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The system of interacting polaritons: Classical versus quantum kinetic equation
Solid State Communications 144 (2007) 378–383 www.elsevier.com/locate/ssc
The system of interacting polaritons: Classical versus quantum kinetic equation I.A. Shelykh a,b , G. Malpuech c,∗ , R. Johne c , A.V. Kavokin d a International Centre for Condensed Matter Physics, Universidade de Brasilia, 70904-970 Brasilia DF, Brazil b St. Petersburg State Polytechnical Univetrsity, 29, Politechnicheskaya, 195251, St-Petersburg, Russia c LASMEA, UMR 6602 CNRS, Universit´e Blaise-Pascal, 24, av des Landais, 63177, Aubi`ere, France d School of Physics and Astronomy, Univeristy of Southampton, SO17 1BJ Southampton, UK
Received 30 January 2007; accepted 1 July 2007 by the Guest Editors Available online 19 August 2007
Abstract We derive the set of quantum kinetic equations for the system of interacting exciton polaritons in microcavities relaxing the Born–Markov approximation. We demonstrate the crucial role played by the nonclassical four particle correlators in the dynamics. The decoherence process is shown to be responsible for the transition from quantum to classical limit. The general formalism is applied for description of the dynamics of the polariton parametric oscillator. c 2007 Elsevier Ltd. All rights reserved.
PACS: 05.30.Jp; 71.35.Lk; 71.36.+c Keywords: D. Quantum kinetics; D. Decoherence; D. Exciton-polaritons
1. Introduction The systems of interacting composite bosons are subject of enhanced interest nowadays. The experimental realization of the Bose–Einstein condensation (BEC) of cold atoms, marked by a Nobel prize in physics of 2004, has stimulated numerous theoretical works in this field. The experimental studies of the solid state composite bosonic systems such as excitons and exciton polaritons have shown that the BCE in semiconductor systems is also possible. The exciton–polaritons can be considered as bosons if their concentration n D is small enough so that the condition is satisfied: n D a BD << 1
(1)
where D is a dimensionality of the system (D = 1, 2, 3), a B is an exciton Bohr radius. The bosonic properties of the exciton systems are in the focus of the condensed matter theory since the pioneering work of Keldysh and Kopaev [1], where ∗ Corresponding author.
E-mail address: [email protected] (G. Malpuech). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.07.042
the possibility of the exciton BEC was theoretically analyzed for the first time. Different types of excitonic systems were subsequently examined for suitability for observation of the BEC, including indirect excitons in coupled quantum wells [2], optically inactive excitons in Cu2 O [3] etc. Recently, it has been shown that exciton polaritons in quantum microcavities (cavity polaritons) are probably the most suitable candidates for the BEC. Being the quasiparticles that combine both exciton and photon properties exciton–polaritons clearly demonstrate a bosonic behaviour [4, 5]. Compared to the bare excitons, cavity polaritons have two main advantages from the point of view of BEC. First, due to their photonic component and light effective mass polaritons are much less affected by localization effects provoked by the disorder. Second, again due the light effective mass of polaritons high occupation numbers of their ground state can be achieved at relatively weak pumping which does not suppress the strong-coupling regime in microcavities. Polariton BEC has been recently experimentally observed in CdTe based microcavities [6] at the temperatures up to 40 K, which would be impossible in any atomic system.
I.A. Shelykh et al. / Solid State Communications 144 (2007) 378–383
As the cavity polaritons have a finite life-time, their BEC is always happening out of thermal equilibrium, strictly speaking. The macroscopically occupied state at k = 0, k being the inplane wave-vector, can be formed only if the scattering rate of exciton–polaritons strongly exceeds their radiative decay rate. Thus, the thermodynamic treatment is not sufficient for the polaritonic systems and the kinetic effects are of crucial importance for the BEC issue in most cases. The low density of states in the vicinity of the ground polariton state prevents the energy relaxation of the polaritons assisted by acoustic phonons (“bottleneck effect”) [7], thus forming an obstacle for the dynamical condensation. The most reliable mechanism for overcoming of the bottleneck effect is polariton–polariton scattering, governed by the Coulomb interactions between the excitonic fractions of two colliding polaritons. The matrix element of the polariton–polariton scattering is dependent on the exciton binding energy E B , exciton Bohr radius a B , and the area occupied by the condensate S. Roughly, it can be estimated as [8] Vk,k0 ;k00 ,k000 = hk, k0 |Vint |k00 , k000 i ≈
6E B a 2B ∗ ∗ X k X k0 X k00 X k000 δk+k0 −k00 −k000 S
(2)
where X k is a Hopfield coefficient giving the percentage of the excitonic fraction in the state k, the Dirac delta function ensures the momentum conservation during the scattering act. The description of the energy relaxation of the system of interacting polaritons is usually performed using a set of semiclassical Boltzmann equations [9], which means that all quantum correlations between different states are neglected. This approach gives satisfactory results for the case of nonresonant pumping. However, in the case of resonant excitation, where the microcavity operates as an optical parametric amplifier (OPO), some effects beyond Born–Markov approximation may arise. In the OPO regime the polaritons are excited at the lower dispersion branch by light incident under the magic angle. Resonant polariton–polariton scattering from the pump state populates so-called signal (k = 0) and idler states. The quantum correlations between the signal and idler states play a strong role in OPO dynamics [10]. The derivation of quantum kinetic equations for the system of interacting polaritons in the general case has not yet been reported, to the best of our knowledge. In this paper we report such a derivation and consider the transition between the classical and quantum limit. Quantum correlations are taken into account via nonclassical off-diagonal four-particle correlators. We apply the obtained formalism to describe the dynamics of the polariton paramatric oscillator. We demonstrate that decoherence processes leading to the decay of the nonclassical correlators lead to the transition between the quantum oscillatory regime and the classical relaxation regime for the OPO. We show that in the limit of very fast dephasing the system of quantum equations transforms into the classical system of Boltzmann equations. The spin degree of freedom of the exciton–polaritons is neglected in this work.
