# Giant synthetic gauge field for spinless microcavity polaritons
in crossed electric and magnetic fields

Igor Y. Chestnov  School of Science, Westlake University, 18 Shilongshan Road,
Hangzhou 310024, Zhejiang Province, China  Institute of Natural Sciences,
Westlake Institute for Advanced Study, 18 Shilongshan Road, Hangzhou 310024,
Zhejiang Province, China  Vladimir State University, Gorkii St. 87, 600000,
Vladimir, Russia  Sergey M. Arakelian  Vladimir State University, Gorkii St.
87, 600000, Vladimir, Russia  Alexey V. Kavokin  School of Science, Westlake
University, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China
Institute of Natural Sciences, Westlake Institute for Advanced Study, 18
Shilongshan Road, Hangzhou 310024, Zhejiang Province, China  Russian Quantum
Center, Skolkovo IC, Bolshoy Bulvar 30, bld. 1, Moscow 121205, Russia  NTI
Center for Quantum Communications, National University of Science and
Technology MISiS, Moscow 119049, Russia<EMAIL_ADDRESS>
###### Abstract

The artificial gauge field for electrically neutral exciton polaritons devoid
from the polarization degree of freedom can be synthesized by means of
applying crossed electric and magnetic fields. The appearance of the gauge
potential can be ascribed to the motional (magneto-electric) Stark effect
which is responsible for the presence of a linear-in-momentum contribution to
the exciton kinetic energy. We study the interplay of this phenomenon with the
competing effect which arises from the Rabi-splitting renormalization due the
reduction of the electron-hole overlap for a moving exciton. Accounting for
this mechanism is crucial in the structures with the high ratio of Rabi
splitting and the exciton binding energy. Besides, we propose an approach
which boosts the gauge field in the considered system. It takes advantage of
the crossover from the hydrogen-like exciton to the strongly dipole-polarized
exciton state at a specific choice of electric and magnetic fields. The strong
sensitivity of the exciton energy to the momentum in this regime leads to the
large values of the gauge field. We consider the specific example of a GaAs
ring-shape polariton Berry phase interferometer and show that the flux of the
effective magnetic field may approach the flux quantum value in the considered
crossover regime.

##  I Introduction

Artificial gauge fields allow for revealing of many peculiar phenomena such as
fractional quantum Hall and Aharonov-Bohm effects, various topological phases
of matter etc. in the systems decoupled from the real gauge fields. The
paradigmatic examples here are photonic systems [1, 2], ultra-cold atomic
gases [3, 4, 5] and solid state media [6]. The synthesis of the artificial
gauge fields for electrically neutral particles is aimed at making them to
behave as if they are charged and affected by the real electromagnetic field.
Synthetic gauge fields can be described by an effective vector potential
$\mathbf{A}_{\rm eff}$ that results is the appearance of an effective magnetic
field $\mathbf{B_{\rm eff}}=\bm{\nabla}\times\mathbf{A_{\rm eff}}$. The
Schröedinger equation for a moving particle of a charge $q$ is modified in
this case by a substitution
$\hat{\mathbf{p}}\rightarrow\hat{\mathbf{p}}-q\mathbf{A}_{\rm eff}$.
Therefore, in the case of a parabolic dispersion
$\hat{H}\propto\hat{\mathbf{p}}^{2}$, the gauge field is manifested by the
presence of the term linear in $\mathbf{p}\mathbf{A}_{\rm eff}$ in the single-
particle dispersion relation.

Recently, the synthesis of the tunable artificial gauge field for electrically
neutral superfluid of exciton polaritons was demonstrated [7]. Microcavity
polaritons are hybrid quasiparticles which arise from the coherent coupling of
photons with the quantum well excitons. The approach employed in [7] takes
advantage of the excitonic magneto-electric Stark effect [8] which occurs in
crossed electric and magnetic fields. The electric field endows exciton with
the static dipole moment $\mathbf{d}$. If the exciton moves in perpendicular
magnetic field, it feels the effective electric field $\bm{\mathcal{E}}_{b}$
which originates from the oppositely directed Lorentz forces acting on the
electron and the hole, see Fig. 1. The strength of this field is governed by
the exciton center-of-mass momentum,
$\bm{\mathcal{E}}_{b}=\mathbf{p}\times\mathbf{B}/M$, where $M$ is the exciton
mass. Therefore the moving dipole-polarized exciton interacts with the
effective electric field with the energy
$-e\mathbf{d}\cdot\bm{\mathcal{E}}_{b}=\mathbf{d}\times\mathbf{B}\cdot\mathbf{p}/M$
which is linear in $\mathbf{p}$. It leads to the appearance of the effective
gauge potential which affects polariton dynamics via their excitonic
component.

In the experiment [7], the presence of the gauge field was evidenced by a non-
reciprocal polariton transport which can be connected with different
geometrical phases accumulated along and opposite to the gauge field
directions. The origin of these phases can be attributed with He-McKellar-
Wilkens (HMW) effect [9, 10, 11]. It predicts the acquisition of the Berry
phase by the electric dipole moving in the magnetic field. This effect
possesses a fundamental similarity with the Aharonov-Bohm and the Aharonov-
Casher effects which are responsible for the relevant phenomena for a charge
moving in a magnetic field and for a magnetic dipole (spin) moving in an
electric field, respectively [12].

The nontrivial manifestations of the HMW effect were initially foreseen in the
closed geometry where the Berry phases acquired by the dipoles travelling
adiabatically along different trajectories lead to a quantum interference.
Such phenomena were predicted in a superfluid helium and dipolar condensate
localized on a torus. Generation of the HMW phase in this case implies the use
of either radial electric and axial magnetic fields [13, 11, 14] or co-axial
electric and radial magnetic fields [9]. The latter configuration can also be
realized with no need of electric field once the excitonic superfluid formed
in the bilayer systems is considered [15, 16, 17]. The typical manifestation
of the quantum interference is an appearance of a circular persistent current
of the dipolar superfluid as the Berry phase accumulated along the
circumference exceeds $\pi$ [14].

Figure 1: The sketch of a micropillar cavity with a ring-shaped polariton
condensate excited by a nonresonant optical pump. The electrodes generate an
electric field with the divergent in-plane component. In combination with the
magnetic field oriented along the pillar growth axis, it generates the
azimuthal effective gauge potential for polaritons [18, 19]. The coupling
between the collective and the internal motions of the exciton which is
responsible for the effective electric field appearance is illustrated on the
left.

