# Enhanced Entanglement in Trion-Micropillars under Artificial Magnetic Fields

T.A. Khudaiberganov  Department of Physics and Applied Mathematics, Vladimir
State University named after A. G. and N. G. Stoletovs, 87 Gorkii st., 600000
Vladimir, Russia  I.Yu. Chestnov  ITMO University, St. Petersburg, 197101,
Russia  S.M. Arakelian  Department of Physics and Applied Mathematics,
Vladimir State University named after A. G. and N. G. Stoletovs, 87 Gorkii
st., 600000 Vladimir, Russia

###### Abstract

We investigate quantum phenomena in a system of three coupled microcavities
each supporting regime of strong coupling with exciton resonance. The
possibility of observing polariton blockade in a dimer and triple micropillar
configuration is discussed. We identify the resonance conditions corresponding
to the onset of the photon blockade. The discovered quantum effects allow
using these systems as versatile sources of individual polaritons. Various
manifestations of the quantum blockade can be tuned with the use of the
pumping laser frequency. We demonstrate that the presence of the cascade
coupling between pillars enhances manifestations of sub-Poissonian statistics.
Application of the effective gauge field in the triangle configuration allows
for manifestation of a collective antibunching effect. In particular, we
identify conditions of a multimode blockade – a phenomenon consisting in
blocking of excitation of the state with $n$ particles distributed over
multiple coupled modes. A new effect of quantum entanglement of a non-
Hermitian dimer with a micropillar in an artificial magnetic field has been
discovered, associated with the simultaneous manifestation of antibunching on
the dimer, the micropillar associated with it, and the triple-micropillar.

##  I introduction

Quantum correlations reflect the non-classical nature of quantum states - the
impossibility of representing the whole as a sum (Cartesian product) of its
parts. Many-particle interactions in multimode bosonic systems provide an
opportunity to implement quantum topological materials [1, 2] and multimode
entanglement [3]. Multimode photon blockade can be utilized to execute
universal operations on qubits and qudits in various bosonic modes within
coupled microcavity systems [4]. Qubits in coupled micropillar system are
realized on top of exciton-polariton condensates by utilizing quantum weak
perturbation of the mean-field value [5]. The exploration of the quantum
statistical properties of nonlinear light-matter systems is highly valuable
due to its applications in the field of quantum information [6, 7, 5] and the
development of single-photon sources [8, 9, 10, 11].

The quantum nature of light is reflected in its statistical properties such as
sub-Poisson statistics and in the violation of classical inequalities (of
Cauchy-Schwartz type) on correlation functions. The interaction of a photon
with a nonlinear system prevents interaction with further photons, resulting
in the emission of a single photon. This effect allows systems to suppress the
emission of photons in the form of pairs ($g^{(2)}(0)=0$), and lead to sub-
Poissonian statistic of radiation [12]. This is an effect of photon blockade
[10, 16, 13, 15, 14] that allows the conversion of coherent laser light into a
stream of individual photons in the Fock state [17], when autocorrelations
arise in the photon count. The quantum anticorrelations of the radiation
emitted by the nonlinear systems typically originates from the spectrum
anharmonicity [10, 18]. The anharmonicity leads to the fact that the first
excited state of an oscillator is not resonant with the second excited state.
So, the appearance of two photons at the same frequency is unlikely (but
becomes possible in the presence of the dissipation-induced broadening).
Usually, a pronounced photon blockade effect requires an inter-particle
interaction strength $U$ comparable to the net loss rate $\gamma$. In
contrast, the unconventional photon blockade (UPB) can occur in weakly
nonlinear systems with $U<\gamma$, which is a typical condition in condensed
matter physics [16, 19, 20, 21]. To maintain stationary population of optical
modes, resonators are usually used. A typical system for studying interaction
of light and matter is low-dimensional systems - quantum wells [22] or quantum
dots [23] embedded in a microcavity.

Semiconductor Bragg resonators can concentrate optical radiation in a small
volume on a micrometers scale [24]. These structures also have high
$Q$-factors, allowing for a control of a photon signal unattainable in other
types of resonators. They are manufactured using molecular beam epitaxy, with
flat resonators being the most popular configuration, resembling a quantum
well between two Bragg mirrors [25]. Chemical etching enables control over the
resonator’s shape. This approach achieves high optical radiation concentration
in the in-plane direction. An example is resonators with circular cross-
section, which are microscopic columns - pillar microcavities (MP) [26, 27,
28, 29] with typical for this system weak nonlinearity [30] $U\approx
0.01\gamma$. Such micropillar systems enable creation of highly bright sources
of single photons with $g^{(2)}$ reaching $0.05$ [31, 32]. The sub-Poisson
statistics of radiation from MPs, including those supporting strong coupling
between exciton and photon modes, was predicted theoretically for weak [33,
34] and strong pumping [35].

The physical basis for studying many-body interactions can be arrays of
micropillars interacting through tunneling photons arranged in a graph
configuration and described by the Bose-Hubbard model on such a graph. The
topological properties of such systems were studied in the works [37, 36]. It
is known that cyclic connections in the graph violate gauge invariance to the
artificial magnetic field [38]. In this work, a triangle of micropillars is
considered and its non-trivial statistical properties are investigated.

Figure 1:  (a) A sketch of a polariton quadruplet which can be reduced to the
non-reciprocal triple-micropillar (see appendix C) by adiabatic elimination of
a rapidly decaying mode $b$ (empty MP, i.e. having no active medium). (b)
Sketch of a triangular connection diagram of MP with Hermitian coupling; (c)
Schematic sample design of non-hermitain trion+empty MP. The device consists
of an trion MPs with cavity are excited via laser and modulated cavity of MP
use surface acoustic wave (SAW).

The article has the following structure: in the first section, a general model
of a polariton trion see 1b consisted of three closely placed MPs is
introduced. In the second section, we consider the statistics of the polariton
dimer, and then move on to the section devoted to the polariton triple-
micropillar. Third section obtain condition for multimode quantum blockade for
the polariton triple-micropillar.

