Nonlinear Effects in Multi-Photon Polaritonics

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.


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][2][3] and superfluidity [4][5][6]; the Josephson effect [7]; and the formation of various topological excitations such as vortices [8][9][10] and solitons [11][12][13][14] Polaritonics stems from the proposal that quantum microcavities can be used as efficient sources of THz radiation [23], 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], 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], 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.

Polaritonics with 2p dark exciton state
We consider the following geometry (Fig. 1): a quantum well (QW) with 2p excitonic transition having energyhω p placed inside a planar microcavity supporting the photonic mode of energy ω 2c ≈ ω p /2. In this configuration the cavity mode can resonantly interact with the 2p excitonic state.
The Hamiltonian of the system reads: where the operatorsp,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: where S is a quantization area of the sample; m 0 , m * and µ are the free electron, effective and reduced exciton masses, respectively; q is the elementary charge of the electron; E g is the band gap of the material; ω is the energy of the photon; E n are the eigenvalues of the 2D Hydrogen atom; Φ n ( r) and R 21 ( r), R 10 ( r) are normalized angular and radial eigenfunctions of the 2D Hydrogen atom, which are written explicitly in the Appendix, Eqs. (30) and (31), respectively, and A is the vector potential, which can be writen in the dipole approximation: where ε 0 and ε are vacuum and material permitivities; L is the length of the cavity.

Energy levels for the case of an ideal cavity
Note, that the Hamiltonian (Eq. (1)) commutes with the excitations' number operatorN =â †â + 2p †p , 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  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 E 0 = 0 and E 1 =hω 2c 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: and the eigenenergies in the case of resonance 2ω 2c = ω p are 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).

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 where |Ψ in is an initial wavefunction in the manifold with N cavity photons and |Ψ out is a final wavefunction in the manifold with (N-1) cavity photons. p(Ψ in ) represents the probability of occupation of the initial state [26]. 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: 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 ∆Ω R,S = 2g s , if g s is the bright exciton-single photon coupling strength. Considering our system with two-photon absorption for large N (N >> 1), and substituting a, a † → √ N one immediately obtains ∆Ω R ≈ 2g √ N. Accounting for finite lifetime of the photons and excitons, one obtains [27]: 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 g 2 N depends on the concentration of the photons n a = N/S only.

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 operatorsp andâ: Then, one should take averages of the above equations thus passing from the operators to the mean values asp → p = Tr{ρp}. We present a mean product of several operators by products of the mean values of the operators (e.g. pâ † → p â * , â 2 → â 2 ). 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: where γ p,a = 1/(2τ p,a ) with τ p,a being lifetimes of the modes, P is the amplitude of the coherent pump and Ω p = g/h. Making a substitutionâ ←âe −iωt ,p ←pe −2iωt , in the stationary regime the equations read: where ∆ p = ω p − 2ω, ∆ a = ω a − ω, which can be reduced to: This result can also be written in terms of the real functions describing the number of the photons, N a = |a| 2 and excitons, N p = |p| 2 : where Equation (15) is a cubic equation for N a . The results are represented in Fig. 3(a) for the following parameters corresponding to the system: γ a = 0.05 meV , γ p = 0.01 meV , and three different values of detunings ∆ a and ∆ p . 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][29][30][31][32][33][34]. 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.
Bistability is a fundamental ingredient of several devices based on semiconductor microcavities, such as memory elements [35], spin switches [36], spin patterns [37, 38], and optical circuit designs [39,40]. However, there is substantial difference between the previous works and the paradigm developed in our paper. Indeed, in most of the previous approaches bistability resulted from the exciton-exciton (Coulomb and exchange) interaction, even in THz emitting systems [41]. Here, the bistability arises due to the nonlinearity of the two-photon absorption process. It should be noted that, still, excitons are present in the system and their interaction can, in principle, change the physical behavior. The exciton-exciton interaction can be accounted for by means of the term: which should be added to Hamiltonian (Eq. (1)). Then, the equations of motion change to: and in the stationary regime we obtain the following solution: n a + 4((γ a γ p −h 2 ∆ a ∆ p )n p −h∆ a α p n 2 p ) (h 2 ∆ 2 a + γ 2 a ) + 4g 2 n p n a (h 2 ∆ 2 a + γ 2 a ) = I a .
To define the constant α p , we use the formula α p ≈ 6E B a 2 B /S, where E B is the exciton binding energy, a B is the exciton Bohr radius and S is the laser spot area [42]. In QWs based on GaAs-based alloys, we can use the parameters: E B ≈ 19.2 meV and a B = 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].