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2. Quantum kinetic equation for the system of the interacting bosons The system of the interacting exciton–polaritons can be described by the following model Hamiltonian: X X + H = εk ak+ ak + Uk,q ak+ ak−q (bq + b−q ) k
+
k,q
1 X Vk,k0 ,q ak+ ak+0 ak−q ak0 +q + H.c. 2 0
(3)
k,k ,q
where the operators ak describe annihilation of the exciton– polaritons, the operators bk are annihilation operators for the acoustic phonons. The first term describes the free particles propagation, the second term describes the polariton–acoustic phonon scattering, the third term describes the polariton–polariton scattering. The phonon field can be treated classically. The usual way to deal with it is to use the Born–Markov approximation for the Liouville–von Neumann equation for the density matrix. As a result the system of the Boltzmann-type kinetic equations appears which would satisfactorily describe the dynamics of polariton relaxation with acoustic phonons. The procedure of its derivation is described in the literature (see e.g. Ref. [11]). Here we shall not repeat it, and will neglect the second term of (3) in the further procedure. Instead we shall consider in detail the dynamics of polariton–polariton interactions described by the third term in Eq. (3). Formally, the Born–Markov approximation can be applied also to this term [12], but it can be hardly justified as there is no classical reservoir that can be traced over in this case. We shall go beyond the Born–Markov approximation in the further development. We start with the Liouville–von Neumann equation written with the Hamiltonian (3) excluding its second term: X dρ = [H ; ρ] = εk [ak+ ak ρ − ρak+ ak ] ih¯ dt k +
1X V 0 [a + a +0 ak−q ak0 +q ρ − ak+ ak+0 ak−q ak0 +q ρ]. (4) 2 k,q k,k ,q k k
It yields the following dynamics of the occupation numbers dNk dρ + = Tr ak ak dt dt X 1 = Im[Vk,k0 ,q hak+ ak+0 ak−q ak0 +q i] h¯ k 0 ,q =
1X Im[Vk,k0 ,q Ak,k0 ,q ]. h¯ k 0 ,q
(5)
The right part of Eq. (5) contains the fourth-order nonclassical correlators Ak,k0 ,q = hak+ ak+0 ak−q ak0 +q i = Tr[ρak+ ak+0 ak−q ak0 +q ]. Note that Eq. (5) is obtained from Eq. (4) without any simplifying assumptions. In order to take into account the finite lifetime of exciton–polaritons, the additional term −Nk /τk should be introduced into Eq. (5).
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To complete the set of the kinetic equations one must add an expression for the temporal derivative of Ak,k0 ,q which reads dAk,k0 ,q dρ + + = Tr a a 0 ak−q ak0 +q dt dt k k i = Tr([ρ; H ]ak+ ak+0 ak−q ak0 +q ) h¯ i = (εk + εk0 − εk−q − εk0 +q )Ak,k0 ,q h¯ i X + + + [−Vk,k00 ,q0 hak00 ak−q + 0 ak00 +q0 ak0 ak−q ak0 +q i h¯ 00 0 k ,q
− Vk0 ,k00 ,q0 hak+ ak00 ak+0 −q0 ak+00 +q0 ak−q ak0 +q i
+ ak−q i] − hak+ ak ak+0 ak0 ak0 +q ak+0 +q i − hak+ ak ak+0 ak0 ak−q
≈
F k/k0 /k−q/k0 +q6=k00 = q=±q0
+ Vk0 +q,k00 ,q0 hak+ ak+0 ak−q ak+00 ak0 +q−q0 ak00 +q0 i] i = (εk + εk0 − εk−q − εk0 +q )Ak,k0 ,q h¯ + Fq0 =0 + F k/k0 /k−q/k0 +q=k00 q=±q0 + F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q6=k00
× +
(Vk0 ,k00 ,−q hak+ ak00 ak+0 +q ak+00 −q ak−q ak0 +q i) (Vk−q,k00 ,−q hak+ ak+0 ak+00 ak ak00 −q ak0 +q i)
k00 6=k0 +q
q6=±q0
where we have divided all the terms provided by the polariton–polariton scattering into four groups
X
−
(Vk0 +q,k00 ,q hak+ ak+0 ak−q ak+00 ak0 ak00 i)
k00 6=k−q
F k/k0 /k−q/k0 +q6=k00 q6=±q0
depending on the possibility of their efficient factorization. Below we write them explicitly. 1. The term Fq0 =0 describes the forward scattering Fq0 =0
i h¯
+ ak+00 +q ak+0 ak−q ak0 +q i) (Vk,k00 ,q hak00 ak−q
X
−
(6)
q=±q0
k00 6=k0
X
q=±q0
k00 6=k
+ F k/k0 /k−q/k0 +q6=k00
q6=±q0
X
q=±q0
Fq0 =0 , F k/k0 /k−q/k0 +q=k00 , q=±q0 F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q6=k00 ,
k00 6=k
−
X
Vk00 ,k−q,q Nk Ak00 ,k0 ,q
k00 6=k0 +q
k ,q
− Vk0 +q,k00 ,q0 )hak+ ak+0 ak0 +q ak−q ak+00 ak00 i X i ≈ Ak,k0 ,q (Vk,k00 ,q0 + Vk0 ,k00 ,q0 − Vk−q,k00 ,q0 h¯ 00 0
F k/k0 /k−q/k0 +q=k00 = q6=±q0
(7) ×
q0
,
(9)
i h¯
+ + + [Vk,k0 ,q0 hak0 ak−q 0 ak0 +q0 ak0 ak−q ak0 +q i
+ + Vk0 ,k,q0 hak+ ak ak+0 −q0 ak+q 0 ak−q ak0 +q i +
− Vk−q,k0 +q,q0 hak+ ak+0 ak+0 +q ak−q−q0 ak0 +q+q0 ak0 +q i + ak0 +q−q0 ak0 −q+q0 i] − Vk0 +q,k0 −q,q0 hak+ ak+0 ak−q ak−q
will show below that it has the most important impact on the dynamics of a polariton OPO. It writes
+ + hak ak+ ak−q ak−q ak+0 +q0 ak0 +q i
X
q=±q0
q=±q0
Vk0 +q,k00 ,q Nk0 Ak,k00 ,q
k00 6=k−q
where in the passage from the second to the third line we have used the mean-field approximation. Thus, the considered term provides nothing more then the energy renormalization of the states coupled by polariton–polariton interactons. In the s-wave approximation Vk,k00 ,q0 = Vk0 ,k00 ,q0 = Vk−q,k00 ,q0 = Vk0 +q,k00 ,q0 and thus Fq0 =0 = 0. 2. The term F k/k0 /k−q/k0 +q=k00 is much more interesting. We
i + = Vk,k0 ,q [hak+0 ak0 ak−q ak−q ak+0 +q ak0 +q i h¯
X
−
k ,q
− Vk0 +q,k00 ,q0 )Nk00
i X ≈ V 00 Nk−q Ak00 +q,k0 ,q h¯ 00 0 k,k ,q k 6=k X + Vk00 ,k0 ,q Nk0 +q Ak,k00 +q,q
i X = (V 00 0 + Vk0 ,k00 ,q0 − Vk−q,k00 ,q0 h¯ 00 0 k,k ,q
F k/k0 /k−q/k0 +q=k00
(8)
where again we used the mean-field approximation. The Eq. (8) shows that the dynamics of the fourth-order nonclassical correlator is governed by the dynamics of the occupation numbers, as might have been expected. The term (8) ensures the spontaneous apparition of the nonclassical correlators in the system where initially they are absent. 3. The term F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q6=k00 is given by q6=±q0
+ Vk−q,k00 ,q0 hak+ ak+0 ak+00 ak−q−q0 ak00 +q0 ak0 +q i
q6=±q0
i V 0 [Nk−q Nk0 +q (Nk + Nk0 + 1) h¯ k,k ,q − Nk Nk0 (Nk−q + Nk0 +q + 1)]
≈
i X V 0 0 [(Nk + Nk0 + 1)Ak−q0 ,k0 +q0 ,q−q0 h¯ q0 k,k ,q − (Nk−q + Nk0 +q + 1)Ak,k0 ,q−q0 ].
