Blast waves in a paraxial fluid of light
Murad Abuzarli, Tom Bienaimé, Elisabeth Giacobino, Alberto Bramati, Quentin Glorieux
BBlast waves in a paraxial fluid of light
Murad Abuzarli, Tom Bienaim´e, Elisabeth Giacobino, Alberto Bramati, and Quentin Glorieux ∗ ALaboratoire Kastler Brossel, Sorbonne Universit´e, CNRS,ENS-Universit´e PSL, Coll`ege de France - Paris, France (Dated: January 25, 2021)We study experimentally blast wave dynamics on a weakly interacting fluid of light. The fluiddensity and velocity are measured in 1D and 2D geometries. Using a state equation arising from theanalogy between optical propagation in the paraxial approximation and the hydrodynamic Euler’sequation, we access the fluid hydrostatic and dynamic pressure. In the 2D configuration, we observea negative differential hydrostatic pressure after the fast expansion of a localized over-density, whichis a typical signature of a blast wave for compressible gases. Our experimental results are comparedto the Friedlander waveform hydrodynamical model[1]. Velocity measurements are presented in 1Dand 2D configurations and compared to the local speed of sound, to identify supersonic region ofthe fluid. Our findings show an unprecedented control over hydrodynamic quantities in a paraxialfluid of light.
INTRODUCTION
In classical hydrodynamics, a blast wave is character-ized by an increased pressure and flow resulting from therapid release of energy from a concentrated source [2].The particular characteristics of a blast wave is that itis followed by a wind of negative pressure, which inducesan attractive force back towards the origin of the shock.Typical blast waves occur after the detonation of trini-trotoluene [3, 4], nuclear fission [5], break of a pressurizedcontainer [6] or heating caused by a focused pulsed laser[7]. The sudden release of energy causes a rapid expan-sion, which in a three dimensional space is analogous to aspherical piston [8] and produces a compression wave inthe ambient gas. For a fast enough piston, the compres-sion wave develops into a shock wave which is character-ized by the rapid increase of all the physical propertiesof the gas, namely, the hydrostatic pressure, density andparticle velocity [9]. In 1946, Friedlander predicted thatimmediately after the shock front the physical propertiesat a given position in space decay exponentially [1, 10]. Inthis model, for 3-dimensional and 2-dimensional spacesthe hydrostatic pressure and the density are expected tofall below the values of the ambient atmosphere leadingto a blast wind [2].Shock waves have been studied in several contexts inphysics, including acoustics, plasma physics, ultra-coldatomic gases [11–13] and non-linear optics [6, 14–17]. Inoptics, the hydrodynamics interpretation relies on theMadelung transforms which identify the light intensityto the fluid density and the phase gradient to the fluidvelocity[18]. Recently several works have studied analyt-ically shock wave formation in one and two dimensions[19, 20]. Optical systems allow for repeatable experi-ments and precise control of the experimental parame-ters. For example dispersive superfluid-like shock waveshave been observed [14], as well as generation of soli-tons [16], shocks in non-local media [15, 21], shocks in ∗ [email protected] disordered media [17], analogue dam break [6] and Rie-mann waves [22]. However, an experimental study ofblast waves has not been done in atomic gases nor innon-linear optics systems. In this work, we demonstratethe generation of a blast wave in a fluid of light. In-terestingly, the prediction of a blast wind with negativepressure and density holds in two dimensional space butnot in 1 dimension [23]. Optical analogue systems allowfor an experimental validation of this prediction.In this letter, we study the formation of blast waves ina paraxial fluid of light. We measure the time evolutionof