# Nonreciprocal Superradiant Phase Transitions and Multicriticality in a
Cavity QED System

Gui-Lei Zhu  Department of Physics, Zhejiang Sci-Tech University, Hangzhou
310018, China  Theoretical Quantum Physics Laboratory, Cluster for Pioneering
Research, RIKEN, Wakoshi, Saitama 351-0198, Japan  Chang-Sheng Hu  Department
of Physics, Anhui Normal University, Wuhu 241000, China  Hui Wang  Theoretical
Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi,
Saitama 351-0198, Japan  Wei Qin  Theoretical Quantum Physics Laboratory,
Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan
Center for Joint Quantum Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300350, China  Xin-You Lü<EMAIL_ADDRESS>School of Physics, Huazhong University of Science and Technology and Wuhan
Institute of Quantum Technology, Wuhan 430074, China  Franco Nori
<EMAIL_ADDRESS>Theoretical Quantum Physics Laboratory, Cluster for Pioneering
Research, RIKEN, Wakoshi, Saitama 351-0198, Japan  Quantum Computing Center,
RIKEN, Wakoshi, Saitama 351-0198, Japan  Department of Physics, The University
of Michigan, Ann Arbor, MI, 48109-1040, USA

###### Abstract

We demonstrate the emergence of nonreciprocal superradiant phase transitions
and novel multicriticality in a cavity quantum electrodynamics (QED) system,
where a two-level atom interacts with two counter-propagating modes of a
whispering-gallery-mode (WGM) microcavity. The cavity rotates at a certain
angular velocity, and is directionally squeezed by a unidirectional parametric
pumping $\chi^{(2)}$ nonlinearity. The combination of cavity rotation and
directional squeezing leads to nonreciprocal first- and second-order
superradiant phase transitions. These transitions do not require ultrastrong
atom-field couplings and can be easily controlled by the external pump field.
Through a full quantum description of the system Hamiltonian, we identify two
types of multicritical points in the phase diagram, both of which exhibit
controllable nonreciprocity. These results open a new door for all-optical
manipulation of superradiant transitions and multicritical behaviors in light-
matter systems, with potential applications in engineering various integrated
nonreciprocal quantum devices.

Phase transitions and critical phenomena are at the heart of understanding the
nature of the matter in condensed matter physics and material science Domb
(2000); Sachdev . One of the most intriguing topics in light-matter systems is
the superradiant phase transition Dicke (1954); Hepp and Lieb (1973); Wang and
Hioe (1973), where increasing the atom-field coupling through a critical value
induces a transition from the normal phase (NP) to the superradiant phase (SP)
Emary and Brandes (2003); Lambert  _et al._ (2004); Dimer  _et al._ (2007);
Baumann  _et al._ (2010); Baden  _et al._ (2014); Nagy  _et al._ (2010); Lü
_et al._ (2018a); Kirton  _et al._ (2019); Xu  _et al._ (2021); Zhao and Hwang
(2022). This superradiant transition typically occurs in the thermodynamic
limit, where the number of atoms $N$ approaches infinity. In the quantum Rabi
model Rabi (1937); Braak (2011); Xie  _et al._ (2017) with $N=1$, a similar
transition can occur, but it requires both ultrastrong light-matter coupling
and an extremely large atomic frequency Hwang  _et al._ (2015); Liu  _et al._
(2017); Lü  _et al._ (2018b); Hwang  _et al._ (2018); Zhang  _et al._ (2021);
Fallas Padilla  _et al._ (2022); Liu  _et al._ (2022); Shen  _et al._ (2022).
The realization of the phase transition in such single-atom models has been
successfully demonstrated in quantum simulation platforms, including nuclear
magnetic resonance (NMR) quantum simulators Georgescu  _et al._ (2014); Chen
_et al._ (2021), driven atoms in trapped ions Puebla  _et al._ (2017); Cai
_et al._ (2021) and superconducting qubits Zheng  _et al._ (2023).

In open systems, the presence of photon loss can have a significant impact on
the critical behavior  of the Hamiltonian part of the system Kirton and
Keeling (2017); Shammah  _et al._ (2018). It can induce multicritical
phenomena Soriente  _et al._ (2018); Zhu  _et al._ (2020a); Lin  _et al._
(2022); Xu and Pu (2019); Zhu  _et al._ (2020b), or even completely suppress
its criticality Larson and Irish (2017). In addition, this class of driven-
dissipative systems allows for the demonstration of richer physics, such as
the breakdown of photon blockade Carmichael (2015); Fink  _et al._ (2017);
Reiter  _et al._ (2020), stable superradiant lasers Meiser  _et al._ (2009);
Bohnet  _et al._ (2012); Norcia and Thompson (2016); Norcia  _et al._ (2016),
time crystals Gong  _et al._ (2018); Mattes  _et al._ (2023); Ferioli  _et
al._ (2023), and atomic synchronization Xu  _et al._ (2014); Masson and
Asenjo-Garcia (2022). However, thus far, the realization of superradiant
transitions has not simultaneously combined the features necessary for
exquisite controllability, such as strong nonreciprocity, tuneability, and
compact integration.

Optical nonreciprocity Lin  _et al._ (2011); Peng  _et al._ (2014); Chang  _et
al._ (2014); Fan  _et al._ (2012); Cao  _et al._ (2017); Manipatruni  _et al._
(2009); Shen  _et al._ (2016); Wang  _et al._ (2013); Ramezani  _et al._
(2018); Zhang  _et al._ (2018); Xia  _et al._ (2018); Tang  _et al._ (2022) is
characterized by the asymmetric behavior of optical signals as they travel
through an optical system in opposite directions. This phenomenon plays a
crucial role in optical information processing and quantum networks Bennett
and DiVincenzo (2000); Buluta  _et al._ (2011); Kimble (2008). Notably, a
theory of nonreciprocal phase transitions in non-equilibrium systems has been
proposed, suggesting that asymmetric couplings of multiple species can give
rise to time-dependent phases Fruchart  _et al._ (2021). Recently, this
concept has been applied to the Dicke model with two spin species Chiacchio
_et al._ (2023). Yet, the technique used to achieve nonreciprocal phase
transition in these schemes is not readily applicable to achieving
controllable nonreciprocal transitions in a light-matter system. In general,
achieving the superradiant transition requires tunability of the dipole
coupling of the radiation field to atoms. To this end, many pioneering
approaches have been proposed, including stimulated Raman transitions Ferri
_et al._ (2021); Klinder  _et al._ (2015); Zhiqiang  _et al._ (2017) and
quantum simulators Feng  _et al._ (2015); Chen  _et al._ (2021); Zheng  _et
al._ (2023). However, breaking the reciprocity of the system, in these
approaches, remains challenging. The exquisite control of superradiant
transitions and multicriticality through external fields is an intriguing
topic, may inspire new applications, such as on-chip unidirectional
superradiant lasers Meiser  _et al._ (2009); Bohnet  _et al._ (2012); Norcia
and Thompson (2016); Norcia  _et al._ (2016) and integrated high-precision
quantum sensing Garbe  _et al._ (2020); Chu  _et al._ (2021); Ying  _et al._
(2022); He  _et al._ (2023).

In this work, inspired by recent experimental advances Aoki  _et al._ (2006);
Alton  _et al._ (2011); Junge  _et al._ (2013); Shomroni  _et al._ (2014,
2014); Scheucher  _et al._ (2016); Bechler  _et al._ (2018); Will  _et al._
(2021); Maayani  _et al._ (2018) and related works Bliokh and Nori (2015);
Bliokh  _et al._ (2015), we propose an experimentally feasible approach for
all-optical control of superradiant phase transitions in a cavity QED system.
Our method focuses on a rotating dual-coupling Jaynes-Cummings (JC) model,
where the clockwise and counterclockwise resonator modes are simultaneously
coupled to a two-level atom. Generally, the JC model does not exhibit
superradiant transitions in the presence of cavity dissipation Larson and
Irish (2017). Here, we revive such transitions by introducing a classical
field to subtly parametrically pump one of the cavity modes. This driven-
dissipative setup enables steady-state superradiant transitions to occur in
the experimentally friendly cavity QED system, which do not require
ultrastrong coupling strengths, and the extremely large atomic detuning can be
easily achieved by tuning the external pump field. This all-optical control
has some main advantages compared to its magnetic and electronic counterparts,
such as compactness, ease of integration, and lower power consumption (see
Sec. S7.B in Supplementary Material SM ).

Interestingly, the combination of cavity rotation Maayani  _et al._ (2018);
Jing  _et al._ (2018); Huang  _et al._ (2018); Jiao  _et al._ (2020) and
directional squeezing causes the critical points of phase transitions to shift
in opposite directions. As a result, the system exhibits nonreciprocal first-
order and second-order superradiant phase transitions. Moreover, we observe a
rich phase diagram featuring controllable tricritical point and multicritical
points, all displaying nonreciprocity. Our work fundamentally combines the
theories of phase transitions and multicriticality with nonreciprocal physics,
and could provide valuable resources for quantum metrology Hotter  _et al._
(2024).

Figure 1: Schematic illustration of the rotating dual-coupling Jaynes-Cummings
(JC) model in the forward (a) and backward (b) pumps. The whispering-gallery-
mode (WGM) resonator with $\chi^{(2)}$ nonlinearity embedded (not shown here),
supports counterclockwise and clockwise cavity modes labeled as $a$ and $b$
respectively; and both modes interact with a two-level atom. The resonator
rotates counterclockwise at an angular velocity $\Omega$. Panels (c,d) show
the boundaries of the normal phase (NP) and superradiant phase (SP) for
forward and backward pumps, respectively. The parameters used here are:
$\Delta=2,\kappa/\Delta=0.05,G/\kappa=1.5$, $J=0$, and $\Delta_{F}/\Delta=0.5$
in (c), and $\Delta_{F}/\Delta=-0.5$ in (d).

Model.—Here, we consider the model of a two-level atom interacting with two
counter-propagating modes of a whispering-gallery-mode (WGM) resonator, as
depicted in Figs. 1(a,b). The resonator is made of materials with second-order
($\chi^{(2)}$) nonlinearity Lu  _et al._ (2019, 2020). A classical field at
frequency $\omega_{p}$ is input from either the forward or backward to drive
the nonlinearity, which generates directional squeezing cavity mode through an
optical parametric amplification (OPA) process Scully and Zubairy (1999);
Agarwal (2012); Huang and Agarwal (2009); Lü  _et al._ (2015); Qin  _et al._
(2018); Tang  _et al._ (2022); Qin  _et al._ (2022) (see Sec. S9.B in SM ).
The resonator rotates counterclockwise with an angular velocity, denoted as
$\Omega$. This rotation can be experimentally achieved by mounting the
resonator on a turbine Maayani  _et al._ (2018) (see Sec. S9.C in SM ).
Therefore, the two cavity modes undergo Sagnac-Fizeau shifts concerning their
static resonance frequency, represented by $\omega_{0}$ Malykin (2000);
Maayani  _et al._ (2018), i.e.,
$\omega_{0}\rightarrow\omega_{0}\pm\Delta_{F}$, with
$\Delta_{F}\approx{nR\Omega\omega_{0}(1-n^{-2})}/{c}$. Here $n$ and $R$
present the refractive index and radius of the resonator, and $c$ is the speed
of light. Note that a positive Sagnac-Fizeau shift ($\Delta_{F}>0$)
corresponds to the forward pump case, while a negative shift ($\Delta_{F}<0$)
corresponds to the backward pump.

In the frame rotating at $\omega_{p}/2$, the dual-coupling JC Hamiltonian with
the forward pump is given by ($\hbar=1$)

$\displaystyle H=H_{0}+H_{\rm int},$ (1) $\displaystyle
H_{0}=(\Delta\\!+\\!\Delta_{F})a^{\dagger}a\\!+\\!(\Delta\\!-\\!\Delta_{F})b^{\dagger}b\\!+\\!\frac{\Delta_{q}}{2}\sigma_{z}\\!+\\!G(a^{\dagger
2}\\!+\\!a^{2}),$ $\displaystyle H_{\rm
int}=\big{[}(g_{a}a+g_{b}b)\sigma_{+}\\!+\\!Ja^{\dagger}b+{\rm H.c.}\big{]},$

where $a$ and $b$ are the annihilation operators of the counterclockwise and
clockwise cavity modes, respectively, and $\sigma_{\pm}=(\sigma_{x}\pm
i\sigma_{y})/2$ are the atomic Pauli matrices. The detunings are defined as
$\Delta=\omega_{0}-\omega_{p}/2$ and $\Delta_{q}=\omega_{q}-\omega_{p}/2$. The
cavity modes $a$ and $b$ are coupled to the atom with strengths $g_{a}$ and
$g_{b}$, respectively, and $G(a^{{\dagger}2}+a^{2})$ is the two-photon drive
term generated from the OPA process, with $G$ representing the pump strength.
In our analytical calculations, the focus is on the forward pump case, and the
backward Hamiltonian can be obtained by replacing the two-photon term with
$G(b^{{\dagger}2}+b^{2})$. The parameter $J$ represents the hopping amplitude
between two cavity modes, and for simplicity, we assume $J=0$ in the following
analytical derivations, as it does not affect the main results (see Sec. S8 in
SM ).

Taking into account both the cavity and atomic dissipation, the system
evolution is described by the master equation,

$\displaystyle\frac{d\rho}{dt}=-{i}[H,\rho]+\kappa\mathcal{D}[a]+\kappa\mathcal{D}[b]+\gamma\mathcal{D}[\sigma_{-}],$
(2)

where $\rho$ is the densitiy matrix of the system, $\mathcal{D}[o]=2o\rho
o^{\dagger}-(o^{\dagger}o\rho+\rho o^{\dagger}o)$ is the Lindblad
superoperator. Here, we have assumed equal decay rates for the two cavity
modes ($\kappa_{a}=\kappa_{b}=\kappa$). To ensure the conservation of the
total pseudo-angular momentum, we focus mainly on the effect of cavity
dissipation and set the atomic decay rate $\gamma=0$.

The presence of dissipation, as discussed in Larson and Irish (2017), can
completely suppress the superradiant transition in the JC model.
Interestingly, in our model, the two-photon drive is capable of resurrecting
this transition. From a symmetry perspective, the two-photon term changes the
system from a continuous $U(1)$ symmetry to a discontinuous $Z_{2}$ symmetry.
This $Z_{2}$ symmetry is defined by $[H,\Pi]=0$ with the parity operator
$\Pi=\exp\\{i\pi N\\}$, where
$N=a^{{\dagger}}a+b^{{\dagger}}b+(\sigma_{z}+1)/2$ is the total number of
excitations in the system Hwang  _et al._ (2015). We define the ratios between
the atomic detuning and cavity detuning as
$\eta_{\pm}=\Delta_{q}/(\Delta\pm\Delta_{F})$, and $\mu=\Delta_{q}/G$. Note
that the limit of $\eta_{\pm}\rightarrow\infty$ is physically similar to the
infinite-frequency limit in the standard Rabi model
($\omega_{q}/\omega_{0}\rightarrow\infty$). Fortunately, in our model, this
limit can be easily achieved by tuning the pump frequency $\omega_{q}$. For
convenience, we introduce the dimensionless atom-field couplings as
$\lambda_{a,b}=2g_{a,b}/\sqrt{\Delta_{q}(\Delta\pm\Delta_{F})}$. These
couplings are determined by the detunings $\Delta_{q}$ and $\Delta$, rather
than the atomic frequency $\omega_{q}$ and resonator frequency $\omega_{0}$.
Thus, our approach relaxes the couplings required for the superradiant phase
transitions from the ultrastrong coupling regime to the strong coupling regime
(see Sec. S7.A in SM ). In the large atomic detuning limit,
$\eta_{\pm}\rightarrow\infty$ and $\mu\rightarrow\infty$, when the couplings
$\lambda_{a},\lambda_{b}$ increase beyond their critical values, the system
undergoes a Rabi-like phase transition from the NP to the SP, indicating the
breaking of $Z_{2}$ symmetry.

