††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Generation of spin squeezing via a fully quantum degenerate parametric
amplifier

Yang Liu  Fujian Key Laboratory of Quantum Information and Quantum Optics,
Fuzhou University, Fuzhou 350108, China  Department of Physics, Fuzhou
University, Fuzhou 350108, China  Jie Song  School of Physics, Harbin
Institute of Technology, Harbin 150001, China  Wei Qin  Theoretical Quantum
Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama
351-0198, Japan  Ye-Hong Chen  Theoretical Quantum Physics Laboratory, RIKEN
Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan  Quantum
Information Physics Theory Research Team, RIKEN Center for Quantum computing
(RQC), Wako-shi, Saitama 351-0198, Japan  Fujian Key Laboratory of Quantum
Information and Quantum Optics, Fuzhou University, Fuzhou 350108, China
Department of Physics, Fuzhou University, Fuzhou 350108, China  Yan Xia
Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou
University, Fuzhou 350108, China  Department of Physics, Fuzhou University,
Fuzhou 350108, China

###### Abstract

Spin squeezing is one of the most attractive methods for realizing high-
precision metrology. In this paper, we propose a protocol for generating spin
squeezing in an atomic ensemble via a fully quantum degenerate parametric
amplifier. We discuss the properties of generating spin squeezing with and
without driving the pump cavity. Numerical simulation results show that the
generated spin squeezing strength is sizable, and is able to be comparable to
that obtained using a two-axis twisting (TAT) model. Moreover, we demonstrate
that the protocol is experimentally feasible by introducing the corresponding
experimental parameters. Therefore, the proposed protocol provides a promising
approach to realize spin squeezing in photon-spin coupling systems.

##  I INTRODUCTION

Spin squeezing, which reduces the fluctuation noise of one quadrature in phase
space but increases the fluctuation noise of the other quadrature, has shown
its advantages in quantum metrology [1, 2, 3, 4]. Up to now, spin squeezing
has been applied in many fields requiring high-precision measurements, such as
Ramsey spectroscopy [5, 6, 7, 8], atomic clocks [9, 10, 11], and
gravitational-wave interferometers [12, 13]. Due to these promising
applications, significant efforts have been devoted to generating spin
squeezing in many physical systems, such as molecules [14, 15] and atomic
ensembles [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33]. Among the proposed protocols for atomic ensembles, the basic methods rely
upon, e.g., quantum nondemolition measurement (QNDM) [23, 24], and nonlinear
one-axis twisting (OAT) [25, 26] or two-axis twisting (TAT) [31, 32, 33] spin-
spin coupling. It has been shown that different methods have different
suppression effects on quantum fluctuations. For an ensemble with $N$ atoms,
the maximum amounts of squeezing, obtained with QNDM, OAT, and TAT, scale as
${N^{-1/2}}$, ${N^{-2/3}}$, and ${N^{-1}}$(the ideal Heisenberg limit),
respectively [1].

Due to the ability to reduce the quantum fluctuation noise to the fundamental
Heisenberg limit, TAT squeezing is considered as superior to other methods.
Thus in the past few decades, many protocols [30, 31, 32, 33] have been
proposed to generate spin squeezing by constructing the TAT interaction. These
protocols include, e.g., exploiting the Raman processes in the the Bose-
Einstein condensates (BEC) [30], transforming from the OAT interaction [31],
coupling the spin ensembles with a parametric driven cavity [32], and
modifying a phase-locked atom-photon coupling [33]. However, there are always
some difficulties in applying these theoretical protocols to experimental
implementations. Indeed, these difficulties include, e.g., the imperfect
cooling [30], the imprecise time control [31], the introduced squeezing noise
[32], and the complex pump drivings [33]. Recently, Macri $et\ al.$ [27]
proposed an interesting protocol for generating spin squeezing via an
effective cavity-induced TAT-like interaction constructed by one-photon-two-
atom excitation processes. This protocol paves a promising way to construct
the TAT model and seems to be able to generate significant amount of
squeezing. However, this protocol replies on a specific atom-cavity coupling
(e.g., a transverse coupling), which cannot be collectively enhanced in atomic
ensembles. This makes the strength of one-photon-two-atom processes extremely
weak in typical ensemble-cavity systems.

To address the problem, we propose a protocol for generating spin squeezing in
atomic ensembles by using a fully quantum degenerate parametric amplifier
(DPA) [34]. Here, the DPA is represented by two parametrically coupled single-
mode cavities, i.e., a pump cavity and a signal cavity. An effective
Hamiltonian describing the cavity-induced TAT-like interaction is obtained
through tuning the system parameters. The strength of the generated spin
squeezing is determined by the properties of the pump cavity, such as the
initial state, driving strength, and cavity decay. Specifically, we study the
generation of spin squeezing with and without driving the pump cavity mode.
Numerical simulations show that with and without driving the pump cavity mode,
a sizable spin squeezing can be generated. In particular, for a strong driving
strength and a strong pump cavity decay, the resulting spin squeezing strength
is comparable with that of the TAT squeezing. Meanwhile, we investigate the
sensitivity of the generated spin squeezing to the decoherence, including the
decay of the cavities, the spontaneous emission of the atoms, and the
collective dephasing of the atomic ensemble. The results show that the
protocol is robust to the spontaneous emission of the atoms and the decay of
the signal cavity. Additionally, the experimental feasibility of the protocol
is also discussed by using current experimental parameters [35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46].

The rest part of the paper is organized as follows. In Sec. II, we give a
brief description about the physical model and derive its effective
Hamiltonian. In Sec. III, we study the generation of spin squeezing in an
atomic ensemble. In Sec. IV, we show the experimental feasibility of the
protocol. The conclusion of the present paper is summarised in Sec. V.

##  II PHYSICAL MODEL AND EFFECTIVE DYNAMICS

In this paper, as shown in Fig. 1, we consider a system consisting of two
parametrically coupled single-mode cavities (a pump cavity and a signal
cavity), and an ensemble of $N$ identical two-level atoms placed in the signal
cavity. The resonance frequencies of the pump cavity and the signal cavity are
assumed to be $\omega_{p}$ and $\omega_{s}$, respectively. The parametric
coupling of strength $J$ describes a nonlinear conversion between a single
pump photon and a pair of signal photons. Note that the strength $J$, which
ranges from some tens of kHz to some tens of MHz, has been realized in recent
experimental advances [35, 36, 37, 38]. Therefore, the pump cavity and the
signal cavity constitute a fully quantum degenerate parametric amplifier
(DPA). We apply a classical driving field of frequency $\omega_{l}$ and
amplitude $\Omega$ to the pump cavity. Meanwhile, the atoms of transition
frequency $\omega_{q}$ couple to the signal cavity with a strength $g$. The
Hamiltonian of the system can accordingly be written as (in the units of
$\hbar$) [47]

$\displaystyle\hat{H}_{\text{sys}}$ $\displaystyle=$
$\displaystyle\omega_{s}\hat{a}^{\dagger}_{s}\hat{a}_{s}+\omega_{p}\hat{a}^{\dagger}_{p}\hat{a}_{p}+\omega_{q}\hat{S}_{z}+g(\hat{a}_{s}\hat{S}_{+}+\hat{a}^{\dagger}_{s}\hat{S}_{-})$
(3)
$\displaystyle+\left(\frac{\Omega^{*}}{2}\hat{a}_{p}e^{i\omega_{l}t}+\frac{\Omega}{2}\hat{a}^{\dagger}_{p}e^{-i\omega_{l}t}\right)+J(\hat{a}_{p}\hat{a}^{\dagger
2}_{s}+\hat{a}^{\dagger}_{p}\hat{a}^{2}_{s}),$

where $\hat{a}^{\dagger}_{s}(\hat{a}^{\dagger}_{p})$ and
$\hat{a}_{s}(\hat{a}_{p})$ are the creation operator and the annihilation
operator of the signal (pump) cavity, respectively. Moreover,
${\hat{S}_{u}=\sum_{k}\hat{\sigma}^{u}_{k}/2}$ $(k=1,2,\dots,N)$ is the
collective spin operator and $\hat{\sigma}^{u}_{k}$ ${(u=x,y,z)}$ is the Pauli
operator of the $k$th atom. ${\hat{S}_{+}=\hat{S}_{x}+i\hat{S}_{y}}$ and
${\hat{S}_{-}=\hat{S}_{x}-i\hat{S}_{y}}$ are the raising and lowering
operators of the collective spin, respectively. For simplicity, we set
hereafter ${g_{c}=\sqrt{N}g}$ as the strength of the collective coupling
between the ensemble and the cavity, and further assume that
$\omega_{p}\simeq\omega_{l}\simeq 2\omega_{q}\simeq 2\omega_{s}$. In the
rotation frame of reference of
$\hat{H}_{0}=\omega_{l}(\hat{a}^{\dagger}_{s}\hat{a}_{s}+2\hat{a}^{\dagger}_{p}\hat{a}_{p}+\hat{S}_{z})$,
when ${\omega_{l}\gg\\{g_{c},\,J\\}}$, one can obtain

$\displaystyle\hat{H}_{I}$ $\displaystyle=$
$\displaystyle\delta_{p}\hat{a}^{\dagger}_{p}\hat{a}_{p}+\delta_{s}\hat{a}^{\dagger}_{s}\hat{a}_{s}+\delta_{q}\hat{S}_{z}$
(8) $\displaystyle+(J\hat{a}_{p}\hat{a}^{\dagger
2}_{s}+g\hat{a}_{s}\hat{S}_{+}+\frac{\Omega^{*}}{2}\hat{a}_{p}+\text{H.c.}),$

where ${\delta_{q}=\omega_{q}-\omega_{l}}$,
${\delta_{s}=\omega_{s}-\omega_{l}}$, and
${\delta_{p}=\omega_{p}-2\omega_{l}}$.

Figure 1: Physical model of the protocol. Two single-mode cavities, the pump
cavity and the signal cavity, are coupled with a parametric coupling of
strength $J$. An ensemble of $N$ identical two-level atoms is located inside
the signal cavity, and is coupled to this cavity with a single-photon single-
atom coupling strength $g$. $\omega_{p}(\omega_{s})$ is the resonance
frequency of the pump (signal) cavity. A driving field of frequency
$\omega_{l}$ and amplitude $\Omega$ is applied to the pump cavity.
$\omega_{q}$ is the transition frequency of atoms, $\kappa_{p}(\kappa_{s})$ is
the single-photon dissipation rate of the pump (signal) cavity, $\gamma_{s}$
is the spontaneous emission rate of atoms, and $\gamma_{c}$ represents the
collective dephasing of the atomic ensemble.

