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arXiv:2609.05287

Fluctuations of a Photon Bose-Einstein Condensate Coupled to a Reservoir: Describing Coherence Properties in a Free-Energy Model

Martin Weitz, Andreas Redmann, Riccardo Panico, Leon Kleeblank, Kevin J. H. Peters, Frank Vewinger, Julian Schmitt

Published Sep 4, 2026• 0 citations
arXivHTMLPDF

Abstract

Photons are mutually nearly noninteracting particles, so thermalized photon ensembles are commonly obtained not from direct particle-particle-interactions but rather from contact with matter, which can constitute a reservoir for the photon gas. We develop a theory model for photons in a material-filled (e.g. liquid dye) optical microcavity, with the aim to study the fluctuation properties using a free-energy description for noninteracting photons coupled to a reservoir of material electronic excitations. To begin with, we use a single mode description for the condensate. For a small relative size of the material reservoir, corresponding to the canonical regime, condensate number fluctuations are small, and the derived free energy landscape takes the usual Mexican-hat shaped form such that spontaneous symmetry breaking occurs. In contrast, for a large relative size of the reservoir, corresponding to the grand canonical regime, fluctuations become as large as the average particle number. We show that the resulting free energy landscape acquires a bowl-shaped form, with a single minimum at the origin. Thus, a macroscopic occupation of the ground state (i.e., Bose-Einstein condensation) in the absence of spontaneous symmetry breaking is expected. We also provide a model for the treatment of a photon gas trapped in a box-shaped potential with spatially distributed coupling to a reservoir. The model predicts, for example, a statistically fluctuating pattern of islands with long-range coherence, resembling transient microcondensates.