analogue physical properties such as the hydrostaticpressure, the density, the particle velocity and the dy-namic pressure at a fixed point for 1 and 2-dimensionalsystems. We report the observation of a negative hy-drostatic differential pressure after a shock wave in 2-dimensional system and we show that the Friedlanderwaveform describes quantitatively our experimental re-sults for all physical parameters. This paper is organizedas follows. We first introduce the analogy between thepropagation equation of a laser beam through non-linearmedium (a warm atomic vapor) and the hydrodynamicsequation and derive the relevant analogue physical prop-erties. In the second section of this work, we describe ourexperimental setup and present our results on the densityand hydrostatic pressure measurements. We highlightthe striking differences between 1 and 2-dimensional sys-tems. Finally, we study the time evolution of the velocityand dynamic pressure. THEORETICAL MODEL
We describe the propagation of a linearly polarizedmonochromatic beam in a local Kerr medium. We sep-arate the electric field’s fast oscillating carrier from theslowly varying (with respect to the laser wavelength) en-velope : E = E ( r , z )e i ( kz − k ct ) + complex conjugate. Un-der the paraxial approximation, the propagation equa-tion for the envelope E is the Non-Linear Schr¨odinger a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Equation (NLSE) [18]: i ∂ E ∂z = (cid:18) − k ∇ ⊥ + g |E| − iα (cid:19) E , (1)where k is the laser wavevector in the medium, α is theextinction coefficient accounting for losses due to absorp-tion, and the g parameter is linked to the intensity depen-dent refractive index variation ∆ n via: g |E| = − k ∆ n (with k the laser wavevector in vacuum).The NLSE is analogous to a 2D Gross-Pitaevskii equa-tion describing the dynamics of a quantum fluid in themean-field approximation. This analogy is possible bymapping the envelope E to the quantum fluid many-bodywavefunction and the axial coordinate z to an effectiveevolution time. The non-linear refractive index variationplays then the role of a repulsive photon-photon interac-tion, since all measurements in this work are done in theself-defocusing regime i.e. ∆ n < g > k = 8 . m − . Using the Madelungtransformation: E = √ ρ e iφ , v = ck ∇ ⊥ φ one can derivefrom the NLSE hydrodynamic equations [14, 19], linkingthe fluid’s density ρ with its velocity v : ∂ρ∂z + ∇ ⊥ . (cid:16) ρ v c (cid:17) = − αρ (2) ∂ v ∂z + 12 c ∇ ⊥ v = −∇ ⊥ (cid:18) cgρk − c k √ ρ ∇ ⊥ √ ρ (cid:19) . (3)Eq. (2) is the continuity equation with a loss term ac-counting for photon absorption. Eq. (3) is similar tothe Euler equation without viscosity, in which the driv-ing force stems from interaction and the so-called quan-tum pressure term due to diffraction. Establishing theformal analogy requires, however, defining an analoguepressure P to be able to re-express the right-hand sideof Eq. (3) as: − /ρ · ∇ ⊥ P . This is possible for thefirst term stemming from interactions. Using the iden-tity: −∇ ⊥ ρ = − / (2 ρ ) ∇ ⊥ ρ one can define the so calledbulk hydrostatic pressure P as: P = c ρ gk = 12 ρc s , (4)where the last equality comes from c s = c · gρ/k . Eq.(4) is the state equation linking the fluid hydrostatic pres-sure P to its density if one neglects the quantum pressureterm. It is the consequence of the mean-field formula-tion of the interaction. It also implies that the fluid oflight is compressible with the compressibility equal to: k/ ( c ρ g ). One then gets the analogue Euler equation: ∂ v ∂ ( z/c ) + 12 ∇ ⊥ v = − ρ ∇ ⊥ P, (5)with a pressure P of dimension [ density ] × [ speed ] . Asalready mentioned, the fluid dynamics can be studiedby accessing its state at different z positions, however this is not recommend practically since imaging insidea non-linear medium is highly challenging task. Alter-natively, one can