# Theoretical Analysis of Heterodyne Rydberg Atomic Receiver Sensitivity Based
on Transit Relaxation Effect and Frequency Detuning

Shanchi Wu, Chen Gong, Shangbin Li, Rui Ni, Jinkang Zhu This work was
supported in part by National Natural Science Foundation of China under Grant
62171428, in part by Key Research Program of Frontier Sciences of CAS under
Grant QYZDY-SSW-JSC003, and Huawei Innovation Project.Shanchi Wu, Chen Gong,
Shangbin Li, and Jinkang Zhu are with University of Science and Technology of
China, Email address: wsc0807@mail.ustc.edu.cn,{cgong821, shbli,
<EMAIL_ADDRESS>Ni is with Huawei Technology, Email address:
<EMAIL_ADDRESS>
###### Abstract

We conduct a theoretical investigation into the impacts of local microwave
electric field frequency detuning, laser frequency detuning, and transit
relaxation rate on enhancing heterodyne Rydberg atomic receiver sensitivity.
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
heterodyne Rydberg atomic receiver can detect weak microwave signals
continuously over a wide frequency range with the same sensitivity or even
more sensitivity than the resonance case. To evaluate the transit relaxation
effect, a modified Liouville equation is used. We find that the transition
relaxation rate increases the time it takes to reach steady state and
decreases the sensitivity of the system detection.

###### Index Terms:

Rydberg atom, frequency detuning, sensitivity optimization, transit
relaxation.

##  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.

Utilizing electromagnetically induced transparency (EIT) 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, predicting the Rydberg atomic
sensor performance under different parameter settings through theoretical
analysis and numerical simulation is 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 our preliminary work, we have evaluated the optimal value of local
microwave frequency detuning under high transmittance case [13]. In this work,
we further characterize the effects of local microwave, probing laser and
coupling laser frequency detuning on the sensitivity of Rydberg atomic
sensors. In addition, we investigate the impact of transit relaxation on the
sensitivity of the system. Assume that the heterodyne scheme can detect the
microwave signal amplitude, phase, and frequency, and the system can be tuned
by changing the local microwave signal frequency. We aim to build on this
heterodyne reception framework.

The reminder of this paper is organized as follows. In Section II, we outline
the basic detection principles of heterodyne schemes. 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
consider the transit relaxation effect caused by the atomic motion, and
numerically characterize the system sensitivity variation. Finally in Section
V, we conclude this work.

##  II Theoretical Model

A schematic diagram of the detection system based on Rydberg atomic sensor is
shown in Fig. 1. The probing laser and coupling laser simultaneously excite
the atoms in the atomic vapor cell, and the photodetector receives the
transmission probing laser. The probing laser and coupling laser are locked on
the EIT transmission resonance. When the microwave signal radiates cross the
atomic vapor cell, the resonant microwave electric field can help improve the
transmission performance and the AC Stark shift caused by off-resonant
microwave electric field also changes the probing laser transmission, thereby
changing the signal intensity received by the photodetector. Because the
system response is sensitive to the direction of polarization, we set that the
system lies in the $xoz$ plane, and the probing laser and coupling laser have
the same polarization in the $yoz$ plane. The incident microwave electric
filed has the $y$-axis polarization and $\beta$ is the angle difference. It
has been demonstrated that the microwave electric field polarization can also
be obtained by atomic system [14]. In this work, we assume that the lasers and
microwave signal have the same polarization, i.e., the polarization angle
$\beta$ between lasers and microwave is zero.

Figure 1: The diagram of the atomic heterodyne detection system.

The diagram for a typical atomic four-level structure is shown in Fig. 2. 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 2: 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 [15]

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

Under resonant conditions, i.e., $\Delta_{c}=\Delta_{p}=\Delta_{L}=0$,
ignoring the atom collision and transit time broadening effect, we have
$\gamma=\gamma_{c}=0$. The steady-state solution $\rho_{21}(t)$ is given by

$\rho_{21}(t)=-\frac{i\gamma_{2}\Omega_{p}\Omega^{2}}{\gamma_{2}^{2}\Omega^{2}+2\Omega_{c}^{2}\Omega_{p}^{2}+2\Omega_{p}^{4}+2\Omega_{p}^{2}\Omega^{2}}.$
(6)

