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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15183–15194
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Nonlinear effects in multi-photon polaritonics

A. A. Pervishko, T. C. H. Liew, V. M. Kovalev, I. G. Savenko, and I. A. Shelykh  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15183-15194 (2013)
http://dx.doi.org/10.1364/OE.21.015183


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Abstract

We consider theoretically nonlinear effects in a semiconductor quantum well embedded inside a photonic microcavity. Two-photon absorption by a 2p exciton state is considered and investigated; the matrix element of two-photon absorption is calculated. We compute the emission spectrum of the sample and demonstrate that under coherent pumping the nonlinearity of the two photon absorption process gives rise to bistability.

© 2013 OSA

1. Introduction

Polaritonics is a rapidly developing branch of science lying at the intersection of the physics of semiconductors, quantum mechanics and nonlinear optics. The interest in this field has been stimulated by a series of fascinating experimental and theoretical discoveries, which demonstrate that a variety of quantum collective phenomena can be observed in polariton systems including: Bose-Einstein condensation (BEC) [1

1. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose-Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006) [CrossRef] [PubMed] .

3

3. 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, “Coherent zero-state and p-state in an exciton-polariton condensate array,” Nature 450(7169), 529–532 (2007) [CrossRef] [PubMed] .

] and superfluidity [4

4. A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnick, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009) [CrossRef] [PubMed] .

6

6. I. Carusotto and C. Ciuti, “Probing microcavity polariton superfluidity through resonant Rayleigh scattering,” Phys. Rev. Lett., 93(16), 166401 (2004) [CrossRef] .

]; the Josephson effect [7

7. K. G. Lagoudakis, B. Pietka, M. Wouters, R. Andre, and B. Deveaud-Pledran, “Coherent oscillations in an exciton-polariton Josephson junction,” Phys. Rev. Lett. 105(12), 120403 (2010) [CrossRef] [PubMed] .

]; and the formation of various topological excitations such as vortices [8

8. K. G. Lagoudakis, T. Ostatnicky, A. V. Kavokin, Y. G. Rubo, R. Andre, and B. Deveaud-Pledran, “Observation of half-quantum vortices in an exciton-polariton condensate,” Science 13(5955), 974–976 (2009) [CrossRef] .

10

10. G. Nardin, G. Grosso, Y. Leger, B. Pietka, F. Morier-Genoud, and B. Deveaud-Pledran, “Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum fluid,” Nature Phys. 7, 635–641 (2011) [CrossRef] .

] and solitons [11

11. A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemnager, R. Houdr, E. Giacobino, C. Ciuti, and A. Bramati, “Polariton superfluids reveal quantum hydrodynamic solitons,” Science 332(6034), 1167–1170 (2011) [CrossRef] [PubMed] .

14

14. R. Hivet, H. Flayac, D. D. Solnyshkov, D. Tanese, T. Boulier, D. Andreoli, E. Giacobino, J. Bloch, A. Bramati, G. Malpuech, and A. Amo, “Half-solitons in a polariton quantum fluid behave like magnetic monopoles,” Nature Phys. 8, 724–728 (2012) [CrossRef] .

], as well as other spatial patterns [15

15. F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. Andre, and B. Deveaud-Pledran, “Spontaneous pattern formation in a polariton condensate,” Phys. Rev. Lett. 107(10), 106401 (2011) [CrossRef] [PubMed] .

17

17. E. Kammann, T. C. H. Liew, H. Ohadi, P. Cilibrizzi, P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, A. V. Kavokin, and P. G. Lagoudakis, “Nonlinear optical spin Hall effect and long-range spin transport in polariton lasers,” Phys. Rev. Lett. 109(3), 036404 (2012) [CrossRef] [PubMed] .

]. While most fundamental studies have been performed at low temperature, experiments in GaN [18

18. S. Christopoulos, G. Baldassarri Höger von Högersthal, A. Grundy, P. G. Lagoudakis, A. V. Kavokin, J. J. Baumberg, G. Christmann, R. Butté, E. Feltin, J.-F. Carlin, and N. Grandjean, “Room-temperature polariton lasing in semiconductor microcavities,” Phys. Rev. Lett. 98(12), 126405 (2007) [CrossRef] [PubMed] .

, 19

19. Ayan Das, Junseok Heo, Marc Jankowski, Wei Guo, Lei Zhang, Hui Deng, and Pallab Bhattacharya, “Room temperature ultralow threshold GaN nanowire polariton laser,” Phys. Rev. Lett. 107(6), 066405 (2011) [CrossRef] [PubMed] .

], ZnO [20

20. R. Schmidt-Grund, B. Rheinlnder, C. Czekalla, G. Benndorf, H. Hochmut, A. Rahm, M. Lorenz, and M. Grundmann, “ZnO based planar and micropillar resonators,” Superlattic. Microstruct. 41(5–6), 360–363 (2007) [CrossRef] .

] and organic [21

21. S. Kena-Cohen and S. R. Forrest, “Room-temperature polariton lasing in an organic single-crystal microcavity,” Nature Photonics 4, 371–375 (2010) [CrossRef] .

] microcavities demonstrate devices working at room temperature. Now it becomes obvious, that polaritonics has a huge potential for technological applications [22

22. T. C. H. Liew, I. A. Shelykh, and G. Malpuech, “Polaritonic devices,” Physica E 43(9), 1543–1568 (2011) [CrossRef] .

].

Polaritonics stems from the proposal that quantum microcavities can be used as efficient sources of THz radiation [23

23. K. V. Kavokin, M. A. Kaliteevski, R. A. Abram, A. V. Kavokin, S. Sharkova, and I. A. Shelykh, “Stimulated emission of terahertz radiation by exciton-polariton lasers,” Appl. Phys. Lett. 97(20), 201111 (2010) [CrossRef] .

], which needs efficient pumping of the 2p excitonic state. It can not be excited under single photon absorption due to optical selection rules. In one of the recent publications [24

24. A. V. Kavokin, I. A. Shelykh, T. Taylor, and M. M. Glazov, “Vertical cavity surface emitting terahertz laser,” Phys. Rev. Lett. 108(19), 197401 (2012) [CrossRef] [PubMed] .

