# Controllable Bistability and Squeezing of Confined Polariton Dark Solitons

Gang Wang1, Kexin Wu1, Yang Liu1, Weibin Li2, and Yan Xue1,∗ 1College of
Physics, Jilin University, Changchun 130012, P. R. China  2School of Physics
and Astronomy, and Centre for the Mathematics and Theoretical Physics of
Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7
2RD, United Kingdom  email: ∗<EMAIL_ADDRESS>
###### Abstract

The generation of squeezed light in semiconductor materials opens
opportunities for building on-chip devices that are operated at the quantum
level. Here we study theoretically a squeezed light source of polariton dark
solitons confined in a geometric potential well of semiconductor microcavities
in the strong coupling regime. We show that polariton dark solitons of odd and
even parities can be created by tuning the potential depth. When driving the
potential depth linearly, a bistability of solitons with the two different
parities can be induced. Strong intensity squeezing is obtained near the
turning point of the bistability due to the large nonlinear interaction, which
can be controlled by Feshbach resonance. The phase diagram of the bistability
and squeezing of the dark solitons is obtained through large scale numerical
calculations. Our study contributes to the current efforts in realizing
topological excitations and squeezed light sources with solid-state devices.

Introduction.—The peculiar light-matter composition of polaritons Sanvitto ;
Kavokin  allows the formation of Bose-Einstein condensation (BEC)
Kasprzak_2006 ; Amo_2009 ; Xiong_2020 ; Schneider_2020  at elevated
temperatures. The two-body interaction produced by the matter component opens
a pathway to explore a broad variety of nonlinear phenomena in the solid-
system setting 2019natphon_Lagoudakis ; 2009prl_Shelykh ; 2015prl_Flach ,
where the nonlinearity is tunable either by modulating the pump light power
Carusotto_2013 ; Valle_2020  or through the Feshbach resonance technology
2019prl_Feshbach ; 2014nphy_Deveaud . In particular, topological excitation of
dark solitons attracts great interest rooted from the fundamental importance
and all-optical signal processing applications Blair_1999 . Due to the giant
nonlinearity available in polariton condensates dark solitons can form at the
sub-millimeter length scales Amo_2011  and on picoseconds time scales
2017prl_Ma . Inspired by the high controllability and on-chip integration of
the semiconductor microcavity, enormous efforts have been spent on the study
of nonlinear phenomena in the semi-classical regime, including dark
2014prl_Yan ; Walker_2017  and half-dark Hivet_2012 ; 2015_spinor-half-
vortex_sanvitto  solitons, dark soliton trains 2014prl_Flayac  and their
bistability 2016Bloch_solitonbistability , Cherenkov radiation of solitons
2017Nc_Cherenkov_Skryabin , dark-soliton molecules Anne2020 ; 2020prr_Bramati
, etc..

Figure 1: (Color online) (a) sketch of the quasi-1D semiconductor microcavity.
The metallic mesa in the central area creates the centrosymmetric finite
square potential well with tunable amplitude $V_{0}$, which can be tuned by
applying an electric field. A homogeneous cw light pumps the semiconductor
microcavity to generate polariton BEC. (b) Quantized energy states in the
potential well with potential depth $V_{0}$.

On the other hand, the Kerr nonlinearity has been used to generate intensity
squeezing that can surpass the quantum noise limit Pan_2020 ; 2020prx_Buisson
. As an effective Kerr medium, the refractive index of the polariton
condensate can be controlled by the light intensity 2010prb_Bramati ;
2020prb_Andreev ; 2000prb_Yamamoto , holding the promise to realize a squeezed
light source. As such, it has been shown recently the intensity squeezing of
polariton condensate at the turning point of the bistability polariton
squeezing_2014 , the periodic squeezing in a Josephson junction polariton
squeezing_2017 , and polariton blockade 2006prb_blockade_Carusotto ;
2019natmat_blockade_volz ; polariton blockade_2019  in a fiber cavity.
However, the squeezing of polariton dark solitons is mainly limited by two
obstacles, i.e., the limited stability 2014Kivshar_solitonstability  and the
excess noise characterized by a continuum of microcavity polariton modes. The
latter is prone to couple with lattice phonons and to reduce the quantum
effect, which could be surpassed via the geometric confinement (in the strong
localization limit) Bamba_2010 . The challenge is to realize polariton dark
soliton squeezing by overcoming the two limitations simultaneously.

In this work, we study the intensity squeezing of polariton dark solitons near
the turning point of bistability, which are stabilized at quantized energy
states of the geometric confining potential. Our setting is a semiconductor
microcavity pumped nonresonantly with a homogeneous cw light beam and
patterned with a central potential [see Fig. 1(a)], leading to localized,
quantized energy states. The interference of the discrete states can form
polariton dark solitons of even and odd parities, which are stabilized by the
strong confinement of the potential well. We show that the two types of dark
solitons exhibit bistable when adiabatically scanning the potential depth up
and down at a constant rate. The phase diagram as a function of the nonlinear
interaction is obtained. With the help of Feshbach resonance 2019prl_Feshbach
; 2014nphy_Deveaud , the squeezing of polariton dark solitons is achieved near
the turning point of the bistability curve. Our study might pave a new route
to develop low-power squeezed light sources in solid-state devices with the
controllable microcavity polaritons.