The linear susceptibility can be written as

$\chi\left(t\right)=-\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\epsilon_{0}\Omega_{p}}\rho_{21}\left(t\right),$
(7)

where $N_{0}$ is the total density of atoms, $\mu_{12}$ is the dipole moment
of transition $|1\rangle\rightarrow|2\rangle$ and $\epsilon_{0}$ is the
permittivity in vacuum. Assuming $\Omega_{s}\ll\Omega_{L}$, it has been proved
that the optimal local microwave filed amplitude can improve the system
sensitivity. The optimal Rabi frequency of $\Omega_{L}$ is given by [4]

$\Omega_{L}=\Omega_{p}\sqrt{\frac{2\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)}{3\left(2\Omega_{p}^{2}+\gamma_{2}^{2}\right)}}.$
(8)

In this work, based on the steady-state solution of the system Liouville
equation, we derive the expression of the linear susceptibility and output
optical power, and then find the optimal values. In general, we expect that
the output optical power has the formalism as follows,

$P\left(t\right)=\bar{P}_{0}+\kappa\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right).$
(9)

We aim to maximize the absolute value of conversion coefficient $\kappa$,
which represents the detection sensitivity.

##  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
[16]. We analyze the influence of laser frequencies and local microwave
electric field frequency detuning on the system sensitivity.

To gain an intuitive understanding, we begin with analytical derivations. 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). The corresponding linear susceptibility of the atomic system can be
written as

$\mathfrak{J}\left(\chi\left(t,\
\Delta\right)\right)=\chi_{0}\left(\Delta\right)+\chi_{1}\left(\Delta\right)\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right),$
(10)

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 signal component at the photodetector, and the second
part reflects the information transmission of signal microwave field.

The probing laser transmission is associated with the imaginary component of
the susceptibility as

$\displaystyle P\left(t\right)$
$\displaystyle=P_{i}e^{-kL\Im\left(\chi\left(t,\ \Delta\right)\right)}$ (11)
$\displaystyle=e^{-kL\chi_{0}\left(\Delta\right)}P_{i}e^{-kL\chi_{1}\left(\Delta\right)\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right)},$

where $k=2\pi/\lambda_{p}$ is the wavevector of probing laser, and $L$ is the
length of atomic vapor cell along the laser beam propagation direction. The
peak-to-peak value of the output signal is

$P_{pp}\left(\Delta\right)=e^{-kL\chi_{0}\left(\Delta\right)}P_{i}\left(e^{kL\chi_{1}\left(\Delta\right)\Omega_{s}}-e^{-kL\chi_{1}\left(\Delta\right)\Omega_{s}}\right).$
(12)

Since the amplitude of microwave signal is linearly dependent on its
corresponding Rabi frequency $\Omega_{s}$, we can optimize coefficients
$\chi_{0}\left(\Delta\right)$ and $\chi_{1}\left(\Delta\right)$ in Eq. (12) to
maximize $P_{pp}\left(\Delta\right)$. Two cases are considered in the
following.

General case: In weak signal regime,
$|kL\chi_{1}\left(\Delta\right)\Omega_{s}|$ is much lower than one. The output
laser power can be approximated as

$\begin{aligned} P\left(t\right)&\approx
e^{-kL\chi_{0}\left(\Delta\right)}P_{i}\left(1-kL\chi_{1}\left(\Delta\right)\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right)\right)\\\
&=\bar{P}_{0}\left(\Delta\right)+\kappa\left(\Delta\right)P_{i}\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right)\\\
\end{aligned},$ (13)

where $\bar{P}_{0}\left(\Delta\right)=P_{i}e^{-kL\chi_{0}\left(\Delta\right)}$
is the DC component of the output optical power and
$\kappa\left(\Delta\right)=-e^{-kL\chi_{0}\left(\Delta\right)}kL\chi_{1}\left(\Delta\right)$
represents the conversion coefficient to signal. The DC component is
associated with the photodetector operation point. Our goal is to maximize the
conversion coefficient $|\kappa(\Delta)|$ through parameter optimization.

