Photon condensation from thermal sources and the limits of heat engines

Abstract

The trapping and cooling of photon gases in microcavities has been used to create Bose-Einstein condensates. We investigate the conditions required for condensation in dye-filled microcavities, with photon populations created either by driving a transition of the dye, or by coupling the cavity modes to a thermal photon reservoir such as sunlight. We find that the threshold pump temperature, above which condensation appears, is determined by the second law of thermodynamics. The minimum achievable threshold is that of a reversible three-level heat engine, which we show arises in the dye-pumped case, and for pumping of the modes of a two-level cavity. For a many-level cavity condensation occurs at a similar but higher temperature. Our results show that photon condensates can be produced by pumping with incoherent thermal sources, opening possibilities for coherent light generation, energy harvesting, and experimental studies of quantum heat engines.

Figures (4)

• Figure 1: Photon condensation in a dye-filled cavity as a heat engine. The cavity photons which condense originate from an external source, corresponding to the hot bath of the engine, while excess energy is dissipated into the cold bath, provided by the solvent. The work output corresponds to the coherent emission from the condensate.
• Figure 2: Ground-state photon occupations as functions of the pump strength expressed as a temperature $T_h$, for different forms of pumping. Solid : pumping of the dye, with $T_h$ the temperature of its electronic transition. Dotted and dot-dashed: pumping of the cavity modes by coupling to an external thermal photon reservoir of temperature $T_h$, considering only two cavity energy levels (dotted) or all of them (dot-dashed). The thresholds for dye pumping and the two-level cavity agree with Eq. \ref{['eq:revthresh']} (left arrow). The threshold for many-level thermal pumping agrees with Eq. \ref{['eq:threshold']} (right arrow).
• Figure 3: Phase diagram of photon condensation in a many-level harmonic trap with external thermal pumping, calculated using Eq. \ref{['eq:threshold']} (black curve) and by numerically evolving Eq. \ref{['eq:nmdotmultimode']} (orange points). The red dashed curve is the corresponding threshold for a reversible heat engine. Regions above the curves are condensed.
• Figure 4: (a,b) Energy currents for dye pumping (a) and external thermal pumping of a many-level trap (b). The black curves are the work outputs, and the red (blue) curves the currents to the hot (cold) bath. (c) Efficiencies for dye pumping (black solid) and external pumping (blue solid). For dye pumping the efficiency is determined by the energy ratio $\omega_0/\omega_d$ (horizontal dashed line). The threshold is where this crosses the Carnot efficiency for the given bath temperatures (black dashed curve).