Multivalley engineering in semiconductor microcavities

We consider exciton-photon coupling in semiconductor microcavities in which separate periodic potentials have been embedded for excitons and photons. We show theoretically that this system supports degenerate ground-states appearing at non-zero inplane momenta, corresponding to multiple valleys in reciprocal space, which are further separated in polarization corresponding to a polarization-valley coupling in the system. Aside forming a basis for valleytronics, the multivalley dispersion is predicted to allow for spontaneous momentum symmetry breaking and two-mode squeezing under non-resonant and resonant excitation, respectively.

whereẼ C (q) =¯h 2 q 2 2m C − ih τ ,Ẽ X (q) =¯h 2 q 2 2m X − ih λ ,Ṽ C (0, −G, G) are the 0th, -1st, 1st order terms of the Fourier series for the periodic potential of the cavity photon.Ṽ X (0, −G, G) are the 0th, -1st, 1st order terms of the Fourier series for the excitonic potential. The matrix in (1) is reduced to the summation of terms in Eq. (1) from −G to G. In our real calculations, we did the summation over the terms from −150 · G to 150 · G.

Simplified model of equilibrium polariton condensation
Following Eqs. (2) and (3) in the main text, for the case of a fixed number of particles and small temperature, we can write the probability of occupation of different modes referred to as the probability distribution function (PDF): where Z = ∑ n 1 ,n 2 e −E(n 1 ,n 2 )/k B T is the partition function, T is the temperature and k B is Boltzmann's constant. The second order correlation function then reads where n 1 , n 2 , and n 1 n 2 can be calculated from the PDF, p(n 1 , n 2 ), see Fig. 2 in the main text. At zero temperature g (2) 12 = 0, confirming our earlier arguments on the choice of the state required for energy minimisation. With increasing temperature, g (2) 12 rises as the system may be excited out of the ground state.
Allowing for the population of many modes in reciprocal space (instead of the only two which we considered), the probability of occupation of any quantum state can be found by straightforward generalisation of Eq. (2). In principle, the full EP intensity distribution can then be obtained by summing over the PDF. However, in practice, the size of the Hilbert space grows exponentially with the number of particles in the system, therefore, it becomes possible to use our simple treatment to evaluate the equilibrium photoluminescence spectrum in the low density regime only with a few particles in the system (see also inset in Fig. 2 in the main text). The spectrum here was phenomenologically broadened in energy and wave vector, accounting for the finite lifetime of polaritons and finite size of a typical condensate, respectively.

Entanglement generation
We treat particles localized at the dispersion minima as two quantum modes, which deterministic evolution is governed by the Hamiltonian (5) in the main text. The last term there,Ĥ CK = 4αâ † 1â † 2â 2â1 , is a typical cross-Kerr interaction term that can be linearised by expanding the total quantum fields asâ j → ξ j + δâ j , where δâ j are the displaced quantum fluctuation fields on top of the classical mean fields, ξ j . Using such a substitution and keeping terms up to the second order in δâ j only, we come up with the linearised interaction term, where we assumed ξ 1,2 ∈ R for clarity and definedẼ 1,2 = ∆ 2,1 |ξ 1,2 | 2 ,F 1,2 = ξ 1,2 |ξ 2,1 | 2 and S = ξ 1 ξ 2 . The first term is an energy shift, the second one is an effective driving term, and crucially, the last term is a typical two-mode squeezing interaction of magnitude S set by the product of the two mean fields, ξ 1,2 , and responsible for the entanglement between the two modes. The squeezing and entanglement magnitude can therefore be adjusted by varying the resonant driving strengths, F 1,2 .

Nonequilibrium model of polariton condensation
The interaction with the reservoir of acoustic phonons of the semiconductor crystal lattice is described by the microscopic Fröhlich Hamiltonian: 1 where parameters G q are the exciton-phonon interaction strengths evaluated elsewhere 2 . The phonon wavevector here is q = e x q x + e y q x + e z q z , where e x , e y and e z are unit vectors: e x is in the direction of the 1D polariton system, e z is in the structure growth direction, and e y is perpendicular to both of those two. The phonon dispersion relation,hω q =hc s q 2 x + q 2 y + q 2 z , is determined by the sound velocity, c s .
The equations of motion for the polariton macroscopic wave function, ψ, and the exciton reservoir occupation number, n R , read 3, 4 : where F −1 stands for the inverse Fourier transform, E k is the free polariton dispersion, ψ k is the Fourier image of the macroscopic wave function; P and γ R are the incoherent reservoir homogenous pumping intensity and inverse lifetime of the reservoir, correspondingly; R is the system-reservoir excitation exchange rate. The term S k (t) corresponds to the emission of phonons by a condensate stimulated by the polariton concentration.  The stochastic term T q x in the last line of (6) is defined by the correlations: where n q is the temperature-dependent density of phonons in the state with a wave vector q. Solving (6) numerically, and averaging over different stochastic realizations of the phonon field, we obtain the results shown in Fig. 3 of the main text.

Energy band structure
Here we present details on the energy band structure calculation for the EP lattice, see Fig. 1a in the main text and one more alternative configuration. Parameters of these plots are: the lattice period: T = 2.0 µm, potential profile for the cavity photons: the sine function −1.1 ∼ 0.25 meV, potential profile for the excitons: the sine function −0.95 ∼ 460.29 meV, exciton-photon coupling constant: Ω = 0.7 meV, decay rate for the cavity photons: γ = 0.42 meV, decay rate for the excitons: 0.04meV, effective mass of the cavity photon: 5 * 10 −5 m e , effective mass of the exciton: 0.22m e , where m e is the free electron mass.
Let us also change the parameters of the potential energies of excitons and photons and see, which dispersion we can achieve in k-space. If we take the lattice period: T = 3.0 µ m, potential profile for cavity photon: square function −0.25 ∼ 0 meV, potential profile for exciton: sine function −0.389 ∼ 697.95 meV, exciton-photon coupling constant: Ω = 2.47 meV, decay rate for cavity photon: γ = 1 meV , decay rate for exciton: 0 meV , effective mass of cavity photon: 5 * 10 −5 m e , effective mass of exciton: 0.22m e , where m e is the free electron mass, then we yield the dispersion presented in Figs. 2 and 3. Figure 3 is the result of the nonequilibrium model. Since the decay rate of the EPs vary significantly with momentum k, it becomes easier for EPs to condensate at k = ±k BZ rather than k = 0. As a result, the blueshifts are different at k = 0 and at 3/4  k = ±k BZ . The minima in the k-space become inequivalent in properties, making it difficult to consider entanglement between them. It should be noted, that, instead, in Fig. 3b in the main text the two minima at k = ±k 0 are equivalent in properties. Figure 4 is the 3D plot of the energy dispersion in 2D lattice corresponding to Fig. 5a in the main text. One can see four degenerate valleys with equivalent properties. It results in the possibility to create spin-valley coupling.