High transmittance case: If $|kL\chi_{0}\left(\Delta\right)|$ is much lower
than one, $e^{-kL\chi_{0}\left(\Delta\right)}$ can be approximated as a
constant. The output laser power can be approximated as

$\displaystyle P\left(t\right)$ $\displaystyle\approx
P_{i}\left(1-kL\chi_{1}\left(\Delta\right)\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right)\right)$
(14)
$\displaystyle=\bar{P}_{0}^{{}^{\prime}}\left(\Delta\right)+\kappa^{\prime}\left(\Delta\right)P_{i}\Omega_{s}\cos\left(2\pi\delta_{s}t+\phi_{s}\right),$

where $\bar{P}_{0}^{{}^{\prime}}\left(\Delta\right)\approx P_{i}$ is the DC
component of the output optical power and
$\kappa^{\prime}\left(\Delta\right)=-kL\chi_{1}\left(\Delta\right)$ represents
the conversion coefficient. In high transmittance case, the optimal problem of
maximizing conversion coefficient $|\kappa^{\prime}(\Delta)|$ is equivalent to
maximizing the coefficient $|\chi_{1}|$.

###  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
Eq. (15).

$\displaystyle\rho_{21}(\Delta_{L})$
$\displaystyle=-\frac{i\Omega_{p}\Omega^{2}\left(\gamma_{2}\Omega^{2}+2i\Delta_{L}\Omega_{c}^{2}\right)}{\gamma_{2}^{2}\Omega^{4}+4\Delta_{L}^{2}\Omega_{c}^{4}+8\Delta_{L}^{2}\Omega_{c}^{2}\Omega_{p}^{2}+4\Delta_{L}^{2}\Omega_{p}^{4}+2\Omega_{c}^{2}\Omega_{p}^{2}\Omega^{2}+2\Omega_{p}^{4}\Omega^{2}+2\Omega_{p}^{2}\Omega^{4}}$
(15)
$\displaystyle=-\frac{i\gamma_{2}\Omega_{p}\Omega^{4}-2\Delta_{L}\Omega_{p}\Omega^{2}\Omega_{c}^{2}}{\gamma_{2}^{2}\Omega^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega^{2}\Omega_{p}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}+\Omega^{2}\right)}.$

In the weak signal regime, $\Omega_{s}\ll\Omega_{L}$, the first order
approximation of $\Omega_{s}/\Omega_{L}$ is given by Eq. (16).

$\displaystyle\Im\left(\rho_{21}(\Delta_{L})\right)\approx-\frac{\gamma_{2}\Omega_{p}\Omega_{L}^{4}}{\gamma_{2}^{2}\Omega_{L}^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}+\Omega_{L}^{2}\right)}$
(16)
$\displaystyle-\frac{4\gamma_{2}\Omega_{L}^{3}\Omega_{p}\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)\left(\Omega_{L}^{2}\Omega_{p}^{2}+4\Delta_{L}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)\right)}{\left[\gamma_{2}^{2}\Omega_{L}^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}+\Omega_{s}^{2}\right)\right]^{2}}\Omega_{s}\cos
S\left(t\right).$

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

$\Im\left(\chi\left(t,\
\Delta_{L}\right)\right)=\chi_{0}\left(\Delta_{L}\right)+\chi_{1}\left(\Delta_{L}\right)\Omega_{s}\cos
S\left(t\right),$ (17)

which has a time-invariant part as Eq. (18),

$\chi_{0}\left(\Delta_{L}\right)=\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\epsilon_{0}}\frac{\gamma_{2}\Omega_{L}^{4}}{\gamma_{2}^{2}\Omega_{L}^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}+\Omega_{L}^{2}\right)},$
(18)

and a time variant part coefficient as Eq. (19).

$\chi_{1}\left(\Delta_{L}\right)=\frac{2N_{0}\mu_{12}^{2}}{{\hbar}\epsilon_{0}}\frac{4\gamma_{2}\Omega_{L}^{3}\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)\left(\Omega_{L}^{2}\Omega_{p}^{2}+4\Delta_{L}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)\right)}{\left[\gamma_{2}^{2}\Omega_{L}^{4}+4\Delta_{L}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)^{2}+2\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}+\Omega_{L}^{2}\right)\right]^{2}}.$
(19)

In the general case, the sensitivity optimization problem is formulated as
follows.