The use of microcavity polaritons instead of the helium or excitonic systems
as proposed in [7] is advantageous because of several aspects. Polariton
condensates can be generated at high temperatures and are convenient for the
optical detection and control. Besides, as polaritons are partially photons,
the presence of the HMW phase paves the way for the realization of the fast
tunable gauge field for photons.

Moreover, polaritons are subject to strong spin–orbit coupling effects which
are a source of the non-Abelian gauge fields [20]. The theoretical proposal of
a polariton Berry phase interferometer by Shelykh et al. [21] has induced a
significant experimental effort towards observation of topological effects in
polaritonics. Some of these works have been crowned by a stunning success, the
polariton topological insulator [22] and the quantum geometric tensor [23]
have been demonstrated. However, the realisation of Shelykh proposal for a
ring-shape Berry phase interferometer is still pending. One of the main reason
is the pinning of polarization of polariton condensates in laterally confined
structures that suppresses the Berry phase originated from the TE-TM splitting
of exciton-polariton modes. The recent works on superfluid circular currents
of exciton polaritons in micropillars did not reveal any significant
polarizations effects [24]. In this context, it is important to study the
topological effects in a scalar (in contrast to spinor) polariton liquid that
is devoid of the polarization degree of freedom. In this work, we discuss the
use of HMW effect for the synthesis of the $U(1)$ gauge field for polaritons.

The generation of the HMW phase with exciton polaritons is sophisticated due
to their hybrid nature. The energy of microcavity polations is strongly
affected by the strength of the coupling between the excitons and photons
known as a Rabi splitting. The effective electric field $\bm{\mathcal{E}}_{b}$
felt by the moving exciton modifies its internal structure and thus alters the
optical oscillator strength and the Rabi splitting as a consequence. In this
paper we demonstrate that this effect can be also used for generation of the
artificial gauge field for polaritons. However it turns out that this field is
oriented oppositely to the one arising from the HMW effect. Therefore,
accounting for both mechanisms is crucial. We study the interplay of theses
effects and discuss the ways to maximize the effective gauge field.

The strength of the effective gauge field is the key characteristic of any
method of synthesis. The quantum interference effects in the circular geometry
1 are expected to occur once the Berry phase acquired along the ring
circumference $\phi=\hbar^{-1}\oint\mathbf{A_{\rm eff}}\cdot d\bm{l}$ is about
$\pi$. However due to the small effective mass of exciton polaritons the gauge
field arising due to the motion Stark effect is rather weak. In this paper we
propose the approach capable of increasing the gauge field strength by at
least an order of magnitude with no need of extremely strong electric or
magnetic fields. It implies an operation in the regime of a crossover from the
hydrogen-like exciton to the strongly decentred exciton state.

Figure 2: The schematic of a double-well potential $W(\mathbf{r})$ for the 2D
electron-hole pair in crossed magnetic and electric fields. (a) The hydrogen-
like exciton at weak in-plane electric and normal magnetic field. (b) The case
of the moderate electric and magnetic fields illustrating a crossover
manifested in the overlap of the wave functions localized in the parabolic
diamagnetic potential and in the Coulomb well (dashed curves). The diamagnetic
potential is shifted with respect to the Coulomb well due to the presence of
electric field. (c) The polarized state possessing large electric dipole at
strong magnetic and electric fields.

The existence of this crossover is relevant to the problem of two oppositely
charged particles placed in crossed electric and magnetic fields which was
addressed in [25, 26, 27, 28]. These studies predict the existence of two
types of bound states of the interacting electron-hole pair. The states of the
first type correspond to the hydrogen-like exciton (HLE) whose electron
average position remains close to the hole so the relative motion wave
function $\chi(\mathbf{r})$ is essentially centred in the Coulomb potential
well, see Fig. 2a. This is the regime employed for the gauge field synthesis
in Ref. [7].

For the states of the second kind, the wave function $\chi(\mathbf{r})$ is
mostly concentrated outside the Coulomb well. The electron motion is strongly
decentred from the hole which is why these states are referred to as
decentred. This is the case of strong electric and magnetic fields when the
exciton structure is dominated by the diamagnetic potential whose position is
shifted apart from the Coulomb well due to the presence of electric field, see
Fig. 2c. These states posses large electric dipole and thus potentially can
feel a strong effective gauge field. However due to the vanishing electron-
hole overlap they are decoupled from the microcavity photons and can not form
polaritons.

Here we focus on the intermediate regime illustrated in Fig. 2b, where a
crossover between the states with the weak and strong decentring occurs [28,
29]. In the crossover regime, the exciton combines a large electric dipole
with the non-vanishing coupling to a photon field. These intermediate states
appear to be strongly sensitive to the variation of the electric field
strength. Therefore, even weak apparent electric field $\bm{\mathcal{E}}_{b}$
felt by the moving exciton leads to the significant modifications of the
exciton energy and structure. It causes a significant enhancement of the
effective gauge field strength as compared to the case of the HLE [7].

The paper is organized as follows. In Sec. II we evaluate the effective gauge
field for polaritons subjected the crossed electric and magnetic fields. Using
a hydrogen-like variational wave function for the ground state exciton we
study the interplay between the HMW effect and the Rabi-splitting
renormalization for a moving polariton. This section is summarized by the
discussion of the strength of the gauge field attainable in this regime.
Section III presents the study of the artificial gauge field for polaritons
formed by non-hydrogenic excitons. Solving the electron-hole bound state
problem numerically, we investigate the properties of the mixed exciton states
arising in the crossover regime. The synthesis of the artificial gauge field
for polaritons in this regime is discussed in Sec. IV. The results are
summarized in the Conclusions.

##  II Hydrogen-like exciton in the crossed electric and magnetic fields

The quantum interference effects in the polariton superfluid are expected to
occur in the ring-shaped polariton condensates affected by the perpendicular
magnetic and radial electric fields [18, 19]. Such a configuration can be
realized in the annual optical traps [30] or in a pillar microcavity [24].
Alternatively one can think about constructing of the Berry phase
interferometer in the design proposed in [21].

To be specific, we consider a ring-shaped condensate of exciton polaritons
excited in the configuration shown in Fig. 1. A combination of a spot-like
inner and a ring outer electrodes creates the radial electric field, which
endows the exciton with a dipole moment $\mathbf{d}$ in a direction
perpendicular to its circular motion. The external magnetic field is oriented
along the growth axis. This configuration guarantees a mutual orthogonality of
the induced electric dipole $\mathbf{d}$, the magnetic field $\mathbf{B}$ and
the azimuthal component of the condensate wave vector.