##  II Model

Consider the following model of four tunnel-connected [39] micropillars (MPs)
see fig.1(a) which are modeled as four non-linear quantum oscillators in the
presence of a drive field of the following form (in the rotating wave
approximation):

$\begin{array}[]{l}{\hat{H}=\sum_{i=1}^{3}\Delta_{i}\hat{a}^{{\dagger}}_{i}\hat{a}_{i}+\sum_{i\neq
j}^{3}g_{ij}\hat{a}^{{\dagger}}_{i}\hat{a}_{j}}+\\\
+{\sum_{i=1}^{3}F_{i}(\hat{a}_{i}^{{\dagger}}+\hat{a}_{i})+U_{i}\hat{a}^{{\dagger}}_{i}\hat{a}^{{\dagger}}_{i}\hat{a}_{i}\hat{a}_{i}}+\\\
+{(\omega_{b}-\omega_{L})\hat{b}^{{\dagger}}\hat{b}+\kappa(\hat{a}_{1}^{{\dagger}}+\hat{a}_{2}^{{\dagger}})\hat{b}+\kappa\hat{b}^{{\dagger}}(\hat{a}_{1}+\hat{a}_{2})},\end{array}$
(1)

Here $\hat{a}_{i}$($\hat{a}_{i}^{{\dagger}}$) being a polariton annihilation
(creation) operator (Bose) in the $i^{\rm th}$ MP while $U_{i}$ quantifies the
strength of the interparticle interaction inside the $i^{\rm th}$ MP [40];
$\Delta_{i}=\omega_{i}-\omega_{L}$ is the detuning of the $i^{\rm th}$ cavity
from the frequency $\omega_{L}$ of the coherent driving field $F_{i}$ which is
injected into $i^{\rm th}$ MP, $g_{ij}$ is the coupling amplitude between
$i^{\rm th}$ and $j^{\rm th}$ MPs.

For the description of the open quantum system we use a general master
equation for the density matrix:

$\begin{array}[]{l}i\frac{\partial\rho}{\partial
t}=\left[\hat{H},\rho\right]+i\sum_{j=1}^{3}{\gamma_{j}\mathbf{D}[\hat{a}_{j};\rho]}+i\gamma_{b}\mathbf{D}[\hat{b};\rho]\end{array},$
(2)

where we define the dissipator super-operator
$\mathbf{D}[a;\rho]=\hat{a}\rho\hat{a}^{{\dagger}}-\frac{1}{2}\\{\hat{a}^{{\dagger}}\hat{a},\rho\\}$
which describes interaction with the environment with the rates
$\gamma_{1,2,3}$ for the first three MPs and $\gamma_{b}>\gamma_{1,2,3}$ \- is
the dissipation parameter of the fourth MP $"b"$. Here $\\{.,.\\}$ stands for
the anticommutator.

Adiabatically eliminating empty fast decay MP $"b"$ [38], – see appendix A, we
obtain an effective the triple-micropillar system shown in Fig. 1b with the
following non-Hermitain Hamiltonian:

$\begin{array}[]{l}{\hat{H}_{t}=\mathbf{\hat{a}^{{\dagger}}}\mathbf{G}\mathbf{\hat{a}}+F_{1}(\hat{a}_{1}^{{\dagger}}+\hat{a}_{1})+\sum_{i}{U_{i}\hat{a}^{{\dagger}}_{i}\hat{a}^{{\dagger}}_{i}\hat{a}_{i}\hat{a}_{i}}},\end{array}$
(3)

Here
$\mathbf{\hat{a}}=\left(\hat{a}_{1},\hat{a}_{2},\hat{a}_{3}\right)^{\intercal}$,
$\mathbf{\hat{a}^{{\dagger}}}=\left(\hat{a}_{1}^{{\dagger}},\hat{a}_{2}^{{\dagger}},\hat{a}_{3}^{{\dagger}}\right)^{\intercal}$
is a vector of annihilation and creation operators.

The following matrix was introduced in (3):

$\mathbf{G}=\begin{bmatrix}\Delta_{1}&g_{12}&e^{-i\Phi_{1}}g_{+}-\frac{i}{2}e^{-i\Phi_{2}}g_{-}\\\
g_{21}&\Delta_{2}&g_{23}\\\
e^{i\Phi_{1}}g_{+}-\frac{i}{2}e^{i\Phi_{2}}g_{-}&g_{32}&\Delta_{3}\end{bmatrix},$
(4)

where $g_{\pm}$ – are Hermitian and non-Hermitian coupling strength, see
appendix A.

We can rewrite master equation (2) as:

$\begin{array}[]{l}i\frac{\partial\rho}{\partial
t}=\left[\hat{H}_{t}^{+},\rho\right]+\\{\hat{H}_{t}^{-},\rho\\}+i\sum_{j=1}^{3}{\gamma_{j}D[\hat{a}_{j}]}+ig_{d}D[\hat{a}_{1}+\hat{a}_{3}]\end{array},$
(5)

where $\hat{H}_{t}^{\pm}$ are the Hermitian and non-Hermitian parts of the
full Hamiltonian (3), see appendix A.

The concept of representing the field as a collection of interconnected
”balls”, using a connectivity matrix to describe their coupling coefficients,
and incorporating a deformation tensor to account for changes in these
couplings, provides a framework for understanding the dynamic nature of field
interactions. In the context of the general theory of relativity, space-time
is characterized by curvature, a product of the mass and energy distribution
within it. This curvature caused by the mass and energy distribution can lead
to the generation of magnetic fields through the motion of charged particles.

