Superfluid Stiffness of a Driven Dissipative Condensate with Disorder

Observations of macroscopic quantum coherence in driven systems, e.g. polariton condensates, have strongly stimulated experimental as well as theoretical efforts during the last decade. We address the question of whether a driven quantum condensate is a superfluid, allowing for the effects of disorder and its nonequilibrium nature. We predict that for spatial dimensions $d<4$ the superfluid stiffness vanishes once the condensate exceeds a critical size, and treat in detail the case $d=2$. Thus a nonequilibrium condensate is not a superfluid in the thermodynamic limit, even for weak disorder, although superfluid behavior would persist in small systems.

Figure 1

Superfluid stiffness as a function of disorder strength, ${\kappa }^2$, for nonequilibrium parameters $\alpha =0.1$, 0.3, 0.6, 1 from highest to lowest curves, respectively, and system sizes $L=64$ (left panel) and 96 (right panel). Points show numerical results, and lines the perturbative expression, Eq. (13). Numerical results are averages over 120 disorder realizations.

Figure 2

Phase (left) and current (right) response due to a phase twist $\theta$ along $x$, in a typical disorder realization. The plotted current response is $\nabla {\varphi }_{\theta }(x)-\nabla \varphi (x)$ averaged along $y$. Note the exponentially decaying tails of the current, c.f. Eq. (14), and the formation of a domain wall in the phase. Parameters used are $\alpha =0.5$, $\kappa =0.5$, $\theta =1$.

Figure 3

Scaling behavior of the decay length $\zeta$ with $\alpha$ and $\kappa$. Points are disorder averaged numerical results for the inverse of $\zeta (\alpha ,\kappa )$ normalized to a reference value ${\zeta }_0\equiv \zeta ({\alpha }_0,{\kappa }_0)$. Lines are linear fits on a double log scale. Left: dependence on $\alpha$ at fixed $\kappa ={\kappa }_0$ shows ${\zeta }^{-1}\sim {\alpha }^2$. Right: dependence on $\kappa$ at fixed $\alpha ={\alpha }_0$ shows ${\zeta }^{-1}\sim \kappa$. The parameters are $L=256a_L$, ${\alpha }_0=0.7$, ${\kappa }_0=0.25$, $\theta =1$, 72 disorder realizations.

Figure 4

The numerically calculated stiffness as a function of $c_2{\alpha }^4{\kappa }^2{L}^2\sim L^2/{\mathcal{L}}_s^2$ shows a clear data collapse. Inset: comparison between the exponential tail and the scaling form, Eq. (15), using the values of $c_2$ and $g$ obtained perturbatively (for details see text). Points are shown for $L=64$ and 96; $\alpha =0.9$, 1, and 1.2. For each data point, we simulated up to 1320 disorder realizations.

Figure 5

Proposed measurement of condensate stiffness via the response of the condensate emission frequency $\omega$ or phase profile $\varphi$ to a phase twist $\theta$ (see text).