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 systems, 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 varied; 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 electrons or photon-photon scattering in a nonlinear resonator configuration. Number-conserving thermalization was experimentally observed 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 gases and new coherent ultraviolet sources.

, the macroscopic accumulation of bosonic particles in the energetic ground state below a critical temperature, has been demonstrated in several physical systems [2][3][4][5][6][7][8] . The perhaps best known example of a bosonic gas, blackbody radiation 9 , however exhibits no Bose-Einstein condensation at low temperatures 10 . Instead of collectively occupying the lowest energy mode, the photons disappear in the cavity walls when the temperature is loweredcorresponding to a vanishing chemical potential. Here we report on evidence for a thermalized two-dimensional photon gas with a freely adjustable chemical potential. Our experiment is based on a dye-filled optical microresonator, 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 provide both an effective photon mass and a confining potential-key prerequisites for the Bose-Einstein condensation of photons. As a striking example of the unusual system properties, we demonstrate a yet unobserved light concentration effect into the centre of the confining potential, an effect with prospects for increasing the efficiency of diffuse solar light collection 11. Following the achievement of atomic Bose-Einstein condensation 2-4 , we have witnessed interest in light sources where a macroscopically populated photon mode is not the consequence of laser-like amplification, but rather results from a thermal equilibrium phase transition. Work in this direction includes the proposal of a superfluid phase transition of photons in a nonlinear cavity [12][13][14] and, albeit in the strong-coupling regime, the demonstration of a quasiequilibrium phase transition of excitonpolariton quasiparticles to 'half matter, half light' condensates 6-8 . In the weak-coupling regime (as in our case), optical cavities have been used to achieve a modified spontaneous emission of atoms and molecules [15][16][17] . The main idea of our experiment is to study thermalization of a photon gas, to a heat bath near room temperature (dye molecules), in a system with reduced spatial dimensionality and an energy spectrum restricted to values far above the thermal energy. The photons are trapped in a curved-mirror optical microcavity, and repeatedly scatter off dye molecules. The longitudinal confinement (along the cavity axis) introduces a large frequency spacing between adjacent longitudinal modes and modifies spontaneous emission coupling such that basically only photons of longitudinal mode number q = 7 (Fig. 1a) are observed to populate the cavity. By this, an effective low frequency cutoff ω cutoff (the eigenfrequency for the corresponding TEM 00 transverse mode) is introduced, withhω cutoff ∼ 2.1 eV, much larger than the thermal energy k B T (∼1/40 eV at room temperature). The two remaining transverse modal degrees of freedom of light thermalize to the (internal rovibrational) temperature of the dye solution, and the photon frequencies will be distributed by an amount ∼k B T/h abov...

The efficient conversion of thermal energy to mechanical work by a heat engine is an ongoing technological challenge. Since the pioneering work of Carnot, it is known that the efficiency of heat engines is bounded by a fundamental upper limit -the Carnot limit. Theoretical studies suggest that heat engines may be operated beyond the Carnot limit by exploiting stationary, nonequilibrium reservoirs that are characterized by a temperature as well as further parameters. In a proof-of-principle experiment, we demonstrate that the efficiency of a nano-beam heat engine coupled to squeezed thermal noise is not bounded by the standard Carnot limit. Remarkably, we also show that it is possible to design a cyclic process that allows for extraction of mechanical work from a single squeezed thermal reservoir. Our results demonstrate a qualitatively new regime of non-equilibrium thermodynamics at small scales and provide a new perspective on the design of efficient, highly miniaturized engines.

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. Contents