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.

Keywords:

• Bose–Einstein condensation
• microcavities
• open quantum systems

Acknowledgements

We thank the UK Engineering and Physical Sciences Research Council for supporting this work through fellowship EP/J017027/1 and the Controlled Quantum Dynamics CDT EP/L016524/1 which was co-directed by Danny for many years.

Notes

No potential conflict of interest was reported by the authors.

1 The formal Penrose-Onsager definition [2] is essentially that the largest eigenvalue of the single-particle density matrix that solves the full many-body problem is extensive with system size. But that’s not the kind of definition that helps the non-expert.

2 The canonical citation of Stepanov is Ref. [5], but we cannot find a copy of it, nor read Russian.

3 For the sake of simplicity we have omitted a factor ${\mathit{ϵ}}^3$ which accounts for the density of states available for spontaneous emission.

Additional information

Funding

This work was supported by Engineering and Physical Sciences Research Council [grant number EP/J017027/1], [grant number EP/L016524/1].