Nonreciprocal superradiant phase transitions.— According to Eq. (2), we obtain
a set of Heisenberg equations of motion for operators (see Sec. S1 in SM ). We
define the renormalized occupation of the cavity modes as $\langle
a\rangle=\alpha\sqrt{\eta_{+}}$ and $\langle b\rangle=\beta\sqrt{\eta_{-}}$,
with $\alpha=\alpha_{\rm re}+i\alpha_{\rm im}$, $\beta=\beta_{\rm
re}+i\beta_{\rm im}$. In the long-time limit, the system reaches a steady
state and we can obtain the mean-field solutions for cavity occupations
$\alpha_{\rm re},\alpha_{\rm im}$ and $\beta_{\rm re}$, $\beta_{\rm im}$ SM .
The validity of the mean-field approach is discussed in Sec. S6 in SM . When
$\alpha_{\rm re},\alpha_{\rm im},\beta_{\rm re},\beta_{\rm im}=0$, the system
is in the NP, while non-zero values of $\alpha_{\rm re},\alpha_{\rm
im},\beta_{\rm re},\beta_{\rm im}$ indicate the SP. Figures 1(c,d) display the
phase diagrams of $\alpha_{\rm re},\alpha_{\rm im}$ and $\beta_{\rm re}$,
$\beta_{\rm im}$, for the forward and backward pumps, respectively. When the
pump strength is fixed at $G/\kappa=1.5$, the first- (and second-) order
transition boundary differs between the two pump directions, leading to the
nonreciprocity of the superradiant transitions (see below). Here, the first-
and second-order transition boundaries are, respectively, characterized by
discontinuous jumps and continuous changes in the cavity occupations (see Fig.
S1 in SM ).

Figure 2: The cavity occupations $\alpha$ and $\beta$ versus the pump strength
$G/\kappa$, for both the forward and backward pumps. The solid (or dotted)
curve denotes the real (or imaginary) part of cavity occupations. In panels
(a,b), where $\lambda=1.5$, the shaded areas indicate the nonreciprocal first-
order phase transition (NR 1st-PT). In panels (c,d), where $\lambda=1.36$, the
shaded areas indicate the nonreciprocal second-order phase transition (NR 2nd-
PT). Here, $\Delta_{F}/\Delta=\pm 0.5$ for the forward and backward pumps,
respectively, $\lambda_{a}=\lambda_{b}=\lambda$, and other parameters are the
same as in Fig. 1. Figure 3: (a)-(c) Phase diagram of the steady-state photon
number fluctuations, with $\langle c^{\dagger}c\rangle$ for the forward [panel
(b)], $\langle d^{\dagger}d\rangle$ for the backward [panel (c)], and $\langle
c^{\dagger}c\rangle$ (or $\langle d^{\dagger}d\rangle$) for the static
microcavity [panel (a)]. The blue (red) bar represents fluctuations on top of
the mean-field solutions in the normal (superradiant) phase. Three regimes,
denoted as I, II, and III, are identified based on the number of stable
coexisting solutions. Subscripts 1 and 2 are employed to distinguish regimes
with different Wigner distributions. The tricritical points are shown by blue
triangles, and the multicritical points are marked by blue circles. The green
curve in panel (b) indicates the unstable region (see details in Sec. S5 of SM
). (d) The Wigner functions in different regimes. Here, we consider the
coordinates of $\lambda$ and $G/\kappa$ as ${\rm I}(1.0,1.1),{\rm
II}_{1}(1.42,2.6),{\rm III}_{1}(1.8,1.98),{\rm II}_{2}(2.04,2.24)$ and ${\rm
III}_{2}(2.2,2.42)$, $\Delta_{q}/\Delta=1000$ and atomic decay
$\gamma=\kappa$. (e) The critical lines at $\lambda=2$ [the black-dotted line
in panel (a)] as a function of $G$. (f,g) Cut plots of ${\rm ln}(\langle
c^{{\dagger}}c\rangle$+1) [black-dotted line in (b)] and ${\rm ln}(\langle
d^{{\dagger}}d\rangle$+1) [black-dotted line in panel (c)] as a function of
$\lambda$. In (e-g), the blue solid and red dotted lines denote cavity
fluctuations in the NP and SP, respectively. We consider $\Delta_{F}=0$ in
panels (a,e), $\Delta_{F}/\Delta=0.5$ in panels (b,d,f) and
$\Delta_{F}/\Delta=-0.5$ in panels (c,g).

To enable all-optical manipulation of superradiant phase transitions using an
external field, we simplify the system by adjusting
$\lambda_{a}=\lambda_{b}=\lambda$. In this configuration, with a fixed
coupling $\lambda$, increasing the pump strength $G$ beyond a critical value
can induce either a first- or second-order phase transition. The respective
critical pump strengths can be analytically calculated as (see Sec. S3 in SM
),

$\displaystyle G_{c}^{\rm 1st}$
$\displaystyle=\frac{1}{4\kappa\Delta_{-}}\big{[}\\!-\\!u\\!+\\!\sqrt{(u\\!+\\!2\Delta_{-}\kappa^{2})^{2}\\!+\\!4\Delta_{+}^{2}\Delta_{-}^{2}\kappa^{2}}~{}\big{]},$
(3) $\displaystyle G_{c}^{\rm 2nd}\\!$
$\displaystyle=\\!\frac{\big{[}\Delta^{4}p^{2}\\!\\!+\\!(2\kappa^{2}\\!\\!-\\!\Delta_{F}^{2}p)^{2}\\!\\!+\\!\Delta^{2}\big{(}qp^{2}\\!+\\!4\kappa^{2}\big{)}\big{]}^{{1}/{2}}}{\big{[}{16\kappa^{2}\\!+\\!\Delta_{-}^{2}(p-2)^{2}}\big{]}^{1/2}},$
(4)

where $\Delta_{\pm}=\Delta\pm\Delta_{F}$,
$u=\Delta_{+}(\Delta_{-}^{2}+\kappa^{2})$,
$q=\kappa^{2}\\!-\\!2\Delta_{F}^{2}$ and $p=\lambda^{2}-2$. Notably, the
first-order critical pump strength $G_{c}^{\rm 1st}$ is irrelevant to the
coupling $\lambda$. By setting $G_{c}^{\rm 1st}=G_{c}^{\rm 2nd}$, the
tricritical point can be determined at $\lambda_{\rm tric}\approx\sqrt{2}$
(see Fig. S2(b) in SM  for details).

Figure 2 shows the cavity occupations $\alpha$ and $\beta$ versus the pump
strength $G$. In the left panel, we consider the case of $\lambda=1.5$. When
the system is driven by a forward pump [Fig. 2(a)], the cavity occupations
exhibit a discontinuous jump from zero to finite values, when $G$ exceeds a
critical value $G_{\rm c,f}^{\rm 1st}=1.96\kappa$ (located near the right edge
of the shaded region). This is evidence of a first-order phase transition.
However, when driven by a backward pump [Fig. 2(b)], the critical value is
shifted to $G_{\rm c,b}^{\rm 1st}=0.66\kappa$, the left edge of the shaded
region. Between these two critical pump strengths (within the shaded areas),
if driven forward, the system is in the NP, while if driven backward, the
system is in the SP. This phenomenon is called nonreciprocal first-order
superradiant phase transition.

In the right panels of Fig. 2, we fix $\lambda=1.36$ and apply the forward and
backward pumps, respectively. In this case, the cavity occupations exhibit
continuous increase from zero to finite values at the critical pump strengths
$G^{\rm 2nd}_{\rm c,f}=2.78\kappa$ and $G^{\rm 2nd}_{\rm c,b}=0.94\kappa$,
respectively, as shown in Figs. 2(c,d). These results indicate a nonreciprocal
second-order superradiant phase transition. Physically, the nonreciprocity
originates from the opposite Sagnac shifts induced by the rotation of the
cavity. This leads to a shift of the critical point for the forward (or
backward) pump towards larger (or smaller) values (see Fig. S2(a) in SM ).

Nonreciprocal multicriticality.— To explore the steady-state fluctuations of
the system, we adopt a semiclassical approach by displacing the cavity fields
as $a\rightarrow\langle a\rangle+c$, $b\rightarrow\langle b\rangle+d$, where
$c$ and $d$ are the annihilation operators that describe cavity fluctuations.
In the large atomic detuning limit, we can obtain the effective low-energy
Hamiltonian (see Sec. S2 in SM )

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{\rm eff}=\,$
$\displaystyle\Lambda_{1}c^{\dagger}c+\Lambda_{2}d^{{\dagger}}d+\big{[}Gc^{2}+\Lambda_{3}cd^{{\dagger}}+\Lambda_{4}e^{2i\phi}c^{2}$
$\displaystyle+\Lambda_{5}e^{2i\phi}d^{2}\\!+\Lambda_{6}e^{2i\phi}cd+{\rm
H.c.}\big{]},$ (5)

where the parameters $\Lambda_{n}$ ($n=1,2,...,6$), $\phi$ are determined by
the Sagnac shift $\Delta_{F}$ and the mean-field solutions $\alpha$ and
$\beta$. In particular, the trivial solutions of $\alpha=\beta=0$ correspond
to the low-energy Hamiltonian in the normal phase. To analyze the nature of
the nonreciprocal transitions in the open system, we combine Eq. (S2.2) with
the master equation (2) and obtain the steady-state solution of quantum
fluctuations (see Sec. S4 in SM ).

Figures 3(a-c) present the steady-state phase diagrams of photon number
fluctuations. For a static cavity (i.e., $\Delta_{F}=0$) [Fig. 3(a)], we
observe two types of multicritical behaviors. There is a tricritical point
denoted by the blue triangle where the first- and second-order critical lines
meet, which is of the same type as in Refs. Soriente  _et al._ (2018); Zhu
_et al._ (2020a); Lin  _et al._ (2022). The other is the multicritical point
denoted by the blue circle, where regimes I, II, and III coexist. Specially,
these two critical lines, exhibiting different asymptotic behaviors Nagy  _et
al._ (2011), can morph into a divergent curve [see Fig. 3(e)]. Remarkably,
this multicritical point shows nonreciprocal characteristics: in the forward
pump, the multicritical point splits into two [see Fig. 3(b)], while in the
backward pump and with the same atom-field coupling, the multicritical point
disappears [see Fig. 3(c)]. Moreover, the cut plots of the phase diagram also
show the nonreciprocity of the photon number fluctuations [see Figs. 3(f,g)].

To distinguish between regimes I, II, and III, in Fig. 3(d), we numerically
calculate the Wigner functions of the cavity field using Hamiltonian in Eq.
(S1) Johansson  _et al._ (2012, 2013), considering the detuning
$\Delta_{q}/\Delta=10^{3}$ and the nonzero atomic decay $\gamma=\kappa$. In
the NP (regime I), the cavity field is almost in a vacuum state. In regime
${\rm II}_{1}$, the Wigner function shows two peaks, reflecting the $Z_{2}$
symmetry breaking of the system. In regimes ${\rm III_{1}}$ and ${\rm
III_{2}}$, the cavity fields are both squeezed, but with an orthogonal
squeezing direction.

Possible implementations.— The proposed protocols could be implemented using
either cold cesium atoms falling onto the surface of a WGM microdisk Aoki  _et
al._ (2006); Alton  _et al._ (2011) or a single trapped 85Rb atom interacts
with a WGM microresonator Will  _et al._ (2021) (see more details in Sec. S9.A
of SM ). We consider the feasible parameters of the microresonator to be
$Q=6.0\times 10^{9}$, $R=1.1$ mm, $n=1.4$, $\lambda_{0}=1550\,{\rm nm}$ and
$\Omega=6.6$ kHz Maayani  _et al._ (2018), which yields a Sagnac shift of
$\Delta_{F}=3.2$ MHz and the intrinsic loss rate of $\kappa_{i}=32$ kHz. The
large atomic detuning required for the transition can be achieved by tuning
the pump frequency $\omega_{p}\approx 2\omega_{0}$. We consider
$\eta_{+}=100$, resulting in $\Delta=6.4$ MHz and $\Delta_{q}=960$ MHz.
Previous studies have shown that a detuning value of
$\omega_{q}/\omega_{0}=50$ is sufficient to observe the superradiant
transitions in NMR simulator Chen  _et al._ (2021) and trapped ion systems
Puebla  _et al._ (2017); Cai  _et al._ (2021). Based on the above parameters,
the nonreciprocal first- and second-order superradiant transitions are
predicted to occur at the atom-field couplings $g_{a}\approx 68$ MHz and
$g_{b}\approx 39$ MHz, which are experimentally feasible by adjusting the
distance between the atom and the surface Will  _et al._ (2021). Additionally,
the squeezed cavity mode can be generated using a periodically poled thin film
of lithium niobate media Lu  _et al._ (2019, 2020). Here, we consider the pump
strength $G/\kappa=2$, the external decay $\kappa_{\rm ex}^{p}=2\kappa=640$
kHz, and the coupling rate of the parametric nonlinear process
($g/\kappa=0.001$ Lu  _et al._ (2020)), which yields the feasible critical
pump power $P_{c}\approx 2.1$ nW.

Conclusions.— In summary, we have proposed a method for all-optical
manipulating superradiant phase transitions and multicritical phenomena in an
open dual-coupling JC model. The nonreciprocal first- and second-order
superradiant transitions can be easily achieved by tuning the external pump
field. Furthermore, the model allows for exquisite manipulation of the
tricritical and multicritical points, both exhibiting controllable
nonreciprocity. This general approach can be extended to the case of $N$
particles ($N\gg 1$), specifically in the context of the dual-coupling Tavis-
Cummings model (see Sec. S10 in SM ). We anticipate that this work will
stimulate further theoretical studies and experimental explorations of a
broader range of physical phenomena, such as superradiant cooling Xu  _et al._
(2016) and atomic synchronization Xu  _et al._ (2014); Masson and Asenjo-
Garcia (2022), and could find applications in modern quantum technology
Braunstein and van Loock (2005).

