0 1 2 0.2 0.4 0.6 0.8 (a) v/c$_{s}$ F/F 0 1 2 −4 −2 0 2 4 Re[k/k ] v/c$_{s}$ (b) 0 1 2 −3 −2 −1 0 1 2 3 Im[k/k ] v/c$_{s}$ (c) −20 0 20 −20 −10 0 10 20 x( m) y( m) (a) −20 0 20 −20 −10 0 10 20 x( m) y( m) (b) −20 0 20 −20 −10 0 10 20 x( m) y( m) (c) 0.98 0.99 1 1.01 1.02 3 the Landau critical velocity v$_{c}$ = min$_{k}$ { Re[ ω o $_{Bog}$( k )] /k } , where ω o $_{Bog}$( k ) is the real part of the Bogoliubov dis- persion for a condensate at rest. For dilute systems, v$_{c}$ coincides with the sound velocity c$_{s}$ = √ gn$_{c}$/m [1]. As a consequence of the diffusive character of the Goldstone mode, a na¨ıve application of this definition to the Bo- goliubov dispersion of non-equilibrium condensates an- ticipated in [12–14] gives a vanishing prediction for the critical velocity, v$_{c}$ = 0: Bogoliubov waves are emitted in a moving condensate hitting a (weak) defect at any value of the condensate speed. A complete picture of non-equilibrium condensates re- quires taking into account the non-trivial dynamics of the imaginary part of the Bogoliubov dispersion. Nu- merical plots of the density perturbation induced by a single weak stationary defect in a (dynamically stable) moving condensate can be calculated from the general- ized GPE (1) in the presence of a suitable defect potential V$_{ext}$ ( r ) and are shown in Fig.3 for different values of the condensate velocity. In contrast to the predictions of the na¨ıve Landau criterion stated above, the induced per- turbation closely resembles the one induced in an equi- librium condensate described by the standard GPE [22]: at high speeds [panel (a)], the defect creates a series of parabolic fringes that propagate away from the defect; at low speeds [panel (c)], the propagating fringes are re- placed by a localized perturbation in the vicinity of the defect. Remarkably, the characteristic speed at which the fringes disappear is of the order of the equilibrium sound speed c$_{s}$ = √ gn$_{c}$/m , but does not correspond to any noticeable feature in the real part of the Bogoliubov dispersion of Fig.1(a,c,e). As it was pointed out in a different context in [20], the emission pattern by a localized monochromatic source is better understood in terms of the (complex) wavevector of the emitted wave at the given (real) frequency. As one can see in Fig. 4 (b), the wavevector ˜ k of the Bogoliubov wave emitted at zero frequency by the static defect starts having a finite real part only after a branching point: extended oscillations in the density are observed as soon as the real part of ˜ k exceeds the imaginary part. This physical picture is confirmed by numerical simu- lations of the time evolution under the generalized GPE (1) starting from an initial condition with a supersonic flow. Periodic boundary conditions are assumed. For a low defect density, the supercurrent state is maintained for very long times with no sign of decay. The char- acteristic Cerenkov-like density patterns in the vicinity Supersonic flows in equilibrium condensates in ring- shaped geometries are generally strongly sensitive to the presence of defects: as soon as nodes appear in the con- densate wavefunction, the topological stability of the su- percurrent state is broken, which leads to a rapid slow down of the condensate motion. The finite damping rate of excitations in polariton condensates introduces a substantial modification to this picture: the perturba- tion created by each defect is not able to propagate on long distances, but rather remains localized in space on a length scale inversely proportional to Im[ ˜ k ]. As a re- sult, we expect it is much harder to break the topological stability of the supercurrents. This behavior is reflected in the drag force F exerted by the moving condensate onto the defect as a function of the condensate velocity [Fig.4(a)]. In terms of the perturbed density profile n ( r ), the drag force is given by F = − ∫ d 2 r n ( r ) ∇$_{r}$ V$_{def}$ ( r ) [21]. As a consequence of the finite lifetime of the Bogoliubov modes, the drag force has a non-vanishing value at all v . In addition, the drag force shows a marked threshold at a velocity value that closely corresponds to the onset of fringes in the density profile: the lower the value of the n$_{c}$r/γ parameter, the sharper the threshold. On the other hand, the role of Ω$_{K}$ on this effect is minor. Even though no experiment has so far investigated the superfluid properties of polariton condensates under non-resonant pumping, it is likely that this generalized form of the Landau criterion provides at least a partial explaination of the recent experimental observation of superfluidity in a OPO regime [10]. FIG. 3: Density perturbation created in a moving condensate by a stationary weak defect for three different values of the condensate velocity v/c$_{s}$ = 1 . 5 , 1 , 0 . 4 across the critical value for superfluidity. Parameters: n$_{c}$g/γ = n$_{c}$r/γ = 1, Ω$_{K}$ /γ = 50. FIG. 4: Panel (a): Force exerted on a weak stationary defect by a moving condensate as a function of the condensate veloc- ity for different values of rn$_{c}$/γ = 0 . 1 , 1 , 2 (blue solid, green dashed, red dotted lines), in units of F$_{γ}$ = ℏ $^{2}$k3 $_{γ}$/m . For com- parison, the same force on an equilibrium condensate is shown as a thin solid black line. Panel (b,c): Real and imaginary parts of the complex wavevector ˜ k of the Bogoliubov emission in the upstream direction as a function of the condensate ve- locity, for rn$_{c}$/γ = gn$_{c}$/γ = 1. The other parameters are the same as in Fig. 3.