The consideration of three coupled oscillators as the simplest configuration
stems from its gauge invariance with respect to an artificially introduced
magnetic field [38]. This artificial magnetic field is introduced in (4) via
an invariant phases $\Phi_{1},\Phi_{2}$ in the coupling strength, as detailed
in appendix B. The two circuits give two invariant phases, see Appendix C, but
it can be shown, see Appendix C, that these two phases can be reduced to a
single phase $\Phi(\Phi_{1},\Phi_{2})$. This phases can be obtained surface
acoustic wave modulated eignfrequency of MPs [41], see fig.1(c).

The master equation (5) is the most accurate description of quantum processes
of quantum systems separated from their environment. However, if we neglect
quantum jumps $a_{i}\rho a_{j}^{{\dagger}}$ (where $i,j=1,2,3$ and $i\neq j$),
we can rewrite the matrix (4):

$\mathbf{G_{eff}}=\begin{bmatrix}\tilde{\Delta_{1}}&g_{12}&(e^{-i\Phi_{1}}g_{+}-\frac{i}{2}e^{-i\Phi_{2}}g_{-})\\\
g_{21}&\tilde{\Delta_{2}}&g_{23}\\\
(e^{i\Phi_{1}}g_{+}-\frac{i}{2}e^{i\Phi_{2}}g_{-})&g_{32}&\tilde{\Delta_{3}}\end{bmatrix},$
(6)

here we take in account non-Hermicity through changing settings to the complex
manifold
$\Delta_{i}\Rightarrow\tilde{\Delta_{1,3}}=\Delta_{1,3}-i\gamma_{1,3}/2-ig_{-}/2,\tilde{\Delta_{2}}=\Delta_{2}-i\gamma_{2}/2$.

The we rewrite the Hamiltonian (3) using (6):

$\begin{array}[]{l}{\hat{H}_{eff}=\mathbf{\hat{a}^{{\dagger}}}\mathbf{G_{eff}}\mathbf{\hat{a}}+\sum_{i}{F_{i}(\hat{a}_{i}^{{\dagger}}+\hat{a}_{i})}+\sum_{i}{U_{i}\hat{a}^{{\dagger}}_{i}\hat{a}^{{\dagger}}_{i}\hat{a}_{i}\hat{a}_{i}}}\end{array}.$
(7)

The corresponding wave function of the system can be written as a ket vector
in the Fock-space basis:

$|\psi\rangle=\sum_{i+j+k<=M}c_{ijk}(t)|ijk\rangle,$ (8)

where $M$ is a size of the truncated Hilbert space which corresponds to the
maximum number of photons in the system. This function is governed by the
Schrödinger equation

$i\frac{d}{dt}|\psi\rangle=H_{eff}|\psi\rangle,$ (9)

which obeys the matrix elements to satisfy the following equations

$\begin{array}[]{l}{\frac{idc_{ijk}}{dt}=\sum_{l}(\Delta_{l}n_{l}+U_{l}n_{l}(n_{l}-1))c_{ijk}+}\\\
{+F_{1}\sqrt{n_{1}+1}c_{(i+1)jk}+F_{2}\sqrt{n_{1}}c_{(i-1)jk}+}\\\
{+g_{12}\sqrt{n_{1}(n_{2}+1)}c_{(i-1)(j+1)k}+g_{21}\sqrt{n_{2}(n_{1}+1)}c_{(i+1)(j-1)k}+}\\\
{+g_{13}\sqrt{n_{1}(n_{3}+1)}c_{(i-1)j(k+1)}+g_{31}\sqrt{n_{2}(n_{1}+1)}c_{(i+1)j(k-1)}+}\\\
{+g_{23}\sqrt{n_{2}(n_{3}+1)}c_{i(j-1)(k+1)}+g_{32}\sqrt{n_{3}(n_{2}+1)}c_{i(j+1)(k-1)}}.\end{array}$
(10)

Here $c_{ijk}$ \- the amplitude of the probability of the system being in one
of the states $|i,j,k>$.

We can rewrite Eq. (10) as a sum of $n-$particles states [42]:

$|\psi\rangle=\sum_{n<=M}|\psi^{(n)}\rangle,$ (11)

here we introduce state with $n$-particle quantum state:

$|\psi^{(n)}\rangle=\sum_{i+j+k=n}c_{ijk}|ijk\rangle,$ (12)

where $i+j+k=n$ is total number of particles in the state $|ijk\rangle$, which
denotes a state with $i$ particles in the first mode, $j$ particles in the
second mode and $k$ particles in the third mode.

Rewrite the effective Hamiltonian (7) as a block-matrix consistent from
$n$-particle Hamiltonians as (see appendix C)

$H_{eff}\approx\mathbf{\hat{a}^{{\dagger}}}diag(G_{eff},G^{(2)},...,G^{(n)})\mathbf{\hat{a}}+\sum_{i}{F_{i}\hat{a}_{i}^{{\dagger}}}$
(13)

the notation $diag(a_{1},a_{2},...)$ represents a diagonal matrix using the
provided elements $a_{1},a_{2},...$ and so on. The first part of the
Hamiltonian is a block one and consists of Hamiltonians acting on $n$-particle
states with matrices $G^{(n)}$ with rank n(1+n)/2, these can be represented as
tensor networks of a certain rank. The second part of the Hamiltonian takes
the system from a lower rank tensor network to a higher one.

