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arXiv:2001.07137 · Physical Review A

Classical field model for arrays of photon condensates

V. N. Gladilin, Michiel Wouters

Published Apr 13, 2020• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.101.043814

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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.
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 develop a nonequilibrium model of condensation and lasing of photons in a dye filled microcavity. We examine in detail the nature of the thermalization process induced by absorption and emission of photons by the dye molecules, and investigate when the photons are able to reach a thermal equilibrium Bose-Einstein distribution. At low temperatures, or large cavity losses, the absorption and emission rates are too small to allow the photons to reach thermal equilibrium and the behavior becomes more like that of a conventional laser.
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.
The experimental realization of a Bose-Einstein condensate of photons, first achieved in 2010 at Bonn University, allowed for new ways to probe the properties of condensates. For thermal light in free space the chemical potential is zero and condensation is not possible. By confining light to a microcavity, the transverse degrees of freedom behave as massive bosons which can have nonzero chemical potential. The gas of thermal photons can be brought to thermalize with a fluorescent dye though absorption and emission cycles. By pumping the dye molecules with a laser, the chemical potential can be increased and the photons will condense. \nWe have built a setup with which we can spatially and spectrally resolve the photon distribution in such a dye-filled microcavity. This allows us to study the shape and occupation of the low energy transverse modes of the cavity. We use this to investigate the thermalization process of the photon gas. A rate equation model is presented and qualitatively compared to experimental data. We find that the photon gas is not in thermal equilibrium.\nWe also investigate effective photon-photon interactions by measuring the spatial extent of the condensate. We find an effective interaction in the form of thermal lensing. As the thermal lensing depends on the interaction between the photon gas and the dye, we find that the model used for thermalization can also make good predictions about effective photon-photon interactions.
Photons can come to thermal equilibrium at room temperature by scattering multiple times from a fluorescent dye. By confining the light and dye in a microcavity, a minimum energy is set and the photons can then show Bose–Einstein condensation. We present here the physical principles underlying photon thermalization and condensation, and review the literature on the subject. We then explore the ‘small’ regime where very few photons are needed for condensation. We compare thermal equilibrium results to a rate-equation model of microlasers, which includes spontaneous emission into the cavity, and we note that small systems result in ambiguity in the definition of threshold.
We examine in detail the mechanisms behind thermalization and Bose-Einstein condensation (BEC) of a gas of photons in a dye-filled microcavity. We derive a microscopic quantum model, based on that of a standard laser, and show how this model can reproduce the behavior of recent experiments. Using the rate-equation approximation of this model, we show how a thermal distribution of photons arises. We go on to describe how the nonequilibrium effects in our model can cause thermalization to break down as one moves away from the experimental parameter values. In particular, we examine the effects of changing cavity length, and of altering the vibrational spectrum of the dye molecules. We are able to identify two measures which quantify whether the system is in thermal equilibrium. Using these, we plot ``phase diagrams'' distinguishing BEC and standard lasing regimes. Going beyond the rate-equation approximation, our quantum model allows us to investigate both the second-order coherence g(2){g}^{(2)}g(2) and the linewidth of the emission from the cavity. We show how the linewidth collapses as the system transitions to a Bose condensed state, and compare the results to the Schawlow-Townes linewidth.