Model.— In our setting, a metallic mesa is embedded in the central area of a
quasi-1D planar semiconductor microcavity [see Fig. 1(a)] to fabricate a
centrosymmetric finite potential well,

$V(x)=\begin{cases}\ V_{0},~{}|x|\leq 1\mu m\\\ \ ~{}0,~{}~{}|x|>1\mu
m\end{cases}$ (1)

where the depth $V_{0}$ is tunable by modulating the electric field applied on
the mesa 2013hofling_electry ; 2020Masumoto . States in the potential well are
quantized, while states out of the potential remain continuous. The deeper the
depth $V_{0}$, the more the quantized states are, as depicted in Fig. 1(b). A
homogeneous cw optical field excites non-resonantly polariton particles to
form two domains of BEC in real space, separated by the potential well.

Dynamics of the mean value $\psi$ of the polariton field $\hat{\Psi}$ is
governed by an open Gross-Pitaevskii (GP) equation coupled to the rate
equation for the exciton reservoir density $n_{\rm R}$ Yamamoto_2012 :

$\displaystyle i\hbar\partial\psi=$
$\displaystyle[\frac{\hbar^{2}}{2m}(iD_{0}n_{\rm R}-1)\nabla^{2}+V(x)+g_{\rm
R}^{\rm 1D}n_{\rm R}+g_{\rm C}^{\rm 1D}|\psi|^{2}$
$\displaystyle+\frac{i\hbar}{2}(R^{\rm 1D}n_{\rm R}-\gamma_{\rm
C})]\psi\partial t+i\hbar dW,$ $\displaystyle\frac{\partial n_{\rm
R}}{\partial t}=$ $\displaystyle P_{0}-\gamma_{\rm R}n_{\rm R}-R^{\rm
1D}n_{\rm R}|\psi|^{2},$ (2)

where $m=5\times 10^{-5}m_{e}$ (${m_{e}}$ to be electron mass) is the
effective mass of polaritons. $D_{0}=10^{-3}\mu$m and $R^{\rm 1D}$ is the
energy relaxation constant, and stimulated scattering factor. $\gamma_{\rm
C}=(50$ps$)^{-1}$ and $\gamma_{\rm R}=2\gamma_{\rm C}$ are the loss rate of
the polariton and exciton reservoirs due to the finite lifetime, respectively.
The polaritons are subject to an effective nonlinear Kerr effect originated
from the repulsive polariton-polariton interaction with coefficient $gc$,
which is controlled by Feshbach resonance 2019prl_Feshbach ; 2014nphy_Deveaud
. Coefficient $g_{R}^{\rm 1D}=2gc$ gives the effective repulsive Coulomb
interaction between polaritons and exciton reservoir. $P_{0}$ is a cw optical
field tuned well above the polariton resonance. $dW$ is a complex stochastic
term that describes the quantum fluctuation. In the truncated Wigner
approximation Michiel_2009 , the correlations of $dW$ are given by

$\displaystyle\langle dW(\mathbf{r},t)dW(\mathbf{r}^{\prime},t)\rangle$
$\displaystyle=0,\ \ \ \langle
dW(\mathbf{r},t)dW(\mathbf{r},t^{\prime})\rangle=0,$ $\displaystyle\langle
dW(\mathbf{r},t)dW^{*}(\mathbf{r}^{\prime},t^{\prime})\rangle$
$\displaystyle=\frac{dt}{2dxdy}(Rn_{R}+\gamma_{c})\delta_{\mathbf{r},\mathbf{r}^{\prime}}\delta_{t,t^{\prime}},$

with $\delta_{\mathbf{r},\mathbf{r}^{\prime}}$ and $\delta_{t,t^{\prime}}$ to
be Kronecker delta function.

Polariton dark solitons.— The quasi-1D polariton condensate is generated when
the pump field is above the threshold $P_{0}>P_{th}=\gamma_{\rm C}\gamma_{\rm
R}/R^{1D}$ 2010Bloch_BEC . Without the potential well ($V_{0}=0$), polariton
density distributes evenly in space with $n_{C}^{0}=(P_{0}-P_{th})/\gamma_{\rm
C}$ in the steady state. With the centrosymmetric potential well (Fig. 1), two
domains of polariton BEC appear in the extended space across the potential
well [see Fig. 2]. Away from the potential well, the polariton density
($\approx n_{C}^{0}$) is nearly homogeneous. In the potential region, both the
polariton density and phase $\phi$ vary dramatically. As the velocity
$v\propto\nabla_{x}\phi$, one notes that the polariton particles in these two
domains accelerate towards the center of the potential, leading to counter-
propagating waves. Due jointly to the high-velocity propagation and phonon-
assisted relaxation, polaritons with opposite wave vectors collide in the
potential center, and induce a sink-type interference. Depending on whether
the interference is constructive or destructive, $\psi$ exhibits odd or odd
parity [see Fig 2(a2), Fig. 2(b2), and discussion below].

