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Superfluidity in its various forms has fascinated scientists since the observation of frictionless flow in liquid helium II. In three spatial dimensions (3D), it is conceptually associated with the emergence of long-range order (LRO) at a critical temperature $T_{\text{c}}$. One of its hallmarks, predicted by the highly successful two-fluid model and observed in both liquid helium and ultracold atomic gases, is the existence of two kinds of sound excitations, the first and second sound. In 2D systems, thermal fluctuations preclude LRO, but superfluidity nevertheless emerges at a nonzero $T_{\text{c}}$ via the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition, which is associated with a universal jump in the superfluid density $n_{\text{s}}$ without any discontinuities in the fluid's thermodynamic properties. BKT superfluids are also predicted to support two sounds, but the observation of this has remained elusive. Here we observe first and second sound in a homogeneous 2D atomic Bose gas, and from the two temperature-dependent sound speeds extract its superfluid density. Our results agree with BKT theory, including the prediction for the universal superfluid-density jump., Comment: 6 pages, 3 figures

Bosonic gases coupled to a particle reservoir have proven to support a regime of operation where Bose-Einstein condensation coexists with unusually large particle-number fluctuations. Experimentally, this situation has been realized with two-dimensional photon gases in a dye-filled optical microcavity. Here, we investigate theoretically and experimentally the open-system dynamics of a grand canonical Bose-Einstein condensate of photons. We identify a regime with temporal oscillations of the second-order coherence function $g^{(2)}(\tau)$, even though the energy spectrum closely matches the predictions for an equilibrium Bose-Einstein distribution and the system is operated deeply in the regime of weak light-matter coupling. The observed temporal oscillations are attributed to the nonlinear, weakly driven-dissipative nature of the system which leads to time-reversal symmetry breaking., Comment: Published version, Physical Review A, Editors' Suggestion

Phase transitions between different states of matter can profoundly modify the order in physical systems, with the emergence of ferromagnetic or topological order constituting important examples. Correlations allow the quantification of the degree of order and the classification of different phases. Here we report measurements of first-order spatial correlations in a harmonically trapped two-dimensional photon gas below, at and above the critical particle number for Bose–Einstein condensation, using interferometric measurements of the emission of a dye-filled optical microcavity. For the uncondensed gas, the transverse coherence decays on a length scale determined by the thermal de Broglie wavelength of the photons, which shows the expected scaling with temperature. At the onset of Bose–Einstein condensation, true long-range order emerges, and we observe quantum statistical effects as the thermal wave packets overlap. The excellent agreement with equilibrium Bose gas theory prompts microcavity photons as promising candidates for studies of critical scaling and universality in optical quantum gases., Phase transitions in quantum matter are related to correlation effects and they can change the ordering of material. Here the authors measure the first-order spatial correlation and the de Broglie wavelength for both thermal and condensed form of a photonic Bose gas in a dye-filled optical microcavity.

Variable micropotentials for light are created by thermo-optic imprinting of a dye–polymer solution within a microcavity. A thermalized photon Bose–Einstein condensate as well as the coupling and eigenstate hybridization of sites are demonstrated. Quantum gases in lattice potentials have been a powerful platform to simulate phenomena from solid-state physics, such as the Mott insulator transition1. In contrast to ultracold atoms, photon-based platforms, such as photonic crystals, coupled waveguides or lasers, usually do not operate in thermal equilibrium2,3,4,5. Advances towards photonic simulators of solid-state equilibrium effects include polariton lattice experiments6,7,8,9,10, and the demonstration of a photon condensate11,12. Here, we demonstrate a technique to create variable micropotentials for light using thermo-optic imprinting of a dye–polymer solution within an ultrahigh-finesse microcavity. We study the properties of single- and double-well potentials, and find the quality of structuring sufficient for thermalization and Bose–Einstein condensation of light. The investigation of effective photon–photon interactions along with the observed tunnel coupling between sites makes the system a promising candidate to directly populate entangled photonic many-body states. The demonstrated scalability suggests that thermo-optic imprinting provides a new approach for variable microstructuring in photonics.