In this section we consider the regime of moderate electric and magnetic
fields, when the exciton relative motion wave function is localised mainly in
the Coulomb potential well, see Fig. 2a.

###  II.1 Guage field for a single exciton

The Hamiltonian describing a motion of the interacting electron-hole pair in
the crossed electric $\bm{\mathcal{E}}$ and magnetic $\mathbf{B}$ fields in
the symmetric gauge reads [29]

$\displaystyle\hat{H}_{\rm
ex}=\frac{{\mathbf{p}}^{2}}{2M}-\frac{\hbar^{2}}{2\mu}\Delta_{\mathbf{r}}+i\frac{e\hbar
B}{2\nu}\bm{\nabla}_{\mathbf{r}}\times\mathbf{r}+\frac{e^{2}B^{2}}{8\mu}r^{2}$
(1)
$\displaystyle+V_{c}(\mathbf{r})-e\mathbf{r}\bm{\mathcal{E}}+e\mathbf{r}\frac{\mathbf{B}\times{\mathbf{p}}}{M},$

where $M=m_{e}+m_{h}$ is the exciton mass associated with its center of mass
motion, while the relative mass $\mu=m_{e}m_{h}\left/{M}\right.$ and the
parameter $\nu=m_{e}m_{h}\left/\left(m_{e}-m_{h}\right)\right.$ are dependent
on the asymmetry between the electron $m_{e}$ and the hole $m_{h}$ masses;
$\mathbf{r}$ is the electron-hole separation vector; $V_{c}=-\varkappa
e^{2}\left/\left|\mathbf{r}\right|\right.$ is a Coulomb potential with
$\varkappa=(4\pi\varepsilon_{0}\varepsilon)^{-1}$. The last term in (1) stands
for the coupling between the exciton internal structure and its center of mass
momentum [31]. The last two terms can be cast as
$-e\mathbf{r}\left(\bm{\mathcal{E}}+\bm{\mathcal{E}}_{b}\right)$, where
$\bm{\mathcal{E}}_{b}={\mathbf{p}}\times\mathbf{B}/M$ is the effective field.

We note that strong magnetic fields are also expected to induce significant
Zeeman splittings. In this work we limit our consideration to single-component
polaritons that are devoid of the polarization degree of freedom. In the
presence of a strong magnetic field it would correspond to a 100% circularly
polarized polariton flow.

The problem of a ground state of the Hamiltonian (1) has the rigorous
solutions in the case of zero [32] or strong magnetic fields [33, 29]. However
in the case where both $\bm{\mathcal{E}}$ and $\mathbf{{B}}$ are non-vanishing
the excitonic problem cannot be resolved analytically. For this reason we
approach this problem variationally.

The trial wave function of the relative motion of the bound electron-hole pair
is taken in the form

$\chi(\mathbf{r})=\sqrt{\frac{2}{\pi}}\frac{(1-\lambda^{2})^{3/4}}{a}\exp\left(-\frac{|\mathbf{r}|-\lambda
x}{a}\right),$ (2)

where the Cartesian $x$-axis is parallel to the electric field direction, $a$
and $\lambda$ are variational parameters. The prefactor of the exponent
satisfies the normalization condition
$\int|\chi(\mathbf{r})|^{2}d\mathbf{r}$=1. The value of $a$ corresponds to the
effective in-plane exciton Bohr radius, while $\lambda$ accounts for the
exciton polarization due to the electric field.

The minimization procedure for the energy $E_{\rm
ex}(\mathbf{k})=\int\hat{H}_{\rm ex}|\chi(\mathbf{r})|^{2}d\mathbf{r}$ of the
exciton moving with a momentum $\mathbf{p}=\hbar\mathbf{k}$ yields:

$E_{\rm ex}(\mathbf{k})=\frac{\hbar^{2}\mathbf{k}^{2}}{2M}+\frac{\hbar}{2\mu
a^{2}}-\frac{2\varkappa
e^{2}}{a}+\frac{3e^{2}B^{2}a^{2}}{16\mu}-\frac{\alpha\left({\bm{\mathcal{E}}}+{\bm{\mathcal{E}}}_{b}\right)^{2}}{2},$
(3)

where we have assumed that the exciton polarization is weak, $\lambda^{2}\ll
1$. Here the fourth term accounts for the Langevin diamagnetic shift while the
last one stands for the DC-Stark shift from the combined action of the real
$\bm{\mathcal{E}}$ and effective $\bm{\mathcal{E}}_{b}$ electric fields. The
DC Stark effect if governed by the exciton polarizability which reads:

$\alpha=\frac{\alpha_{0}\tau^{3}}{1+\alpha_{0}\frac{5\tau^{3}}{12\mu}B^{2}},$
(4)

where $\alpha_{0}=9/8\varkappa^{-1}a_{\rm B}^{3}$ defines the 2D exciton
polarizability at zero magnetic field with $a_{\rm
B}={\hbar^{2}}\left/\left({2\varkappa e^{2}\mu}\right)\right.$ being the 2D
exciton Bohr radius. The dimensionless parameter $\tau(B)=a(B)/a_{\rm B}$
quantifies the reduction of the 2D exciton size in the presence of magnetic
field [34].

Taking into account that the effective field
$\bm{\mathcal{E}}_{b}={\hbar}\mathbf{k}\times\mathbf{B}/M$ is linear in the
exciton momentum, one can recast equation (3) in the form

$E_{\rm ex}=E_{\rm hl}+\frac{\left(\hbar\mathbf{k}-\mathbf{A}_{\rm
HMW}\right)^{2}}{2m_{\rm ex}}-\frac{{A}_{\rm HMW}^{2}}{m_{\rm ex}},$ (5)

where the effective gauge potential

$\mathbf{A}_{\rm HMW}=e\mathbf{d}\times\mathbf{B}\frac{m_{\rm ex}}{M}$ (6)

is proportional to the dipole moment $-e\mathbf{d}=\alpha\bm{\mathcal{E}}$
induced by the external electric field, $m_{\rm ex}^{-1}=M^{-1}\left(1-\alpha
B^{2}/M\right)$ is the exciton effective mass renormalized due to
magnetoelectric Stark effect. The zero-momentum energy of the hydrogen-like
exciton is given by $E_{\rm hl}=\frac{\hbar}{2\mu a^{2}}-\frac{2\varkappa
e^{2}}{a}+\frac{3e^{2}B^{2}a^{2}}{16\mu}-\frac{\alpha\mathcal{E}}{2}$.

