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Publication# Higher order coherence of exciton-polariton condensates

Abstract

The second- and third-order coherence functions g((n))(0) (n= 2 and 3) of an exciton-polariton condensate are measured and compared to the theory. Contrary to an ideal photon laser, deviation from unity in the second- and third-order coherence functions is observed, thus showing a bunching effect, but not the characteristics of a standard thermal state with g((n))(0) = n!. The increase in bunching with the order of the coherence function, g((3))(0) > g((2))(0) > 1, indicates that the polariton condensate is different from a coherent state, a number state, or a thermal state. The measurement of third-order coherence has the advantage, compared to the second-order one, that the difference between a thermal state and a coherent state is more pronounced. The experimental results are in agreement with the theoretical model where polariton-polariton and polariton-phonon interactions are responsible for the loss of temporal coherence.

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Hadis Abbaspour, Albert Adiyatullin, Mitchell David Anderson, Benoît Marie Joseph Deveaud, Marcia Portella Oberli

Second-order time correlation measurements with a temporal resolution better than 3 ps were performed on a CdTe microcavity where spontaneous Bose-Einstein condensation is observed. After the laser pulse, the nonresonantly excited thermal polariton population relaxes into a coherent polariton condensate. Photon statistics of the light emitted by the microcavity evidences a clear phase transition from the thermal state to a coherent state, which occurs within 3.2 ps after the onset of stimulated scattering. Following this very fast transition, we show that the emission possesses a very high coherence that persists for more than 100 ps after the build-up of the condensate. (C) 2015 AIP Publishing LLC.

This thesis presents the coherence properties of polaritons in semiconductor microcavities. Semiconductor microcavities are microstructures in which the exciton ground state of a semiconductor quantum well is coupled to a photonic mode of a microresonator. The strong coupling mixes the character of excitons and photons, giving rise to the lower and upper polariton branches, quasiparticles with an unusual energetic dispersion relation due to the extreme mass difference between exciton and photon. Particularly special is the dispersion of the lower polariton, which forms a dip in the 2-dimensional k-space around the lowest energy state with zero in-plane momentum. In this dip, which can be seen as a trap in momentum space, the polaritons are efficiently isolated from dephasing mechanisms involving phonons. Polaritons can be resonantly excited at desired points on the polariton dispersion by shining on the microcavity laser light at the appropriate angle and wavelength. Polaritons can interact and scatter pairwise with each other conserving energy and in plane momentum k, a process similar to parametric scattering of photons in a nonlinear crystal. One polariton from a pump reservoir scatters down to the signal state at k = 0 (corresponding to normal incidence) and a second takes away the excess energy and momentum of the first and scatters up to the idler position ({kP, kP} → {0, 2kP }). This process can be stimulated by a small amount of signal polaritons injected with a probe laser beam at normal incidence. Here the coherence properties of the polariton parametric scattering have been investigated using spectroscopy techniques sensitive to the optical phase, for example coherent control with phase-locked femtosecond probe pulses. Just above the threshold for the stimulated parametric scattering, the parametric amplification process is given by the linear superposition of the individual amplification processes of each probe pulse. The emission of signal, pump, and idler can be controlled by tuning the relative phase of the 150fs-long probe pulses, which are separated by a few picoseconds in time. Experiments are presented that deal with the real-time dynamics of the parametric scattering in the spontaneous and the stimulated regime. It is shown, that in the spontaneous regime the scattering is started by a small amount of polaritons which have relaxed to the band bottom by emitting phonons. In the regime where polariton scattering is stimulated by an external probe, the rise of the signal intensity is delayed with respect to the arrival time of both pump and probe, a feature that can be attributed to the complex phase-matching mechanism for the parametric scattering. In the second part of the thesis, the spontaneous build up of a macroscopic coherence in a CdTe microcavity under non-resonant laser excitation is analysed. The build up of a long-range spatial coherence easily exceeding the thermal wavelength of the polaritons is shown. This is the hallmark of Bose-Einstein condensation and the proof of a macroscopic wavefunction. Experimental data on the statistical distribution of the polaritons in time, the polarisation of the non-linear emission, and the quantum transition from a thermal to a coherent state1 confirm that Bose-Einstein condensation of microcavity polaritons has been observed. We regard these observations as the first bullet-proof evidence for spontaneous Bose-Einstein condensation in a solid state system, a phenomenon that has been the subject to many investigations and controversies during the past four decades. ------------------------------ 1 The data about the statistical distribution, the polarisation, and the transition from a thermal to a coherent state is by courtesy of Jacek Kasprzak of the University of Grenoble.