(9a)
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It describes the mutual influence of the correlators corresponding to various pairs of the initial and final states in the reciprocal space. 4. Finally, the term F k/k0 /k−q/k0 +q6=k00 contains the correlator q6=±q0
of the operators corresponding to six different states in the reciprocal space. It cannot be factorized in the satisfactory manner. However, it rapidly vanishes as the processes of the decoherence in the system lead to the faster decay of this correlator than of anyone of the six-order correlators involving the same states twice or more times. In the following we shall assume F k/k0 /k−q/k0 +q6=k00 = 0. q6=±q0
The set of Eqs. (5)–(9) describes the dynamics of the bosonic system accounting for particle–particle interactions. Note that the Markov approximation has not been used while deriving Eqs. (5)–(9). Consequently, Eqs. (5)–(9) allow for the energynonconservative processes and may predict a qualitatively different dynamics of the system with respect to the Boltzmann equations. We remind that the Boltzmann equations only contain the occupation numbers, while in the system (5)–(9) the correlations between different states in the reciprocal space are described by means of the fourth-order nonclassical correlators. 3. Decoherence and classical limit
q6=q0
Ak,k0 ,k−q,k0 +q = =
q=±q0
1
τdec
−
i h¯ (εk
+ εk0 − εk−q − εk0 +q ) iVk,k0 ,q
h¯ (εk + εk0 − εk−q − εk0 +q ) + i τdec × [Nk−q Nk0 +q (Nk + Nk0 + 1) − Nk Nk0 (Nk−q + Nk0 +q + 1)].
(11)
Substituting the expression for Ak,k0 ,q into the equation for the occupation numbers (5) one readily obtains a familiar set of the semiclassical Boltzmann equations: X dNk = Wk,k0 ,q [Nk−q Nk0 +q (Nk + Nk0 + 1) dt 0 − Nk Nk0 (Nk−q + Nk0 +q + 1)],
(12)
where the scattering rates are given by Wk,k0 ,q =
|Vk,k0 ,q |2 h¯ ×
1/τdec 2 (εk + ε − εk−q − εk0 +q )2 / h¯ 2 +1/τdec
.
(13)
k0
|V
0
|2 τdec
given by the Fermi golden rule, Wk,k0 ,q = k,k ,qh¯ . For the energy nonconserving processes the probability of scattering is reduced by a Lorentzian factor.
q6=±q0
q=±q0
F k/k0 /k−q/k0 +q=k00
One can see from Eq. (13), that the scattering rate is the fastest for the energy-conserving processes, where it is simply
equation for Ak,k0 ,q in the following form dAk,k0 ,q i 1 = (εk + εk0 − εk−q − εk0 +q ) − Ak,k0 ,q dt h¯ τdec + F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q6=k00
q=±q0
k ,q
Until now we did not discuss the decoherence processes specifically. These processes are necessarily present and important in all realistic systems. In particular they come from the interaction of polaritons with a classical phonon reservoir and from the polariton–polariton forward scattering beyond the Born approximation [13]. Both these effects contribute to the temporal decay of the correlators. Introducing the decoherence time τdec and neglecting F k/k0 /k−q/k0 +q6=k00 one can rewrite the
q=±q0
in the chemical kinetics problems, where all the intermediate products are considered to be in the quasi-equilibrium (in our case the fourth order correlator is indeed an “intermediate product”!). These simplifying assumptions allow us to estimate the correlators Ak,k0 ,q as follows dAk,k0 ,k−q,k0 +q i 1 (εk + εk0 − εk−q − εk0 +q ) − = h¯ dt τdec × Ak,k0 ,k−q,k0 +q + F k/k0 /k−q/k0 +q=k00 = 0
4. Application: Polariton parametric oscillator
.
