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arXiv:1310.3641 · Physical Review B

Quantum theory of multimode polariton condensation

David Racine, P. R. Eastham

Published Aug 18, 2014• 34 citations
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Abstract

We develop a theory for the dynamics of the density matrix describing a multimode polariton condensate. In such a condensate several single-particle orbitals become highly occupied, due to stimulated scattering from reservoirs of high-energy excitons. A generic few-parameter model for the system leads to a Lindblad equation which includes saturable pumping, decay, and condensate interactions. We show how this theory can be used to obtain the population distributions, and the time-dependent first- and second-order coherence functions, in such a multimode condensate. As a specific application, we consider a polaritonic Josephson junction, formed from a double-well potential. We obtain the population distributions, emission line shapes, and widths (first-order coherence functions), and predict the dephasing time of the Josephson oscillations.

DOI
10.1103/physrevb.90.085308

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We develop a theory for the dynamics of the density matrix describing a multimode polariton condensate. In such a condensate several single-particle orbitals become highly occupied, due to stimulated scattering from reservoirs of high-energy excitons. A generic few-parameter model for the system leads to a Lindblad equation which includes saturable pumping, decay, and condensate interactions. We show how this theory can be used to obtain the population distributions, and the time-dependent first- and second-order coherence functions, in such a multimode condensate. As a specific application, we consider a polaritonic Josephson junction, formed from a double-well potential. We obtain the population distributions, emission line shapes, and widths (first-order coherence functions), and predict the dephasing time of the Josephson oscillations.
A second-order coherence function of an exciton-polariton condensate is calculated in the time domain by the quantum Monte-Carlo wavefunction method.We fully consider polariton-polariton interactions within the five lowest quantized exciton-polariton branch in a one-dimensional model.Our calculated second-order coherence function qualitatively agrees with the experimental results obtained by H. Deng et.al, 1) and it approaches to one at well above threshold like an ideal photon laser.However, the polariton number distribution is broad compared with Poissonian distribution, which indicates that the polariton condensate is different from a coherent state.
Dynamics of stimulated scattering of quasi-two-dimensional exciton polaritons in a semiconductor microcavity is considered theoretically. A system possessing a bistable response to coherent excitation is studied. Being a system of weakly interacting Bose particles under strongly nonequilibrium conditions, it exhibits a complex dynamics of the development of condensate modes. It is shown that multimode instability can arise in a threshold way in the system under a smooth increase in pump intensity. In this case, rapid populating of a multitude of unstable modes leads to stochastization of scattering, and evolution to a quasi-steady macrostate is performed as self-organization in an open nonequilibrium system. Arising fluctuations lead to the occurrence of a spatial nonuniformity of the distribution of scattered states.
Bose-Einstein condensate of exciton polaritons in a semiconductor microcavity is a macroscopically populated coherent quantum state subject to concurrent pumping and decay. Debates about the fundamental nature of the condensed phase in this open quantum system still persist. Here, we gain a new insight into the spontaneous condensation process by imaging long-lifetime exciton polaritons in a high-quality inorganic microcavity in the single-shot optical excitation regime, without averaging over multiple condensate realisations. In this highly non-stationary regime, a condensate is strongly influenced by the `hot' incoherent reservoir, and reservoir depletion is critical for the transition to the ground energy and momentum state. Condensates formed by more photonic exciton polaritons exhibit dramatic reservoir-induced density filamentation and shot-to-shot fluctuations. In contrast, condensates of more excitonic quasiparticles display smooth density and are second-order coherent. Our observations show that the single-shot measurements offer a unique opportunity to study formation of macroscopic phase coherence during a quantum phase transition in a solid state system.
