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arXiv:1106.0682 · Physical Review Letters

Superfluid Density of an Open Dissipative Condensate

Jonathan Keeling

Published Aug 16, 2011• 49 citations
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Abstract

I calculate the superfluid density of a nonequilibrium steady state condensate of particles with finite lifetime. Despite the absence of a simple Landau critical velocity, a superfluid response survives, but dissipation reduces the superfluid fraction. I also suggest an idea for how the superfluid density of an example of such a system, i.e., microcavity polaritons, might be measured.

DOI
10.1103/physrevlett.107.080402

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I calculate the superfluid density of a nonequilibrium steady state condensate of particles with finite lifetime. Despite the absence of a simple Landau critical velocity, a superfluid response survives, but dissipation reduces the superfluid fraction. I also suggest an idea for how the superfluid density of an example of such a system, i.e., microcavity polaritons, might be measured.
We theoretically study the superfluidity properties of a nonequilibrium Bose-Einstein condensate of exciton polaritons in a semiconductor microcavity under incoherent pumping. The dynamics of the condensate is described at mean-field level in terms of a generalized Gross-Pitaevskii equation. The drag force on a small moving object and the onset of fringes in the density profile are shown to have a sharp threshold as a function of the velocity; a generalized Landau criterion is developed to explain this behavior in terms of the dispersion of elementary excitations. Metastability of supercurrents in multiply-connected geometries is shown to persist up to higher flow speeds.
We study theoretically superfluidity in a driven-dissipative Bose gas out of thermal equilibrium, and discuss the relation with conventional superfluids. We show how the superfluid behavior is characterized by a dramatic increase in the lifetime of a quantized vortex and point out the influence of the spatial geometry of the condensate. We apply our study to a condensate of polaritons in a semiconductor microcavity, whose properties can be directly inferred from optical spectroscopy. We propose three different experimental schemes to measure the vorticity of the polariton condensate.
We review recent results on the coherence and superfluidity of driven dissipative condensates, i.e., systems of weakly interacting nonconserved bosons, such as polariton condensates. The presence of driving and dissipation has dramatically different effects depending on dimensionality and anisotropy. In three dimensions, equilibrium behaviour is recovered at large scales for static correlations, while the dynamical behaviour is altered by the microscopic driving. In two dimensions, for an isotropic system, drive and dissipation destroy the algebraic order that would otherwise exist; however, a sufficiently anisotropic system can still show algebraic phase correlations. We discuss the consequences of this behaviour for recent experiments measuring phase coherence and outline potential measurements that might directly probe superfluidity. Introduction This chapter is dedicated to superfluidity and its relation to Bose-Einstein condensation (BEC), a topic with a long history. Many reviews of the concepts of condensation and superfluidity in thermal equilibrium can be found; see for example Refs. [1, 2, 3, 4]. The focus of this chapter is on how these concepts apply (or fail to apply) to driven dissipative condensates – systems of bosons with a finite lifetime, in which loss is balanced by continuous pumping. We focus entirely on the steady state of such systems, neglecting transient, time-dependent behaviour. Experimentally, the most studied example of a driven dissipative condensate has been microcavity polaritons, an overview of which is given in Ref. [5] and Chapter 4. However, similar issues can arise in many other systems, most obviously photon condensates [6] (see also Chapter 19), magnon condensates [7] (see also Chapters 25–26), and potentially exciton condensates (although typical exciton lifetimes are much longer than for polaritons). Even experiments on cold atoms (see, e.g., Chapter 3) could be driven into a regime in which such physics occurs, when considering continuous loading of atoms balancing threebody losses [8] or atom laser setups [9, 10, 11]. Experiments on polaritons are two-dimensional, and in two dimensions it is particularly important to clearly distinguish three concepts often erroneously treated as equivalent: superfluidity, condensation, and phase coherence. This is because no true Bose-Einstein condensate exists in a homogeneous two-dimensional system.
The flow of a uniform Bose gas at speeds greater than the Landau critical velocity vc{v}_{c}vc​ does not necessarily destroy superfluidity, but rather needs only to lead to a decrease of the superfluid mass density ${\ensuremath{\rho}}_{s}$. Analyzing a weakly interacting Bose gas with a finite-range interparticle interaction that leads to a Landau critical velocity at nonzero quasiparticle momentum, we explicitly construct the (nonuniform) condensate for fluid flow faster than vc{v}_{c}vc​ and calculate the accompanying decrease in ${\ensuremath{\rho}}_{s}$. We briefly comment on the relation of the physics to other problems in superfluids, e.g., solitons and vortices in Bose-Einstein condensates, and critical currents in superconductors.
We show that in a nonequilibrium, two-dimensional superfluid near the critical point, the energy for dissipative fluctuations is greatly reduced as a result of screening by the gas of bound vortex pairs, leading to an enhanced decay of the superfluid velocity. We also show that the superfluid density is reduced by a velocity-dependent amount as Tc{T}_{c}Tc​ is approached from below.
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.