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Physical Review Letters

Evidence for Bose-Einstein condensation of a two-component exciton gas

David W. Snoke, James P. Wolfe, A. Mysyrowicz

Published May 21, 1990• 110 citations
Publisher

Abstract

Time-resolved spectral and spatial measurements of photoluminescence in ${\mathrm{Cu}}_{2}$O at low temperatures suggest that the gas of paraexcitons undergoes Bose-Einstein condensation at high densities. As a result of the two-component nature of the gas (orthoexcitons and paraexcitons with different ground-state energies and spin multiplicities) and the nonequilibrium nature of photoproduced excitons, the condensation process is quite different from the simple ideal-gas case envisioned by Einstein.

DOI
10.1103/physrevlett.64.2543

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Evidence for Bose-Einstein condensation of excitons in Cu2 O

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Pulsed-laser excitation of naturally grown crystals of ${\mathrm{Cu}}_{2}$O produces a dense gas of excitons, which exhibits Bose-Einstein quantum statistics. Both orthoexcitons and paraexcitons, split by a 12-meV electron-hole exchange energy, are formed. At high excitation levels, the time-resolved luminescence of orthoexcitons yields a density-temperature relation that follows the phase boundary for Bose-Einstein condensation, n=${\mathit{CT}}^{3/2}$, but no striking evidence for a condensed fraction is observed. We have now measured the time-resolved spectrum of the lower-lying paraexcitons whose luminescence is 1500 times weaker than the orthoexciton luminescence. At intermediate power levels, the paraexciton distribution is more degenerate than the orthoexcitons. The relative intensities of the two species indicate that at high excitation levels the paraexciton density exceeds the critical density for condensation. At these high densities the paraexciton spectrum develops a structure that cannot be explained in terms of the thermalized Bose-Einstein distribution. Time-resolved spatial measurements of the luminescence indicate a rapid nondiffusive expansion of the gas at supersonic velocities that approach the expected ballistic velocities of the excitons. The rapid transport and the concurrent changes in the spectral distribution of paraexcitons may be the first observation of excitonic Bose-Einstein condensation and superfluidity.

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Quantum Kinetics of the Exciton Bose-Einstein Condensation

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Photo-excited excitons, i.e. closely bound electron-hole pairs, are in the low-density regime approximately bosons. For sufficiently low temperatures and sufficiently high densities of the exciton gas, a nonequilibrium phase transition in the form of a Bose-Einstein condensation (BEC) can be expected. Various levels of the description of the kinetics of relaxation and condensation due to the coupling of the excitons to a cold reservoir of acoustic phonons are discussed. For noninteracting excitons a rate equation approach with the exciton Boltzmann equation can be used. For interacting excitons the nonmarkovian quantum kinetics is required for the description of the BEC in order to incorporate the changes of the energy spectrum during the condensation self-consistently. Finally colored Langevin fluctuations can be used to calculate the phase diffusion of the condensate.