(10)
Then, assuming that the decoherence time is short enough, 1 so that V Ntot << τdec , where Ntot is a total number of the polaritons in the system and V is a mean value of the matrix element, one can neglect the term F k/k0 /k−q/k0 +q=k00 + q6=±q0
F k/k0 /k−q/k0 +q6=k00 in (10). This term can also be neglected q=±q0
because it contains a sum of the fourth-order correlators corresponding to the different quantum states having random phases, in general, so that altogether they give a vanishing contribution. One can also assume that if the decoherence processes are strong the correlators Ak,k0 ,q reach their equilibrium values much faster than occupation numbers. This assumption allows us to separate the variables in our system into slowly changing ones (occupation numbers) and fast changing ones (correlators). Such a separation is routinely used
To illustrate the general theory developed above, let us consider a simple example of the bosonic system for which the number of states in the reciprocal space is sufficiently reduced. In the polariton OPO configuration, due to the strong nonparabolicity of the lower polariton branch, a pair of exciton–polaritons created by the pump pulse incident under the magic anlge scatters into the non-degenerate signal and idler states with both energy and momentum conservation [14]. The scattering to all other states can be neglected in the first order as it does not conserve energy. A reliable model considering the polariton OPO may account for only three quantum states, namely the pump, signal and idler states. The Hamiltonian of this system can be written in the following form: + H = (εs as+ as + εi a + p a p + εi ai ai ) + + U (X s as+ as + X p a + p a p + X i ai ai )
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I.A. Shelykh et al. / Solid State Communications 144 (2007) 378–383 + × (X s as+ as + X p a + p a p + X i ai ai ) + + + + V (a + p a p as ai + a p a p as ai )
(14)
where the indices p, s and i correspond to the pump, signal and idler respectively. The first term in the right part of Eq. (14) describes the free particles propagation, the second term describes the polariton energy blueshifts for the three states under consideration (X is a percentage of the exciton fraction in each of the states), √ the third term describes the parametric process, V ≈ X p X i X s U . The previous models of the polariton OPO either consider all three states involved in the parametric process as the classical fields coupled by a four-wave mixing process [15] or consider the case of cw pumping neglecting the pump depletion [10]. The exception is the recent work of Glazov and Kavokin [16] where the hyperspin formalism was applied for the analysis of the parametric amplifier. While it allows up to some point for description of quantum correlations in a threelevel system, its extension to more complex systems would require extremely heavy analytics computations. Besides, the role of the decoherence processes as well as a reduction of the quantum kinetic equation to semiclassical Boltzmann equation in the limit of strong decoherence is not straightforward within the hyperspin approach. On the other hand, the complete set of quantum kinetic equations for a system of interacting bosons presented in this work allows for description of the transition from quantum to classical limit. Let us apply the formalism developed above for the description of the cavity OPO. The Eqs. (5)–(9) in this case reduce to dNs Ns 2 = − + Im{V has+ ai+ a p a p i} dt τs h¯ Ns 2 = − + Im{V A}, τs h¯ Ni 2 dNi =− + Im{V A}, dt τi h¯ Np dN p 4 =− − Im{V A}, dt τp h¯ 1 1 1 1 dA A = − + + + dt 2τs 2τi τp τdec i + [(εs + εi − 2ε p ) + U (X i + X s − 2X p ) h¯ × (X i Ni + X s Ns + X p N p )]A+ iV + [N p2 (Ns + Ni + 1) − 4Ns Ni (N p + 1)], h¯
(15) (15a) (15b)
(15c)
where we have introduced the polariton lifetimes τi . The system of Eq. (15) is equivalent to one describing a damped nonlinear oscillator. It takes into account the depletion of the pump and the spontaneous scattering and thus can be used both above and below the stimulation threshold, unlike the models of Refs. [10,11]. The dynamics of the occupation numbers given by Eq. (15) also differ sufficiently from those given by a set of the semiclassical Boltzmann equations. The Boltzmann equations
predict the monotonous temporal dependence of the occupation numbers, while the system (15) allows for their oscillatory behaviour. The period of the oscillations depends on the total number of the particles in the system Ntot , T ∼ V N1 tot . If the 1 V Nh¯ tot , these oscillations are washed damping is strong, τdec out and the dynamics of the occupation numbers become classical.
5. Conclusions In conclusion, we have derived the system of quantum kinetic equations for the system of interacting bosons which goes beyond the Markov approximation. The dynamics of the occupation numbers is shown to be closely lined with the build-up of the nonclassical four-particle correlators. The decoherence process leading to the fast decay of these correlators is shown to provoke the transition from the quantum to the classical limit. The system of the kinetic equations we obtain can be applied for description of the dynamics of the polariton OPO. Acknowledgements This work has been supported by the STREP project “STIMSCAT” 517769 and by the Chair of Excellence of ANR. I. A. Shelykh acknowledges Brazilian Ministry of Science and technology and IBEM, Brazil for support. References [1] L.V. Keldysh, Yu.V. Kopaev, Fiz. Tverd. Tela (Leningrad) 6 (1964) 2791; Sov. Phys. Solid State 6 (1965) 2219. [2] Yu.E. Lozovik, V.I. Yudson, Pis’ma Zh. Eksp. Teor. Fiz. 22 (1975) 26; Sov. Phys. JETP Lett. 22 (1975) 26; Yu.E. Lozovik, V.I. Yudson, Solid State Comm. 18 (1976) 628; A.L. Larionov, V.B. Timofeev, J. Hvam, K. Soerensen, Zh. Eksp. Teor. Fiz. 117 (2000) 1255; JETP 90 (2000) 1093; L.V. Butov, C.W. Lai, A.L. Ivanov, A.C. Gossard, D.S. Chemla, Nature 417 (2002) 47; D. Snoke, S. Denev, Y. Liu, L. Pfeiffer, K. West, Nature 418 (2002) 754; O. Berman, Yu.E. Lozovik, D.W. Snoke, R.D. Coalson, Phys. Rev. B 70 (2004) 235310. [3] D. Snoke, J.P. Wolfe, A. Mysyrowicz, Phys. Rev. Lett. 89 (1987) 527; D. Snoke, J.P. Wolfe, A. Mysyrowicz, Phys. Rev. B 41 (1990) 11171; Jia Ling Lin, J.P. Wolfe, Phys. Rev. Lett. 71 (1993) 1222; ´ S´uilleabh´ain, J.P. Wolfe, Phys. Rev. B 60 (1999) 10565; K.E. O’Hara, L. O G.M. Kavoulakis, Phys. Rev. B 65 (2001) 035204. [4] P.G. Savvidis, J.J. Baumberg, R.M. Stevenson, M.S. Skolnick, D.M. Whittaker, J.S. Roberts, Phys. Rev. Lett. 84 (2000) 1547. [5] H. Deng, G. Weihs, C. Santori, J. Bloch, Y. Yamamoto, Science 298 (2002) 199. [6] J. Kasprzak, et al., Nature 443 (2006) 409. [7] F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani, P. Schwendimann, Phys. Rev. B 56 (1997) 7554. [8] A. Kavokin, G. Malpuech, Cavity Polaritons, Elsevier North Holland, ISBN: 0-125-33032-4, 2003. [9] D. Porras, C. Ciuti, J.J. Baumberg, C. Tejedor, Phys. Rev. B 66 (2002) 085304. [10] C. Ciuti, P. Schwendimann, A. Quattropani, Phys. Rev. B 63 (2001) 041303. [11] H.J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equation and Fokker–Planck Equations, Springer, ISBN: 3540548823, 2003.