The non-classical effects can be detected by measuring the second (or higher)
order correlation or cross-correlation functions. The second-order correlation
function for the $i^{\rm th}$ microcavity is defined as:

$\begin{array}[]{l}{{g^{\left(2\right)}_{i}\left(0\right)=\frac{\left\langle\hat{a}^{{\dagger}2}_{i}\hat{a}^{2}_{i}\right\rangle}{\left\langle\hat{a}^{{\dagger}}_{i}\hat{a}_{i}\right\rangle^{2}}}}\end{array}.$
(14)

##  III Methods

The task was to find the optimal parameters when the blockade effect occurs.
To do this, the stationary Schrödinger equation with the effective Hamiltonian
$\eqref{Eq15_}$ was solved, see Appendix C. Next, the master equation
$\eqref{Eq5_}$ was solved numerically using the QuTiP package version 4.7.1, a
Quantum Toolbox in Python [43] with the found parameters. Codes are available
upon request by email<EMAIL_ADDRESS>
##  IV Results

###  IV.1 Dimer

Figure 2: (a) The minimum value of the second order correlation function for
the first micropillar (which are pump) $g^{(2)}_{2}$ for different values of
the coupling amplitude: blue squared – two coupled micropillars with the
coupling constant $g=5\gamma$ and the green dots with the coupling amplitude
$g=10\gamma$ and red diamonds denote one micropillar. Other parameters are:
[44]:
$U_{1}=U_{2}=U_{3}=U_{LP}=0.01\gamma_{1},\gamma_{1}=\gamma_{2}=\gamma_{3}=\gamma,F_{1}=\gamma_{1},F_{2}=0,F_{3}=0,\hbar\gamma=0.0274$
meV. (b) The $g^{2}_{1}$ \- correlation function map for the first
micropillar. The blue disk (with coordinates
$\Delta_{1}/\gamma=0.3,\Delta_{2}/\gamma=0.1$) on the map corresponds to the
minimum value of the function $g^{(2)}_{1}$ for the optimal coupling constant
$g=6.2\gamma$ obtained using (15). (c-e) The $g^{(2)}$-correlation function
map for the first, second and third micropillars for Hermitian triple
micropillar as functions of the coupling amplitudes between the second and the
third micropillars for parameters (c-e panels)
$\Delta_{1}/\gamma=0.4,\Delta_{2}/\gamma=0.3,\Delta_{3}/\gamma=0.4,F_{1}=0.1\gamma,F_{2,3}=0,g_{12}=g_{21}=5\gamma$.

With a dimer, sub-Poissonian statistics can be observed at small
nonlinearities via unconventional scenario while a single pillar exhibits only
weakly non-classical behavior, see fig.2a in this regime. The optimal
conditions for the minimum $g^{(2)}_{1}$ have been found; for the quantum
blockade regime, see fig. 2b.

For a dimer the conditions of quantum blockade by coupling [21] (for resonance
condition):

$g_{opt}/\gamma\approx\sqrt{\frac{2}{27}}\sqrt{\sqrt{3+4\left(\frac{U}{\gamma}\right)^{2}}\left(\frac{U}{\gamma}+3\frac{\gamma}{U}\right)-2\left(\frac{U}{\gamma}\right)^{2}}$
(15)

When the third micropillar is connected to the dimer, the antibunching effect
observed on the first micropillar remains relatively constant and shows a weak
correlation with the connection strength $g_{13}$ between the first and third
micropillars (Fig. 2c). However, when the connection strength $g_{23}$ between
the second and third micropillars is increased, the antibunching effect
vanishes, as illustrated in Fig.2c. In such a scenario, the statistics for the
second micropillar, with a strong coupling represented by $g_{23}$, exhibit
non-classical behavior with the onset of the antibunching effect shown in Fig.
2d. The antibunching effect transitions to the second micropillar as the
connection with the third micropillar is enhanced, while there is minimal
impact on the statistical behavior of the third micropillar, as depicted in
Fig. 2e. By introducing the third micropillar, additional pathways for quantum
blockade interference are established, enabling quantum destructive
interference with the blockade of higher states - a phenomenon that is
exclusive to the micropillar under excitation. Notably, if both coupling
parameters $g_{13}$ and $g_{23}$ are simultaneously increased, the
antibunching effect disappears.

###  IV.2 Trion

Analysis of the quantum statistics of the Hermitian triple micropillar
revealed the disappearance of the effect of quantum blockade. In $PT$-symmetry
non-Hermitian system, the quantum blocked effects within use non-reciprocal
coupling were discovered [45]. For obtained non-reciprocal coupling, we added
a fourth fast-decaying micropillar $\gamma_{b}>\gamma_{i}$ without a quantum
dot coupled with fist and third MPs with a coupling coefficient $~{}\kappa$,
see Fig.1a. Then this MP was adiabatically excluded, see Appendix A, and an
effective nonlinear three-mode system was obtained - a triple micropillar with
non-Hermitain coupling between the first and the third MPs with Hamiltonian
(3).

When we do not distinguish from which MP the photon came (assuming equal
radiation frequencies), we can consider three or two MPs as a single source of
photons, wherein the optical mode’s radiation pattern should remain consistent
regardless of polarization, so as to prevent any ’which-path’ information are
obtained from angle radiation [46].

If someone blocks the signal from any micropillar, the radiation statistics of
a collective emission of the triple micropillar changes significantly. The
output mixed second order for the first and the third MPs (non-reciprocal
polariton dimer) is defined as:

$\begin{array}[]{l}{g_{1+3}^{\left(2\right)}(0)=\frac{\left\langle\hat{a}_{1+3}^{{\dagger}2}\hat{a}_{1+3}^{2}\right\rangle}{\left\langle\hat{a}_{1+3}^{{\dagger}}\hat{a}_{1+3}\right\rangle^{2}}\approx}\\\
{\approx
2\mathcal{N}\frac{|c_{200}+c_{002}+\sqrt{2}c_{101}|^{2}}{|c_{100}+c_{001}|^{4}}}\end{array}$
(16)

This way, the output mixed second order for the signal from the whole triple-
micropillar structure is:

$\begin{array}[]{l}{g_{1+2+3}^{\left(2\right)}(0)=\frac{\left\langle\hat{a}_{1+2+3}^{{\dagger}2}\hat{a}_{1+2+3}^{2}\right\rangle}{\left\langle\hat{a}_{1+2+3}^{{\dagger}}\hat{a}_{1+2+3}\right\rangle^{2}}\approx}\\\
{\approx
2\mathcal{N}\frac{|c_{200}+\sqrt{2}c_{110}+c_{020}+\sqrt{2}c_{101}+c_{002}+\sqrt{2}c_{011}|^{2}}{|c_{100}+c_{010}+c_{001}|^{4}}}\end{array}$
(17)

where $\mathcal{N}=\sum_{n=0}^{M}{\sum_{i+j+k=n}{|c_{ijk}^{(n)}|^{2}}}$ \- is
normalization constant. Also we introduced the following notation
$a_{\sum{i}}=\sum{a_{i}}$ is the collective output mode subspace of the
multimode space state [47]. Also here the approximation of the truncated wave
function (11) on two particles is used.