Acknowledgments.— We acknowledge valuable discussions with Ye-Xiong Zeng, Ran
Huang, and Zi-Yong Ge. We also appreciate the valuable comments from Ye-Hong
Chen, Zongping Gong, Keyu Xia, Bo Wang, and Zeliang Xiang. X.-Y.L. is
supported by the National Key Research and Development Program of China (Grant
No. 2021YFA1400700). F.N. is supported in part by: Nippon Telegraph and
Telephone Corporation (NTT) Research, the Japan Science and Technology Agency
(JST) [via the Quantum Leap Flagship Program (Q-LEAP), and the Moonshot R&D
Grant Number JPMJMS2061], the Asian Office of Aerospace Research and
Development (AOARD) (via Grant No. FA2386-20-1-4069), and the Office of Naval
Research (ONR) (via Grant No. N62909-23-1-2074). The computation is completed
in the HPC Platform of Huazhong University of Science and Technology.

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Supplementary Material for “Nonreciprocal Superradiant Phase Transitions and
Multicriticality in a Cavity QED System”

Gui-Lei Zhu1,2, Chang-Sheng Hu3, Hui Wang2,Wei Qin2,4, Xin-You Lü${}^{5,^{*}}$
and Franco Nori${}^{2,6,7,^{\dagger}}$

1Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China
2Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research,
RIKEN, Wakoshi, Saitama 351-0198, Japan
3Department of Physics, Anhui Normal University, Wuhu 241000, China
4Center for Joint Quantum Studies and Department of Physics, School of
Science, Tianjin University, Tianjin 300350, China
5School of Physics, Huazhong University of Science and Technology and Wuhan
Institute of Quantum Technology, Wuhan 430074, China
6Quantum Computing Center, RIKEN, Wakoshi, Saitama 351-0198, Japan
7Department of Physics, The University of Michigan, Ann Arbor, MI, 48109-1040,
USA

###### Contents

  1. S1 Mean-field solutions of $\alpha$ and $\beta$
  2. S2 Effective Hamiltonian in the infinite-detuning limit ($\Delta_{q}/(\Delta\pm\Delta_{F})\rightarrow\infty$)
    1. S2.1 Low-energy Hamiltonian in the normal phase
    2. S2.2 Low-energy Hamiltonian in the superradiant phase
  3. S3 Phase transition boundaries
    1. S3.1 First-order phase transition boundary
    2. S3.2 Second-order phase transition boundary
  4. S4 Dynamic evolution of two-operator correlators
    1. S4.1 Equations of motion in the normal phase
    2. S4.2 Equations of motion in the superradiant phase
  5. S5 Stability analysis
  6. S6 The validity of the semiclassical approach
    1. S6.1 Overview
    2. S6.2 Analytical derivations
    3. S6.3 Squeezing parameter
  7. S7 Discussions on the dual-coupling JC model
    1. S7.1 Comparison between the dual-coupling JC model and the Rabi model
    2. S7.2 Comparison between all-optical controls and its magnetic or electronic counterpart
  8. S8 Discussion on the assumption of system parameters
    1. S8.1 The case of nonzero cavity hopping rate $J\neq 0$
    2. S8.2 Numerical simulations on steady-state mean photon number
  9. S9 Possible experimental implementations
    1. S9.1 Strong coupling between an atom and microresonator
    2. S9.2 Directionally squeezing of the cavity mode
    3. S9.3 Rotating resonator
    4. S9.4 Discussion on the effect of unexpected noises and perturbations
  10. S10 Extending the model to the case of $N$ particles ($N\gg 1$)

##  S1 Mean-field solutions of $\alpha$ and $\beta$

We recall that the system Hamiltonian in the forward pump reads,

$\displaystyle H=$
$\displaystyle(\Delta+\Delta_{F})a^{\dagger}a+\frac{\Delta_{q}}{2}\sigma_{z}+g_{a}(a\sigma_{+}+a^{\dagger}\sigma_{-})+G(a^{\dagger
2}+a^{2})$
$\displaystyle+(\Delta-\Delta_{F})b^{\dagger}b+g_{b}(b\sigma_{+}+b^{\dagger}\sigma_{-})+J(a^{\dagger}b+b^{\dagger}a),$
(S1)

where $a$ and $b$ are the annihilation operators for the counterclockwise and
clockwise cavity modes, respectively. Also,
$\sigma_{\pm}=(\sigma_{x}+i\sigma_{y})/2$, and $\sigma_{x,y,z}$ are Pauli
matrices of the two-level atom. Here, $\Delta_{F}$ is the Sagnac shift caused
by the rotation of the cavity. The counterclockwise and clockwise cavity modes
are coupled to the atom with the strengths $g_{a}$ and $g_{b}$, respectively.
By employing the master equation (2) shown in the main text, we can derive the
Heisenberg equations of motion for operators,

$\displaystyle\frac{da}{dt}$ $\displaystyle=-i(\Delta+\Delta_{F})a-\kappa
a-2iGa^{{\dagger}}-iJb-\frac{i}{2}g_{a}\sigma_{x}-\frac{1}{2}g_{a}\sigma_{y},$
(S2) $\displaystyle\frac{db}{dt}$ $\displaystyle=-i(\Delta-\Delta_{F})b-\kappa
b-iJa-\frac{i}{2}g_{b}\sigma_{x}-\frac{1}{2}g_{b}\sigma_{y},$ (S3)
$\displaystyle\frac{d\sigma_{x}}{dt}$
$\displaystyle=-\Delta_{q}\sigma_{y}+ig_{a}(a-a^{{\dagger}})\sigma_{z}+ig_{b}(b-b^{{\dagger}})\sigma_{z},$
(S4) $\displaystyle\frac{d\sigma_{y}}{dt}$
$\displaystyle=\Delta_{q}\sigma_{x}-g_{a}(a+a^{{\dagger}})\sigma_{z}-g_{b}(b+b^{{\dagger}})\sigma_{z},$
(S5) $\displaystyle\frac{d\sigma_{z}}{dt}$
$\displaystyle=ig_{a}(a^{{\dagger}}-a)\sigma_{x}+ig_{b}(b^{{\dagger}}-b)\sigma_{x}+g_{a}(a+a^{{\dagger}})\sigma_{y}+g_{b}(b+b^{{\dagger}})\sigma_{y}.$
(S6)

In these equations, we assumed that the decay rates of cavity modes $a$ and
$b$ are equal, namely, $\kappa_{a}=\kappa_{b}=\kappa$. Additionally, we
neglected the atomic loss in the system. Similar to the standard Rabi model,
in the steady-state limit, we set the following expectation values:
$\langle\sigma_{x}\rangle=X$, $\langle\sigma_{y}\rangle=Y$,
$\langle\sigma_{z}\rangle=Z$, $\langle a\rangle=\alpha\sqrt{\eta_{+}}$,
$\langle b\rangle=\beta\sqrt{\eta_{-}}$, where

$\displaystyle\eta_{+}=\frac{\Delta_{q}}{\Delta+\Delta_{F}},\,\,\,\eta_{-}=\frac{\Delta_{q}}{\Delta-\Delta_{F}}.$
(S7)

The mean-field approximation holds in the infinite detuning limit, where
$\eta_{\pm}\rightarrow\infty$. The validity of the mean-field approximation is
discussed in Sec. S6. Under the spin-conservation law, $X^{2}+Y^{2}+Z^{2}=1$,
the following relations are obtained:

$\displaystyle C$ $\displaystyle=\Gamma_{1}\alpha_{\rm
re}^{2}+\Gamma_{2}\alpha_{\rm re}\alpha_{\rm im}+\Gamma_{3}\alpha_{\rm
im}^{2},$ (S8) $\displaystyle C$ $\displaystyle=K(\alpha_{\rm
re}^{2}+\alpha_{\rm im}^{2}),$ (S9)

where $C=1-Z^{2}$,

$\displaystyle\Gamma_{1,3}=\frac{[16(\Delta+\Delta_{F}\pm
2G)^{2}+\kappa^{2}]}{(\Delta+\Delta_{F})^{2}\lambda_{a}^{2}},\,\,\,\Gamma_{2}=\frac{128G\kappa}{(\Delta+\Delta_{F})^{2}\lambda_{a}^{2}},\,\,\,K=\frac{16[(\Delta-\Delta_{F})^{2}+\kappa^{2}]\lambda_{a}^{2}Z^{2}}{16\kappa^{2}+(\Delta-\Delta_{F})^{2}(4+\lambda_{b}^{2}Z)^{2}}.$
(S10)

In these equations, we have separated the real and imaginary parts of $\alpha$
and $\beta$ as $\alpha=\alpha_{\rm re}+i\alpha_{\rm im}$ and $\beta=\beta_{\rm
re}+i\beta_{\rm im}$. The constraint parameter $Z$ satisfies

$\displaystyle\\!\\!\\!\\!Z_{\pm}=4\frac{\\!-\\!\Delta_{+}h_{1}-\Delta_{-}h_{2}\pm\sqrt{\big{[}h_{1}(2G-\kappa)-\kappa
h_{2}\big{]}\big{[}h_{1}(2G+\kappa)+\kappa
h_{2}\big{]}}}{\Delta_{+}h_{1}\lambda_{a}^{2}+2\Delta_{+}\Delta_{-}(\Delta_{+}\Delta_{-}+\kappa^{2})\lambda_{a}^{2}\lambda_{b}^{2}+\Delta_{-}h_{2}\lambda_{b}^{2}},$
(S11)

where we have denoted $\Delta_{\pm}=\Delta\pm\Delta_{F}$,
$h_{1}=\Delta_{+}(\Delta_{-}^{2}+\kappa^{2})\lambda_{a}^{2}$ and
$h_{2}=\Delta_{-}(\Delta_{+}^{2}-4G^{2}+\kappa^{2})\lambda_{b}^{2}$ for
simplification purposes.

The mean-field solutions of $\alpha$ and $\beta$ in the steady state can be
obtained by solving Eqs. (S8) and (S9). The solutions are given by

$\displaystyle\alpha_{\rm re}$
$\displaystyle=\pm\sqrt{C}\sqrt{\frac{\Gamma_{2}^{2}-2(\Gamma_{1}-\Gamma_{3})(\Gamma_{3}-K)+\Gamma_{2}\sqrt{\Gamma_{2}^{2}+4(\Gamma_{1}-K)(K-\Gamma_{3})}}{2(\Gamma_{2}^{2}+(\Gamma_{1}-\Gamma_{3})^{2})K}},$
(S12) $\displaystyle\alpha_{\rm im}$
$\displaystyle=\frac{\Gamma_{2}-\sqrt{\Gamma_{2}^{2}+4(\Gamma_{1}-K)(K-\Gamma_{3})}}{2(K-\Gamma_{3})}\alpha_{\rm
re},$ (S13) $\displaystyle\beta_{\rm re}$
$\displaystyle=-4\frac{(\Delta+\Delta_{F}+2G)\alpha_{\rm re}+\kappa\alpha_{\rm
im}}{(\Delta+\Delta_{F})\lambda_{a}\lambda_{b}Z}-\frac{\lambda_{a}}{\lambda_{b}}\alpha_{\rm
re},$ (S14) $\displaystyle\beta_{\rm im}$
$\displaystyle=-4\frac{(\Delta+\Delta_{F}-2G)\alpha_{\rm im}-\kappa\alpha_{\rm
re}}{(\Delta+\Delta_{F})\lambda_{a}\lambda_{b}Z}-\frac{\lambda_{a}}{\lambda_{b}}\alpha_{\rm
im}.$ (S15)

where we have defined the rescaled atom-field coupling strengths as
$\lambda_{a}=2g_{a}/\sqrt{(\Delta+\Delta_{F})\Delta_{q}}$ and
$\lambda_{b}=2g_{b}/\sqrt{(\Delta-\Delta_{F})\Delta_{q}}$.

##  S2 Effective Hamiltonian in the infinite-detuning limit
($\Delta_{q}/(\Delta\pm\Delta_{F})\rightarrow\infty$)

In Sec. S1, we used the mean-field approximation to obtain the steady-state
solutions of $\alpha$ and $\beta$, which neglected quantum fluctuations. In
this section, we provide a full quantum description of the Hamiltonian.

###  S2.1 Low-energy Hamiltonian in the normal phase

In the large detuning limit, where the atomic energy scale is much larger than
the field part, we can decouple the atomic subspaces and focus on the lowest
atomic subspace. This allows us to eliminate the atomic part and obtain the
effective low-energy Hamiltonian. To achieve this, we rewrite Hamiltonian (S1)
as

$\displaystyle H$ $\displaystyle=H_{c}+g_{a}V_{a}+g_{b}V_{b},$ (S16)

with

$\displaystyle
H_{c}=(\Delta+\Delta_{F})a^{\dagger}a+\frac{\Delta_{q}}{2}\sigma_{z}+G(a^{\dagger
2}+a^{2})+(\Delta-\Delta_{F})b^{\dagger}b+J(a^{{\dagger}}b+b^{{\dagger}}a),$

and

$\displaystyle
V_{a}=(a\sigma_{+}\\!+\\!a^{\dagger}\sigma_{-}),\,\,\,\,\,\,\,\,\,V_{b}=(b\sigma_{+}\\!+\\!b^{\dagger}\sigma_{-}).$

The Hamiltonian $H_{c}$ has two decoupled spin subspaces
$\mathcal{H}_{\downarrow}$ and $\mathcal{H}_{\uparrow}$, where
$\lvert\uparrow\rangle$ and $\lvert\downarrow\rangle$ are the eigenstates of
$\sigma_{z}$. However, the Hamiltonian $g_{a}V_{a}+g_{b}V_{b}$ introduces the
interactions between these two subspaces. To eliminate the interaction terms,
we apply the Schrieffer-Woff transformation with the unitary operator $S_{\rm
np}=g_{a}/\Delta_{q}(a^{{\dagger}}\sigma_{-}-a\sigma_{+})+g_{b}/\Delta_{q}(b^{{\dagger}}\sigma_{-}-b\sigma_{+})$
to the master equation given in Eq. (2) of the main text. We keep terms up to
$g_{a,b}^{2}/\Delta_{q}$ and neglect higher-order terms in the large detuning
limit, i.e., $\Delta_{q}/(\Delta\pm\Delta_{F})\rightarrow\infty$ and
$\Delta_{q}/G\rightarrow\infty$. Under the projection of the
$\mathcal{H}_{\downarrow}$ space, we obtain the effective master equation in
the normal phase,

$\displaystyle\dot{\rho}_{\rm np}=-i[H_{\rm np},\rho_{\rm
np}]+\kappa(2a\rho_{\rm np}a^{{\dagger}}-a^{{\dagger}}a\rho_{\rm np}-\rho_{\rm
np}a^{{\dagger}}a)+\kappa(2b\rho_{\rm np}b^{{\dagger}}-b^{{\dagger}}b\rho_{\rm
np}-\rho_{\rm np}b^{{\dagger}}b),$ (S17)

where $\rho_{\rm np}=\langle\downarrow|e^{-S_{\rm np}}\rho e^{S_{\rm
np}}|\downarrow\rangle$, and the effective low-energy Hamiltonian in the
normal phase becomes,

$\displaystyle H_{\rm np}$ $\displaystyle\equiv\langle\downarrow|e^{-S_{\rm
np}}He^{S_{\rm np}}|\downarrow\rangle$
$\displaystyle=\Delta_{a}a^{{\dagger}}a+\Delta_{b}b^{{\dagger}}b+G(a^{{\dagger}2}+a^{2})+J_{s}(ab^{{\dagger}}+a^{{\dagger}}b)-\frac{\Delta_{q}}{2},$
(S18)

where
$\Delta_{a}=(\Delta+\Delta_{F})(1-{\lambda_{a}^{2}}/{4}),\Delta_{b}=(\Delta-\Delta_{F})(1-{\lambda_{b}^{2}}/{4}),J_{s}=J-\\!\lambda_{a}\lambda_{b}\sqrt{(\Delta+\Delta_{F})(\Delta-\Delta_{F})}/{4}$
and the renormalized atom-field couplings
$\lambda_{a}={2g_{a}}/{\sqrt{(\Delta+\Delta_{F})\Delta_{q}}}$ and
$\lambda_{b}={2g_{b}}/{\sqrt{(\Delta-\Delta_{F})\Delta_{q}}}$.

