Abstract
Quantum fluids of light merge manybody physics and nonlinear optics, revealing quantum hydrodynamic features of light when it propagates in nonlinear media. One of the most outstanding evidence of light behaving as an interacting fluid is its ability to carry itself as a superfluid. Here, we report a direct experimental detection of the transition to superfluidity in the flow of a fluid of light past an obstacle in a bulk nonlinear crystal. In this cavityless alloptical system, we extract a direct optical analog of the drag force exerted by the fluid of light and measure the associated displacement of the obstacle. Both quantities drop to zero in the superfluid regime characterized by a suppression of longrange radiation from the obstacle. The experimental capability to shape both the flow and the potential landscape paves the way for simulation of quantum transport in complex systems.
Introduction
Superfluidity was originally discovered in 1938^{1} when a ^{4}He fluid cooled below a critical temperature flowed in a nonclassical way along a capillary^{2}. This was the trigger for the development of many experiments genuinely realized with quantum matter, as with ^{3}He fluids^{3} or ultracold atomic vapors^{4,5}. The superfluid behavior of mixed lightmatter cavity gases of excitonpolaritons was also extensively studied^{6,7}, leading to the emergent field of “quantum fluids of light”^{8}. Before being theoretically developed for cavity lasers^{9,10}, the idea of a superfluid motion of light originates from pioneering studies in cavityless alloptical configurations^{11} in which the hydrodynamic nucleation of quantized vortices past an obstacle when a laser beam propagates in a bulk nonlinear medium was investigated^{12}. In such a cavityless geometry, the paraxial propagation of a monochromatic optical field in a nonlinear medium may be mapped onto a twodimensional GrossPitaevskiitype evolution of a quantum fluid of interacting photons in the plane transverse to the propagation^{4}. The intensity, the gradient of the phase and the propagation constant of the optical field assume respectively the roles of the density, the velocity, and the mass of the quantum fluid. The photon–photon interactions are mediated by the optical nonlinearity. It took almost twenty years for this idea to spring up again^{13,14,15,16}, driven by the emergence of advanced laserbeamshaping technologies allowing to precisely tailor both the shape of the flow and the potential landscape.
The ways of tracking light superfluidity are manifold. Recently, superfluid hydrodynamics of a fluid of light has been studied in a nonlocal nonlinear liquid through the measurement of the dispersion relation of its elementary excitations^{17} and the detection of a vortex nucleation in the wake of an obstacle^{18}. The stimulated emission of dispersive shock waves in nonlinear optics was also studied in the context of light superfluidity^{13}. However, one of the most striking manifestations of superfluidity—which is the ability of a fluid to move without friction^{19}—has never been directly observed in a cavityless nonlinearoptics platform. A direct consequence of this feature is the absence of longrange radiation in a slow fluid flow past a localized obstacle. In optical terms, this corresponds to the absence of light diffraction from a local modification of the underlying refractive index in the plane transverse to the propagation. On the contrary, in the “frictional”, nonsuperfluid regime, light becomes sensitive to such an index modification and diffracts while hitting it.
Here, we report a direct observation of a superfluid regime characterized by the absence of longrange radiation from the obstacle. This regime is usually associated with the cancellation of the drag force experienced by the obstacle, as studied for ^{4}He^{20}, ultracold atomic gases^{21,22,23,24,25}, or cavity excitonpolaritons^{26,27,28,29}. In our cavityless alloptical system, we extract on the one hand a quantity corresponding to the optical analog of this force and measure on the other hand the associated obstacle displacement. For the first time, at least within the framework of fluids of light, we observe that this displacement is nonzero in the nonsuperfluid case and tends to vanish while reaching the superfluid regime.