], some theoretical aspects of THz radiation emission under two-photon excitation were considered. The consideration was based on the master equation approach. The two-photon absorption and THz emission were investigated as incoherent processes. This kind of consideration, however, is not always applicable. One can imagine the situation when the two-photon transition is tuned in resonance with the energy of the confined electromagnetic mode of a cavity. In this case, the coupling is of resonant character and the theoretical description should be performed along another route. Theoretical consideration of this regime is the subject of the present paper.

We consider a cavity which is subject to two-photon resonant excitation, in which takes place, pumping of the dark (2p) exciton state. Instead of considering the strong coupling between near resonant photons and 1s excitons [25

25. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69(23), 3314–3317 (1992) [CrossRef] [PubMed] .

], we uncover a different kind of exciton-polariton generated by the non-linear coupling of photon pairs with 2p excitons. Similiar to the conventional case, a characteristic splitting of the photoluminescence spectrum is observed. However, the value of the Rabi splitting now becomes dependent on the intensity of the pump due to the nonlinear nature of the two photon absorption. This nonlinearity also results in the onset of bistable behavior.

2. Polaritonics with 2p dark exciton state

We consider the following geometry (Fig. 1): a quantum well (QW) with 2p excitonic transition having energy h̄ωp placed inside a planar microcavity supporting the photonic mode of energy ω2cωp/2. In this configuration the cavity mode can resonantly interact with the 2p excitonic state.

Fig. 1 Geometry of the structure. We consider a microcavity, which is made by Bragg mirrors, with the quantum well inside, where a 2p excitonic state can be excited by two photons(red curves) with energy h̄ω2c each.

The Hamiltonian of the system reads:
^=h¯ωpp^p^+h¯ω2ca^a^+g(p^a^a^+p^a^a^),
(1)
where the operators , â correspond to 2p dark excitons and photons respectively and satisfy bosonic commutation relations. The first two terms in Eq. (1) describe free excitons and photons and the last term corresponds to the resonant interaction between them, where g is the two photon-exciton coupling constant calculated in Appendix using second-order perturbation theory:
g=S2(qA0μ)2niEgm022m*Φn(0)drR21(r)rR10(r)2ω(EgEn+ω)im0h¯2(E2pEns),
(2)
where S is a quantization area of the sample; m0, m* and μ are the free electron, effective and reduced exciton masses, respectively; q is the elementary charge of the electron; Eg is the band gap of the material; ω is the energy of the photon; En are the eigenvalues of the 2D Hydrogen atom; Φn(r⃗) and R21(r⃗), R10(r⃗) are normalized angular and radial eigenfunctions of the 2D Hydrogen atom, which are written explicitly in the Appendix, Eqs. (30) and (31), respectively, and A⃗ is the vector potential, which can be writen in the dipole approximation:
A=eA0,A0=h¯2εε0ωLS,
(3)
where ε0 and ε are vacuum and material permitivities; L is the length of the cavity.

2.1. Energy levels for the case of an ideal cavity

Note, that the Hamiltonian (Eq. (1)) commutes with the excitations’ number operator = ââ+ 2, that means that the eigenvectors of these two operators coincide and the Hilbert space of the problem can be represented as a direct product of the manifolds corresponding to given numbers of N. Each manifold consists of ⌊N/2⌋ basis vectors: |N, 0〉, |N − 2, 1〉,....,|0, N/2〉 for even N and |N, 0〉, |N − 2, 1〉,....,|1, N − 1/2〉 for odd N. Here the first number in kets corresponds to the number of photons, and the second number to the number of 2p excitons. The matrix element of Hamiltonian (Eq. (1)) between any two vectors belonging to manifolds with different N is zero. This means that determination of the spectrum of the problem consists in diagonalization of N × N matrices, each of which has a finite size.

For N = 0 and N = 1 the result is trivial and the corresponding energies are E0 = 0 and E1 = h̄ω2c respectively. The cases of N = 2, 3 are also straightforward to consider, as the size of the reduced Hamiltonian for the manifold is 2×2 in these cases:
2=(2h¯ω2cg2g2h¯ωp),3=(3h¯ω2cg6g6h¯(ωp+ω2c)),
(4)
and the eigenenergies in the case of resonance 2ω2c = ωp are
E2=2h¯ωc±g2,E3=3h¯ωc±g6.
(5)
Consideration of the states with larger numbers of photons requires diagonalization of matrices of higher size. The energy spectrum is represented in Fig. 2(a).

Fig. 2 (a) Energy levels of the QW with 2p excitonic transition placed inside the micro-cavity. We consider the case when the exciton energy is h̄ωp and the photon energy is ω2cωp/2. Possible transitions for N = 1, 2, 3, 4 manifolds for the ideal cavity are represented on the right side of the figure (a). (b) Emission spectrum of the system. The intensity of transitions from individual levels is presented by the blue lines. We consider the case when ω2cωp/2 = 0.71eV, g = 0.025meV and the average number of the excitations in the system 〈N〉 = 30 and the statistics of photons is Poissonian. Including a Lorentzian broadening of each transition line, gives the double peak spectrum of emission shown by the red curve. We consider broadening equals 0.05meV. One can define the value of the Rabi splitting in the system as the distance between the two peaks.

2.2. Spectrum of emission

If the mirrors of the cavity are not perfect and some photons tunnel through them, the system can emit in the outside world. The spectrum of emission is determined by the energy distances between the levels corresponding to N and N − 1 manifolds and the intensities of the transitions will be proportional to squares of the matrix elements of the photon annihilation operator Iif ∼ |〉Ψout|âin〉|2pin), where |Ψin〉 is an initial wavefunction in the manifold with N cavity photons and |Ψout〉 is a final wavefunction in the manifold with (N-1) cavity photons. pin) represents the probability of occupation of the initial state [26

26. I. G. Savenko, O. V. Kibis, and I. A. Shelykh, “Asymmetric quantum dot in a microcavity as a nonlinear optical element,” Phys. Rev. A 85(5), 053818 (2012) [CrossRef] .

