Bose-Einstein condensation of photons from the thermodynamic limit to small photon numbers

Abstract

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. FOREWORDThis article is written in memory of Danny Segal, who was a colleague of one of us (Rob Nyman) in the Quantum Optics and Laser Science group at Imperial College for many years. The topic of this article touches on the subject of dye lasers, the stuff of nightmares for any AMO physicist of his generation, but a stronger connection to Danny is that he was very supportive of my application for the fellowship that pushed my career forward, and funded this research. One of Danny's quirks was a strong dislike of flying. As a consequence, I had the pleasure of joining him on a 24 hour, four-train journey from London to Italy to a conference. That's a lot of time for story telling and forging memories for life. Danny was one of the good guys, and I sorely miss his good humour and advice.This article presents a gentle introduction to thermalization and Bose-Einstein condensation (BEC) of photons in dye-filled microcavities, followed by a review of the state of the art. We then note the similarity to microlasers, particularly when there are very few photons involved. We compare a simple non-equilibrium model for microlasers with an even simpler thermal equilibrium model for BEC and show that the models coincide for similar values of a 'smallness' parameter.

Statistical theory of photon gas in plasma

“…Experimental observations of photon BEC in the microcavity have been reported in [23]. Further studies of the photon BEC based on variations of the previous ideas can be found in [24][25][26][27][28][29].…”

Collective modes of a photon Bose–Einstein condensate with thermo-optic interaction

“…These particles possess a mass w = ( ) m n c cutoff 0 2 , where the cavity-cutoff frequency is denoted by ω cutoff and the light velocity in the dye solution is c/n 0 . Furthermore, these particles are trapped in a harmonic potential with frequency W = ( ) c L R n 2 0 0 , that is determined by the cavity length L 0 and the radius of curvature R of the mirror [6,13,29]. Thus, the evolution of the condensate wave function y ( ) t r, , i.e.the electric field normalised to the photon number, is described by an opendissipative Schrödinger equation of the form [23, 29-31] Figure 1.…”

Bose condensation of squeezed light

“…A coherent coupling between a laser-driven optomechanical membrane and a Bose-Einstein condensate has been achieved as well [30,31] which shows strong squeezing in the mechanical mode [31]. The optical cavity has been used to demonstrate variable potentials leading to a small critical photon number of N 0 = 68 for condensation [32] which physical principles are discussed in [33].…”