Bogoliubov excitations of a polariton condensate in dynamical equilibrium with an incoherent reservoir

Abstract

The classic Bogoliubov theory of weakly interacting Bose gases rests upon the assumption that nearly all the bosons condense into the lowest quantum state at sufficiently low temperatures. Here we develop a generalized version of Bogoliubov theory for the case of a driven-dissipative excitonpolariton condensate with a large incoherent uncondensed component, or excitonic reservoir. We argue that such a reservoir can consist of both excitonic high-momentum polaritons and optically dark superpositions of excitons across different optically active layers, such as multiple quantum wells in a microcavity. In particular, we predict interconversion between the dark and bright (lightcoupled) excitonic states that can lead to a dynamical equilibrium between the condensate and reservoir populations. We show that the presence of the reservoir fundamentally modifies both the energy and the amplitudes of the Bogoliubov quasiparticle excitations due to the non-Galileaninvariant nature of polaritons. Our theoretical findings are supported by our experiment, where we directly detect the Bogoliubov excitation branches of an optically trapped polariton condensate in the high-density regime. By analyzing the measured occupations of the excitation branches, we extract the Bogoliubov amplitudes across a range of momenta and show that they agree with our generalized theory.

Dark and thermal reservoir contributions to polariton sound velocity

“…This controversy indicates that the effects of finite temperature and the optically-dark reservoir on the elementary excitations of polariton condensates are poorly understood. The aim of the current Letter is to provide a stitching of the finitetemperature theory [19] with a simple phenomenological rate-equations model that allows addressing the situations with large reservoir-to-condensate ratios [29,30].…”

“…Pump enters the equations for n in and n D as P and P . All the coefficients in (4) can be defined from comparison with experiment (here [29,30]). The main difference of the model (4) compared to the previously developed phenomenological approaches is that it describes the dynamical feeding of the polariton system with the total density n, wherein the equilibration between the condensate n 0 and non-condensate n particles is assumed to be governed by temperature.…”

High-resolution coherent probe spectroscopy of a polariton quantum fluid

“…In this Letter, we present a high-resolution angleresolved coherent probe spectroscopy technique that allows for the measurement of the Bogoliubov dispersion in the linear and nonlinear regimes of interactions. In contrast to previous work [16], we continuously pump the polariton fluid near resonance to increase the spectral resolution; and unlike in photoluminescence experiments [17][18][19], we directly monitor the resonant response of the fluid to a coherent probe, thus efficiently isolating the signal on top of background emission from the fluid. We obtain a dramatic increase in the spectral resolution and resolve the dispersion at all wave-numbers (down to k = 0) and reveal the sound-like behaviour at low wave-number.…”

“…Interestingly, this value is reduced by a factor α with respect to the theoretical prediction c th s = δ/m LP . This can be explained in terms of an incoherent reservoir of long-lived dark excitons providing an additional contribution to the blue-shift and to the bistability curve but not to the speed of sound [17,19,22]. In our experiment, an estimate of the reservoir contribution to the blue-shift g r n r = 0.75 δ is obtained from the measured values of α 0.49 and of the total blue-shift gn + g r n r = δ at the turning point of the bistability.…”

“…Unlike the ω B+ branch, the ghost branch does not resonate with the cavity, and is spontaneously populated by quantum depletion of the fluid. This renders its direct observation in photo-luminescence difficult [17][18][19]. Here, we stimulate the coherent conversion of the fluid of polaritons toward the ghost branch via the coherent probe [30]: this configuration induces a four-wave mixing (FWM) process (ω p , ω p ) → (ω + , ω − ), and the probe at ω − < ω p directly seeds Bogoliubov excitations on the ghost branch which manifest as an amplification peak R > 1 above the baseline of the probe's reflectivity spectrum.…”