Figure 2: (Color online) (a) Single dark soliton and (b) a pair of sink-type
dark solitons. Both solitons are localized in the potential well region. Top
row shows the dynamical evolution of polariton density, and the formation of
the polarton dark solitons. Middle row shows polariton amplitude $|\psi(x)|$
and phase $\phi(x)$ at t=10ns, which clearly shows the formation of the dark
soliton. Bottom row gives the corresponding spectra of these states in the
potential well. Other parameters are $V_{0}=-0.6$meV in (a) and
$V_{0}=-1.0$meV in (b), while the pumping power $P_{0}=2.3P_{th}$ and
$gc=0.4787\mu$eV $\mu$m.

Dark solitons are stabilized by the nonlinear interaction [see Figs. 2(a1) and
(b1)]. As low-energy excitations, the dark soliton is additionally affected by
the pumping field power and potential depth. To illustrate this dependence, we
first consider a scenario with fixed $P_{0}=2.3P_{th}$ and $gc=0.4787\mu$eV
$\mu$m. When -0.8meV$\leq V_{0}\leq$ -0.5meV, a dark soliton with odd parity
forms, depicted in Fig. 2(a). Deepening the potential depth down to
-1.6mev$\leq V_{0}\leq-1.0$meV, a dark soliton of even parity is found,
originated from the constructive interference between the counter-propagating
waves. Note that the two groups of dark solitons might be regarded as complex
modes of the bound states $\ket{1}$ and $\ket{2}$. When projected to the bound
states, the odd (even) parity soliton maintains the maximum population in
state $|1\rangle|$ ($|2\rangle$). This is in sharp contrast to the typically
equilibrium case where polariton particles condense on the lowest energy
state. As shown in Fig. 2, the destructive and constructive interference
result in a neat separation of polariton condensate into two domains by a wavy
junction in the potential well. The steep jump of the phase across the
junction, together with the corresponding density depletion, demonstrates the
spontaneous creation of these two types (odd or even) of dark solitons acting
as an insulating barrier Sanvitto_2019 .

Figure 3: (Color online) (a-b) transition between the even and odd parity
polariton dark solitons. The arrow shows the direction when changing $V_{0}$.
(c) bistability phase diagram of the even and odd parity polariton dark
solitons by scanning $V_{0}$ with rate $5\times 10^{-5}$meV/ps. One dark
soliton is found in the odd parity and two dark solitons are found in the even
parity. The arrows indicate the direction of bistability transitions. (d)
snake instability between the two phases at the parameters denoted by O in (a)
and (c).

Emergent bistabilities—. In the odd parity case we obtain a single polariton
dark soliton, while the even parity could be regarded as two dark solitons,
whose phase changes by $-\pi$ and $\pi$, respectively [from left to right in
Fig. 2(b2)]. These two types of dark solitons can transit from one to the
other when the potential depth $V_{0}$ is adiabatically scanned up and down,
as shown in Figs. 3(a) and (b). Uniquely the bistability is controlled
entirely by the potential depth $V_{0}$, shown in Fig. 3(c). When $V_{0}$ is
scanned down to the value between the two phases [Fig. 3(a) and (c)], the
polariton condensate is not in a steady state, whose density oscillates
periodically [see Fig. 3(d)]. This leads to a snake instability due to the
superposition between the odd and even parity solitons. Note that the
appearance of the snake instability is directional, i.e. only found in the
odd-even transition (decreasing $V_{0}$). This is largely due to the fact that
the sudden birth of the even parity soliton corresponds to a high energy
state. The relaxation of this excitation is very slow as a significant
fraction of the population in the state $|1\rangle$ and $|2\rangle$.