###  S2.2 Low-energy Hamiltonian in the superradiant phase

To address the superradiant phase where the cavity fields are macroscopically
occupied, we apply displacement transformations to cavity modes. Specifically,
we have

$\displaystyle
D^{{\dagger}}[\alpha\sqrt{\eta_{+}}]aD[\alpha\sqrt{\eta_{+}}]=c+\alpha\sqrt{\eta_{+}},\,\,\,D^{{\dagger}}[\beta\sqrt{\eta_{-}}]bD[\beta\sqrt{\eta_{-}}]=d+\beta\sqrt{\eta_{-}}$
(S19)

with $D[\alpha\sqrt{\eta_{+}}]=\exp[\sqrt{\eta_{+}}(\alpha
a^{{\dagger}}-\alpha^{*}a)]$ and
$D[\beta\sqrt{\eta_{-}}]=\exp[\sqrt{\eta_{-}}(\beta
b^{{\dagger}}-\beta^{*}b)]$ . Notably, the displacements $\alpha$ and $\beta$
correspond to the mean-field solutions discussed in Sec. S1. Here, we use $c$
and $d$ to denote the fluctuation operators of the cavity modes. By applying
the displacement transformation, the master equation becomes

$\displaystyle\dot{\tilde{\rho}}=-i[H(\alpha,\beta),\tilde{\rho}]+\kappa(2c\tilde{\rho}c^{{\dagger}}-c^{{\dagger}}c\tilde{\rho}-\tilde{\rho}c^{{\dagger}}c)+\kappa(2d\tilde{\rho}d^{{\dagger}}-d^{{\dagger}}d\tilde{\rho}-\tilde{\rho}d^{{\dagger}}d),$
(S20)

where
$\tilde{\rho}=D^{{\dagger}}[\alpha\sqrt{\eta_{+}}]D^{{\dagger}}[\beta\sqrt{\eta_{-}}]\rho
D[\alpha\sqrt{\eta_{+}}]D[\beta\sqrt{\eta_{-}}]$ and $H(\alpha,\beta)$ reads,

$\displaystyle H(\alpha,\beta)=$
$\displaystyle\Delta_{+}c^{{\dagger}}c+\frac{\Delta_{q}}{2}\sigma_{z}+\Delta_{-}d^{{\dagger}}d+\frac{1}{2}\lambda_{a}\sqrt{\Delta_{q}\Delta_{+}}(c\sigma_{+}+c^{{\dagger}}\sigma_{-})+\frac{1}{2}\lambda_{b}\sqrt{\Delta_{q}\Delta_{-}}(d^{{\dagger}}\sigma_{-}+d\sigma_{+})\\!$
$\displaystyle+G(c^{{\dagger}2}+c^{2})+J(c^{{\dagger}}d+d^{{\dagger}}c)+\frac{\Delta_{q}\lambda_{a}}{2}(\alpha\sigma_{+}+\alpha^{*}\sigma_{-})+\frac{\Delta_{q}\lambda_{b}}{2}(\beta\sigma_{+}+\beta^{*}\sigma_{-}),$
(S21)

where $\Delta_{\pm}=\Delta\pm\Delta_{F}$, $\alpha=\alpha_{\rm re}+i\alpha_{\rm
im},\beta=\beta_{\rm re}+i\beta_{\rm im}$ and we have omitted the linear terms
and constant terms in the Hamiltonian.

Next, we focus on the atomic part of Hamiltonian, which includes terms such as
${\Delta_{q}}\sigma_{z}/{2}+{\Delta_{q}\lambda_{a}}(\alpha\sigma_{+}+\alpha^{*}\sigma_{-})/{2}+{\Delta_{q}\lambda_{b}}(\beta\sigma_{+}+\beta^{*}\sigma_{-})/{2}$.
The eigenvalues of this atomic part are $\pm{\tilde{\Delta}_{q}}/{2}$, with

$\displaystyle\tilde{\Delta}_{q}=\Delta_{q}\sqrt{1+\lambda_{a}^{2}(\alpha_{\rm
re}^{2}+\alpha_{\rm im}^{2})+2\lambda_{a}\lambda_{b}(\alpha_{\rm im}\beta_{\rm
im}+\alpha_{\rm re}\beta_{\rm re})+\lambda_{b}^{2}(\beta_{\rm
im}^{2}+\beta_{\rm re}^{2})},$ (S22)

and the corresponding eigenstates are given by

$\displaystyle\lvert\tilde{\downarrow}\rangle$ $\displaystyle=\cos\theta
e^{i\gamma}\lvert\uparrow\rangle+\sin\theta e^{i\phi}\lvert\downarrow\rangle,$
(S23) $\displaystyle\lvert\tilde{\uparrow}\rangle$ $\displaystyle=-\sin\theta
e^{-i\phi}\lvert\uparrow\rangle+\cos\theta
e^{-i\gamma}\lvert\downarrow\rangle,$ (S24)

where

$\displaystyle e^{i\phi}=\frac{\lambda_{a}(\alpha_{\rm im}+i\alpha_{\rm
re})+\lambda_{b}(\beta_{\rm im}+i\beta_{\rm
re})}{\sqrt{\lambda_{a}^{2}(\alpha_{\rm re}^{2}+\alpha_{\rm
im}^{2})+2\lambda_{a}\lambda_{b}(\alpha_{\rm im}\beta_{\rm im}+\alpha_{\rm
re}\beta_{\rm re})+\lambda_{b}^{2}(\beta_{\rm re}^{2}+\beta_{\rm im}^{2})}},$
(S25)

and $e^{i\gamma}=-i$. The angle $\theta$ satisfies

$\displaystyle\tan 2\theta=-\sqrt{\lambda_{a}^{2}(\alpha_{\rm
re}^{2}+\alpha_{\rm im}^{2})+2\lambda_{a}\lambda_{b}(\alpha_{\rm im}\beta_{\rm
im}+\alpha_{\rm re}\beta_{\rm re})+\lambda_{b}^{2}(\beta_{\rm
re}^{2}+\beta_{\rm im}^{2})}.$ (S26)

Now, we define the Pauli matrices in the
$\\{\lvert\tilde{\downarrow}\rangle,\lvert\tilde{\uparrow}\rangle\\}$ basis as
$\tau_{x}=\lvert\tilde{\uparrow}\rangle\langle\tilde{\downarrow}\rvert+\lvert\tilde{\downarrow}\rangle\langle\tilde{\uparrow}\rvert$,
$\tau_{y}=-i(|\tilde{\uparrow}\rangle\langle\tilde{\downarrow}\rvert-\lvert\tilde{\downarrow}\rangle\langle\tilde{\uparrow}\rvert)$
and
$\tau_{z}=\lvert\tilde{\uparrow}\rangle\langle\tilde{\uparrow}\rvert-\lvert\tilde{\downarrow}\rangle\langle\tilde{\downarrow}\rvert$.
Then, $\sigma_{\pm,z}$ in terms of $\tau_{\pm,z}$ can be expressed as

$\displaystyle\sigma_{+}$ $\displaystyle=-\sin\theta\cos\theta
e^{i(\phi-\gamma)}\tau_{z}+\cos^{2}\theta e^{-2i\gamma}\tau_{-}-\sin^{2}\theta
e^{2i\phi}\tau_{+},$ (S27) $\displaystyle\sigma_{z}$ $\displaystyle=-\cos
2\theta\tau_{z}-\sin 2\theta e^{i(\phi+\gamma)}\tau_{+}-\sin 2\theta
e^{-i(\phi+\gamma)}\tau_{-}.$ (S28)

By substituting the expression for $\sigma_{\pm}$ and $\sigma_{z}$ from Eqs.
(S27) and (S28) into the Hamiltonian equation (S2.2),we obtain the transformed
Hamiltonian as follows:

$\displaystyle H(\alpha,\beta)$
$\displaystyle=\Delta_{+}c^{{\dagger}}c+\Delta_{-}d^{{\dagger}}d+\frac{\tilde{\Delta}_{q}}{2}\tau_{z}+G(c^{{\dagger}2}+c^{2})+J(c^{{\dagger}}d+d^{{\dagger}}c)$
$\displaystyle-\frac{1}{2}\lambda_{a}\sqrt{\Delta_{q}\Delta_{+}}\sin^{2}\theta(c\tau_{+}e^{2i\phi}+c^{{\dagger}}\tau_{-}e^{-2i\phi})-\frac{1}{2}\lambda_{a}\sqrt{\Delta_{q}\Delta_{+}}\cos^{2}\theta(c\tau_{-}+c^{{\dagger}}\tau_{+})$
$\displaystyle-\frac{1}{2}\lambda_{b}\sqrt{\Delta_{q}\Delta_{-}}\sin^{2}\theta(d\tau_{+}e^{2i\phi}+d^{{\dagger}}\tau_{-}e^{-2i\phi})-\frac{1}{2}\lambda_{b}\sqrt{\Delta_{q}\Delta_{-}}\cos^{2}\theta(d\tau_{-}+d^{{\dagger}}\tau_{+}).$
(S29)

Next, we apply the unitary transformation $H^{\prime}_{\rm sp}=e^{-S_{\rm
sp}}H(\alpha,\beta)e^{S_{\rm sp}}$ with the operator $S_{\rm sp}$ given by:

$\displaystyle S_{\rm sp}=$
$\displaystyle\frac{\lambda_{a}\sqrt{\Delta_{q}(\Delta+\Delta_{F})}}{2\bar{\Delta}_{q}}\left[\sin^{2}\theta(c\tau_{+}e^{2i\phi}-c^{{\dagger}}\tau_{-}e^{-2i\phi})+\cos^{2}(c^{{\dagger}}\tau_{+}-c\tau_{-})\right]$
$\displaystyle+\frac{\lambda_{b}\sqrt{\Delta_{q}(\Delta-\Delta_{F})}}{2\bar{\Delta}_{q}}\left[\sin^{2}\theta(d\tau_{+}e^{2i\phi}-d^{{\dagger}}\tau_{-}e^{-2i\phi})+\cos^{2}\theta(d^{{\dagger}}\tau_{+}-d\tau_{-})\right].$
(S30)

Projecting $H^{\prime}_{\rm sp}$ onto $|\tilde{\downarrow}\rangle$ basis, we
obtain the effective Hamiltonian in the superradiant phase

$\displaystyle H_{\rm sp}$
$\displaystyle=\Lambda_{1}c^{\dagger}c+\Lambda_{2}d^{{\dagger}}d\\!+\\!G(c^{2}\\!+\\!c^{{\dagger}2})\\!+\\!\Lambda_{3}(cd^{{\dagger}}\\!+\\!c^{{\dagger}}d)+\Lambda_{4}(e^{2i\phi}c^{2}\\!+\\!e^{-2i\phi}c^{{\dagger}2})\\!$
$\displaystyle+\\!\Lambda_{5}(e^{2i\phi}d^{2}\\!+\\!e^{-2i\phi}d^{{\dagger}2})+\Lambda_{6}(e^{2i\phi}cd\\!+\\!e^{-2i\phi}c^{{\dagger}}d^{{\dagger}}),$
(S31)

where

$\displaystyle\Lambda_{1}=(\Delta+\Delta_{F})\left(1-\frac{\lambda_{a}^{2}\Delta_{q}(\sin^{4}\theta+\cos^{4}\theta)}{4\tilde{\Delta}_{q}}\right),\,\,\,\,\Lambda_{2}=(\Delta-\Delta_{F})\left(1-\frac{\lambda_{b}^{2}\Delta_{q}(\sin^{4}\theta+\cos^{4}\theta)}{4\tilde{\Delta}_{q}}\right),$
$\displaystyle\Lambda_{3}=J-\frac{\sqrt{(\Delta+\Delta_{F})(\Delta-\Delta_{F})}\lambda_{a}\lambda_{b}\Delta_{q}(\sin^{4}\theta+\cos^{4}\theta)}{4\tilde{\Delta}_{q}},\,\,\,\,\Lambda_{4}=-\frac{(\Delta+\Delta_{F})\lambda_{a}^{2}\Delta_{q}\sin^{2}\theta\cos^{2}\theta}{4\tilde{\Delta}_{q}},$
$\displaystyle\Lambda_{5}=-\frac{(\Delta-\Delta_{F})\lambda_{b}^{2}\Delta_{q}\sin^{2}\theta\cos^{2}\theta}{4\tilde{\Delta}_{q}},\,\,\,\,\Lambda_{6}=-\frac{\sqrt{(\Delta+\Delta_{F})(\Delta-\Delta_{F})}\lambda_{a}\lambda_{b}\Delta_{q}\sin^{2}\theta\cos^{2}\theta}{2\tilde{\Delta}_{q}}.$
(S32)

##  S3 Phase transition boundaries

###  S3.1 First-order phase transition boundary

In this section, we derive the first-order phase transition boundary for the
superradiant phase transition. Starting from Eq. (S11), we impose a constraint
that the square root term must be greater than or equal to zero, resulting in
the condition:

$\displaystyle h_{1}^{2}(4G^{2}-\kappa^{2})-h_{2}^{2}\kappa^{2}\geq
2h_{1}h_{2},$ (S33)

where $h_{1}=\Delta_{+}(\Delta_{-}^{2}+\kappa^{2})\lambda_{a}^{2}$ and
$h_{2}=\Delta_{-}(\Delta_{+}^{2}-4G^{2}+\kappa^{2})\lambda_{b}^{2}$. We can
set this condition with an equal sign to obtain the critical ratio between the
atom-field couplings $\lambda_{ac}$ and $\lambda_{bc}$ as follows:

$\displaystyle\chi_{c}=\frac{\lambda_{bc}}{\lambda_{ac}}=\frac{\sqrt{2G-\kappa}\sqrt{(\Delta+\Delta_{F})((\Delta-\Delta_{F})^{2}+\kappa^{2})}}{\sqrt{(\Delta-\Delta_{F})\kappa((\Delta+\Delta_{F})^{2}-4G^{2}+\kappa^{2})}}.$
(S34)

This ratio $\chi_{c}$ defines the first-order superradiant phase transition
boundary in the $\lambda_{a}$–$\lambda_{b}$ phase space. To ensure that
$\chi_{c}$ is a real number, the pump strength $G$ must satisfy the condition:

$\displaystyle\frac{\kappa}{2}<G<\frac{\sqrt{(\Delta+\Delta_{F})^{2}+\kappa^{2}}}{2}.$
(S35)

From the condition in Eq. (S33), we can derive the critical pump strength for
the first-order phase transition, denoted as $G_{\rm c}^{\rm 1st}$,

$\displaystyle G_{\rm c}^{\rm
1st}=\frac{1}{4\Delta_{-}\kappa\lambda_{b}^{2}}\left\\{-\Delta_{+}(\Delta_{-}^{2}+\kappa^{2})\lambda_{a}^{2}+\sqrt{\Big{[}\Delta_{+}(\Delta_{-}^{2}+\kappa^{2})\lambda_{a}^{2}+2\Delta_{-}\kappa^{2}\lambda_{b}^{2}\Big{]}^{2}+4\Delta_{+}^{2}\Delta_{-}^{2}\kappa^{2}\lambda_{b}^{4}}\right\\}.$
(S36)

In the special case where $\lambda_{a}=\lambda_{b}=\lambda$, the above
equation simplifies to

$\displaystyle G_{\rm c}^{\rm
1st}=\frac{1}{4\Delta_{-}\kappa}\left\\{-\Delta_{+}(\Delta_{-}^{2}+\kappa^{2})+\sqrt{\Big{[}\Delta_{+}(\Delta_{-}^{2}+\kappa^{2})+2\Delta_{-}\kappa^{2}\Big{]}^{2}+4\Delta_{+}^{2}\Delta_{-}^{2}\kappa^{2}}\right\\},$
(S37)

which is Eq. (3) in the main text.