]. Likewise assume that the number of excitations in the system is sufficiently to apply the central limit theorem. In this case the occupation of states with different numbers of particles is described by a Poisson distribution:
p(λ;N)=λNeλN!.
(6)
Figure 2(b) shows the total photoluminescence emission spectrum.

It is well-known, that in the case of one photon and a resonant 1s bright exciton in a standard semiconductor microcavity the energy splitting does not depend on the occupation number and equals ΔΩR,S = 2gs, if gs is the bright exciton- single photon coupling strength. Considering our system with two-photon absorption for large N (N >> 1), and substituting a, aN one immediately obtains ΔΩR2gN. Accounting for finite lifetime of the photons and excitons, one obtains [27

27. A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, (2007)) [CrossRef] .

]:
ΔΩR4g2N(γaγp)24.
(7)

Varying the number of the excitations, which can be done by increasing of the pump power, one can thus change the Rabi splitting in the system. Note that coupling constant g scales with the quantization area as g ∼ (S)−1/2, so g2N depends on the concentration of the photons na = N/S only.

3. Classical field equations for coherent pump

To investigate the dynamics of the system under coherent pump with account for finite lifetimes of the photons and dark excitons in the cavity, the approach should be modified. We will use a system of coupled nonlinear equations for the dynamics of the macroscopic wavefunctions describing cavity photons and dark excitons. The latter can be obtained in the following way. First, let us write the Heisenberg equations of motion for the operators and â:
ih¯dp^dt=[p^,^]=h¯ωpp^+ga^2,
(8)
ih¯da^dt=[a^,^]=h¯ωaa^+2gp^a^.
(9)

Then, one should take averages of the above equations thus passing from the operators to the mean values as → 〈〉 = Tr{ρp̂}. We present a mean product of several operators by products of the mean values of the operators (e.g. 〈p̂â〉 → 〈〉〈â*, 〈â2〉 → 〈â2). This approximation is relevant for coherent statistics. Finally, we phenomenologically introduce the lifetimes of the modes and external coherent pumping of cavity photons (by a coherent laser beam) with frequency ω. As a result, Eqs. (8) and (9) transform into the following system of nonlinear differential equations:
dp^dt=(iωp+γph¯)p^iΩpa^2,
(10)
da^dt=(iωa+γah¯)a^2iΩpp^a^*ih¯Peiωt.
(11)
where γp,a = 1/(2τp,a) with τp,a being lifetimes of the modes, P is the amplitude of the coherent pump and Ωp = g/h̄. Making a substitution ââeiωt, p̂e−2iωt, in the stationary regime the equations read:
(ih¯Δp+γp)p+ih¯Ωpa2=0,
(12)
(ih¯Δa+γa)a+2ih¯Ωppa*=iP,
(13)
where Δp = ωp − 2ω, Δa = ωaω, which can be reduced to:
(ih¯Δa+γa)a+2h¯2Ωp2ih¯Δp+γp|a|2a+iP=0.
(14)
This result can also be written in terms of the real functions describing the number of the photons, Na = |a|2 and excitons, Np = |p|2:
Na[1+c1Na+c2Na2]=Ia,
(15)
Np=g2Na2γp2+h¯2Δp2,
(16)
where
c1=4g2(γaγph¯2ΔaΔp)(h¯2Δp2+γp2)(h¯2Δa2+γa2),
(17)
c2=4g4(h¯2Δp2+γp2)(h¯2Δa2+γa2),
(18)
Ia=|P|2h¯2Δa2+γa2.
(19)

Equation (15) is a cubic equation for Na. The results are represented in Fig. 3(a) for the following parameters corresponding to the system: γa = 0.05 meV, γp = 0.01 meV, and three different values of detunings Δa and Δp. These curves reveal an S-shaped dependence of the mode populations on the intensity of the pump, which characterizes the phenomenon of bistability, which is well-known in semiconductor microcavities [28

28. V. D. Kulakovskii, A. I. Tartakovskii, D. N. Krizhanovskii, A. Armitage, J. S. Roberts, and M. S. Skolnick, “Two-dimensional excitonic polaritons and their interaction,” Phys. Usp. 43(8), 853–857 (2000) [CrossRef] .

34

34. D. Bajoni, E. Semenova, A. Lematre, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R. Kuszelewicz, and J. Bloch, “Optical bistability in a GaAs-based polariton diode,” Phys. Rev. Lett. 101(26), 266402 (2008) [CrossRef] [PubMed] .

]. If the value of the pumping intensity lies in the bistable zone, the system can, in principle, occupy more than one possible state, and the particular choice of this state depends on the history of the evolution.

Fig. 3 Considering the finite lifetime of the photons and external coherent pumping of the cavity, we observed a bistable behaviour of the number of photons on intensity of the coherent pump. (a) The parameters are varied for detuning of the system: (red dashed line) Δa = 0.375meV, Δp = 0.75meV; (green dot line) Δa = 0.25meV, Δp = 0.5meV; (blue bold line) Δa = 0.125meV, Δp = 0.25meV. The influence of the exciton-exciton interaction on the hysteresis curve is illustrated in figure (b), where the cases of αp=0 (red dashed curve) and αp=6EBaB2/S (red solid curve) are presented. The detuning parameters are Δa = 0.375meV, Δp = 0.75meV.

To define the constant αp, we use the formula αp6EBaB2/S, where EB is the exciton binding energy, aB is the exciton Bohr radius and S is the laser spot area [42

42. F. Tassone and Y. Yamamoto, “Exciton-exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons,” Phys. Rev. B 59(16), 10830–10842 (1999) [CrossRef] .

]. In QWs based on GaAs-based alloys, we can use the parameters: EB ≈ 19.2 meV and aB = 5.8 nm. The result is presented in Fig. 3(b).