The expression (6) for the effective gauge field for dipole-polarized exciton
provides an instructive insight into the origin of this phenomenon and its
connection with the HMW effect. According to the original proposal of He,
McKellar and Wilkens [9, 10] the phase accumulated by the electric dipole
$q\mathbf{d}$ moving on a path $\mathcal{D}$ is $\phi_{\rm
HMW}=q\hbar^{-1}\int_{\mathcal{D}}\mathbf{d}\times\mathbf{B}\cdot d{\bm{l}}$,
where $\bm{l}$ is the coordinate along the pass. Thus, the value
$\mathbf{A}_{\rm HMW}=q\mathbf{d}\times\mathbf{B}$ can be associated with the
effective gauge potential, $\phi_{\rm
HMW}=\hbar^{-1}\int_{\mathcal{D}}\mathbf{A}_{\rm HMW}\cdot d\bm{l}$, which
obeys the standard pulse rescaling rule
$\mathbf{p}\rightarrow\mathbf{p}-\mathbf{A}_{\rm HMW}$. This expression for
$\mathbf{A}_{\rm HMW}$ is identical to Eq. (6) up to the mass renormalization
factor. Note that the presence of the gauge field for dipole-polarized
excitons governed by the Hamiltonian (1) was pointed out by Sonin in [16, 17].

According to (6), the gauge field strength $A_{\rm HMW}$ scales linearly with
the polarizing electric field while its magnetic field dependence is primarily
governed by the exciton polarizability (4). Note that at $B=0$ our variational
solution only slightly underestimates the 2D exciton polarizability whose
exact value is $\alpha_{0}=21/16\varkappa^{-1}a_{\rm B}^{3}$ [32].

At weak magnetic fields $B<\sqrt{12\mu\alpha_{0}/5}$ and $\tau\approx 1$, the
exciton polarizability is approximately constant and the gauge field grows
linearly with the magnetic field [35], see Fig. 3a. At the same time in the
limit of strong magnetic field polarizability decreases as $\alpha\propto
B^{-2}$ and then $A_{\rm HMW}\propto B^{-1}$. This behaviour can be connected
with the exciton size reduction induced by the magnetic field [34] and the
gradual transition to the magnetoexciton regime [33]. The maximum value of the
gauge field strength is achieved in the intermediate regime. For the
considered parameters which are typical for GaAs structures, the magnetic
field that maximizes the gauge field would be about 20 T for the case of an
idealized 2D exciton. In realistic quantum wells (QWs) the exciton radius is
larger than one of a 2D exciton and thus the polarizability (4) decreases
faster as the magnetic field grows. In particular, for GaAs QW of $L=10$ nm
thickness the gauge field maximum corresponds to $B\approx 10$ T, see Fig. 3a.

To check the validity of our variational approach (2), we calculate the ground
state of the Hamiltonian (1) numerically using the finite element method
(FEM). The corresponding gauge potential is shown in Fig. 3a with the solid
line. The results of simulations demonstrate a high degree of reliability of
our variational approach.

Figure 3: (a) The dependence of the exciton gauge field strength $A_{\rm HMW}$
on the magnetic field for the 2D GaAs exciton (blue lines) at the external
electric field $\mathcal{E}=1$ $\rm kV/cm$. The analytical solution (6) is
shown with the dashed line while the results extracted from numerical
simulations correspond to the solid line. The dash-dotted line corresponds to
the case of the QW well of the thickness $L=10$ nm. (b) The magnitudes of the
polariton gauge potentials. (c) $A_{\rm HMW}/A_{\rm EPD}$ relation for
experimentally feasible range of $B$. The dashed line corresponds to Eqs.
(12a) and (12b) while the solid lines show the data obtained numerically. (d)
The geometric phase difference $\phi$ between polaritons travelling in
opposite arms of a ring-shaped Berry-phase interferometer. The right axis
shows the energy splitting between the clockwise and the anti-clockwise
currents of polariton superfluid excited on the thin ring, see Fig. 1. The
ring radius is $R_{0}=15$ $\mu$m. The material parameters under consideration
are typical for the GaAs medium: $m_{e}=0.067m_{\rm fe}$, $m_{h}=0.45m_{\rm
fe}$, $\varepsilon=12.9$ where $m_{\rm fe}$ is the free-electron mass.

###  II.2 Gauge field for exciton polaritons

In the regime of strong light-matter coupling, the excitons impart their
sensitivity to the effective gauge field to polaritons. The interaction of the
exciton with the microcavity mode can be described by the model of two coupled
oscillators. It yields that the energy of the lower branch polariton reads

$E_{\rm LP}=\frac{1}{2}\left(E_{\rm ex}+E_{\rm ph}-\sqrt{\left[E_{\rm
ex}-E_{\rm ph}\right]^{2}+\Omega^{2}}\right),$ (7)

where the microcavity mode dispersion is $E_{\rm
ph}=E_{\rm{c}}+\hbar^{2}k^{2}\left/2m_{\rm ph}\right.$, $\Omega$ is the
exciton-photon coupling strength (resonant Rabi splitting) and $m_{\rm ph}$ is
the microcavity photon mass. For the sake of self-consistency, we add the
semiconductor band gap energy $E_{g}$ to the exciton energy (5).

According to its definition, the Rabi splitting $\Omega$ is dependent on the
electron-hole relative motion wave function taken at the coincidence of the
electron and hole positions, $\Omega\propto\chi(0)$. With the trial function
(2) we have $\chi(0)=\sqrt{2/\pi a^{2}}\left(1-\lambda^{2}\right)^{3/4}$. Then
assuming $\lambda^{2}\ll 1$, we expand the coupling strength to the first
order in $\lambda^{2}$:

$\Omega\approx(1-3/4\lambda^{2})\Omega_{0},$ (8)

where $\Omega_{0}\equiv\Omega_{0}(B)$ defines the coupling strength for a non-
polarized exciton at $\bm{\mathcal{E}}=0$. Minimization of the exciton energy
thus yields $\lambda=\lambda_{0}(\mathcal{E}+\mathcal{E}_{b})$, where

$\lambda_{0}=-\frac{3a_{\rm
B}^{2}}{4e\varkappa}\frac{\tau^{2}}{1+\alpha_{0}\frac{5\tau^{3}}{12\mu}B^{2}}.$
(9)

We assume that all the momentum-dependent terms under the square root in (7)
are small compared to the Rabi splitting
$\Omega_{R}=\sqrt{\Delta^{2}+\Omega_{0}^{2}}$, where $\Delta=E_{g}+E_{\rm
hl}-E_{c}$. Then after a simple algebra one obtains:

$E_{\rm LP}=E_{\rm LP}^{(0)}+\frac{\left(\hbar\mathbf{k}-\mathbf{A}_{\rm
pol}\right)^{2}}{2m_{\rm pol}}-\frac{A_{\rm pol}^{2}}{2m_{\rm pol}}.$ (10)

Here $E_{\rm LP}^{(0)}=1/2\left(E_{g}+E_{\rm hl}+E_{c}-\Omega_{R}\right)$, the
polariton mass is $m_{\rm pol}^{-1}=C_{x}^{2}m_{\rm ex}^{-1}+C_{p}^{2}m_{\rm
ph}^{-1}-\frac{9\Omega_{0}^{2}}{8\Omega_{R}}B^{2}\lambda_{0}^{4}\mathcal{E}^{2}M^{-2}$
and $C_{x,p}=2^{-1/2}\left(1\mp\Delta\left/\Omega_{R}\right.\right)^{1/2}$ are
the Hopfield coefficients. The effective gauge potential accounts for the
contribution from two effects:

$\mathbf{A}_{\rm pol}=\mathbf{A}_{\rm HMW}+\mathbf{A}_{\rm EPD},$ (11)

where the first term

$\mathbf{A}_{\rm HMW}=C_{x}^{2}e\mathbf{d}\times\mathbf{B}\frac{m_{\rm
pol}}{M}$ (12a) corresponds to the gauge field associated with the HMW phase.
It is crucial that the gauge field strength for polaritons is reduced compared
to the excitons (6) by the factor of $m_{\rm pol}/M$ which is typically of the
order of $10^{-4}$. This is because the HMW gauge field originates from the
motion of excitons. Polaritons in contrast to excitons are light
quasiparticles, which inherit their transport properties mainly from their
photonic part. Therefore, the exciton gauge field has a suppressed impact on
the effective dynamics of polaritons, see Figs. 3a,b.

The second term in (11) describes the gauge field arising due to the dipole-
polarization-induced decoupling of the exciton and the microcavity mode:

$\mathbf{A}_{\rm
EPD}\simeq-\frac{3\Omega_{0}^{2}}{4\Omega_{R}}\lambda_{0}^{2}{\mathbf{B}\times\bm{\mathcal{E}}}\frac{m_{\rm
pol}}{M}.$ (12b)

Note that vectors $\mathbf{A}_{\rm HMW}$ and $\mathbf{A}_{\rm EPD}$ are anti-
collinear. Indeed, the reduction of a coupling strength increases polariton
energy (7), while the $\mathbf{k}$-dependent DC Stark shift of the exciton
level is always negative, Eq. (3). So, the strength of the resulting
artificial gauge field (11) is determined by the competition of these effects.
At the weak magnetic field, $\tau\approx 1$, and zero exciton-photon detuning
$\Delta=0$, Eqs. (12a) and (12b) give the ratio:

$\left|\frac{{A}_{\rm HMW}}{{A}_{\rm
EPD}}\right|=\frac{4E_{b0}}{3\Omega_{0}},$ (13)

where $E_{b0}=2\varkappa^{2}e^{4}\mu\hbar^{-2}$ stands for the 2D exciton
binding energy at zero magnetic field. For GaAs-based microcavities considered
here, the Rabi splitting can be as high as $10$ meV while the 2D exciton
binding energy is $E_{b0}\simeq 19$ meV. In this case the effect associated
with the HMW phase dominates over the mechanism of Rabi splitting
renormalization, see Fig. 3b. The ratio $|{A}_{\rm HMW}/{A}_{\rm EPD}|$ in the
experimentally feasible range of magnetic field $B$ is shown in Fig. 3c. The
imbalance between the gauge potentials gradually grows as the magnetic field
increases.

The numerical solution for the 2D exciton ground state (the solid lines in
Fig. 3b) demonstrates that our variational ansatz underestimates the effect of
the exciton deformation responsible for the Rabi-splitting renormalization.
The actual ratio $\left|{{A}_{\rm HMW}}\left/{{A}_{\rm EPD}}\right.\right|$ is
close to unity at weak magnetic field when both mechanisms responsible for the
gauge field generation must be taken into account, see Fig. 3c.

In QWs the exciton binding energy is significantly smaller than the two-
dimensional exciton binding energy because of the finite widths of the QWs
that allows for a significant spread of electron and hole envelope functions
in normal to the QW plane direction. The exciton radius increases with the
increase of the QW width, which leads to the decrease in the exciton-photon
coupling strength. However, the Rabi-splitting reduction can be compensated by
an increase of the number of QWs embedded in the microcavity. As a result, the
Rabi splitting in GaAs-based microcavities can be comparable with the exciton
binding energy [30]. In this case the Rabi-splitting renormalization dominates
at weak magnetic field. It is demonstrated in Fig. 3c which shows the ratio
$\left|{{A}_{\rm HMW}}\left/{{A}_{\rm EPD}}\right.\right|$ for QWs of
thicknesses $L=5,\,10$ and $20$ nm for which the exciton binding energy is
$13$, $10.6$ and $8$ meV, respectively. Here we use the same value of the Rabi
splitting at zero magnetic and electric fields $\Omega_{0}(B=0)=10$ meV. At
strong magnetic field the gauge potential coming from the HMW effect dominates
in accordance with the predictions of our variational approach.

The solutions for the QW excitons were found numerically as the eigenstates of
the Hamiltonian (1) with the effective Coulomb potential
$V_{c}(r)=-e^{2}\varkappa\iint|v_{e}(z_{e})|^{2}|v_{h}(z_{h})|^{2}\left/\sqrt{(z_{e}-z_{h})^{2}+r^{2}}\right.dz_{e}dz_{h}$.
Here the electron and hole wave functions in the growth direction were taken
as $v_{e,h}(z_{e,h})=\sqrt{2/L}\sin\left(z_{e,h}\pi/L\right)$.

Several strategies for detecting the presence of the polariton gauge field in
a ring geometry can be applied. In the case of a ring-shaped polariton
condensate shown in Fig. 1, the radial electric field creates the angular
gauge potential $\mathbf{A}_{\rm pol}={A}_{\rm pol}\hat{\varphi}$, where
$\hat{\varphi}$ is the polar angle. It imposes chirality on the polariton
superfluid in a sense of the energetically favourable direction for a circular
polariton flow. Therefore, in the case of polariton condensate localised on a
thin ring with the radius $R_{0}$, the energies of the counter-propagating
circular currents with unit vorticities are split by

$\Delta E=\frac{2\hbar A_{\rm pol}}{m_{\rm pol}R_{0}}.$ (14)

Due to the strong imbalance between the exciton and polariton masses discussed
above, the gauge field is weak. So, the energy splitting $\Delta E$ turns out
to be very low for the feasible experimental parameters, see Fig. 3d. In
particular, even if the electric field is as high as several kV$/$cm, the
splitting remains below $10^{-4}$ meV. It is expected to be hidden behind the
spectral broadening of polariton states which is about tens of $\mu$eV [30].
Increasing the splitting above this value requires the electric fields about
$10^{3}$ kV$/$cm at which the exciton dissociates.

