Barland, Azam, Lippi, Nyman and Kaiser (Optics Express 29, 8368 (2021)) observed photon thermalization and a condensation phase transition at room temperature in an electrically pumped broad-area semiconductor microresonator (a planar vertical-cavity surface-emitting laser (VCSEL) cavity with no engineered confinement), and posed the classification problem this transition raises: in the strongly driven regime, with residual spatial inhomogeneity and finite planar boundaries, does the dense photon gas display Bose-Einstein condensation, Berezinskii-Kosterlitz-Thouless correlations, or Kardar-Parisi-Zhang correlations, and do grand-canonical ensemble conditions hold with respect to its variable-number electron-hole reservoir? Later literature has substantially developed the missing thermalization theory for semiconductor reservoirs and confirmed semiconductor photon Bose-Einstein condensation spectroscopically, but spatially resolved correlation measurements and the phase classification on this electrically driven platform remain undone.

Background

Photons in a suitably designed optical microcavity can form a Bose-Einstein condensate (BEC), a state in which a macroscopic fraction of bosons occupies the lowest-energy quantum state. In the first realization, a dye-filled microcavity (Klaers, Schmitt, Vewinger and Weitz, Nature 468, 545 (2010)), two ingredients turned the intracavity photon gas into a massive Bose gas at room temperature: the cavity cut-off bends the dispersion relation into a parabolic shape near its minimum, so the photons behave as a two-dimensional gas with a finite effective mass, and repeated absorption and re-emission by dye molecules exchanges energy and particles with the medium many times before a photon is lost, so the medium acts as a heat and particle bath that gives the photon gas both a temperature and a chemical potential μ\mu. When μ\mu approaches the ground-mode energy, the phase-space density (photon density times thermal de Broglie area, ρ2DΛ2\rho_{2D}\Lambda^2) reaches order unity and the lowest mode becomes macroscopically occupied. Because the molecular reservoir is far larger than the photon gas, the gas realizes a grand-canonical ensemble, in which the particle number of the condensed mode fluctuates by amounts comparable to its mean; this was confirmed experimentally via grand-canonical number statistics (Schmitt et al., 2014), and the first-order spatial coherence of the thermalized photon gas has been mapped directly (Damm et al., 2017).

Barland et al. (Optics Express 29, 8368 (2021)) reported the electrically pumped counterpart: a broad-area vertical-cavity surface-emitting laser (VCSEL, a semiconductor microresonator with two distributed-Bragg reflectors (periodic multilayer mirrors of very high reflectivity) and electrical carrier injection, operating in the weak light-matter coupling regime where cavity photons and carriers are distinct excitations). They measured the photon dispersion and effective mass, found a spectrum close to a room-temperature Boltzmann distribution over a broad pump range, and observed a transition, at phase-space density of order unity, with saturation of excited-state populations and superlinear growth of the lowest-energy occupation, interpreted as photon BEC in equilibrium with a reservoir. Two features distinguish this platform from the dye experiments. First, no parabolic trapping potential is engineered: the device is planar. In an ideally infinite, perfectly homogeneous two-dimensional driven system, thermal long-wavelength phase fluctuations destroy true long-range order, so the expected correlated states are a Berezinskii-Kosterlitz-Thouless (BKT) phase, meaning quasi-long-range order with correlations decaying algebraically, destroyed at a critical point by unbinding of vortex-antivortex pairs, or, when drive and dissipation dominate the phase dynamics, a state with Kardar-Parisi-Zhang (KPZ) correlations, the universality class of the stochastic nonlinear growth equation that governs phase fluctuations of driven-dissipative condensates (Altman, Sieberer, Chen, Diehl and Toner, 2015); in that analysis the long-wavelength phase dynamics of an isotropic driven condensate map onto the KPZ equation, so asymptotic algebraic (BKT-type) order requires anisotropy, and algebraic correlations in an isotropic device appear only as an intermediate-scale crossover, with true long-distance correlations decaying exponentially. In the actual device, finite-size effects are known to be negligible in the strongly nonlinear regime, but residual deviations from perfect homogeneity remain relevant. Second, the reservoir is not a finite set of discrete dye colour centres but electron and hole populations in variable numbers, so the thermalization theories available at the time did not directly apply. The paper therefore concluded that the role of finite boundary conditions in the planar geometry still required careful theoretical treatment; that spatially resolved amplitude, phase and intensity correlation measurements should be performed to confirm the grand-canonical conditions of the experiment and to discriminate among possible nonequilibrium photonic phases; and that a theory of thermalization and condensation in such a variable-number carrier reservoir was still needed. The distinction from ordinary lasing matters throughout: a laser also macroscopically occupies a single mode, but through stimulated emission alone, without the photon gas passing through a thermalized distribution on the way; the correlation measurements requested in the source are precisely what would separate a thermalized condensate from an out-of-equilibrium laser in this device.

Problem Statement

For the dense photon gas of an electrically pumped broad-area semiconductor microresonator (VCSEL-type, weak light-matter coupling, planar geometry without engineered confinement, with residual spatial inhomogeneity and finite boundaries), determine which correlated phase the strongly driven condensed regime realizes. The target is this platform class, with the device of Barland et al. (reference 1) as its concrete realization: the quantitative parameters reported there (device diameter, photon effective mass, fitted reservoir temperature, photon lifetime, and phase-space density at threshold) anchor the descriptors "dense" and "strongly driven" and the drive and dissipation ranges over which the classification is sought.

(1) Theoretically: construct the treatment of this planar, finite, weakly inhomogeneous, driven-dissipative photon gas, building on the now-available thermalization theories for its variable-number electron-hole reservoir, and determine whether the observed condensation transition corresponds to Bose-Einstein condensation with trapped-like macroscopic ground-state occupation, Berezinskii-Kosterlitz-Thouless quasi-long-range order, or Kardar-Parisi-Zhang scaling of the first-order correlation function, as a function of drive and dissipation strength and of the degree of residual inhomogeneity. Because macroscopic ground-state occupation is a one-point occupation diagnostic while BKT and KPZ are two-point spatial-correlation classes, the classification must state, where macroscopic occupation coexists with algebraic or KPZ-type correlations in the finite device, which asymptotic correlation class applies and under which crossover conditions. The treatment must include the role of the finite boundary conditions of the planar geometry, which the source identified as requiring careful analysis, and must identify, consistently with the negligibility of finite-size effects in the strongly nonlinear regime, the conditions under which boundaries and residual inhomogeneity affect the classification.

(2) Experimentally: the source proposes spatially resolved amplitude, phase and intensity correlation measurements (for example g(1)(r)g^{(1)}(r) and g(2)(r)g^{(2)}(r) below, at and above the condensation threshold) on the electrically driven semiconductor platform as the route to confirm or refute grand-canonical ensemble conditions of the photon gas with respect to its carrier reservoir and to discriminate among Bose-Einstein-condensed, BKT and KPZ spatial correlation behavior (algebraic versus exponential spatial decay, coherence exponents, vortex content, KPZ scaling).

A complete answer fixes the universality class of the condensation transition first reported by Barland et al. and states the spatial correlation structure of the condensed phase together with the boundary, drive, dissipation and inhomogeneity conditions under which that classification holds, and states whether the photon gas obeys grand-canonical ensemble conditions with respect to its variable-number carrier reservoir; this outcome, rather than the performance of any particular measurement or calculation, defines completion, and it may be established theoretically, experimentally, or in combination.

References

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