Role of geometry in the superfluid flow of nonlocal photon fluids

Recent work has unveiled a new class of optical systems that can exhibit the characteristic features of superfluidity. One such system relies on the repulsive photon-photon interaction that is mediated by a thermal optical nonlinearity and is therefore inherently nonlocal due to thermal diffusion. Here we investigate how such a nonlocal interaction, which at a first inspection would not be expected to lead to superfluid behavior, may be tailored by acting upon the geometry of the photon fluid itself. Our models and measurements show that restricting the laser profile and hence the photon fluid to a strongly elliptical geometry modifies thermal diffusion along the major beam axis and reduces the effective nonlocal interaction length by two orders of magnitude. This in turn enables the system to display a characteristic trait of superfluid flow: the nucleation of quantized vortices in the flow past an extended physical obstacle. These results are general and apply to other nonlocal fluids, such as dipolar Bose-Einstein condensates, and show that “thermal” photon superfluids provide an exciting and novel experimental environment for probing the nature of superfluidity, with applications to the study of quantum turbulence and analog gravity.

Figure 1

Experimental setup. A 532-nm laser beam with an elliptical spatial beam profile is launched into a sample filled with a methanol-graphene solution. A knife blade is mounted inside the sample and slightly tilted with respect to the propagation direction at an angle $\theta$ that determines the flow of the photon fluid [inset (a)]. Inset (b) shows the measured nonlocal dispersion relation of the photon fluid from Ref. [13]. Solid red line: Bogoliubov dispersion relation, Eq. (4) with a local nonlinearity $\left|\mathrm{\Delta }n\right|=7.6\times {10}^{-6}$. Solid blue line: nonlocal dispersion relation with ${\widehat{R}}_{1D}\left(K_x\right)={[1+{\overline{\sigma }}^2{K}_x^2]}^{-1}$ and $\overline{\sigma }=110\mu \mathrm{m}$ and $\left|\mathrm{\Delta }n\right|=7.6\times {10}^{-6}$ determined from the best fit to the experimental data (black squares).

Figure 2

Cross section of the real-space, thermally induced refractive index variation, $\left|\mathrm{\Delta }n\right(x,y=0\left)\right|$. Red curve: experimental measurement using a methanol-graphene medium with $L=12\mathrm{cm}, W=1\mathrm{cm}, \alpha \sim 0.013{\mathrm{cm}}^{-1}$, similar to that used in the experiments in Fig. 3. Dashed blue curve: the numerical solution to Eq. (6) with a physical circular boundary at $W=1\mathrm{cm}$. Black curve: the analytical solution from the DLM model with $\sigma =W=1\mathrm{cm}$. All curves are normalized to 1.

Figure 3

Experimental measurements of the near-field beam profile at the sample output $I(x,y)/I_0$, normalized by input beam intensity $I_0$, showing superfluid instability at high flow speed $v_f\sim 1.3\times {10}^6$ m/s: (a) $I_0=4.0{\mathrm{W}/\mathrm{cm}}^2$, i.e., $\mathrm{\Delta }n=-3.0\times {10}^{-5}$ and $v_f/c_s\sim 1.2$, and (b) $I_0=6.8{\mathrm{W}/\mathrm{cm}}^2$, i.e., $\mathrm{\Delta }n=-5.2\times {10}^{-5}$ and $v_f/c_s\sim 0.9$. White circles indicate the position of the vortex singularities obtained from the corresponding phase diagrams shown in (c) and (d), respectively.

Figure 4

Numerical simulations. Circular input pump beam with 1-mm $1/e^2$ beam radius: (a) intensity profile at the output of the nonlinear medium, zoomed in around the obstacle tip. The inset shows the full beam profile. (b) Phase profile for the circular beam (wrapped between $-\pi$ and $+\pi$). The smoothness and absence of singularities indicate a complete absence of vortex nucleation around the obstacle. Elliptical input pump beam ($w_x=1$ mm, $w_y=0.15$ mm): (c) intensity profile with a white circle indicating the position of a vortex nucleated from the obstacle and (d) the phase profile. A clear phase singularity can be observed, confirming the presence of the vortex. Simulations for both round and elliptical input beam profiles are performed under identical conditions, aside from the input beam profile, with input intensity $I_0=2.65{\mathrm{W}/\mathrm{cm}}^2$, propagation length $L=18\mathrm{cm}$, fluid flow speed $v_f=1.0\times {10}^6$ m/s, and nonlocal interaction length $\sigma =1\mathrm{cm}$.

Figure 5

Numerical results. (a) Plot of vortex core $1/e^2$ radii, $a$ and $b$, normalized to the local healing length $\xi$, versus $\sigma$. The inset, corresponding to the white region in (e), shows a vortex core with labeled major axis $a$ and minor axis $b$. (b) normalized beam intensity $I(x,y)/I_0$, (c) phase $\varphi (x,y)$, and (d) nonlinear potential $\mathrm{\Delta }n(x,y)/\mathrm{\Delta }n$ for a purely local superfluid after 20 cm of propagation, with $v_f/c_s=0.56, \mathrm{\Delta }n=-1.2\times {10}^{-5}$, and $1/e^2$ beam radii $w_{\mathrm{x}}=w_{\mathrm{y}}=1$ mm. (e), (f), and (g) show the corresponding profiles for a nonlocal superfluid with $\sigma =110\mu \mathrm{m}$ and $\mathrm{\Delta }n=-2.4\times {10}^{-5}$.

Figure 6

Bogoliubov dispersion in a fluid at rest: (a) and (b) are for a local fluid $\sigma /\epsilon =0$ and (c) and (d) are for a nonlocal fluid with a value of $\sigma /\xi =4.76$ inspired by the experiment. The dotted curve in the right panels repeats the local fluid dispersion for comparison. The red dashed line is the straight line $\mathrm{\Omega }=v_{c}K$ corresponding to the critical speed $v_c$. The panels in the bottom row show the same curves in log-log scale.

Figure 7

Plots of the dependence of $v_c$ (a) on the nonlocal length $\sigma$ and (b) on the density $\rho$. In (b), the density is normalized to ${\rho }_{\sigma }={\hslash }^2/mg{\sigma }^2$, and the velocity is normalized to $v_{\sigma }=\hslash /m\sigma$. The vertical dot-dashed lines indicate the transition points between the two $\sigma \lessgtr \xi /2$ regimes. The vertical dashed line in (a) indicates the experimental conditions. The dotted line in the bottom panel shows how $\sqrt{\rho }$ behavior would extend to the whole domain.