As one can see the exciton-exciton interaction term changes the shape of the bistable behavior of the system. It becomes more extended on intensity of the coherent pump. The density of the excitons compared with the non-interacting case decreases as well. We therefore conclude that the influence of the nonlinearity of the two-photon absorption on the bistability of the system is comparable with the contribution of exciton-exciton interactions. For simplicity, here we have considered a single spin degree of freedom, which corresponds to the case in which a circularly polarized optical pump excites 2p excitons with a single spin polarization. Indeed, we do expect a variety of spin sensitive phenomena to take place in the system with different polarizations of pumping. Note that the polarization of 2p excitons can still be controlled via the polarization of the optical pumping, allowing analogous effects to those in conventional microcavities [43

43. I. A. Shelykh, A. V. Kavokin, Yuri G. Rubo, T. C. H. Liew, and G. Malpuech, “Polariton polarization-sensitive phenomena in planar semiconductor microcavities”, Semicond. Sci. Technol. 25(1), 013001 (2010) [CrossRef] .

].

4. Conclusions

We developed a theoretical approach for the description of strong light-matter interaction in a system with two-photon absorption processes. We calculated the photoluminescence spectrum, where a splitting occurs due to the coupling between photons and 2p dark-excitons in the two photon absorption process. We developed the mean-field equations, valid under coherent excitation of the system, and demonstrated that the nonlinearity of the two-photon absorption process gives rise to the phenomenon of bistability. The two-photon to 2p exciton matrix element was accurately calculated. This formalism is not only directly applicable for device construction, but also highlights the nonlinear nature of the system, which may be useful for the realization of quantum optical devices under coherent pumping [44

44. A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73(19), 193306 (2006) [CrossRef] .

, 45

45. B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Generation of entangled photon holes using quantum interference,” Phys. Rev. A 74(4), 010303()R (2006).

].

5. Appendix

In the present appendix we calculate the matrix element of two-photon absorption using second order perturbation theory. The excitation of the 2p exciton state can occur, mediated by a virtual state representing the state of the system after the absorption of a photon. The virtual states of s-type are the only ones allowed by selection rules and the transition from an s-type state to the 2p exciton state occurs with the absorption of the second photon. The total two-photon absorption matrix element is given by summing over all virtual s-type states.

The Hamiltonian in the case of exciton-photon interaction can be written in the form:
^=(p^eqeAe)22me+(p^hqhAh)22mh+U(rerh),
(24)
where qe and qh are the elementary charges of an electron and hole, respectively; me and mh are the electron and hole effective masses, respectively; U(r⃗er⃗h) is the electron-hole interaction potential; p^e and p^h are the quasi-momenta; A⃗ is the vector potential of the electromagnetic field. In the dipole approximation, the vector potential does not depend on coordinate and reads:
Ae=Ah=eA0,A0=h¯2εε0ωV,
(25)
where e⃗ is a unit vector describing the light polarization. Here ε0 and ε are the vacuum and the relative permittivity, respectively; ω is the frequency of exciting light; V is the cavity volume, V ≈ (λ/2n)S, where λ is the wavelength of the fundamental cavity mode; n is the refractive index of the material and S is the area of the laser spot which roughly defines the area in which excitons exist. Keeping the terms linear in A⃗ and introducing the center of mass coordinates, we obtain the equation for the perturbation operator:
W^=qA0μep^,
(26)
where p^ is the quasi-momentum of the relative electron-hole motion, q = |qe| = |qh| and μ = memh/(me + mh) is the reduced mass. The intrinsic motion of the exciton can be described by the Hamiltonian:
^=p^22μ+U(r).
(27)
We will not consider the motion of the center of mass, as in dipole approximation it does not feel any influence of light at normal incidence. The eigenfunctions and eigenstates will be similar in form to eigenfunctions and eigenstates for the 2D Hydrogen atom with the difference that the reduced mass ’μ’ replaces the free electron mass, the material permitivity replaces the vacuum permitivity and energies are taken with respect to the band-gap [46

46. X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys.Rev. A 43(3), 1186–1196 (1991) [CrossRef] [PubMed] .

]. The eigenstates for a 2D systen are described by the Schrödinger equation, which in polar coordinates reads:
(h¯22μ[2r2+1rr+1r22ϕ2]q24πε0εr)ψ(r,ϕ)=(EEg)ψ(r,ϕ),
(28)
where Eg is the semiconductor band-gap.

Using separation of variables
ψ(r,ϕ)=R(r)Φ(ϕ),
(29)
we obtain the normalized angular and radial eigenfunctions:
Φ(ϕ)=1(2π)1/2eilϕ,l=0,±1,±2,
(30)
Rnl(r)=βn(2|l|)![(n|l|1)!(2n1)(n|l|1)!]1/2(βnr)|l|eβnr/2F11(n+|l|+1,2|l|+1,βnr),
(31)
where n and l are principle and angular momentum quantum numbers, respectively;
βn=24πε0ε(n1/2)μe2h¯2,
(32)
and 1F1 is the confluent hypergeometric function.

The 2D eigenvalues are:
En=Eg12(4πε0ε)2(n1/2)μq4h¯2.
(33)

The process we are interested in goes through the virtual state, and the transition can be described by means of second-order perturbation theory. The corresponding matrix element is:
Mfi=ξf|W^|ξξ|V^(1)|iEiEη+f|V^(2)|i.
(34)
Here (1) and (2) are perturbation operators of the first and second order, respectively. In Eq. (34) summation over ξ means summation over η (the virtual levels) and integration over momentum. Here i is the initial state: there are 0 excitons, 2 photons in the system. It can be described by:
|i=δ(rerh),Ei=2ω,
(35)
η is a virtual state with one exciton and one photon:
|η=Fη(re,rh)=eiPch¯RcSψη(r),Eη=EgEn+ω.
(36)
Here f denotes a final state, which describes excitation of the 2p level. Substituting the wave-functions into the equation for the matrix elements, one can obtain:
η|V^(1)|i=qA0μ(ep^)cvdredrhδ(rerh)Fη(re;rh)==qA0μ(ep^)cvRη(0)1S(2πh¯)2δ(Pc),
(37)
f|W^|η=qA0μdRcdrFη*(re;rh)ep^Fη((re;rh))==qA0μδ(PcPc)1S(2πh¯)2drRη*(r)ep^Rη(r)==qA0μ(ep^)ηη1S(2πh¯)2δ(PcPc),
(38)
where (ep^)cv is the matrix element of transition between valence and conduction band Bloch functions. The matrix element 〈f|(2)|i〉 is equal to zero as a consequence of orthogonality of the Bloch amplitudes of the conduction and valence bands.