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

$\Delta_{L}^{*}=\underset{\Delta_{L}}{\text{argmax
}}\kappa(\Delta_{L})=\underset{\Delta_{L}}{\text{argmax
}}e^{-kL\chi_{0}\left(\Delta_{L}\right)}\chi_{1}\left(\Delta_{L}\right).$ (20)

Adopting the following variable substitution,

$\frac{\Omega_{p}^{2}}{\gamma_{2}^{2}}\rightarrow x,\
\frac{\Omega_{c}^{2}}{\gamma_{2}^{2}}\rightarrow
y,\,\,\frac{\Omega_{L}^{2}}{\gamma_{2}^{2}}\rightarrow
z,\,\,\frac{\Delta_{L}^{2}}{\gamma_{2}^{2}}\rightarrow w,$ (21)

and we can get an equivalent optimization problem as Eq. (22),

$w^{*}=\underset{w}{\text{arg}\max}\left\\{\begin{array}[]{c}\exp[-\frac{Cz^{2}}{z^{2}+4w\left(x+y\right)^{2}+2xz\left(x+y+z\right)}]\cdot\frac{z^{3/2}\left(x+y\right)\left(xz+4w\left(x+y\right)\right)}{\left[z^{2}+4w\left(x+y\right)^{2}+2xz\left(x+y+z\right)\right]^{2}}\\\
\end{array}\right\\},$ (22)

where $C=2kLN_{0}\mu_{12}^{2}/{(\hbar\epsilon_{0}{\gamma_{2}})}$ is a
constant. Thus, the optimal value of $w^{*}$ is the maximum point of function
$g(x,y,z,w)$ as Eq. (23).

$g\left(x,y,z,w\right)=\exp[-\frac{Cz^{2}}{z^{2}+4w\left(x+y\right)^{2}+2xz\left(x+y+z\right)}]\cdot\frac{z^{3/2}\left(x+y\right)\left(xz+4w\left(x+y\right)\right)}{\left[z^{2}+4w\left(x+y\right)^{2}+2xz\left(x+y+z\right)\right]^{2}}.$
(23)

Letting $u=x(x+y)$, and $v=z(2x+1)$, the optimal value for $w$ is

$w^{*}=\frac{x^{2}z\left(Cz-2u+\sqrt{4\left(u+v\right)^{2}+\left(Cz\right)^{2}}\right)}{8u^{2}}.$
(24)

The corresponding microwave field detuning is obtained as

$\Delta_{L}^{*}=\frac{\Omega_{p}^{2}\Omega_{L}}{2U}\sqrt{\frac{C\gamma_{2}^{2}\Omega_{L}^{2}-2U+\sqrt{4\left(U+V\right)^{2}+\left(C\gamma_{2}^{2}\Omega_{L}^{2}\right)^{2}}}{2}},$
(25)

where $U=\Omega_{p}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)$, and
$V=\Omega_{L}^{2}\left(2\Omega_{p}^{2}+\gamma_{2}^{2}\right)$.

In high transmittance case, the optimization problem in Eq. (20) is simplified
as follows.

(P2): Sensitivity Maximization in High Transmittance Case via Local Microwave
Detuning

$\Delta_{L}^{**}=\underset{{\Delta_{L}}}{\text{argmax
}}\chi_{1}\left(\Delta_{L}\right).$ (26)

According to Eq. (19) and Eq. (21), we can get the following equivalent
optimization problem,

$w^{**}=\underset{w}{\text{argmax
}}\left\\{\begin{array}[]{c}\frac{z^{3/2}\left(x+y\right)\left(xz+4w\left(x+y\right)\right)}{\left[z^{2}+4w\left(x+y\right)^{2}+2xz\left(x+y+z\right)\right]^{2}}\\\
\end{array}\right\\}.$ (27)

Thus, the optimal value is the maximum point of function with respect to $w$,

$h\left(x,y,z,w\right)=\frac{z^{3/2}\left(x+y\right)\left(xz+4w\left(x+y\right)\right)}{\left[z^{2}+4w\left(x+y\right)^{2}+2xz\left(x+y+z\right)\right]^{2}}.$
(28)

Deriving $h(x,y,z,w)$ with respect to $w$, we have

$\frac{\partial h}{\partial
w}=\frac{4z^{3/2}\left(x+y\right)^{2}\left(\left(2x+1\right)z^{2}-4w\left(x+y\right)^{2}\right)}{\left(4w\left(x+y\right)^{2}+z\left(2x\left(x+y+z\right)+z\right)\right)^{3}}.$
(29)

Since $x,y,z>0$, the optimal value for $w$ is

$w^{**}=\frac{\left(2x+1\right)z^{2}}{4\left(x+y\right)^{2}},$ (30)

and the corresponding local microwave frequency detuning is

$\Delta_{L}^{**}=\frac{\Omega_{L}^{2}}{2\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)}\sqrt{2\Omega_{p}^{2}+\gamma_{2}^{2}}.$
(31)

###  III-C Probing Laser 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
Eq. (32).

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

In weak signal regime, $\Omega_{s}\ll\Omega_{L}$, the first order
approximation of $\Omega_{s}/\Omega_{L}$ is given by Eq. (33).

