Condensation of cavity polaritons in a disordered environment

A model for direct two band excitons in a disordered quantum well coupled to light in a cavity is investigated. In the limit in which the exciton density is high, we assess the impact of weak “pair-breaking” disorder on the feasibility of condensation of cavity polaritons. The mean-field phase diagram shows a “lower density” region, where the condensate is dominated by electronic excitations and where disorder tends to close the condensate and quench coherence. Increasing the density of excitations in the system, partially due to the screening of Coulomb interaction, the excitations contributing to the condensate become mainly photonlike and coherence is re-established for any value of disorder. In contrast, in the photon dominated region of the phase diagram, the energy gap of the quasiparticle spectrum still closes when the disorder strength is increased. Above mean-field, thermal, quantum, and fluctuations induced by disorder are considered and the spectrum of the collective excitations is evaluated. In particular, it is shown that the angle resolved photon intensity exhibits an abrupt change in its behavior, going from the condensed to the noncondensed region.

Figure 1

Schematic picture of the valence $\left(a\right)$ and conductance $\left(b\right)$ band, and the interactions included in the model (1).

Figure 2

Relative number of (total) electrons $n_{\mathrm{el}}={\rho }_{\mathrm{el}}∕{\rho }_{\mathrm{ex}}$ and (condensed) photons $n_{\mathrm{ph}}={\rho }_{\mathrm{ph}}∕{\rho }_{\mathrm{ex}}$ are plotted against ${\rho }_{\mathrm{ex}}{a}_0^2$ for fixed values of the coupling constants $\tilde{g}=1$ and ${\lambda }_c=10$ and for $d=200$. Note that, for ${\rho }_{\mathrm{ex}}{a}_0^2>25$, the system enters a strong coupling regime, where our model cannot be applied. In the inset, the number of photons ${\rho }_{\mathrm{ph}}{a}_0^2$ is compared with the number of electrons in the condensate, $N_{\mid \Sigma \mid }=\mid \Sigma \mid \nu a_0^2$ [see Eq. (21)]. For ${\rho }_{\mathrm{ex}}{a}_0^2>d∕4\pi \simeq 16$, the excitations in the condensate are dominated by photons.

Figure 3

Mean-field phase diagram for $\tilde{g}=1$, ${\lambda }_c=10$, and $d=200$; ${\rho }_{\mathrm{ex}}{a}_0^2$ represents the excitation density, while $\xi =1∕{\tau }_c\mathcal{R}\mathrm{y}$ the disorder strength.

Figure 4

The order parameter $\mid \Delta \mid ∕\mid {\Delta }_0\mid$ (solid line) and energy gap ${ϵ}_{\text{gap}}∕\mid {\Delta }_0\mid$ (dashed line) versus the strength of disorder $1∕{\tau }_c\mid {\Delta }_0\mid$. Here, $\mid {\Delta }_0\mid$ represents the value of the order parameter in absence of disorder. The critical value at which the order parameter disappear is given by $1∕{\tau }_c\mid {\Delta }_0\mid =1∕2$, while the gapless region starts at $1∕{\tau }_c\mid {\Delta }_0\mid =e^{-\pi ∕4}\simeq 0.46$ (or equivalently at $\zeta =1∕{\tau }_c\mid \Delta \mid =1$).

Figure 5

Normalized density of excitations ${\rho }_{\mathrm{ex}}{a}_0^2$ as a function of the rescaled chemical potential $x=({\omega }_c-\mu )∕\mathcal{R}\mathrm{y}$ (in logarithmic scale) for $d=({\omega }_c-E_g)∕\mathcal{R}\mathrm{y}=200$, $\tilde{g}=1$, ${\lambda }_c=10$ (bold line) and for $\tilde{g}=0$ (dashed line). The value of the density at $x=0$ for $\tilde{g}=0$, $d∕4\pi \simeq 16$, has been marked. In this limit, all excitations appear as electrons and holes, allowing the population of states with energies in excess of ${\omega }_c$. The dependence from of the dimensionless effective coupling constant $\nu L^2{g}_{\mathrm{eff}}$ on ${\rho }_{\mathrm{ex}}{a}_0^2$ in both cases is shown in the inset. Note that the weak coupling limit imposes the constraint ${\rho }_{\mathrm{ex}}{a}_0^2<25$.

Figure 6

Order parameter $\mid \Delta \mid ∕\mathcal{R}\mathrm{y}$ versus ${\rho }_{\mathrm{ex}}{a}_0^{2}S$ for $\tilde{g}=1$, ${\lambda }_c=10$, and $d=200$ and different values of the disorder strength $\xi =1∕{\tau }_c\mathcal{R}\mathrm{y}$.

Figure 7

Three-dimensional plot of the critical temperature, $T_c∕\mathcal{R}\mathrm{y}$, versus the disorder strength, $\xi =1∕{\tau }_c\mathcal{R}\mathrm{y}$, and the excitation density, ${\rho }_{\mathrm{ex}}{a}_6^2$ ($\tilde{g}=1$, ${\lambda }_c=10$, and $d=200$). The contour line where the critical temperature goes to zero separates the zero-temperature condensed phase from the noncondensed one and it coincides with the contour given in Fig. 3.

Figure 8

Zero temperature plot of the zero momentum longitudinal ${\Pi }_{0,0}^{\mathrm{L}}-{\Pi }_{{\omega }_h,0}^{\mathrm{L}}$ and transverse ${\Pi }_{0,0}^{\mathrm{T}}-{\Pi }_{{\omega }_h,0}^{\mathrm{T}}$ components versus the rescaled frequency ${\omega }_h∕\mid \Delta \mid$, for different values of the disorder strength $\zeta =1∕{\tau }_c\mid \Delta \mid$.

Figure 9

Dispersion of the collective excitation spectrum in the gapped $\zeta <1$ (left) and gapless $\zeta >1$ (right) regions. Here, we have chosen the particular values $c_1=0.8$, $a_1=7.1$, $a_2=7.7$, $b_1=1$, $b_2=1.2$, and $f=0.6$ (corresponding to ${\rho }_{\mathrm{ex}}{a}_0^2=16$, $\xi =1∕{\tau }_c\mathcal{R}\mathrm{y}=1$, and $D{\omega }_c∕c^2=0.1$) in the first case, and $c_1=0.1$, $a_1=7.6$, $a_2=7.6$, $b_1=0.4$, $b_2=1.1$, and $f=0.7$ (corresponding to ${\rho }_{\mathrm{ex}}{a}_0^2=16$, $\xi =4$ and $D{\omega }_c∕c^2=0.1$) in the second (note that this choice masks the actual linear dispersion of the massless mode). When $\zeta >1$, both the massive and massless excitations acquire a finite lifetime with, respectively, the first finite and almost constant, while the second exhibits a quadratic dispersion and therefore null at zero momentum. Their linewidths, $\mathfrak{R}{\tilde{E}}_{1,2}\pm \mathfrak{J}{\tilde{E}}_{1,2}$, are explicitly plotted.