Superfluid motion and drag-force cancellation in a fluid of light

ABSTRACT Quantum fluids of light merge many-body 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 all-optical 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 long-range

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. SIMILAR

CONTENT BEING VIEWED BY OTHERS THE PISTON RIEMANN PROBLEM IN A PHOTON SUPERFLUID Article Open access 06 June 2022 DYNAMICS OF THE BEREZINSKII–KOSTERLITZ–THOULESS TRANSITION IN A PHOTON FLUID

Article 13 July 2020 DIMENSIONAL CROSSOVER IN A QUANTUM GAS OF LIGHT Article 06 September 2024 INTRODUCTION Superfluidity was originally discovered in 19381 when a 4He fluid cooled below a

critical temperature flowed in a nonclassical way along a capillary2. This was the trigger for the development of many experiments genuinely realized with quantum matter, as with 3He fluids3

or ultracold atomic vapors4,5. The superfluid behavior of mixed light-matter cavity gases of exciton-polaritons was also extensively studied6,7, leading to the emergent field of “quantum

fluids of light”8. Before being theoretically developed for cavity lasers9,10, the idea of a superfluid motion of light originates from pioneering studies in cavityless all-optical

configurations11 in which the hydrodynamic nucleation of quantized vortices past an obstacle when a laser beam propagates in a bulk nonlinear medium was investigated12. In such a cavityless

geometry, the paraxial propagation of a monochromatic optical field in a nonlinear medium may be mapped onto a two-dimensional Gross-Pitaevskii-type evolution of a quantum fluid of

interacting photons in the plane transverse to the propagation4. 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

again13,14,15,16, driven by the emergence of advanced laser-beam-shaping 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 excitations17 and the detection of a vortex nucleation in the wake of an obstacle18. The stimulated emission of dispersive shock waves in nonlinear optics was also

studied in the context of light superfluidity13. However, one of the most striking manifestations of superfluidity—which is the ability of a fluid to move without friction19—has never been

directly observed in a cavityless nonlinear-optics platform. A direct consequence of this feature is the absence of long-range 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 long-range radiation from the obstacle. This regime is usually associated with the cancellation of the drag force experienced by the obstacle, as studied for

4He20, ultracold atomic gases21,22,23,24,25, or cavity exciton-polaritons26,27,28,29. In our cavityless all-optical 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 light13,30,31,32. As sketched in Fig. 1a and detailed in Fig. 1c, a local

drop of the optical index is photo-induced 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 fluid-of-light beam in the paraxial approximation is ruled by a two-dimensional Gross-Pitaevskii-type equation (also known as a nonlinear

Schrödinger-type equation): $$i\partial _zE_{\mathrm{f}} = - \frac{1}{{2n_{\mathrm{e}}k_{\mathrm{f}}}}\nabla ^2E_{\mathrm{f}} - k_{\mathrm{f}}{\mathrm{\Delta

}}n(I_{{\mathrm{ob}}})E_{\mathrm{f}} - k_{\mathrm{f}}{\mathrm{\Delta }}n(I_{\mathrm{f}})E_{\mathrm{f}}$$ (1) The propagation coordinate _z_ plays the role of time. The transverse-plane

coordinates R = (_x_, _y_) span the two-dimensional space in which the fluid of light evolves. The propagation constant _n_e_k_f = _n_e × 2_π_/_λ_f of the fluid-of-light 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 fluid-of-light 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 self-defocusing nonlinear

contribution Δ_n_(_I_f) < 0 to the total refractive index provides repulsive photon–photon interactions and ensures robustness against modulational instabilities33. 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 light4,16 (see Supplementary Note 1). The photorefractive nonlinear response of the material, Δ_n_(_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 superfluidity19 applies and the so-called 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_s4,34. 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

Δ_n_[_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 self-defocusing 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 fluid-of-light 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 obstacle14,16,35. 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 fluid-of-light 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.

