Observation of the Bogoliubov Dispersion in a Fluid of Light

Quantum fluids of light are a photonic counterpart to atomic Bose gases and are attracting increasing interest for probing many-body physics quantum phenomena such as superfluidity. Two different configurations are commonly used: the confined geometry where a nonlinear material is fixed inside an optical cavity and the propagating geometry where the propagation direction plays the role of an effective time for the system. The observation of the dispersion relation for elementary excitations in a photon fluid has proved to be a difficult task in both configurations with few experimental realizations. Here, we propose and implement a general method for measuring the excitations spectrum in a fluid of light, based on a group velocity measurement. We observe a Bogoliubov-like dispersion with a speed of sound scaling as the square root of the fluid density. This Letter demonstrates that a nonlinear system based on an atomic vapor pumped near resonance is a versatile and highly tunable platform to study quantum fluids of light.

Figure 1

Experimental setup: polarized beam splitter (PBS) and half wave plate (HWP). $\theta$ is the angle between the probe (orange beam) and the optical axis defined by the pump (red beam). The probe interferes with the pump and slightly modulates its intensity. (Blue inset) Integrated intensity profile at the input of the medium ($z=0$). The wavelength $\mathrm{\Lambda }$ of the density modulation is given by $2\pi /k_{\perp }$, where $k_{\perp }=k_0\sin \theta$. (Orange inset) Integrated intensity profile at the output of the medium ($z=L$). The distance $D$ between the two wave packets gives access to the group velocity of the elementary excitations in the transverse plane. The output plane is imaged on a CMOS camera. (Inset, top left) Background-subtracted image obtained for $\theta \approx 0\mathrm{rad}$ and associated integrated envelope profile (blue: original; red dotted: high frequency filtered).

Figure 2

(a) Simulation of the propagation of a weak intensity modulation through a nonlinear medium, with zero transverse speed ($\theta =0\mathrm{rad}$). The modulation generates two counterpropagating Bogoliubov modes at the medium interface that get amplified until they separate from each other. The wave packet is not spreading along propagation due to the nondispersive regime (soundlike behavior). (b) Same as (a), for an incident probe at $\theta =5\times {10}^{-3}\mathrm{rad}$ (high transverse speed). Interference fringes appear and the wave packet spreads. (c) Simulation of the intensity profile envelope in the output plane for different probe wave vector. (Dashed black lines) Group velocity given by Eq. (4). (d) Experimental data. The upper wave packet amplitude decreases with $k_{\perp }$, as the efficiency of the degenerate four-wave mixing processes depends on the phase matching conditions, which is optimal for $k_{\perp }=0$. The parameters in (a)–(c) are those used experimentally for (d): ${\lambda }_0=780\mathrm{nm}$, $\mathrm{\Delta }n=1.3\times {10}^{-5}$, and ${\omega }_x^p=180\mu \mathrm{m}$ ($\alpha =0$ in numerical simulations).

Figure 3

(a) Group velocity as a function of $k_{\perp }$. The circles represent the experimental data obtained for $P=175\mathrm{mW}$. The theoretical model is plotted in black (no free parameter—see text for details). The dashed lines figure the asymptotic behaviors: constant group velocity at small $k_{\perp }$ and linear increase at large $k_{\perp }$. (b) Dispersion relation obtained after integration of the group velocity. Linear (blue) and parabolic (red) dispersion curves are plotted as a reference.

Figure 4

Speed of sound $c_s$ as a function of the pump intensity. Because of the $2\mathrm{D}+1$ geometry, the speed of sound has the dimension of an angle. Data are plotted in blue dots. The light intensity corresponds to the fluid density; therefore, a scaling as a square root is expected as plotted in black solid. No free parameters are needed, as the nonlinearity has been measured independently. Uncertainty area in light blue is extracted from this independent measurement.