Exciton-polariton condensate

An exciton-polariton condensate has coherent macroscopic occupation of an exciton–photon hybrid mode. Finite photon lifetime makes it an open system: pumping a noncondensed reservoir can replenish condensate losses. A condensate equation coupled to a reservoir-rate equation describes gain saturation, interaction shifts and reservoir feedback; it need not possess a conserved Hamiltonian.

A condensation claim needs macroscopic low-energy occupation and spontaneous coherence together with a model of cooling, lifetime and pumping. Spectral narrowing, a nonlinear intensity threshold or long-range transport alone can have competing explanations.

For a stationary pumped condensate in a closed cross-section, integrate the gain-loss Madelung equations. Vanishing lateral condensate number current and constant integrated axial current imply the displayed balance. It fixes a density-weighted mean, not the pointwise density. If the integrated axial current $J(z)$ varies, the exact relation instead is ${\int }_S{n}^{2}dS=(\gamma /\Gamma )N(z)-[\hslash /(2\Gamma )]J^{\prime }(z)$. Thus lateral no-flux conditions alone are insufficient for a slice-by-slice balance in an axially inhomogeneous state.

For the cubic complex Ginzburg–Landau equation with $\alpha ,\beta >0$, $\Psi =\sqrt{\alpha /\beta }{e}^{-i\mu t}$ is spatially uniform, with $\mu =g\alpha /\beta -s$. In the reservoir reduction of a polariton condensate, its pump threshold is $P_{\mathrm{th}}={\gamma }_C{\gamma }_R/R_R$ and its number density is $({\gamma }_R/R_R)(1-P_{\mathrm{th}}/P)$. The full two-field homogeneous model instead gives $(P-P_{\mathrm{th}})/{\gamma }_C$. These agree near threshold only to the expansion's accuracy. Existence of the homogeneous solution does not imply stability for arbitrary interaction coefficients.

For positive gain and nonlinear-loss coefficients, a constant-density exciton-polariton condensate with physical velocity $v\hat{x}$ has $\psi =\sqrt{\gamma /\Gamma }\exp [i(mvx-\mu t)/\hslash ]$. Its chemical potential obeys the displayed relation. For prescribed real $\mu$, a real velocity requires $\mu \ge g\gamma /\Gamma$. If chemical potential is determined by the solution, any real velocity is allowed. Linear density-phase perturbations in the comoving frame satisfy $\lambda (\lambda +2\gamma /\hslash )+{ϵ}_k({ϵ}_k+2g{n}_{\ast })/{\hslash }^2=0$, where ${ϵ}_k={\hslash }^2{k}^2/(2m)$. Repulsive coupling has no growing modes in this simple model; attractive coupling has a long-wavelength instability even though the plane wave exists.

For reservoir kinetics ${\mathcal{R}}_t=-({\gamma }_R+R_{R}n)\mathcal{R}+P$, adiabatic elimination gives $\mathcal{R}=P/({\gamma }_R+R_{R}n)$. Expanding for $R_{R}n/{\gamma }_R\ll 1$ and inserting into the condensate equation gives a complex Ginzburg–Landau equation withFast reservoir response and weak depletion are separate assumptions. Without the number density expansion the gain is saturable, not cubic. The coefficients depend on position when the pump does.