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From Polariton Condensates to Highly Photonic Quantum Degenerate States of Bosonic Matter

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From Polariton Condensates to Highly Photonic Quantum Degenerate States of Bosonic Matter From polariton condensates to highly photonic quantum degenerate states of bosonic matter Marc Aßmanna,1, Jean-Sebastian Tempela, Franziska Veita, Manfred Bayera, Arash Rahimi-Imanb, Andreas Löfflerb, Sven Höflingb, Stephan Reitzensteinb, Lukas Worschechb, and Alfred Forchelb aExperimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany; and bTechnische Physik, Physikalisches Institut and Wilhelm Conrad Röntgen Research Center for Complex Material Systems, Universität Würzburg, D-97074 Würzburg, Germany Edited by Peter Littlewood, University of Cambridge, Cambridge, United Kingdom, and accepted by the Editorial Board November 3, 2010 (received for review July 7, 2010) Bose–Einstein condensation (BEC) is a thermodynamic phase tran- tion and thermalized polariton gases to have condensation sition of an interacting Bose gas. Its key signatures are remarkable effects. It should be pointed out that the polariton system as a quantum effects like superfluidity and a phonon-like Bogoliubov whole is almost never in thermal equilibrium. Usually it will con- excitation spectrum, which have been verified for atomic BECs. sist of the low-energy part of the polariton distribution which may In the solid state, BEC of exciton–polaritons has been reported. be described by an effective temperature if this part reaches a Polaritons are strongly coupled light-matter quasiparticles in semi- local self-equilibrium, the high-energy exciton-like part therma- conductor microcavities and composite bosons. However, they are lized at the lattice temperature, and the intermediate bottleneck subject to dephasing and decay and need external pumping to regime which is almost always out of equilibrium. It has been sta- reach a steady state. Accordingly the polariton BEC is a nonequili- ted that such a local self-equilibrium condensed state is directly brium process of a degenerate polariton gas in self-equilibrium, related to BEC, whereas one which does not obtain a local self- but out of equilibrium with the baths it is coupled to and therefore equilibrium can hardly be described in the framework of BEC deviates from the thermodynamic phase transition seen in atomic (15). Although this statement is true in terms of thermodynamic BECs. Here we show that key signatures of BEC can even be ob- properties, it is not immediately clear which of the signatures seen served without fulfilling the self-equilibrium condition in a highly in local self-equilibrium polariton condensates will still prevail far photonic quantum degenerate nonequilibrium system. from even local self-equilibrium. Recent theoretical analysis (16) predicts condensation effects even when the local self-equili- photon statistics ∣ quantum optics ∣ semiconductor photon sources brium condition is not fulfilled and also not only for the equili- brium Bose–Einstein distribution, but also for a wide range of icrocavity polaritons are composite bosons, which are partly nonequilibrium distribution functions. However, although there Mphotons and partly excitons as quantified by the Hopfield are theoretical studies classifying the condensate ground state coefficients jC2j and jX 2j giving the relative photonic and exci- for different detunings and mean polariton separations (17), from tonic content (1), respectively, and are expected to condense at an experimental point of view, detailed studies of this regime high temperatures because of their light mass (2). Moreover, the without local self-equilibrium are still missing. photonic and excitonic contents of polaritons can be precisely We realize a system far from equilibrium using polaritons Δ ¼ − adjusted by changing the detuning Ec Ex between the bare with high-photonic content, characterized by a negative detuning cavity mode and the bare exciton mode. Unlike Bose–Einstein Δ < 0 between the bare cavity and exciton modes, and excited by condensates (BECs) in atomic gases, solid-state systems are short laser pulses with duration of 1.5 ps. The fast decay times subject to strong dephasing and decay on timescales on the order because of the high-photonic content of the polaritons ensure of the particle lifetimes. As a consequence, external pumping is that the polariton distribution is neither in thermal equilibrium required to achieve a steady state. Despite this nonequilibrium with the lattice nor in local self-equilibrium and a stationary character, degenerate polariton systems show several textbook situation is never reached. The thermalization times for our sam- features of BECs (3), including spontaneous build up of coher- ple agree well with values previously reported (18) where the ence (4) and polarization (5), quantized vortices (6, 7), spatial polariton thermalization time becomes shorter than their lifetime condensation (8), and superfluidity (9, 10). Usually this behavior above the degeneracy threshold for positive detunings, but is is attributed to the system undergoing