instead re-scale the effective time byincorporating fluid interaction [20, 24]. Fluid interactioncan then be varied experimentally and the fluid dynam-ics can be studied while imaging only the state at themedium output plane. Re-scaling the time is based ondefining following quantities: z NL = 1 gρ (0 , L ) , non-linear axial length (6) ξ = (cid:114) z NL k , transverse healing length (7) c s = ckξ , speed of sound (8) ψ = E (cid:112) ρ (0 , L ) , (9)and substituting the time and space variables as: τ = z/z NL , ˜ r = r /ξ , ˜ ∇ ⊥ = ξ ∇ ⊥ . L is the non-linearmedium length. The propagation equation then reads: i ∂ψ∂τ = (cid:18) −
12 ˜ ∇ ⊥ + | ψ | (cid:19) ψ. (10)One can note that dynamics of ψ is not anymore dissipa-tive, due to the normalization with respect to the expo-nentially decaying density: ρ (0 , L ) = ρ (0 , − αL ),measured at at the medium exit plane. This formula-tion is necessary to describe accurately the experimen-tal results of this work probing temporal dynamics of afluid of light by varying the strength of the optical non-linearity and not the imaged z plane. The effective time τ = | ∆ n ( r ⊥ = 0 , L ) | k L equals to the maximal accumu-lated non-linear phase. Rewriting the Madelung trans-formation with the new variables, we obtain: ψ = (cid:112) ˜ ρ e iφ = (cid:114) ρρ (0 , L ) e iφ , ˜ v = v c s = ˜ ∇ ⊥ φ. (11)One gets dimensionless Euler and the continuity equa-tions: ∂ ˜ ρ∂τ + ˜ ∇ ⊥ . (˜ ρ ˜ v ) = 0 (12) ∂ ˜ v ∂τ + 12 ˜ ∇ ⊥ ˜ v = − ˜ ∇ ⊥ (cid:18) ˜ ρ − √ ˜ ρ ˜ ∇ ⊥ (cid:112) ˜ ρ (cid:19) , (13)where the link between Eq. (13) and the Euler equationis made by neglecting the quantum pressure and definingthe the dimensionless hydrostatic pressure as:˜ P = 12 ˜ ρ . (14)Finally, the dynamic pressure is defined as a vector quan-tity by: ˜ P d = 12 ˜ ρ ˜ v | ˜ v | , (15)The dynamic pressure is the fluid kinetic energy flux andaccounts for the amount of pressure due to fluid mo-tion. The impact force on an obstacle hit by a shock-wave is proportional to its dynamic pressure. Expressedin dimensionless units, the dynamic pressure gives thestrength of the convection term normalized by the pres-sure due to the interactions in the Eq. (13). It can becomputed directly from the density and velocity mea-surements. SHOCK WAVES AND BLAST WIND
In this work, we study the dynamics of a fluid oflight disturbed by a localized Gaussian over-density δρ ( r ,
0) = ρ exp (cid:0) − r /ω (cid:1) . ρ is of the same mag-nitude as the background fluid density ρ and ω quanti-fies the perturbation width. We can write ρ ( r , L ) = ρ + δρ ( r , L ) . Normalizing the total density by its maximalundisturbed value one gets: ˜ ρ ( r , τ ) = ρ ( r , L ) /ρ (0 , L ).Extending this definition to ρ and ρ , we obtain ˜ ρ bound between 0 and 1 and having a Gaussian shape,and ˜ ρ expressing the perturbation strength with respectto the fluid background density. To take into account theGaussian profile of the density ρ , we define the over-pressure from the pressure difference between the casewith and without perturbation: δ ˜ P ( r , τ ) = ˜ P ( r , τ ) − ˜ P ( r , τ ) . (16)To evaluate the differential pressure ∆ ˜ P ( τ ), showing theinstantaneous difference in pressure between the pertur-bation center and the external undisturbed area, we de-fine: ∆ ˜ P ( τ ) = ˜ P (0 , τ ) − P ( r ext , τ ) . (17)The differential pressure ∆ ˜ P ( τ ) is the most importantquantity we study in this work and we expect major dif-ferences in the non-linear perturbation dynamics betweenthe 1D and the 2D geometries. Finally, the fluid velocitycan be measured experimentally. It requires a measure-ment of the beam wavefront which is realized using off-axis interferometry. Calculating numerically the gradientof the phase, we obtain the background fluid velocity v and the perturbation velocity v by analyzing the imageswithout and with the perturbation, respectively.Several