Bose-Einstein condensates of photons in dye filled microcavities are known to feature many properties of the ideal bose gas at thermal equilibrium. Nevertheless, the finite photon life time renders them driven dissipative systems where losses are balanced by continuous driving by a pump laser. We show here with simulations based on a classical field model that their driven-dissipative nature causes their first order coherence to feature Kardar-Parisi-Zhang (KPZ) scaling, a phenomenon well known in the related condensates of exciton-polaritons. Remarkably, we find that the KPZ scaling even occurs in a regime with large density fluctuations. We discuss the modification of the scaling by the occupation of excited states and how to recover the universal scaling behavior by spectral filtering.
Bose-Einstein condensates of photons in dye filled microcavities are known to feature many properties of the ideal bose gas at thermal equilibrium. Nevertheless, the finite photon life time renders them driven dissipative systems where losses are balanced by continuous driving by a pump laser. We show here with simulations based on a classical field model that their driven-dissipative nature causes their first order coherence to feature Kardar-Parisi-Zhang (KPZ) scaling, a phenomenon well known in the related condensates of exciton-polaritons. Remarkably, we find that the KPZ scaling even occurs in a regime with large density fluctuations. We discuss the modification of the scaling by the occupation of excited states and how to recover the universal scaling behavior by spectral filtering.
We introduce a classical phasor model for the description of multimode photon condensates that thermalize through repeated absorptions and reemissions by dye molecules. Thermal equilibrium is expressed through the fluctuation-dissipation relation that connects the energy damping to spontaneous emission fluctuations. We apply our model to a photonic Josephson junction (two coupled wells) and to one- and two-dimensional arrays of photon condensates. In the limit of zero pumping and cavity losses, we recover the thermal equilibrium result, but in the weakly driven-dissipative case in the canonical regime, we find suppressed density and phase fluctuations with respect to the ideal Bose gas.
Abstract Photon condensation was first experimentally realized in 2010 within a dye-filled microcavity at room temperature. Since then, interest in the field has increased significantly, as a photon Bose-Einstein condensate (BEC) represents a prototypical driven-dissipative system. Here, we investigate how its inherent open nature influences the condensation process both quantitatively and qualitatively. To this end, we consider a mean-field model, which can be derived microscopically from a Lindblad master equation. The underlying rate equations depend on various external parameters such as emission and absorption rates of the dye molecules as well as the cavity photon loss rate. In steady state, we obtain an open-dissipative Bose-Einstein distribution for the mode occupations. The chemical potential of this distribution depends on the occupations of the dye molecules in both their ground and excited state and must therefore be determined self-consistently. We find that the resulting photon distribution is strongly influenced by the driven-dissipative parameters. Based on this result, we identify the main differences between a photonic BEC, an atomic BEC, and a laser.
We review recent results on the coherence and superfluidity of driven dissipative condensates, i.e., systems of weakly interacting nonconserved bosons, such as polariton condensates. The presence of driving and dissipation has dramatically different effects depending on dimensionality and anisotropy. In three dimensions, equilibrium behaviour is recovered at large scales for static correlations, while the dynamical behaviour is altered by the microscopic driving. In two dimensions, for an isotropic system, drive and dissipation destroy the algebraic order that would otherwise exist; however, a sufficiently anisotropic system can still show algebraic phase correlations. We discuss the consequences of this behaviour for recent experiments measuring phase coherence and outline potential measurements that might directly probe superfluidity. Introduction This chapter is dedicated to superfluidity and its relation to Bose-Einstein condensation (BEC), a topic with a long history. Many reviews of the concepts of condensation and superfluidity in thermal equilibrium can be found; see for example Refs. [1, 2, 3, 4]. The focus of this chapter is on how these concepts apply (or fail to apply) to driven dissipative condensates – systems of bosons with a finite lifetime, in which loss is balanced by continuous pumping. We focus entirely on the steady state of such systems, neglecting transient, time-dependent behaviour. Experimentally, the most studied example of a driven dissipative condensate has been microcavity polaritons, an overview of which is given in Ref. [5] and Chapter 4. However, similar issues can arise in many other systems, most obviously photon condensates [6] (see also Chapter 19), magnon condensates [7] (see also Chapters 25–26), and potentially exciton condensates (although typical exciton lifetimes are much longer than for polaritons). Even experiments on cold atoms (see, e.g., Chapter 3) could be driven into a regime in which such physics occurs, when considering continuous loading of atoms balancing threebody losses [8] or atom laser setups [9, 10, 11]. Experiments on polaritons are two-dimensional, and in two dimensions it is particularly important to clearly distinguish three concepts often erroneously treated as equivalent: superfluidity, condensation, and phase coherence. This is because no true Bose-Einstein condensate exists in a homogeneous two-dimensional system.
Through trapping light in a dye-filled optical microcavity, one can engineer a gapped band structure for photons, causing them to behave as massive bosons moving in two dimensions. Repeated absorption and emission by the dye can lead to a thermal distribution and Bose- Einstein condensation. In this thesis, I study photonic condensates beyond the equilibrium thermodynamic Bose-Einstein condensation limit in three ways. First, I describe a method for fine control of the trapping potential for light allowing for future study of thermalised light in box potentials and 1D potentials as well as others, expected to show quasi-condensation and possibly superfluid behaviour, rather than ideal Bose-Einstein condensation. Second, I show how this fabrication technique has been used to define tight trapping potentials with only a few thermally accessible energy levels. In these systems, dominance of the ground- state population and build-up of coherence occur at photon numbers less than ten, raising interesting questions about how to define condensation. I discuss these questions through the relevant chapter. Third, pulsed-pump experiments show interesting temporal dynamics including jitter in the formation time of a condensate after a pump pulse due to the stochastic nature of the emission processes involved. I show how the non-stationary two-time correlation function g(2)(t1, t2) is a powerful tool for characterising these temporal dynamics and I interpret the results in a more general framework of the evolution of a probability density function through an effective free energy landscape, driven by drift terms and stochastic forces.
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.
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.