###  S3.2 Second-order phase transition boundary

In order to explore the second-order phase transition boundary of the system,
we start with Eq. (S17) and derive the dynamic equation for the first-order
bosonic moments

$\displaystyle\dot{\textbf{L}}_{\rm np}=\Sigma_{\rm np}{\textbf{L}}_{\rm np},$
(S38)

where ${\textbf{L}}_{\rm np}=[{\langle a\rangle},{\langle
a^{{\dagger}}\rangle},{\langle b\rangle},{\langle b^{{\dagger}}\rangle}]^{\rm
T}$ and

$\displaystyle\\!\\!\\!\Sigma_{\rm
np}=\left(\begin{array}[]{cccc}\\!-i\Delta_{a}\\!-\\!\kappa&\\!-2iG&\\!-iJ_{s}&\\!0\\\
\\!2iG&i\Delta_{a}\\!-\\!\kappa&\\!0&\\!iJ_{s}\\\
\\!-iJ_{s}&\\!0&\\!-i\Delta_{b}-\kappa&\\!0\\\
\\!0&\\!iJ_{s}&\\!0&\\!i\Delta_{b}-\kappa\end{array}\right),$ (S43)

where $\Delta_{a},\Delta_{b}$ and $J_{s}$ have been defined in Eq. (S2.1).
Diagonalizing the matrix $\Sigma_{\rm np}$ and setting the real part of the
eigenvalues to zero, we obtain that for a certain $G$, the two critical atom-
field couplings $\lambda_{ac}$ and $\lambda_{bc}$ satisfy:

$\displaystyle\\!\\!\lambda_{bc}={\frac{\Big{\\{}4q_{1}\Delta_{-}-\Delta_{+}(\Delta_{+}\Delta_{-}\\!+\\!\kappa^{2})\lambda_{ac}^{2}\\!-\\!2\Big{[}4\kappa^{2}q_{1}(\Delta_{F}\Delta_{+}\lambda_{ac}^{2}-q_{1})\\!+\\!\Delta_{+}^{2}\Big{(}\Delta_{-}^{2}G^{2}+(G^{2}-\Delta_{F}^{2})\kappa^{2}\Big{)}\lambda_{ac}^{4}\Big{]}^{1/2}~{}\Big{\\}}^{1/2}}{q_{1}\Delta_{-}}},$
(S44)

where $q_{1}=\Delta_{+}^{2}-4G^{2}+\kappa^{2}$. Equation (S44) gives the
second-order phase transition boundary in the $\lambda_{a}$–$\lambda_{b}$
phase space.

In Fig. S1 we plot the order parameters $\alpha_{\rm re},\alpha_{\rm
im},\beta_{\rm re},\beta_{\rm im}$ for the forward [panels (a)-(d)] and
backward [panels (e)-(h)] pump cases. For each fixed pump direction, the order
parameters exhibit similar features and share the same boundary. The system
undergoes phase transitions from the normal phase (NP) to the superradiant
phase (SP) through two distinct paths: one is through the first-order phase
transition boundary (indicated by the gray dashed line) where the order
parameters undergo an abrupt and discontinuous change from zero to a finite
value (as given by Eq. (S34)); the other path is through the second-order
phase transition boundary (magenta dash-dotted curve), where the order
parameters vary continuously from zero to a finite value as $\lambda_{a}$ or
$\lambda_{b}$ increases [as given by Eq. (S44)]. The first- and second-order
phase transition boundaries meet at the tricritical point (see below). It is
worth noting that the phase diagram of the order parameters exhibits different
boundaries for different pump directions at a fixed pump strength. This
nonreciprocal nature of the phase transition highlights the dependence of the
system’s behavior on the pump direction.

Figure S1: Order parameters $\alpha_{\rm re},\alpha_{\rm im},\beta_{\rm
re},\beta_{\rm im}$ as functions of the atom-field couplings $\lambda_{a}$ and
$\lambda_{b}$ for the forward (a-d) and backward (e-h) pumps. The gray dotted
line and magenta dash-dotted curves denote the first-order [see Eq. (S34)] and
second-order [see Eq. (S44)] phase transition boundaries, respectively. We
consider $G/\kappa=1.5$ for both cases, and $\Delta_{F}/\Delta=0.5$ for the
forward pump, and $\Delta_{F}/\Delta=-0.5$ for the backward pump,
respectively. All other parameters are the same as in Fig. 1 of the main text.

In the case of tuned atom-field couplings where
$\lambda_{a}=\lambda_{b}=\lambda$, the eigenvalues of $\Sigma_{\rm np}$ are
given by

$\displaystyle\\!\\!E_{\rm np,\pm}^{(1)}$
$\displaystyle=-\kappa\pm\frac{\Big{\\{}16G^{2}+4\Delta_{F}^{2}p-\Delta^{2}(p^{2}+4)-\Big{[}(\Delta^{4}\lambda^{4}-32\Delta\Delta_{F}G^{2}-8\Delta^{2}\Delta_{F}^{2}p)(p-2)^{2}-16G^{2}\lambda^{4}\Delta_{+}\Delta_{-}+256G^{4}\Big{]}^{1/2}\Big{\\}}^{1/2}}{2\sqrt{2}},$
(S45) $\displaystyle\\!\\!E_{\rm np,\pm}^{(2)}$
$\displaystyle=-\kappa\pm\frac{\Big{\\{}16G^{2}+4\Delta_{F}^{2}p-\Delta^{2}(p^{2}+4)+\Big{[}(\Delta^{4}\lambda^{4}-32\Delta\Delta_{F}G^{2}-8\Delta^{2}\Delta_{F}^{2}p)(p-2)^{2}-16G^{2}\lambda^{4}\Delta_{+}\Delta_{-}+256G^{4}\Big{]}^{1/2}\Big{\\}}^{1/2}}{2\sqrt{2}},$
(S46)

with $p=\lambda^{2}-2$. Setting the real part of the eigenvalues equal to
zero, i.e., ${\rm Re}[E_{\rm np}]=0$, we obtain the critical pump strength for
the second-order phase transition

$\displaystyle G_{c}^{\rm
2nd}=\frac{\Big{\\{}\Delta^{4}(\lambda^{2}-2)^{2}+\big{(}2\kappa^{2}-\Delta_{F}^{2}(\lambda^{2}-2)\big{)}^{2}+\Delta^{2}\Big{[}\kappa^{2}((\lambda^{2}-2)^{2}+4)-2\Delta_{F}^{2}(\lambda^{2}-2)^{2}\Big{]}\Big{\\}}^{1/2}}{\Big{[}16\kappa^{2}+(\Delta-\Delta_{F})^{2}(\lambda^{2}-4)^{2}\Big{]}^{1/2}},$
(S47)

which is Eq. (4) of the main text. In Fig. S2(a), we illustrate $G_{c}^{\rm
1st}$ and $G_{c}^{\rm 2nd}$ as a function of the Sagnac shift $\Delta_{F}$.
The interplay of cavity rotation and directional pumping induces opposite
Sagnac shifts, consequently causing the critical points moving towards larger
(or smaller) values for the forward (or backward) pump.

The tricritical point is the intersection point where the first- and second-
order boundaries meet. By setting $G_{c}^{\rm 1st}=G_{c}^{\rm 2nd}$, we can
determine the atom-field coupling of the tricritical point,

$\displaystyle\\!\\!\lambda_{\rm
tric}\\!=\\!\Big{(}\frac{\Delta_{-}^{2}\\!+\\!\kappa^{2}}{\Delta_{-}^{3}(\Delta_{+}\Delta_{-}\\!+\\!\kappa^{2})}\Big{)}^{{1}/{2}}\Big{[}3\Delta_{-}^{2}\Delta_{+}\\!-\\!(\Delta_{F}\\!-\\!3\Delta)\kappa^{2}\\!-\\!\Big{(}\\!\Delta_{-}^{4}\Delta_{+}^{2}\\!+\\!2\Delta_{-}^{2}\Delta_{+}(5\Delta\\!+\\!\Delta_{F})\kappa^{2}\\!+\\!(\Delta_{F}\\!-\\!3\Delta)^{2}\kappa^{4}\Big{)}^{1/2}\Big{]}^{1/2}.$
(S48)

If the fixed coupling $\lambda>\lambda_{\rm tric}$ (or $\lambda<\lambda_{\rm
tric}$), surpassing the critical pump $G_{c}^{\rm 1st}$ (or $G_{c}^{\rm 2nd}$)
can trigger a first- (or second-) order phase transition. In our considered
parameter regime, where $\kappa/\Delta\sim 0.01$ and
$-\Delta<\Delta_{F}<\Delta$, the equation for the atom-field coupling at the
tricritical point can be simplified as:

$\displaystyle\lambda_{\rm
tric}\approx\sqrt{\frac{2(\Delta+\Delta_{F})[(\Delta-\Delta_{F})^{2}+\kappa^{2}]}{(\Delta-\Delta_{F})[(\Delta+\Delta_{F})(\Delta-\Delta_{F})+\kappa^{2}]}}\approx\sqrt{2}.$
(S49)

In Fig. S2, we plot the difference between $\lambda_{\rm tric}$ and
$\sqrt{2}$. It can be seen that this difference is on the order of $10^{-3}$
in our considered parameter regime, which is consistent with the analytical
result given in Eq. (S49). In other words, in our model, the atom-field
coupling at the tricritical point is insensitive to the Sagnac shift.

Figure S2: (a) The first-order and second-order critical pumps $G_{c}^{\rm
1st}$ and $G_{c}^{\rm 2nd}$ versus the Sagnac shift $\Delta_{F}$. (b) The
atom-field coupling strength $\lambda_{\rm tric}$ of the tricritical point
minus $\sqrt{2}$ as a function of the Sagnac shift $\Delta_{F}$. The
parameters used $\Delta_{F}/\Delta=\pm 0.5$ are labeled with the dotted
curves.

##  S4 Dynamic evolution of two-operator correlators

###  S4.1 Equations of motion in the normal phase

To be consistent with the operator representation used in the main text, we
perform a displacement on the cavity fields, i.e., $a\rightarrow
c+\alpha\sqrt{\eta_{+}}$, $b\rightarrow d+\beta\sqrt{\eta_{-}}$, where $c$ and
$d$ are cavity fluctuation operators. In the normal phase, $\alpha=\beta=0$.
Using the equation $\partial\langle o\rangle/\partial t={\rm
tr}\\{o\partial\rho/\partial t\\}$ for the expectation value $\langle
o\rangle={\rm tr}\\{o\rho\\}$ of any observable $o$ of interest, and utilizing
Eq. (S17), we can obtain a set of closed equations of the motion for the two-
operator fluctuation correlators. These equations take the following form:

$\displaystyle\frac{d{\textbf{v}}_{\rm np}}{dt}={W}_{\rm np}{\textbf{v}_{\rm
np}}+R_{\rm np},$ (S50)

where ${\textbf{v}}_{\rm np}=[{\langle c^{{\dagger}}c\rangle},{\langle
c^{{\dagger}2}\rangle},{\langle c^{2}\rangle},{\langle cd\rangle},{\langle
c^{{\dagger}}d^{{\dagger}}\rangle},{\langle cd^{{\dagger}}\rangle},{\langle
c^{{\dagger}}d\rangle},{\langle d^{{\dagger}}d\rangle},{\langle
d^{2}\rangle},{\langle d^{{\dagger}2}\rangle}]^{\rm T}$, $R_{\rm
np}=[0,2iG,-2iG,0,0,0,0,0,0,0]^{T}$, and

$\displaystyle W_{\rm np}=\left(\begin{array}[]{cccc}W_{\rm np}^{(1,1)}&W_{\rm
np}^{(1,2)}\\\ W_{\rm np}^{(2,1)}&W_{\rm np}^{(2,2)}\\\ \end{array}\right)$
(S53)

with

$\displaystyle\\!\\!\\!W_{\rm
np}^{(1,1)}=\left(\begin{array}[]{ccccc}\\!-2\kappa\\!&-\\!2iG&\\!2iG&\\!0&\\!0\\\
\\!4iG&2i\Delta_{a}\\!-\\!2\kappa&\\!0&\\!0&\\!2iJ_{s}\\\
\\!-4iG&\\!0&\\!-2i\Delta_{a}\\!-\\!2\kappa&-2iJ_{s}&\\!0\\\
\\!0&0&\\!-iJ_{s}&-i(\Delta_{a}\\!+\\!\Delta_{b})\\!-\\!2\kappa&\\!0\\\
0&iJ_{s}&0&0&i(\Delta_{a}\\!+\\!\Delta_{b})\\!-\\!2\kappa\\\
\end{array}\right),$ (S59) $\displaystyle\\!\\!\\!W_{\rm
np}^{(1,2)}=\left(\begin{array}[]{ccccc}iJ_{s}&-iJ_{s}&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\\\ 0&-2iG&0&-iJ_{s}&0\\\ 2iG&0&0&0&iJ_{s}\\\
\end{array}\right),W_{\rm
np}^{(2,1)}=\left(\begin{array}[]{ccccc}iJ_{s}&0&0&0&-2iG\\\
-iJ_{s}&0&0&2iG&0\\\ 0&0&0&0&0\\\ 0&0&0&-2iJ_{s}&0\\\ 0&0&0&0&2iJ_{s}\\\
\end{array}\right),$ (S70) $\displaystyle\\!\\!\\!W_{\rm
np}^{(2,2)}=\left(\begin{array}[]{ccccc}\\!-i(\Delta_{a}\\!-\\!\Delta_{b})\\!-\\!2\kappa\\!&\\!0&\\!-iJ_{s}&\\!0&\\!0\\\
\\!0&i(\Delta_{a}\\!-\\!\Delta_{b})\\!-\\!2\kappa&\\!iJ_{s}&\\!0&\\!0\\\
\\!-iJ_{s}&\\!iJ_{s}&\\!-2\kappa&0&\\!0\\\
\\!0&0&\\!0&-2i\Delta_{b}\\!-\\!2\kappa&\\!0\\\
0&0&0&0&2i\Delta_{b}\\!-\\!2\kappa\\\ \end{array}\right),$ (S76)

In the steady state, we can solve Equation (S50) by setting the left-hand side
to zero, yielding the following expression for the ten two-operator
correlators in the steady state:

$\textbf{v}_{\rm np}=-W_{\rm np}^{-1}R_{\rm np}.$

The analytical expressions of these correlators are complex and not presented
here, but their numerical results are shown in Figs. 3(a)-3(c) in the main
text.