To further illustrate the directional excitation in the bistable transition,
we examine the energy of the polariton dark soliton. In the perturbative
regime, we assume that the dark soliton wave function $\psi_{d}$ propagates on
a background of spatially homogeneous condensate, given by
$\psi=\sqrt{n_{C}^{0}}\times\psi_{d}$ and $n_{\rm R}=n_{\rm R}^{0}+m_{R}$.
Here $n_{C}^{0}$ and $n_{\rm R}^{0}$ are spatially homogeneous densities
respectively for polaritons and exciton reservoir. $m_{R}$ is the perturbation
for $n_{\rm R}^{0}$. Using Eq. (Controllable Bistability and Squeezing of
Confined Polariton Dark Solitons), we obtain the energy of the dark soliton
2014prb_elena :

$\displaystyle E_{s}=$ $\displaystyle\frac{1}{\hbar
n_{C}^{0}}\int[\frac{\hbar^{2}}{2m}(1-iD_{0}\frac{\gamma_{\rm C}}{R^{\rm
1D}})|\frac{\partial\psi_{d}}{\partial x}|^{2}-V(x)|\psi_{d}|^{2}$
$\displaystyle+\frac{1}{2}g_{\rm C}^{\rm
1D}(n_{C}^{0}-|\psi_{d}|^{2})^{2}]dx.$ (3)

As shown in Fig. 4(a) and (b), the perturbative energy exhibits different
profiles, depending on how the potential depth is varied. It has a similar
shape to the bistability transition found in Fig. 3(c). The oscillation of the
energy signifies the snake instability, as seen in the density pattern shown
in fig. 3(d). Note that it is found only when $V_{0}$ is changed across the
bistable transition.

We want to emphasize that the bistability discussed in this work is realized
in an off-resonantly coupled polariton condensate. The bistability is driven
by the potential depth $V_{0}$, and the two phases are stabilized by the
nonlinear interaction and spatial confinement. Such bistability is essentially
different from the one found in polariton condensates with resonant pumping
2016prl_Bloch , which is achieved by scanning the phase twist of the pump
field with a constant laser power. In both situations, on the other hand, the
interference between polariton condensates 2016prl_Lagoudakis  plays key roles
in obtaining the different types (parities) of the dark solitons.

Figure 4: (Color online) (a)-(b) The perturbative energy and (c)-(d) intensity
noise (squeezing parameter) when scanning $V_{0}$. In (a) the density
dependent nonlinear effect dominates, such that squeezing is absent in (c).
Increasing $gc$, the nonlinear interaction suppresses the quantum
fluctuations, leading to squeezing, shown in (d). The directional dependent
phase diagram for both the bistability and squeezing is shown in (e) by
decreasing $V_{0}$, and (f) by increasing $V_{0}$ across the bistable region,
respectively. In (a) and (c) only stability is found [corresponding to point A
in (e) and (f) with BT phase], where the parameter $P_{0}=2.3P_{th}$, $g_{\rm
C}^{\rm 1D}=0.4787\mu$eV$\cdot\mu$m [the same as in Fig. 3(c)]. In (b) and (d)
both bistability and squeezing are found [corresponding to point B in (e) and
(f) with BTS phase]. The related parameters are $P_{0}=1.6P_{th}$ and $g_{\rm
C}^{\rm 1D}=1.5958\mu$eV$\cdot\mu$m. In all panels, the arrows indicate the
scanning direction of $V_{0}$.

Squeezing of dark solitons—. The Kerr-like nonlinear interaction allows to
achieve quadrature-squeezed light. Typically the strongest squeezing is found
around the turning points of the bistability squeezing_1992 . We characterize
the squeezing of intensity noise through parameter $\xi$ polariton
squeezing_2014 ; polariton squeezing_2017 :

$\displaystyle\xi=\frac{\langle\Delta
N\rangle^{2}}{\langle\hat{\Psi}^{\dagger}\hat{\Psi}\rangle}\approx\frac{2\langle|\psi|^{2}\delta\hat{\psi}^{\dagger}\delta\hat{\psi}\rangle+2\langle
Re[\psi^{*2}\delta\hat{\psi}\delta\hat{\psi}]\rangle}{\langle|\psi|^{2}+\delta\hat{\psi}^{\dagger}\delta\hat{\psi}\rangle}$
(4)

where $\delta\hat{\psi}=\hat{\Psi}-\psi$ and
$\delta\hat{\psi}^{\dagger}=\hat{\Psi}^{\dagger}-\psi^{*}$ represent the
fluctuation (noise) operator fulfilling
$\langle\delta\hat{\psi}\rangle=\langle\delta\hat{\psi}^{\dagger}\rangle=0$,
with $\hat{\Psi}$ and $\hat{\Psi}^{\dagger}$ to be the annihilation and
creative operators of polariton field. The correlation of fluctuation
operators satisfies the following relation,