Results
Hydrodynamics of light
We make use of a biased photorefractive crystal which is, thanks to its controllable nonlinear optical response, convenient for probing the hydrodynamic behavior of light^{13,30,31,32}. As sketched in Fig. 1a and detailed in Fig. 1c, a local drop of the optical index is photoinduced by a narrow beam in the crystal and creates the obstacle. Simultaneously, a second, larger monochromatic beam is sent into the crystal and creates the fluid of light. The propagation of the fluidoflight beam in the paraxial approximation is ruled by a twodimensional GrossPitaevskiitype equation (also known as a nonlinear Schrödingertype equation):
The propagation coordinate z plays the role of time. The transverseplane coordinates r = (x, y) span the twodimensional space in which the fluid of light evolves. The propagation constant n_{e}k_{f} = n_{e} × 2π/λ_{f} of the fluidoflight beam propagating in the crystal of refractive index n_{e} is equivalent to a mass; the associated Laplacian term describes light diffraction in the transverse plane. The density of the fluid is given by the intensity \(I_{\mathrm{f}} \propto \left {E_{\mathrm{f}}} \right^2\). Its velocity corresponds to the gradient of the phase of the optical field. At the input, it is simply given by \(v \simeq \theta _{{\mathrm{in}}}{\mathrm{/}}n_{\mathrm{e}}\), with θ_{in} the angle between the fluidoflight beam and the z axis (see Supplementary Note 1 for more details). The local refractive index depletion \({\mathrm{\Delta }}n\left[ {I_{{\mathrm{ob}}}({\bf{r}})} \right] < 0\) is induced by the obstacle beam of intensity I_{ob}(r). The selfdefocusing nonlinear contribution Δn(I_{f}) < 0 to the total refractive index provides repulsive photon–photon interactions and ensures robustness against modulational instabilities^{33}. From the latter, we define an analog healing length ξ = \(\left[ {n_{\mathrm{e}}k_{\mathrm{f}} \times k_{\mathrm{f}}\left {{\mathrm{\Delta }}n(I_{\mathrm{f}})} \right} \right]^{  1/2}\), which corresponds to the smallest length scale for intensity modulations, and an analog sound velocity c_{s} = (n_{e}k_{f} × ξ)^{−1} = \(\left[ {\left {{\mathrm{\Delta }}n(I_{\mathrm{f}})} \right{\mathrm{/}}n_{\mathrm{e}}} \right]^{1/2}\) for the fluid of light^{4,16} (see Supplementary Note 1). The photorefractive nonlinear response of the material, Δn(I), is plotted in blue in Fig. 1b as a function of the laser intensity I (see the Methods section for details). In the same figure, the red dashed curve represents the speed of sound c_{s}(I).
When the obstacle is infinitely weakly perturbing, Landau’s criterion for superfluidity^{19} applies and the socalled Mach number v/c_{s} mediates the transition around v/c_{s} = 1 from a nonsuperfluid regime at large v/c_{s} to a superfluid regime at low v/c_{s}. Generally this condition is not fulfilled and the actual critical velocity is lower than the sound velocity c_{s}^{4,34}. This is the case in the present work for two main reasons. First, we consider a weakly but finite perturbing obstacle. It means a small variation of the refractive index Δn[I_{ob}(r)] = −2.2 × 10^{−4} and a radius of 6 μm comparable to ξ (see Methods section and Supplementary Note 2). Note however that the perturbation is weak enough for the transition not to be blurred by the emission of nonlinear excitations like vortices or solitons. Second, remaining within Landau’s picture, the speed of sound is here defined for I_{f} measured at its maximum value, at z = 0, whereas the latter naturally suffers from linear absorption and selfdefocusing along the z axis.