When the pump cavity suffers a cavity decay, the dynamics of the system can be
estimated by the master equation in the Lindblad form [48]

$\displaystyle\dot{\rho}=i[\rho,\hat{H}_{I}]+\kappa_{p}\mathcal{L}(\hat{a}_{p})\hat{\rho},$
(9)

where for an arbitrary operator $\hat{o}$, the standard Lindblad superoperator
is defined as

$\displaystyle\mathcal{L}(\hat{o})\rho=\hat{o}\rho\hat{o}^{\dagger}-\frac{1}{2}(\hat{o}^{\dagger}\hat{o}\rho+\rho\hat{o}^{\dagger}\hat{o}).$
(10)

Here, $\kappa_{p}$ is the single-photon dissipation rate (i.e., the cavity
decay rate) of the pump cavity. Due to the presence of the driving $\Omega$ of
the pump mode, we first set ${\hat{c}_{p}=\hat{a}_{p}+d}$ to derive the
effective Hamiltonian of the system. Thereafter, the Hamiltonian in Eq. (8) is
transformed to

$\displaystyle\hat{H}_{h}$ $\displaystyle=$
$\displaystyle\delta_{p}(\hat{c}^{\dagger}_{p}-d^{*})(\hat{c}_{p}-d)+\delta_{s}\hat{a}^{\dagger}_{s}\hat{a}_{s}+\delta_{q}\hat{S}_{z}$
(13) $\displaystyle+\left[J(\hat{c}_{p}-d)\hat{a}^{\dagger
2}_{s}+g\hat{a}_{s}\hat{S}_{+}+\frac{\Omega^{*}}{2}\hat{c}_{p}+\text{H.c.}\right].$

Meanwhile, the second term of the master equation in Eq. (9) becomes

$\displaystyle\kappa_{p}\mathcal{L}(\hat{a}_{p})\rho$ $\displaystyle=$
$\displaystyle\kappa_{p}\mathcal{L}(\hat{c}_{p})\rho-i[\hat{H}_{\text{me}},\rho],$
(14)

where
${\hat{H}_{\text{me}}=-i\kappa_{p}(d^{*}\hat{c}_{p}-d\hat{c}^{\dagger}_{p})/2}$.
Then, the total Hamiltonian can be described as

$\displaystyle\hat{H}_{\text{tot}}$ $\displaystyle=$
$\displaystyle\hat{H}_{h}+\hat{H}_{\text{me}}$ (15) $\displaystyle=$
$\displaystyle\delta_{p}\hat{c}^{\dagger}_{p}\hat{c}_{p}+\delta_{s}\hat{a}^{\dagger}_{s}\hat{a}_{s}+\delta_{q}\hat{S}_{z}{-J}\left(d\hat{a}^{\dagger
2}_{s}+d^{*}\hat{a}^{2}_{s}\right)$ (19)
$\displaystyle+\left(\Omega_{d}\hat{c}_{p}+J\hat{c}_{p}\hat{a}^{\dagger
2}_{s}+g\hat{a}_{s}\hat{S}_{+}+\text{H.c.}\right),$

and the dynamics of the system can be modelled by the master equation

$\displaystyle\dot{\rho}=-i[\hat{H}_{\text{tot}},\rho]+\kappa_{p}\mathcal{L}(\hat{c}_{p})\rho,$
(20)

where $\Omega_{d}=(\Omega^{*}-2d^{*}\delta_{p}-i\kappa_{p}d^{*})/2$. For
simplicity, we have set ${d=\Omega/(2\delta_{p}-i\kappa_{p})}$ to eliminate
the driving term with $\Omega_{d}$.

According to the second and fourth terms in Eq. (15), the signal cavity is
squeezed [49, 50, 51, 52, 53, 54]. Applying the Bogoliubov transformation [29,
55, 56, 52, 53, 54]

$\hat{a}_{\texttt{S}}=\cosh(r)\hat{a}_{s}+\sinh(r)e^{i\theta_{\texttt{S}}}\hat{a}^{\dagger}_{s},$

one can diagonalize the signal-cavity Hamiltonian in Eq. (15), and obtain the
Hamiltonian

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\delta_{p}\hat{c}^{\dagger}_{p}\hat{c}_{p}+\delta_{\texttt{S}}\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}+\delta_{q}\hat{S}_{z}$
(25)
$\displaystyle+\Big{\\{}J\big{[}\cosh(r)\hat{a}_{\texttt{S}}+\sinh(r)e^{i\theta_{\texttt{S}}}\hat{a}^{\dagger}_{\texttt{S}}\big{]}^{2}\hat{c}^{\dagger}_{p}$
$\displaystyle+g\big{[}\cosh(r)\hat{a}_{\texttt{S}}+\sinh(r)e^{i\theta_{\texttt{S}}}\hat{a}^{\dagger}_{\texttt{S}}\big{]}\hat{S}_{+}+\text{H.c.}\Big{\\}},$

where ${\delta_{\texttt{S}}=\sqrt{\delta^{2}_{s}-(2J|d|)^{2}}}$,
$\theta_{\texttt{S}}=-\arctan(\text{Im}(d)/\text{Re}(d))$, and
$r=(1/4)\ln[(\delta_{s}+2J|d|)/(\delta_{s}-2J|d|)]$. Here, we set
${\delta_{s}\gg 2J|d|}$, which makes the cavity squeezing parameter $r$
approach $0$. Then, we have ${\cosh(r)\rightarrow 1}$, ${\sinh(r)\rightarrow
r}$, and ${\sinh(2r)\rightarrow 2r}$. Thereafter, the Hamiltonian
$\hat{\mathcal{H}}$ in Eq. (25) becomes approximated by

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle\simeq$
$\displaystyle\delta_{\texttt{S}}\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\mathtt{S}}+\delta_{q}\hat{S}_{z}+\delta_{p}\hat{c}^{\dagger}_{p}\hat{c}_{p}$
(28)
$\displaystyle+\Big{[}J(\hat{a}_{\texttt{S}}+re^{i\theta_{\texttt{S}}}\hat{a}^{\dagger}_{\texttt{S}})^{2}\hat{c}^{\dagger}_{p}+g(\hat{a}_{\texttt{S}}+re^{i\theta_{\texttt{S}}}\hat{a}^{\dagger}_{\texttt{S}})\hat{S}_{+}+\text{H.c.}\Big{]}.$

Furthermore, when the condition ${\delta_{\texttt{S}}\gg\\{J,\,g_{c}\\}}$ is
satisfied, according to the works in Refs. [57, 58], the dynamics of the
system can, up to the third order, be described by the effective Hamiltonian

$\displaystyle\hat{\mathcal{H}}_{\text{eff}}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{H}}^{(1)}_{\text{eff}}+\hat{\mathcal{H}}^{(2)}_{\text{eff}}+\hat{\mathcal{H}}^{(3)}_{\text{eff}},$
(31) $\displaystyle\hat{\mathcal{H}}^{(1)}_{\text{eff}}$ $\displaystyle=$
$\displaystyle\delta_{p}\hat{c}^{\dagger}_{p}\hat{c}_{p}+\delta_{q}\hat{S}_{z}+Jr(2\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}+1)(e^{-i\theta_{\texttt{S}}}\hat{c}_{p}+\text{H.c.}),$
(33) $\displaystyle\hat{\mathcal{H}}^{(2)}_{\text{eff}}$ $\displaystyle=$
$\displaystyle-\frac{J^{2}}{2\delta_{\texttt{S}}}(2\hat{c}^{\dagger}_{p}\hat{c}_{p}+4\hat{c}^{\dagger}_{p}\hat{c}_{p}\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}-\hat{a}^{\dagger}_{\texttt{S}}\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}\hat{a}_{\texttt{S}})$
(37)
$\displaystyle-\frac{g^{2}}{\delta_{\texttt{S}}}(2\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}\hat{S}_{z}+\hat{S}_{+}\hat{S}_{-}),$
$\displaystyle\hat{\mathcal{H}}^{(3)}_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{\delta_{\texttt{S}}}\left[\left(\frac{J}{\delta_{\texttt{S}}}\hat{c}_{p}+re^{i\theta_{\texttt{S}}}\right)\hat{S}_{+}^{2}+\text{H.c.}\right],$
(39)

where $\hat{\mathcal{H}}^{(\mathcal{J})}_{\text{eff}}$ (${\mathcal{J}=1,2,3}$)
represents the $\mathcal{J}$th order processes of the effective Hamiltonian.
Here, we also have neglected the fast-oscillating terms by the rotating wave
approximation. Note that the Hamiltonian
$\hat{\mathcal{H}}^{(3)}_{\text{eff}}$ in Eq. (31) contains the nonlinear
interactions of the atomic ensembles, which is crucial for generating spin
squeezing in the protocol. However, these nonlinear interactions are third-
order processes whose effect would be masked by that of the lower-order
interactions.

To eliminate the influence of the lower-order interactions, we tune the
parameters and make some proper assumptions as shown in the following. First,
we assume that the signal cavity is well cooled so that the signal cavity is
in the vacuum state, i.e.,
${\langle\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}\rangle\simeq 0}$.
Therefore, all the interactions involving
$\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}$, like the terms
$\hat{a}^{\dagger}_{p}\hat{a}_{p}\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}$
and $\hat{a}^{\dagger}_{\texttt{S}}\hat{a}_{\texttt{S}}\hat{S}_{z}$, can be
neglected. Meanwhile, ${\delta_{p}=J^{2}/\delta_{\texttt{S}}}$ is set to
eliminate the Stark shift of the pump cavity. Furthermore, we assume that in
the ensemble, the number of excited atoms is much smaller than the total
number of atoms. Under this condition, the spin operator $\hat{S}_{z}$ can be
divided into two parts, i.e.,

$\displaystyle\hat{S}_{z}=-\frac{N}{2}+\Delta\hat{S}_{z},$ (40)

where $\Delta\hat{S}_{z}$ is a small fluctuation around the ground state.
After substituting Eq. (40) into the relation
$\hat{S}_{z}^{2}-\hat{S}_{z}+\hat{S}_{+}\hat{S}_{-}=N(N+1)/4$, we obtain
(ignoring the constant terms)

$\displaystyle\hat{S}_{+}\hat{S}_{-}\simeq(N+1)\Delta S_{z}=(N+1)\hat{S}_{z}.$
(41)

This means that, when the atomic ensemble is in the low-excitation regime, the
effect of the coupling $\hat{S}_{+}\hat{S}_{-}$ is to produce a Stark shift
given by $(N+1)\hat{S}_{z}$. Therefore, one can choose
${\delta_{q}=(N+1)g^{2}/\delta_{\texttt{S}}}$ to eliminate he influence of the
term $\hat{S}_{+}\hat{S}_{-}$ effectively. Meanwhile, the effect of this term
can be eliminated completely by introducing an auxiliary atomic level and a
driven optical cavity [59, 60, 61] (see more details in Appendix A). In the
rest of paper and numerical simulations, we assume the effect of
$\hat{S}_{+}\hat{S}_{-}$ has been fully compensated and $\delta_{q}=0$.
Correspondingly, the effective Hamiltonian of the system becomes

$\displaystyle\hat{\mathcal{H}}_{\text{eff}}\approx
g_{\text{eff}}(\hat{c}_{p}-d)\hat{S}^{2}_{+}+\frac{J^{2}}{\delta_{\mathtt{S}}}d^{*}\hat{c}_{p}+\text{H.c.},$
(42)

where we have made the approximation ${r\rightarrow J|d|/\delta_{\texttt{S}}}$
and assumed ${g_{\text{eff}}=g^{2}J/{\delta_{\mathtt{S}}^{2}}}$. Then, by
taking corresponding reverse transformations, one can convert the effective
Hamiltonian in Eq. (42) back to the original frame and obtain

$\displaystyle\hat{H}_{\text{eff}}=g_{\text{eff}}(\hat{a}_{p}\hat{S}^{2}_{+}+\hat{a}^{\dagger}_{p}\hat{S}^{2}_{-})+\left(\frac{\Omega^{*}}{2}\hat{a}_{p}+\frac{\Omega}{2}\hat{a}^{\dagger}_{p}\right).$
(43)

Meanwhile, the effective Lindblad-type master equation in Eq. (20) becomes

$\displaystyle\dot{\rho}=i[\rho,\hat{H}_{\text{eff}}]+\kappa_{p}\mathcal{L}(\hat{a}_{p})\rho.$
(44)

The first term in Eq. (43) indicates that the system involves a cavity-induced
TAT-like interaction which can be used to generate spin squeezing [27]. The
built-in mechanism is that when the pump cavity is in a coherent state with
${\langle\hat{a}_{p}\rangle=\beta}$, the effective Hamiltonian in Eq. (43) can
be approximated as

$\displaystyle\hat{H}^{\prime}_{\text{eff}}=g_{\text{eff}}(\beta\hat{S}^{2}_{+}+\beta^{*}\hat{S}^{2}_{-}),$
(45)

i.e., the TAT interaction. Moreover, the TAT model in Eq. (45) can be
established through keeping the pump cavity in a state with
$\langle\hat{a}_{p}\rangle\not=0$ during the evolution. According to Eqs. (43)
and (44), keeping the state in the pump cavity unchanged can be achieved
effectively by increasing the ratio between $\Omega$ and $g_{\text{eff}}$. The
detail is that for a larger ratio $\Omega/g_{\text{eff}}$, the pump cavity
stays in a coherent state with amplitude $d_{0}=i\Omega/\kappa_{p}$
efficiently for a longer period (see below for numerical demonstrations). For
convenience, we take the coherent state as the quasi-steady state of the pump
cavity in the remaining of the text.

