that weaker
local microwave electric field strength corresponds to larger conversion
coefficient maximum. At the same time, the detuning value at the maximum point
will also become smaller.

Figure 5: Conversion coefficient $|\chi_{1}(\Delta_{L})\Omega_{s}|$ with
respect to local microwave detuning $\Delta_{L}$ in the general case,
$\Omega_{s}=2\pi\times 10^{5}$ Hz (P1).

Probing laser frequency detuning case: The relationship between $\Delta_{p}$
and conversion coefficient $|\chi_{1}{\Delta_{p}}\Omega_{s}|$ is shown in Fig.
6. The maximum values of $|\chi_{1}{\Delta_{p}}\Omega_{s}|$ at
$\Omega_{L}/2\pi=2$ MHz, 4 MHz, 6 MHz are $7.309\times 10^{-8}$, $1.144\times
10^{-7}$, $1.430\times 10^{-7}$, respectively. In contrast to local microwave
frequency detuning case, stronger local microwave electric field strength
corresponds to larger conversion coefficient maximum in this case. Besides, we
can obtain bigger conversion coefficient by setting optimal probing laser
frequency.

Figure 6: Conversion coefficient $|\chi_{1}(\Delta_{p})\Omega_{s}|$ with
respect to probing laser detuning $\Delta_{p}$ in the general case,
$\Omega_{s}=2\pi\times 10^{5}$ Hz (P2).

Coupling laser frequency detuning case: The relationship between $\Delta_{c}$
and conversion coefficient $|\chi_{1}{\Delta_{c}}\Omega_{s}|$ is shown in Fig.
7. The maximum values of $|\chi_{1}{\Delta_{p}}\Omega_{s}|$ at
$\Omega_{L}/2\pi=2$ MHz, 4 MHz, 6 MHz are $1.112\times 10^{-9}$, $2.743\times
10^{-9}$, $5.737\times 10^{-7}$, respectively. This case has similar gain than
local microwave frequency detuning case.

Figure 7: Conversion coefficient $|\chi_{1}(\Delta_{c})\Omega_{s}|$ with
respect to coupling laser detuning $\Delta_{c}$ in the general case,
$\Omega_{s}=2\pi\times 10^{5}$ Hz (P3).

##  V Sensitivity Estimation

###  V-A Calculation

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

$P_{\min\text{\,\,}}=\left<|\vec{S}|\right>\cdot
A_{\text{eff}}=\frac{1}{2}\varepsilon_{0}cE_{\min\text{\,\,}}^{2}\cdot
A_{\text{eff}},$ (44)

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.

###  V-B Atomic shot noise limit sensitivity

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

$E_{\min}=\frac{h}{\mu_{\text{E}}T\sqrt{N}},$ (45)

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

$E_{\min}=\frac{h}{\mu_{\text{E}}\sqrt{N_{\text{at}}T^{{}^{\prime}}T_{2}}}.$
(46)

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

$\frac{E_{\min}}{\sqrt{\text{Hz}}}=\frac{h}{\mu_{\text{E}}\sqrt{N_{\text{at}}T_{2}}}.$
(47)

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.

###  V-C Photon shot noise limit

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

$\phi_{N}=\frac{2\mu_{21}}{d\Omega_{p}}\sqrt{\frac{k_{p}}{h\varepsilon_{0}}},$
(48)

where detection efficiency assumed to be unity in this work.

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

$\frac{\Delta\phi}{\phi_{N}}=\frac{k_{p}Ld\Omega_{p}\Omega_{s}}{8\mu_{21}}\sqrt{\frac{h\varepsilon_{0}}{k_{p}}}\chi_{1}\left(\Delta\right).$
(49)

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

$\frac{E_{\min}}{\sqrt{\text{Hz}}}=\frac{4\mu_{21}}{\pi
Ld\mu_{E}\Omega_{p}\chi_{1}\left(\Delta\right)}\sqrt{\frac{h}{k_{p}\varepsilon_{0}}}.$
(50)

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.

##  VI Conclusion

Based on the atomic superheterodyne receiver and Mach-Zehnder interferometer,
we evaluate the detection sensitivity of phase-based readout method. We have
derived the steady-state solutions of the density matrix of the Rydberg atomic
system operating with frequency detuning. Given the input microwave signal, we
have established an optimization problem of the linear susceptibility of
atomic medium to maximize the amplitude of the output phase signal. We have
numerically analyzed the effects of electric field, probing laser, and
coupling laser frequency detuning on the optimization target. Numerical
results show that the detection sensitivity can be improved by setting
appropriate frequency detuning, especially for probing laser. The photon shot
noise limit sensitivity of the phase-based readout method has 36.6 dB gain
compared with that of experimental result of the intensity-based readout
method.

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