We find the following optimal parameters from steady state solution (9) with
effective Hamiltonian (13):
$\gamma=0.1ps^{-1},\gamma_{1}=0.128\gamma,\gamma_{2}=\gamma,\gamma_{3}=0.28\gamma,U_{1}=U_{2}=U_{3}=0.01\gamma,F_{1}=0.1\gamma,F_{2}=0,F_{3}=0,g_{12}=g_{21}=g_{23}=g_{32}=g=0.39\gamma,g_{+}=4.86\gamma,g_{-}=4.97\gamma$
and $\Phi_{1}=7\pi/20,\Phi_{2}=7\pi/10,\Delta_{1,2,3}=\Delta=0$.

Radiation from whole triple micropillar is almost coherent
$g_{1+2+3}^{(2)}(0)=0.969$. However, if we block the light from the second
micropillar which is connected to the non-reciprocal polariton dimer, then the
radiation from the triple micropillar becomes more ordered than the coherent
radiation. An antibunching effect will occur with minimum value
$g_{1+3}^{(2)}(0)=0.058$. Figure 3a,b demonstrate this correlation functions
in the dependency of the detuning $\Delta$. Note that for individual MPs
$g_{1,2}^{(2)}(0)\approx 1$, $g_{3}^{(2)}(0)\approx 0.8$.

Let’s adjust the pump frequency by $\Delta_{1,2,3}=1.8\gamma$ and find the
phases values $\Phi_{1}=9\pi/20,\Phi_{2}=11\pi/10$. Then we have
${g^{(2)}_{1+2+3}}_{min}=0.057$,${g^{(2)}_{1+3}}_{min}=0.202$. How the
correlation functions change for this case are shown in the figures 3c,d.
Let’s adjust the pump frequency by $\Delta_{1,2,3}=1.8\gamma$ and find the
phase values $\Phi_{1}=9\pi/20,\Phi_{2}=11\pi/10$. Then we have antibunching
for full triple micropillar ${g^{(2)}_{1+2+3}}_{min}=0.057$ and
${g^{(2)}_{1+3}}_{min}=$0.202. Changes in the correlation functions when
varying the detuning from the optimal value are shown in Figures 3c, d.

Figure 3: (a,c) Correlation characteristics for individual MPs, (b,d) mixed
correlation function $\eqref{Eq16a_}$,$\eqref{Eq16b_}$ of $\Delta$. Parameters
are:
$\gamma=0.1ps^{-1},\gamma_{1}=0.128\gamma,\gamma_{2}=\gamma,\gamma_{3}=0.28\gamma,U_{1}=U_{2}=U_{3}=0.01\gamma,F_{1}=0.1\gamma,F_{2}=0,F_{3}=0,g_{12}=g_{21}=g_{23}=g_{32}=g=0.39\gamma,g_{+}=4.86\gamma,g_{-}=4.97\gamma$
(a,b) $\Phi_{1}=7\pi/20,\Phi_{2}=7\pi/10$ and (c,d)
$\Phi_{1}=9\pi/20,\Phi_{2}=11\pi/10$. These results were obtained from the
solution of the master equation (5) using the QuTiP package version 4.7.1, a
Quantum Toolbox in Python [43].

The conditions for multimode blockade are achieved if the sum of the
amplitudes of the corresponding probabilities is equal to zero, otherwise if
the amplitudes are represented as vectors, then the blockade condition is
reduced to the condition of the closedness of the polygon of state vectors,
see Fig. 4c.

Figure 4: (a). The heat map of output mixed second order for the signal from
the whole triple-micropillar structure in dependency artificial magnetic
fields $\phi_{1,2}$; (b) first eigenvalue of matrix $G^{(2)}$; (c) Graphic
representation of the summation amplitude of probability for blockade case
(blue cross on panel (a)) and (d) not collective blockade $\Phi=0$; (e) The
second order correlation function for each MP; (f) The output mixed second
order correlation function from whole structure and non-hermitian coupled MP
depending on the magnetic field $\Phi(\phi_{1},\phi_{2})=k\phi_{2}+b$ along
the dashed line in panel (a), here $k=0.69,b=0.67$; (g) Entanglement
criterion. Parameters are:
$g_{+}=g_{-}=5\gamma,g=0.4\gamma,\Delta=-4.76\gamma$.

Quantum blockade of the sum mode of a triple micropillar occurs in the
presence of an artificial magnetic field, while in the non-Hermitian triple
micropillar there are two phases (see Appendix C), the effects of bunching and
antibunching appear along the dashed line shown $\Phi(\phi_{1},\phi_{2}),$see
fig. 4a, it passes at an angle to the spectrum of the matrix $G^{(2)}$, see
fig. 4b. In this magnetic field, a region with a giant bunching effect
appears, on both sides of which there are two regions with a strong
antibunching effect with $g^{(2)}_{1+2+3}=0.006$.

For the polariton blockade effect to occur, it is necessary that the
probability amplitudes of two-particle (with certain multipliers) states add
up to zero, i.e. If you imagine this graphically, they were closed, see fig.
4c. In particular, if, with these parameters, the phase is removed from the
Hermitian coupling, then there will be no short circuit - see the green arrow
on fig. 4d.

When blockade is simultaneously present on the triple-micropillar fig. 4f on a
non-Hermitian dimer fig. 4f and antibunching is observed on the second
micropillar fig. 4e (attached), then quantum entanglement between the dimer
and the micropillar is observed in the system, while there is no three-mode
quantum entanglement throughout the entire triple-micropillar, see fig. 4g.

