Power-law decay of the spatial correlation function in exciton-polariton condensates

Abstract

We create a large exciton-polariton condensate and employ a Michelson interferometer setup to characterize the short-and longdistance behavior of the first order spatial correlation function. Our experimental results show distinct features of both the twodimensional and nonequilibrium characters of the condensate. We find that the gaussian short-distance decay is followed by a powerlaw decay at longer distances, as expected for a two-dimensional condensate. The exponent of the power law is measured in the range 0.9-1.2, larger than is possible in equilibrium. We compare the experimental results to a theoretical model to understand the features required to observe a power law and to clarify the influence of external noise on spatial coherence in nonequilibrium phase transitions. Our results indicate that Berezinskii-KosterlitzThouless-like phase order survives in open-dissipative systems.quantum well excitons | semiconductor microcavities T he spatial correlation function quantifies the coherence properties of a system (1). In a 3D Bose-condensed gas, long range order is observed, and the correlation function decays toward a plateau at large distances (2, 3). In the homogeneous 2D Bose gas (4), however, no long range order can be established (5). Instead, Berezinskii-Kosterlitz-Thouless (BKT) theory of the equilibrium interacting gas predicts a transition to a low-temperature superfluid phase, which shows a power-law decay of the correlation function (6, 7). Unfortunately, it is frequently hard to directly measure this, and only very recently (8) was indication of the power-law decay of coherence seen in a 2D atomic gas. It has been theoretically predicted (9, 10) that power-law decay of coherence survives in the nonequilibrium problem, and it is this prediction that the current experiment sets out to test.Exciton polaritons are short-lived quasiparticles formed in a semiconductor quantum well strongly coupled to a planar microcavity (11). Each one is a superposition of a quantum well exciton and a microcavity photon, and they behave as 2D bosons below the Mott density. Above a threshold particle density, condensation is observed (12). Due to the nonequilibrium nature of polariton condensation, understanding its coherence properties is quite revealing regarding the different roles of fluctuations in the equilibrium and nonequilibrium problems.Previous measurements on polariton condensates have demonstrated coherence at large distances but were limited by large experimental uncertainties (13) or highly disordered samples (14,15), and the long-distance behavior could not be fully extracted. Recently, the correlation function at large distances was studied in 1D condensates confined in a quantum wire (16) and in a valley of the disorder potential (17). In ref. 16, the data was energy-resolved so that excited states were filtered out, while in ref. 17 a rare area on the sample was chosen in which a single mode condensate is seen. The purpose of both those experiments was to investigate how long the coherence length o...

Keldysh field theory for driven open quantum systems

“…Thus, approaching the transition from above by lowering the pump power, the critical value x KT is reached first, and the system loses algebraic order through unbinding of vortices [300][301][302]. On the other hand, the crossover to the disordered regime will be controlled by KPZ physics once x * ≥ x KT , which can be achieved by increasing the loss rate (i.e., reducing the cavity Q) toγ ≈ 0.5.…”

“…Both in experiments on incoherently pumped polariton condensates [300,301] and simulations of parametrically pumped systems [302], spatial correlations have been found to decay algebraically within the confines of the system. This, however, is not in contradiction to the present analysis based on the KPZ equation: indeed, if the microscopic value g 0 of the rescaled non-linearity (190) is small, which is actually the case in current experiments [19], a renormalized value of g = 1 is reached in the RG flow only at the exponentially large scale…”

Observation of the Berezinskii-Kosterlitz-Thouless Phase Transition in an Ultracold Fermi Gas

“…We arrive at g 1 (r) ∼ r −η phase (T ) f ( r R TF ) (10) with the function f being shown in Fig. S 3a.…”

“…Several key signatures of BKT physics have been experimentally observed in a variety of systems such as exciton-polariton condensates [10], layered magnets [11,12], liquid 4 He films [13], and trapped Bose gases [14][15][16][17][18][19][20]. Particularly in the context of superfluidity, the universal jump in the superfluid density was measured in thin films of liquid 4 He [13].…”

Quantum Langevin model for nonequilibrium condensation

“…On the other hand, it is an experimentally accessible quantity, easily measured from the far-field angular distribution of emitted light. By Fourier transform, it is directly related to the twopoints, one-time coherence function of the cavity field, a quantity which is of widespread use in experiments [3,4,31,32]. Grouping in the V k = v s k v s † k variance matrix the steadystate variances of all operator pairs, from a straightforward integration of the quantum Langevin equations [55], we obtain a Lyapunov equation:…”