The full matrix element takes the form:
Mfi=S(qA0μ)2η(ep^)ηη(ep^)cvψη(0)2ω(EgEn+ω).
(39)
First, let us find the interband matrix element 〈η|(1)|i〉. Built on the Bloch functions, this matrix element can be estimated from the k · p method:
1m*2|(ep^)cv|2Egm02,
(40)
or using rudimentary considerations,
ih¯dx^dt=ih¯p^m0=[x^;^]c|p^|vih¯m0aEg.
(41)
Thus,
(ep^)cvih¯Egam0,
(42)
where Eg is the bandgap and a is the lattice constant.

Second, one should find the term (e⃗p⃗)η′η. In polar coordinates, the momentum operator can be presented as:
p^=ih¯^=ih¯(r^r+ϕ^1rϕ),
(43)
and for circularly polarized light:
ep^=ih¯2eiϕ(r+i1rϕ).
(44)
Due to angular momentum conservation, the possible η states have the orbital momentum equal to zero. The radial wavefunction is then:
Rn0=βn12n1eβnr2F11(n1,1,βnr)==βn12n1eβnr2Ln10(βnr).
(45)
We are interested in the case when the final state is the 2p state, n = 2, l = 1 :
R21=β226reβ2r2.
(46)
Using definition of momentum we can switch from the momentum to coordinate operator:
2p|p^|1s=2p|m0ih¯[^,r^]|1s=im0h¯(E2pE1s)2p|r^|1s.
(47)
As a result, the matrix element can be presented in form:
f|W^|η=qA0μδ(PcPc)(2πh¯)2Sδ(PcPc)drRη*(r)ep^Rη(r)==qA0μδ(PcPc)(2πh¯)2SdrRη*(r)eiϕ2(ih¯r)Rη(r)==qA0μδ(PcPc)(2πh¯)2Sdreiϕ2im0h¯(EηEη)Rη*(r)rRη(r),
(48)
2p|W^|ns=qA0μδ(PcPc)eiϕ2im0h¯(E2pEns)(2πh¯)2SdrR2p*(r)rR1s(r).
(49)

The overlap integral can be calculated as:
drR21(r)rRn0(r)=243(n2)n3(n+1)n2(2n1)5(μq2/(πε0εh¯2))4β22βn6(2n1).
(50)
We did not take into account n = 2, since the energy splitting between the 2s and 2p levels is insignificant compared to the other levels (causing (E2pE2s) to vanish). Finally,
M2p,i=S(qA0μ)2niEgm022m*Φn(0)drR21(r)rR10(r)2ω(EgEn+ω)im0h¯2(E2pEns).
(51)
For a GaAs structure, a = 5.6 Å, S = πR2, R = 0.1 μm, Eg = 1.42 eV, ε = 12.9, n = 3.312, λ = 800 nm, μ = 0.058m0, m* = 0.067m0. This matrix element is connected with the interaction constant ’g’, which describes interaction between photons and exciton in the Hamiltonian = gp̂â+â+ + gp̂+ââ.
M2p,i=2p|^|2ω=g2p|p^a^+a++p^+a^a^|2ω=g2.
(52)
It is approximately equal to 0.025 meV.

Acknowledgments

The work was supported by IRSES “POLATER” AND “POLAPHEN” projects. T. C. H. Liew was supported by the EU Marie-Curie project EPOQUES. V. M. Kovalev was supported by RFBR 12-02-31012. I. G. Savenko acknowledges support of the Eimskip foundation.

References and links

1.

J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose-Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006) [CrossRef] [PubMed] .

2.

R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, “Bose-Einstein condensation of microcavity polaritons in a trap,” Science 316(5827), 1007–1010 (2007) [CrossRef] [PubMed] .

3.

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, “Coherent zero-state and p-state in an exciton-polariton condensate array,” Nature 450(7169), 529–532 (2007) [CrossRef] [PubMed] .

4.

A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnick, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009) [CrossRef] [PubMed] .

5.

A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdre, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,” Nature Phys. , 5, 805–810 (2009) [CrossRef] .

6.

I. Carusotto and C. Ciuti, “Probing microcavity polariton superfluidity through resonant Rayleigh scattering,” Phys. Rev. Lett., 93(16), 166401 (2004) [CrossRef] .

7.

K. G. Lagoudakis, B. Pietka, M. Wouters, R. Andre, and B. Deveaud-Pledran, “Coherent oscillations in an exciton-polariton Josephson junction,” Phys. Rev. Lett. 105(12), 120403 (2010) [CrossRef] [PubMed] .

8.

K. G. Lagoudakis, T. Ostatnicky, A. V. Kavokin, Y. G. Rubo, R. Andre, and B. Deveaud-Pledran, “Observation of half-quantum vortices in an exciton-polariton condensate,” Science 13(5955), 974–976 (2009) [CrossRef] .

9.

K. G. Lagoudakis, F. Manni, B. Pietka, M. Wouters, T. C. H. Liew, V. Savona, A. V. Kavokin, R. Andre, and B. Deveaud-Pledran, “Probing the dynamics of spontaneous quantum vortices in polariton superfluids,” Phys. Rev. Lett. 106(11), 115301 (2011) [CrossRef] [PubMed] .

10.

G. Nardin, G. Grosso, Y. Leger, B. Pietka, F. Morier-Genoud, and B. Deveaud-Pledran, “Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum fluid,” Nature Phys. 7, 635–641 (2011) [CrossRef] .

11.

