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Physical review. B, Condensed matter

Many-photon coherence of Bose-condensed excitons: Luminescence and related nonlinear optical phenomena

Yu. E. Lozovik, I. V. Ovchinnikov

Published Aug 30, 2002• 54 citations
Publisher

Abstract

We consider many-photon coherent emission governed by coherent recombination of many excitons from Bose condensate. Momentum conservation makes photons simultaneously created in the coherent recombination of several excitons from the condensate have zero sum momentum. Many-photon correlations in the processes of simultaneous many-photon production (photons squeezing) could be detected by photon counting experiments with several detectors spatially arranged in an appropriate way, i.e., by Hanbury-Brown-Twiss-like experiments. We analyze the stimulated processes of N-exciton coherent recombination from the condensate resonantly induced by $N\ensuremath{-}1$ external laser beams with momenta ${\mathbf{k}}_{1},{\mathbf{k}}_{2},\dots{},{\mathbf{k}}_{N\ensuremath{-}1}.$ Such stimulated processes must reveal themselves through a unidirectional beam with recoil momentum ${\mathbf{k}}_{N}=\ensuremath{-}{\ensuremath{\sum}}_{i=1}^{N\ensuremath{-}1}{\mathbf{k}}_{i}.$ In particular, two-exciton coherent recombination from the condensate stimulated by a laser beam effectively acts in a three-dimensional (3D) exciton system as a laser beam backscattering from exciton Bose condensate. For 2D coherent exciton systems besides the backscattering stimulated two-exciton coherent recombination has an additional manifestation---anomalous beam transmission with the only change in the sign of in-plane momentum component of inducing laser beam. We estimate the rates of many-photon coherent emission from a 3D system of ${\mathrm{Cu}}_{2}\mathrm{O}$ excitons and from a 2D exciton system in GaAs/AlGaAs coupled quantum wells. The estimations show that these nonlinear optical effects are experimentally observable.