$\displaystyle\Im(\rho_{21}(\Delta_{p}))$
$\displaystyle\approx\frac{\gamma_{2}\Omega_{p}\left(4\Delta_{p}^{2}-\Omega_{L}^{2}\right)^{2}}{D_{p}(\Omega_{L})}-\frac{4\gamma_{2}\Omega_{p}\Omega_{L}A_{p}\Omega_{s}\cos
S\left(t\right)}{D_{p}^{2}(\Omega_{L})},$ (33) $\displaystyle A_{p}$
$\displaystyle=\left(4\Delta_{p}^{2}-\Omega_{L}^{2}\right)\left(\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{c}^{2}+\Omega_{p}^{2}\right)-16\Delta_{p}^{4}\Omega_{c}^{2}+4\Delta_{p}^{2}\left(\Omega_{c}^{2}\left(\Omega_{c}^{2}+\Omega_{L}^{2}\right)\right)\right)$
$\displaystyle+3\Omega_{p}^{2}\left(4\Delta_{p}^{2}-\Omega_{L}^{2}\right)\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right).$

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

$\Im\left(\chi\left(t,\Delta_{p}\right)\right)=\chi_{0}(\Delta_{p})+\chi_{1}(\Delta_{p})\Omega_{s}\cos
S(t),$ (34)

which has a time-invariant part

$\chi_{0}(\Delta_{p})=-\frac{2N_{0}\mu_{12}^{2}}{\epsilon_{0}{\hbar}\Omega_{p}}\frac{\gamma_{2}\Omega_{p}\left(4\Delta_{p}^{2}-\Omega_{L}^{2}\right)^{2}}{D_{p}(\Omega_{L})},\\\
$ (35)

and a time variant part

$\chi_{1}(\Delta_{p})=\frac{2N_{0}\mu_{12}^{2}}{\epsilon_{0}{\hbar}\Omega_{p}}\frac{4\gamma_{2}\Omega_{p}\Omega_{L}A_{p}}{D_{p}^{2}(\Omega_{L})}.$
(36)

The sensitivity optimization problem in the general case is formulated as
follows.

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

$\Delta_{p}^{*}=\underset{{\Delta_{p}}}{\text{argmax
}}\kappa(\Delta_{p})=\underset{{\Delta_{p}}}{\text{argmax
}}e^{-kL\chi_{0}\left(\Delta_{p}\right)}\chi_{1}\left(\Delta_{p}\right).$ (37)

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

###  III-D Coupling Laser 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
Eq. (38).

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

In weak signal regime, $\Omega_{s}\ll\Omega_{L}$, the first order
approximation of $\Omega_{s}/\Omega_{L}$ is given by Eq. (39).

$\displaystyle\Im(\rho_{21}(\Delta_{c}))$
$\displaystyle\approx\frac{\gamma_{2}\Omega_{p}\left(4\Delta_{c}^{2}-\Omega_{L}^{2}\right)^{2}}{D_{c}(\Omega_{L})}+\frac{4\gamma_{2}\Omega_{p}\Omega_{L}A_{c}\Omega_{s}\cos\left(S\left(t\right)\right)}{D_{c}^{2}(\Omega_{L})},$
(39) $\displaystyle A_{c}$
$\displaystyle=\left(4\Delta_{c}^{2}-\Omega_{L}^{2}\right)\left(\Omega_{L}^{2}\Omega_{p}^{2}\left(\Omega_{p}^{2}+\Omega_{c}^{2}\right)+4\Delta_{c}^{2}\left(\Omega_{c}^{4}+3\Omega_{c}^{2}\Omega_{p}^{2}+3\Omega_{p}^{4}\right)\right).$

The corresponding imaginary 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}(\Delta_{c})=-\frac{2N_{0}\mu_{12}^{2}}{\epsilon_{0}{\hbar}\Omega_{p}}\frac{\gamma_{2}\Omega_{p}\left(4\Delta_{c}^{2}-\Omega_{L}^{2}\right)^{2}}{D_{c}(\Omega_{L})},\\\
$ (41)

and a time variant part

$\chi_{1}(\Delta_{c})=-\frac{2N_{0}\mu_{12}^{2}}{\epsilon_{0}{\hbar}\Omega_{p}}\frac{4\gamma_{2}\Omega_{p}\Omega_{L}A_{c}}{D_{c}^{2}(\Omega_{L})}.$
(42)

The sensitivity optimization problem in the general case is formulated as
follows.