A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemnager, R. Houdr, E. Giacobino, C. Ciuti, and A. Bramati, “Polariton superfluids reveal quantum hydrodynamic solitons,” Science 332(6034), 1167–1170 (2011) [CrossRef] [PubMed] .

12.

G. Grosso, G. Nardin, F. Morier-Genoud, Y. Léger, and B. Deveaud-Plédran, “Soliton instabilities and vortex street formation in a polariton quantum fluid,” Phys. Rev. Lett. 107(24), 245301 (2011) [CrossRef] .

13.

M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gorbach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mndez, K. Biermann, R. Hey, and P. V. Santoset, “Observation of bright polariton solitons in a semiconductor microcavity,” Nature Photon. 6, 50–55 (2012) [CrossRef] .

14.

R. Hivet, H. Flayac, D. D. Solnyshkov, D. Tanese, T. Boulier, D. Andreoli, E. Giacobino, J. Bloch, A. Bramati, G. Malpuech, and A. Amo, “Half-solitons in a polariton quantum fluid behave like magnetic monopoles,” Nature Phys. 8, 724–728 (2012) [CrossRef] .

15.

F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. Andre, and B. Deveaud-Pledran, “Spontaneous pattern formation in a polariton condensate,” Phys. Rev. Lett. 107(10), 106401 (2011) [CrossRef] [PubMed] .

16.

G. Christmann, G. Tosi, N. G. Berloff, P. Tsotsis, P. S. Eldridge, Z. Hatzopoulos, P. G. Savvidis, and J. J. Baumberg, “Polariton ring condensates and sunflower ripples in an expanding quantum liquid,” Phys. Rev. B 85(23), 235303 (2012) [CrossRef] .

17.

E. Kammann, T. C. H. Liew, H. Ohadi, P. Cilibrizzi, P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, A. V. Kavokin, and P. G. Lagoudakis, “Nonlinear optical spin Hall effect and long-range spin transport in polariton lasers,” Phys. Rev. Lett. 109(3), 036404 (2012) [CrossRef] [PubMed] .

18.

S. Christopoulos, G. Baldassarri Höger von Högersthal, A. Grundy, P. G. Lagoudakis, A. V. Kavokin, J. J. Baumberg, G. Christmann, R. Butté, E. Feltin, J.-F. Carlin, and N. Grandjean, “Room-temperature polariton lasing in semiconductor microcavities,” Phys. Rev. Lett. 98(12), 126405 (2007) [CrossRef] [PubMed] .

19.

Ayan Das, Junseok Heo, Marc Jankowski, Wei Guo, Lei Zhang, Hui Deng, and Pallab Bhattacharya, “Room temperature ultralow threshold GaN nanowire polariton laser,” Phys. Rev. Lett. 107(6), 066405 (2011) [CrossRef] [PubMed] .

20.

R. Schmidt-Grund, B. Rheinlnder, C. Czekalla, G. Benndorf, H. Hochmut, A. Rahm, M. Lorenz, and M. Grundmann, “ZnO based planar and micropillar resonators,” Superlattic. Microstruct. 41(5–6), 360–363 (2007) [CrossRef] .

21.

S. Kena-Cohen and S. R. Forrest, “Room-temperature polariton lasing in an organic single-crystal microcavity,” Nature Photonics 4, 371–375 (2010) [CrossRef] .

22.

T. C. H. Liew, I. A. Shelykh, and G. Malpuech, “Polaritonic devices,” Physica E 43(9), 1543–1568 (2011) [CrossRef] .

23.

K. V. Kavokin, M. A. Kaliteevski, R. A. Abram, A. V. Kavokin, S. Sharkova, and I. A. Shelykh, “Stimulated emission of terahertz radiation by exciton-polariton lasers,” Appl. Phys. Lett. 97(20), 201111 (2010) [CrossRef] .

24.

A. V. Kavokin, I. A. Shelykh, T. Taylor, and M. M. Glazov, “Vertical cavity surface emitting terahertz laser,” Phys. Rev. Lett. 108(19), 197401 (2012) [CrossRef] [PubMed] .

25.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69(23), 3314–3317 (1992) [CrossRef] [PubMed] .

26.

I. G. Savenko, O. V. Kibis, and I. A. Shelykh, “Asymmetric quantum dot in a microcavity as a nonlinear optical element,” Phys. Rev. A 85(5), 053818 (2012) [CrossRef] .

27.

A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, (2007)) [CrossRef] .

28.

V. D. Kulakovskii, A. I. Tartakovskii, D. N. Krizhanovskii, A. Armitage, J. S. Roberts, and M. S. Skolnick, “Two-dimensional excitonic polaritons and their interaction,” Phys. Usp. 43(8), 853–857 (2000) [CrossRef] .

29.

N. A. Gippius, S. G. Tikhodeev, V. D. Kulakovskii, D. N. Krizhanovskii, and A. I. Tartakovskii, “Nonlinear dynamics of polariton scattering in semiconductor microcavity: Bistability vs. stimulated scattering,” Europhys. Lett. 67(6), 997 (2004) [CrossRef] .

30.

D. M. Whittaker, “Effects of polariton-energy renormalization in the microcavity optical parametric oscillator,” Phys. Rev. B 71(11), 115301 (2005) [CrossRef] .

31.

A. Baas, J. P. Karr, M. Romanelli, A. Bramati, and E. Giacobino, “Optical bistability in semiconductor microcavities in the nondegenerate parametric oscillation regime: Analogy with the optical parametric oscillator,” Phys. Rev. B 70(16), 161307(R) (2004) [CrossRef] .

32.

N. A. Gippius, I. A. Shelykh, D. D. Solnyshkov, S. S. Gavrilov, Yuri G. Rubo, A. V. Kavokin, S. G. Tikhodeev, and G. Malpuech, “Polarization multistability of cavity polaritons,” Phys. Rev. Lett. 98(23), 236401 (2007) [CrossRef] [PubMed] .

33.

