PHYSICAL REVIEW B 97, 041302(R) (2018)
Rapid Communications
Quantum exciton-polariton networks through inverse four-wave mixing
T. C. H. Liew1 and Y. G. Rubo2,3
1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University,
21 Nanyang Link, Singapore 637371
2Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34051, Republic of Korea
3Instituto de Energías Renovables, Universidad Nacional Autónoma de México, Temixco, Morelos 62580, Mexico
(Received 24 May 2017; published 10 January 2018)
We demonstrate the potential of quantum operation using lattices of exciton-polaritons in patterned semicon-
ductor microcavities. By introducing an inverse four-wave mixing scheme acting on localized modes, we show
that it is possible to develop nonclassical correlations between individual condensates. This allows a concept of
quantum exciton-polariton networks, characterized by the appearance of multimode entanglement even in the
presence of realistic levels of dissipation.
DOI: 10.1103/PhysRevB.97.041302
Introduction. Recently, there has been significant attention
devoted to the study of exciton-polaritons in lattices [1–7].
As systems of nonlinear interacting bosons, they have often
been suggested as potential candidates for quantum simulators
[8,9] and indeed the minimization of the energy of a particular
Hamiltonian on a graph was a problem considered recently
[10]. While the majority of studies of exciton-polaritons have
been restricted to the classical regime [11,12], the quantum
nature of polaritons has received revived attention recently
[13]. Therefore, it is natural to question whether exciton-
polaritons can be used to form lattices of entangled modes.
Here we must be aware that a lattice or graph of polaritons
does not behave as a system of qubits. Instead each node of
a polariton network could be described by the quantum field
amplitude aˆn or the continuous amplitude and phase variables
associated with the operators:
qˆn = aˆn + aˆ
†
n√
2
, pˆn = aˆn − aˆ
†
n
i
√
2
. (1)
Since continuous variable modes can be entangled, networks
of continuous variable modes are highly relevant for quantum
applications. As an example, cluster state computation [14]
based on continuous variables [15] is a potential route toward
universal quantum computation. It relies on producing a highly
entangled state from an arbitrary lattice or graph of modes
coupled by two-mode squeezing type interactions, with a
Hamiltonian of the form
HS =
∑
nm
wnm(aˆnaˆm + aˆ†naˆ†m), (2)
where wnm describes the weights of different connections in
the graph. Arranging such a Hamiltonian is already a problem
and it must be done by making use of some interaction
process that is stronger than any detrimental processes in
the system (dissipation, dephasing, etc.). While evidence of
strongly interacting polaritons [16] was reported recently, it is
not clear if any nonlinear interaction process in microcavities
is sufficiently strong for the generation of quantum resources.
In the absence of strong interactions, exciton-polaritons tend
to only demonstrate nonlinear effects at high densities, when
they are well described by the classical physics corresponding
to the mean-field approximation. For this reason only a handful
of experimental reports of quantum exciton-polariton effects
have appeared in the literature [17,18].
Two-mode squeezing. Before considering how to build a
polariton network, it is instructive to consider the effect of the
two-mode squeezing type Hamiltonian:
ˆH = − iα
2
(aˆ1aˆ2 − aˆ†1aˆ†2). (3)
Such a Hamiltonian generates entanglement, which can be
characterized by the violation of the inequality [19,20]
1  S12 = 12 [V (qˆ1 − qˆ2) + V (pˆ1 + pˆ2)], (4)
where the variances are defined by V ( ˆO) = 〈 ˆO2〉 − 〈 ˆO〉2.
The Heisenberg equations of motion give the evolution of
the quantum field operators aˆ1,2(t) ≡ ei ˆHt aˆ1,2e−i ˆHt (we set
h¯ = 1)
aˆ1,2(t) = cosh(αt/2)aˆ1,2 + sinh(αt/2)aˆ†2,1. (5)
To calculate the second-order correlators, we consider oper-
ators ˆK = 1 + aˆ†1aˆ1 + aˆ†2aˆ2, ˆL = aˆ1aˆ2 + aˆ†1aˆ†2, ˆM = i(aˆ1aˆ2 −
aˆ
†
1aˆ
†
2) and use the Lie algebra [ ˆM, ˆK] = 2i ˆL, [ ˆM, ˆL] = 2i ˆK to
obtain
ˆK(t) = cosh(αt) ˆK + sinh(αt) ˆL, (6a)
ˆL(t) = cosh(αt) ˆL + sinh(αt) ˆK. (6b)
Using these relations and taking the vacuum state as an
initial condition, one arrives at
S12 = 〈 ˆK(t) − ˆL(t)〉 = e−αt < 1, (7)
indicating evolution of the system toward the entangled co-
eigenstate of the EPR pair of operators qˆ1 − qˆ2 and pˆ1 + pˆ2.