Probing the transition to superfluidity
The ratio v/c_{s} is controlled in the experiment both by the incidence angle θ_{in} and the input intensity I_{f} of the fluidoflight beam. Figure 2 presents typical experimental results for the spatial distribution of the light intensity observed at the output of the crystal for various input conditions. Figure 2a displays the output spatial distributions of intensity for different fluid velocities v at a fixed speed of sound, c_{s} = 3.2 × 10^{−3}. This allows to vary v/c_{s} from 0 to 3.1. As v increases, diffraction appears in the transverse plane, and progressively manifests as a characteristic cone of fringes upstream from the obstacle^{14,16,35}. Another way to probe the transition is to fix the transverse velocity v and to vary the sound velocity c_{s} by changing the intensity of the fluidoflight beam. Although the two ways of varying v/c_{s} are not equivalent, as we shall discuss later, the results shown in Fig. 2b are similar with the interference pattern becoming more and more pronounced as v/c_{s} increases. Figure 2c represents the intensity distribution at the output of the crystal for v/c_{s} = 0.4. Longrange radiation upstream from the obstacle is no longer present in this case, indicating a superfluid motion of light. The lack of uniformity of the intensity upstream from the obstacle is due to the intrinsic linear absorption of the material^{29}.
Dragforce and obstacle displacement
In the supersonic regime, the intensity modulation of the fluid of light flowing around the obstacle induces a local opticalindex modification of the material. This modification influences the propagation of the beam responsible for the obstacle, for which a transverse displacement is expected. On the contrary, in the superfluid regime, the absence of longrange intensity perturbations implies no local variation of the optical index and then one does not await for any displacement of the obstacle beam.
As theoretically investigated in^{36} for a material obstacle (here, we rather consider an alloptical obstacle), the local intensity difference for the fluid of light between the front (I_{+}) and the back (I_{−}) of the obstacle, I_{+} − I_{−}, is proportional to the dielectric force experienced by the obstacle. This force turns out to be closely analogous to the drag force that an atomic BoseEinstein condensate exerts onto some obstacle. Figure 3a–e depicts the variation of I_{+} − I_{−}, measured at the output of the crystal, as a function of v/c_{s} for various initial conditions. As illustrated in the inset of Fig. 3e, both intensities are integrated over a typical distance of the order of ξ surrounding the obstacle. For all intensities, we observe a rather smooth, but net transition for v slightly smaller than c_{s}. The increasing tendency for low Mach numbers is associated to linear absorption, as discussed in the context of cavity quantum fluids of light^{26,27,29}. The wellknown decreasing tendency at large Mach numbers is also observed. Indeed, the obstacle can always be treated as a perturbation at large velocities and the associated drag force resultingly decreases^{37}. As the intensities increase, one can see that the local intensity difference sticks to zero for nonzero values of v/c_{s}, as predicted for the drag fore in a superfluid regime. Moreover, Fig. 3a–e shows that the curves with different intensities I_{f}, although renormalized by the respective sound velocity c_{ s }, do not fall on a single universal curve. This is due to the fact that changing the intensity also affects crucial quantities like the healing length ξ and the relative strength of the obstacle with respect to the nonlinear term, Δn(I_{ob})/Δn(I_{f}). While the drop of this force is among the main signatures of superfluidity in material fluids, so far this is the first experiment on fluids of light investigating it.
To go one step further, we probe the corresponding transverse displacement of the obstacle, independently on the measurement of I_{+} − I_{−}. By assuming that the transverse component of the fluidoflight beam is nonzero only along the x axis, we denote by \(\left\langle x \right\rangle = {\int} {x\left {E_{{\mathrm{ob}}}} \right^2{\mathrm{d}}x}\) the position of the centroid of the obstacle beam. Using an optical equivalent of the Ehrenfest relations, one can derive the following equation of motion (see Supplementary Note 3 for full derivation):
This means that the alloptical obstacle is sensitive to the surrounding refractive index potential that mainly results from the spatial distribution of intensity of the beam creating the fluid of light. It thus might move of a distance \(d = \left\langle x \right\rangle  x_0\) from its initial position x_{0} in the transverse plane. The measurement of d for various conditions in the case of an obstacle evolving in a fluid of light at rest allows to validate such an experimental approach and to extract experimental parameters as I_{sat} and Δn_{max} (see Methods section and Supplementary Note 3).