We can use Hillery-Zubairi two mode entanglement criteria [47, 48] for one
mode ”2” and sum mode ”1+3”:

$E^{(1+3,2)}=\frac{\left\langle(\hat{a}_{1}^{{\dagger}}+\hat{a}_{3}^{{\dagger}})(\hat{a}_{1}+\hat{a}_{3})\hat{a}_{2}^{{\dagger}}\hat{a}_{2}\right\rangle}{|\left\langle\hat{a}_{2}^{{\dagger}}(\hat{a}_{1}+\hat{a}_{3})\right\rangle|^{2}}\stackrel{{\scriptstyle\text{ent}}}{{<}}1,$
(18)

and for three mode [47, 48]:

$E^{(1,2,3)}=\frac{\langle\hat{a}_{1}^{{\dagger}}\hat{a}_{1}\hat{a}_{2}^{{\dagger}}\hat{a}_{2}\hat{a}_{3}^{{\dagger}}\hat{a}_{3}\rangle}{|\langle
a_{1}^{{\dagger}}a_{2}a_{3}\rangle|^{2}}\stackrel{{\scriptstyle\text{ent}}}{{<}}1$
(19)

Let us present simple considerations of the connection between quantum
blockade of collective modes and quantum entanglement, with simultaneous
blockade on the entire trion and on the non-Hermitian dimer, we obtain from
the numerator $\eqref{Eq16b_}$ that $c_{020}+\sqrt{2}(c_{110}+c_{011})\approx
0$, and since $E^{(1+3,2)}\sim|c_{110}+c_{011}|^{2}$, then if $c_{020}\approx
0$, therefore $c_{110}+c_{011}\approx 0$ and numerator of $E^{(1+3,2)}\approx
0$.

##  V Conclusion

The study demonstrates that quantum blockade occurs of the sum mode of a
triple micropillar in the presence of an artificial magnetic field, and the
effects of bunching and antibunching are observed along a specific magnetic
line. For the polariton blockade effect to manifest, it is necessary for the
probability amplitudes of two-particle states to sum to zero, and quantum
entanglement is observed between the dimer and the micropillar while being
absent in the three-mode entanglement not have. It is known that for
indistinguishable bosons, the probability of occupying a single state
increases with the population. Interparticle interactions can prevent such
population and lead to the effect of quantum unconventional blockade. Unlike
the Pauli exclusion principle (which are fundamental law), such blockade
occurs only under certain conditions of system parameters. The study
identified conditions under which multimode blockade is observed in several
micropillars connected in a triangular configuration.

##  VI Acknowledgment

The research was carried out within the state assignment in the field of
scientific activity of the Ministry of Science and Higher Education of the
Russian Federation (theme FZUN-2024-0018, state assignment of the VlSU).

##  VII Appendix

###  VII.1 The master equation for dissipative coupled system

Let’s consider the circuit from the work [38] \- two coupled harmonic
oscillators (or odes) with frequencies $\omega_{1}-i\gamma_{1}/2$ and
$\omega_{2}-i\gamma_{2}/2$, which are connected by a Hermitian coupling
$g_{3}$ and each of which is associated with the rapidly decaying mode $b$
with the coupling coefficient $\kappa$ with the dissipation parameter
$\gamma_{b}$, which are $\gamma_{b}\gg\gamma_{1,3}$, see fig5.

We can write the follow system equation for complex amplitude:

$\begin{array}[]{l}{\frac{da_{1}}{dt}=-i(\omega_{1}-i\gamma_{1}/2)a_{1}-i\kappa
b-ig_{3}a_{3},}\\\
{\frac{db}{dt}=-i(\omega_{b}-\gamma_{b})b-i\kappa(a_{1}+a_{3})}\\\
{\frac{da_{3}}{dt}=-i(\omega_{3}-i\gamma_{3}/2)a_{3}-i\kappa
b-ig_{3}a_{1}.}\end{array}$ (20)

Adiabatic eliminate the mode $"b"$, as $\frac{db}{dt}=0$, we obtained the
equation for two modes:

$\begin{array}[]{l}{i\frac{da_{1}}{dt}=((\tilde{\omega}_{1}-i\gamma_{1}/2)-\Gamma)a_{1}-(g_{3}-i\Gamma)a_{3},}\\\
{i\frac{da_{3}}{dt}=((\tilde{\omega}_{3}-i\gamma_{3}/2)-\Gamma)a_{3}-(g_{3}-i\Gamma)a_{1}},\end{array}$
(21)

here $\Gamma=|\kappa|^{2}/\gamma_{b}$. The dissipator of ”$b$”-mode ae the
follow form

$D_{b}[b]\approx|\kappa|^{2}D[a_{1}+a_{3}].$ (22)

in our notation $g_{d}=|\kappa|^{2}$, $\gamma_{1,3}=\gamma_{LP1,3}-i\Gamma$,
$g_{+}=g_{3},g_{-}=\Gamma$.

Figure 5: (a) The scheme of trion (three connected micropillar) with a rapidly
decaying polariton ”b”, we transformation this trion to to the non-hermitian
dimer (b) with dissipative coupling [38].

###  VII.2 Invariant phase $\Phi$

The Hamiltonian of the cyclic triangle circuit with (see fig. 5c) with the
natural frequency of harmonic oscillators $\omega_{i}$ modulated (example,
acoustic modulated) at frequency $\omega_{i}$.