So far, we have considered a model, which contains only the photon loss of the
pump mode. In reality, there always exists some dissipative processes for the
atomic ensemble and the signal cavity. Here, we assume that the system suffers
from dissipation induced by the ensemble collective dephasing, the atomic
spontaneous emission, and the signal cavity single-photon loss. The dynamics
of the system can therefore be described by the following master equation

$\displaystyle\dot{\rho}_{I}{=}i[\rho_{I},\hat{H}_{I}]{+}\\!\\!\left[\sum_{v=s,p}\kappa_{v}\mathcal{L}(\hat{a}_{v}){+}\sum_{k=1}^{N}\gamma_{s}\mathcal{L}(\hat{\sigma}^{-}_{k}){+}\gamma_{c}\mathcal{L}(\hat{S}_{z})\right]\\!\rho_{I},$
(46) (47)

where $\rho_{I}$ is the density operator under the full dynamics of the
system. The parameters $\kappa_{s}$, $\gamma_{s}$, and $\gamma_{c}$ are the
single-photon loss rate of the signal cavity, the spontaneous emission rate of
atoms, and the collective dephasing rate of the atomic ensemble, respectively.
In general, the third term in Eq. (46), i.e.,
$\sum_{k=1}^{N}\gamma_{s}\mathcal{L}(\hat{\sigma}^{-}_{k})\rho_{I}$, makes
numeral simulations of a large-size ensemble dynamics extremely difficult
because the demanded computation resources increase exponentially with the
total number of atoms $N$. However, the system dynamics involves only the zero
momentum mode of the atomic ensemble, and also does not mix it with other
nonzero momentum modes. Thus according to Refs. [27, 62, 63, 29], the third
term in Eq. (46) can be reduced to

$\displaystyle\sum_{k}\mathcal{L}(\hat{\sigma}^{-}_{k})\rho_{I}=\frac{1}{N}\mathcal{L}(\hat{S}_{-})\rho_{I}.$
(49)

Then, the full master equation in Eq. (46) becomes

$\displaystyle\dot{\rho}_{I}=i[\rho_{I},\hat{H}_{I}]+\\!\\!\left[\sum_{v=p,s}\\!\\!\kappa_{v}\mathcal{L}(\hat{a}_{v})+\frac{\gamma_{s}}{N}\mathcal{L}(\hat{S}_{-})+\gamma_{c}\mathcal{L}(\hat{S}_{z})\right]\\!\rho_{I}.$
(50) (51)

Accordingly, the effective master equation in Eq. (44) is transformed to

$\displaystyle\dot{\rho}_{\text{eff}}$ $\displaystyle=$ $\displaystyle
i[\rho_{\text{eff}},\hat{H}_{\text{eff}}]$ (55)
$\displaystyle+\\!\left[\sum_{v=s,p}\kappa_{v}\mathcal{L}(\hat{a}_{v}){+}\frac{\gamma_{s}}{N}\mathcal{L}(\hat{S}_{-}){+}\gamma_{c}\mathcal{L}(\hat{S}_{z})\right]\rho_{\text{eff}},$

where $\rho_{\text{eff}}$ represents the density operator under the effective
dynamics of the system. Note that, the part
$\kappa_{s}\mathcal{L}(\hat{a}_{s})$ in Eq. (44) can be subtracted since the
signal cavity has been decoupled from the dynamics of the effective
Hamiltonian in Eq. (43).

As demonstrated above, we have obtained the cavity-induced TAT-like
interaction effectively, and the Lindblad-type master equation to simulate the
dynamics of the system. These enable us to generate spin squeezing and study
the properties of the generated spin squeezing under different decoherence
noise sources.

##  III GENERATING SPIN SQUEEZING

In this section, we investigate the generation of spin squeezing through
theoretical analyses and numerical simulations. First, we need to introduce
the spin squeezing parameter $\xi^{2}_{R}$ proposed by Wineland $et\ al.$ [5,
6, 1]

$\displaystyle\xi^{2}_{R}=N\frac{\langle(\hat{\bm{S}}\cdot\bm{n}_{\bot})^{2}\rangle-\langle\hat{\bm{S}}\cdot\bm{n}_{\bot}\rangle^{2}}{|\langle\hat{\bm{S}}\rangle|^{2}},$
(56)

where
$\hat{\bm{S}}=\hat{S}_{x}\bm{e}_{x}+\hat{S}_{y}\bm{e}_{y}+\hat{S}_{z}\bm{e}_{z}$
and $\bm{e}_{u}(u=x,y,z)$ is the unit vector along the $u$ direction. The unit
vector $\bm{n}_{\bot}$ is along the direction minimizing the numerator. The
spin squeezing parameter represents the ratio of the fluctuations between a
quantum state of interest and a coherent spin state (CSS) in Ramsey
spectroscopy. Here, the CSS acts as a noise-reference state. It is seen from
Eq. (56) that, when $\xi^{2}_{R}<1$, the phase sensitivity of the state of
interest is improved over the standard quantum limit (SQL), i.e., this state
is squeezed. Note that, a smaller $\xi_{R}^{2}$ indicates a stronger spin
squeezing. In this paper, we choose the parameter $\xi^{2}_{R}$ to
characterize the strength of the generated spin squeezing.

To numerically simulate the evolution of the system, we use the Monte Carlo
approach (i.e., the quantum-jump method) [64], where individual quantum
trajectories of the system evolve under a non-Hermitian Hamiltonian, and then
are randomly interrupted by quantum jumps. Moreover, the dynamics of the
system is regarded as an ensemble average over these trajectories of the
system wave functions. The number of the involved trajectories is larger and
the description on the dynamics of the system is more precise. However,
simulating a large number of trajectories also requires many computation
resources. Therefore, after balancing the precise of the numerical simulation
and the requirement of the computation resources, we take the average over
$1000$ trajectories to calculate the dynamics of the system.

In the following two subsections, we assume that initially, the pump cavity is
in a coherent state $|\alpha\rangle_{p}$, where
${\alpha=|\alpha|e^{i\varphi}}$ is the complex amplitude with an argument
${\varphi=\arg(\alpha)}$. At the same time, the signal cavity and the atomic
ensemble are well cooled to their ground states, i.e., $|0\rangle_{s}$ and
$|l,-l\rangle_{e}$, respectively, where ${l=N/2}$. Here, $|j,m_{z}\rangle_{e}$
represents a collective spin state of an ensemble of $N$ spin-1/2 atoms, with
an orbital angular momentum quantum number and a magnetic quantum number
${m_{z}\in\\{-l,-l+1,\dots,l-1,l\\}}$.

###  III.1 WITH DRIVING THE PUMP CAVITY

In this subsection, we study the generation of spin squeezing in the case
where the pump cavity is driven by external fields and is initialized in the
quasi-steady coherent state, i.e., $\alpha=d_{0}=i\Omega/\kappa_{p}$.

Figure 2: Time evolution of (a) the parameters $F_{I(\text{eff})}$,
$|\langle\hat{a}_{p}\rangle|$, and (b) the spin squeezing parameter
$\xi^{2}_{R}$, given by the full master equation in Eq. (50) (solid curve) and
the effective master equation in Eq. (55) (dashed curve) for
${\kappa_{p}=\Omega=g_{c}}$. All other parameters are ${N=50}$,
${\delta_{s}=15g_{c}}$, $J=\sqrt{2}g_{c}$, and
${\kappa_{s}=\gamma_{c}=\gamma_{s}=0}$. (c) Husimi-$Q$ function and spin
squeezing direction (red dashed curve) for the strongest spin squeezing
(corresponding to the dots in (b)). Here, $\rho_{I}$ and $\rho_{\text{eff}}$
represent the states given by the full master equation in Eq. (50) and the
effective master equation in Eq. (55), respectively.

First, we assume that the dynamics of the system can be well described by the
master equation in Eq. (9). From the discussion below Eq. (44), one can find
that the strength of the generated spin squeezing is determined by the state
of the pump cavity during the evolution. Therefore, the evolution of the pump
cavity is worth studying. According to Eqs. (43) and (44), the pump cavity
seems to stay in a coherent state effectively during the evolution when one
sets ${\\{\kappa_{p},\Omega\\}\gg g_{\text{eff}}}$. Thus, we introduce a
coherent state $|\beta_{p}\rangle_{p}$ whose amplitude is
$\beta_{p}\equiv\langle\hat{a}_{p}\rangle_{p}$ (time-dependent), as a
reference state, and define the fidelity
$F_{I(\text{eff})}={{}_{p}\langle}\beta_{p}|\rho_{I(\text{eff})}|\beta_{p}\rangle_{p}$
to study the state of the pump cavity. From Fig. 2, we can obverse that both
$F_{I}$ and $F_{\text{eff}}$ are always greater than $0.995$ within the
evolution time of $t=300/g_{c}$. In other words, the pump cavity stays in the
coherent state effectively during the evolution. Therefore, according to Eq.
(45), the dynamics of the ensemble can be well described by the TAT model if
the pump cavity is initially in a coherent state. Therefore, the protocol
could give rise to a strong squeezing strength. According to Fig. 2, we find
that under the chosen parameters, the protocol is able to generate a spin
squeezing of $\xi^{2}_{R}\sim-9.24\,\text{dB}$ in an ensemble of $N=50$ atoms.

Figure 3: (a) Time evolution of the spin squeezing parameter $\xi^{2}_{R}$ for
an ensemble of ${N=50}$ atoms, given by the full master equation in Eq. (50).
(b) Minimum values of the spin squeezing parameter $\xi^{2}_{R,{\text{min}}}$
for different $N$, given by the effective master equation in Eq. (55). We
assumed that ${\kappa_{p}=g_{c}}$ (red solid curve) and ${\kappa_{p}=5g_{c}}$
(blue dashed curve) for the present protocol, and that
$g_{\text{tat}}=d_{0}g_{\text{eff}}=ig_{\text{eff}}$ (black dotted curve) for
the TAT protocol. Other parameters are ${\Omega=\kappa_{p}}$,
${\delta_{s}=15g_{c}}$, $J=\sqrt{2}g_{c}$, and
$\kappa_{s}=\gamma_{c}=\gamma_{s}=0$.