DOI
10.48550/arxiv.2609.05287

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Photons are mutually nearly noninteracting particles, so thermalized photon ensembles are commonly obtained not from direct particle-particle-interactions but rather from contact with matter, which can constitute a reservoir for the photon gas. We develop a theory model for photons in a material-filled (e.g. liquid dye) optical microcavity, with the aim to study the fluctuation properties using a free-energy description for noninteracting photons coupled to a reservoir of material electronic excitations. To begin with, we use a single mode description for the condensate. For a small relative size of the material reservoir, corresponding to the canonical regime, condensate number fluctuations are small, and the derived free energy landscape takes the usual Mexican-hat shaped form such that spontaneous symmetry breaking occurs. In contrast, for a large relative size of the reservoir, corresponding to the grand canonical regime, fluctuations become as large as the average particle number. We show that the resulting free energy landscape acquires a bowl-shaped form, with a single minimum at the origin. Thus, a macroscopic occupation of the ground state (i.e., Bose-Einstein condensation) in the absence of spontaneous symmetry breaking is expected. We also provide a model for the treatment of a photon gas trapped in a box-shaped potential with spatially distributed coupling to a reservoir. The model predicts, for example, a statistically fluctuating pattern of islands with long-range coherence, resembling transient microcondensates.
We review recent experiments on the Bose-Einstein condensation of photons in a dye-filled optical microresonator. The most well-known example of a photon gas, photons in blackbody radiation, does not show Bose-Einstein condensation. Instead of massively populating the cavity ground mode, photons vanish in the cavity walls when they are cooled down. The situation is different in an ultrashort optical cavity imprinting a low-frequency cutoff on the photon energy spectrum that is well above the thermal energy. The latter allows for a thermalization process in which both temperature and photon number can be tuned independently of each other or, correspondingly, for a nonvanishing photon chemical potential. We here describe experiments demonstrating the fluorescence-induced thermalization and Bose-Einstein condensation of a two-dimensional photon gas in the dye microcavity. Moreover, recent measurements on the photon statistics of the condensate, showing Bose-Einstein condensation in the grand-canonical ensemble limit, will be reviewed. Introduction Quantum statistical effects become relevant when a gas of particles is cooled, or its density is increased, to the point where the associated de Broglie wavepackets spatially overlap. For particles with integer spin (bosons), the phenomenon of Bose-Einstein condensation (BEC) then leads to macroscopic occupation of a single quantum state at finite temperatures [1]. Bose-Einstein condensation in the gaseous case was first achieved in 1995 by laser and subsequent evaporative cooling of a dilute cloud of alkali atoms [2, 3, 4], as detailed in Chapter 3 of this volume. The condensate atoms can be described by a macroscopic singleparticle wavefunction, similar to the case of liquid helium [1]. Bose-Einstein condensation has also been observed for exciton-polaritons, which are hybrid states of matter and light [5, 6, 7] (see Chapter 4), magnons [8] (see Chapter 25), and other physical systems. Other than material particles, photons usually do not show Bose-Einstein condensation [9]. In blackbody radiation, the most common Bose gas, photons at low temperature disappear, instead of condensing to a macroscopically occupied ground-state mode. In this system, photons have a vanishing chemical potential, meaning that the number of photons is determined by the available thermal energy and cannot be tuned independently from temperature.
We examine the phase evolution of a Bose-Einstein condensate of photons generated in a dye microcavity by temporal interference with a phase reference. The photoexcitable dye molecules constitute a reservoir of variable size for the condensate particles, allowing for grand canonical statistics with photon bunching, as in a lamp-type source. We directly observe phase jumps of the condensate associated with the large statistical number fluctuations and find a separation of correlation time scales. For large systems, our data reveal phase coherence and a spontaneously broken symmetry, despite the statistical fluctuations.
We report measurements of particle number correlations and fluctuations of a photon Bose-Einstein condensate in a dye microcavity using a Hanbury Brown-Twiss experiment. The photon gas is coupled to a reservoir of molecular excitations, which serve as both heat bath and particle reservoir to realize grand-canonical conditions. For large reservoirs, we observe strong number fluctuations of the order of the total particle number extending deep into the condensed phase. Our results demonstrate that Bose-Einstein condensation under grand-canonical ensemble conditions does not imply second-order coherence.
Bose-Einstein condensation is a state of matter for which massive bosons develop a macroscopic occupation of the ground state. After the theoretical discovery by Bose and Einstein, it took approximately 70 years to be experimentally realized using Rubidium-87 atoms. Ever since, many different species have been shown to Bose condense. The key in these experiments is cooling down atoms temperatures below 200 nK, while maintaining a large density. Despite all these species, it was thought that photons could not Bose condense. This changed in 2010 when the first Bose-Einstein condensate of photons (phBEC) was achieved. In a table top experiment photons are confined to a dye-filled microcavity where under the right conditions a thermalized photon gas is achieved. Using our setup, we quantitatively show from both the spectral and spatial distribution of photons in the microcavity that a macroscopic occupation of the ground state is achieved, i.e. a condensate of photons. The achievement of phBEC in a dye-filled microcavity has led to a renewed interest in the density distribution of the ideal Bose gas in a two-dimensional harmonic oscillator. We present measurements of the radial profile of photons inside the microcavity below and above the critical point for phBEC with a large signal-to-noise ratio. We obtain good agreement with theoretical profiles obtained using exact summation of eigenstates. An interesting open question is, whether the photons in the condensate are interacting or non-interacting and as a consequence, whether the condensate is also a superfluid or only forms a macroscopically occupied ground state, as predicted by Einstein. When increasing the photon density, we observe an increase in the radius of the condensate which we attribute to effective repulsive interactions. For several dye concentrations we accurately determine the radius of the condensate as a function of the number of condensate photons, and derive from the measurements an effective interaction strength. In many systems ranging from condensed matter physics and particle physics to cosmology, phase transitions and spontaneous symmetry breaking play a crucial role. The information of a polarized condensate from an unpolarized thermal cloud constitutes a directly observable prototype of spontaneous symmetry breaking. For every single-shot of the pump laser, we determine the Stokes parameters of the photons inside the microcavity which fully characterizes the polarization state of the sample. By varying experimental parameters, we investigate if the polarization is randomly chosen from shot-to-shot of the experiment. The dye-filled microcavity is a beautiful tool to achieve phBEC. However, it cannot provide the periodic potential required to achieve quantum phase transitions, such as the Mott insulator that have been so successfully exploited in the field of ultra-cold atoms. We use a local density approach to simulate large-area photonic crystal cavities. Clever interplay with both the electronic and photonic bandgap of these cavities has the potential of achieving phBEC in a periodic potential. We show that when the hole radius varies quadratically as a function of position, the eigenmodes of the photonic crystal can be described by the corresponding eigenmodes of the quantum harmonic oscillator.
We theoretically analyze the temperature behavior of paraxial light in thermal equilibrium with a dye-filled optical microcavity. At low temperatures the photon gas undergoes Bose-Einstein condensation, and the photon number in the cavity ground state becomes macroscopic with respect to the total photon number. Owing to a grand-canonical excitation exchange between the photon gas and the dye molecule reservoir, a regime with unusually large fluctuations of the condensate number is predicted for this system that is not observed in present atomic physics Bose-Einstein condensation experiments.
Bose-Einstein condensation in the grand canonical ensemble admits a formulation in terms of a phase-density decomposition of the condensate mode operator ψ^0\hatψ_{\bf 0}ψ^​0​. In the presence of macroscopic condensate number fluctuations this representation presents nontrivial implications. In particular, we show that, for the ideal gas, under the assumption of a well-defined phase and a fluctuating condensate density, the square modulus of the anomalous average ⟨ψ^0⟩\langle \hatψ_{\bf 0}\rangle⟨ψ^​0​⟩ can provide only a fraction of the whole condensate density ρ0ρ_{\bf 0}ρ0​ and for the grand canonical statistics of the ideal Bose gas one obtains the value ∣⟨ψ^0⟩∣2=(π/4)ρ0|\langle {\hat ψ}_{\bf 0}\rangle|^2 =(π/4) ρ_{\bf 0}∣⟨ψ^​0​⟩∣2=(π/4)ρ0​. The remaining part is supplemented by the (macroscopic) fluctuations of ψ^0\hatψ_{\bf 0}ψ^​0​, which become a distinctive feature of the BEC in this setting. This provides a transparent physical picture of a condensate of photons with a well-defined phase but large number fluctuations, as observed in dye-filled microcavity photon experiments. We also propose a way to access the square modulus of the anomalous average to test theoretical predictions.