###  S4.2 Equations of motion in the superradiant phase

To derive the dynamic equation for the two-operator fluctuation correlators in
the superradiant phase, we start with the equation of motion for the reduced
density matrix in the superradiant phase, given by

$\displaystyle\dot{{\rho}}_{\rm sp}=-i[H_{\rm sp},{\rho}_{\rm
sp}]+\kappa(2c{\rho}_{\rm sp}c^{{\dagger}}-c^{{\dagger}}c{\rho}_{\rm
sp}-{\rho}_{\rm sp}c^{{\dagger}}c)+\kappa(2d{\rho}_{\rm
sp}d^{{\dagger}}-d^{{\dagger}}d{\rho}_{\rm sp}-{\rho}_{\rm
sp}d^{{\dagger}}d).$ (S77)

Here, ${\rho}_{\rm
sp}=\langle\tilde{\downarrow}\lvert\tilde{\rho}\rvert\tilde{\downarrow}\rangle$
represents the reduced density matrix in the superradiant phase. Using the
equation $\partial\langle o\rangle/\partial t={\rm
tr}\\{o\partial\rho/\partial t\\}$ for the expectation value $\langle
o\rangle={\rm tr}\\{o\rho\\}$, we can express the time derivative of the two-
operator fluctuation correlators as follows:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle c^{{\dagger}}c\rangle}{dt}=$
$\displaystyle\,2i(G+\Lambda_{4}e^{2i\phi})\langle
c^{2}\rangle-2i(G+\Lambda_{4}e^{-2i\phi})\langle
c^{{\dagger}2}\rangle+i\Lambda_{3}(\langle cd^{{\dagger}}\rangle-\langle
c^{{\dagger}}d\rangle)+i\Lambda_{6}(e^{2i\phi}\langle
cb\rangle-e^{-2i\phi}\langle c^{{\dagger}}d^{{\dagger}}\rangle)-2\kappa\langle
c^{{\dagger}}c\rangle,$ (S78)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle c^{{\dagger}2}\rangle}{dt}=$
$\displaystyle\,2i(\Lambda_{1}+i\kappa)\langle
c^{{\dagger}2}\rangle+2i(G+\Lambda_{4}e^{2i\phi})(2\langle
c^{{\dagger}}c\rangle+1)+2i\Lambda_{3}\langle
c^{{\dagger}}d^{{\dagger}}\rangle+2i\Lambda_{6}e^{2i\phi}\langle
c^{{\dagger}}d\rangle,$ (S79)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle c^{2}\rangle}{dt}=$
$\displaystyle-2i(\Lambda_{1}-i\kappa)\langle
c^{2}\rangle-2i(G+\Lambda_{4}e^{-2i\phi})(2\langle
c^{{\dagger}}c\rangle+1)-2i\Lambda_{3}\langle
cd\rangle-2i\Lambda_{6}e^{-2i\phi}\langle cd^{{\dagger}}\rangle,$ (S80)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle cd\rangle}{dt}=$
$\displaystyle\\!-i(\Lambda_{1}\\!+\\!\Lambda_{2}\\!-\\!2i\kappa)\langle
cd\rangle\\!-\\!2i(G\\!+\\!\Lambda_{4}e^{-2i\phi})\langle
c^{{\dagger}}d\rangle\\!-\\!i\Lambda_{3}(\langle c^{2}\rangle\\!+\\!\langle
d^{2}\rangle)\rangle\\!-\\!2i\Lambda_{5}e^{-2i\phi}\langle
cd^{{\dagger}}\\!-\\!i\Lambda_{6}e^{-2i\phi}\langle
d^{{\dagger}}d\rangle\\!-\\!i\Lambda_{6}e^{-2i\phi}(\langle
c^{{\dagger}}c\rangle\\!+\\!1),$ (S81)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle
c^{{\dagger}}d^{{\dagger}}\rangle}{dt}=$
$\displaystyle\,i(\Lambda_{1}\\!+\\!\Lambda_{2}\\!+\\!2i\kappa)\langle
c^{{\dagger}}d^{{\dagger}}\rangle\\!+\\!2i(G\\!+\\!\Lambda_{4}e^{2i\phi})\langle
cd^{{\dagger}}\rangle\\!+\\!i\Lambda_{3}(\langle
c^{{\dagger}2}\rangle\\!+\\!\langle
d^{{\dagger}2}\rangle)\rangle\\!+\\!2i\Lambda_{5}e^{2i\phi}\langle
c^{{\dagger}}d\\!+\\!i\Lambda_{6}e^{2i\phi}(\langle
d^{{\dagger}}d\rangle\\!+\\!1)\\!+\\!i\Lambda_{6}e^{2i\phi}\langle
c^{{\dagger}}c\rangle,$ (S82)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle cd^{{\dagger}}\rangle}{dt}=$
$\displaystyle\\!-\\!i(\Lambda_{1}\\!-\\!\Lambda_{2}\\!-\\!2i\kappa)\langle
cd^{{\dagger}}\rangle\\!-\\!2i(G\\!+\\!\Lambda_{4}e^{-2i\phi})\langle
c^{{\dagger}}d^{{\dagger}}\rangle\\!+\\!i\Lambda_{3}(\langle
c^{{\dagger}}c\rangle\\!-\\!\langle
d^{{\dagger}}d\rangle)\\!+\\!2i\Lambda_{5}e^{2i\phi}\langle
cd\rangle\\!-\\!i\Lambda_{6}(e^{-2i\phi}\langle
d^{{\dagger}2}\rangle\\!-\\!e^{2i\phi}\langle c^{2}\rangle),$ (S83)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle c^{{\dagger}}d\rangle}{dt}=$
$\displaystyle\,i(\Lambda_{1}\\!-\\!\Lambda_{2}\\!+\\!2i\kappa)\langle
c^{{\dagger}}d\rangle\\!+\\!2i(G\\!+\\!\Lambda_{4}e^{2i\phi})\langle
cd\rangle\\!-\\!i\Lambda_{3}(\langle c^{{\dagger}}c\rangle\\!-\\!\langle
d^{{\dagger}}d\rangle)\\!-2i\Lambda_{5}e^{-2i\phi}\langle
c^{{\dagger}}d^{{\dagger}}\rangle+i\Lambda_{6}(e^{2i\phi}\langle
d^{2}\rangle-e^{-2i\phi}\langle c^{{\dagger}2}\rangle),$ (S84)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle d^{{\dagger}}d\rangle}{dt}=$
$\displaystyle-i\Lambda_{3}(\langle cd^{{\dagger}}\rangle-\langle
c^{{\dagger}}d\rangle)+i\Lambda_{6}(e^{2i\phi}\langle
cd\rangle-e^{-2i\phi}\langle
c^{{\dagger}}d^{{\dagger}}\rangle)+2i\Lambda_{5}(e^{2i\phi}\langle
d^{2}\rangle-e^{-2i\phi}\langle d^{{\dagger}2}\rangle)-2\kappa\langle
d^{{\dagger}}d\rangle,$ (S85)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle d^{2}\rangle}{dt}=$
$\displaystyle-2i(\Lambda_{2}+i\kappa)\langle
d^{2}\rangle-2i\Lambda_{3}\langle cd\rangle-2i\Lambda_{6}e^{-2i\phi}\langle
c^{{\dagger}}d\rangle-2i\Lambda_{5}e^{-2i\phi}(2\langle d^{{\dagger}}d+1),$
(S86) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{d\langle
d^{{\dagger}2}\rangle}{dt}=$ $\displaystyle\,2i(\Lambda_{2}+i\kappa)\langle
d^{{\dagger}2}\rangle+2i\Lambda_{3}\langle
c^{{\dagger}}d^{{\dagger}}\rangle+2i\Lambda_{6}e^{2i\phi}\langle
cd^{{\dagger}}\rangle+2i\Lambda_{5}e^{2i\phi}(2\langle
d^{{\dagger}}d\rangle+1).$ (S87)

We solve these ten coupled equations to obtain the cavity fluctuation $\langle
c^{{\dagger}}c\rangle$ or $\langle d^{{\dagger}}d\rangle$ in the steady state,
as shown in Figs. 3(a)-3(c) in the main text.

##  S5 Stability analysis

In this section, we discuss the stability of the system. Starting with the
Heisenberg equations (S2-S6), we perform a semiclassical analysis.
Specifically, we set $\langle\sigma_{x}\rangle=X$,
$\langle\sigma_{y}\rangle=Y$, $\langle\sigma_{z}\rangle=Z$, $\langle
a\rangle=\alpha\sqrt{\eta_{+}}$, $\langle b\rangle=\beta\sqrt{\eta_{-}}$.
Furthermore, we expand the order parameters as
$\alpha\rightarrow\alpha+\delta\alpha,\beta\rightarrow\beta+\delta\beta$ and
$X\rightarrow X+\delta X,Y\rightarrow Y+\delta Y,Z\rightarrow Z+\delta Z$,
where $\alpha,\beta,X,Y,Z$ are the mean-field steady-state solutions shown in
Sec. S1, and $\delta\alpha,\delta\beta,\delta X,\delta Y,\delta Z$ are quantum
fluctuations. These expansions are valid in the limits of
$\Delta_{q}/(\Delta\pm\Delta_{F})$ and $\Delta_{q}/G\rightarrow\infty$. The
equations of motion for the quantum fluctuations satisfy

$\dot{\textbf{u}}={\rm{M}}{\textbf{u}},$

with ${\textbf{u}}=[\delta\alpha_{\rm re},\delta\alpha_{\rm
im},\delta\beta_{\rm re},\delta\beta_{\rm im},\delta X,\delta Y]^{\rm T}$ and
the stability matrix $\rm{M}$ is given by

Figure S3: Stability of the system for the forward and backward pumps in the
normal phase (NP) and $Z_{+}$ branch of the superradiant phase (SP). The upper
panels show the stability of the forward pump case, while the lower panels
show the stability of the backward pump case. Panels (c,f) show the stability
of the combination of the NP and SP. In the blank regions, the system is
unstable. The parameter $\Delta_{F}/\Delta=0.5$ for the forward pump and
$\Delta_{F}/\Delta=-0.5$ for the backward pump. The values of $\Delta_{q}$ is
$10^{4}$, and the other parameters are the same as in Fig. 3 of the main text.

$\displaystyle\\!\\!\\!{\rm
M}=\left(\begin{array}[]{cccccc}\\!-\kappa&\\!\Delta_{+}\\!-\\!2G&\\!0&\\!J\sqrt{\frac{\Delta_{+}}{\Delta_{-}}}&\\!0&\\!-\frac{1}{4}\lambda_{a}\Delta_{+}\\\
\\!-(\Delta_{+}\\!+\\!2G)&\\!-\kappa&\\!-J\sqrt{\frac{\Delta_{+}}{\Delta_{-}}}&\\!0&\\!-\frac{1}{4}\lambda_{a}\Delta_{+}&\\!0\\\
\\!0&\\!J\sqrt{\frac{\Delta_{-}}{\Delta_{+}}}&\\!-\kappa&\\!\Delta_{-}&\\!0&\\!-\frac{1}{4}\lambda_{b}\Delta_{-}\\\
\\!-J\sqrt{\frac{\Delta_{-}}{\Delta_{+}}}&\\!0&\\!-\Delta_{-}&\\!-\kappa&\\!-\frac{1}{4}\lambda_{b}\Delta_{-}&\\!0\\\
\\!0&\\!-\lambda_{a}\Delta_{q}Z&\\!0&\\!-\lambda_{b}\Delta_{q}Z&\\!\frac{X}{Z}\Delta_{q}(\lambda_{a}\alpha_{\rm
im}\\!+\\!\lambda_{b}\beta_{\rm
im})&\\!\Delta_{q}[-1\\!+\\!\frac{Y}{Z}(\lambda_{a}\alpha_{\rm
im}\\!+\\!\lambda_{b}\beta_{\rm im})]\\\
\\!-\lambda_{a}\Delta_{q}Z&\\!0&\\!-\lambda_{b}\Delta_{q}Z&\\!0&\\!\Delta_{q}[1\\!+\\!\frac{X}{Z}(\lambda_{a}\alpha_{\rm
re}\\!+\\!\lambda_{b}\beta_{\rm
re})]&\\!\Delta_{q}\frac{Y}{Z}(\lambda_{a}\alpha_{\rm
re}\\!+\\!\lambda_{b}\beta_{\rm re})\end{array}\right),$ (S94)

where $\Delta_{\pm}=\Delta\pm\Delta_{F}$. During the stability analysis, we
separated the mean-field solutions and quantum fluctuations into real and
imaginary parts, namely, $\alpha=\alpha_{\rm re}+i\alpha_{\rm im}$,
$\beta=\beta_{\rm re}+i\beta_{\rm im}$ and $\delta\alpha=\delta\alpha_{\rm
re}+i\delta\alpha_{\rm im}$, $\delta\beta=\delta\beta_{\rm
re}+i\delta\beta_{\rm im}$. The stability of the system is determined by the
eigenvalues of the matrix $\rm{M}$. If all eigenvalues have negative real
parts, the system is stable and evolves into its steady state for
$t\rightarrow\infty$.

Figure S3 shows the stable regions in the normal and superradiant phases for
both forward and backward pump cases at $\Delta_{q}=10^{4}$. We find that for
the forward pump, there is an unstable region located in the vicinity of
$\lambda=2$ [see Fig. S3(c)]. However, for the backward pump case, this
unstable region disappears, and both the normal and $Z_{+}$ branch of the
superradiant phases are stable, as illustrated in Fig. S3(f). It is worth
noting that the $Z_{-}$ branch solution in the superradiant phase is unstable
(not shown here).

