-5 0 5 -10 -5 0 5 10 Re[ $_{ω}$Bog (k)] -5 0 5 -10 -5 0 5 10 -5 0 5 -10 -5 0 5 10 -5 0 5 q / k -1 0 Im[ $_{ω}$Bog (k)] -5 0 5 q / k -1 0 -5 0 5 q / k -1 0 -5 0 5 -0.2 -0.1 0 0.1 (a) (c) (e) (b) (d) (f) decreases for increasing k . This fact is in agreement with the experimental observation of condensation naturally occurring around k = 0 as soon as the sample is suffi- ciently large and free from disorder [5]. In spite of this natural preference, condensation can be forced to occur in finite momentum state by seeding the system with a short resonant pulse at the desired k$_{c}$ as proposed and numerically assessed in Ref. [8]. A related configuration was experimentally demonstrated for the OPO pump scheme in [11]. The effect of the frequency- dependent pumping is then visible in the state equation relating the pumping strength P to the condensate den- sity n$_{c}$ = | ψ | 2 in the stationary state: As a consequence of the frequency dependent pumping, the interaction-induced blue-shift of the condensate fre- quency ω$_{c}$ = k 2 $_{c}$/ 2 m + g | ψ | 2 is responsible for a slower increase of density with pumping strength P . The dynamical stability of a moving condensate is to be assessed by linearization of the CGLE around the stationary solution. Stability requires that the imagi- nary part of the frequency is negative for all Bogoliubov modes. Examples of the Bogoliubov dispersion ω$_{Bog}$ ( q ) for moving condensates are shown in Fig.1. In the left- most panels (a,b), the limit Ω$_{K}$ → ∞ of a negligible frequency-dependence of pumping is considered. Apart for the global Doppler tilting of the real part due to the finite condensate velocity, the plotted dispersion fully re- covers the diffusive character first discussed in [12–14]: a It is of crucial importance to note that this instability mechanism has no direct counterpart in standard equi- librium superfluids and is effective even in the absence of any defect potential. Of course, the present theory is directly applicable only to infinite, homogeneous conden- sates or in multiply connected ring-shaped geometries. In finite systems, assessing the stability of moving conden- sates would be made harder by the flow of condensate polaritons outside the pump regions [15]. In the presence of weak defects, the frictionless flow of equilibrium condensates is limited to flow speeds below The right panels (e,f) refer to the case of a stronger pumping well above the threshold. In this case, the damping rate of all modes other than the Goldstone one is comparable to the polariton lifetime γ . In this case, the frequency-dependence of pumping is not able to desta- bilize the moving condensate. The stability domain as a function of the momentum k$_{c}$ and the density n$_{c}$ of the condensate is summarized in Fig.2: as the density is increased, stable condensates survive up to larger mo- menta. FIG. 1: Elementary excitation spectrum of a moving, spa- tially homogeneous, non-equilibrium condensate as described by the linearized CGLE around the steady-state solution. Parameters: k$_{c}$/k$_{γ}$ = 4, r/γ = 1, g/γ = 1, Ω$_{K}$ /γ = 50, n$_{c}$g/γ = 0 . 1 (c,d) and n$_{c}$g/γ = 1 (e,f). Inset: magnified view of (d). Panels (a,b): same as panels (e,f) in the limit of a frequency-independent pumping Ω$_{K}$ = ∞ . Momenta are measured in units of k$_{γ}$ = √ mγ/ ℏ . FIG. 2: Growth rate (in units of γ ) of maximally unstable mode as a function of condensate momentum k$_{c}$ and density n$_{c}$ . The dashed line indicates the boundary of the stability region. Same system parameters as in Fig.1(c-f). flat dispersion around the condensate wavevector and an imaginary part quadratically growing in q . In this limit, stable condensates exist for any value of k$_{c}$ . The effect of a frequency-selective pumping is ad- dressed in the other panels. The central panels (c,d) refer to the case of a pumping intensity just above the thresh- old. In this regime, the condensate density is very small and the real part of the Bogoliubov mode frequency re- duces to the single-particle one. On the other hand, the characteristic damping rate of density fluctuations (the gapped mode at q = 0) and of the high-momentum modes is suppressed by a critical slowing down phenomenon [13]. As a result, the low energy modes around q ≃ − k$_{c}$ be- come dynamically unstable. Eventually, this mechanism leads to the disappearence of the original condensate and the formation of another -stable- condensate in a lower momentum state. 2 | ψ | 2 = [ 1 − 1 Ω$_{K}$ ( ℏ k 2 c 2 m + g | ψ | 2 )] P r . (2)