Figure 3f–j shows the transverse displacement measured in a moving fluid of light varying the Mach number v/c_{s} for different initial conditions. To take into account the gaussian shape of I_{f}, we subtract, for each data point, the displacement measured when the influence of the obstacle on the fluid of light is negligible (i.e., very low I_{ob}), as illustrated in Supplementary Note 3. The white points in Fig. 3f correspond to the displacement along the y axis and is expected to be zero. The grey boxes thus define the typical uncertainty in the measured quantities. The fluctuation are attributed to the inherent imperfections of the fluidoflight beam. We observe that the transverse displacement of the obstacle behaves very similarly to the intensity difference I_{+} − I_{−} displayed in Fig. 3a–e. That is, an increasing displacement from almost zero in the deeply subsonic regime to maximum signal, and then a decreasing tendency in the supersonic regime. We also measured an opposite transverse displacement for negative v/c_{ s }. Note that in this case, due to cavity effects, large interference patterns blurred the signal and did not allow to perform quantitative analysis (see Supplementary Note 4). The fact that the displacement is not purely zero in the superfluid regime is likely due to the displacement acquired during the nonstationary regime at early stage of the propagation (see Supplementary Note 5 for qualitative discussion supported by numerical simulations). This is, to the best of our knowledge, the first observation of the displacement of an alloptical obstacle in a fluid of light.
Discussion
We reported a direct experimental observation of the transition from a “frictional” to a superfluid regime in a cavityless alloptical propagating geometry. We performed a quantitative study by extracting an optical equivalent of the drag force that the fluid of light exerts on the obstacle. This result is in very good agreement with an independent measurement that consists in studying the transverse displacement of the obstacle surrounded by the fluid of light. We restricted the present study to the case of a weakly perturbing obstacle but our experimental setup allows to reach the turbulent regime associated to vortex generation through the induction of a greater opticalindex depletion. On the other hand, a different shaping of the beam creating the obstacle will allow to generate any kind of optical potential and to extend the study to imaging through disordered environments.
Methods
Experimental setup
The nonlinear medium consists in a 5 × 5 × 10 mm^{3} strontium barium niobate (SBN:61) photorefractive crystal additionally doped with cerium (0.01%) to enhance its photoconductivity^{38} albeit it induces linear absorption (3.2 dB/cm). The basic mechanism of the photorefractive effect remains in the photogeneration and displacement of mobile charge carriers driven by an external electric field E_{0}. The induced permanent spacecharge electric field thus implies a modulation of the refractive index of the crystal^{39}, Δn(I, r) = \( 0.5n_{\mathrm{e}}^3r_{33}E_0{\mathrm{/}}\left[ {1 + I\left( {\bf{r}} \right){\mathrm{/}}I_{{\mathrm{sat}}}} \right]\), where n_{e} is the optical refractive index and r_{33} the electrooptic coefficient of the material along the extraordinary axis, I(r) is the intensity of the optical beam in the transverse plane r(x, y), and I_{sat} is the saturation intensity which can be adjusted with a white light illumination of the crystal. The blue curve in Fig. 1b shows the saturable nonlinear response of the material Δn(I) against the laser intensity I. The red dashed curve represents the sound velocity c_{s}(I) for the saturable nonlinear response of the material Δn(I). The maximum value of the optical index variation is theoretically Δn_{max} = −2.32 × 10^{−4} for E_{0} = 1.5 kV cm^{−1}.