Moreover, the squeezing direction is also an essential property of spin
squeezing. From the cavity-induced TAT-like interaction and the initial
coherent state of the pump cavity,
${\langle\hat{S}_{x}\rangle=\langle\hat{S}_{y}\rangle=0}$ is maintained during
the evolution of the system. Therefore, the mean-spin direction is along the
$z$ direction and the spin squeezing only occurs in the ${x{-}y}$ plane. To
find the spin squeezing direction $\bm{n}_{\perp}$, we randomly choose a
direction of the $x{-}y$ plane
${\bm{n}_{r}=\cos\theta\,\bm{e}_{x}+\sin\theta\,\bm{e}_{y}}$, where $\theta$
is the polar angle. As discussed above, the argument $\varphi$ is maintained
effectively during the evolution. According to Eq. (45) and Ref. [1], the
squeezing direction $\bm{n}_{\perp}$ satisfies $\theta=\pi/4+\varphi/2$. Here,
we demonstrate the spin squeezing direction at the moment where the strongest
squeezing occurs (the dots in Fig. 2). To show the spin squeezing direction
intuitively, we introduce the Husimi-$Q$ function which represents the
quasiprobability distribution of any spin states [1]. The Husimi-$Q$ function
is defined as

$\displaystyle
Q_{I(\text{eff})}=\langle\text{CSS}|\hat{R}^{\dagger}(\theta_{Q},\phi_{Q})\rho_{I(\text{eff})}\hat{R}(\theta_{Q},\phi_{Q})|\text{CSS}\rangle,$

where $|\text{CSS}\rangle$ represents a CSS with all the atoms in their
excited states and
$\hat{R}(\theta_{Q},\phi_{Q})=\exp[i\theta_{Q}(\hat{S}_{x}\sin\phi_{Q}-\hat{S}_{y}\cos\phi_{Q})]$
is a rotation operator [1]. From Fig. 2, we find that the spin squeezing
direction under the full dynamics of the system deviates slightly from the
effective prediction (i.e., $y$ axis). In other words, the analysis about the
spin squeezing direction is effective. Moreover, from Figs. 2(a-c), we have
demonstrated the validity of the effective master equation to describe the
dynamics of the full system. These results show that the assumptions and the
approximations we have made in Sec. II are suitable.

Figure 4: Time evolution of the spin squeezing parameter $\xi^{2}_{R}$ for
different $\gamma_{s}$. The involved spontaneous emission rates $\gamma_{s}$
are set to be: ${10^{-3}g_{c}}$ (for $\rho_{I}$, red solid curve, and for
$\rho_{\text{eff}}$, blue dashed curve) and $0$ (for $\rho_{I}$, black dotted
curve). All other parameters are same as those in Fig. 2. Figure 5: Time
evolution of the spin squeezing parameter $\xi^{2}_{R}$ for different
$\gamma_{c}$. The involved dephasing rates $\gamma_{c}$ are set to be:
${10^{-3}g_{c}}$ (for $\rho_{I}$, red solid curve, and for
$\rho_{\text{eff}}$, blue dashed curve) and ${0}$ (for $\rho_{I}$, black
dotted curve). All other parameters are same as those in Fig. 2.

According to the mechanism of the cavity-induced TAT-like interaction in Sec.
II, when the condition $\\{\kappa_{p},\Omega\\}\gg g_{\text{eff}}$ is
satisfied, the dynamics of the system is consistent with that of the TAT
model. To confirm this, we now introduce the TAT protocol, as a reference. The
Hamiltonian in the TAT protocol is given by

$\displaystyle\hat{H}_{\text{TAT}}=g_{\text{tat}}\hat{S}^{2}_{+}+g^{*}_{\text{tat}}\hat{S}^{2}_{-},$
(57)

where ${g_{\text{tat}}=d_{0}g_{\text{eff}}}$. Compared to the present
protocol, the TAT protocol is equivalent to a case where the pump cavity stays
in the quasi-steady state during the evolution. Thereafter, we numerically
compare the present protocol and the TAT protocol. Figure 3 plots the
evolution of $\xi_{R}^{2}$ verse $t$, showing that a larger $\kappa_{p}$
increases the similarity between the present protocol and the TAT protocol. In
particular, all the curves are in clear agreement with each other at the
beginning of the evolution. The reason is that, from Fig. 2, we can obtain
that the pump cavity is in the quasi-steady state effectively at the beginning
of the evolution. From Fig. 3, one can find that for a small ensemble, the
present protocol can generate the same spin squeezing strength as the TAT
protocol, e.g., ${\Omega=\kappa_{p}=5g_{c}}$ for ${N=10}$. Moreover, it is
also seen that, as the number of atoms increases, the minimum value of the
spin squeezing parameter $\xi^{2}_{R,\min}$ is decreased for both the present
protocol and the TAT protocol. However, an increasing in the number of atoms
causes the strongest spin squeezing generated with the present protocol to
gradually deviate from that generated with the TAT protocol. The reason is
that the generation of spin squeezing in an ensemble with more atoms generally
leads more atoms to be excited, which would cause the state of the pump cavity
to deviate more from the quasi-steady coherent state. Moreover, according to
Figs. 3 and 3, a larger $\kappa_{p}$ results in a stronger spin squeezing.
This means that a setup with a larger single-photon dissipation of the pump
cavity and a stronger external driving field is more efficient for the present
protocol to generate a stronger spin squeezing.

Then, we study the influence of the spontaneous emission of the atoms and the
collective dephasing of the ensemble. From Fig. 4, we find that the effect of
spontaneous emission prolongs the duration of the squeezing at the expense of
reducing the squeezing strength slightly. Meanwhile, it can be seen from Fig.
5 that, at the same intensity, the influence of the collective dephasing of
the ensemble on the generated spin squeezing is much greater than that caused
by the atomic spontaneous emission. The essential reason is that the
collective dephasing would affect the relative phases between the energy
levels of the ensemble, so that the direction of the squeezing gradually
becomes chaotic during the evolution.

Figure 6: Time evolution of the spin squeezing parameter $\xi^{2}_{R}$ for
different $\kappa_{s}$. The involved signal cavity decay rates $\kappa_{s}$
are set to be: ${0.1g_{c}}$ (for $\rho_{I}$, red solid curve, and for
$\rho_{\text{eff}}$, blue dashed curve) and ${0}$ (for $\rho_{I}$, black
dotted curve). All other parameters are same as those in Fig. 2.

Though the signal cavity is decoupled from the effective dynamics described by
Eq. (55), the signal cavity decay affects inevitably the full dynamics of the
system in reality. Here, we discuss the influence of the signal cavity decay
on the system. With the help of the theory of the adiabatic elimination [65,
66], we can replace $\gamma_{s}\mathcal{L}(\hat{a}_{s})\rho$ with
$J^{2}p_{\gamma}\mathcal{L}({\hat{a}_{p}})\rho+g^{2}p_{\gamma}\mathcal{L}(\hat{S}_{-})\rho$,
where ${p_{\gamma}=4\gamma_{s}/(4\delta^{2}_{s}+\gamma^{2}_{s})}$, in the
master equation in Eqs. (50) and (55) (see more details in Appendix B).
Therefore, the effect of the signal cavity decay causes an additional single-
photon dissipation of the pump cavity and an additional collective spontaneous
emission of the atoms. From Sec. II, the condition ${\kappa_{p}\gg\
J^{2}p_{\gamma}}$ is well satisfied, which means that the adiabatic effect on
the pump cavity may be not obvious. Then, the signal cavity decay mainly
produces an additional collective spontaneous emission. This analysis can be
verified indirectly by comparing the evolution of the spin squeezing parameter
$\xi^{2}_{R}$ in Figs. 4 and 6.

###  III.2 WITHOUT DRIVING THE PUMP CAVITY

Figure 7: Time evolution of (a) the parameters $F_{I(\text{eff})}$,
$|\langle\hat{a}_{p}\rangle|$, and (b) the spin squeezing parameter
$\xi^{2}_{R}$, given by the full master equation in Eq. (50) (the solid curve)
and the effective master equation in Eq. (55) (the dashed curve) for
${\alpha=i}$. All other parameters are ${N=50}$, ${\delta_{s}=15g_{c}}$,
$J=\sqrt{2}g_{c}$, and $\Omega=\kappa_{p}=\kappa_{s}=\gamma_{c}=\gamma_{s}=0$.
(c) Husimi-$Q$ function and spin squeezing direction (the red dashed curve)
for the strongest spin squeezing (corresponding to the dots in (b)).

In this subsection, we discuss the property of the generated spin squeezing
without driving the pump cavity. Here, the effective Hamiltonian in Eq. (43)
is given by

$\hat{H}^{\prime}_{\text{eff}}=g^{\prime}_{\text{eff}}(\hat{a}_{p}\hat{S}^{2}_{+}+\hat{a}^{\dagger}_{p}\hat{S}^{2}_{-}),$
(58)

where $g^{\prime}_{\text{eff}}=g^{2}J/{\delta_{s}^{2}}$.

Figure 8: Time evolution of the spin squeezing parameter $\xi^{2}_{R}$ for
different $\alpha$, given by the full master equation in Eq. (50). The
involved amplitudes of the initial coherent state $\alpha$ are set to be:
${i}$ (red solid curve), ${1.5i}$ (blue dashed curve), and ${2i}$ (black
dotted curve). All other parameters are same as those in Fig. 7.

First, we study the generation of spin squeezing in the ideal case of no
decoherence involved. Similar to the Sec. III.1, we start from studying the
coherence properties of the pump cavity with a coherent state as a reference
state. Without the driving, the amplitude of the reference state is decreased
sharply in the early stage, as shown in Fig. 7. This indicates that the
initial coherent state of the pump cavity is destroyed, which, according to
Eq. (45), limits the strength of the generated spin squeezing. As shown in
Fig. 7, the protocol is able to generate a spin squeezing of
${\xi^{2}_{R}\sim-5.19\,\text{dB}}$ in an ensemble of ${N=50}$ atoms.
Meanwhile, from Figs. 7 and 7, we find that the fidelity of the reference
coherent state exceeds $0.98$ before the strongest squeezing occurs. In other
words, the pump cavity is in a coherent state approximately. Therefore, the
prediction of the spin squeezing direction along $\theta=\pi/4+\varphi/2$ in
the $x{-y}$ plane is still valid. From Fig. 7, the spin squeezing direction
based on the full master equation only has a small deviation from the
effective prediction (i.e., along the $y$ axis). Moreover, from Figs. 7 and 7,
the effective master equation is also still valid to describe the system
dynamics. However, the effective master equation, when used to predict the
evolution of $\xi^{2}_{R}$, has a big deviation due to the sensitivity of
$\xi^{2}_{R}$ to the parameters. Therefore, the following numerical
simulations in this subsection are all based on the full master equation.

According to Figs. 7 and 7, an increase in the amplitude of the initial
coherent state of the pump cavity might lead to a faster and stronger spin
squeezing. Here, we numerically study this deduction and plot the
corresponding results in Fig. 8. From Fig. 8, a larger-amplitude coherent
state prepared in the pump cavity accelerates the process of generating spin
squeezing. The reason is that according to Eq. (45), an increase in the
amplitude of the coherent state in the pump cavity increases the coupling
strength of the nonlinear interaction between the atoms. But at the same time,
a larger-amplitude coherent state in the pump cavity also involves a violation
of the assumption $\delta_{s}\gg 2|\alpha|J$. This results in a decrease of
the generated squeezing strength, as shown in Fig. 8.