Note that due to the interplay between the HMW effect and the Rabi-splitting
renormalization QWs, the effective gauge field changes its sign as the
magnetic field increases, see Fig. 3d. This effect is pronounced in the wide
QWs for which the Rabi splitting exceeds the binding energy.

The alternative approach consists in using of a ring Berry-phase
interferometer [21] designed in such a way that interferometer arms encircle
the radial electric field divergency point. In this case the gauge potential
is responsible for the difference of geometrical phases gained by polaritons
travelling in different arms. The resulted phase difference is

$\phi=\hbar^{-1}\oint\mathbf{A_{\rm pol}}\cdot d\bm{l}=2\pi
R_{0}\hbar^{-1}A_{\rm pol}.$ (15)

While the interference effects are pronounced at $\phi\sim\pi$, the expecting
geometric phase accumulated in the polariton ring interferometer is smaller by
two orders of magnitude even for the path circumference as high as few hundred
micrometers, see Fig. 3d.

##  III Crossover between the hydrogenic and the dipole-polarized states

###  III.1  Synthetic gauge field for polaritons formed by a non-hydrogen-like
exciton

Since the $\mathbf{A}_{\rm pol}(B)$-dependence has a maximum corresponding to
the optimal magnetic field, the only remaining strategy for increasing of the
gauge field strength is to amplify the polarizing electric field
$\bm{\mathcal{E}}$. However the strong electric field substantially modifies
the exciton internal structure, so the variational approach based on the
hydrogen-like exciton wave function (2) fails. In this regime, the effect of
polarizing field can not be treated perturbatively and the expressions (3),
(5) and (6) do not describe the linear-in-$\mathbf{k}$ response of the exciton
energy.

For this reason, in the non-hydrogen exciton regime, the required linear-
in-$\mathbf{k}$ dependencies of the exciton energy and the Rabi splitting can
be extracted from the expansion of $E_{\rm ex}(\mathcal{E})$\- and
$\Omega(\mathcal{E})$-dependencies around the external electric field
$\mathcal{E}$:

$\displaystyle E_{\rm ex}(\mathcal{E}+\mathcal{E}_{b})$ $\displaystyle\simeq$
$\displaystyle E_{\rm ex}(\mathcal{E})+\mathcal{E}_{b}\frac{\partial E_{\rm
ex}}{\partial\mathcal{E}}+\mathcal{O}\left[\mathcal{E}_{b}^{2}\right],$ (16a)
$\displaystyle\xi(\mathcal{E}+\mathcal{E}_{b})$ $\displaystyle\simeq$
$\displaystyle\xi(\mathcal{E})+\mathcal{E}_{b}\frac{\partial\xi}{\partial\mathcal{E}}+\mathcal{O}\left[\mathcal{E}_{b}^{2}\right],$
(16b)

where $\xi=\Omega/\Omega_{0}$ quantifies the coupling strength reduction. This
approach takes advantage from the notion that the artificial gauge field stems
from the modification of the exciton’s energy and internal structure by the
effective electric field $\bm{\mathcal{E}}_{b}$ associated with its motion.
For the exciton states with small momenta situated within the light cone and
at feasible magnetic field, the effective field is typically weak and does not
exceed $0.1$ $\rm kV/cm$. Therefore, one can consider ${\mathcal{E}}_{b}$ as a
small parameter $\mathcal{E}_{b}\ll\mathcal{E}_{0}$ of the expansions (16a)
and (16b).

The factors at the terms linear in effective field in (16a) and (16b) can be
associated with the gauge fields magnitudes. Following the procedure from the
previous section, one obtains:

$\displaystyle{A}_{\rm HMW}$ $\displaystyle=$ $\displaystyle-
C_{x}^{2}\frac{\partial E_{\rm ex}}{\partial{\mathcal{E}}}B\frac{m_{\rm
pol}}{M},$ (17a) $\displaystyle{A}_{\rm EPD}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{0}^{2}}{4\Omega_{R}}\frac{\partial\xi^{2}}{\partial{\mathcal{E}}}B\frac{m_{\rm
pol}}{M},$ (17b)

where the polariton effective mass is $m_{\rm pol}^{-1}=C_{x}^{2}m_{\rm
ex}^{-1}+C_{p}^{2}m_{\rm
ph}^{-1}-\frac{\Omega_{0}^{2}}{2\Omega_{R}}\left(\partial\xi/\partial\mathcal{E}\right)^{2}B^{2}/M^{2}$.
Note that these expressions are consistent with Eqs. (12a) and (12b) once
$E_{\rm ex}$ is given by (5) and $\xi$ is defined by Eqs. (8) and (9).
Therefore, the polariton gauge field magnitude can be connected with the
sensitivity of the exciton energy and the Rabi splitting to the electric field
strength variation. In the next section, we demonstrate that this sensitivity
can be sufficiently increased when the exciton state experiences a crossover
to the strongly decentred solution.

###  III.2 The crossover regime

The conditions for the crossover existence can be determined using the
effective potential representation. The terms accounting for the diamagnetic
shift, attractive electron-hole interaction and the electric field in the
Hamiltonian (1) can be regrouped as
$W(\mathbf{r})=V_{d}(\mathbf{r})+V_{c}(\mathbf{r})$, where

$V_{d}(\mathbf{r})=\frac{\mu\omega_{c}^{2}}{8}(\mathbf{r}+\bm{\rho}_{0})^{2}-\frac{e\bm{\rho}_{0}\bm{\mathcal{E}}}{2}$
(18)

stands for the parabolic diamagnetic potential shifted along the electric
field direction by $\bm{\rho}_{0}=4\mu\bm{\mathcal{E}}/(eB^{2})$. Here
$\omega_{c}=eB/\mu$ is the electron cyclotron frequency. The negative energy
offset $-e\bm{\rho}_{0}\bm{\mathcal{E}}/{2}$ is caused by the presence of
electric field.