I.A. Shelykh et al. / Solid State Communications 144 (2007) 378–383 [12] M.M. Glazov, I.A. Shelykh, G. Malpuech, K.V. Kavokin, A.V. Kavokin, Solid State Comm. 134 (2005) 117. [13] F. Laussy, I.A. Shelykh, A.V. Kavokin, G. Malpuech, Phys. Rev. B 73 (2006) 035315. [14] P.G. Savvidis, J.J. Baumberg, R.M. Stevenson, M.S. Skolnick, D.M.
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The system of interacting polaritons: Classical versus quantum kinetic equation I.A. Shelykh a,b , G. Malpuech c,∗ , R. Johne c , A.V. Kavokin d a International Centre for Condensed Matter Physics, Universidade de Brasilia, 70904-970 Brasilia DF, Brazil b St. Petersburg State Polytechnical Univetrsity, 29, Politechnicheskaya, 195251, St-Petersburg, Russia c LASMEA, UMR 6602 CNRS, Universit´e Blaise-Pascal, 24, av des Landais, 63177, Aubi`ere, France d School of Physics and Astronomy, Univeristy of Southampton, SO17 1BJ Southampton, UK
Received 30 January 2007; accepted 1 July 2007 by the Guest Editors Available online 19 August 2007
Abstract We derive the set of quantum kinetic equations for the system of interacting exciton polaritons in microcavities relaxing the Born–Markov approximation. We demonstrate the crucial role played by the nonclassical four particle correlators in the dynamics. The decoherence process is shown to be responsible for the transition from quantum to classical limit. The general formalism is applied for description of the dynamics of the polariton parametric oscillator. c 2007 Elsevier Ltd. All rights reserved.
PACS: 05.30.Jp; 71.35.Lk; 71.36.+c Keywords: D. Quantum kinetics; D. Decoherence; D. Exciton-polaritons
1. Introduction The systems of interacting composite bosons are subject of enhanced interest nowadays. The experimental realization of the Bose–Einstein condensation (BEC) of cold atoms, marked by a Nobel prize in physics of 2004, has stimulated numerous theoretical works in this field. The experimental studies of the solid state composite bosonic systems such as excitons and exciton polaritons have shown that the BCE in semiconductor systems is also possible. The exciton–polaritons can be considered as bosons if their concentration n D is small enough so that the condition is satisfied: n D a BD << 1
(1)
where D is a dimensionality of the system (D = 1, 2, 3), a B is an exciton Bohr radius. The bosonic properties of the exciton systems are in the focus of the condensed matter theory since the pioneering work of Keldysh and Kopaev [1], where ∗ Corresponding author.
E-mail address: [email protected] (G. Malpuech). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.07.042
the possibility of the exciton BEC was theoretically analyzed for the first time. Different types of excitonic systems were subsequently examined for suitability for observation of the BEC, including indirect excitons in coupled quantum wells [2], optically inactive excitons in Cu2 O [3] etc. Recently, it has been shown that exciton polaritons in quantum microcavities (cavity polaritons) are probably the most suitable candidates for the BEC. Being the quasiparticles that combine both exciton and photon properties exciton–polaritons clearly demonstrate a bosonic behaviour [4, 5]. Compared to the bare excitons, cavity polaritons have two main advantages from the point of view of BEC. First, due to their photonic component and light effective mass polaritons are much less affected by localization effects provoked by the disorder. Second, again due the light effective mass of polaritons high occupation numbers of their ground state can be achieved at relatively weak pumping which does not suppress the strong-coupling regime in microcavities. Polariton BEC has been recently experimentally observed in CdTe based microcavities [6] at the temperatures up to 40 K, which would be impossible in any atomic system.
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As the cavity polaritons have a finite life-time, their BEC is always happening out of thermal equilibrium, strictly speaking. The macroscopically occupied state at k = 0, k being the inplane wave-vector, can be formed only if the scattering rate of exciton–polaritons strongly exceeds their radiative decay rate. Thus, the thermodynamic treatment is not sufficient for the polaritonic systems and the kinetic effects are of crucial importance for the BEC issue in most cases. The low density of states in the vicinity of the ground polariton state prevents the energy relaxation of the polaritons assisted by acoustic phonons (“bottleneck effect”) [7], thus forming an obstacle for the dynamical condensation. The most reliable mechanism for overcoming of the bottleneck effect is polariton–polariton scattering, governed by the Coulomb interactions between the excitonic fractions of two colliding polaritons. The matrix element of the polariton–polariton scattering is dependent on the exciton binding energy E B , exciton Bohr radius a B , and the area occupied by the condensate S. Roughly, it can be estimated as [8] Vk,k0 ;k00 ,k000 = hk, k0 |Vint |k00 , k000 i ≈
6E B a 2B ∗ ∗ X k X k0 X k00 X k000 δk+k0 −k00 −k000 S
(2)
where X k is a Hopfield coefficient giving the percentage of the excitonic fraction in the state k, the Dirac delta function ensures the momentum conservation during the scattering act. The description of the energy relaxation of the system of interacting polaritons is usually performed using a set of semiclassical Boltzmann equations [9], which means that all quantum correlations between different states are neglected. This approach gives satisfactory results for the case of nonresonant pumping. However, in the case of resonant excitation, where the microcavity operates as an optical parametric amplifier (OPO), some effects beyond Born–Markov approximation may arise. In the OPO regime the polaritons are excited at the lower dispersion branch by light incident under the magic angle. Resonant polariton–polariton scattering from the pump state populates so-called signal (k = 0) and idler states. The quantum correlations between the signal and idler states play a strong role in OPO dynamics [10]. The derivation of quantum kinetic equations for the system of interacting polaritons in the general case has not yet been reported, to the best of our knowledge. In this paper we report such a derivation and consider the transition between the classical and quantum limit. Quantum correlations are taken into account via nonclassical off-diagonal four-particle correlators. We apply the obtained formalism to describe the dynamics of the polariton paramatric oscillator. We demonstrate that decoherence processes leading to the decay of the nonclassical correlators lead to the transition between the quantum oscillatory regime and the classical relaxation regime for the OPO. We show that in the limit of very fast dephasing the system of quantum equations transforms into the classical system of Boltzmann equations. The spin degree of freedom of the exciton–polaritons is neglected in this work.