DOI
10.1103/physrevb.66.075124

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The coherent recombination of several (N) Bose-condensed excitons with simultaneous creation of N photons is considered. Due to the momentum conservation law, the total momentum of created photons is zero because of the zero momentum of excitons in Bose condensate. This requirement, in conjunction with the fact that the photon wavenumbers are fixed and equal to approximately E g /c (E g is the semiconductor gap and c is the speed of light), determines the mutual orientation of the wavevectors of emitted photons. This can be seen from the photon angular correlation in the experiments with several appropriately oriented detectors operating in the time-coincidence mode. It is shown that, if these processes are induced by N− 1 external laser beams (each with wavevector k i ), then a unidirectional radiation with recoil wavevector k=−∑i k i should emerge from the exciton system. The intensities of coherent three-and four-exciton recombination are estimated for the exciton system in Cu2O.
A new effect—light backscattering from exciton Bose-condensate—is considered. This effect is connected with the photoinduced coherent recombination of two excitons in the condensate with the production of two photons with opposite momenta. The effect of two-exciton coherent recombination leads also to the appearance of the second-order coherence in exciton luminescence connected with squeezing between photon states with opposite momenta. The estimations given for Cu 2 O and GaAs excitons show that the effect of stimulated light backscattering can be detected experimentally. Moreover, in the system of 2D excitons in coupled quantum wells, the effect of stimulated anomalous light transmission must also take place. Anologous effects can also take place in systems of Bose-condensed atoms in excited (but metastable) states.
ABSTRACT The new effect stimulated light backstattering and anomalous light transmission in Bose-condensed excitonsystem are considered. The effects are connected with photoinduced coherent recombination of two excitonsfrom the condensate with production of two photons with opposite momenta. The effect of two-exciton coherentrecombination leads also to the appearence of the second order coherence in exciton luminescence connectedwith squeezing between photon states with opposite momenta. Coherent 3- and 4-photon emission processesstimulted by respectively two and three laser beams are also considered.Keywords: nanostructure, quantum well, nanooptics, exciton, Bose condensationThe great progress has been made in the study of exciton Bose-condensation of exciton systems in quantumwells (QW) and Cu20 (see, e. g. [1—7] and references therein). However, the study of various ways which excitoncondensate should unambiguously reveal itself by its optical properties is still vital [3, 4].We analyze the coherent coupling of photons with opposite momenta, originating from coherent recombina-tion of two excitons from the condensate. We reveal that this process leads to new effect — stimulated lightbackscattering on exciton condensate. This effect can be viewed as a photoinduced coherent recombination oftwo condensate excitons with production of two photons with opposite momenta. We also consider analagouscoherent many-photon processes.Before we analyse the system of quasi-2D excitons in QW let us first consider the simpler case of 3D excitonsin Cu20. Direct recombination of electron and hole in Cu20 is a forbidden process and an exciton decays mainlywith production of a photon as well as of an optical phonon (Fig. 1 (a)). The process of two exciton recombination( Fig. 1(b)) has extra vertex of exciton—photon interaction and, therefore, is substantially weaker.
An investigation is reported of stimulated two-photon emission by Bose-condensed excitons accompanied by a coherent two-exciton recombination, i.e., by simultaneous recombination of two excitons with opposite momenta leaving unchanged the occupation numbers of exciton states with momenta p≠0. Raman light scattering (RLS) accompanied by a similar two-exciton recombination (or production of two excitons) is also analyzed. The processes under consideration can occur only if a system contains Bose condensate, so their detection can be used as a new method to reveal Bose condensation of excitons. The recoil momentum, which corresponds to a change in the momentum of the electromagnetic field in the processes, is transferred to phonons or impurities. If the recoil momentum is transmitted to optical phonons with frequency ω 0 s , whose occupation numbers are negligible, and the incident light frequency satisfies ω<2Ω, where Ω_=Ω−ω 0 s is the difference frequency and Ω is the light frequency corresponding to the recombination of an exciton with zero momentum, stimulated two-photon emission and RLS with coherent two-exciton recombination give rise to a line at 2Ω_ −ω and an anti-Stokes component at ω+2Ω_, respectively. For ω2Ω_ the RLS spectrum contains Stokes and anti-Stokes components at frequencies ω±2Ω_, whereas stimulated two-photon emission is impossible. Formulas for the cross sections at finite temperatures are obtained for the processes under consideration. Our estimates indicate that a spectral line at 2Ω_−ω, corresponding to the stimulated two-photon emission accompanied by coherent optical phonon-assisted two-exciton recombination can be experimentally detected in Cu2O.
The unusual two-photon emission by Bose-condensed excitons caused by simultaneous recombination of two excitons with opposite momenta leaving the occupation numbers of excitonic states with momenta $\mathbf{p}\ensuremath{\ne}0$ unchanged (below coherent two-exciton recombination) is investigated. Raman scattering accompanied by the analogous two-exciton recombination (or creation) is also analyzed. The excess momentum equal to the change of the electromagnetic field momentum in these processes can be transferred to phonons or impurities. The processes under consideration take place if there is Bose condensation in the interacting exciton system, and, therefore, can be used as a method to reveal exciton Bose condensation. If the frequency of the incident light $\ensuremath{\omega}<2\ensuremath{\Omega}$ $(\ensuremath{\Omega}$ is the frequency corresponding to the recombination of an exciton with p=0), the coherent two-exciton recombination with the excess momentum elastically transferred to impurities leads to the appearance of the spectral line $2\ensuremath{\Omega}\ensuremath{-}\ensuremath{\omega}$ corresponding to the induced two-photon emission. In this case the anti-Stokes line on frequency $\ensuremath{\omega}+2\ensuremath{\Omega}$ also appears in the Raman spectrum. If $\ensuremath{\omega}>2\ensuremath{\Omega},$ there are both Stokes and anti-Stokes lines on frequencies $\ensuremath{\omega}\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}$ in the Raman spectrum. The induced two-photon emission is impossible in this case. The spectral lines mentioned above have phonon replicas on frequencies $|\ensuremath{\omega}\ifmmode\pm\else\textpm\fi{}(2\ensuremath{\Omega}\ensuremath{-}n{\ensuremath{\omega}}_{0}^{s})|$ corresponding to the transmission of the excess momentum (partially or as a whole) to optical phonons of frequency ${\ensuremath{\omega}}_{0}^{s}$ (n(n(n is an integer number). The quantitative estimation shows that the light corresponding to the coherent two-exciton recombination can be experimentally observed in Cu2O.{\mathrm{Cu}}_{2}\mathrm{O}.Cu2​O.
A theoretical study is reported of stimulated light scattering, including wave-vector reversal and anomalous transmission, by a coherent phase in electron-hole (e-h) systems of low and high charge-carrier density. For these two cases the coherent phase is taken to be a Bose-Einstein condensate of excitons or a BCS-like state of e-h pairs, respectively. The scattering mechanism is laser-induced coherent recombination of two excitons or two coherent e-h pairs, respectively. The e-h system is assumed to exist within a GaAs/AlGaAs double quantum well or bulk GaAs. The emission rate of two photons with antiparallel momenta is estimated. Multiphoton emission due to multiexciton coherent recombination is covered. Methods for detecting the effects predicted are proposed.