In the absence of any external field the electron-hole relative motion is
bounded in the Coulomb potential well $V_{c}(\mathbf{r})$. This is the case of
the HLE whose energy $E_{\rm hl}$ and structure are described by Eqs. (5) and
(2), respectively. When the shift $\rho_{0}$ of the diamagnetic potential
position is small, i.e. at weak electric and strong magnetic fields, the
Coulomb $V_{c}$ and the diamagnetic $V_{d}$ potentials coalesce, see Fig. 2a.
In this case magnetic field is responsible for the squeezing of the exciton
wave function [34], while the electric field has a merely perturbative effect
causing formation of the dipole-polarized exciton.

At strong electric and weak magnetic field so that the shift $\rho_{0}$ of the
diamagnetic well position is large, the total potential $W(\mathbf{r})$ has a
double-well shape, Fig. 2c. In this case each potential well can be
characterized by the independent set of energy levels. For the states
localized close to the diamagnetic well minimum, the presence of the Coulomb
interaction can be accounted perturbatively. These states thus resemble the
states of a free particle in the magnetic field [29] with the characteristic
localization scale equal to the magnetic length $l_{c}=\sqrt{\hbar/(eB)}$ and
with the energy
$E_{d}\approx\hbar\omega_{c}/2-e{\rho}_{0}{\mathcal{E}}/2-\varkappa
e^{2}/\left|{\rho_{0}}\right|$. Note that the negative energy offset in this
case accounts for the interaction of the dipole $e\rho_{0}$ with the electric
field. For large $\rho_{0}$ it typically means that the lowest level of the
diamagnetic potential lies below the hydrogen-like exciton and corresponds to
the ground state. However, because of the strongly decentred relative
electron-hole position, this state does not couple to the photon field. The
polaritons are formed by the hydrogenic excitons. Due to the vanishing overlap
between the states located in the potential minima at
$\rho_{0}\gg\\{a,l_{c}\\}$, the HLE state remains unaffected by the presence
of the second potential well.

Figure 4:  (a) Anticrossing of two least energy exciton states at $B=10$ T
calculated numerically. The false color demonstrates the effect of the Rabi-
splitting renormalization. The line brightness encodes the exciton oscillator
strength variation quantified by the parameter $\xi=\Omega/\Omega_{0}$. The
green dashed line shows the energy of the hydrogen-like exciton (5) at zero
momentum, while the red dashed curve corresponds to the least energy state of
the diamagnetic potential well, $E_{d}$. (b) The anticrossing of the energy
levels for different values of the magnetic field. At the bottom, the effect
of exciton oscillator strength reduction at the crossover is illustrated. The
red facet indicates the position of the crossover estimated by equating
$E_{d}$ and $E_{\rm hl}$. (c) The relative motion wave function
$|\chi(\mathbf{r})|^{2}$ of the lowest energy state in the crossover regime at
four different values of the magnetic field shown in panel (b).

For the intermediated values of the shift parameter $\rho_{0}$, the minima of
the effective potential $W(\mathbf{r})$ get closer to each other so the
tunnelling through the barrier between the potential wells becomes possible,
see Fig. 2b. This leads to the mixing of the HLE and the strongly polarized
states. The crossover manifests itself in the anticrossing in the exciton
energy spectra. In particular, Fig. 4a demonstrates the numerically calculated
energies of the two least energy states as a function of electric field at
$B=10$ T. The splitting of the energy levels occurs at the point at which the
energy of the decentred state $E_{d}$ (red dashed curve) is equal to the HLE
energy $E_{\rm hl}$ (green dashed line). This condition provides a simple
estimate of the crossover position.

The energy of the splitting is governed by the overlap of the relative wave
functions. As the magnetic field decreases, the inter-well separation
$\rho_{0}$ grows gradually and the splitting at the crossover position weakens
as it is shown in Fig. 4b. As a result, at weak magnetic field (below
$B\approx 3$ T for the considered material parameters), the anticrossing
disappears since the potential wells are separated too far for providing an
efficient coupling between them. The effect of separation of the effective
potential wells is illustrated in Fig. 4c where the structure of the lowest
energy states at the crossover position is shown at four values of the
magnetic field.

The crossover from the hydrogenic to the dipole-polarized state is accompanied
by a significant modification of the exciton oscillator strength which governs
the value of the polariton Rabi splitting, see Fig. 4a. This is because the
hydrogenic wave function is pushed out form the Coulomb well as the electric
field increases beyond the region of anticrossing. Eventually it is superposed
by the decentred state which is almost decoupled from the photon field. This
transition is illustrated in Fig. 4b where the variation of ground state
oscillator strength is shown in the parameter plane $(\mathcal{E},B)$. The
regions of the weak and strong oscillator strength of the lowest energy
exciton state are separated by the red line indicating the crossover position.
This boundary is obtained by equating the energies of the dipole-polarized
state $E_{d}$ and the zero-momentum hydrogen-like exciton $E_{\rm hl}$. Note
that the oscillator strength of the upper state gradually grows due to the
mixing with the hydrogenic exciton states as the electric field approaches the
anticrossing region. This important feature of the crossover regime affects
the properties of the polariton states as it will be considered in the next
section.

##  IV The giant artificial gauge field for polaritons in the crossover regime

According to Eqs. (17a) and (17b), the magnitude of the polariton artificial
gauge field is determined by the slope of the $E(\mathcal{E})$\- and
$\Omega(\mathcal{E})$-dependencies. For both the slope increases in the
vicinity of the crossover to the strongly polarized exciton state. Thus one
can expect a significant enhancement of the gauge field strength in this
parameter domain. However, in the vicinity of the anticrossing the optical
response of the medium is determined by the presence of several exciton
resonances. In this case, the polariton energies can be found by the solution
of the eigenvalue problem for the following three-mode Hamiltonian:

$\hat{H}=\left(\begin{array}[]{ccc}E_{\rm
ex}^{(1)}(\mathbf{k})&0&\Omega_{1}/2\\\ 0&E_{\rm
ex}^{(2)}(\mathbf{k})&\Omega_{2}/2\\\ \Omega_{1}/2&\Omega_{2}/2&E_{\rm
ph}(\mathbf{k})\end{array}\right),$ (19)

where $E_{\rm ex}^{(1,2)}$ and $\Omega_{1,2}$ stand for the energies and the
exciton-photon coupling strengths for the two lowest exciton states. Here we
neglected by the presence of the upper exciton states since their oscillator
strengths are typically weak near the crossover. The diagonalization of the
Hamiltonian (19) yields three polariton branches. We focus on the lowest
polariton state whose energy is denoted as $E_{\rm pol}$.