The advent of controlled experimental accessibility of Bose-Einstein condensates, as realized with e.g. cold atomic gases, exciton-polaritons, and more recently photons in a dye-filled optical microcavity, has paved the way for new studies and tests of a plethora of fundamental concepts in quantum physics. We here describe recent experiments studying a transition between laser-like dynamics and Bose-Einstein condensation of photons in the dye microcavity system. Further, measurements of the second-order coherence of the photon condensate are presented. In the condensed state we observe photon number fluctuations of order of the total particle number, as understood from effective particle exchange with the photo-excitable dye molecules. The observed intensity fluctuation properties give evidence for Bose-Einstein condensation occurring in the grand-canonical statistical ensemble regime.

The Tutorial reports recent experimental advances in studies of the dynamics as well as the number and phase correlations of a Bose-Einstein condensed photon gas confined in a high-finesse dye-filled microcavity. Repeated absorption-emission-processes of photons on dye molecules here establish a thermal coupling of the photonic quantum gas to both a heat bath and a particle reservoir comprised of dye molecules. In this way, for the first time Bose-Einstein condensation under grand-canonical statistical ensemble conditions becomes experimentally accessible., Comment: 38 pages, 30 figures, Tutorial article, J. Phys. B: At. Mol. Opt. Phys. 2018

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 photo-excitable 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 timescales. For large systems, our data reveals phase coherence and a spontaneously broken symmetry, despite the statistical fluctuations., 11 pages, 4 figures

We report a time-resolved study of the thermalization dynamics and the lasing to photon Bose-Einstein condensation crossover by in-\textit{situ} monitoring the photon kinetics in a dye microcavity. When the equilibration of the light to the dye temperature by absorption and re-emission is faster than photon loss in the cavity, the optical spectrum becomes Bose-Einstein distributed and photons accumulate at low-energy states, forming a Bose-Einstein condensate. The thermalization of the photon gas and its evolution from nonequilibrium initial distributions to condensation is monitored in real-time. In contrast, if photons leave the cavity before they thermalize, the system operates as a laser., 11 pages, 4 figures

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 both as heat bath and particle reservoir to realize grand-canonical conditions. For large reservoirs, we observe strong number fluctuations of 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., 11 pages, 4 figures

Bose-Einstein condensation has in the last two decades been observed in cold atomic gases and in solid-state physics quasiparticles, exciton-polaritons and magnons, respectively. The perhaps most widely known example of a bosonic gas, photons in blackbody radiation, however exhibits no Bose-Einstein condensation, because the particle number is not conserved and at low temperatures the photons disappear in the system's walls instead of massively occupying the cavity ground mode. This is not the case in a small optical cavity, with a low-frequency cutoff imprinting a spectrum of photon energies restricted to values well above the thermal energy. The here reported experiments are based on a microscopic optical cavity filled with dye solution at room temperature. Recent experiments of our group observing Bose-Einstein condensation of photons in such a setup are described. Moreover, we discuss some possible applications of photon condensates to realize quantum manybody states in periodic photonic lattices and photonic Josephson devices.

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.

Photons, due to the virtually vanishing photon-photon interaction, constitute to very good approximation an ideal Bose gas, but owing to the vanishing chemical potential a (free) photon gas does not show Bose-Einstein condensation. However, this is not necessarily true for a lower-dimensional photon gas. By means of a fluorescence induced thermalization process in an optical microcavity one can achieve a thermal photon gas with freely adjustable chemical potential. Experimentally, we have observed thermalization and subsequently Bose-Einstein condensation of the photon gas at room temperature. In this paper, we give a detailed description of the experiment, which is based on a dye-filled optical microcavity, acting as a white-wall box for photons. Thermalization is achieved in a photon number-conserving way by photon scattering off the dye molecules, and the cavity mirrors both provide an effective photon mass and a confining potential - key prerequisites for the Bose-Einstein condensation of photons. The experimental results are in good agreement with both a statistical and a simple rate equation model, describing the properties of the thermalized photon gas., Experiment Review, 17 pages with 15 figures

