Are non-equilibrium Bose-Einstein condensates superfluid? Michiel Wouters 1 and Iacopo Carusotto 2 $^{1}$Institute of Theoretical Physics, Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland $^{2}$CNR INFM-BEC Center and Dipartimento di Fisica, Universit` a di Trento, I-38123 Povo, Italy We theoretically study the superfluidity properties of a non-equilibrium Bose-Einstein condensate of exciton-polaritons in a semiconductor microcavity. The dynamics of the condensate is described at mean-field level in terms of a modified Complex Ginzburg Landau equation. A generalized Landau criterion is formulated which estimates the onset of the drag force on a small moving defect. Metastability of supercurrents in multiply connected geometries persists up to higher flow speeds. PACS numbers: 03.75.Kk, 05.70.Ln, 71.36.+c. arXiv:1001.0660v1 [cond-mat.mes-hall] 5 Jan 2010 Superfluidity is among the most remarkable conse- quences of macroscopic quantum coherence in condensed matter systems and manifests itself in a number of fasci- nating effects [1, 2]. A unified description of these phe- nomena is obtained in the framework of the so-called two- fluid hydrodynamics, in which the macroscopic conden- sate wavefunction adds up to the standard hydrodynamic variables [3]. The phenomenon of macroscopic coherence is not restricted to systems at (or close to) thermodynam- ical equilibrium such as liquid Helium, ultracold atomic gases, or superconducting materials, but has been ob- served also in systems far from thermodynamical equi- librium, whose state is determined by a dynamical bal- ance of driving and losses. Most remarkable examples are lasers and, more recently, Bose-Einstein condensates of magnons in magnetic solids [4] and exciton-polaritons in semiconductor microcavities [5, 6]. In particular, the issue of superfluidity in this latter system has attracted a significant interest from both the theoretical [7, 8] and experimental [9–11] points of view. pumping and losses [13, 14]. This mean-field descrip- tion has been able to explain a number of experimen- tal observations on polariton condensates, e.g. the ring- shaped momentum distribution of spatially narrow con- densates [6, 15], the synchronization/desynchronization transition [16], the spontaneous appearance of vor- tices [17]. Nonetheless, the implicit assumption that the pumping mechanism is not frequency-selective can lead to unphysical predictions, e.g. that in a spatially ho- mogeneous or ring-like geometry condensation is equally likely to occur in any momentum state. Kinetic calcu- lations [18] have pointed out the significant energy de- pendence of the polariton-polariton scattering processes that are responsible for replenishing the condensate. In- cluding this feature as an energy-dependent amplification mechanism turned out to be crucial in order to extract physically meaningful predictions for the condensate fluc- tuations in the Wigner Monte Carlo simulations of [19]. A simplest generalization of the GPE to include frequency-dependent pumping has the form: Recent experiments with resonantly pumped polari- ton condensates [9] have demonstrated superfluidity as a dramatic reduction in the intensity of resonant Rayleigh scattering as originally predicted in [7]. The situation is less clear in the case of non-resonant [5] or parametrical (OPO) [10] pumping schemes: recent experiments [10] have observed propagation of polariton bullets without apparent friction, which is in contrast with the predic- tions of a na¨ıve Landau criterion based on the elementary excitation spectrum predicted in [12]. Another aspect of superfluidity, namely metastability of supercurrent in multiply-connected geometries was investigated theoret- ically in [8] and experimentally confirmed in [11]. The present paper reports a comprehensive theoretical inves- tigation of the meaning of superfluidity for polariton con- densates under a non-resonant pumping. Emphasis will be given to the novel features that originate from their non-equilibrium character. At the mean-field level, the condensate dynamics can be described in terms of the so-called Gross-Pitaevskii equation (GPE) [1], which was recently generalized to non-equilibrium condensates by including the effect of The energy zero has been set for convenience at the bot- tom of the lower polariton branch; the efficiency of am- plification (proportional to the pumping strength P ) de- creases to zero a frequency interval Ω$_{K}$ above it. As- suming a linear form of the frequency dependence of the amplification, the generalized GPE (1) maintains a tem- porally local form. The others terms describing gain sat- uration ( r ), losses ( γ ), polariton mass ( m ), polariton- polariton interactions ( g ), external potential ( V$_{ext}$ ) have the same meaning as in previous works [13, 14]. i dψ dt = { − ℏ 2 m ∇ 2 + V$_{ext}$ + i 2 [ P ( 1 − i Ω$_{K}$ d dt ) − r | ψ | 2 − γ ] + g | ψ | 2 } ψ. (1) We first consider the evolution of the system starting from an initial state with no condensate ψ = 0. If the strength of pumping is enough to overcome losses P > γ , the ψ = 0 state is dynamically unstable against the cre- ation of a finite condensate amplitude in all the low- momentum modes for which ℏ k $^{2}$/ 2 m < Ω$_{K}$ (1 − γ/P ). The rate of this instability is maximum at k = 0 and