T. K. Paraïso, M. Wouters, Y. Léger, F. Morier-Genoud, and B. Deveaud-Plédran, “Multistability of a coherent spin ensemble in a semiconductor microcavity,” Nature Mater. 9, 655–660 (2010) [CrossRef] .

34.

D. Bajoni, E. Semenova, A. Lematre, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R. Kuszelewicz, and J. Bloch, “Optical bistability in a GaAs-based polariton diode,” Phys. Rev. Lett. 101(26), 266402 (2008) [CrossRef] [PubMed] .

35.

T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, “Optical circuits based on polariton neurons in semiconductor microcavities,” Phys. Rev. Lett. 100(11), 116401 (2008) [PubMed] .

36.

A. Amo, T. H. C. Liew, C. Adrados, R. Houdre, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nature Photon. 4, 361–366 (2010) [CrossRef] .

37.

D. Sarkar, S. S. Gavrilov, M. Sich, J. H. Quilter, R. A. Bradley, N. A. Gippius, K. Guda, V. D. Kulakovskii, M. S. Skolnick, and D. N. Krizhanovskii, “Polarization bistability and resultant spin rings in semiconductor microcavities,” Phys. Rev. Lett. 105(21), 216402 (2010) [CrossRef] .

38.

C. Adrados, A. Amo, T. C. H. Liew, R. Hivet, R. Houdr, E. Giacobino, A. V. Kavokin, and A. Bramati, “Spin rings in bistable planar semiconductor microcavities,” Phys. Rev. Lett. 105(21), 216403 (2010) [CrossRef] .

39.

T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, “Optical circuits based on polariton neurons in semiconductor microcavities,” Phys. Rev. Lett. 101(1), 016402 (2008) [CrossRef] [PubMed] .

40.

T. C. H. Liew, A. V. Kavokin, T. Ostatnický, M. Kaliteevski, I. A. Shelykh, and R. A. Abram, “Exciton-polariton integrated circuits,” Phys. Rev. B 82(3), 033302 (2010) [CrossRef] .

41.

I. G. Savenko, I. A. Shelykh, and M. A. Kaliteevski, “Nonlinear terahertz emission in semiconductor microcavities,” Phys. Rev. Lett. 107(2), 027401 (2011) [CrossRef] [PubMed] .

42.

F. Tassone and Y. Yamamoto, “Exciton-exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons,” Phys. Rev. B 59(16), 10830–10842 (1999) [CrossRef] .

43.

I. A. Shelykh, A. V. Kavokin, Yuri G. Rubo, T. C. H. Liew, and G. Malpuech, “Polariton polarization-sensitive phenomena in planar semiconductor microcavities”, Semicond. Sci. Technol. 25(1), 013001 (2010) [CrossRef] .

44.

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73(19), 193306 (2006) [CrossRef] .

45.

B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Generation of entangled photon holes using quantum interference,” Phys. Rev. A 74(4), 010303()R (2006).

46.

X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys.Rev. A 43(3), 1186–1196 (1991) [CrossRef] [PubMed] .

OCIS Codes
(190.1450) Nonlinear optics : Bistability
(190.4180) Nonlinear optics : Multiphoton processes

ToC Category:
Nonlinear Optics

History
Original Manuscript: April 8, 2013
Revised Manuscript: May 25, 2013
Manuscript Accepted: May 27, 2013
Published: June 18, 2013

Citation
A. A. Pervishko, T. C. H. Liew, V. M. Kovalev, I. G. Savenko, and I. A. Shelykh, "Nonlinear effects in multi-photon polaritonics," Opt. Express 21, 15183-15194 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15183


