Superfluidity of a nonequilibrium Bose-Einstein condensate of polaritons

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.

Figure 1

(Color online) The left column show the first-order spatial coherence between inversion symmetric points $g^{\left(1\right)}\left(x\right)=⟨{\psi }^{†}\left(-\mathbf{x}\right)\psi \left(\mathbf{x}\right)⟩/\sqrt{n\left(\mathbf{x}\right)n\left(-\mathbf{x}\right)}$ for values of the density increasing from top to bottom ($n=1.3$, 5.7, and $32\mu {\text{m}}^{-2}$, respectively), before the arrival of the Gauss-Laguerre pulse; the insets show the values at the latest simulated times. The right column shows the temporal evolution of the total polariton density (thin dashed line) and angular momentum (thick full line) after a 1 ps Gauss-Laguerre probe pulse with a waist of $20\mu \text{m}$ has perturbed the polariton condensates. The additional density introduced by the resonant probe pulse is, from top to bottom, four, forty, and twenty times the condensate density. The thin line shows the observable $\Lambda$ that we propose to use for a time-resolved measurement of the vorticity. The thick dashed line shows the probability that the vortex core is still inside the condensate region. The inset in the lower right figure shows the response of the condensate to a probe pulse that is too weak to create a vortex. The polariton condensate is created with a ring-shaped excitation laser (inner radius $7\mu \text{m}$ and outer radius $20\mu \text{m}$).

Figure 2

(Color online) The same as Fig. 1, but for a condensate excited with a top-hat excitation laser with a radius of $20\mu \text{m}$. The density in the center is $n=48\mu {\text{m}}^{-2}$. The inset in the left-hand panel shows the spatial coherence at $t=100\text{ps}$. The pump intensity is the same as in the lower panels of Fig. 1.

Figure 3

(Color online) Representative vortex trajectories (left-hand panels) for the top hat condensate from Fig. 2 on top of the polariton density (in gray scale). The right-hand panels show the angular momentum of the corresponding Monte Carlo simulation. The vertical line shows the time when the vortex leaves the condensate. Typically, the angular momentum is very close to zero when the vortex leaves.

Figure 4

(Color online) The same as in Fig. 3, but under excitation with a ring-shaped laser spot.

Figure 5

(Color online) Interferogram as a function of the phase difference $\alpha$ and rotation angle $\theta$ between the copies of the condensate that are interfered, before ($t=0\text{ps}$, left-hand panel) and after ($t=250\text{ps}$, right-hand panel) the arrival of the Gauss-Laguerre pulse. The black lines show the contour lines $I\left(\alpha ,\theta \right)=1$ and the white line shows $\alpha =\pi /2+⟨{\widehat{L}}_z⟩\theta$, where $⟨{\widehat{L}}_z⟩$ is taken from Fig. 1. The presence of angular momentum is signaled by the tilting of the interference fringes in the $\left(\alpha ,\theta \right)$ plane.

Figure 6

(Color online) Sketch of the scheme to detect the extra velocity in the polariton condensate due to the presence of a vortex. The momentum distribution from a slice selected in real space is shifted in the presence of a vortex (full line) with respect to the momentum distribution in its absence (dashed line).