$\begin{array}[]{l}{H=\sum_{i=1}^{3}{\left(\omega_{i}+A_{i}\cos\left(w_{i}t+\varphi_{i}\right)\right){\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{i}}+}\\\
{+\sum_{i\neq
j}^{3}{g_{ij}{\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{j}+h.c.}+\sum_{i=1}^{3}{\frac{U}{2}{\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{i}{\widehat{a}}_{i}}}\end{array}$
(23)

Let’s transform Hamiltonian (23) using the unitary matrix

$U(t)=\exp\left\\{\sum_{i=1}^{N}{\left(\omega_{i}t+A_{i}\sin\left(w_{i}t+\varphi_{i}\right)\right){\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{i}}\right\\}$
(24)

Then using the transformation

$H^{{}^{\prime}}=UHU^{{\dagger}}+i\frac{dU}{dt}U^{{\dagger}}$ (25)

We get the following Hamiltonian

$H^{{}^{\prime}}=\sum_{i\neq
j}^{N}{g_{ij}e^{i{\overline{\mathrm{\Delta}}}_{ij}(t)}{\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{j}+h.c.}+\sum_{i=1}^{N}{\frac{U}{2}{\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{i}{\widehat{a}}_{i}}$
(26)

where
${\overline{\mathrm{\Delta}}}_{ij}(t)=\left(\omega_{i}-\omega_{j}\right)t+A_{i}\sin\left(w_{i}t+\varphi_{i}\right)-A_{j}\sin\left(w_{j}t+\varphi_{j}\right)=\mathrm{\Delta}_{ij}t+A\sin\left({\mathrm{\Delta}w}_{ij}t+{\mathrm{\Delta}\varphi}_{ij}\right)$;
here
$\mathrm{\Delta}_{ij}=\left(\omega_{i}-\omega_{j}\right),{\mathrm{\Delta}w}_{ij}=\frac{\left(w_{i}-w_{j}\right)}{2},{\mathrm{\Delta}\varphi}_{ij}=\frac{\left(\varphi_{i}-\varphi_{j}\right)}{2},$
and
$\cos\left(\frac{\left(w_{i}+w_{j}\right)}{2}t+\frac{\left(\varphi_{i}+\varphi_{j}\right)}{2}\right)\approx\frac{1}{2}$;
we consider it a slow variable, we also assumed that $A_{i}=A_{j}=A$.

The nonlinear part of the Hamiltonian (23) does not change in (26), since it
commutes with the unitary operator (26), we can expand the exponential in a
series in Bessel functions

$e^{i{\overline{\mathrm{\Delta}}}_{ij}(t)}=e^{i\mathrm{\Delta}_{ij}t}\sum_{n=-\infty}^{\infty}{J_{n}(-A)e^{-in{\mathrm{\Delta}w}_{ij}t}e^{-in{\mathrm{\Delta}\varphi}_{ij}}}.$
(27)

Take only the first term in the expression 27 and ${\mathrm{\Delta}w}_{ij}=$
$\mathrm{\Delta}_{ij}$, then the linear part of Hamiltonian 23 takes the form

${H^{{}^{\prime}}}_{l}=\sum_{i\neq
j}^{3}{{g^{{}^{\prime}}}_{ij}e^{-i{\mathrm{\Delta}\varphi}_{ij}}{\widehat{a}}_{i}^{{\dagger}}{\widehat{a}}_{j}+h.c.}$
(28)

It can be easily shown that such a Hamiltonian is invariant to the gauge
transformation $a\rightarrow ae^{i\phi}$ only if it is not closed in a cycle.

###  VII.3 Schrodinger equation for trion

Following [11] and neglecting quantum jumps, we can write the equation
governing for the time dependency n-particles state $|\psi^{(n)}(t)\rangle$
for the non-Hermitian Hamiltonian:

$i\frac{d}{dt}|\psi^{(n)}(t)\rangle\approx
H_{0}|\psi^{(n)}(t)\rangle+H_{+}|\psi^{(n-1)}(t)\rangle,$ (29)

Here we introduce the Hamiltonian that does not change the number of particles
$H_{0}=\hat{a}^{+}\mathbf{G_{eff}}\hat{a}+\sum_{i}{U_{i}\hat{a}^{+}_{i}\hat{a}^{+}_{i}\hat{a}_{i}\hat{a}_{i}}$
and increases the number of particles by one
$H_{+}=\sum_{i}{F_{i}\hat{a}_{i}^{+}}$ and neglecting the annihilation part of
the pump Hamiltonian for weak pump limit due to sub-leading ordered
$c^{(n+1)}\ll c^{(n)}$ [11].

We can obtained the recursive system equations for the $n$-state amplitude
vector
$c^{(n)}=(c_{n,0,0},c_{n-1,1,0},c_{n-1,0,1},...,c_{n-i,i,0},c_{n-i,i-1,1},...,c_{n-i,i-j,j},\\\
...,c_{n-i,0,i},...,c_{0,0,n})^{T}$ (which are contain $T_{n+1}$ elements,
where $T_{n}=\frac{(1+n)n}{2}$ triangular numbers) by projection of
Schrödinger equation (29) on bra-vector $\langle\psi^{(n)}(t)|$,

$c^{(n)}=(-1)^{n}\prod\limits_{i=n}^{1}G^{{(i)}^{-1}}f^{(i)}$ (30)

Where we introduce the matrix $f^{(n)}$ size $T_{n}$ by $T_{n+1}$ and the
follow matrix elements

$\begin{array}[]{l}{f^{(n)}_{s,s^{\prime}}=F_{1}\sqrt{n-m_{1}}\delta_{s_{1},s^{\prime}}+F_{2}\sqrt{m_{1}-m_{2}}\delta_{s_{2},s^{\prime}}+}\\\
{+F_{3}\sqrt{m_{2}}\delta_{s_{3},s^{\prime}}}\end{array}$ (31)

And the matrix $G^{(n)}$ size $T_{n+1}$ by $T_{n+1}$ with the follow matrix
elements