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References

  1. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose-Einstein condensation of exciton polaritons,” Nature443, 409–414 (2006). [CrossRef] [PubMed]
  2. R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, “Bose-Einstein condensation of microcavity polaritons in a trap,” Science316(5827), 1007–1010 (2007). [CrossRef] [PubMed]
  3. 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, “Coherent zero-state and p-state in an exciton-polariton condensate array,” Nature450(7169), 529–532 (2007). [CrossRef] [PubMed]
  4. A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnick, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature457, 291–295 (2009). [CrossRef] [PubMed]
  5. A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdre, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,” Nature Phys., 5, 805–810 (2009). [CrossRef]
  6. I. Carusotto and C. Ciuti, “Probing microcavity polariton superfluidity through resonant Rayleigh scattering,” Phys. Rev. Lett.,93(16), 166401 (2004). [CrossRef]
  7. K. G. Lagoudakis, B. Pietka, M. Wouters, R. Andre, and B. Deveaud-Pledran, “Coherent oscillations in an exciton-polariton Josephson junction,” Phys. Rev. Lett.105(12), 120403 (2010). [CrossRef] [PubMed]
  8. K. G. Lagoudakis, T. Ostatnicky, A. V. Kavokin, Y. G. Rubo, R. Andre, and B. Deveaud-Pledran, “Observation of half-quantum vortices in an exciton-polariton condensate,” Science13(5955), 974–976 (2009). [CrossRef]
  9. K. G. Lagoudakis, F. Manni, B. Pietka, M. Wouters, T. C. H. Liew, V. Savona, A. V. Kavokin, R. Andre, and B. Deveaud-Pledran, “Probing the dynamics of spontaneous quantum vortices in polariton superfluids,” Phys. Rev. Lett.106(11), 115301 (2011). [CrossRef] [PubMed]
  10. G. Nardin, G. Grosso, Y. Leger, B. Pietka, F. Morier-Genoud, and B. Deveaud-Pledran, “Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum fluid,” Nature Phys.7, 635–641 (2011). [CrossRef]
  11. A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemnager, R. Houdr, E. Giacobino, C. Ciuti, and A. Bramati, “Polariton superfluids reveal quantum hydrodynamic solitons,” Science332(6034), 1167–1170 (2011). [CrossRef] [PubMed]
  12. G. Grosso, G. Nardin, F. Morier-Genoud, Y. Léger, and B. Deveaud-Plédran, “Soliton instabilities and vortex street formation in a polariton quantum fluid,” Phys. Rev. Lett.107(24), 245301 (2011). [CrossRef]
  13. M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gorbach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mndez, K. Biermann, R. Hey, and P. V. Santoset, “Observation of bright polariton solitons in a semiconductor microcavity,” Nature Photon.6, 50–55 (2012). [CrossRef]
  14. R. Hivet, H. Flayac, D. D. Solnyshkov, D. Tanese, T. Boulier, D. Andreoli, E. Giacobino, J. Bloch, A. Bramati, G. Malpuech, and A. Amo, “Half-solitons in a polariton quantum fluid behave like magnetic monopoles,” Nature Phys.8, 724–728 (2012). [CrossRef]
  15. F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. Andre, and B. Deveaud-Pledran, “Spontaneous pattern formation in a polariton condensate,” Phys. Rev. Lett.107(10), 106401 (2011). [CrossRef] [PubMed]
  16. G. Christmann, G. Tosi, N. G. Berloff, P. Tsotsis, P. S. Eldridge, Z. Hatzopoulos, P. G. Savvidis, and J. J. Baumberg, “Polariton ring condensates and sunflower ripples in an expanding quantum liquid,” Phys. Rev. B85(23), 235303 (2012). [CrossRef]
  17. E. Kammann, T. C. H. Liew, H. Ohadi, P. Cilibrizzi, P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, A. V. Kavokin, and P. G. Lagoudakis, “Nonlinear optical spin Hall effect and long-range spin transport in polariton lasers,” Phys. Rev. Lett.109(3), 036404 (2012). [CrossRef] [PubMed]
  18. S. Christopoulos, G. Baldassarri Höger von Högersthal, A. Grundy, P. G. Lagoudakis, A. V. Kavokin, J. J. Baumberg, G. Christmann, R. Butté, E. Feltin, J.-F. Carlin, and N. Grandjean, “Room-temperature polariton lasing in semiconductor microcavities,” Phys. Rev. Lett.98(12), 126405 (2007). [CrossRef] [PubMed]
  19. Ayan Das, Junseok Heo, Marc Jankowski, Wei Guo, Lei Zhang, Hui Deng, and Pallab Bhattacharya, “Room temperature ultralow threshold GaN nanowire polariton laser,” Phys. Rev. Lett.107(6), 066405 (2011). [CrossRef] [PubMed]
  20. R. Schmidt-Grund, B. Rheinlnder, C. Czekalla, G. Benndorf, H. Hochmut, A. Rahm, M. Lorenz, and M. Grundmann, “ZnO based planar and micropillar resonators,” Superlattic. Microstruct.41(5–6), 360–363 (2007). [CrossRef]
  21. S. Kena-Cohen and S. R. Forrest, “Room-temperature polariton lasing in an organic single-crystal microcavity,” Nature Photonics4, 371–375 (2010). [CrossRef]
  22. T. C. H. Liew, I. A. Shelykh, and G. Malpuech, “Polaritonic devices,” Physica E43(9), 1543–1568 (2011). [CrossRef]
  23. K. V. Kavokin, M. A. Kaliteevski, R. A. Abram, A. V. Kavokin, S. Sharkova, and I. A. Shelykh, “Stimulated emission of terahertz radiation by exciton-polariton lasers,” Appl. Phys. Lett.97(20), 201111 (2010). [CrossRef]
  24. A. V. Kavokin, I. A. Shelykh, T. Taylor, and M. M. Glazov, “Vertical cavity surface emitting terahertz laser,” Phys. Rev. Lett.108(19), 197401 (2012). [CrossRef] [PubMed]
  25. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett.69(23), 3314–3317 (1992). [CrossRef] [PubMed]
  26. I. G. Savenko, O. V. Kibis, and I. A. Shelykh, “Asymmetric quantum dot in a microcavity as a nonlinear optical element,” Phys. Rev. A85(5), 053818 (2012). [CrossRef]
  27. A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, (2007)). [CrossRef]
  28. V. D. Kulakovskii, A. I. Tartakovskii, D. N. Krizhanovskii, A. Armitage, J. S. Roberts, and M. S. Skolnick, “Two-dimensional excitonic polaritons and their interaction,” Phys. Usp.43(8), 853–857 (2000). [CrossRef]
  29. N. A. Gippius, S. G. Tikhodeev, V. D. Kulakovskii, D. N. Krizhanovskii, and A. I. Tartakovskii, “Nonlinear dynamics of polariton scattering in semiconductor microcavity: Bistability vs. stimulated scattering,” Europhys. Lett.67(6), 997 (2004). [CrossRef]
  30. D. M. Whittaker, “Effects of polariton-energy renormalization in the microcavity optical parametric oscillator,” Phys. Rev. B71(11), 115301 (2005). [CrossRef]
  31. A. Baas, J. P. Karr, M. Romanelli, A. Bramati, and E. Giacobino, “Optical bistability in semiconductor microcavities in the nondegenerate parametric oscillation regime: Analogy with the optical parametric oscillator,” Phys. Rev. B70(16), 161307(R) (2004). [CrossRef]
  32. N. A. Gippius, I. A. Shelykh, D. D. Solnyshkov, S. S. Gavrilov, Yuri G. Rubo, A. V. Kavokin, S. G. Tikhodeev, and G. Malpuech, “Polarization multistability of cavity polaritons,” Phys. Rev. Lett.98(23), 236401 (2007). [CrossRef] [PubMed]
  33. T. K. Paraïso, M. Wouters, Y. Léger, F. Morier-Genoud, and B. Deveaud-Plédran, “Multistability of a coherent spin ensemble in a semiconductor microcavity,” Nature Mater.9, 655–660 (2010). [CrossRef]
  34. D. Bajoni, E. Semenova, A. Lematre, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R. Kuszelewicz, and J. Bloch, “Optical bistability in a GaAs-based polariton diode,” Phys. Rev. Lett.101(26), 266402 (2008). [CrossRef] [PubMed]
  35. T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, “Optical circuits based on polariton neurons in semiconductor microcavities,” Phys. Rev. Lett.100(11), 116401 (2008). [PubMed]
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