Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1604.08747 · Nature Communications

Calorimetry of a Bose–Einstein-condensed photon gas

Tobias Damm, Julian Schmitt, Qi Liang, David Dung, Frank Vewinger, Martin Weitz, Jan Klaers

Published Apr 19, 2016• 38 citations
arXivHTMLPDF

Abstract

Phase transitions, as the condensation of a gas to a liquid, are often revealed by a discontinuous behaviour of thermodynamic quantities. For liquid helium, for example, a divergence of the specific heat signals the transition from the normal fluid to the superfluid state. Apart from liquid helium, determining the specific heat of a Bose gas has proven to be a challenging task, for example, for ultracold atomic Bose gases. Here we examine the thermodynamic behaviour of a trapped two-dimensional photon gas, a system that allows us to spectroscopically determine the specific heat and the entropy of a nearly ideal Bose gas from the classical high temperature to the Bose-condensed quantum regime. The critical behaviour at the phase transition is clearly revealed by a cusp singularity of the specific heat. Regarded as a test of quantum statistical mechanics, our results demonstrate a quantitative agreement with its predictions at the microscopic level.

DOI
10.1038/ncomms11340

Related papers

Observation of Critical Scaling in a Nearly Ideal Bose Gas

Leon Espert Miranda

The study of second-order phase transitions is a broad research theme that provides fundamental insights into the physical nature of complex systems. A central aspect deals with the concept of universal behavior. Universality governs, for example, the diverging correlated fluctuations of an order parameter, characterized by a macroscopic correlation length ξ, as well as the divergence of other thermodynamic quantities near the critical point. Extracting the critical exponents associated with these diverging quantities shows that quite different physical systems can exhibit identical scaling behavior, placing them into common universality classes. Bose-Einstein condensation of an ideal (i.e., non-interacting) gas of bosons is a second-order phase transition, and thus provides a valuable platform to investigate universal critical behavior in the quantum gas regime, driven purely by quantum statistics rather than inter-particle interactions. While historically studied in ultracold atomic and molecular gases, Bose-Einstein condensation can now be realized in a two-dimensional photonic gas at room temperature using dye-filled microcavities. The microcavity imposes a low-energy cutoff, providing a non-vanishing ground state and an effective mass for the intracavity photons, which thermalize via repeated absorption and emission processes with the dye reservoir rather than inter-particle interactions. Consequently, this platform establishes an almost perfectly ideal, two-dimensional quantum gas of light. In combination with versatile trapping techniques, it provides a highly tunable platform to explore the thermodynamics of quantum gases. This thesis presents a quantitative study of spatial correlations in a uniformly trapped two-dimensional photon gas. The uniform confinement is achieved by a custom mirror-structuring technique, resulting in box-like trapping potentials. The temporal and spatial coherence properties of the gas are validated in both the thermal and condensed regimes using an interferometric measurement technique. Because the applicability of scanning interferometry limits measurements of critical scaling behavior, an alternative method is introduced to extract the first-order spatial coherence directly from the far-field angular distribution of the cavity emission in a single experimental shot. This technique enables the extraction of the spatial correlation length ξ and a precise evaluation of its scaling behavior in the vicinity of the critical point Nc. The measurements presented here demonstrate, for the first time, the determination of a critical exponent in the (ideal) Bose gas universality class. The obtained critical exponent of ν = 0.52 ± 0.04 for the two-dimensional ideal Bose gas is consistent with theoretical expectations. Crucially, this critical exponent is found to be robust and independent of the system size within experimental uncertainties, demonstrating the universality of the scaling behavior. Finally, the limits of this universal behavior are explored by demonstrating the breakdown of critical scaling when the thermalizing coupling between the photons and dye molecules in the cavity is reduced, driving the photon gas out of thermal equilibrium.