$\displaystyle\langle\delta\hat{\psi}\delta\hat{\psi}\rangle=$
$\displaystyle\gamma_{\rm
C}M_{11}M_{12}(\langle\hat{\Psi}_{in}^{\dagger}\hat{\Psi}_{in}\rangle+\langle\hat{\Psi}_{in}\hat{\Psi}_{in}^{\dagger}\rangle)$
$\displaystyle\langle\delta\hat{\psi}^{\dagger}\delta\hat{\psi}\rangle=$
$\displaystyle\gamma_{\rm
C}M_{21}M_{12}\langle\hat{\Psi}_{in}\hat{\Psi}_{in}^{\dagger}\rangle+\gamma_{\rm
C}M_{22}M_{11}\langle\hat{\Psi}_{in}^{\dagger}\hat{\Psi}_{in}\rangle$

where $M_{11}=M_{22}^{*}=\frac{\hbar(i+D_{0}n_{\rm R})}{2m}\nabla^{2}-ig_{\rm
R}^{\rm 1D}n_{\rm R}-2ig_{\rm C}^{\rm 1D}|\psi|^{2}+\frac{1}{2}(R^{\rm
1D}n_{\rm R}-\gamma_{c})-iV(x)$ and $M_{12}=M_{21}^{*}=-ig_{\rm C}^{\rm
1D}\psi^{2}$. $\hat{\Psi}_{in}^{(\dagger)}$ is the input annihilation
(correlation) operator of polaritons, which has the form of
$\langle\hat{\Psi}_{in}^{\dagger}(t)\hat{\Psi}_{in}(t^{\prime})\rangle=\sigma
P_{0}\delta(t-t^{\prime})$ and
$\langle\hat{\Psi}_{in}(t)\hat{\Psi}_{in}(t^{\prime})\rangle=0$ ($\sigma$ is a
probability constant) squeezing_1992 . The squeezing parameter determines
statistical properties of fluctuations, which is a Poissonian when $\xi=1$,
super-Poissonian (bunched) when $\xi>1$, and squeezed (antibunched) when
$\xi<1$, correspondingly.

Both the nonlinear interaction coefficient $g_{\rm C}^{\rm 1D}$ and pump power
plays crucial roles in realizing squeezing ($\xi<1$). To demonstrate this, we
first examine the squeezing parameter corresponding to the situation discussed
in Fig. 4(a). As shown in Fig. 4(c), the intensity noise becomes weaker at the
turning points of bistability, where $\xi>1$ is relatively smaller (larger)
along the odd-even (even-odd) transition. However irrelevant to the direction,
squeezing is not found in either case. Both the topological excitation of
polariton dark solitons and the intensity squeezing require the strong
nonlinear interaction. We note that the polariton dark solitons are classical
excitations that depend strongly on the pump field. The stronger the pump
power, the higher the polariton density. Though this could generate stronger
density-dependent nonlinear effects (e.g., stabilizing the dark soliton),
quantum fluctuations also increases significantly at the same time [Fig.
4(c)], such that squeezing of the noise is not seen.

Thanks to the Feshbach resonance 2019prl_Feshbach ; 2014nphy_Deveaud , we can
maintain strong nonlinearities (via tuning $g_{\rm C}^{\rm 1D}$) even at low
polariton densities. In this case, bistability is still clearly visible, as
shown in Fig. 4(b). Squeezing, however, is drastically improved, as can be
seen in Fig. 4(d). It is found that $\xi$ becomes smaller round the bistable
transition region. More importantly, squeezing, i.e. $\xi<1$, is achieved
around the turning point of the bistability, which is independent of the
direction when changing $V_{0}$. The squeezing strength, however, depending on
the direction. Our numerical simulations show that $\xi=0.667$ at the turning
point of the bistability when decreasing $V_{0}$. In contrast, a slightly
weaker squeezing with $\xi=0.768$ is obtained when increasing $V_{0}$.

Through large scale numerical simulations, we obtain phase diagrams to capture
both bistability and squeezing of polariton dark solitons, shown in Fig. 4(e)
and (f). To obtain the phase diagram, dynamics of the system is solved by
scanning $V_{0}$ downwards [Fig. 4(e)] and upwards [Fig. 4(f)] with a given
set of $g_{\rm C}^{\rm 1D}$ and $P_{0}$. By examining stability behaviors and
squeezing parameters, three distinctive phases, one without bistability (blue
region), one with only bistability (BT phase, grey region), and one with both
bistability and squeezing (BTS phase, colors other than blue or grey), are
identified. For a given $g_{\rm C}^{\rm 1D}$, the BT phase forms when the pump
field is strong enough, such that the density (Kerr) nonlinearity can
stabilize the two types of solitons. Squeezing, on the other hand, occurs when
the nonlinear interaction coefficient $g_{\rm C}^{\rm 1D}$ is large enough
while the polariton density is relatively low. Hence the BT phase occupies
larger areas than the BTS phase. Such feature is independent of the changing
direction of $V_{0}$, which is consistent with the example shown in Fig. 4(c)
and (d). In the phase diagram, the strongest squeezing is $\xi=0.516$, which
is obtained at $g_{\rm C}^{\rm 1D}=2.473\mu$eV$\cdot\mu$m and
$P_{0}=1.56P_{th}$ when $V_{0}$ is decreasing.