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2. Quantum kinetic equation for the system of the interacting bosons The system of the interacting exciton–polaritons can be described by the following model Hamiltonian: X X + H = εk ak+ ak + Uk,q ak+ ak−q (bq + b−q ) k
+
k,q
1 X Vk,k0 ,q ak+ ak+0 ak−q ak0 +q + H.c. 2 0
(3)
k,k ,q
where the operators ak describe annihilation of the exciton– polaritons, the operators bk are annihilation operators for the acoustic phonons. The first term describes the free particles propagation, the second term describes the polariton–acoustic phonon scattering, the third term describes the polariton–polariton scattering. The phonon field can be treated classically. The usual way to deal with it is to use the Born–Markov approximation for the Liouville–von Neumann equation for the density matrix. As a result the system of the Boltzmann-type kinetic equations appears which would satisfactorily describe the dynamics of polariton relaxation with acoustic phonons. The procedure of its derivation is described in the literature (see e.g. Ref. [11]). Here we shall not repeat it, and will neglect the second term of (3) in the further procedure. Instead we shall consider in detail the dynamics of polariton–polariton interactions described by the third term in Eq. (3). Formally, the Born–Markov approximation can be applied also to this term [12], but it can be hardly justified as there is no classical reservoir that can be traced over in this case. We shall go beyond the Born–Markov approximation in the further development. We start with the Liouville–von Neumann equation written with the Hamiltonian (3) excluding its second term: X dρ = [H ; ρ] = εk [ak+ ak ρ − ρak+ ak ] ih¯ dt k +
1X V 0 [a + a +0 ak−q ak0 +q ρ − ak+ ak+0 ak−q ak0 +q ρ]. (4) 2 k,q k,k ,q k k
It yields the following dynamics of the occupation numbers dNk dρ + = Tr ak ak dt dt X 1 = Im[Vk,k0 ,q hak+ ak+0 ak−q ak0 +q i] h¯ k 0 ,q =
1X Im[Vk,k0 ,q Ak,k0 ,q ]. h¯ k 0 ,q
(5)
The right part of Eq. (5) contains the fourth-order nonclassical correlators Ak,k0 ,q = hak+ ak+0 ak−q ak0 +q i = Tr[ρak+ ak+0 ak−q ak0 +q ]. Note that Eq. (5) is obtained from Eq. (4) without any simplifying assumptions. In order to take into account the finite lifetime of exciton–polaritons, the additional term −Nk /τk should be introduced into Eq. (5).
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To complete the set of the kinetic equations one must add an expression for the temporal derivative of Ak,k0 ,q which reads dAk,k0 ,q dρ + + = Tr a a 0 ak−q ak0 +q dt dt k k i = Tr([ρ; H ]ak+ ak+0 ak−q ak0 +q ) h¯ i = (εk + εk0 − εk−q − εk0 +q )Ak,k0 ,q h¯ i X + + + [−Vk,k00 ,q0 hak00 ak−q + 0 ak00 +q0 ak0 ak−q ak0 +q i h¯ 00 0 k ,q
− Vk0 ,k00 ,q0 hak+ ak00 ak+0 −q0 ak+00 +q0 ak−q ak0 +q i
+ ak−q i] − hak+ ak ak+0 ak0 ak0 +q ak+0 +q i − hak+ ak ak+0 ak0 ak−q
≈
F k/k0 /k−q/k0 +q6=k00 = q=±q0
+ Vk0 +q,k00 ,q0 hak+ ak+0 ak−q ak+00 ak0 +q−q0 ak00 +q0 i] i = (εk + εk0 − εk−q − εk0 +q )Ak,k0 ,q h¯ + Fq0 =0 + F k/k0 /k−q/k0 +q=k00 q=±q0 + F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q6=k00
× +
(Vk0 ,k00 ,−q hak+ ak00 ak+0 +q ak+00 −q ak−q ak0 +q i) (Vk−q,k00 ,−q hak+ ak+0 ak+00 ak ak00 −q ak0 +q i)
k00 6=k0 +q
q6=±q0
where we have divided all the terms provided by the polariton–polariton scattering into four groups
X
−
(Vk0 +q,k00 ,q hak+ ak+0 ak−q ak+00 ak0 ak00 i)
k00 6=k−q
F k/k0 /k−q/k0 +q6=k00 q6=±q0
depending on the possibility of their efficient factorization. Below we write them explicitly. 1. The term Fq0 =0 describes the forward scattering Fq0 =0
i h¯
+ ak+00 +q ak+0 ak−q ak0 +q i) (Vk,k00 ,q hak00 ak−q
X
−
(6)
q=±q0
k00 6=k0
X
q=±q0
k00 6=k
+ F k/k0 /k−q/k0 +q6=k00
q6=±q0
X
q=±q0
Fq0 =0 , F k/k0 /k−q/k0 +q=k00 , q=±q0 F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q6=k00 ,
k00 6=k
−
X
Vk00 ,k−q,q Nk Ak00 ,k0 ,q
k00 6=k0 +q
k ,q
− Vk0 +q,k00 ,q0 )hak+ ak+0 ak0 +q ak−q ak+00 ak00 i X i ≈ Ak,k0 ,q (Vk,k00 ,q0 + Vk0 ,k00 ,q0 − Vk−q,k00 ,q0 h¯ 00 0
F k/k0 /k−q/k0 +q=k00 = q6=±q0
(7) ×
q0
,
(9)
i h¯
+ + + [Vk,k0 ,q0 hak0 ak−q 0 ak0 +q0 ak0 ak−q ak0 +q i
+ + Vk0 ,k,q0 hak+ ak ak+0 −q0 ak+q 0 ak−q ak0 +q i +
− Vk−q,k0 +q,q0 hak+ ak+0 ak+0 +q ak−q−q0 ak0 +q+q0 ak0 +q i + ak0 +q−q0 ak0 −q+q0 i] − Vk0 +q,k0 −q,q0 hak+ ak+0 ak−q ak−q
will show below that it has the most important impact on the dynamics of a polariton OPO. It writes
+ + hak ak+ ak−q ak−q ak+0 +q0 ak0 +q i
X
q=±q0
q=±q0
Vk0 +q,k00 ,q Nk0 Ak,k00 ,q
k00 6=k−q
where in the passage from the second to the third line we have used the mean-field approximation. Thus, the considered term provides nothing more then the energy renormalization of the states coupled by polariton–polariton interactons. In the s-wave approximation Vk,k00 ,q0 = Vk0 ,k00 ,q0 = Vk−q,k00 ,q0 = Vk0 +q,k00 ,q0 and thus Fq0 =0 = 0. 2. The term F k/k0 /k−q/k0 +q=k00 is much more interesting. We
i + = Vk,k0 ,q [hak+0 ak0 ak−q ak−q ak+0 +q ak0 +q i h¯
X
−
k ,q
− Vk0 +q,k00 ,q0 )Nk00
i X ≈ V 00 Nk−q Ak00 +q,k0 ,q h¯ 00 0 k,k ,q k 6=k X + Vk00 ,k0 ,q Nk0 +q Ak,k00 +q,q
i X = (V 00 0 + Vk0 ,k00 ,q0 − Vk−q,k00 ,q0 h¯ 00 0 k,k ,q
F k/k0 /k−q/k0 +q=k00
(8)
where again we used the mean-field approximation. The Eq. (8) shows that the dynamics of the fourth-order nonclassical correlator is governed by the dynamics of the occupation numbers, as might have been expected. The term (8) ensures the spontaneous apparition of the nonclassical correlators in the system where initially they are absent. 3. The term F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q6=k00 is given by q6=±q0
+ Vk−q,k00 ,q0 hak+ ak+0 ak+00 ak−q−q0 ak00 +q0 ak0 +q i
q6=±q0
i V 0 [Nk−q Nk0 +q (Nk + Nk0 + 1) h¯ k,k ,q − Nk Nk0 (Nk−q + Nk0 +q + 1)]
≈
i X V 0 0 [(Nk + Nk0 + 1)Ak−q0 ,k0 +q0 ,q−q0 h¯ q0 k,k ,q − (Nk−q + Nk0 +q + 1)Ak,k0 ,q−q0 ].