(P4): Sensitivity Maximization in General Case via Coupling Laser Detuning

$\Delta_{c}^{*}=\underset{{\Delta_{c}}}{\text{argmax
}}\kappa(\Delta_{c})=\underset{{\Delta_{c}}}{\text{argmax
}}e^{-kL\chi_{0}\left(\Delta_{c}\right)}\chi_{1}\left(\Delta_{c}\right).$ (43)

We also numerically characterize the coupling laser detuning effect on the
weak signal response in Section IV.

###  III-E Transit Relaxation Effect

Our approach adpots each non-interacting and independent atom that
participates as an identical and stable atomic sensor of the microwave
electric field. Except for spontaneous radiation and atoms collision, transit
relaxation is an important type of relaxation to thermal vapor Rydberg atomic
sensors. Due to random motion of atoms, atoms continuously enter or leave the
area covered by the laser beam. We assume that atoms in the ground state enter
the interaction region from outside the beam at a rate $\gamma$, while the
atoms in different states in the interaction region leave at the same rate.
When the atoms leave, the number of excited atoms and the coherence between
the different states are destroyed. The characteristic time for this process
to occur is determined by the average time of flight through the cross section
of the laser beams, which defines the effective relaxation rate [16]

$\gamma=\sqrt{\frac{8k_{B}T}{\pi m}}\frac{1}{w\sqrt{2\ln 2}},$ (44)

where $w$ is the $1/e^{2}$ beam waist, $m$ is the atom mass, $T$ the ensemble
temperature, and $k_{B}$ is Boltzmann’s constant. Fig. 3 shows the
relationship between the $1/e^{2}$ beam waist and transit relaxation rate for
Cs atoms at $T=300$ K.

Figure 3: Transit relaxation rate to the $1/e^{2}$ beam waist at $T=300$ K.

For non-zero transit relaxation rate $\gamma$, we assume that
$\Delta_{p}=\Delta_{L}=\Delta_{p}=\gamma_{3}=\gamma_{4}=\gamma_{c}=0$. We also
expect that the corresponding imaginary component of linear susceptibility has
a time-invariant part and a time variant part,

$\Im\left(\chi\left(t,\gamma\right)\right)=\chi_{0}(\gamma)+\chi_{1}(\gamma)\Omega_{s}\cos\left(S(t)\right).$
(45)

The sensitivity optimization problem is formulated as follows.

(P5): Sensitivity Maximization in General Case via Transit Relaxation Rate

$\gamma^{*}=\underset{{\gamma}}{\text{argmax
}}\kappa(\gamma)=e^{-kL\chi_{0}\left(\gamma\right)}\chi_{1}\left(\gamma\right).$
(46)

We numerically characterize the transit relaxation 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 $~{}1/400$ for the scheme shown in Fig. 2 [17]. The
effective atoms density for atomic vapor is approximated as
$N_{0,\text{eff}}\approx 10^{8}$ cm-3. The four-level configuration in Fig. 2
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 ($F=4\rightarrow
F=5$) [18]. 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 in
numerical results.

In Section III, we assume that $|kL\chi_{1}(\Delta)\Omega_{s}|\ll 1$ and solve
the optimization problem (P1) to maximize the conversion coefficient
$|\kappa(\Delta)|$. In high transmittance case, the optimization problem is
simplified as (P2) to maximize $|\kappa^{\prime}(\Delta)|$. According to Eq.
(6), the valid range for the general case is estimated as $\Omega_{s}<10$ kHz.
The high transmittance case corresponds to thin medium with $L\leq 1$ cm. For
$L=1$ cm and $\Omega_{s}=2\pi\times 1$ kHz, the numerical values of
$kL\chi_{0}(\Delta_{L})$ and $kL\chi_{1}(\Delta_{L})\Omega_{s}$ are shown in
Fig. 4, which means that the assumptions in general case and high
transmittance case can be satisfied.