This result, S12 = e−αt , remains unchanged for initial coherent
states of the fields. We note that the EPR pair of operators [21]
needed to demonstrate the nonclassical correlations depends on
the Hamiltonian. Other possible pairs can be obtained with the
2469-9950/2018/97(4)/041302(5) 041302-1 ©2018 American Physical Society
T. C. H. LIEW AND Y. G. RUBO PHYSICAL REVIEW B 97, 041302(R) (2018)
a
aL
aU
Laser
Laser
a1 a2
J
(a) (b)
(c)
FIG. 1. (a) Scheme of inverse four-wave mixing. (b) Coupling
of spatially separated modes, each driven by the inverse four-wave
mixing scheme of panel (a). (c) Potential generalization into lattices
and arbitrary graphs.
gauge transformation of operators in Eq. (4), aˆ1,2 → aˆ1,2eiφ1,2 ,
with subsequent optimization over the phases φ1,2.
Inverse four-wave mixing. Let us now consider a single
cavity with a four-wave mixing (parametric) type resonance,
described with the Hamiltonian
H0 = α02 (aˆ
†aˆ†aˆLaˆU + aˆ†Laˆ†U aˆaˆ), (8)
where α0 describes the strength of the four-wave mixing pro-
cess. Physical realizations of the above Hamiltonian could be
made in exciton-polariton micropillars [22] or a Kerr nonlinear
photonic crystal cavity [23]. Finding a parametric resonance
would, however, require careful tuning [24], which suggests
that systems compatible with postgrowth tuning would be the
most realistic choices. For example, dipolariton-based setups
allow electrical control of mode energies [25]. Regardless of
the mechanism of introducing Hamiltonian (8), it is typically
the case that α0 will be weak compared to the system losses
, that is, typical optical systems are only weakly nonlinear
(α0  ).
The Hamiltonian (8) is usually considered for generating the
fields aˆL and aˆU from initial excitation of the field aˆ; however,
we can also consider the inverse process illustrated in Fig. 1(a).
Namely, if the modes aˆL and aˆU are driven by coherent laser
fields then particles scatter in pairs from aˆL and aˆU to the
mode aˆ. It is true that under such conditions the modes aˆL
and aˆU should behave only classically, such that their physics
cannot go beyond what is expected from making the mean-
field approximation on these modes, but doing so leaves still a
reduced quantum Hamiltonian acting on the mode aˆ:
H0 = α2 (aˆ
†aˆ† + aˆaˆ), (9)
where α = α0〈aU 〉〈aL〉. It is implicit in writing this Hamilto-
nian that the basis of the field operator aˆ has been rotated to
account for any phases of the fields driving the modes 〈aU 〉
and 〈aL〉, which can be chosen freely. While Eq. (9) is just
the Hamiltonian of two-particle creation, by considering its
introduction via the aforementioned inverse four-wave mixing
process we have a way to make this a strong effect: Since 〈aL〉
and 〈aU 〉 can be increased by the resonant driving intensity,
one can reach the regime α  .
Considering exciton-polariton systems, the regime α 
 has essentially been realized previously under different
conditions, where the blueshift due to polariton-polariton in-
teractions may exceed the linewidth and cause bistability [26].
It is worth mentioning that four-wave mixing experiments also
revealed an interesting polarization dependence [27,28], which
allow the signal mode aˆ to have a different linear polarization
than that of the others (aˆU and aˆL), useful for better resolution
and limiting other scattering processes.
Coupled cavities. If we now consider a pair of coupled
cavities, which could be made with the techniques of Ref. [22],
the model Hamiltonian becomes [see Fig. 1(b)]
H = α
2
(
aˆ21 + aˆ22 + aˆ†21 + aˆ†22
) − J (aˆ†1aˆ2 + aˆ†2aˆ1), (10)
where J is the coupling constant between the cavities and we
can set α > 0 without loss of generality. We show below that
this Hamiltonian results in entanglement between modes aˆ1
and aˆ2.