•Aug 12, 2026•0 citations

Observation of critical scaling in the Bose gas universality class

Leon Kleebank, Frank Vewinger, Arturo Camacho-Guardian +4 more

Critical exponents characterize the divergent scaling of thermodynamic quantities near phase transitions and allow for the classification of physical systems into universality classes. While quantum gases thermalizing by interparticle interactions fall into the XY model universality class, the ideal Bose gas has been predicted to form a distinct universality class whose signatures have not yet been revealed experimentally. Here, we report the observation of critical scaling in a two-dimensional trapped quantum gas of essentially noninteracting photons, which thermalize by radiative contact to a reservoir of molecules inside a microcavity. By measuring the spatial correlations near the condensation transition, we determine the critical exponent for the correlation length to be $ν= 0.52(4)$. Our results constitute a first experimental test of the long-standing scaling predictions for the Bose gas universality class.

cond-mat.quant-gasCross-listedarXiv:2511.14754•Science Advances•Nov 19, 2025•0 citations

Bose-Einstein condensation of photons in an ideal atomic gas

Alexander Kruchkov, Yu. V. Slyusarenko

We study peculiarities of Bose-Einstein condensation of photons that are in thermodynamic equilibrium with atoms of noninteracting gases. General equations of the thermodynamic equilibrium of the system under study are obtained. We examine solutions of these equations in the case of high temperatures, when the atomic components of the system can be considered as nondegenerated ideal gases of atoms, and the photonic component can form a state with the Bose condensate. Transcendental equation for transition temperature and expression for the density of condensed photons in the considered system are derived. We also obtain analytical solutions of the equation for the critical temperature in a number of particular cases. The existence of two regimes of Bose condensation of photons, which differ significantly in nature of transition temperature dependence on the total density of photons pumped into the system, is revealed. In one case, this dependence is a traditional fractional-power law, and in another one it is the logarithmic law. Applying numerical methods, we determine boundaries of existence and implementation conditions for different regimes of condensation depending on the physical parameters of the system under study. We also show that for a large range of physical systems that are in equilibrium with photons (from ultracold gases of alkali metals to certain types of ideal plasma), the condensation of photons should occur according to the logarithmic regime.

arXiv:1305.1210•Physical Review A•Jul 9, 2013•23 citations

Energy fluctuation and discontinuity of specific heat

Shyamal Biswas, Joydip Mitra, Saugata Bhattacharyya

The specific heat per particle ( c v ) of an ideal gas, on many occasions, is interpreted as energy fluctuation per particle (▵ 2 ) of the ideal gas through the relation ▵ 2 = kT 2 c v , where k is the Boltzmann constant and T is the temperature. This relationship is true only in the classical limit and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3D free Bose and Fermi gases. We have explored the same for harmonically trapped cases. We have obtained a hump of around its condensation point for 3D harmonically trapped Bose gas. We have discussed the possibility of an occurring phase transition with the discontinuity of heat capacity from the existence of such a hump for other Bose and Fermi systems.

arXiv:1502.05542•Journal of Statistical Mechanics Theory and Experiment•Mar 11, 2015•4 citations

On Some Properties of Bose-Einstein Gases

Louis Goldstein

The condition which has to be fulfilled in order that the Bose-Einstein condensation be accompanied by a discontinuity of the heat capacity at the condensation temperature is discussed. The nature of the transition depends on the distribution of the energy levels of the individual particles forming the gas. The condensation is studied for several types of individual energy spectra. It is shown that all those Bose-Einstein (B.E.) assemblies which admit a non-ambiguous definition of the statistical distribution function present, beside interesting analogies, a qualitatively different behavior from that observed in liquid helium. It appears that as long as the transition in momentum space of non-ideal B.E. systems is not understood, the possibility of connection between B.E. condensation and the transition observed in liquid helium remains an open problem.