Phase Transitions•Jan 1, 2002•1 citation
Time-resolved spectral and spatial measurements of photoluminescence in Cu2{\mathrm{Cu}}_{2}Cu2​O at low temperatures suggest that the gas of paraexcitons undergoes Bose-Einstein condensation at high densities. As a result of the two-component nature of the gas (orthoexcitons and paraexcitons with different ground-state energies and spin multiplicities) and the nonequilibrium nature of photoproduced excitons, the condensation process is quite different from the simple ideal-gas case envisioned by Einstein.
Pulsed-laser excitation of naturally grown crystals of Cu2{\mathrm{Cu}}_{2}Cu2​O produces a dense gas of excitons, which exhibits Bose-Einstein quantum statistics. Both orthoexcitons and paraexcitons, split by a 12-meV electron-hole exchange energy, are formed. At high excitation levels, the time-resolved luminescence of orthoexcitons yields a density-temperature relation that follows the phase boundary for Bose-Einstein condensation, n=CT3/2{\mathit{CT}}^{3/2}CT3/2, but no striking evidence for a condensed fraction is observed. We have now measured the time-resolved spectrum of the lower-lying paraexcitons whose luminescence is 1500 times weaker than the orthoexciton luminescence. At intermediate power levels, the paraexciton distribution is more degenerate than the orthoexcitons. The relative intensities of the two species indicate that at high excitation levels the paraexciton density exceeds the critical density for condensation. At these high densities the paraexciton spectrum develops a structure that cannot be explained in terms of the thermalized Bose-Einstein distribution. Time-resolved spatial measurements of the luminescence indicate a rapid nondiffusive expansion of the gas at supersonic velocities that approach the expected ballistic velocities of the excitons. The rapid transport and the concurrent changes in the spectral distribution of paraexcitons may be the first observation of excitonic Bose-Einstein condensation and superfluidity.
In experiments on excitons and spin-polarized atomic hydrogen, one deals with a gas involving two Bose branches (for example, paraexcitons and orthoexcitons). Based on a simple model for the coupling between these two components, we discuss how interconversion processes affect Bose condensation in such a system. Our analysis is based on an exact transformation to renormalized Bose particles which are uncoupled. We give results for the condensate fraction as a function of the temperature in the case when the two original Bose particles (a and b) are not in chemical equilibrium (i.e., the chemical potentials ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}}$ and ${\mathrm{\ensuremath{\mu}}}_{\mathit{b}}$ are not equal) as well as in the case of chemical equilibrium (${\mathrm{\ensuremath{\mu}}}_{\mathit{a}}$=${\mathrm{\ensuremath{\mu}}}_{\mathit{b}}$). Our results are of interest in connection with current attempts to observe Bose condensation in atomiclike gases.
The case of excitons as candidates for Bose–Einstein condensation is discussed, and experimental results in CuCl and Cu 2 O are presented. In CuCl, spectral analysis of the luminescence from biexcitons as a function of their density reveals a gradual evolution from classical statistics towards a quantum degenerate regime. The appearance of a sharp emission line below a critical temperature and above a critical density is attributed to the presence of a laser-induced Bose–Einstein condensate of excitonic molecules. This interpretation is supported by pump-probe experiments which show that additional particles injected in the presence of a biexciton condensate are drawn into it. In Cu 2 O, free exciton luminescence spectral analysis of ortho- and paraexcitons reveals a gradual evolution from a classical to a Bose quantum degenerate regime with increasing particle densities. Orthoexciton densities close to the critical density for condensation are obtained at high incoherent excitation. Under similar pumping, paraexciton densities exceeding the critical value are inferred from luminescence intensity ratios. Anomalous transport properties of paraexcitons, such as ballistic propagation over macroscopic distances and formation of soliton-like excitonic packets are discussed as evidence for excitonic superfluidity. Introduction Excitons The lowest electronically excited state of a non-metallic crystal corresponds to the promotion of one electron from the top of the highest fully occupied valence band to the bottom of the next empty conduction band. A correct evaluation of the required energy must include the Coulomb correlation between the promoted electron and all other electrons left behind in the valence band.
The system under consideration is a multicomponent gas of interacting paraexcitons and orthoexcitons confined in a three-dimensional potential trap. We calculate the spatially resolved optical emission spectrum due to interband transitions involving weak direct and phonon-mediated exciton-photon interactions. For each component, the occurrence of a Bose-Einstein condensate changes the spectrum in a characteristic way so that it directly reflects the constant chemical potential of the excitons and the renormalization of the quasiparticle excitation spectrum. Moreover, the interaction between the components leads, in dependence on temperature and particle number, to modifications of the spectra indicating phase separation of the subsystems. Typical examples of density profiles and luminescence spectra of ground-state paraexcitons and orthoexcitons in Cu2O{\text{Cu}}_{2}\text{O}Cu2​O are given.
Abstract An unambiguous observation of the Bose-Einstein condensation (BEC) of excitons in a photoexcited bulk semiconductor and elucidation of its inherent nature have been longstanding problems in condensed matter physics. Here, we observe the quantum phase transition and a Bose-Einstein condensate appearing in a trapped gas of 1s paraexcitons in bulk Cu2O below 400 mK, by directly visualizing the exciton cloud in real space using mid-infrared induced absorption imaging that we realized in a dilution refrigerator. Our study shows that the paraexciton condensate is undetectable by conventional luminescence spectroscopy. We find an unconventionally small condensate fraction of 0.016 with the spatial profile of the condensate well described by mean-field theory. Our discovery of this new type of BEC in the purely matter-like exciton system interacting with a cold phonon bath could pave the way for the classification of its long-range order, and for essential understanding of quantum statistical mechanics of non-equilibrium open systems.
Photo-excited excitons, i.e. closely bound electron-hole pairs, are in the low-density regime approximately bosons. For sufficiently low temperatures and sufficiently high densities of the exciton gas, a nonequilibrium phase transition in the form of a Bose-Einstein condensation (BEC) can be expected. Various levels of the description of the kinetics of relaxation and condensation due to the coupling of the excitons to a cold reservoir of acoustic phonons are discussed. For noninteracting excitons a rate equation approach with the exciton Boltzmann equation can be used. For interacting excitons the nonmarkovian quantum kinetics is required for the description of the BEC in order to incorporate the changes of the energy spectrum during the condensation self-consistently. Finally colored Langevin fluctuations can be used to calculate the phase diffusion of the condensate.