It is convenient to define new operators, representing a
symmetric-antisymmetric basis
aˆ1 = aˆ+ + aˆ−√
2
, aˆ2 = aˆ+ − aˆ−√
2
, (11)
decoupling the Hamiltonian into two parts:
ˆH = 1
2
∑
σ=±
[
α
(
aˆ2σ + aˆ†2σ
) − 2σJ aˆ†σ aˆσ
]
. (12)
We can then consider the Bogoliubov transform
aˆσ = cosh(x/2) ˆbσ + σ sinh(x/2) ˆb†σ , (13)
which in the case |J | > α and tanh(x) = α/J reduces the
Hamiltonian into the simple form
H = ω( ˆb†+ ˆb+ − ˆb†− ˆb−), (14)
where ω = √J 2 − α2.
If we take the vacuum state as the initial condition, then
only the second-order correlators of a fields contribute to the
inequality (4). It is easy to show that
〈 ˆb†σ ˆbσ (t)〉 = sinh2(x/2), (15a)
〈 ˆb2σ (t)〉 = −
σ
2
sinh(x)e−2iσωt , (15b)
which results in
〈aˆ†σ aˆσ (t)〉 = sinh2(x) sin2(ωt), (16a)
〈aˆ2σ (t)〉 = sinh(x) sin(ωt)[σ cosh(x) sin(ωt) + i cos(ωt)].
(16b)
In the symmetric-antisymmetric basis, we have
S12 = 1 + 〈aˆ†+aˆ+ + aˆ†−aˆ− − Re(aˆ2+ − aˆ2−)〉, (17)
where Re denotes the real part and the first-order correlators
vanish in our case. Substituting the explicit form of the
correlators, we obtain
S12 = 1 + 2 sinh2(x) sin2(ωt) − sinh(2x) sin2(ωt)
= J + α cos(2ωt)
J + α . (18)
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0.1
1
5
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
JΤ
S 1
2
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
ΑΤ
S 1
2
(a) (b)
FIG. 2. (a) Dependence of S12 on J . Here S12 is evaluated at t =
τ and it can be seen that there is an optimal value of J ≈ α. The
minimum value of S12 can become smaller for increasing values of α.
(b) Dependence of the optimum value of S12 (with optimally chosen
J ) on α. The dashed curve shows the function S12 = e−ατ obtained
from the ideal squeezing Hamiltonian (3) for comparison.
While this expression can never reach the value of zero,
for the case J > α, one can reach the value (J − α)/(J + α)
for the specific time when the cosine function evaluates to −1.
Since J − α can be tuned to be small, one can then in principle
reach arbitrarily small values of S12. To give a visualization,
Fig. 2(a) shows the variation in S12 at some fixed time as a
function of J . Figure 2(b) then shows the minimal value of S12
obtainable for increasing values of α.
The formation of entanglement in the above scheme might
be seen as a round about way to create entanglement from
four-wave mixing, which could be obtained already from
Hamiltonian (8). Indeed, the usual method of exciting the
central mode aˆ and looking at correlations between aˆL and
aˆU has been considered before, in different contexts [29–33].
It should be stressed, however, that the conventional method
requires α0 to be significant compared to the dissipation rate
and also α0 should be stronger than other scattering processes
(e.g., scattering with acoustic phonons) that may resonantly
couple the modes to be entangled. In the scheme that we
consider here, α can become the dominant interaction in the
system as it is enhanced by the density of modes aL and
aU . Furthermore, local interactions, such as scattering with
phonons and sample disorder, are not able to couple spatially
separated modes aˆ1 and aˆ2.
It should be noted that formation of entanglement between
two condensate modes in this system does not lead to ap-
pearance of quantum correlations for each individual mode.
The fourth-order correlators can be evaluated similarly to the
second-order ones, resulting in
g(2) =
〈
aˆ
†2
1 aˆ
2
1
〉
〈aˆ†1aˆ1〉2
=
〈
aˆ
†2
2 aˆ
2
2
〉
〈aˆ†2aˆ2〉2
= cos(2ωt) + cosh(2x) sin
2(ωt)
sinh2(x) sin2(ωt) .
(19)
One can see that the second-order coherence g(2) > 2, so
that no antibunching in photon statistics is expected. This is
contrary to the case of a Josephson coupled pair of condensates
with onsite interactions (the Bose-Hubbard dimer) [34].