The polariton artificial gauge field can be determined by analogy with the
approach used in Sec .III.1:

$\mathbf{A}_{\rm pol}=-\frac{\partial E_{\rm
pol}}{\partial{\mathcal{E}}}B\frac{m_{\rm pol}}{M},$ (20)

Note that Eq. (20) accounts simultaneously for the gauge field arising from
the Rabi-splitting renormalization and the one associated with the HMW phase.
In the two-mode approximation, $\Omega_{2}\approx 0$ and $|E_{\rm
ex}^{(2)}-E_{\rm ph}|\gg\Omega_{1}$, (20) reduces to the definitions (17a) and
(17b).

Figure 5: The Berry phase difference gained by polaritons travelling in
opposite directions in the ring-shaped interferometer with the radius
$R_{0}=15$ $\mu$m. The exciton is formed in the 10 nm-thick QW. The polariton
Rabi coupling strength for a non-polarized exciton at zero magnetic field is
$\Omega_{0}(B=0)=5$ meV for both panels. The bare cavity frequency $E_{c}$ is
resonant with the lower exciton state energy. The energy splitting between the
clockwise and the anti-clockwise currents of a polariton condensate localised
on the ring is shown in the right axis. Here for simplicity we neglected the
weak dependence of the lower-branch polariton mass $m_{\rm pol}$ on the
electric field.

The enhancement of the polariton gauge field strength in the crossover region
is demonstrated in Fig. 5, which shows the geometric phase $\phi$ gained by
polaritons in the ring Berry-phase interferometer, see (15), as a function of
the polarising electric field. In the HLE regime (below few kV$/$cm for the
considered case) the gauge field grows linearly with the increase of the
electric field strength as it is described in II. In this regime the Berry-
phase difference remains below hundredth of $2\pi$. Typically, it does not not
allow for the observation of the effects connected with the geometric phase.
However, the value of $\phi$ increases sharply as the polarizing electric
field strength approaches the crossover.

For the ring GaAs-based interferometer of $R_{0}=15$ $\mu$m radius, the Berry
phase approaches $2\pi$ before the lowest polariton state becomes strongly
exciton-like and decouples from the external light. In this case the
artificial gauge field can generate the effective magnetic field whose flux
$\Phi=e^{-1}\oint\mathbf{A}_{\rm pol}\cdot d\bm{l}$ is close to the flux
quantum $\Phi_{0}=h/e$. This is the value required for the observation of the
quantum interference effects [14] in the polariton condensate localised on a
ring of the respective radius.

Note that when the Rabi splitting is small in comparison to the exciton
binding energy (see Eq. (13)), the gauge field associated with the HMW phase
dominates over the field arising from the exciton-photon coupling strength
renormalization. This is the case shown in Fig. 5. The mixing of the exciton
states in the crossover region endows the optically active exciton with the
large dipole moment typical for the decentred state. Therefore the HMW phase
(12a) which polaritons accumulate via their excitonic component increases
drastically. The concurrent decrease of the oscillator strength causes an
opposite effect whose strength scales linearly in the Rabi splitting
$\Omega_{0}$ according to (17a) and (17b). So, for small $\Omega_{0}$ and a
smooth crossover, this effect is suppressed.

When the overlap between the exciton states localized in the different
potential minima is small, the effect of mixing is weak as well as the impact
of the HMW effect. In this case the crossover occurs in the narrow region of
$\mathcal{E}$ as it shown in Fig. 4b,c for the case of $B=4$ T. It means that
the exciton oscillator strength is strongly sensitive to the value of the
effective electric field associated with the exciton motion. Therefore, the
impact of the Rabi-coupling renormalization effect is strong. In this case the
magnitude of the resulting gauge field decreases, see the orange curve in Fig.
5 corresponding to $B=4$ T.

Our numerical simulations demonstrate that for the Rabi splittings which are
comparable or larger than the exciton binding energy, the effect of exciton-
photon decoupling can dominate over the HMW effect. However for the GaAs based
structures in which the condition $\Omega_{0}\lesssim E_{b0}$ is typically
valid, the strongest value of the resulted gauge field magnitude can be
obtained at the moderate and weak Rabi splittings, where the gauge field is
mainly governed by the HMW effect. The results shown in Fig. 5 correspond to
$\Omega=5$ meV while for larger Rabi splittings the competition between the
two sources of the gauge field leads to the reductions of the resulted gauge
potential.

##  V Conclusion

The synthesis of the artificial gauge potential for polaritons in crossed
electric and magnetic fields is studied. We demonstrate that for the hydrogen-
like exciton the gauge field associated with the HMW phase typically dominates
over the contribution from the exciton-photon decoupling in the structures for
which the exciton binding energy exceeds the Rabi coupling. However, in the
system where the value of the polariton Rabi splitting becomes comparable to
the binding energy of the unperturbed exciton, the impact of the Rabi
splitting renormalization is significant. In this case the reduction of the
exciton oscillator strength leads to the screening of the HMW gauge field
potential. Accounting for both mechanisms is thus crucial for the efficient
generation of the gauge field for exciton polaritons.

Besides, we propose the approach which substantially enhances the efficiency
of the gauge field synthesis. It implies the use of the partially decentred
excitons which combine the large electric dipole with the strong coupling to
the photon field. In this regime, the gauge field can generate the flux
approaching the flux quantum for the ring-shaped polariton condensate having a
radius of several tens of micrometers [36]. This is the necessary prerequisite
for observation of the quantum interference effects such as the gauge-field
induced formation of the persistent current [14].

Such a strong synthetic gauge field can be used for the fast coherent
manipulation of the polariton condensate excited in the ring geometry. This
particular problem is important for realization of the logic operation with
macroscopic polariton currents [19]. The artificial gauge field causes the
energy splitting $\Delta E$ between the contra-rotating polariton flows. It
approaches tens of $\mu$eV in the crossover regime, see Fig. 5. This is of the
same order with the values of the splitting between the level of spatial
quantization in the annular optical traps [30]. The predicted enhancement of
the gauge field synthesis paves the way to the realization and investigation
of the new topological phases of matter with exciton polaritons.

###### Acknowledgements.

We thank Project No. 041020100118 supported by Westlake University and Program
2018R01002 supported by Leading Innovative and Entrepreneur Team Introduction
Program of Zhejiang. I.C. also acknowledges funding from NSFC (Grant No.
12050410250). The support from RFBR grants 21-52-10005, 20-02-00515, from the
Grant of the President of the Russian Federation for state support of young
Russian scientists No. MK-5318.2021.1.2 and from the state task for VlSU in
the scientific activity project 0635-2020-0013 are acknowledged.

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