Figure 9: Time evolution of the spin squeezing parameter $\xi^{2}_{R}$ for
different $\gamma_{s}$, given by the full master equation in Eq. (50). The
involved atomic spontaneous emission rates $\gamma_{s}$ are set to be:
${10^{-3}g_{c}}$ (red solid curve) and ${0}$ (blue dashed curve). All other
parameters are same as those in Fig. 7. Figure 10: Time evolution of the spin
squeezing parameter $\xi^{2}_{R}$ for different $\gamma_{c}$, given by the
full master equation in Eq. (50). The involved collective dephasing rates
$\gamma_{c}$ are set to be: ${10^{-3}g_{c}}$ (red solid curve) and ${0}$ (blue
dashed curve). All other parameters are same as those in Fig. 7.

It is worth studying further the effects of the spontaneous emission of the
atoms and the collective dephasing of the ensemble on the system. As shown in
Fig. 9, the spontaneous emission of the atoms slightly influence on the
evolution of spin squeezing. Compared to the case with driving the pump cavity
in Sec. III.1, the atomic spontaneous emission cannot prolong the duration of
spin squeezing in the case of no pump cavity driving. According to Figs. 9 and
10, when setting $\gamma_{s}=\gamma_{c}$, the dephasing of the ensemble has a
greater effect on the generation of spin squeezing than the atomic spontaneous
emission. Meanwhile, from Figs. 5 and 10, it can be found that in the case of
no pump cavity driving, the collective dephasing of the ensemble has a weaker
influence on the generation of spin squeezing than that in the case with
driving the pump cavity.

Figure 11: Time evolution of the spin squeezing parameter $\xi^{2}_{R}$ for
different $\kappa_{p}$, given by the full master equation in Eq. (50). The
involved single-photon dissipation rates of the pump cavity $\kappa_{p}$ are
set to be: $0.01g_{c}$ (red solid curve), $0.1g_{c}$ (blue dashed curve), and
$0$ (black dotted curve). All other parameters are same as those in Fig. 7.
Figure 12: Time evolution of the spin squeezing parameter $\xi^{2}_{R}$ for
different $\kappa_{s}$, given by the full master equation in Eq. (50). The
involved single-photon dissipation rates of the signal cavity $\kappa_{s}$ are
set to be: $0.1g_{c}$ (red solid curve) and $0$ (blue dashed curve). All other
parameters are same as those in Fig. 7.

Furthermore, the effect of the cavity decay on the system also needs to be
investigated. Here, the pump cavity decay becomes an undesired physical
process. According to Fig. 11, the pump cavity decay can significantly prolong
the duration of squeezing, but at the same time, it can also clearly reduce
the strength of squeezing. This indicates that the protocol in the case
without driving the pump cavity is particularly suitable for a system with a
high-quality pump cavity. Meanwhile, as discussed in Sec. III.1, the influence
of the signal cavity decay is equivalent to introducing an extra pump cavity
decay and an extra atomic collective spontaneous emission. It is worth noting
that the extra pump cavity decay cannot be ignored like in the case with
driving the pump cavity. Thus, the robustness of the generated spin squeezing
to the signal cavity decay would be reduced, which is shown in Fig. 12.

##  IV EXPERIMENTAL FEASIBILITY

Figure 13: Schematic for a possible architecture for the protocol. The CPWR I
and the CPWR II are coupled to the SQUID. The atomic ensemble is placed above
the CPWR II and is coupled to the magnetic mode of the CPWR II. An extra
driving is applied to the CPWR I. Table 1: Experimental feasible parameters
and the minimum of spin squeezing parameters

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | Type of ensemble
| Parameter |  |  | Rb atoms |  |  | NV centers |
| encoded states |  |  | $|1,\,-1\rangle\leftrightarrow|2,\,1\rangle$ |  |  | $|m_{s}=0\rangle\leftrightarrow|m_{s}=-1\rangle$ |
| number of spins |  |  | $10^{6}$ |  |  | $10^{12}$ |
| spin frequency |  |  | 6.8324 |  |  | 2.6899 |
|  $\omega_{q}/2\pi$ (GHz) |  |  |  |  |
| spin spontaneous emission rate |  |  | — |  |  | — |
|  $\gamma_{s}/2\pi$ (kHz) |  |  |  |  |
| spin collective dephasing rate |  |  | — |  |  | 0.26 |
|  $\gamma_{c}/2\pi$ (kHz) |  |  |  |  |
| collective coupling strength |  |  | 40 |  |  | 1.200$\times 10^{4}$ |
|  $g_{c}/2\pi$ (kHz) |  |  |  |  |
| signal cavity frequency |  |  | 6.8330 |  |  | 2.8691 |
|  $\omega_{s}/2\pi$ (GHz) |  |  |  |  |
| signal cavity decay rate |  |  | 7 |  |  | 3 |
|  $\kappa_{s}/2\pi$ (kHz) |  |  |  |  |
| parameter coupling strength |  |  | 56.569 |  |  | 1.697$\times 10^{4}$ |
|  $J/2\pi$ (kHz) |  |  |  |  |
| SQUID’s pumping frequency |  |  | 10 |  |  | 2 |
|  $\omega_{\text{SQUID}}/2\pi$ (GHz) |  |  |  |  |
| pump cavity frequency |  |  | 3.6660 |  |  | 3.7389 |
|  $\omega_{p}/2\pi$ (GHz) |  |  |  |  |
| external field |  |  | driving |  |  | driving |  |  | no driving |  |  | driving |  |  | driving |  |  | no driving |
| driving amplitude |  |  | 10 |  |  | 10 |  |  | — |  |  | 10 |  |  | 10 |  |  | — |
|  $\Omega/2\pi$ (MHz) |  |  |  |  |
| pump cavity decay rate |  |  | 10 |  |  | 10 |  |  | $0.003$ |  |  | 10 |  |  | 10 |  |  | $0.003$ |
|  $\kappa_{p}/2\pi$ (MHz) |  |  |  |  |
| amplitude of the initial |  |  | $0$ |  |  | $i$ |  |  | $i$ |  |  | $0$ |  |  | $i$ |  |  | $i$ |
| coherent state $\alpha$ |  |  |  |  |  |  |
| minimum of the spin squeezing |  |  | $-15.13$ |  |  | $-14.91$ |  |  | $-2.34$ |  |  | $-13.58$ |  |  | $-13.58$ |  |  | $-9.51$ |
| parameter $\xi^{2}_{\mathcal{R},\min}$ (dB)1 |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

  * 1

Parameter $\xi^{2}_{\mathcal{R},\min}$ differs from the parameter
$\xi^{2}_{R,\min}$ in Sec. III. The details are shown in Appendix C.

In order to demonstrate further the performance of the protocol, we discuss
the experimental feasibility by combining the theoretical model in the
protocol with the current experiments.

Here, we consider that a setup consisting of two coplanar waveguide resonators
(CPWRs), a superconducting quantum interference device (SQUID), and a rubidium
(Rb) atomic ensemble, as shown in Fig. 13. The SQUID is used to mediate the
parametric conversion between a single photon of the pump cavity (CPWR I) and
a pair of photons of the signal cavity (CPWR II). In other words, the SQUID
constructs the parametric coupling between two cavities. The pumping frequency
of the SQUID satisfies ${\omega_{\text{SQUID}}=2\omega_{s}-\omega_{p}}$. The
SQUID has been designed and realized in many experiments, like the
asymmetrically threaded SQUID (ATS) [37] and the rf-SQUID [38]. The strength
of the parametric coupling constructed by the SQUID has been reported with a
range of ${J/2\pi=0.1\sim 17.7}$ MHz [35, 36, 37, 38]. Meanwhile, we take the
atomic clock states ${\\{5S^{1/2},\,F=1,\,m_{F}=-1\\}:=|1,\,-1\rangle}$ and
${\\{5S^{1/2},\,F=2,\,m_{F}=1\\}:=|2,\,1\rangle}$ as the information carrier
[39, 40]. To date, coupling these two states to the CPWR has been
experimentally implemented with an additional rf field [39]. However, the
reported coupling strength is too weak to implement efficient quantum coherent
operations [41]. Fortunately, several approaches can be used to increase the
coupling strength. Such approaches include decreasing the space between the
atomic ensemble and the CPWR [39], and introducing novel CPWRs of being able
to provide some strong magnetic fields [42]. Thus, here, we take
$g_{c}/2\pi=40$ kHz with $N=10^{6}$ [40]. Note that, both the relaxation time
$T_{1}$ and the coherent time $T_{2}$ of the Rb atomic ensemble are of the
order sec [43]. This indicates that one can ignore the effect of the atomic
spontaneous emission and the collective dephasing on the dynamics of the
system. In addition, for the CPWRs, the quality factor over $10^{6}$ has been
realized experimentally [44, 45, 46]. Thus, we take ${\kappa_{p}\sim
10^{-6}\omega_{p}}$ and ${\kappa_{s}\sim 10^{-6}\omega_{s}}$. According to
these parameters mentioned above, we list a group of feasible parameters and
then estimate the minimum of the spin squeezing parameter in Table 1. Note
that, for the case with driving (discussed in Sec. III.1), it is not necessary
that the initial state of the pump cavity is the quasi-steady coherent state.
As shown in Fig. 14, the generation of spin squeezing is also able to be
achieved when the pump cavity is in the ground state initially. Meanwhile,
from Fig. 14, one can find that the generations of spin squeezing between
these two cases with different initial states of the pump cavity are almost
the same. The reason of this similarity is that the time cost of constructing
the quasi-steady coherent state from the ground state in the pump cavity is
too short to change the generation of spin squeezing significantly.

Figure 14: Time evolution of (a) the parameter $|\langle\hat{a}_{p}\rangle|$,
and (b) the spin squeezing parameter $\xi^{2}_{R}$ for the pump cavity
initiated in the ground state (the blue solid curve) and the quasi-steady
state (the red dotted curve), given by the full master equation in Eq. (50).
All other parameters are ${N=50}$, ${\delta_{s}=15g_{c}}$,
$\Omega=\kappa_{p}$, $J=\sqrt{2}g_{c}$, and
$\kappa_{s}=\gamma_{c}=\gamma_{s}=0$.

Furthermore, the protocol can also be implemented with ensembles of other
particles, such as nitrogen-vacancy (NV) centers in diamond [67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77]. In this case, a feasible setup is the same as the
one shown in Fig. 13, but with replacing the atomic ensemble by the NV
centers. We encode the internal states of the NV centers, ${|m_{s}=0\rangle}$
and ${|m_{s}=-1\rangle}$, as the information carrier [71]. The coupling
strength between a single NV center and a single CPWR photon $g/2\pi$ exceeds
$12\,\text{Hz}$ [67, 71]. To date, for a typical ensemble of NV centers, the
relaxation time of ${T_{1}\sim 40\,\text{s}}$ (${\gamma_{s}\ll 1\,\text{Hz}}$)
have been demonstrated experimentally [67] and, with the spin-echo sequences,
the coherent time of ${T_{2}>600\,\mu\text{s}}$
(${\gamma_{c}<0.26\,\text{kHz}}$) has also been reached [72, 29]. Thus, we
also give a group of feasible parameters and the minimum of the spin squeezing
parameter in Table 1.