We can also note that the entanglement is achieved only
for sufficiently strong Josephson coupling, i.e., when J > α.
The system in this case does not behave classically even at
0
1
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
JΤ
S 1
2
0.1
1
2
5
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
S 1
2
(a) (b)
0.5
FIG. 3. (a) Dependence of S12 on J . As in Fig. 2(a), S12 is
evaluated at t = τ , but here we consider a fixed value of ατ = 5
and consider different values of dissipation . (b) Dependence of the
optimum value of S12 (with optimally chosen J ) on .
large occupations of the modes, which can be obtained for
0 < J − α  J , i.e., when the individual mode squeezing is
comparable to the Josephson coupling between the modes.
Here there is no contradiction with classical limit. Indeed, the
classical limit is established by increasing the intensity of U
and L fields. This corresponds to increase of the parameter α in
the Hamiltonian (10) and entanglement disappears for α > J .
Dissipation. We have shown so far that the system of
coupled cavities driven by parametric resonance can generate
entangled states, which become asymptotically close to the
level of entanglement expected from a two-mode squeezing
type operation, as measured by the violation of inequality (4).
As we have noted in the previous section, α can be controlled
by the intensity of external lasers. In principle, J can also be
controlled by external fields, for example, by using external
electric [35] or optical fields [36] to modify the potential
between lattice points.
While we expect the regime α   to be experimentally
accessible, given the parametric driving scheme, it is still
instructive to consider the influence of dissipation in the
system. This is readily introduced by modification of the
Heisenberg equations:
d〈 ˆO〉
dt
= i〈[ ˆH, ˆO]〉 + 
2
∑
n
〈2aˆ†n ˆOaˆn − aˆ†naˆn ˆO − ˆOaˆ†naˆn〉.
(20)
This introduces additional dissipation terms in our equations
of motion, which are solved in the Supplemental Material [37].
The resulting effect of dissipation is illustrated in Figs. 3(a) and
3(b). As one would expect, too much dissipation results in a
loss of entanglement. However, given the parametric pumping
scheme, it is in principle possible to work in the limit where
  α. At some short time such that τ  1/ one then
obtains a high degree of entanglement despite the presence
of dissipation.
Let us recall that Hamiltonian (10) relied on treating the
coherently excited modes aˆL and aˆU as mean fields. We have
further tested the validity of this approximation with a fully
quantum treatment [37].
Multimode entanglement. In comparison to conventional
methods of entanglement generation with respect to four-wave
mixing, the main advantage of our scheme that entangles
polariton modes separated in real space is that it is in principle
041302-3
T. C. H. LIEW AND Y. G. RUBO PHYSICAL REVIEW B 97, 041302(R) (2018)
scalable; by coupling more cavities in space, arbitrary networks
could be considered such as the one illustrated in Fig. 1(c).
As an example, let us consider a system of four identical
cavities, which are subjected to the Hamiltonian:
H4 =
4∑
n=1
α
2
(aˆ†naˆ†n + aˆnaˆn)
− JA(t)(aˆ†1aˆ2 + aˆ†2aˆ1 + aˆ†3aˆ4 + aˆ†4aˆ3)
− JB (t)(aˆ†1aˆ3 + aˆ†3aˆ1 + aˆ†2aˆ4 + aˆ†4aˆ2). (21)
This Hamiltonian is a generalization of Hamiltonian (10),
where we assume that it is possible to control the linear
coupling in time. For simplicity, we will consider (JA(t) = J ,
JB(t) = 0) for the time 0 < t < τ and (JA(t) = 0, JB(t) =
J ) for time τ < t < 2τ . It is possible to write Heisenberg
equations of motion and their time-dependent solution can
be obtained analytically [37]. Alternatively, in the absence
of dissipation, it is more efficient to solve for the operator
evolution in the Heisenberg picture [37].
While violation of inequality (4) is a sufficient condition
for entanglement, the definition given of S12 is not ideal
for all states. In particular, varying the phases of modes aˆ1
and aˆ2 changes the value of S12 and thus to demonstrate
the entanglement we should minimize S12 over all choices
of local phases. As we mentioned above, this is equivalent
to finding the best EPR pair of operators. The procedure is
detailed in Ref. [37], where we define ˜S12 as the value of
S12 minimized over phase rotations. The result is shown in
Fig. 4(a), where, in addition to characterizing the entanglement
between modes aˆ1 and aˆ2, we find also entanglement between
other pairs of modes, using similar definitions for S13 and S14
(other combinations of modes display identical entanglement
characteristics due to symmetry).