$\begin{array}[]{l}{G^{(n)}_{s,s^{\prime}}=\sum_{i=1,2,3}{(\tilde{\Delta_{i}}n_{i}+U_{i}n_{i}(n_{i}-1))\delta_{s,s^{\prime}}}}\\\
{+g_{i,j}\sqrt{n_{i}(n_{j}-1)}\delta_{s_{ij},s^{\prime}}},\end{array}$ (32)

where

$\begin{array}[]{l}{s_{12}=1+k+\frac{1}{2}(1-n_{1}+n)(2-n_{1}+n),}\\\
{s_{21}=1+k+\frac{1}{2}(n_{1}-n)(1+n_{1}-n),}\\\
{s_{23}=2+k+\frac{1}{2}(n-n_{1})(1-n_{1}+n),}\\\
{s_{32}=k+\frac{1}{2}(n-n_{1})(1-n_{1}+n),}\\\
{s_{31}=k+\frac{1}{2}(n_{1}-n)(1+n_{1}-n),}\\\
{s_{13}=2+k+\frac{1}{2}(1-n_{1}+n)(2-n_{1}+n),}\\\
{s_{1}=1+k+\frac{1}{2}(n-n_{1})(1-n_{1}+n),}\\\
{s_{2}=1+k+\frac{1}{2}(n-1-n_{1})(n-n_{1}),}\\\
{s_{3}=k+\frac{1}{2}(n-1-n_{1})(n-n_{1}),}\\\ {n_{1}=n-m_{1},}\\\
{n_{2}=m_{1}-m_{2},}\\\ {n_{3}=m_{2}.}\end{array}$ (33)

here
$m_{1}=[\frac{1}{2}(-1+\sqrt{8s-7})],m_{2}=[\frac{1}{2}(-1+\sqrt{8s-7-4m_{1}(m_{1}+1)})]$,
where $[.]$ stands for the integer part of number in notation of Iverson.

Example,

$f^{(2)}=\begin{bmatrix}\sqrt{2}F_{1}&0&0\\\ F_{2}&F_{1}&0\\\ F_{3}&0&F_{1}\\\
0&\sqrt{2}F_{2}&0\\\ 0&F_{3}&F_{2}\\\ 0&0&\sqrt{2}F_{3}\end{bmatrix},$ (34)

$G^{(2)}=\begin{bmatrix}2U+2\Delta_{1}&\sqrt{2}g_{12}&\sqrt{2}g_{13}&0&0&0\\\
\sqrt{2}g_{21}&\Delta_{1}+\Delta_{2}&g_{23}&\sqrt{2}g_{12}&g_{13}&0\\\
\sqrt{2}g_{31}&g_{32}&\Delta_{1}+\Delta_{3}&0&g_{12}&\sqrt{2}g_{13}\\\
0&\sqrt{2}g_{21}&0&2U+2\Delta_{2}&\sqrt{2}g_{23}&0\\\
0&g_{31}&g_{21}&\sqrt{2}g_{32}&\Delta_{2}+\Delta_{3}&0\\\
0&0&\sqrt{2}g_{31}&0&\sqrt{2}g_{32}&2U+2\Delta_{3}\end{bmatrix}$ (35)

###  VII.4 Number of invariant phases on the graph

Imagine the Hamiltonian cycle

$H=\sum_{|j-i|=1}^{N}g_{ij}e^{-i\phi_{ij}}\hat{a}_{i}^{\dagger}\hat{a}_{j}+h.c.$
(36)

where $\phi_{ij}$ \- is a phase coupling.

The invariant gauge $a_{i}\rightarrow a_{i}e^{is_{i}}$ cannot eliminate the
phase coupling if the determinant of the graph’s adjacency matrix is zero. The
determinant of the adjacency matrix of a closed graph is zero.

By calibration, all phases can be grouped into one phase
$\Phi=\sum_{|j-i|=1}^{N}\phi_{ij}$ via the follow procedure:

$s_{1}=-\phi_{N},s_{2}=s_{1}-\phi_{1},s_{3}=s_{2}-\phi_{2},...s_{N-1}=s_{N-2}-\phi_{N-2},s_{N}=0.$
(37)

where $\phi_{ii-1}\equiv\phi_{i}$ is independent phase.

Let’s connect two arbitrary non-adjacent nodes $1$ and $j$ in a cyclic graph -
we get two cyclic graphs. The sum of the phases on each of them may differ,
for first subset 1,…,j with $\Phi_{1}=\sum_{i=1}^{j-1}\phi_{i}+\phi_{N+1}$ and
second subset j+1,…,N with $\Phi_{2}=\sum_{i=j-1}^{N}\phi_{i}+\phi_{N+1}$. It
can be shown that it is possible to calibrate the phases to different edges
for each graph, but it is impossible to calibrate to a common edge. Let’s
carry out the $\eqref{Eq19_}$ procedure for the first graph and obtain for
common vertices (for the second graph): $s_{1}=-\phi_{N+1},s_{j}=0$. The same
result will be obtained if we apply $\eqref{C9_}$ to the second graph, see
fig. 2a. Therefore, the $\eqref{C9_}$ procedure works for calibrations of this
type and no contradictions arise.

If we assume a phase on a common edge, see fig. 2b, we get the follow systems
equations:

$\begin{array}[]{l}{s_{2}-s_{1}+\phi_{1}=0}\\\ {s_{3}-s_{2}+\phi_{2}=0}\\\
{...}\\\ {s_{j}-s_{j-1}+\phi_{j}=0}\\\
{s_{1}-s_{j}+\phi_{N+1}=\Phi}\end{array}$ (38)

and

$\begin{array}[]{l}{s_{N}-s_{1}+\phi_{N}=0}\\\ {s_{N-1}-s_{N}+\phi_{N-1}=0}\\\
{...}\\\ {s_{j}-s_{j+1}+\phi_{j+1}=0}\\\
{s_{1}-s_{j}+\phi_{N+1}=\Phi},\end{array}$ (39)

adding equations in each the systems $\eqref{C10_}$ and $\eqref{C11_}$ and
equating them, obtain:

$\sum_{i=1}^{j-1}\phi_{i}=\sum_{i=j-1}^{N}\phi_{i},$ (40)

which contradicts the independence of the phases.

Figure 6: graph scheme (a) different invariant phases and (b) common phase.

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