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Physical Review A

Tight-binding model for exciton-polariton condensates in external potentials

P. Stępnicki, Michał Matuszewski

Published Sep 23, 2013• 13 citations
Publisher

Abstract

We propose a mean-field model to describe exciton-polariton condensates in deep periodic external potentials. We derive a set of coupled discrete equations for both condensed and uncondensed components, with interaction and tunneling coefficients obtained within the tight-binding approximation. We use the model to explain the intriguing phenomenon of increasing density modulation in a one-dimensional valley with disorder, reported by F. Manni et al. [Phys. Rev. Lett. 106, 176401 (2011)].

DOI
10.1103/physreva.88.033626

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We propose a mean-field model to describe exciton-polariton condensates in deep periodic external potentials. We derive a set of coupled discrete equations for both condensed and uncondensed components, with interaction and tunneling coefficients obtained within the tight-binding approximation. We use the model to explain the intriguing phenomenon of increasing density modulation in a one-dimensional valley with disorder, reported by F. Manni et al. [Phys. Rev. Lett. 106, 176401 (2011)].
In this work, we study the exciton-polariton condensate phase transition in a microcavity matter-light system in which electron-hole Coulomb interaction and matter-light coupling effects are treated on an equal footing. In the framework of the unrestricted Hartree-Fock approximation applying the two-dimensional exciton-polariton model, we derive the self-consistent equations determining simultaneously the excitonic and the photonic condenstate order parameters. In the thermal-equilibrium limit, we find a condensed state of the exciton-polariton systems and phase diagrams are then constructed. At a given low temperature, the condensate by its nature shows a crossover from an excitonic to a polaritonic and finally photonic condensed state as the excitation density increases at large detuning. Without the detuning, the excitonic condensed state disappears whereas the polaritonic or photonic phases dominate. The crossover is also found by lowering the Coulomb interaction at a finite matter-light coupling. Lowering the Coulomb interaction or increasing the temperature, the excitonic Mott transition occurs, at which the exciton-polariton condensates dissociate to free electron-hole/photon. Depending on temperature and excitation density, the phase transition of the exciton-polariton condensates is also addressed in signatures of photoluminescence mapping to the photonic momentum distribution.
A condensed-matter platform for analog simulation of complex two-dimensional molecular bonding configurations, based on optically trapped exciton-polariton condensates is proposed. The stable occupation of polariton condensates in the excited states of their optically configurable potential traps permits emulation of excited atomic orbitals. A classical mean-field model describing the dissipative coupling mechanism between ppp-orbital condensates is derived, identifying lowest-threshold condensation solutions as a function of trap parameters corresponding to bound and antibound $\ensuremath{\pi}$ and $\ensuremath{\sigma}$ bonding configurations, similar to those in quantum chemistry.
The dynamics of Bose-Einstein condensates trapped in a deep optical lattice is governed by a discrete nonlinear equation (DNL). Its degree of nonlinearity and the intersite hopping rates are retrieved from a nonlinear tight-binding approximation taking into account the effective dimensionality of each condensate. We derive analytically the Bloch and the Bogoliubov excitation spectra and the velocity of sound waves emitted by a traveling condensate. Within a Lagrangian formalism, we obtain Newtonian-like equations of motion of localized wave packets. We calculate the ground-state atomic distribution in the presence of a harmonic confining potential, the frequencies of small amplitude dipole, and quadrupole oscillations. We finally quantize the DNL, recovering an extended Bose-Hubbard model.
Bose-Einstein condensates of exciton-polaritons in inorganic semiconductor microcavities are known to possess strong interparticle interactions attributed to their excitonic component. The interactions play a crucial role in the nonlinear dynamics of such systems and can be witnessed as the high energy shift of polariton states. However, the localised nature of Frenkel excitons in strongly coupled organic microcavities precludes interparticle Coulomb exchange-interactions that affect nonlinear dynamics and change mechanisms of polariton energy shifts accordingly. We have scrutinized the origins of energy shifts in connection with nonlinear dynamics in Frenkel exciton-polariton condensates and examined the possible contributions: intracavity optical Kerr-effect, gain-induced frequency pulling, polariton interactions and effects related to saturation of optical transitions for "dark"- and "bright" molecules [1]. Unlike the conventional strongly coupled semiconductor microcavities, we have shown that nonlinear interactions within the condensate do not rely on polariton interactions but instead originated from Pauli-blocking principle forbidding double excitation of the same molecular states. We have developed a theoretical model explaining the omnipresent energy shift of the condensate wavefunction together with its spectral and polarization features at the BEC transition in a consistent way. The crucial role of intermolecular energy transfer and “dark” exciton reservoir has been demonstrated for the first time. We believe the principles explored in this work are relevant to other systems exhibiting strong coupling of Frenkel excitons with a cavity mode, regardless of a cavity type, whether one dealing with Fabry-Perot cavities or plasmonic nanocavities etc and; therefore, provide general insight on nonlinear phenomena in composite light-matter condensates. [1] T. Yagafarov, D. Sannikov, A. Zasedatelev, K. Georgiou, A. Baranikov, O. Kyriienko, I. Shelykh, L. Gai, Z. Shen, D. G. Lidzey, P. Lagoudakis, Mechanisms of blueshifts in organic polariton condensates, Commun Phys 3, 18 (2020). https://doi.org/10.1038/s42005-019-0278-6
We study the behavior of exciton polaritons in an optical microcavity with an embedded semiconductor quantum well. We use a two-component exciton-photon approach formulated in terms of path integral formalism. In order to describe spatial distributions of the exciton and photon condensate densities, the two coupled equations of the Gross-Pitaevskii type are derived. For a homogeneous system, we find the noncondensate photon and exciton spectra, calculate the coefficients of transformation from the exciton-photon basis to the lower-upper polariton basis, and obtain the exciton and photon occupation numbers of the lower and upper polariton branches for nonzero temperatures. For an inhomogeneous system, the set of coupled equations of the Bogoliubov--de Gennes type is derived. The equations govern the spectra and spatial distributions of noncondensate photons and excitons.
We study theoretically Bloch oscillations of half-matter, half-light quasiparticles: exciton-polaritons. We propose an original structure for the observation of this phenomenon despite the constraints imposed by the relatively short lifetime of the particles and the limitations on the engineered periodic potential. First, we focus on the linear regime in a perfect lattice where regular oscillations are obtained. Second, we take into account a realistic structural disorder known to localize noninteracting particles, which is quite dramatic for propagation-related phenomena. In the nonlinear condensed regime the renormalization of the energy provided by interactions between particles allows us to screen efficiently the disorder and to recover oscillations. This effect is useful only in a precise range of parameters outside of which the system becomes dynamically unstable. For a large chemical potential of the order of the potential's amplitude, a strong Landau-Zener tunneling tends to completely delocalize particles.