In addition to entanglement between pairs of modes, multi-
mode entanglement, simultaneously between all four modes of
the system can be evidenced by the violation of the inequality
[38],
1
2 [V (qˆ1 − qˆ2) + V (pˆ1 + pˆ2 + gpˆ3 + gpˆ4)] = I  1, (22)
S12
S13
S14
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
ΑΤ
S
0.1
0.2
0.5
1
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
I
(a) (b)
FIG. 4. (a) Dependence of the optimum values of ˜S12, ˜S13, and
˜S14 (with optimally chosen J ) on α. Parameters:  = 0, tA = tB = τ .
(b) Dependence of the optimum value of I (with optimally chosen J )
on , for different values of α.
where g is an arbitrary real parameter that should be chosen so
as to optimize the violation of the inequality. In the general case
of four modes, one should also break two other inequalities to
evidence an entangled state, obtained by permuting the modes
[38]. However, given the symmetry of our four-mode example
in a ring, these inequalities are equivalent and the violation of
inequality (22) is a sufficient condition. Following the correct
choice of the parameter g and optimization over the phases of
the modes [37], we indeed find that the quantity I can drop
below one and even reach zero, as shown in Fig. 4 for different
values of α and .
Conclusion. The evolution of polariton networks from the
classical to quantum regime implies finding a mechanism of
generating quantum correlations that can overcome the dissi-
pation of the system. Nonlinearity, in the form of polariton-
polariton interactions, is traditionally weak; however, here
we have shown theoretically that an inverse four-wave
mixing geometry allows enhancement to an effective strongly
nonlinear regime. Local nonlinearity and standard Josephson
coupling between spatially separated modes is then sufficient to
generate quantum entanglement both between pairs of modes
and multiple modes. We hope this will stimulate further discus-
sions on polariton simulators, which have begun recently [10].
Acknowledgments. This research was supported by the
Ministry of Education (Singapore) Grant No. 2015-T2-1-055
and by CONACYT (Mexico) Grant No. 251808.
[1] C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H. Deng, M.
D. Fraser, T. Byrnes, P. Recher, N. Kumada, T. Fujisawa, and Y.
Yamamoto, Nature (London) 450, 529 (2007).
[2] E. A. Cerda-Méndez, D. N. Krizhanovskii, M. Wouters, R.
Bradley, K. Biermann, K. Guda, R. Hey, P. V. Santos, D. Sarkar,
and M. S. Skolnick, Phys. Rev. Lett. 105, 116402 (2010).
[3] N. Y. Kim, K. Kusudo, C. Wu, N. Masumoto, A. Löffler,
S. Höfling, N. Kumada, L. Worschech, A. Forchel, and Y.
Yamamoto, Nat. Phys. 7, 681 (2011).
[4] N. Y. Kim, K. Kusudo, A. Löffler, S. Höfling, A. Forchel, and
Y. Yamamoto, New J. Phys. 15, 035032 (2013).
[5] E. A. Cerda-Méndez, D. Sarkar, D. N. Krizhanovskii, S. S.
Gavrilov, K. Biermann, M. S. Skolnick, and P. V. Santos, Phys.
Rev. Lett. 111, 146401 (2013).
[6] T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Sol-
nyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and
A. Amo, Phys. Rev. Lett. 112, 116402 (2014).
[7] K. Winkler, O. A. Egorov, I. G. Savenko, X. Ma, E. Estecho, T.
Gao, S. Müller, M. Kamp, T. C. H. Liew, E. A. Ostrovskaya,
S. Höfling, and C. Schneider, Phys. Rev. B 93, 121303(R)
(2016).
[8] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
[9] S. Lloyd, Science 273, 1073 (1996).
[10] N. G. Berloff, M. Silva, K. Kalinin, A. Askitopoulos, J. D.
Töpfer, P. Cilibrizzi, W. Langbein, and P. G. Lagoudakis, Nat.
Mater. 16, 1120 (2017).
[11] S. Utsunomiya, K. Takata, and Y. Yamamoto, Opt. Express 19,
18091 (2011).
041302-4
QUANTUM EXCITON-POLARITON NETWORKS THROUGH … PHYSICAL REVIEW B 97, 041302(R) (2018)
[12] A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto,
Nat. Photon. 8, 937 (2014).