The demonstration of coherent polariton emission from a CdTe microcavity structure [1] has provided a new system for the study of condensate physics. Unlike atomic condensates, the microcavity system is not in equilibrium; the polariton life-time in the cavity is only ∼ 2ps, so the population has to be maintained by pumping. This raises interesting questions about how the properties of such a non-equilibrium condensate differ from those of an equilibrium system. Recent experimental measurements of the first (g(1)) and second (g(2)) order correlation functions of the emitted light have shown that the decay times of both are ∼ 100−200ps, much longer than the cavity life-time. In this work, we present a theroetical treatment which provides a quantitative explanation of these results, and shows that the ultimate cause of the coherence decay is interactions between polaritons in the condensate. The microcavity polariton condensate is a mesoscopic quantum system, consisting of N ∼ 102 po-laritons within the finite size excitation spot. As a result, number fluctuations of ∼ N 1 2 are expected. For a finite equilibrium condensate of interacting particles, it is straight forward to show that the decay
Exciton-polariton Bose-Einstein condensates (BECs) are newly emerged quantum systems that are capable of showing macroscopic quantum phenomena with intrinsic open-dissipative nature. The spatial distribution of the polariton density, without any external potential, can be controlled by the geometric shape of the pumping laser, enabling the investigation of polariton dynamics with topologically non-trivial configurations. Meanwhile, exciton-polaritons have spin degrees of freedom inherited from excitons and photons, making it a candidate for the realization of quantum logic gates. In this thesis, we will investigate theoretically the nonlinear dynamics of exciton-polariton BECs involving both polaritons' spatial degrees of freedom and spin degrees of freedom, and interactions between them. This thesis is organised as follows: In Chapter 1, we will present an overall review of exiton-polariton systems and important properties of polariton BECs and then introduce the dynamical equations with various interactions that will serve as the main theoretical tool for subsequent chapters. Several polariton pumping and trapping techniques appearing in later chapters will also be introduced. In Chapter 2, we will investigate the superfluidity properties of a single-component polariton condensate under an incoherent annular pumping configuration. By studying the stability properties of polariton persistent currents, we find that the persistent currents can exhibit dynamical instability and energetic-like instability according to different parameter region. A stability phase diagram will be given and its relation with the Landau's criterion will be discussed. In Chapter 3, we will investigate the spin dynamics of a two-component polariton condensate under a homogeneous pumping configuration. Owing to the Josephson coupling, there exist multiple steady state solutions that allow of controlled spin state switching. A desynchronized region where there exists no stable steady solution is found. In the desynchronized region, a desynchronized state beating periodically over time can exist, which will serve as a building block of spin waves presented in the next chapter. In Chapter 4, by combining results from the previous two chapters we will investigate generally the nonlinear dynamics of polariton condensates under an annular pumping configuration. The spin-orbit interaction provided by the Josephson coupling supports azimuthon states that have simultaneous modulations in both amplitude and phase. The azimuthon states, when viewed in a different polarization basis, form rotating spin waves that can be referred to as the optical ferris wheel. In Chapter 5, results from previous chapters will be extended to micocavities that support the anisotropic TE-TM splitting interaction. Rotating singularities (small-scale vortices) are found as a result. Their properties and experimental observation techniques will be discussed. Chapter 2-5 provide a theoretical framework for the nonlinear dynamics of polariton condensates. They rely mostly on optical trapping techniques and are ready to be tested in experiments. In Chapter 6, polaritons trapped by an engineered periodic mesa potential will be discussed. We will investigate the band structure of polaritons under the influence of the periodic potential together with discussions on the phase-modulated interference pattern which corresponds to the polariton Talbot patterns observed in experiments.
A full quantum theory beyond the mean-field regime is developed for an exciton-polariton condensate, for a complete understanding of quantum fluctuations. We find an analytical solution for the polariton density matrix, showing the nonlinearity causing fast relaxation correlated with the pump so as to yield the quantum condensation above threshold. Increasing the pump intensity, a nonequilibrium phase transition towards the condensation of the lower polaritons emerges, with a statistics transiting from a thermal, through a super-Poissonian, and towards a nonclassical distribution; the latter two possess off-diagonal long-range order but the last one is beyond. The results signify the role of dark states for polariton fluctuations, and lead to a nonclassical counting statistics of emitted photons, which elaborates the role of the key parameters, e.g., pump, detuning, and temperature.