Long-range 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 material29. DRAG-FORCE AND OBSTACLE DISPLACEMENT In the supersonic regime, the intensity modulation of the fluid of light flowing around the

obstacle induces a local optical-index 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 long-range 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 in36 for a material obstacle (here, we rather consider an all-optical 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 Bose-Einstein 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 light26,27,29. The well-known 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 decreases37. As the intensities increase, one can see that the local intensity

difference sticks to zero for non-zero 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, Δ_n_(_I_ob)/Δ_n_(_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 fluid-of-light beam is non-zero 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): $$n_{\mathrm{e}}\partial _{zz}\left\langle x \right\rangle = \partial _x\left[

{{\mathrm{\Delta }}n(I_{\mathrm{f}})} \right].$$ (2) This means that the all-optical 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 Δ_n_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_o_b_), 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 fluid-of-light 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 non-stationary 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 all-optical 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

all-optical 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 optical-index

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 mm3 strontium barium niobate (SBN:61) photorefractive crystal additionally doped with cerium (0.01%)

to enhance its photoconductivity38 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 space-charge electric field thus implies a modulation of the refractive index of the crystal39,

Δ_n_(_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 electro-optic 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 Δ_n_(_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 Δ_n_(_I_). The maximum value of the optical index variation

is theoretically Δ_n_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 diffraction-free 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 _z_-direction. From Fig. 1b, the propagation of the obstacle beam into the crystal induces a local drop

Δ_n_(_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 fluid-of-light beam. Both laser beams are linearly-polarized along the extraordinary axis to maximize the photorefractive effect. We vary the flow velocity _v_ by changing the input

angle _θ_in of the fluid-of-light 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 half-waveplate 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 near-field 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 Δ_n_(_I_f)

photo-induced by the fluid-of-light beam at rest (_θ_in = 0). In the paraxial approximation, the propagation equation reads $$i\partial _zE_{{\mathrm{ob}}} = -

\frac{1}{{2n_{\mathrm{e}}k_{{\mathrm{ob}}}}}\nabla ^2E_{{\mathrm{ob}}} - k_{{\mathrm{ob}}}{\mathrm{\Delta }}n(I_{\mathrm{f}})E_{{\mathrm{ob}}},$$ (3) with notations similar to the ones used

in Eq. (1). By assuming that the transverse component of the fluid-of-light beam is non-zero 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_obΔ_n_(_I_f)]. Assuming that Δ_n_ is _z_-independent, which is valid in the here-considered linear propagation of the obstacle beam, we readily obtain $$d = \left\langle {x(z)}

\right\rangle - x_0 = \frac{1}{2}\left[ {\partial _x{\mathrm{\Delta }}n(I_{\mathrm{f}}){\mathrm{/}}n_{\mathrm{e}}} \right]{\kern 1pt} z^2$$ (4) 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_obΔ_n_(_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 Δ_n_max = 2.5 ± 0.4 × 10−4. It is worth mentioning that the value of _I_sat is

used for the calculation of Δ_n_(_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

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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 UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01. P.-É.L. was funded by the Centre National de la

Recherche Scientifique (CNRS), the ANR under Grant No. ANR-14-CE26-0032 LOVE, and the Universit?é de Cergy-Pontoise. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Institut de Physique de

Nice, Université Côte d’Azur, CNRS, Nice, France Claire Michel, Omar Boughdad, Mathias Albert & Matthieu Bellec * Laboratoire de Physique Théorique et Modélisation, Université de

Cergy-Pontoise, CNRS, 2 Avenue Adolphe-Chauvin, 95302, Cergy-Pontoise CEDEX, France Pierre-Élie Larré * Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de

France, 4 Place Jussieu, 75252, Paris, CEDEX 05, France Pierre-Élie Larré Authors * Claire Michel View author publications You can also search for this author inPubMed Google Scholar * Omar

Boughdad View author publications You can also search for this author inPubMed Google Scholar * Mathias Albert View author publications You can also search for this author inPubMed Google

Scholar * Pierre-Élie Larré View author publications You can also search for this author inPubMed Google Scholar * Matthieu Bellec View author publications You can also search for this

author inPubMed Google Scholar CONTRIBUTIONS 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. CORRESPONDING AUTHORS Correspondence to Claire Michel or Matthieu Bellec. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no

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http://creativecommons.org/licenses/by/4.0/. Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Michel, C., Boughdad, O., Albert, M. _et al._ Superfluid motion and drag-force

cancellation in a fluid of light. _Nat Commun_ 9, 2108 (2018). https://doi.org/10.1038/s41467-018-04534-9 Download citation * Received: 12 January 2018 * Accepted: 04 May 2018 * Published:

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