Interferences between Bogoliubov excitations in superfluids of light

Paraxial fluids of light represent an alternative platform to atomic Bose-Einstein condensates and superfluid liquids for the study of the quantum behavior of collective excitations. A key step in this direction is the precise characterization of the Bogoliubov dispersion relation, as recently shown in two experiments. However, the predicted interferences between the phonon excitations that would be a clear signature of the collective superfluid behavior have not been observed to date. Here, by analytically, numerically, and experimentally exploring the phonon phase velocity, we observe the presence of interferences between counterpropagating Bogoliubov excitations and demonstrate their critical impact on the measurement of the dispersion relation. These results are evidence of a key signature of light superfluidity and provide a characterization tool for quantum simulations with photons.

Figure 1

Paraxial fluid of light. In the paraxial and scalar approximations, a laser beam propagates along the $z$ axis in a ${\chi }^{\left(3\right)}$ nonlinear medium according to the effective Gross-Pitaevskii equation (1). The field profile on each transverse ${\mathbf{r}}_{\perp }=(x,y)$ plane along the propagation direction $z$ is equivalent to a snapshot of the evolution of the paraxial fluid of light.

Figure 2

(a) Nonlinear phase shift ${\mathrm{\Phi }}_{\mathrm{NL}}\left({\mathbf{k}}_{\perp }\right)$ and (b) relative fringes displacement $\mathrm{\Delta }S\left(\mathrm{\Lambda }\right)$ as functions of, respectively, the Bogoliubov wave number $k_{\perp }=\left|{\mathbf{k}}_{\perp }\right|$ and the Bogoliubov wavelength $\mathrm{\Lambda }=2\pi /k_{\perp }$ for different values of the optical nonlinearity $\mathrm{\Delta }n$. (a) The phase shift ${\mathrm{\Phi }}_{\mathrm{NL}}\left({\mathbf{k}}_{\perp }\right)$ generally follows ${\mathrm{\Omega }}_{\mathrm{B}}\left({\mathbf{k}}_{\perp }\right)L$ on average (dashed lines), except for small $k_{\perp }$, where it saturates at a nonzero value for all values of $\mathrm{\Delta }n\ne 0$. According to (30), the $k_{\perp }=0$ limit ${\mathrm{\Phi }}_{\mathrm{NL}}({\mathbf{k}}_{\perp }=0)$ is a growing function of $\mathrm{\Delta }n$ and tends towards $\pi /2$ for large $\mathrm{\Delta }n$. (b) The staircase structure of ${\mathrm{\Phi }}_{\mathrm{NL}}\left({\mathbf{k}}_{\perp }\right)$ translates into oscillations in $\mathrm{\Delta }S\left(\mathrm{\Lambda }\right)$, most visible for large interactions. At long $\mathrm{\Lambda }$, $\mathrm{\Delta }S\left(\mathrm{\Lambda }\right)$ increases linearly according to (31). This trend is present even for weak interactions ($\mathrm{\Delta }n\to 0$). Solid lines correspond to the full model and the black dash-dotted line shows for comparison the displacement obtained from the geometric approach of (18) with $\mathrm{\Delta }n={10}^{-4}$.

Figure 3

(a) Experimental setup. A laser is shaped with two cylindrical lenses (CL). It is split and recombined (BS1) within an unbalanced Mach-Zehnder (MZ) interferometer to create a low contrast fringes pattern. Two sets of fringes (low and high intensity) are vertically shifted using a 90:10 beam-splitter (BS2) before going into an atomic vapor cell. The cell output is imaged on a camera after filtering in the Fourier space (FS). (b), (c) Background-subtracted images reveal the small amplitude density modulation which propagates on a low (b) and a high intensity background fluid (c). The blue and red points in (d) are obtained by integrating the intensity in between the white dashed lines in (b) and (c), respectively. We first filter out the high-frequency noise (dashed lines) and then normalize the envelopes (solid lines). The shift is computed by measuring the nearest peak-to-peak distance between the solid lines. (e) Fourier space image obtained by inserting a microscope objective (MO).

Figure 4

(a) Displacement $\mathrm{\Delta }S$ as a function of the modulation wavelength $\mathrm{\Lambda }$ for a fluid intensity of 1.3 W ${\mathrm{cm}}^{-2}$. The laser is 2.6-GHz red-detuned with respect to the $F=3\to F^{\prime }$ transition of the ${}^{85}\mathrm{Rb}$ $D_2$ line and the cell length is 7.5 cm. The experimental data (blue circle) are fitted with the full theory (blue line) for $\mathrm{\Delta }n=1.3\times {10}^{-6}$. For comparison, the displacement obtained using (18) has been plotted (black dash-dotted line). (b) Slope of the asymptotic linear increase of $\mathrm{\Delta }S$ at large $\mathrm{\Lambda }$ as a function of the fluid intensity. (c) $\mathrm{\Delta }n$ extracted with Eq. (30) from the slope of $\mathrm{\Delta }S$ at large $\mathrm{\Lambda }$ as a function of the fluid intensity plotted in (b). The linear scaling of $\mathrm{\Delta }n$ with $I$ confirms that we are not saturating the nonlinearity, i.e., $\mathrm{\Delta }n=n_{2}I$ with $n_2=1\times {10}^{-10}{\text{m}}^2$/W.

Figure 5

Numerical simulations (symbols) and analytical solutions (solid and dashed line) of $\mathrm{\Delta }S$ as a function of $\mathrm{\Lambda }$. (a) Different cell transmissions $t$. The simulations and the theory model are similar as long as the transmission remains large ($t>0.5$). (b) Different background widths $w_{0,y}$. (c) Different nonlocal transport length scales $l_b$. The oscillations are smoothed by nonlocality. In our system $l_b<10\mu \mathrm{m}$. (d) Different saturation intensities ${\mathcal{I}}_s$ of the nonlinear Kerr interaction. Once again oscillations are smoothed by a saturation of the medium. For all the simulations $\mathrm{\Delta }n=1.0\times {10}^{-5}$.

Figure 6

Evolution along the $z$ axis of the transverse field intensity in a given $y$ plane for (a) $k_{\perp }\xi =0.5$ and (b) $k_{\perp }\xi =1$. The background intensity is subtracted on both images. The black dashed curves follow the center of a bright fringe. The blue solid line is a trajectory of a Bogoliubov mode at the speed of sound. (c) Visibility of the interference fringes at the output plane $z=L$ as a function of $k_{\perp }$ (solid blue line). Visibility maxima are shown by green diamonds and minima are shown by black circles. The dispersion relation (solid red line, right axis) is reconstructed using a sampling based on the position of the maxima $[{\mathrm{\Omega }}_{\mathrm{B}}\left({\mathbf{k}}_{\perp }\right)=p\pi /L]$ and minima $[{\mathrm{\Omega }}_{\mathrm{B}}\left({\mathbf{k}}_{\perp }\right)=(p+1/2)\pi /L]$. Here $\mathrm{\Delta }n=1.0\times {10}^{-5}$.