##  S6 The validity of the semiclassical approach

###  S6.1 Overview

In Sec. S1, we derive the mean-field solutions of cavity occupations by
applying a semiclassical approach. Here, we offer a comprehensive review of
the semiclassical approach, focusing on scenarios where the field interacts
with a single atom. The semiclassical approach has proven to be effective in
investigating various phase-transition-like behaviors within the Rabi model.
These include the ground-state quantum phase transition (QPT) Ashhab and Nori
(2010); Ashhab (2013); Hwang  _et al._ (2015); Liu  _et al._ (2017), the
excited-state QPT (ESQPT) indicated by the divergence of the semiclassical
density of states Puebla  _et al._ (2016), dissipative QPT Hwang  _et al._
(2018), and the breakdown of the photon blockade Carmichael (2015). It is
worth noting that, although the semiclassical approach may not precisely
capture the diverging quantum fluctuation of the cavity field in the limit of
infinite detuning ($\Omega/\omega_{0}\rightarrow\infty$) in the Rabi model, it
remains a reliable method for describing mean-field quantities such as the
photon population and the atomic population of the ground state.

###  S6.2 Analytical derivations

Here, we present an alternative yet intuitive derivation to illustrate that
applying the semiclassical approximation solely to the cavity field leads to
the semiclassical equations, aligning with the results presented in Sec. S1.
We initiate the derivation with the master equation (2) in the main text and
utilize the equation $\partial\langle O\rangle/\partial t={\rm
Tr}\\{O\partial\rho/\partial t\\}$, where the expectation value is denoted as
$\langle O\rangle={\rm Tr}\\{O\rho\\}$. Subsequently, we obtain the equations
of motion as follows:

$\displaystyle\dot{\langle a\rangle}$
$\displaystyle=-i(\Delta+\Delta_{F})\langle a\rangle-\kappa\langle
a\rangle-2iG\langle a^{\dagger}\rangle-iJ\langle
b\rangle-\frac{i}{2}g_{a}\langle\sigma_{x}\rangle-\frac{1}{2}g_{a}\langle\sigma_{y}\rangle,$
(S96) $\displaystyle\dot{\langle b\rangle}$
$\displaystyle=-i(\Delta-\Delta_{F})\langle b\rangle-\kappa\langle b\rangle-
iJ\langle
a\rangle-\frac{i}{2}g_{b}\langle\sigma_{x}\rangle-\frac{1}{2}g_{b}\langle\sigma_{y}\rangle,$
(S97) $\displaystyle\dot{\langle\sigma_{x}\rangle}$
$\displaystyle=-\Delta_{q}\langle\sigma_{y}\rangle+ig_{a}(\langle
a\sigma_{z}\rangle-\langle a^{\dagger}\sigma_{z}\rangle)+ig_{b}(\langle
b\sigma_{z}\rangle-\langle b^{\dagger}\sigma_{z}\rangle),$ (S98)
$\displaystyle\dot{\langle\sigma_{y}\rangle}$
$\displaystyle=\Delta_{q}\langle\sigma_{x}\rangle-g_{a}(\langle
a\sigma_{z}\rangle+\langle a^{\dagger}\sigma_{z}\rangle)-g_{b}(\langle
b\sigma_{z}\rangle+\langle b^{\dagger}\sigma_{z}\rangle),$ (S99)
$\displaystyle\dot{\langle\sigma_{z}\rangle}$ $\displaystyle=ig_{a}(\langle
a^{\dagger}\sigma_{x}\rangle-\langle a\sigma_{x}\rangle)+ig_{b}(\langle
b^{\dagger}\sigma_{x}\rangle-\langle b\sigma_{x}\rangle)+g_{a}(\langle
a\sigma_{y}\rangle+\langle a^{\dagger}\sigma_{y}\rangle)$
$\displaystyle\,\,\,\,+g_{b}(\langle b\sigma_{y}\rangle+\langle
b^{\dagger}\sigma_{y}\rangle).$ (S100)

The semiclassical (or mean-field) equations can be derived from the Eqs.
(S96-S100) by assuming the vanishing of second cumulants. For instance, this
assumption implies $\langle a\sigma_{z}\rangle=\langle
a\rangle\langle\sigma_{z}\rangle$. Specifically, we express the cavity field
operators as $a=\langle a\rangle+\delta a$ and $b=\langle b\rangle+\delta b$,
where the fluctuation $\delta a$ (or $\delta b$) is much smaller than the mean
values $\langle a\rangle$ (or $\langle b\rangle$). When inserting the operator
$a$ into terms involving the second cumulants, for example, $\langle
a\sigma_{z}\rangle=\langle(\langle a\rangle+\delta a)\sigma_{z}\rangle=\langle
a\rangle\langle\sigma_{z}\rangle+\langle\delta a\sigma_{z}\rangle$, neglecting
the quantum fluctuation $\langle\delta a\sigma_{z}\rangle$, then we obtain
$\langle a\sigma_{z}\rangle=\langle a\rangle\langle\sigma_{z}\rangle$. That
is, regarding the atomic part, we retain the terms
$\langle\sigma_{x,y,z}\rangle$ because they are finite constants and follow
the spin-conservation law
$\langle\sigma_{x}\rangle^{2}+\langle\sigma_{y}\rangle^{2}+\langle\sigma_{z}\rangle^{2}=1$.
This approach allows us to derive the same semiclassical equations as
discussed in Sec. S1.

This approach can be validated through the low-energy Hamiltonian. As
discussed in Sec. S2.B, in the large detuning limit where the atomic energy
scale significantly exceeds the field part, we can decouple the atomic
subspaces and concentrate on the lowest atomic subspace. Based on Eq. (S23),
we can calculate the mean value of the operator $\sigma_{z}$ in the lowest-
energy state (or ground state) as follows:

$\displaystyle\langle\tilde{\downarrow}|\sigma_{z}|\tilde{\downarrow}\rangle=-\frac{1}{\sqrt{1+(\alpha_{\rm
im}^{2}+\alpha_{\rm re}^{2})\lambda_{a}^{2}+2(\alpha_{\rm im}\beta_{\rm
im}+\alpha_{\rm re}\beta_{\rm re})\lambda_{a}\lambda_{b}+(\beta_{\rm
im}^{2}+\beta_{\rm re}^{2})\lambda_{b}^{2}}},$ (S101)

where $\alpha$ and $\beta$ represent the mean-field solutions of the cavity
occupations, and
$\lambda_{a,b}=2g_{a,b}/\sqrt{\Delta_{q}(\Delta\pm\Delta_{F})}$ are
dimensionless atom-field couplings of the Hamiltonian. When calculating
$\langle\tilde{\downarrow}|\sigma_{z}|\tilde{\downarrow}\rangle$, the atomic
part does not involve semiclassical approximation; only the cavity field is
subjected to such an approximation.

In Fig. S4, we present the mean value of $\langle\sigma_{z}\rangle$ obtained
from the mean-field approach (depicted by red circles) and
$\langle\tilde{\downarrow}|\sigma_{z}|\tilde{\downarrow}\rangle$ derived from
the full quantum description (illustrated by the black curve). Clearly, the
mean values obtained from these two different methods are consistent,
confirming the validity of applying semiclassical approximation solely to the
cavity fields.

Figure S4: The atomic expectation values as a function of the pump strength
$G$. The red circles represent the atomic expectation value in the steady
state $\langle\sigma_{z}\rangle$, and the black curve illustrates the atomic
expectation value in the lowest-energy state
$\langle\tilde{\downarrow}|\sigma_{z}|\tilde{\downarrow}\rangle$. The
parameters considered here are
$\Delta=2,\Delta_{F}/\Delta=0.5,\lambda_{a}=\lambda_{b}=1.7,\kappa/\Delta=0.05$.

Based on the semiclassical equations (S96-S100), we can derive the steady-
state mean amplitudes of the cavity fields $\langle a\rangle,\langle b\rangle$
and the atomic expectation values
$\langle\sigma_{x}\rangle,\langle\sigma_{y}\rangle$ and
$\langle\sigma_{z}\rangle$, see Sec. S1. Interestingly, we find that all of
these steady-state quantities exhibit a bifurcation when the system parameter,
specifically the pump strength (or atom-field coupling strength), is
increased. We refer to the point where bifurcation occurs as the critical
point for the second-order superradiant phase transition, denoted as
$G_{c}^{\rm 2nd}$. It is worth noting that the critical point obtained through
the semiclassical calculation aligns well with those derived from the full
quantum description of the low-energy Hamiltonian in the infinite-detuning
limit $\Delta_{q}/(\Delta\pm\Delta_{F})\rightarrow\infty$, as discussed in
Sections S2 and S3.B.

Figure S5: Numerical calculation of the position-squeezing quantifier
$s_{x}+1$ as a function of $G/G_{c}^{\rm 2nd}$ for $\Delta_{q}=50$ (a), 100
(b), 500 (c). Here, the analytical value of critical pump strength provided by
the semiclassical approach, i.e., $G_{c}^{\rm 2nd}$, is denoted by the black
dash–dotted line. Other parameters are same as Fig. 2(a,b) in the main text.

###  S6.3 Squeezing parameter

To further illustrate the alignment of critical points obtained from the
semiclassical approach and the full quantum approach, we begin with the master
equation (2) in the main text and proceed to numerically calculate the
position-squeezing parameter Ashhab and Nori (2010); Ashhab (2013),

$s_{x}=2\langle(X-\langle X\rangle)^{2}\rangle-1,$

where the quadrature variable is defined as $X=(a+a^{\dagger})/\sqrt{2}$. In
Figure S5, we plot the squeezing quantifier $s_{x}+1$ as a function of the
pump strength $G$ (measured in comparison to the second-order critical pump
strength $G_{c}^{\rm 2nd}$, which is the theoretical value obtained from the
mean-field approach). As we increase the atomic detuning $\Delta_{q}$ (i.e.,
getting closer to the infinite detuning limit $\eta\rightarrow\infty$), the
point where the squeezing quantifier reaches its minimum gradually approaches
the theoretical critical pump $G_{c}^{\rm 2nd}$. These numerical results
indicate that by increasing the atomic detuning $\Delta_{q}$, the critical
pump converges towards the theoretical value provided by the mean-field
approach Ashhab and Nori (2010); Ashhab (2013).

##  S7 Discussions on the dual-coupling JC model

###  S7.1 Comparison between the dual-coupling JC model and the Rabi model

Generally, the occurrence of the superradiant phase transition in the standard
Rabi model, typically hinges on two critical conditions: (i) attaining an
exceedingly large atomic frequency, such that
$\omega_{q}/\omega\rightarrow\infty$, and (ii) achieving ultra-strong coupling
between the atom and the field. Our model successfully overcomes the two
critical challenging conditions present in the standard Rabi model.
Specifically, for condition (i), in our model, the large atomic frequency was
transformed into a large atomic detuning (i.e.,
$\Delta_{q}/\Delta\rightarrow\infty$, where
$\Delta_{q}=\omega_{q}-\omega_{p}/2$ and $\Delta=\omega_{0}-\omega_{p}/2$). By
tuning the frequency of the pump field, one can readily achieve this large
atomic detuning, which serves the same purpose as
$\omega_{q}/\omega\rightarrow\infty$ in the standard Rabi model. Thus,
condition (i) can be easily met in our model. Simultaneously, the critical
atom-field coupling strength required for the onset of superradiant phase
transitions is determined by the detunings $\Delta_{q}$ and $\Delta$, rather
than the atomic frequency $\omega_{q}$ and resonator frequency $\omega_{0}$.
Consequently, our proposed scheme can alleviate constraint (ii) from ultra-
strong coupling to strong coupling.

###  S7.2 Comparison between all-optical controls and its magnetic or
electronic counterpart

In our model, the control over superradiant phase transitions and
multicriticality is achieved through all-optical manipulation using external
fields. This all-optical control presents several advantages in comparison to
its magnetic and electronic counterparts. Firstly, in terms of integration,
traditional methods for achieving optical nonreciprocity often rely on the
magneto-optical Faraday effect Wang  _et al._ (2009); Khanikaev  _et al._
(2010). These devices tend to be bulky and necessitate large magnetic fields,
making them inconvenient for integration. Issues such as crosstalk induced by
the magnetic field and lattice mismatches between magneto-optic materials and
silicon further complicate integration Dai  _et al._ (2012). In contrast, our
all-optical method for breaking system reciprocity demonstrates high
compatibility and ease of integration into photonic systems. This
characteristic not only enhances the potential for diverse applications in
quantum communication but also provides essential building blocks for a
quantum network Kurizki  _et al._ (2015). Secondly, all-optical systems are
more compact than their magnetic and electronic counterparts. In our model,
the microcavity is at the millimeter scale, and there have been reports of
micrometer-scale optical cavities Ward and Benson (2011). This characteristic
facilitates the development of on-chip nonreciprocal devices. Thirdly, all-
optical system can be easily reconfigured without need of the complex
electronic components. This characteristic makes opportunity to easily adjust
the external pump strength for manipulating superradiant phase transitions and
multicriticality. Fourthly, compared to the electronic counterparts, all-
optical system, in general, can operate with lower power consumption Jinno and
Matsumoto (1990).

On the other hand, these advantages may stimulate further theoretical and
experimental explorations, potentially advancing on-chip nonreciprocal device
development. Our research reveals nonreciprocal superradiant phase transitions
and a rich phase diagram featured by controllable multicritical points. These
findings not only hold fundamental research significance but also provide
quantum resources for quantum metrology Hotter  _et al._ (2024). Therefore,
our work may inspire the development of integrated high-precision quantum
sensing. Moreover, extending our model to $N$\- particle case may inspire many
applications in optical field. Recently, steady-state superradiance has been
demonstrated in a bad cavity regime, yielding lasing with a linewidth in the
millihertz range Bohnet  _et al._ (2012). From this perspective, the
combination of superradiance with our all-optical system may open up new
avenues for designing on-chip unidirectional laser. Additionally, our system
presents opportunities to explore and manipulate a broader range of physical
phenomena, including superradiant cooling Xu  _et al._ (2016) and atomic
synchronization Xu  _et al._ (2014). This broadened scope enhances the
versatility of our platform, offering unique opportunities for advancing our
understanding and practical utilization of these intriguing phenomena.

##  S8 Discussion on the assumption of system parameters

###  S8.1 The case of nonzero cavity hopping rate $J\neq 0$

Figure S6: (a-c) Phase diagram of the order parameter $\alpha_{\rm re}$ for
different cavity hopping rate $J$. (d-e) Cavity fluctuation $\langle
c^{{\dagger}}c\rangle$ as functions of $\lambda$ and $G$. In all plots, we
consider $\Delta_{F}/\Delta=0.5$.

In this section, we investigate how the cavity hopping rate $J$ affects the
superradiant transitions in the dual-coupling JC model. While the solutions
for $J\neq 0$ have complicated mathematical expressions, we present here only
the numerical results. In Figs. S6(a)- S6(c) we plot the phase diagram of
$\alpha_{\rm re}$ for different $J$. We observe that the non-zero value of $J$
induces a continuous increase of $\alpha_{\rm re}$ in the bulk of the
superradiant phase. Comparing with Fig. S1, we find that $J$ causes only a
small correction to the boundaries of the first- and second-order phase
transitions. Moreover, from Figs. S6(d)- S6(f) we observe that the cone-shaped
critical curve on the left shifts towards larger values of $\lambda$, which
causes the critical point of the second-order transition to shift toward a
larger pump strength $G$. Overall, we conclude that $J\neq 0$ does not alter
the main physics discussed in our model.

