Optomechanical signature of a frictionless flow of superfluid light

We propose an experimental setup that should make it possible to reveal the frictionless flow of a superfluid of light from the suppression of the drag force that it exerts on a material obstacle. In the paraxial-propagation geometry considered here, the photon-fluid dynamics is described by a wave equation analogous to the Gross-Pitaevskii equation of dilute Bose-Einstein condensates and the obstacle consists in a solid dielectric slab immersed into a nonlinear optical liquid. By means of an ab initio calculation of the electromagnetic force experienced by the obstacle, we anticipate that superfluidity is detectable in state-of-the-art experiments from the disappearance of the optomechanical deformation of the obstacle.

Figure 1

Sketch of the considered experimental setup, viewed from above.

Figure 2

One-dimensional plate geometry. Upper left panel: Domain of existence of the stationary ($z$-independent) solutions in the $(V_0/s_{\infty },M_{\infty }=v_{\infty }/s_{\infty })$ plane (gray-shaded area); the red tags indicate the parameters used for plotting the intensity patterns (a)–(d) in the upper right of the figure (the vertical dashed line indicates the position of the $\delta$-peak obstacle). Lower panels: Nonmonotonic behavior—at a fixed asymptotic photon-fluid velocity $v_{\infty }=k_{\infty }/\beta$—of the radiation pressure (17) and (18) as a function of the far-downstream intensity ${\mathcal{I}}_{\infty }$, normalized to the critical Landau intensity ${\mathcal{I}}_{\mathrm{crit}}=\beta v_{\infty }^2/g=k_{\infty }^2/\left(\beta g\right)$, in the two regimes $V_0\lessgtr 0$ (${\chi }_s\gtrless {\chi }_{\ell }$); in each case, the white region indicates the window in which Eq. (6) does not admit a stable stationary solution.

Figure 3

One-dimensional plate geometry. Snapshots of the normalized intensity profile $\mathcal{I}(x,z)/{\mathcal{I}}_0$ at different propagation distances $z$, from the incident spot [panels (a.1) and (b.1)] towards the stationary state [panels (a.3) and (b.3)] showing a (nonlinear) Bogoliubov-Cherenkov modulation [column (a), $M_0=v_0/s_0=2.2]$ or a superfluid behavior [column (b), $M_0=0.5]$. The vertical dashed line at the origin ($x=0$) indicates the position of the $\delta$-peak obstacle. The propagation distance $z$ is such that $k_0^{2}z/\beta =0$ [panels (a.1) and (b.1)], 25 [panels (a.2) and (b.2)], and 125 [panels (a.3) and (b.3)].

Figure 4

Two-dimensional rod geometry. Panels (a)–(c): Stationary (that is, $z$-independent) light-intensity profiles $\mathcal{I}(x,y)$ (normalized to the incident intensity ${\mathcal{I}}_0$) in the deeply nonsuperfluid regime [(a), $M_0=2.98]$, in the vortex-nucleation regime [(b), $M_0=0.75]$, and in the superfluid regime [(c), $M_0=0.25]$; the transverse cross section of the rod-shaped obstacle is indicated by the dashed rectangle; propagation distance: $z/\lambda =9\times {10}^4$ ($\lambda =2\pi /\beta$ is the wavelength of the laser in the liquid). Panel (d): Behavior—at a fixed far-upstream velocity $v_0=k_0/\beta$—of the electromagnetic force ${\mathcal{F}}_{\mathrm{tot}}$ (defined in the first paragraph of Sec. 4b) as a function of the input intensity ${\mathcal{I}}_0$ normalized to the critical Landau intensity ${\mathcal{I}}_{\mathrm{crit}}=\beta v_0^2/g=k_0^2/\left(\beta g\right)$ (as $v_0,k_0=v_{\infty },k_{\infty }$ in the two-dimensional rod configuration, this corresponds to the same normalization as the one used in the lowest panels of Fig. 2); the points (vertical error bars) correspond to the average (standard deviation) of the force over the propagation-distance window $z/\lambda \in [4.5\times {10}^4,9\times {10}^4]$ and the blue straight line indicates the linear behavior of the radiation force in the low-${\mathcal{I}}_0$ regime. Obstacle's amplitude: $({\chi }_s-{\chi }_{\ell })/(1+{\chi }_{\ell })=5\times {10}^{-5}$; obstacle's size: $L_x/\lambda =40, L_y/\lambda =200$.