BCS-BEC crossover in a system of microcavity polaritons

We investigate the thermodynamics and signatures of a polariton condensate over a range of densities, using a model of microcavity polaritons with internal structure. We determine a phase diagram for this system including fluctuation corrections to the mean-field theory. At low densities the condensation temperature $T_c$ behaves like that for point bosons. At higher densities, when $T_c$ approaches the Rabi splitting, $T_c$ deviates from the form for point bosons, and instead approaches the result of a BCS-like mean-field theory. This crossover occurs at densities much less than the Mott density. We show that current experiments are in a density range where the phase boundary is described by the BCS-like mean-field boundary. We investigate the influence of inhomogeneous broadening and detuning of excitons on the phase diagram.

Figure 1

(Color online) Excitation spectra in the condensed (grey) and uncondensed states, superimposed on the mean-field phase diagram, to show choice of density and temperature. Panel A is for the photon and exciton resonant, and the two spectra are for $T=2g\sqrt{n}$ and $\mu ∕g\sqrt{n}=-1.4$ and $-0.24$. Panel B has the exciton detuned by $3g\sqrt{n}$ below the photon, and the spectra are for $T=0.1g\sqrt{n}$ and $\mu ∕g\sqrt{n}=-3.29$, $-3.01$, $-2.54$, and $-0.37$. The photon mass is $m^{*}=0.01$. Temperatures and energies plotted in units of $g\sqrt{n}$, densities in units of $n$, and wave vectors in units of $\sqrt{n}$.

Figure 2

Incoherent luminescence vs momentum ($x$ axis) and energy ($y$ axis) calculated above [panels (A) and (C)] and below [panels (B) and (D)] phase transition. Panels (A) and (B) are for inhomogeneous broadening of $0.1g\sqrt{n}$, and (C) and (D) are for $0.3g\sqrt{n}$. All other parameters are the same as in the unbroadened case shown in panel (A) of Fig. 1. In order to map the infinite range of intensities to a finite scale, the color is allocated as the hyperbolic tangent of intensity. Energies are in units of $g\sqrt{n}$, and wave vectors in units of $\sqrt{n}$.

Figure 3

(Color online) Momentum distribution of photons, from which follows the angular distribution. These are plotted for low-temperature condensed systems, where the approximation of including only phase fluctuations is valid, and for the simpler, uncondensed case. The inset illustrates the choices of density and temperature. (Parameters are ${\Delta }^{*}=1$, $m^{*}=0.01$, wave vector plotted in units of $g\sqrt{n}$, temperature in units of $g\sqrt{n}$, density in units of $n$, and $N\left(k\right)$ in arbitrary units.)

Figure 4

(Color online) Mean-field phase boundary, and phase boundaries including fluctuation correction for four values of photon mass, on a logarithmic scale (Insets are on a linear scale). Panel (A) is the resonant case, $\Delta =0$, adapted from Ref. 15 and shown for comparison. In panels (B) and (C), the exciton is detuned below the photon band by $\Delta =1.5g\sqrt{n}$ and $\Delta =2.5g\sqrt{n}$, respectively. Temperature plotted in units of $g\sqrt{n}$ and density in units of $n$.

Figure 5

(Color online) A schematic picture of how the relevant excitations change between the polariton BEC, the BCS-like mean-field, and the photon BEC regimes. In the low-density limit, a shallow sound mode exists. At higher densities, this becomes steeper, and the relevant excitations are gapped single-particle excitations. At yet higher densities, these modes are saturated, and the high $k$ photon modes become relevant.

Figure 6

(Color online) Chemical potential vs density and temperature at the phase boundary, for the mean-field phase boundary, and $m^{*}=0.50$ fluctuation corrections. Plotted for detunings of $\Delta =1.5g\sqrt{n}$ [panel (A)] and $\Delta =2.5g\sqrt{n}$ [panel (B)], with all other parameters as in Fig. 4. Temperature and chemical potential plotted in units of $g\sqrt{n}$, and density in units of $n$.