Shaping the fluid of light and obstacle beams
Making use of a spatial light modulator, we produce a diffractionfree Bessel beam (λ_{ob} = 532 nm, I_{ob} = 7.6 W cm^{−2} \(\gg I_{{\mathrm{sat}}}\), green path in Fig. 1c). The latter creates the obstacle with a radius of 6 μm (comparable to ξ = 6.2 μm obtained for I_{f} = 349 mW cm^{−2}) that is constant all along the crystal and aligned with the zdirection. From Fig. 1b, the propagation of the obstacle beam into the crystal induces a local drop Δn(I_{ob}) = −2.2 × 10^{−4} in the refractive index. A second laser (λ_{f} = 633 nm, red path in Fig. 1c) delivers a gaussian beam whose radius is extended to 270 μm and which corresponds to the fluidoflight beam. Both laser beams are linearlypolarized along the extraordinary axis to maximize the photorefractive effect. We vary the flow velocity v by changing the input angle θ_{in} of the fluidoflight beam with respect to the propagation axis z (see Fig. 1a). The accessible range, tuned by rotating a mirror imaged at the input of the crystal via a telescope, goes from θ_{in} = 0 to ±23 mrad, corresponding to v ranging from v = 0 to v = ±1.3 × 10^{−2}. The sound velocity c_{s} is controlled by the input intensity of the beam which can be tuned from I_{f} = 0 to 350 mW cm^{−2} via a halfwaveplate and a polarizer. The maximum value for c_{s} is 6.8 × 10^{−3}, as plotted in Fig. 1b. For the detection part, a ×20 microscope objective and a sCMOS camera allow to get the spatial distribution of the nearfield intensity of the beams at the output of the crystal.
Displacement of the obstacle beam in the fluid of light beam at rest
In order to validate our experimental approach, we consider the linear propagation of the green beam creating the obstacle in the optical potential Δn(I_{f}) photoinduced by the fluidoflight beam at rest (θ_{in} = 0). In the paraxial approximation, the propagation equation reads
with notations similar to the ones used in Eq. (1). By assuming that the transverse component of the fluidoflight beam is nonzero only along the x axis, we denote by \(\left\langle x \right\rangle = {\int} {\kern 1pt} x\left {E_{{\mathrm{ob}}}} \right^2{\mathrm{d}}x\) the position of the centroid of the obstacle beam. Using an optical equivalent of the Ehrenfest relations (see Supplementary Note 3 for full derivation), one can derive from Eq. (3) the following equation of motion: \(\left( {n_{\mathrm{e}}k_{{\mathrm{ob}}}} \right)\partial _{zz}\left\langle x \right\rangle\) = −∂_{ x }[−k_{ob}Δn(I_{f})]. Assuming that Δn is zindependent, which is valid in the hereconsidered linear propagation of the obstacle beam, we readily obtain
where x_{0} is the initial position of the obstacle. This displacement is interpreted as the consequence of a force deriving from the optical potential −k_{ob}Δn(I_{f}), and acting on the obstacle.
The experimental measurement of d, for various intensities I_{f} and positions x_{0}, is presented in Supplementary Fig. 2. The experimental data are fitted, using the above expression, the saturation intensity and the maximum refractive index modification being the fitting parameters. We extract I_{sat} = 380 ± 50 mW cm^{−2} and Δn_{max} = 2.5 ± 0.4 × 10^{−4}. It is worth mentioning that the value of I_{sat} is used for the calculation of Δn(I) and its deriving quantities (i.e., c_{s} and ξ).
Data availability
The data supporting the findings of this study are available within the article and the associated Supplementary Information. Any other data is available from the corresponding author upon request.
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Acknowledgements
The authors acknowledge helpful contributions from M. Garsi during the early stage of this work. We also thank I. Carusotto, V. Doya, F. Mortessagne, N. Pavloff, and P. Vignolo for helpful discussions. M.A. is grateful to P. Leboeuf who was very enthusiastic about the idea of superfluid motion of light. This work has been supported by the the Region PACA and the French government, through the UCA^{JEDI} Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR15IDEX01. P.É.L. was funded by the Centre National de la Recherche Scientifique (CNRS), the ANR under Grant No. ANR14CE260032 LOVE, and the Universit?é de CergyPontoise.
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C.M., O.B., and M.B. performed the experiments and analyzed the data. C.M., M.A., P.É.L., and M.B. developed the theory. All authors participated in the discussions and in writing the paper.
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Michel, C., Boughdad, O., Albert, M. et al. Superfluid motion and dragforce cancellation in a fluid of light. Nat Commun 9, 2108 (2018). https://doi.org/10.1038/s41467018045349
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