(9a)
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It describes the mutual influence of the correlators corresponding to various pairs of the initial and final states in the reciprocal space. 4. Finally, the term F k/k0 /k−q/k0 +q6=k00 contains the correlator q6=±q0
of the operators corresponding to six different states in the reciprocal space. It cannot be factorized in the satisfactory manner. However, it rapidly vanishes as the processes of the decoherence in the system lead to the faster decay of this correlator than of anyone of the six-order correlators involving the same states twice or more times. In the following we shall assume F k/k0 /k−q/k0 +q6=k00 = 0. q6=±q0
The set of Eqs. (5)–(9) describes the dynamics of the bosonic system accounting for particle–particle interactions. Note that the Markov approximation has not been used while deriving Eqs. (5)–(9). Consequently, Eqs. (5)–(9) allow for the energynonconservative processes and may predict a qualitatively different dynamics of the system with respect to the Boltzmann equations. We remind that the Boltzmann equations only contain the occupation numbers, while in the system (5)–(9) the correlations between different states in the reciprocal space are described by means of the fourth-order nonclassical correlators. 3. Decoherence and classical limit
q6=q0
Ak,k0 ,k−q,k0 +q = =
q=±q0
1
τdec
−
i h¯ (εk
+ εk0 − εk−q − εk0 +q ) iVk,k0 ,q
h¯ (εk + εk0 − εk−q − εk0 +q ) + i τdec × [Nk−q Nk0 +q (Nk + Nk0 + 1) − Nk Nk0 (Nk−q + Nk0 +q + 1)].
(11)
Substituting the expression for Ak,k0 ,q into the equation for the occupation numbers (5) one readily obtains a familiar set of the semiclassical Boltzmann equations: X dNk = Wk,k0 ,q [Nk−q Nk0 +q (Nk + Nk0 + 1) dt 0 − Nk Nk0 (Nk−q + Nk0 +q + 1)],
(12)
where the scattering rates are given by Wk,k0 ,q =
|Vk,k0 ,q |2 h¯ ×
1/τdec 2 (εk + ε − εk−q − εk0 +q )2 / h¯ 2 +1/τdec
.
(13)
k0
|V
0
|2 τdec
given by the Fermi golden rule, Wk,k0 ,q = k,k ,qh¯ . For the energy nonconserving processes the probability of scattering is reduced by a Lorentzian factor.
q6=±q0
q=±q0
F k/k0 /k−q/k0 +q=k00
One can see from Eq. (13), that the scattering rate is the fastest for the energy-conserving processes, where it is simply
equation for Ak,k0 ,q in the following form dAk,k0 ,q i 1 = (εk + εk0 − εk−q − εk0 +q ) − Ak,k0 ,q dt h¯ τdec + F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q=k00 + F k/k0 /k−q/k0 +q6=k00
q=±q0
k ,q
Until now we did not discuss the decoherence processes specifically. These processes are necessarily present and important in all realistic systems. In particular they come from the interaction of polaritons with a classical phonon reservoir and from the polariton–polariton forward scattering beyond the Born approximation [13]. Both these effects contribute to the temporal decay of the correlators. Introducing the decoherence time τdec and neglecting F k/k0 /k−q/k0 +q6=k00 one can rewrite the
q=±q0
in the chemical kinetics problems, where all the intermediate products are considered to be in the quasi-equilibrium (in our case the fourth order correlator is indeed an “intermediate product”!). These simplifying assumptions allow us to estimate the correlators Ak,k0 ,q as follows dAk,k0 ,k−q,k0 +q i 1 (εk + εk0 − εk−q − εk0 +q ) − = h¯ dt τdec × Ak,k0 ,k−q,k0 +q + F k/k0 /k−q/k0 +q=k00 = 0
4. Application: Polariton parametric oscillator
.