##  V CONCLUSIONS

In this paper, we have proposed a protocol to generate spin squeezing in
atomic ensembles via a fully quantum degenerate parametric amplifier. By
adjusting the parameters, an effective cavity-induced TAT-like interaction can
be achieved. The strength of the generated spin squeezing is determined by
some properties of the pump cavity, such as the initial state, the driving
strength, and the cavity decay. We mainly discuss the generated spin squeezing
in two cases of the pump cavity. For the first case, the pump cavity is
initially in the quasi-steady coherent state by a driving field and the cavity
decay. Meanwhile, for the second case, the pump cavity is initiated in an
arbitrary coherent state and there is no pump cavity driving.

In the first case, theoretical analyses and numerical simulations show that
the present protocol can generate a strong spin squeezing whose strength is
even comparable to that of the TAT model. The reason is that, for a fixed
$d_{0}$ and an extremely large ratio of $\kappa_{p}$ to $g_{\text{eff}}$, the
effect of the cavity-induced TAT-like interaction is equivalent to applying
the TAT interaction to the atomic ensemble. Meanwhile, according to numerical
simulations, the present protocol is robust to the atomic spontaneous emission
and the signal cavity decay.

In the second case, according to numerical simulations, the present protocol
can generate an observable spin squeezing. A properly large amplitude of the
initial coherent state can accelerate the generation of spin squeezing.
Meanwhile, the present protocol is also robust to the collective dephasing of
the ensemble, in addition to the atomic spontaneous emission and the signal
cavity decay. It is worth noting that the pump cavity decay is able to prolong
the duration of squeezing significantly.

After combining the experimental results, we show that the present protocol is
feasible experimentally. A group of realistic parameters, as shown in Table 1,
has been given to predict some experimentally feasible results. Moreover, the
present protocol can be extended to generate spin squeezing in various
ensembles, like ensembles of NV centers. We hope that the present protocol
provides a novel approach for generating spin squeezing in photon-spin
coupling systems.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
under Grants No. 11575045, No. 11874114, and No. 11674060, the Natural Science
Funds for Distinguished Young Scholar of Fujian Province under Grant No.
2020J06011 and Project from Fuzhou University under Grant No. JG202001-2.
Y.-H.C. was supported by the Japan Society for the Promotion of Science (JSPS)
KAKENHI Grant No. JP19F19028. W.Q. is supported in part by the Incentive
Research Project of RIKEN.

##  Appendix Appendix A THE APPROACH FOR COMPENSATING THE TERM
$(g^{2}/\delta_{\texttt{S}})\hat{S}_{+}\hat{S}_{-}$ COMPLETELY

In this appendix, we give the detailed derivation of the approach for
compensating $(g^{2}/\delta_{\texttt{S}})\hat{S}_{+}\hat{S}_{-}$ completely.
We first assume the information of spin squeezing is carrier by the levels of
the atoms $|0\rangle_{a}\equiv|\downarrow\rangle_{a}$ and
$|1\rangle_{a}\equiv|\uparrow\rangle_{a}$. Here, we introduce an optical
cavity, a laser pulse, and an auxiliary level of the atoms $|2\rangle_{a}$
[59, 60, 61]. The optical cavity (pulse) is far-off resonant with the
transition $|0\rangle_{a}\leftrightarrow|2\rangle_{a}$
($|1\rangle_{a}\leftrightarrow|2\rangle_{a}$) with the coupling strength
$g_{d}$ ($\Omega_{o}$) and the detuning $\Delta_{d}$ ($\Delta_{o}$). In the
interaction picture, the Hamiltonian described these interactions is given as

$\displaystyle\hat{H}_{o}$ $\displaystyle=$
$\displaystyle\Delta_{d}\hat{d}^{\dagger}\hat{d}+g_{c}\sum_{k=1}^{N}(|2\rangle^{k}_{a}{{}^{k}_{a}\langle
0|}\hat{c}+|0\rangle^{k}_{a}{{}^{k}_{a}\langle 2|}\hat{c}^{\dagger})$ (A3)
$\displaystyle+\Omega_{o}\sum_{k=1}^{N}(|2\rangle^{k}_{a}{{}^{k}_{a}\langle
1|}e^{-i\Delta_{o}t}+|1\rangle^{k}_{a}{{}^{k}_{a}\langle
2|}e^{i\Delta_{o}t}),$

where $\hat{d}^{\dagger}$ and $\hat{d}$ are the creation operator and the
annihilation operator of the optical cavity mode, respectively.
$|\varepsilon\rangle^{k}_{a}\,(\varepsilon=0,1,2\ \text{and}\ k=1,2,\dots,N)$
represents that the $k$th atom is in $|\varepsilon\rangle_{a}$. When the
conditions $\Delta_{o}-\Delta_{d}=\Delta$ and
$\\{\Delta_{d},\Delta_{o}\\}\gg\\{g_{d},\Omega_{o},\Delta\\}$ are satisfied,
the auxiliary level $|2\rangle_{a}$ can be eliminated adiabatically and then
the effective Hamiltonian of the system is

$\displaystyle\hat{H}^{(1)}_{o,\text{eff}}$ $\displaystyle=$
$\displaystyle\dfrac{g_{d}\Omega_{o}}{\Delta^{\prime}}\sum_{k=1}^{N}(|1\rangle^{k}_{a}{{}^{k}_{a}\langle
0|}\hat{d}e^{i\Delta t}+|0\rangle^{k}_{a}{{}^{k}_{a}\langle
1|}\hat{d}^{\dagger}e^{-i\Delta t})$ (A6)
$\displaystyle+\dfrac{\Omega_{o}^{2}}{\Delta_{o}}\sum_{k=1}^{N}|1\rangle^{k}_{a}{{}^{k}_{a}\langle
1|}+\dfrac{g_{d}^{2}}{\Delta_{d}}\sum_{k=1}^{N}|0\rangle^{k}_{a}{{}^{k}_{a}\langle
0|}\hat{c}^{\dagger}\hat{c},$

where $\Delta^{\prime}=2\Delta_{d}\Delta_{o}/(\Delta_{o}+\Delta_{d})$. To
adiabatically eliminate the optical cavity mode, we set $\Delta\gg
g_{d}\Omega_{o}/\Delta^{\prime}$. Then, the above effective Hamiltonian in Eq.
(A6) is reduced to

$\displaystyle\hat{H}^{(2)}_{o,\text{eff}}$ $\displaystyle=$
$\displaystyle\dfrac{g^{2}_{d}\Omega^{2}_{o}}{\Delta^{\prime
2}\Delta}\hat{S}_{+}\hat{S}_{-}+\dfrac{\Omega_{o}^{2}}{\Delta_{o}}\hat{S}_{z},$
(A7)

where
$\hat{S}_{+}\hat{S}_{-}=\sum_{k=1}^{N}\sum_{k^{\prime}=1}^{N}(|1\rangle^{k}_{a}{{}^{k}_{a}\langle
0|})(|0\rangle^{k^{\prime}}_{a}{{}^{k^{\prime}}_{a}\langle 1|})$ and
$\hat{S}_{z}=1/2\sum_{k=1}^{N}(|1\rangle^{k}_{a}{{}^{k}_{a}\langle
1|}-\sum_{k=1}^{N}|0\rangle^{k}_{a}{{}^{k}_{a}\langle 0|})$. This means that,
when one takes

$\displaystyle\dfrac{g^{2}_{d}\Omega^{2}_{o}}{\Delta^{\prime
2}\Delta}+\dfrac{g^{2}}{\delta_{\texttt{S}}}=\delta_{q}+\dfrac{\Omega_{o}^{2}}{\Delta_{o}}=0,$
(A8)

the effect of the undesired term
$(g^{2}/\delta_{\texttt{S}})\hat{S}_{+}\hat{S}_{-}$ is compensated completely.

##  Appendix Appendix B THE ADIABATIC ELIMINATION OF THE SIGNAL CAVITY

Due to few photons and strong cavity loss, the signal cavity can be considered
as an ambience and can be adiabatically eliminated. Here, we give a detailed
derivation on the adiabatic elimination of the signal cavity. According to the
Hamiltonian in Eq. (8), one can obtain the interactions which excite signal
cavity photons

$\hat{V}_{1}=J\hat{a}_{p}\hat{a}^{\dagger 2}_{s},\
\hat{V}_{2}=g\hat{S}_{-}\hat{a}^{\dagger}_{s},$ (B1)

and the free Hamiltonian of signal cavity photons

$\hat{H}_{e}=\delta_{s}\hat{a}^{\dagger}_{s}\hat{a}_{s}.$ (B2)

We introduce a Lindblad operator $L_{s}=\sqrt{\gamma_{s}}\hat{a}_{s}$ which
satisfies $\mathcal{L}(L_{s})\rho=\gamma_{s}\mathcal{L}(\hat{a}_{s})\rho$.
According to the work in Ref. [65], the Lindblad operators of the effective
master equation can be described as

$\hat{L}_{h}=\hat{L}_{s}\left(\hat{H}_{e}-\dfrac{i}{2}\hat{L}^{\dagger}_{s}\hat{L}_{s}\right)^{-1}\hat{V}_{h},$
(B3)

where, $h=1,2$. Then, one would obtain

$\displaystyle\hat{L}_{1}$ $\displaystyle=$
$\displaystyle\dfrac{2\sqrt{\gamma_{s}}J}{2\delta_{s}-i\gamma_{s}}\hat{a}_{p}\hat{a}^{\dagger}_{s},$
(B4) $\displaystyle\hat{L}_{2}$ $\displaystyle=$
$\displaystyle\dfrac{2\sqrt{\gamma_{s}}g}{2\delta_{s}-i\gamma_{s}}\hat{S}_{-}.$
(B6)

This means that the Lindblad superoperator which describes the signal cavity
decay $\gamma_{s}\mathcal{L}(\hat{a}_{s})\rho$ can be replaced by
$J^{2}p_{\gamma}\mathcal{L}(\hat{a}_{p}\hat{a}^{\dagger}_{s})\rho+g^{2}p_{\gamma}\mathcal{L}(\hat{S}_{-})\rho$,
where $p_{\gamma}=4\gamma_{s}/(4\delta_{s}^{2}+\gamma_{s}^{2})$. Moreover,
since the signal cavity remains in vacuum during the evolution of the system
effectively, the term
$J^{2}p_{\gamma}\mathcal{L}(\hat{a}_{p}\hat{a}^{\dagger}_{s})\rho$ is able to
reduce to $J^{2}p_{\gamma}\mathcal{L}(\hat{a}_{p})\rho$. Therefore, the effect
of the signal cavity decay $\gamma_{s}\mathcal{L}(\hat{a}_{s})\rho$ is
equivalent to introducing an extra pump cavity decay
$J^{2}p_{\gamma}\mathcal{L}(\hat{a}_{p})\rho$ and an extra atomic collective
spontaneous emission $g^{2}p_{\gamma}\mathcal{L}(\hat{S}_{-})\rho$. The
physics mechanics is that, when the signal cavity is eliminated adiabatically,
the pump cavity and the ensemble can be considered to be coupled to a new
vacuum bath (i.e., the vacuum signal cavity).

