Coherence is a defining feature of quantum condensates. These condensates are inherently multimode phenomena and in the macroscopic limit it becomes extremely difficult to resolve populations of individual modes and the coherence between them. In this work we demonstrate non-equilibrium Bose-Einstein condensation (BEC) of photons in a sculpted dye-filled microcavity, where threshold is found for 8 ± 2 photons. With this nanocondensate we are able to measure occupancies and coherences of individual energy levels of the bosonic field. Coherence of individual modes generally increases with increasing photon number, but at the breakdown of thermal equilibrium we observe multimode-condensation phase transitions wherein coherence unexpectedly decreases with increasing population, suggesting that the photons show strong inter-mode phase or number correlations despite the absence of a direct nonlinearity. Experiments are well-matched to a detailed non-equilibrium model. We find that microlaser and Bose-Einstein statistics each describe complementary parts of our data and are limits of our model in appropriate regimes, which informs the debate on the differences between the two [1, 2].

We report on the observation of quantum coherence of Bose-Einstein condensed photons in an optically pumped, dye-filled microcavity. We find that coherence is long-range in space and time above condensation threshold, but short-range below threshold, compatible with thermalequilibrium theory. Far above threshold, the condensate is no longer at thermal equilibrium and is fragmented over non-degenerate, spatially overlapping modes. A microscopic theory including cavity loss, molecular structure and relaxation shows that this multimode condensation is similar to multimode lasing induced by imperfect gain clamping. BEC means that interferometry is one of the most urgent measurements to be made with a condensate [4,5]. Where thermal equilibrium is not completely reached, coherence is the defining characteristic of non-BEC quantum condensation, e.g for semiconductor exciton-polaritons [6-9] and organic polaritons [10,11]. In nonideal Bose gases, such as ultracold atoms, interactions tend to reduce but not destroy the coherence [12][13][14].The long range coherence behaviour of two-dimensional (2D) microcavity condensates is currently an open question. The coherence of interacting, equilibrium 2D Bose gases decays with a power law at large distances. The exponent is no greater than 1/4, which is reached at the threshold for the Berezinskii-Kosterlitz-Thouless transition [15]. It is known that the equation of motion for the local phase of a non-equilibrium drivendissipative 2D condensate is in the universality class of the Kardar-Parisi-Zhang (KPZ) equation [16], and nonpower-law decays are possible. Since the long-range coherence only shows non-equilibrium behaviour for systems which are very large compared to interaction length scales (such as the healing length), it has proven difficult to observe the true long-range behaviour, mainly due to unavoidable pumping inhomogeneities [17].Photon condensates in dye-filled microcavities are weakly interacting [18][19][20][21], inhomogeneous [22, 23], dissipative Bose gases close to thermal equilibrium at room temperature [24][25][26][27][28]. It is worth noting that the physical system has some similarities to a dye laser, with the decisive difference being that lasing is necessarily a non-equilibrium effect whereas photons can also undergo BEC in thermal equilibrium. Consequently BEC implies macroscopic occupation of the ground state independently of the loss and gain properties, whereas a laser is characterised by a large occupation of exactly the mode that has the most gain [29].Unique among physical realisations of BEC, in dye-microcavity photon BEC the particles thermalise only with a bath and not directly among themselves. This implies that the establishment of phase coherence in the OPEN ACCESS RECEIVED

We investigate the steady state of a system of photons in a pumped dye-filled microcavity. By varying pump and thermalization the system can be tuned between Bose-Einstein condensation, multimode condensation, and lasing. We present a rich nonequilibrium phase diagram which exhibits transitions between these phases, including decondensation of individual modes under conditions that would typically favor condensation.

Photonic condensates are complex systems exhibiting phase transitions due to the interaction with their molecular environment. Given the macroscopic number of molecules that form a reservoir of excitations, numerical simulations are expensive, even for systems with few cavity modes. We present a systematic construction of molecular excitation profiles with a clear hierarchical ordering, such that only modes in the lowest order in the hierarchy directly affect the cavity photon dynamics. In addition to a substantial gain in computational efficiency for simulations of photon dynamics, the explicit spatial shape of the mode profiles offers a clear physical insight into the competition among the cavity modes for access to molecular excitations.

We investigate the response of a photonic gas interacting with a reservoir of pumped dye molecules to quenches in the pump power. In addition to the expected dramatic critical slowing down of the equilibration time around phase transitions, we find extremely slow equilibration even far away from phase transitions. This noncritical slowing down can be accounted for quantitatively by fierce competition among cavity modes for access to the molecular environment, and we provide a quantitative explanation for this noncritical slowing down.