(10)
Then, assuming that the decoherence time is short enough, 1 so that V Ntot << τdec , where Ntot is a total number of the polaritons in the system and V is a mean value of the matrix element, one can neglect the term F k/k0 /k−q/k0 +q=k00 + q6=±q0
F k/k0 /k−q/k0 +q6=k00 in (10). This term can also be neglected q=±q0
because it contains a sum of the fourth-order correlators corresponding to the different quantum states having random phases, in general, so that altogether they give a vanishing contribution. One can also assume that if the decoherence processes are strong the correlators Ak,k0 ,q reach their equilibrium values much faster than occupation numbers. This assumption allows us to separate the variables in our system into slowly changing ones (occupation numbers) and fast changing ones (correlators). Such a separation is routinely used
To illustrate the general theory developed above, let us consider a simple example of the bosonic system for which the number of states in the reciprocal space is sufficiently reduced. In the polariton OPO configuration, due to the strong nonparabolicity of the lower polariton branch, a pair of exciton–polaritons created by the pump pulse incident under the magic anlge scatters into the non-degenerate signal and idler states with both energy and momentum conservation [14]. The scattering to all other states can be neglected in the first order as it does not conserve energy. A reliable model considering the polariton OPO may account for only three quantum states, namely the pump, signal and idler states. The Hamiltonian of this system can be written in the following form: + H = (εs as+ as + εi a + p a p + εi ai ai ) + + U (X s as+ as + X p a + p a p + X i ai ai )
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I.A. Shelykh et al. / Solid State Communications 144 (2007) 378–383 + × (X s as+ as + X p a + p a p + X i ai ai ) + + + + V (a + p a p as ai + a p a p as ai )
(14)
where the indices p, s and i correspond to the pump, signal and idler respectively. The first term in the right part of Eq. (14) describes the free particles propagation, the second term describes the polariton energy blueshifts for the three states under consideration (X is a percentage of the exciton fraction in each of the states), √ the third term describes the parametric process, V ≈ X p X i X s U . The previous models of the polariton OPO either consider all three states involved in the parametric process as the classical fields coupled by a four-wave mixing process [15] or consider the case of cw pumping neglecting the pump depletion [10]. The exception is the recent work of Glazov and Kavokin [16] where the hyperspin formalism was applied for the analysis of the parametric amplifier. While it allows up to some point for description of quantum correlations in a threelevel system, its extension to more complex systems would require extremely heavy analytics computations. Besides, the role of the decoherence processes as well as a reduction of the quantum kinetic equation to semiclassical Boltzmann equation in the limit of strong decoherence is not straightforward within the hyperspin approach. On the other hand, the complete set of quantum kinetic equations for a system of interacting bosons presented in this work allows for description of the transition from quantum to classical limit. Let us apply the formalism developed above for the description of the cavity OPO. The Eqs. (5)–(9) in this case reduce to dNs Ns 2 = − + Im{V has+ ai+ a p a p i} dt τs h¯ Ns 2 = − + Im{V A}, τs h¯ Ni 2 dNi =− + Im{V A}, dt τi h¯ Np dN p 4 =− − Im{V A}, dt τp h¯ 1 1 1 1 dA A = − + + + dt 2τs 2τi τp τdec i + [(εs + εi − 2ε p ) + U (X i + X s − 2X p ) h¯ × (X i Ni + X s Ns + X p N p )]A+ iV + [N p2 (Ns + Ni + 1) − 4Ns Ni (N p + 1)], h¯
(15) (15a) (15b)
(15c)
where we have introduced the polariton lifetimes τi . The system of Eq. (15) is equivalent to one describing a damped nonlinear oscillator. It takes into account the depletion of the pump and the spontaneous scattering and thus can be used both above and below the stimulation threshold, unlike the models of Refs. [10,11]. The dynamics of the occupation numbers given by Eq. (15) also differ sufficiently from those given by a set of the semiclassical Boltzmann equations. The Boltzmann equations
predict the monotonous temporal dependence of the occupation numbers, while the system (15) allows for their oscillatory behaviour. The period of the oscillations depends on the total number of the particles in the system Ntot , T ∼ V N1 tot . If the 1 V Nh¯ tot , these oscillations are washed damping is strong, τdec out and the dynamics of the occupation numbers become classical.
5. Conclusions In conclusion, we have derived the system of quantum kinetic equations for the system of interacting bosons which goes beyond the Markov approximation. The dynamics of the occupation numbers is shown to be closely lined with the build-up of the nonclassical four-particle correlators. The decoherence process leading to the fast decay of these correlators is shown to provoke the transition from the quantum to the classical limit. The system of the kinetic equations we obtain can be applied for description of the dynamics of the polariton OPO. Acknowledgements This work has been supported by the STREP project “STIMSCAT” 517769 and by the Chair of Excellence of ANR. I. A. Shelykh acknowledges Brazilian Ministry of Science and technology and IBEM, Brazil for support. References [1] L.V. Keldysh, Yu.V. Kopaev, Fiz. Tverd. Tela (Leningrad) 6 (1964) 2791; Sov. Phys. Solid State 6 (1965) 2219. [2] Yu.E. Lozovik, V.I. Yudson, Pis’ma Zh. Eksp. Teor. Fiz. 22 (1975) 26; Sov. Phys. JETP Lett. 22 (1975) 26; Yu.E. Lozovik, V.I. Yudson, Solid State Comm. 18 (1976) 628; A.L. Larionov, V.B. Timofeev, J. Hvam, K. Soerensen, Zh. Eksp. Teor. Fiz. 117 (2000) 1255; JETP 90 (2000) 1093; L.V. Butov, C.W. Lai, A.L. Ivanov, A.C. Gossard, D.S. Chemla, Nature 417 (2002) 47; D. Snoke, S. Denev, Y. Liu, L. Pfeiffer, K. West, Nature 418 (2002) 754; O. Berman, Yu.E. Lozovik, D.W. Snoke, R.D. Coalson, Phys. Rev. B 70 (2004) 235310. [3] D. Snoke, J.P. Wolfe, A. Mysyrowicz, Phys. Rev. Lett. 89 (1987) 527; D. Snoke, J.P. Wolfe, A. Mysyrowicz, Phys. Rev. B 41 (1990) 11171; Jia Ling Lin, J.P. Wolfe, Phys. Rev. Lett. 71 (1993) 1222; ´ S´uilleabh´ain, J.P. Wolfe, Phys. Rev. B 60 (1999) 10565; K.E. O’Hara, L. O G.M. Kavoulakis, Phys. Rev. B 65 (2001) 035204. [4] P.G. Savvidis, J.J. Baumberg, R.M. Stevenson, M.S. Skolnick, D.M. Whittaker, J.S. Roberts, Phys. Rev. Lett. 84 (2000) 1547. [5] H. Deng, G. Weihs, C. Santori, J. Bloch, Y. Yamamoto, Science 298 (2002) 199. [6] J. Kasprzak, et al., Nature 443 (2006) 409. [7] F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani, P. Schwendimann, Phys. Rev. B 56 (1997) 7554. [8] A. Kavokin, G. Malpuech, Cavity Polaritons, Elsevier North Holland, ISBN: 0-125-33032-4, 2003. [9] D. Porras, C. Ciuti, J.J. Baumberg, C. Tejedor, Phys. Rev. B 66 (2002) 085304. [10] C. Ciuti, P. Schwendimann, A. Quattropani, Phys. Rev. B 63 (2001) 041303. [11] H.J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equation and Fokker–Planck Equations, Springer, ISBN: 3540548823, 2003.
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