##  Appendix Appendix C THE MINIMUM OF THE SPIN SQUEEZING PARAMETER IN TABLE 1

Note that, in typical ensembles, the number of particles is of the order of
millions (e.g., Rb atoms) or trillions (e.g., NV centers). This means that it
is extremely difficult to exactly estimate the full dynamics of the ensemble.
In the following, we simplify the dynamics of the system and then obtain the
minimum of spin squeezing effectively. With the Holstein-Primakoff
transformation (HPT) [78] and in the limit of ${N\rightarrow\infty}$, the
collective spin operators can be transformed to the bosonic operators as

$\hat{S}_{-}\rightarrow\sqrt{N}\hat{b},\,\hat{S}_{z}\rightarrow-N/2+\hat{b}^{\dagger}\hat{b},$
(C1)

where $\hat{b}$ is the bosonic annihilation operator. Accordingly, the spin
squeezing parameter in Eq. (56) is rewritten as

$\xi^{2}_{\mathcal{R}}=\frac{N^{2}}{(N-2\langle\hat{b}^{\dagger}\hat{b}\rangle)^{2}}(1+2\langle\hat{b}^{\dagger}\hat{b}\rangle-2|\langle\hat{b}^{2}\rangle|).$
(C2)

Meanwhile, from the effective master equation in Eq. (55) and the HPT, we
obtain the Heisenberg equation [27]

$\displaystyle\partial_{t}\hat{a}_{p}$ $\displaystyle=$ $\displaystyle-
iNg_{\text{eff}}\hat{b}^{2}-i\Omega^{*}/2-(\kappa_{p}+J^{2}p_{\gamma})\hat{a}_{p}/2,$
(C3a) $\displaystyle\partial_{t}\hat{b}^{2}$ $\displaystyle=$
$\displaystyle-2iNg_{\text{eff}}\hat{a}_{p}(2\hat{b}^{\dagger}\hat{b}+1)-(\gamma_{s}+g_{c}^{2}p_{\gamma}+2\gamma_{c})\hat{b}^{2},$
(C3c) $\displaystyle\partial_{t}\hat{b}^{\dagger}\hat{b}$ $\displaystyle=$
$\displaystyle-2iNg_{\text{eff}}(\hat{a}_{p}\hat{b}^{\dagger
2}-\hat{a}_{p}^{\dagger}\hat{b}^{2})-(\gamma_{s}+g_{c}^{2}p)\hat{b}^{\dagger}\hat{b},$
(C3e)

where $\partial_{t}=\partial/\partial t$. Applying the mean-field
approximation, the instantaneous evolution of the average values of the
operators is described as

$\displaystyle\partial_{t}\langle\hat{a}_{p}\rangle$ $\displaystyle=$
$\displaystyle-
iNg_{\text{eff}}\langle\hat{b}^{2}\rangle-i\Omega^{*}/2-(\kappa_{p}+J^{2}p_{\gamma})\langle\hat{a}_{p}\rangle/2,$
(C4a) $\displaystyle\partial_{t}\langle\hat{b}^{2}\rangle$ $\displaystyle=$
$\displaystyle-2iNg_{\text{eff}}\langle\hat{a}_{p}\rangle(2\langle\hat{b}^{\dagger}\hat{b}\rangle{+}1)-(\gamma_{s}{+}g_{c}^{2}p_{\gamma}{+}2\gamma_{c})\langle\hat{b}^{2}\rangle,$
(C4c) $\displaystyle\partial_{t}\langle\hat{b}^{\dagger}\hat{b}\rangle$
$\displaystyle=$
$\displaystyle-4iNg_{\text{eff}}\text{Im}(\langle\hat{a}_{p}\rangle\langle\hat{b}^{\dagger
2}\rangle)-(\gamma_{s}+g_{c}^{2}p)\langle\hat{b}^{\dagger}\hat{b}\rangle.$
(C4e)

Then, the time evolutions of $\langle\hat{b}^{2}\rangle$ and
$\langle\hat{b}^{\dagger}\hat{b}\rangle$ can be obtained by solving the
coupled equations in Eq. (C4). After substituting $\langle\hat{b}^{2}\rangle$
and $\langle\hat{b}^{\dagger}\hat{b}\rangle$ into Eq. (C2), the evolution of
the spin squeezing parameter $\xi^{2}_{\mathcal{R}}$ is also obtained and then
the minimum of the spin squeezing parameter $\xi^{2}_{\mathcal{R},\min}$ can
be achieved accordingly.