Bose–Einstein condensation has been observed in several physical systems, but is not predicted to occur for blackbody radiation such as photons. However, it becomes theoretically possible in the presence of thermalization processes that conserve photon number. Martin Weitz and colleagues have now realized such conditions experimentally, observing Bose–Einstein condensation of photons in a dye-filled optical microcavity. The effect is of interest for fundamental studies and may lead to new coherent ultraviolet sources. Bose–Einstein condensation has been observed in several physical systems, but is not predicted to occur for blackbody radiation such as photons. However, it becomes theoretically possible in the presence of thermalization processes that conserve photon number. These authors experimentally realise such conditions, observing Bose–Einstein condensation of photons in a dye-filled optical microcavity. The effect is of interest for fundamental studies and may lead to new coherent ultraviolet sources. Bose–Einstein condensation (BEC)—the macroscopic ground-state accumulation of particles with integer spin (bosons) at low temperature and high density—has been observed in several physical systems1,2,3,4,5,6,7,8,9, including cold atomic gases and solid-state quasiparticles. However, the most omnipresent Bose gas, blackbody radiation (radiation in thermal equilibrium with the cavity walls) does not show this phase transition. In such systems photons have a vanishing chemical potential, meaning that their number is not conserved when the temperature of the photon gas is varied10; at low temperatures, photons disappear in the cavity walls instead of occupying the cavity ground state. Theoretical works have considered thermalization processes that conserve photon number (a prerequisite for BEC), involving Compton scattering with a gas of thermal electrons11 or photon–photon scattering in a nonlinear resonator configuration12,13. Number-conserving thermalization was experimentally observed14 for a two-dimensional photon gas in a dye-filled optical microcavity, which acts as a ‘white-wall’ box. Here we report the observation of a Bose–Einstein condensate of photons in this system. The cavity mirrors provide both a confining potential and a non-vanishing effective photon mass, making the system formally equivalent to a two-dimensional gas of trapped, massive bosons. The photons thermalize to the temperature of the dye solution (room temperature) by multiple scattering with the dye molecules. Upon increasing the photon density, we observe the following BEC signatures: the photon energies have a Bose–Einstein distribution with a massively populated ground-state mode on top of a broad thermal wing; the phase transition occurs at the expected photon density and exhibits the predicted dependence on cavity geometry; and the ground-state mode emerges even for a spatially displaced pump spot. The prospects of the observed effects include studies of extremely weakly interacting low-dimensional Bose gases9 and new coherent ultraviolet sources15.

We investigate thermodynamic properties of a two-dimensional photon gas confined by a dye-filled optical microcavity. A thermally equilibrated state of the photon gas is achieved by radiative coupling to a heat bath that is realized with dye molecules embedded in a polymer at room temperature. The chemical potential of the gas is freely adjustable. The optical microcavity consisting of two curved mirrors induces both a non-vanishing effective photon mass and a harmonic trapping potential for the photons. While previous experiments of our group have used liquid dye solutions, the measurements described here are based on dye molecules incorporated into a polymer host matrix. We describe studies of fluorescence properties of dye-doped polymers, and discuss the applicability of Kennard-Stepanov theory in this system. We observe a thermalized two-dimensional photon gas in the solid state based microresonator system. In the future, dye-based solid state systems hold promise for the realization of single-mode light sources in thermal equilibrium based on Bose-Einstein condensation of photons, as well as for solar energy concentrators., 19 pages, 6 figures

Bose-Einstein condensation, the macroscopic ground state occupation of a system of bosonic particles below a critical temperature, has been observed in cold atomic gases and solid-state physics quasiparticles. In contrast, photons do not show this phase transition usually, because in Planck's blackbody radiation the particle number is not conserved and at low temperature the photons disappear in the walls of the system. Here we report on the realization of a photon Bose-Einstein condensate in a dye-filled optical microcavity, which acts as a white-wall photon box. The cavity mirrors provide a trapping potential and a non-vanishing effective photon mass, making the system formally equivalent to a two-dimensional gas of trapped massive bosons. Thermalization of the photon gas is reached in a number conserving way by multiple scattering off the dye molecules. Signatures for a BEC upon increased photon density are: a spectral distribution that shows Bose-Einstein distributed photon energies with a macroscopically populated peak on top of a broad thermal wing, the observed threshold of the phase transition showing the predicted absolute value and scaling with resonator geometry, and condensation appearing at the trap centre even for a spatially displaced pump spot.

The versatility of quantum gas experiments greatly benefits from the ability to apply variable potentials. Here we describe a method which allows the preparation of potential structures for microcavity photons via spatially selective deformation of optical resonator geometries with a heat induced mirror surface microstructuring technique. We investigate the thermalization of a two-dimensional photon gas in a dye-filled microcavity composed of the custom surface-structured mirrors at wavelength-scale separation. Specifically, we describe measurements of the spatial redistribution of thermal photons in a coupled double-ridge structure, where photons form a Bose-Einstein condensate in a spatially split ground state, as a function of different pumping geometries.