The Journal of Chemical Physics•Jun 1, 1941•4 citations

The equation of state of ultracold Bose and Fermi gases: a few examples

Sylvain Nascimbène, Nir Navon, Frédéric Chevy +1 more

We describe a powerful method for determining the equation of state of an ultracold gas from in situ images. The method provides a measurement of the local pressure of a harmonically trapped gas and we give several applications to Bose and Fermi gases. We obtain the grand-canonical equation of state of a spin-balanced Fermi gas with resonant interactions as a function of temperature (Nascimbène et al 2010 Nature 463 1057). We compare our equation of state with an equation of state measured by the Tokyo group (Horikoshi et al 2010 Science 327 442), which reveals a significant difference in the high-temperature regime. The normal phase, at low temperature, is well described by a Landau Fermi liquid model, and we observe a clear thermodynamic signature of the superfluid transition. In a second part, we apply the same procedure to Bose gases. From a single image of a quasi-ideal Bose gas, we determine the equation of state from the classical to the condensed regime. Finally, the method is applied to a Bose gas in a three-dimensional optical lattice in the Mott insulator regime. Our equation of state directly reveals the Mott insulator behavior and is suited to investigate finite-temperature effects.

arXiv:1006.4052•New Journal of Physics•Oct 15, 2010•49 citations
Phase transitions, as the condensation of a gas to a liquid, are often revealed by a discontinuous behaviour of thermodynamic quantities. For liquid helium, for example, a divergence of the specific heat signals the transition from the normal fluid to the superfluid state. Apart from liquid helium, determining the specific heat of a Bose gas has proven to be a challenging task, for example, for ultracold atomic Bose gases. Here we examine the thermodynamic behaviour of a trapped two-dimensional photon gas, a system that allows us to spectroscopically determine the specific heat and the entropy of a nearly ideal Bose gas from the classical high temperature to the Bose-condensed quantum regime. The critical behaviour at the phase transition is clearly revealed by a cusp singularity of the specific heat. Regarded as a test of quantum statistical mechanics, our results demonstrate a quantitative agreement with its predictions at the microscopic level.
The study of second-order phase transitions is a broad research theme that provides fundamental insights into the physical nature of complex systems. A central aspect deals with the concept of universal behavior. Universality governs, for example, the diverging correlated fluctuations of an order parameter, characterized by a macroscopic correlation length ξ, as well as the divergence of other thermodynamic quantities near the critical point. Extracting the critical exponents associated with these diverging quantities shows that quite different physical systems can exhibit identical scaling behavior, placing them into common universality classes. Bose-Einstein condensation of an ideal (i.e., non-interacting) gas of bosons is a second-order phase transition, and thus provides a valuable platform to investigate universal critical behavior in the quantum gas regime, driven purely by quantum statistics rather than inter-particle interactions. While historically studied in ultracold atomic and molecular gases, Bose-Einstein condensation can now be realized in a two-dimensional photonic gas at room temperature using dye-filled microcavities. The microcavity imposes a low-energy cutoff, providing a non-vanishing ground state and an effective mass for the intracavity photons, which thermalize via repeated absorption and emission processes with the dye reservoir rather than inter-particle interactions. Consequently, this platform establishes an almost perfectly ideal, two-dimensional quantum gas of light. In combination with versatile trapping techniques, it provides a highly tunable platform to explore the thermodynamics of quantum gases. This thesis presents a quantitative study of spatial correlations in a uniformly trapped two-dimensional photon gas. The uniform confinement is achieved by a custom mirror-structuring technique, resulting in box-like trapping potentials. The temporal and spatial coherence properties of the gas are validated in both the thermal and condensed regimes using an interferometric measurement technique. Because the applicability of scanning interferometry limits measurements of critical scaling behavior, an alternative method is introduced to extract the first-order spatial coherence directly from the far-field angular distribution of the cavity emission in a single experimental shot. This technique enables the extraction of the spatial correlation length ξ and a precise evaluation of its scaling behavior in the vicinity of the critical point Nc. The measurements presented here demonstrate, for the first time, the determination of a critical exponent in the (ideal) Bose gas universality class. The obtained critical exponent of ν = 0.52 ± 0.04 for the two-dimensional ideal Bose gas is consistent with theoretical expectations. Crucially, this critical exponent is found to be robust and independent of the system size within experimental uncertainties, demonstrating the universality of the scaling behavior. Finally, the limits of this universal behavior are explored by demonstrating the breakdown of critical scaling when the thermalizing coupling between the photons and dye molecules in the cavity is reduced, driving the photon gas out of thermal equilibrium.