Conclusions—. We have shown that the odd and even parity dark solitons can be
generated in a controlled fashion in a nonresonantly pumped semiconductor
microcavity containing a central potential well. The solitons are formed due
to the strong geometric confinement and nonlinear Kerr like interaction. By
varying the potential depth, a bistable phase of the two dark solitons is
found. We have revealed that strong intensity squeezing can be achieved at the
turning points of the bistability. Through large scale numerical simulations,
we have demonstrated that the level of the intensity squeezing can be
controlled by changing either $P_{0}$ or $g_{\rm C}^{\rm 1D}$, where the
latter is achieved via the Feshbach resonance 2019prl_Feshbach ;
2014nphy_Deveaud . Our study might open a way to create ultra-low threshold,
on-chip integrated topological excitations and squeezed light sources for
quantum information-processing applications.

###### Acknowledgements.

This work is supported by National Natural Science Foundation of China (Grant
No. 12074147). W. Li acknowledges support from the UKIERI-UGC Thematic
Partnership (IND/CONT/G/16-17/73), and the Royal Society through the
International Exchanges Cost Share award No.
IEC$\backslash$NSFC$\backslash$181078\.

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##  I Supplementary to ”Controllable Bistability and Squeezing of Confined
Polariton Dark Solitons”

###  I.1 Soliton energy

In the perturbative regime, we assume that the dark soliton wave function
$\psi_{d}$ propagates on a background of spatially homogeneous condensate,
given by $\psi=\sqrt{n_{C}^{0}}\times\psi_{d}$ and $n_{\rm R}=n_{\rm
R}^{0}+m_{R}$. Here $n_{C}^{0}$ and $n_{\rm R}^{0}$ are spatially homogeneous
densities respectively for polaritons and exciton reservoir. $m_{R}$ is the
perturbation for $n_{\rm R}^{0}$. Substituting these definitions into Eq.
(Controllable Bistability and Squeezing of Confined Polariton Dark Solitons)
in the main text, we obtain the following dynamical equations:

$\displaystyle i\frac{\partial\psi_{d}}{\partial
t}=-\frac{\hbar}{2m}(1-D_{0}\frac{\gamma_{\rm
C}}{R}i)\frac{\partial^{2}\psi_{d}}{\partial x^{2}}-g_{\rm C}^{\rm
1D}n_{C}^{0}(1-|\psi_{d}|^{2})\psi_{d}+V(x)\psi_{d}+(iD_{0}\frac{\partial^{2}\psi_{d}}{\partial
x^{2}}+\frac{i}{2}R^{\rm 1D}+g_{\rm R}^{\rm 1D})m_{R}\psi_{d}$
$\displaystyle\frac{\partial m_{R}}{\partial
t}=(P_{0}-P_{th})(1-|\psi_{d}|^{2})-\gamma_{\rm
R}(1-|\psi-d|^{2})m_{R}-\gamma_{\rm R}\frac{P_{0}}{P_{th}}|\psi-d|^{2}m_{R}$
(5)

According to the Lagrange variational method:

$\displaystyle\frac{\delta L}{\delta f}=\frac{\partial}{\partial
z}\frac{\partial L}{\partial_{z}f}+\frac{\partial}{\partial\tau}\frac{\partial
L}{\partial_{\tau}f}-\frac{\partial L}{\partial
f}=\int(R\frac{\partial\psi^{*}_{d}}{\partial
f}+R^{*}\frac{\partial\psi_{d}}{\partial f})dx$ (6)

The Lagrangian could be constructed from Eq.(I.1):

$\displaystyle L=\frac{i}{2}(\psi_{d}\frac{\partial\psi^{*}_{d}}{\partial
t}+\psi^{*}_{d}\frac{\partial\psi_{d}}{\partial
t})+\frac{\hbar}{2m}(1-D_{0}\frac{\gamma_{\rm
C}}{R}i)|\frac{\partial\psi_{d}}{\partial x}|^{2}+\frac{1}{2}g_{\rm C}^{\rm
1D}n_{C}^{0}(1-|\psi_{d}|^{2})^{2}-V(x)|\psi_{d}|^{2}$ (7)

So, the energy of the dark soliton is expressed as:

$\displaystyle E_{s}=\int dx[\frac{\hbar}{2m}(1-iD_{0}\frac{\gamma_{\rm
C}}{R})|\frac{\partial\psi_{d}}{\partial x}|^{2}+\frac{1}{2}g_{\rm C}^{\rm
1D}n_{C}^{0}(1-|\psi_{d}|^{2})^{2}-V(x)|\psi_{d}|^{2}]$ (8)

###  I.2 Intensity noise

Dynamics of the polariton field is modelled by the non-Hermine Hamiltonian of
polariton condensate in the vicinity of the ground state, as can be described
as polariton squeezing_2017 :