[13] A. Cuevas, B. Silva, J. C. L. Carreño, M. de Giorgi, C. S. Muñoz,
A. Fieramosca, D. G. S. Forero, F. Cardano, L. Marrucci, V.
Tasco et al., arXiv:1609.01244.
[14] R. Raussendorf and H. Briegel, Phys. Rev. Lett. 86, 5188
(2001).
[15] N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T.
C. Ralph, and M. A. Nielsen, Phys. Rev. Lett. 97, 110501
(2006).
[16] Y. Sun, Y. Yoon, M. Steger, G. Liu, L. N. Pfeiffer, K. West,
D. W. Snoke, and K. A. Nelson, Nat. Phys. 13, 870 (2017).
[17] J. Ph. Karr, A. Baas, R. Houdré, and E. Giacobino, Phys. Rev.
A 69, 031802(R) (2004).
[18] S. Savasta, O. Di Stefano, V. Savona, and W. Langbein, Phys.
Rev. Lett. 94, 246401 (2005).
[19] Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev.
Lett. 84, 2722 (2000).
[20] R. Simon, Phys. Rev. Lett. 84, 2726 (2000).
[21] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777
(1935).
[22] S. M. de Vasconcellos, A. Calvar, A. Dousse, J. Suffczyn´ski,
N. Dupuis, A. Lemaître, I. Sagnes, J. Bloch, P. Voisin, and P.
Senellart, Appl. Phys. Lett. 99, 101103 (2011).
[23] D. Gerace, H. E. Türeci, A. Imamoglu, V. Giovannetti, and R.
Fazio, Nat. Phys. 5, 281 (2009).
[24] C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaître, J.
Bloch, P. Roussignol, and C. Delalande, Nature (London) 440,
904 (2006).
[25] P. Cristofolini, G. Christmann, S. I. Tsintzos, G. Deligeorgis,
G. Konstantinidis, Z. Hatzopoulos, P. G. Savvidis, and J. J.
Baumberg, Science 336, 704 (2012).
[26] A. Baas, J. Ph. Karr, H. Eleuch, and E. Giacobino, Phys. Rev. A
69, 023809 (2004).
[27] D. N. Krizhanovskii, D. Sanvitto, I. A. Shelykh, M. M. Glazov,
G. Malpuech, D. D. Solnyshkov, A. Kavokin, S. Ceccarelli, M.
S. Skolnick, and J. S. Roberts, Phys. Rev. B 73, 073303 (2006).
[28] C. Leyder, T. C. H. Liew, A. V. Kavokin, I. A. Shelykh, M.
Romanelli, J. Ph. Karr, E. Giacobino, and A. Bramati, Phys.
Rev. Lett. 99, 196402 (2007).
[29] S. Savasta, R. Girlanda, and G. Martino, Phys. Status Solidi A
164, 85 (1997).
[30] P. Schwendimann, C. Ciuti, and A. Quattropani, Phys. Rev. B
68, 165324 (2003).
[31] J. Ph. Karr, A. Baas, and E. Giacobino, Phys. Rev. A 69, 063807
(2004).
[32] S. Portolan, O. Di Stefano, S. Savasta, and V. Savona, Europhys.
Lett. 88, 20003 (2009).
[33] M. Romanelli, J. Ph. Karr, C. Leyder, E. Giacobino, and A.
Bramati, Phys. Rev. B 82, 155313 (2010).
[34] H. Flayac and V. Savona, Phys. Rev. A 95, 043838 (2017).
[35] G. Christmann, C. Coulson, J. J. Baumberg, N. T. Pelekanos, Z.
Hatzopoulos, S. I. Tsintzos, and P. G. Savvidis, Phys. Rev. B 82,
113308 (2010).
[36] A. Amo, S. Pigeon, C. Adrados, R. Houdré, E. Giacobino, C.
Ciuti, and A. Bramati, Phys. Rev. B 82, 081301(R) (2010).
[37] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.97.041302 for analytic solutions to the prob-
lem of two and four coupled modes, both with and without
dissipation; minimization of the quantities S12 and I ; and
comparison to numerical simulations where all modes are treated
as quantum, without any mean-field approximation.
[38] S. L. W. Midgley, A. S. Bradley, P. Pfister, and M. K. Olsen,
Phys. Rev. A 81, 063834 (2010).
041302-5