This thesis presents a theoretical description of the phase transition, with formation of long-range spatial coherence, occurring in a gas of exciton-polaritons in a semiconductor microcavity structure. The results and predictions of the theories developed in this thesis suggest that this phase transition, recently observed in experiments, can be interpreted as the Bose-Einstein Condensation (BEC) of microcavity polaritons. Our theoretical framework is conceived as a generalization to the microcavity polariton system of the standard theories describing the BEC of a weakly interacting Bose gas. These latter are reviewed in Chapter 2, where an introduction to the physics of polaritons is also given. The polariton system is peculiar, basically due to three main features, i.e. the composite nature of polaritons, which are a linear superposition of photon and exciton states, their intrinsic 2-D nature, and the presence of two-body interactions, arising both from the mutual interaction between excitons and from the saturation of the exciton oscillator strength. Therefore it is not clear whether the observed phase transition can be properly described in terms of BEC of a trapped gas. To clarify this point, one has to describe self-consistently the linear exciton-photon coupling giving rise to polariton quasiparticles, and the exciton-nonlinearities. This is made in Chapter 3, where a bosonic theory is developed by generalizing the Hartree-Fock-Popov description of BEC to the case of two coupled Bose fields at thermal equilibrium. Hence, we derive the classical equations describing the condensate wave function and the Dyson-Beliaev equations for the field of collective excitations. In this way, for each value of the temperature and of the total polariton density, a self-consistent solution can be obtained, fixing the populations of the condensate and of the excited states. In particular, the theory allows to describe simultaneously the properties of the polariton, the exciton and the photon fields, this latter being directly investigated in the typical optical measurements. The predicted phase diagram, the energy shifts, the population energy distribution and the behavior of the resulting first order spatial correlation function agree with the recent experimental findings [Kasprzak 06, Balili 07]. These results thus support the idea that the observed experimental signatures are a clear evidence of polariton BEC. However, from a quantitative pint of view, the measured coherence amount in the condensed regime is significantly lower than the predicted one. This discrepancy could be due to deviations from the weakly interacting Bose gas picture and/or to deviations from the thermal equilibrium regime. In particular, these latter are expected to be strong in current experiments, because polaritons have a short radiative lifetime, while the rate of the energy-relaxation mechanisms is very slow. To investigate how the deviations from equilibrium could affect the condensate fraction and the formation of off-diagonal long-range correlations, in Chapter 4, we develop a kinetic theory of the polariton condensation, accounting for both the relaxation mechanisms and for the field dynamics of fluctuations. Within the Hartree-Fock-Bogoliubov limit, we derive a set of coupled equations of motion for the one-particle populations and for the two particle correlations describing quantum fluctuations. We account for the relaxation processes due both to the polariton-phonon coupling and to the exciton-exciton scattering. The actual spectrum of the system is evaluated within the Popov limit, during the relaxation kinetics. Within this model, we solve self-consistently the populations kinetics and the dynamics of the excitation field, for typical experimental conditions. In particular, we show that the role of quantum fluctuations is amplified by non-equilibrium, resulting in a significant condensate depletion. This behavior could explain the partial suppression of off-diagonal long-range coherence reported in experiments [Kasprzak 06, Balili 07]. We complete the analysis, by studying how the deviations from equilibrium depend on the system parameters. Our results show that the polariton lifetime plays a crucial role. In particular, we expect that the increase of the polariton lifetime above 10 ps would lead to thermal-equilibrium polariton BEC in realistic samples. In Chapter 5, devoted to the conclusions, we discuss which issues of BEC could be clarified, by achieving polariton BEC at thermal-equilibrium, and which extensions of the present work would be most promising in this respect.