###  S8.2 Numerical simulations on steady-state mean photon number

In Fig. S7, we conducted numerical simulations to analyze the steady-state
mean photon number $\langle a^{\dagger}a\rangle$ as a function of the pump
strength $G$. These simulations are based on the full Hamiltonian (1) and the
master equation (2), carried through the Qutip. The chosen system parameters
for the simulations are as follows: cavity detuning $\Delta=2$, cavity decay
rate $\kappa/\Delta=0.05$, atomic decay rate $\gamma=\kappa$, cavity hopping
rate $J=\kappa$, and thermal photon number $\bar{n}=7.4\times 10^{-3}$. It is
observed in Fig. S7 that for larger values of $\Delta_{q}$, the growth in
$\langle a^{\dagger}a\rangle$ occurs at a faster rate. As $\Delta_{q}/\Delta$
increases, the tendency of $\langle a^{\dagger}a\rangle$ gradually approaches
the case of infinite detuning limit, i.e.,
$\Delta_{q}/\Delta\rightarrow\infty$. In the infinite detuning limit, $\langle
a^{\dagger}a\rangle$ diverges at the critical point. Note that due to the
limited capabilities of computer, here, we have simulated only the case where
$\Delta_{q}/\Delta$ is maximally set to 50, with the cavity modes $a$ and $b$
truncated to 18. Even though, within reasonable experimental parameters, as we
increase the atomic detuning value $\Delta_{q}/\Delta$, our numerical
simulations, conducted without any assumptions and approximations,
consistently converge towards the analytical results obtained for
$\Delta_{q}/\Delta\rightarrow\infty$. This observation implies that our
assumptions for the sake of analytical simplification are justified.

Figure S7: Numerical simulation of the mean photon number $\langle
a^{{\dagger}}a\rangle$ as a function of the pump strength $G$ for finite
detuning cases: $\Delta_{q}/\Delta=25$ (depicted by the blue dotted curve) and
$\Delta_{q}/\Delta=50$ (illustrated by the red dash-dotted curve). The
analytical result obtained for the infinite detuning case,
$\Delta_{q}/\Delta\rightarrow\infty$, is shown by the black-solid curve. Here,
we considered cavity decay $\kappa=0.1$, atomic decay $\gamma=0.1$, cavity
hopping rate $J=0.1$, and thermal photon number $\bar{n}=7.4\times 10^{-3}$.
Due to computational limitations, simulations were conducted only for the case
where $\Delta_{q}/\Delta$ is maximally set to 50, with both cavity modes $a$
and $b$ truncated to 18. Additional parameters include $\Delta=2$,
$\Delta_{F}=0$, $\lambda_{a}=\lambda_{b}=1.4$.

##  S9 Possible experimental implementations

To achieve nonreciprocal superradiant phase transitions and multicriticality,
the possible experimental implementation relies on three techniques: first,
reaching the strong-coupling regime in the interaction between the atom and
cavity fields; second, unidirectionally squeezing one of the cavity modes;
third, the capability to rotate the resonator. Recently, strong-coupling
cavity QED has been realized using WGM microcavities, involving trapped atoms
Aoki  _et al._ (2006); Dayan  _et al._ (2008); Aoki  _et al._ (2009); Alton
_et al._ (2011); Junge  _et al._ (2013); Scheucher  _et al._ (2016); Will  _et
al._ (2021), quantum dots Kiraz  _et al._ (2001); Peter  _et al._ (2005);
Srinivasan and Painter (2007a, b) and nitrogen-vacancy (NV) centers Park  _et
al._ (2006); Barbour  _et al._ (2010). Combining these achievements with
recent experiments on rotating resonator Maayani  _et al._ (2018) and optical
squeezing Fürst  _et al._ (2011); Lu  _et al._ (2019, 2020), we discuss the
feasibility of implementing our model with two potential setups: (i) cold
cesium atoms falling onto the surface of a WGM microdisk cavity Aoki  _et al._
(2006); Dayan  _et al._ (2008); Aoki  _et al._ (2009); Alton  _et al._ (2011)
and (ii) a single trapped 85Rb atom interacting with a WGM microresonator
Junge  _et al._ (2013); Scheucher  _et al._ (2016); Will  _et al._ (2021).
Next, we provide detailed analysis of these experimental realization.

###  S9.1 Strong coupling between an atom and microresonator

#### (i) cold cesium atoms falling onto the surface of a WGM microdisk

A possible experimental implementation platform could involve single cesium
atoms radiatively coupled to a high-quality toroidal microresonator and in
close proximity to the resonator’s dielectric surface Aoki  _et al._ (2006);
Dayan  _et al._ (2008); Aoki  _et al._ (2009); Alton  _et al._ (2011).
Specifically, a cloud of cold caesium atoms is located $\sim 800\,\mu$m above
the surface of the resonator. Several caesium atoms are released from an
optical dipole-force trap and fall within the evanescent field of cavity mode,
establishing strong coupling to the resonator’s field. Due to the aerodynamic
process caused by the cavity rotation, the falling atom hovers a short
distance above the rapidly rotating resonator instead of crashing into its
surface [see Sec. S7. B]. This is distinct from the scenario involving a
static resonator, as described by Refs. Aoki  _et al._ (2006); Alton  _et al._
(2011). A real-time detection scheme, relying on strong radiative interactions
between individual atoms and the evanescent cavity field, can be employed to
select atomic trajectories passing within 300 nm from the resonator’s surface.
At this scale, the radiative atom-field interactions are characterized by
strong coupling, evident in the observed vacuum-Rabi splitting. This
experimental configuration can be theoretically modeled using the dual-
coupling Jaynes-Cummings (JC) Hamiltonian, as demonstrated in Refs. Aoki  _et
al._ (2006); Dayan  _et al._ (2008); Alton  _et al._ (2011). In these schemes,
the average atom-photon coupling has been achieved at approximately 40 MHz,
surpassing the dissipative rates of both the atom and the cavity, thereby
reaching the strong-coupling regime.

#### (ii) a single trapped atom interacting with a WGM microresonator

Another possible experimental implementation platform could be trapping a
single 85Rb atom at a small distance from the surface of the WGM bottle
microresonator Junge  _et al._ (2013); Scheucher  _et al._ (2016); Will  _et
al._ (2021). The stable and controlled interaction between a single atom and
the resonator in the strong-coupling regime ($\sim 10$ MHz) is demonstrated in
Will  _et al._ (2021). Specifically, a single 85Rb atom is trapped at a
distance of about 200 nm from the resonator surface. This is achieved through
a deep standing-wave optical dipole trap created by retroreflecting a focused
trapping light field from the resonator surface. Additionally, a second,
detuned compensation light field is employed to counteract the position-
dependent detuning of the atomic resonance from the resonator mode. Resonant
detection light is sent through the coupling fiber to detect the presence of a
single atom in the resonator mode in real-time, and the transmitted power is
monitored using a single-photon counting module. The transmission signal is
detected using a field-programmable gate array-based system. Upon detecting an
atom, the detection light is switched off, and the dipole trap is switched on,
with the overall delay between detection and trapping being approximately 250
ns. This duration is significantly shorter than the average transit time of an
atom through the evanescent field of the resonator mode, enabling us to
capture a detected atom if it is located inside the trapping volume.

###  S9.2 Directionally squeezing of the cavity mode

With the advancement of nanofabrication techniques, a range of crystalline
materials has been employed in the fabrication of WGM microresonators Ilchenko
_et al._ (2004); Fürst  _et al._ (2011); Beckmann  _et al._ (2011); Förtsch
_et al._ (2013); Guo  _et al._ (2016); Lu  _et al._ (2019, 2020). A notable
advantage of crystalline cavities is their ability to support nonlinear
optical processes, including optical parametric amplification (OPA). In a
parametric amplifier, a pump beam interacting with a $\chi^{(2)}$ nonlinearity
generates signal and idler beams. This OPA process is considered a crucial
source of squeezed states of the radiative field.

Inspired by recent experiments, here we consider a crystalline WGM
microresonator with high second-order nonlinearity. To ensure phase matching
for parametric down-conversion along the entire circumference of the
resonator, an uniaxial crystal with the optical axis along the symmetry axis
of the cavity should be used. The microdisk can be fabricated from a 5% MgO-
doped z-cut lithium niobate wafer, as demonstrated in Ref. Fürst  _et al._
(2011), or periodically poled lithium niobate, as demonstrated in Refs. Lu
_et al._ (2019, 2020). The pump field interacts with the $\chi^{(2)}$
nonlinearity through a tapered fiber, generating a squeezed cavity mode. This
is an optical parametric amplification process. The forward external light
unidirectionally interacts with the clockwise $\chi^{(2)}$-nonlinearity,
thereby directionally squeezing the clockwise cavity mode Tang  _et al._
(2022). In other words, the forward propagating mode through the waveguide
selectively squeezes the copropagating mode in the resonator. Due to the
rotation of the system, the light circulating in the resonator experiences a
Sagnac-Fizeau shift, denoted by $\Delta_{F}$.

###  S9.3 Rotating resonator

The rotating scheme can be implemented by mounting the WGM resonator on a
turbine, causing the resonator to rotate with a certain angular velocity. This
setup has been demonstrated in a recent experiment by Maayani et al. Maayani
_et al._ (2018). Positioning the rotating microresonator near a single-mode
telecommunications fiber allows light to be evanescently coupled into or out
of the resonator through the tapered region. It is worth noting that in this
rotating scheme, the aerodynamic process plays a crucial role: a rapidly
rotating resonator can drag air into the region between the taper and the
microdisk, forming a boundary layer of air. Due to the air pressure on the
surface of the taper facing the resonator, the taper hovers a short distance
above the rapidly spinning resonator, avoiding direct contact or adherence to
the resonator. This distinguishes it from the situation with a stationary
resonator. If any disturbance causes the taper to ascend beyond its stable
equilibrium height, it naturally returns to its original position, a
phenomenon referred to as ‘self-adjustment’.

###  S9.4 Discussion on the effect of unexpected noises and perturbations

Based on the potential experimental implementations discussed above, we
discuss how possible unexpected noises or perturbations might affect the
performance of our system. The unexpected noises and perturbations may be
attributed to the thermal noise of the cavity and the perturbations caused by
the rotation scheme.

According to the experimentally feasible parameters, the cavity frequency is
$\omega=1.93\times 10^{14}$ Hz. At room temperature $T=300$ K, the thermal
cavity number is $\bar{n}=1/[\exp(\hbar\omega/k_{B}T)-1]\approx 7.4\times
10^{-3}$, which is significantly less than 1 and has been omitted in our
analytical considerations. However, in plotting Fig. S7, we have considered
non-zero thermal cavity number ($\bar{n}=7.4\times 10^{-3}$).

On the other hand, the rapidly rotating cavity drags a boundary layer of air
around the resonator, which may induce the vibrations of the atom. This could
alter the overlap between the atom and the mode volume of the resonator,
modifying the atom-resonator coupling strength. However, as discussed in Sec.
S9.A, the coupling strength between the WGM resonator and an atom in current
experiments is sufficient to achieve nonreciprocal phase transitions in our
model. Moreover, in our model, we achieve phase transitions by effectively
enhancing the pump strength of the external field, and the Sagnac shift
remains robust against modifications in coupling. Therefore, it is safe to say
that this perturbation would not change the nonreciprocal behaviors of our
model.

##  S10 Extending the model to the case of $N$ particles ($N\gg 1$)

Our approach can be extended to scenarios involving an infinite number of
atoms ($N\rightarrow\infty$), particularly within the context of the dual-
coupling Tavis-Cummings (TC) model. Here, we give a brief discussion in this
extension.

We consider $N$ two-level atoms interacting with two counter-propagating modes
of WGM resonator. The resonator is made of materials with second-order
nonlinearity. A classical field with frequency $\omega_{p}$ input from the
forward (or backward) can directionally generate squeezing cavity modes $a$
(or $b$) through an optical parametric amplification process. The system
Hamiltonian can be described by the dual-coupling TC model (in the forward
pump):

$\displaystyle
H=\omega_{0}a^{{\dagger}}a+\omega_{0}b^{{\dagger}}b+{\omega_{q}}J_{z}+\frac{g_{a}}{\sqrt{N}}(aJ_{+}+a^{{\dagger}}J_{-})+\frac{g_{b}}{\sqrt{N}}(bJ_{+}+b^{{\dagger}}J_{-})+G(a^{\dagger
2}e^{-i\omega_{p}t}+a^{2}e^{i\omega_{p}t}).$ (S102)

Here $J_{z}=(1/2)\sum_{i=1}^{N}\sigma_{z}$ and
$J_{\pm}=\sum_{i=1}^{N}\sigma_{\pm}$ represent collective angular-momentum
operators. There are $N$ atoms simultaneously coupled to cavity modes $a$ and
$b$ with collective coupling strengths $g_{a}$ and $g_{b}$, respectively. The
cavity field is unidirectionally pumped by an external field with the pump
strength of $G$. Additionally, we consider the microresonator rotates
counterclockwise with an angular velocity $\Omega$, causing the two cavity
modes to experience Sagnac-Fizeau shifts with respect to their static
resonance frequency $\omega_{0}$, i.e.,
$\omega_{0}\rightarrow\omega_{0}\pm\Delta_{F}$. In the frame rotating at
$\omega_{p}/2$, Hamiltonian (S102) is transformed into

$\displaystyle H$
$\displaystyle=(\Delta+\Delta_{F})a^{{\dagger}}a+(\Delta-\Delta_{F})b^{{\dagger}}b+\Delta_{q}J_{z}+\frac{g_{a}}{\sqrt{N}}(aJ_{+}+a^{{\dagger}}J_{-})$
$\displaystyle+\frac{g_{b}}{\sqrt{N}}(bJ_{+}+b^{{\dagger}}J_{-})+G(a^{\dagger
2}+a^{2}),$ (S103)

where the detunings are defined as $\Delta=\omega_{0}-\omega_{p}/2$ and
$\Delta_{q}=\omega_{q}-\omega_{p}/2$. The form of Hamiltonian (S10) closely
resembles Hamiltonian (1) presented in the main text. Utilizing Hamiltonian
(S10), we can derive mean field equations for $\langle a\rangle,\langle
b\rangle,\langle J_{x,y,z}\rangle$ and subsequently calculate the critical
points for both first-order and second-order superradiant phase transitions.
This process aligns with the method employed in the manuscript.

The dual-coupling TC model maintains the advantages of the dual-coupling JC
model. Firstly, from Hamiltonian (S10), we infer that the critical atom-field
coupling strength necessary for the occurrence of superradiant phase
transitions corresponds to the detunings $\Delta_{q}$ and $\Delta$ rather than
the atomic frequency $\omega_{q}$ and resonator frequency $\omega_{0}$. Thus,
superradiant phase transitions in dual-coupling TC does not require ultra-
strong atom-field coupling. Secondly, the dual-TC model, the condition of an
extremely large detuning is transformed into the limit of an extremely large
number of atoms, i.e., in the thermodynamic limit as $N\rightarrow\infty$.
Thirdly, the control of superradiant phase transitions in the dual-TC model
can also be achieved by adjusting the pump strength.