an equilibrium phase always longer than their lifetime for negative detunings. Although transition toward a condensed state: Although the polariton gas there have been experiments where this regime was called out of is not necessarily in equilibrium with the lattice, it is in self- equilibrium condensation (19) or metastable condensate (15), so equilibrium, if the relaxation kinetics of excited carriers is fast far no systematic studies characterizing the condensate-like prop- enough compared to the leakage of the photonic component erties in this regime where it is not possible to define an effective out of the cavity, which is usually the case for positive detunings temperature have been performed. We will refer to highly photo- Δ ≥ 0. In this case jX2j is larger than 50%, an effective tempera- nic (HI-P) polariton states when discussing this nonequilibrium ture can be defined and the polariton gas can be considered as a regime in order to stress that it is different from common polar- thermodynamic equilibrium state, which is out of equilibrium iton condensates which are also nonequilibrium states, but can be with the baths it is coupled to. This degenerate polariton gas is described by an effective temperature. In the following we test distinguishable from a simple photon laser (11). It is often pointed out that this intrinsic nonequilibrium situation and the Author contributions: M.A. and M.B. designed research; M.A., J.-S.T., F.V., and A.R.-I. two-dimensional order parameter of polariton BECs give rise performed research; A.R.-I., A.L., S.H., S.R., L.W., and A.F. contributed new reagents/ to interesting phenomena like half-vortices (12) and a diffusive analytic tools; M.A., J.-S.T., F.V., and A.R.-I. analyzed data; and M.A. and M.B. wrote Goldstone mode (13, 14), which do not occur in equilibrium the paper. condensates. Accordingly, the next interesting questions are The authors declare no conflict of interest. whether the same or even unique phenomena occur, if the system This article is a PNAS Direct Submission. P.L. is a guest editor invited by the Editorial Board. is driven even further from equilibrium into a regime that cannot Freely available online through the PNAS open access option. be described by an effective temperature and whether it is indeed 1To whom correspondence should be addressed. E-mail: [email protected]. necessary to consider a thermodynamic equilibrium phase transi- uni-dortmund.de. 1804–1809 ∣ PNAS ∣ February 1, 2011 ∣ vol. 108 ∣ no. 5 www.pnas.org/cgi/doi/10.1073/pnas.1009847108 Downloaded by guest on September 27, 2021 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HI-P states for several common signatures of condensation like Γ Γ2 ω ð Þ¼ω ð0Þ − Æ ½ω ð Þ − ω ð0Þ2 − build up of a macroscopic ground-state occupation, suppressed Æ k∥ Bog i 2 Bog k∥ Bog 4 ; quantum fluctuations, and linearization of the excitation spectrum. αβ where Γ ¼ γ 1þαβ is the effective decay rate of the Bogoliubov Basic characteristics of condensation and the excitation spec- mode, γ is the bare cavity decay rate, α ¼ P − 1, where thr means trum are manifested in the polariton dispersion. Above threshold Pthr threshold, gives the relative pump strength compared to the a blueshift of the kjj ¼ 0 lower polariton (LP) is expected and the threshold value, and β quantifies the dependence of the conden- LP dispersion is supposed to change from a parabolic shape in sate amplification rate because of stimulated scattering of polar- the uncondensed case toward a different dispersion in the low- pffiffiffiffiffiffiffiffiffiffiffiffiffiℏ itons on the reservoir population. In the following we assume the momentum condensed regime jk∥ξj < 1, where ξ ¼ is β 2m gn effect of to be negligible above threshold, so that the variation LP c Γ γ the healing length of the condensate (20). In standard homoge- of between zero at threshold and at large pump rates is gov- α neous equilibrium Bogoliubov theory, the expected dispersion is erned by alone. This diffusive spectrum is expected to manifest itself as a flat region in the low-momentum region of the real a phonon-like linear one at small k given by part of the excitation spectrum. The range of this flat region is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expected to increase with α, starting from an unmodified Bogo- ω ð Þ¼ω ð Þþ þ þ ð ξÞ2½ð ξÞ2 þ 2 Bog k∥ LP k∥ gnc grnr gnc k∥ k∥ : liubov dispersion exactly at the threshold, where α is exactly zero. In Fig. 1 we compare the LP dispersion of the emission copolar- ized with the excitation for detunings of 0, −2, −4, and −7 meV to Here g and gr are coupling constants describing the interaction the theoretical predictions below and above threshold. Below between two condensate polaritons and between condensate A E I M polaritons and reservoir excitons, respectively, and n and n are threshold (Fig. 1 , , , and ) the standard quadratic LP dis- c r persion is observed in all cases. Increasing the excitation power their densities. Using resonant excitation, the reservoir exciton near the threshold value (Fig. 1 B, F, J, and N), a blueshift of contributions are expected to be negligible.
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