$\displaystyle\hat{H}=$ $\displaystyle\int
dx[\hat{\Psi}^{\dagger}(x)(-\frac{\hbar^{2}}{2m_{eff}}(1-iD_{0}n_{\rm
R})\nabla^{2}+\hbar V(x))\hat{\Psi}(x)+\hbar g_{\rm C}^{\rm
1D}\hat{\Psi}^{\dagger}(x)\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)\hat{\Psi}(x)+\hbar
g_{\rm R}^{\rm 1D}n_{\rm R}\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)$
$\displaystyle+\frac{i\hbar}{2}(R^{\rm 1D}n_{\rm R}-\gamma_{\rm
C})\hat{\Psi}(x)]$ (9)

where $\hat{\Psi}(x)$ and $\hat{\Psi}^{\dagger}(x)$ are, respectively, the
annihilation and creative operators of a polariton field on the lower-
polariton branch. Governed by the Heisenberg equation
$i\hbar\frac{\partial{\hat{\Psi}(x)}}{\partial{t}}=-[\hat{H},\hat{\Psi}(x)]$,
the Langevin equation for the polariton field is derived as follows:

$\displaystyle i\hbar\frac{\partial\hat{\Psi}(x)}{\partial t}=$
$\displaystyle[-\frac{\hbar^{2}}{2m}(1-iD_{0}n_{R})\nabla^{2}+\hbar
V(x)]\hat{\Psi}(x)+\hbar g_{\rm C}^{\rm
1D}\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)\hat{\Psi}(x)+\hbar g_{\rm R}^{\rm
1D}n_{\rm R}\hat{\Psi}(x)+\frac{i\hbar}{2}(R^{\rm 1D}n_{\rm R}-\gamma_{\rm
C})\hat{\Psi}(x)$ $\displaystyle-i\hbar\sqrt{\gamma_{\rm
C}}\hat{\Psi}_{in}(x)$ (10)

here $\hat{\Psi}_{in}(x)$ is the input operator from the polariton field and
assumed to have a zero average $\langle\hat{\Psi}_{in}(x)\rangle=0$
(incoherent input).

To allow for the large populations involved in polariton condensate, we expand
the polariton field operator as $\hat{\Psi}(x)=(\psi+\delta\hat{\psi})$, where
$\psi=\langle\hat{\Psi}(x)\rangle$ is the coherent mean field component,
$\delta\hat{\psi}$ represents the fluctuation (noise) operator fulfilling
$\langle\delta\hat{\psi}\rangle=0$. Substituting
$\langle\hat{\Psi}(x)\rangle=\psi$ into the Langevin equation, we first
calculate the mean field component of polariton condensate through the Gross-
Pitaevskii equation, coupled to a rate equation for the density of excitation
reservoir, which are expressed in Eq. (Controllable Bistability and Squeezing
of Confined Polariton Dark Solitons) in the main text.

Next Let’s derive the expression of intensity noise with the consideration of
fluctuation operator $\delta\hat{\psi}=\hat{\Psi}-\psi$. Substituting it into
the Eq. (I.2), the quantum Langevin equation with the assumption of linear
approximation is expressed as following:

$\displaystyle i\hbar\frac{\partial\delta\hat{\psi}}{\partial t}$
$\displaystyle=[-\frac{\hbar^{2}}{2m}(1-iD_{0}n_{\rm R})\nabla^{2}+\hbar
V(x)+\hbar g_{\rm R}^{\rm 1D}n_{\rm R}+2\hbar g_{\rm C}^{\rm
1D}|\psi|^{2}+\frac{i\hbar}{2}(R^{\rm 1D}n_{\rm R}-\gamma_{\rm
C})]\delta\hat{\psi}+\hbar g_{\rm C}^{\rm
1D}|\psi|^{2}\delta\hat{\psi}^{\dagger}-i\hbar\sqrt{\gamma_{\rm
C}}\hat{\Psi}_{in}$ (11)

According to Eq. (11), we define the following matrix:

$A=\left(\begin{array}[]{c}\delta\hat{\psi}{}\\\
\delta\hat{\psi}^{\dagger}{}\\\
\end{array}\right)~{}~{}~{}~{}~{}~{}B=\left(\begin{array}[]{c}\sqrt{\gamma_{\rm
C}}~{}\hat{\Psi}^{in}\\\ \sqrt{\gamma_{\rm C}}~{}\hat{\Psi}^{\dagger in}\\\
\end{array}\right)~{}~{}~{}~{}~{}~{}N=\left(\begin{array}[]{cccccc}(i+D_{0}n_{\rm
R})f_{0}+f_{1}-if_{2}&-ig_{\rm C}^{\rm 1D}~{}\psi^{2}\\\ \\\ ig_{\rm C}^{\rm
1D}~{}(\psi^{*})^{2}&(-i+D_{0}n_{\rm R})f_{0}+f_{1}+if_{2}\\\ \\\
\end{array}\right)$ (12)

with $f_{0}=\frac{\hbar}{2m}\nabla^{2}$, $f_{1}=\frac{1}{2}(R^{\rm 1D}n_{\rm
R}-\gamma_{c})$, $f_{2}=g_{\rm R}^{\rm 1D}n_{\rm R}+2g_{\rm C}^{\rm
1D}|\psi|^{2}+V(x)$. Then Eq. (11)can be rewirtten as:

$\displaystyle\frac{\partial}{\partial t}A=NA-B$ (13)

Set $\frac{\partial}{\partial t}A=0$, we have $A=N^{-1}B=MB$ as follows:

$\left(\begin{array}[]{c}\delta\hat{\psi}{}\\\ \delta\hat{\psi}^{\dagger}{}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}M_{11}&M_{12}\\\
M_{21}&M_{22}\\\ \end{array}\right)\left(\begin{array}[]{c}\sqrt{\gamma_{\rm
C}}~{}\hat{\Psi}^{in}\\\ \sqrt{\gamma_{\rm C}}~{}\hat{\Psi}^{\dagger in}\\\
\end{array}\right)$ (14)

with $M_{11}=M_{22}^{*}=\frac{(-i+D_{0}n_{\rm R})f_{0}+f_{1}+if_{2}}{a}$,
$M_{12}=M_{21}^{*}=\frac{ig_{\rm C}^{\rm 1D}~{}\psi^{2}}{a}$ and
$a=(f_{2}-f_{0})^{2}+(f_{1}+D_{0}n_{\rm R}f_{0})^{2}-(g_{\rm C}^{\rm
1D}|\psi|^{2})^{2}$. The correlation of fluctuation operator can be derived
as:

$\displaystyle\langle\delta\hat{\psi}^{\dagger}\delta\hat{\psi}\rangle=$
$\displaystyle\gamma_{\rm
C}M_{21}M_{12}\langle\hat{\Psi}_{in}\hat{\Psi}_{in}^{\dagger}\rangle+\gamma_{\rm
C}M_{22}M_{11}\langle\hat{\Psi}_{in}^{\dagger}\hat{\Psi}_{in}\rangle$
$\displaystyle\langle\delta\hat{\psi}\delta\hat{\psi}\rangle=$
$\displaystyle\gamma_{\rm
C}M_{11}M_{12}(\langle\hat{\Psi}_{in}^{\dagger}\hat{\Psi}_{in}\rangle+\langle\hat{\Psi}_{in}\hat{\Psi}_{in}^{\dagger}\rangle)$
(15)

where $M_{11}=M_{22}^{*}=\frac{\hbar(i+D_{0}n_{\rm R})}{2m}\nabla^{2}-ig_{\rm
R}^{\rm 1D}n_{\rm R}-2ig_{\rm C}^{\rm 1D}|\psi|^{2}+\frac{1}{2}(R^{\rm
1D}n_{\rm R}-\gamma_{c})-iV(x)$ and $M_{12}=M_{21}^{*}=-ig_{\rm C}^{\rm
1D}~{}\psi^{2}$. Assume that the correlation of input operator of polariton
has the form of
$\langle\hat{\Psi}_{in}^{\dagger}(t)\hat{\Psi}_{in}(t^{\prime})\rangle=\sigma
P_{0}\delta(t-t^{\prime})$ and
$\langle\hat{\Psi}_{in}(t)\hat{\Psi}_{in}(t^{\prime})\rangle=0$ ($\sigma$ is a
probability constant). With this assumption and the input-output relation, we
could have the results for the intensity noise according to the following
expression:

$\displaystyle\frac{\langle\Delta
N\rangle^{2}}{\langle\hat{\Psi}^{\dagger}\hat{\Psi}\rangle}$
$\displaystyle=\frac{\langle\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\rangle-\langle\hat{\Psi}^{\dagger}\hat{\Psi}\rangle^{2}}{\langle\hat{\Psi}^{\dagger}\hat{\Psi}\rangle}=\frac{\langle|\psi|^{4}+4|\psi|^{2}\delta\hat{\psi}^{\dagger}\delta\hat{\psi}+2Re[\psi^{*2}\delta\hat{\psi}\delta\hat{\psi}]\rangle-\langle|\psi|^{2}+\delta\hat{\psi}^{\dagger}\delta\hat{\psi}\rangle^{2}}{\langle|\psi|^{2}+\delta\hat{\psi}^{\dagger}\delta\hat{\psi}\rangle}$
$\displaystyle\approx\frac{2\langle|\psi|^{2}\delta\hat{\psi}^{\dagger}\delta\hat{\psi}+Re[\psi^{*2}\delta\hat{\psi}\delta\hat{\psi}]\rangle}{\langle|\psi|^{2}+\delta\hat{\psi}^{\dagger}\delta\hat{\psi}\rangle}$
(16)

Generally, the intensity noise characterizes the statics of light: it is
Poissonian when equals to $1$, super-Poissonian(bunched) when larger than $1$,
and quantum (antibunched) when smaller than $1$.