Figure 4: The conversion coefficient $kL\chi_{0}(\Delta_{L})$ and
$kL\chi_{1}(\Delta_{L})\Omega_{s}$ with respect to local microwave detuning
$\Delta_{L}$ ($\Omega_{s}=2\pi\times 1$ kHz, $L=1$ cm).

For local microwave frequency detuning, the conversion coefficient
$|\kappa(\Delta_{L})|$ in the general case is shown in Fig. 5, and conversion
coefficient $|\kappa^{\prime}(\Delta_{L})|$ in high transmittance case is
shown in Fig. 6. According to Eq. (25) and Eq. (31), the theoretical optimal
values are marked as dashed red lines. It is seen that the solution in high
transmittance case is numerically close to the original one in the general
case. In the general case, the sensitivity gains at $\Omega_{L}/2\pi=2$ MHz, 4
MHz, 6 MHz by local microwave frequency detuning are 0.06 dB, 0.71 dB, and
1.85 dB, respectively. In high transmittance case, the sensitivity gains at
$\Omega_{L}/2\pi=2$ MHz, 4 MHz, 6 MHz by local microwave frequency detuning
are 0.09 dB, 0.77 dB, and 1.96 dB, respectively. It is predictable that
stronger local microwave power can result in greater sensitivity gains.

Figure 5: Conversion coefficient $|\kappa(\Delta_{L})|$ with respect to local
microwave detuning $\Delta_{L}$ in the general case (P1). Figure 6: Conversion
coefficient $|\kappa^{\prime}(\Delta_{L})|$ with respect to local microwave
detuning $\Delta_{L}$ in high transmittance case (P2).

For probing and coupling laser frequency detunings, conversion coefficients
$|\kappa(\Delta_{p})|$ and $|\kappa(\Delta_{c})|$ in the general case are
shown in Fig. 7 and Fig. 8, respectively. The ideal optimization problem is to
maximize the absolute value of $\kappa(\Delta_{p})$ and $\kappa(\Delta_{c})$,
but the analytical optimal values are obtained by the normal form. Thus, we
can find another local maximal value, corresponding to the minimal value of
$\kappa(\Delta_{p})$ and $\kappa(\Delta_{c})$. In the general case, the
sensitivity gains at $\Omega_{L}/2\pi=2$ MHz, 4 MHz, 6 MHz by probing laser
frequency detuning are 0.85 dB, 1.27 dB, and 2.96 dB, respectively. And the
sensitivity gains at $\Omega_{L}/2\pi=2$ MHz, 4 MHz, 6 MHz by coupling laser
frequency detuning are 1.05 dB, 1.59 dB, and 3.41 dB, respectively.

Figure 7: Conversion coefficient $|\kappa(\Delta_{p})|$ with respect to
probing laser detuning $\Delta_{p}$ in the general case (P3). Figure 8:
Conversion coefficient $|\kappa(\Delta_{c})|$ with respect to coupling laser
detuning $\Delta_{c}$ in the general case (P4).

As the transit relaxation rate increases, the conversion coefficient
decreases, as shown in Fig. 9. We can reduce the influence of the transit
relaxation effect by reducing the size of the atomic cavity and the atomic
number density. Correspondingly, the number of effective atoms involved in the
interaction decreases, and the detection sensitivity reduces.

Figure 9: Conversion coefficient $|\kappa(\gamma)|$ with respect to transit
relaxation rate $\gamma$ in the general case (P5).

##  V Conclusion

Based on the atomic four-level model, 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 signal. We have theoretically obtained the optimal
value of local microwave electric field frequency detuning and numerically
analyzed the effects of probing laser and coupling laser frequency detuning on
the optimization target. In addition, considering the existence of the atomic
transit relaxation effect, we have modified the system Liouville equation, and
derived and shown the influence of the transit relaxation rate on the system
detection sensitivity and the time to reach steady state. Such results show
that setting the appropriate frequency detuning can improve the system
detection sensitivity, resulting in extra several dB sensitivity gain through
this method when the device performance cannot be further improved.
Considering the relationship between the local microwave electric field and
the microwave signal in the heterodyne scheme, such results also mean that the
heterodyne Rydberg atomic receiver can provide the same or better sensitivity
as the resonance case in continuous weak microwave signal detection over a
large frequency range [19]. The relaxation effect reduces the sensitivity of
the system, which should be eliminated or reduced in experiments.

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