## References

  * Ma  _et al._ [2011] J. Ma, X.-G. Wang, C.-P. Sun, and F. Nori, Quantum spin squeezing, Phys. Rep. 509, 89 (2011).
  * Pezzè  _et al._ [2018] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).
  * Perlin  _et al._ [2020] M. A. Perlin, C.-L. Qu, and A. M. Rey, Spin squeezing with short-range spin-exchange interactions, Phys. Rev. Lett. 125, 223401 (2020).
  * Bai and An [2021] S.-Y. Bai and J.-H. An, Generating stable spin squeezing by squeezed-reservoir engineering, Phys. Rev. Lett. 127, 083602 (2021).
  * Wineland  _et al._ [1992] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, Spin squeezing and reduced quantum noise in spectroscopy, Phys. Rev. A 46, R6797 (1992).
  * Wineland  _et al._ [1994] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Squeezed atomic states and projection noise in spectroscopy, Phys. Rev. A 50, 67 (1994).
  * Ulam-Orgikh and Kitagawa [2001] D. Ulam-Orgikh and M. Kitagawa, Spin squeezing and decoherence limit in ramsey spectroscopy, Phys. Rev. A 64, 052106 (2001).
  * Döring  _et al._ [2010] D. Döring, G. McDonald, J. E. Debs, C. Figl, P. A. Altin, H.-A. Bachor, N. P. Robins, and J. D. Close, Quantum-projection-noise-limited interferometry with coherent atoms in a ramsey-type setup, Phys. Rev. A 81, 043633 (2010).
  * Kruse  _et al._ [2016] I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, Improvement of an atomic clock using squeezed vacuum, Phys. Rev. Lett. 117, 143004 (2016).
  * Polzik [2008] E. S. Polzik, The squeeze goes on, Nature 453, 45 (2008).
  * Leroux  _et al._ [2010] I. D. Leroux, M. H. Schleier-Smith, and V. Vuletić, Orientation-dependent entanglement lifetime in a squeezed atomic clock, Phys. Rev. Lett. 104, 250801 (2010).
  * Goda  _et al._ [2008] K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, A quantum-enhanced prototype gravitational-wave detector, Nat. Phys. 4, 472 (2008).
  * Walls and Zoller [1981] D. Walls and P. Zoller, Enhanced sensitivity of a gravitational wave detector, Phys. Lett. A 85, 118 (1981).
  * Bilitewski  _et al._ [2021] T. Bilitewski, L. De Marco, J.-R. Li, K. Matsuda, W. G. Tobias, G. Valtolina, J. Ye, and A. M. Rey, Dynamical generation of spin squeezing in ultracold dipolar molecules, Phys. Rev. Lett. 126, 113401 (2021).
  * Bhattacharya [2015] M. Bhattacharya, Spin squeezing a cold molecule, Phys. Rev. A 92, 063823 (2015).
  * Estève  _et al._ [2008] J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Squeezing and entanglement in a bose–einstein condensate, Nature 455, 1216 (2008).
  * Fadel  _et al._ [2018] M. Fadel, T. Zibold, B. Décamps, and P. Treutlein, Spatial entanglement patterns and einstein-podolsky-rosen steering in bose-einstein condensates, Science 360, 409 (2018).
  * Luo  _et al._ [2017] X.-Y. Luo, Y.-Q. Zou, L.-N. Wu, Q. Liu, M.-F. Han, M. K. Tey, and L. You, Deterministic entanglement generation from driving through quantum phase transitions, Science 355, 620 (2017).
  * Chen  _et al._ [2020] L. Chen, Y.-B. Zhang, and H. Pu, Spin squeezing in a spin-orbit-coupled bose-einstein condensate, Phys. Rev. A 102, 023317 (2020).
  * Kuzmich  _et al._ [1997] A. Kuzmich, K. Mølmer, and E. S. Polzik, Spin squeezing in an ensemble of atoms illuminated with squeezed light, Phys. Rev. Lett. 79, 4782 (1997).
  * Hald  _et al._ [1999] J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, Spin squeezed atoms: A macroscopic entangled ensemble created by light, Phys. Rev. Lett. 83, 1319 (1999).
  * Julsgaard  _et al._ [2001] B. Julsgaard, A. Kozhekin, and E. S. Polzik, Experimental long-lived entanglement of two macroscopic objects, Nature 413, 400 (2001).
  * Kuzmich  _et al._ [2000] A. Kuzmich, L. Mandel, and N. P. Bigelow, Generation of spin squeezing via continuous quantum nondemolition measurement, Phys. Rev. Lett. 85, 1594 (2000).
  * Koschorreck  _et al._ [2010] M. Koschorreck, M. Napolitano, B. Dubost, and M. W. Mitchell, Quantum nondemolition measurement of large-spin ensembles by dynamical decoupling, Phys. Rev. Lett. 105, 093602 (2010).
  * Chalopin  _et al._ [2018] T. Chalopin, C. Bouazza, A. Evrard, V. Makhalov, D. Dreon, J. Dalibard, L. A. Sidorenkov, and S. Nascimbene, Quantum-enhanced sensing using non-classical spin states of a highly magnetic atom, Nat. Commun. 9, 4955 (2018).
  * Evrard  _et al._ [2019] A. Evrard, V. Makhalov, T. Chalopin, L. A. Sidorenkov, J. Dalibard, R. Lopes, and S. Nascimbene, Enhanced magnetic sensitivity with non-gaussian quantum fluctuations, Phys. Rev. Lett. 122, 173601 (2019).
  * Macrì  _et al._ [2020] V. Macrì, F. Nori, S. Savasta, and D. Zueco, Spin squeezing by one-photon–two-atom excitation processes in atomic ensembles, Phys. Rev. A 101, 053818 (2020).
  * Tucker  _et al._ [2020] K. Tucker, D. Barberena, R. J. Lewis-Swan, J. K. Thompson, J. G. Restrepo, and A. M. Rey, Facilitating spin squeezing generated by collective dynamics with single-particle decoherence, Phys. Rev. A 102, 051701 (2020).
  * Qin  _et al._ [2020] W. Qin, Y.-H. Chen, X. Wang, A. Miranowicz, and F. Nori, Strong spin squeezing induced by weak squeezing of light inside a cavity, Nanophotonics (Berlin) 9, 4853 (2020).
  * Helmerson and You [2001] K. Helmerson and L. You, Creating massive entanglement of bose-einstein condensed atoms, Phys. Rev. Lett. 87, 170402 (2001).
  * Liu  _et al._ [2011] Y.-C. Liu, Z.-F. Xu, G.-R. Jin, and L. You, Spin squeezing: Transforming one-axis twisting into two-axis twisting, Phys. Rev. Lett. 107, 013601 (2011).
  * Groszkowski  _et al._ [2020] P. Groszkowski, H.-K. Lau, C. Leroux, L. C. G. Govia, and A. A. Clerk, Heisenberg-limited spin squeezing via bosonic parametric driving, Phys. Rev. Lett. 125, 203601 (2020).
  * Zhang  _et al._ [2017] Y.-C. Zhang, X.-F. Zhou, X.-X. Zhou, G.-C. Guo, and Z.-W. Zhou, Cavity-assisted single-mode and two-mode spin-squeezed states via phase-locked atom-photon coupling, Phys. Rev. Lett. 118, 083604 (2017).
  * Qin  _et al._ [2022] W. Qin, A. Miranowicz, and F. Nori, Beating the 3 db limit for intracavity squeezing and its application to nondemolition qubit readout, Phys. Rev. Lett. 129, 123602 (2022).
  * Leghtas  _et al._ [2015] Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J. Hatridge, M. Reagor, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, Confining the state of light to a quantum manifold by engineered two-photon loss, Science 347, 853 (2015).
  * Chang  _et al._ [2020] C. W. S. Chang, C. Sabín, P. Forn-Díaz, F. Quijandría, A. M. Vadiraj, I. Nsanzineza, G. Johansson, and C. M. Wilson, Observation of three-photon spontaneous parametric down-conversion in a superconducting parametric cavity, Phys. Rev. X 10, 011011 (2020).
  * Lescanne  _et al._ [2020] R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Exponential suppression of bit-flips in a qubit encoded in an oscillator, Nat. Phys. 16, 509 (2020).
  * Vrajitoarea  _et al._ [2019] A. Vrajitoarea, Z.-W. Huang, P. Groszkowski, J. Koch, and A. A. Houck, Quantum control of an oscillator using a stimulated josephson nonlinearity, Nat. Phys. 16, 211 (2019).
  * Hattermann  _et al._ [2017] H. Hattermann, D. Bothner, L. Y. Ley, B. Ferdinand, D. Wiedmaier, L. Sárkány, R. Kleiner, D. Koelle, and J. Fortágh, Coupling ultracold atoms to a superconducting coplanar waveguide resonator, Nat. Commun. 8, 2254 (2017).
  * Verdú  _et al._ [2009] J. Verdú, H. Zoubi, C. Koller, J. Majer, H. Ritsch, and J. Schmiedmayer, Strong magnetic coupling of an ultracold gas to a superconducting waveguide cavity, Phys. Rev. Lett. 103, 043603 (2009).
  * Kaiser  _et al._ [2022] M. Kaiser, C. Glaser, L. Y. Ley, J. Grimmel, H. Hattermann, D. Bothner, D. Koelle, R. Kleiner, D. Petrosyan, A. Günther, and J. Fortágh, Cavity-driven rabi oscillations between rydberg states of atoms trapped on a superconducting atom chip, Phys. Rev. Research 4, 013207 (2022).
  * Kroll  _et al._ [2019] J. G. Kroll, F. Borsoi, K. L. van der Enden, W. Uilhoorn, D. de Jong, M. Quintero-Pérez, D. J. van Woerkom, A. Bruno, S. R. Plissard, D. Car, E. Bakkers, M. C. Cassidy, and L. P. Kouwenhoven, Magnetic-field-resilient superconducting coplanar-waveguide resonators for hybrid circuit quantum electrodynamics experiments, Phys. Rev. Applied 11, 064053 (2019).
  * Bernon  _et al._ [2013] S. Bernon, H. Hattermann, D. Bothner, M. Knufinke, P. Weiss, F. Jessen, D. Cano, M. Kemmler, R. Kleiner, D. Koelle, and J. Fortágh, Manipulation and coherence of ultra-cold atoms on a superconducting atom chip, Nat. Commun. 4, 2380 (2013).
  * Bruno  _et al._ [2015] A. Bruno, G. de Lange, S. Asaad, K. L. van der Enden, N. K. Langford, and L. DiCarlo, Reducing intrinsic loss in superconducting resonators by surface treatment and deep etching of silicon substrates, Appl. Phys. Lett. 106, 182601 (2015).
  * Mirhosseini  _et al._ [2018] M. Mirhosseini, E. Kim, V. S. Ferreira, M. Kalaee, A. Sipahigil, A. J. Keller, and O. Painter, Superconducting metamaterials for waveguide quantum electrodynamics, Nat. Commun. 9, 3706 (2018).
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  * Qin  _et al._ [2021] W. Qin, A. Miranowicz, H. Jing, and F. Nori, Generating long-lived macroscopically distinct superposition states in atomic ensembles, Phys. Rev. Lett. 127, 093602 (2021).
  * Ficek and Swain [2005] Z. Ficek and S. Swain,  _Quantum Interference and Coherence_ (Springer New York, NY, USA, 2005).
  * Ran  _et al._ [2020] D. Ran, W.-J. Shan, Z.-C. Shi, Z.-B. Yang, J. Song, and Y. Xia, Generation of nonclassical states in nonlinear oscillators via lyapunov control, Phys. Rev. A 102, 022603 (2020).
  * Liu  _et al._ [2020] S. Liu, D. Ran, Y.-H. Kang, Z.-C. Shi, J. Song, and Y. Xia, Accelerated and robust generation of w state by parametric amplification and inverse hamiltonian engineering, Ann. Phys. (Berlin) 532, 2000002 (2020).
  * Kang  _et al._ [2022] Y.-H. Kang, J. Song, and Y. Xia, Error-resistant nonadiabatic binomial-code geometric quantum computation using reverse engineering, Opt. Lett. 47, 4099 (2022).
  * Wang  _et al._ [2019a] Y. Wang, C. Li, E. M. Sampuli, J. Song, Y. Jiang, and Y. Xia, Enhancement of coherent dipole coupling between two atoms via squeezing a cavity mode, Phys. Rev. A 99, 023833 (2019a).
  * Wang  _et al._ [2019b] Y. Wang, J.-L. Wu, J. Song, Y.-Y. Jiang, Z.-J. Zhang, and Y. Xia, Squeezing-enhanced atom-cavity interaction in coupled cavities with high dissipation rates, Ann. Phys. (Berlin) 531, 1900220 (2019b).
  * Chen  _et al._ [2022] D.-L. Chen, Y.-H. Chen, Y. Liu, Z.-C. Shi, J. Song, and Y. Xia, Detecting a single atom in a cavity using the $\chi^{(2)}$ nonlinear medium, Front. Phys. (Beijing) 17, 52501 (2022).
  * Chen  _et al._ [2021] Y.-H. Chen, W. Qin, X. Wang, A. Miranowicz, and F. Nori, Shortcuts to adiabaticity for the quantum rabi model: Efficient generation of giant entangled cat states via parametric amplification, Phys. Rev. Lett. 126, 023602 (2021).
  * Qin  _et al._ [2019] W. Qin, V. Macrì, A. Miranowicz, S. Savasta, and F. Nori, Emission of photon pairs by mechanical stimulation of the squeezed vacuum, Phys. Rev. A 100, 062501 (2019).
  * James and Jerke [2007] D. F. James and J. Jerke, Effective hamiltonian theory and its applications in quantum information, Can. J. Phys. 85, 625 (2007).
  * Shao  _et al._ [2017] W.-J. Shao, C.-F. Wu, and X.-L. Feng, Generalized james’ effective hamiltonian method, Phys. Rev. A 95, 032124 (2017).
  * Williamson  _et al._ [2014] L. A. Williamson, Y.-H. Chen, and J. J. Longdell, Magneto-optic modulator with unit quantum efficiency, Phys. Rev. Lett. 113, 203601 (2014).
  * Lauk  _et al._ [2020] N. Lauk, N. Sinclair, S. Barzanjeh, J. P. Covey, M. Saffman, M. Spiropulu, and C. Simon, Perspectives on quantum transduction, Quantum Science and Technology 5, 020501 (2020).
  * Liu  _et al._ [2021] T. Liu, J.-L. Zhao, B.-Q. Guo, Q.-C. Wu, Y.-H. Zhou, and C.-P. Yang, One-step implementation of a coherent conversion between microwave and optical cavities via an ensemble of nitrogen-vacancy centers, Phys. Rev. A 103, 023706 (2021).
  * Shammah  _et al._ [2018] N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori, Open quantum systems with local and collective incoherent processes: Efficient numerical simulations using permutational invariance, Phys. Rev. A 98, 063815 (2018).
  * Gelhausen  _et al._ [2017] J. Gelhausen, M. Buchhold, and P. Strack, Many-body quantum optics with decaying atomic spin states: ($\gamma,\kappa$) dicke model, Phys. Rev. A 95, 063824 (2017).
  * Dalibard  _et al._ [1992] J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992).
  * Reiter and Sørensen [2012] F. Reiter and A. S. Sørensen, Effective operator formalism for open quantum systems, Phys. Rev. A 85, 032111 (2012).
  * Chen  _et al._ [2018] Y.-H. Chen, Z.-C. Shi, J. Song, Y. Xia, and S.-B. Zheng, Accelerated and noise-resistant generation of high-fidelity steady-state entanglement with rydberg atoms, Phys. Rev. A 97, 032328 (2018).
  * Amsüss  _et al._ [2011] R. Amsüss, C. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner, S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer, and J. Majer, Cavity qed with magnetically coupled collective spin states, Phys. Rev. Lett. 107, 060502 (2011).
  * Kubo  _et al._ [2010] Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, Strong coupling of a spin ensemble to a superconducting resonator, Phys. Rev. Lett. 105, 140502 (2010).
  * Kubo  _et al._ [2012] Y. Kubo, I. Diniz, A. Dewes, V. Jacques, A. Dréau, J.-F. Roch, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, Storage and retrieval of a microwave field in a spin ensemble, Phys. Rev. A 85, 012333 (2012).
  * Astner  _et al._ [2017] T. Astner, S. Nevlacsil, N. Peterschofsky, A. Angerer, S. Rotter, S. Putz, J. Schmiedmayer, and J. Majer, Coherent coupling of remote spin ensembles via a cavity bus, Phys. Rev. Lett. 118, 140502 (2017).
  * Putz  _et al._ [2014] S. Putz, D. O. Krimer, R. Amsüss, A. Valookaran, T. Nöbauer, J. Schmiedmayer, S. Rotter, and J. Majer, Protecting a spin ensemble against decoherence in the strong-coupling regime of cavity QED, Nat. Phys. 10, 720 (2014).
  * Stanwix  _et al._ [2010] P. L. Stanwix, L. M. Pham, J. R. Maze, D. Le Sage, T. K. Yeung, P. Cappellaro, P. R. Hemmer, A. Yacoby, M. D. Lukin, and R. L. Walsworth, Coherence of nitrogen-vacancy electronic spin ensembles in diamond, Phys. Rev. B 82, 201201 (2010).
  * Cheng  _et al._ [2013] L.-Y. Cheng, H.-F. Wang, S. Zhang, and K.-H. Yeon, Quantum state engineering with nitrogen-vacancy centers coupled to low-q microresonator, Opt. Express 21, 5988 (2013).
  * Han  _et al._ [2017] X. Han, Q. Guo, A.-D. Zhu, S. Zhang, and H.-F. Wang, Effective w-state fusion strategies in nitrogen-vacancy centers via coupling to microtoroidal resonators, Opt. Express 25, 17701 (2017).
  * Jin  _et al._ [2019] Z. Jin, S. L. Su, and S. Zhang, Preparation of a steady entangled state of two nitrogen-vacancy centers by simultaneously utilizing two dissipative factors, Phys. Rev. A 100, 052332 (2019).
  * Cheng  _et al._ [2019] L.-Y. Cheng, Q. Guo, H.-F. Wang, and S. Zhang, Direct entanglement measurement of werner state with cavity-assisted spin–photon interaction system, Quantum Information Processing 18, 214 (2019).
  * Cheng  _et al._ [2016] L.-Y. Cheng, G.-H. Yang, Q. Guo, H.-F. Wang, and S. Zhang, Direct measurement of nonlocal entanglement of two-qubit spin quantum states, Scientific Reports 6, 19482 (2016).
  * Holstein and Primakoff [1940] T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev. 58, 1098 (1940).