Critical exponents characterize the divergent scaling of thermodynamic quantities near phase transitions and allow for the classification of physical systems into universality classes. While quantum gases thermalizing by interparticle interactions fall into the XY model universality class, the ideal Bose gas has been predicted to form a distinct universality class whose signatures have not yet been revealed experimentally. Here, we report the observation of critical scaling in a two-dimensional trapped quantum gas of essentially noninteracting photons, which thermalize by radiative contact to a reservoir of molecules inside a microcavity. By measuring the spatial correlations near the condensation transition, we determine the critical exponent for the correlation length to be ν=0.52(4)ν= 0.52(4)ν=0.52(4). Our results constitute a first experimental test of the long-standing scaling predictions for the Bose gas universality class.
We study peculiarities of Bose-Einstein condensation of photons that are in thermodynamic equilibrium with atoms of noninteracting gases. General equations of the thermodynamic equilibrium of the system under study are obtained. We examine solutions of these equations in the case of high temperatures, when the atomic components of the system can be considered as nondegenerated ideal gases of atoms, and the photonic component can form a state with the Bose condensate. Transcendental equation for transition temperature and expression for the density of condensed photons in the considered system are derived. We also obtain analytical solutions of the equation for the critical temperature in a number of particular cases. The existence of two regimes of Bose condensation of photons, which differ significantly in nature of transition temperature dependence on the total density of photons pumped into the system, is revealed. In one case, this dependence is a traditional fractional-power law, and in another one it is the logarithmic law. Applying numerical methods, we determine boundaries of existence and implementation conditions for different regimes of condensation depending on the physical parameters of the system under study. We also show that for a large range of physical systems that are in equilibrium with photons (from ultracold gases of alkali metals to certain types of ideal plasma), the condensation of photons should occur according to the logarithmic regime.
The specific heat per particle ( c v ) of an ideal gas, on many occasions, is interpreted as energy fluctuation per particle (▵ 2 ) of the ideal gas through the relation ▵ 2 = kT 2 c v , where k is the Boltzmann constant and T is the temperature. This relationship is true only in the classical limit and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3D free Bose and Fermi gases. We have explored the same for harmonically trapped cases. We have obtained a hump of around its condensation point for 3D harmonically trapped Bose gas. We have discussed the possibility of an occurring phase transition with the discontinuity of heat capacity from the existence of such a hump for other Bose and Fermi systems.
The condition which has to be fulfilled in order that the Bose-Einstein condensation be accompanied by a discontinuity of the heat capacity at the condensation temperature is discussed. The nature of the transition depends on the distribution of the energy levels of the individual particles forming the gas. The condensation is studied for several types of individual energy spectra. It is shown that all those Bose-Einstein (B.E.) assemblies which admit a non-ambiguous definition of the statistical distribution function present, beside interesting analogies, a qualitatively different behavior from that observed in liquid helium. It appears that as long as the transition in momentum space of non-ideal B.E. systems is not understood, the possibility of connection between B.E. condensation and the transition observed in liquid helium remains an open problem.
We describe a powerful method for determining the equation of state of an ultracold gas from in situ images. The method provides a measurement of the local pressure of a harmonically trapped gas and we give several applications to Bose and Fermi gases. We obtain the grand-canonical equation of state of a spin-balanced Fermi gas with resonant interactions as a function of temperature (Nascimbène et al 2010 Nature 463 1057). We compare our equation of state with an equation of state measured by the Tokyo group (Horikoshi et al 2010 Science 327 442), which reveals a significant difference in the high-temperature regime. The normal phase, at low temperature, is well described by a Landau Fermi liquid model, and we observe a clear thermodynamic signature of the superfluid transition. In a second part, we apply the same procedure to Bose gases. From a single image of a quasi-ideal Bose gas, we determine the equation of state from the classical to the condensed regime. Finally, the method is applied to a Bose gas in a three-dimensional optical lattice in the Mott insulator regime. Our equation of state directly reveals the Mott insulator behavior and is suited to investigate finite-temperature effects.