Measurement of the Static Structure Factor in a Paraxial Fluid of Light Using Bragg-like Spectroscopy

We implement Bragg-like spectroscopy in a paraxial fluid of light by imprinting analogues of short Bragg pulses on the photon fluid using wavefront shaping with a spatial light modulator. We report a measurement of the static structure factor, $S(k)$, and we find a quantitative agreement with the prediction of the Feynman relation revealing indirectly the presence of pair-correlated particles in the fluid. Finally, we improve the resolution over previous methods and obtain the dispersion relation including a linear phononic regime for weakly interacting photons and low sound velocity.

Figure 1

Principle of the experiment and simplified setup. A 780 nm laser beam is elongated in the $x$ direction and impinges on a spatial light modulator (SLM). The SLM displays a vertical grating, which imprints a sinusoidal phase modulation of wave vector $k_x$. The SLM plane is imaged at the input of the 6.7 cm rubidium vapor cell. This phase modulation creates two counterpropagating left ($L$) and right ($R$) phonons at $+k_x$ and $-k_x$ represented in green and orange. Initially in phase opposition (constant input density), the phonons constructively interfere after some effective time $\tau =z/c$, giving a maximum of density contrast. The output plane of the nonlinear medium (a rubidium vapor cell) is imaged on a camera to study the fringe contrast at $z=L$.

Figure 2

(a),(b) The simulated density modulation for medium of 7.5 cm and a transverse wave vector of $k_x=36.0{\mathrm{mm}}^{-1}$, with $\mathrm{\Delta }n=0$ (a) and $\mathrm{\Delta }n=2.1\times {10}^{-5}$ (b). (c),(d) Experimental images of the density obtained respectively for $k_x=27.8{\mathrm{mm}}^{-1}$ (minimum of contrast) and $k_x=43.2{\mathrm{mm}}^{-1}$ (maximum of contrast) in the noninteracting case ($\mathrm{\Delta }n=0$). These images are taken with a large modulation depth for illustration. (e) An inset of (d).

Figure 3

Contrast for the off-resonance case (grey) with $\mathrm{\Delta }n=0$ and for a close-to-resonance set (red) with an input power of 90 mW, a cell temperature of 106 °C, and an absorption of 40%. For the weakly interacting case, the maximum contrast in the density standing wave is 0.25, which is within the Bogoliubov approximation. The shift of the minima of contrast toward the smaller $k_x$ for the $-2.5\mathrm{GHz}$ set, indicated by the black arrow, is evidence of the nonlinear effect taking place.

Figure 4

Static structure factor measurement. The experimental conditions are the same as the red curve of Fig. 3. The dashed line at $S(k_x)=1$ is the structure factor of a noninteracting gas (coherent state). The solid red line is the Feynman model given by Eq. (7) with the parameter $\mathrm{\Delta }n=4.6\times {10}^{-6}$ extracted from the measurement of the dispersion relation shown in the inset. Inset: dispersion relation for the noninteracting case (black) and for the weakly interacting case (red). Solid lines are fits using Eq. (3).