studies have been performed in both ρ (cid:28) ρ and ρ (cid:29) ρ regimes, observing the Bogoliubov disper-sion of the linearized waves created by the perturbation[25–27], and the shock waves [14, 20], respectively. Inthis work we investigate the case ρ ∼ ρ by analyzingthe fluid density, velocity and pressure both in the 1Dand 2D geometries. The NLSE is known to give rise tosound-like dispersion to the low amplitude waves, gov-erned by the Bogoliubov theory. Here, a perturbation ofthe same order (or larger) than the background resultsin the sound velocity variation following the local den-sity inside the perturbation. This is the prerequisite for observing shock waves, a special type of waves changingtheir shape during propagation towards a steepening pro-file. In hydrodynamics, shock waves are usually reportedas a time evolution measurement of a physical quantity(pressure, density...) at a fixed point in space. After thepassage of a the shock wave front, a blast wind (a nega-tive differential pressure) should be observed in 2 and 3dimensional space. A direct physical consequence of thiswind in classical hydrodynamics is observed for exampleafter an explosion inside an edifice: the presence of glasspieces within the building is the signature of the blastwind . In the next section we report the time evolutionas well as the time snapshots (spatial map of a physicalquantity at fixed time) typically not accessible in classicalhydrodynamics experiments. EXPERIMENTAL SETUP
In our experiment, we investigate the propagation ofa near-resonance laser beam through a warm rubidiumvapor cell, which induces effective photon-photon inter-actions [28]. Two configurations are studied: the 2D ge-ometry with a radially symmetric dynamics and the 1Dgeometry with a background much larger along x thanalong y which allows for a 1D description of non-linearwave dynamics [20]. A tapered amplified diode laser issplit into a background, a reference and a perturbationbeams (see supplementary materials for details). Thebackground beam is enlarged with a telescope up to awaist of 2.5 ± x and 0.8 ± y in the 1D geometry, and 1.8 ± ± µm diameter pinhole intoa photodiode to stabilize the interferometer. The con-trol is realized by locking on local minimum acting on apiezoelectric mirror mount with a RedPitaya hardwarerun by the PyRPL software [29]. Cell temperature is149(2) ° C leading to an atomic density of 8.3 ± × cm − . The cell output is imaged with a × P ranging from50 to 600 mW and different laser detunings ∆ from the Rb D2 line F = 3 → F (cid:48) transition are taken (see sup-plementary materials for details). The reference beamis superimposed with other beams with an angle of 30milli-radians, giving rise to interferogramms with verti-cal fringes of average periodicity of 25 ± µ m. FIG. 1. Density data: The left column corresponds to the 1Dconfiguration and the right column to the 2D case. a) andb) are over-density maps at time τ = 31, obtained by sub-tracting the images with no perturbation from the ones withperturbation in the 1D and 2D geometry, respectively. c), d)are density profiles without (blue) and with (red) perturba-tion in the 1D and 2D geometry, respectively. The profilesare shifted vertically (spacing of 2) for better visibility. DENSITY
The density is an important physical parameter neededto compute the static and hydrodynamic pressure. It isdirectly given by the intensity measurement. In figure 1a) and b), we present the experimental maps of the over-density δ ˜ ρ at time τ = 31, after subtracting the back-ground fluid, in the 1D and 2D geometries respectively.By changing the laser intensity and detuning, we canmodify the effective time τ . The associated time τ is cal-culated from the nonlinear index ∆ n via the off-axis in-terferometric measurement for each experimental config-uration ( P , ∆) (see supplementary materials for details).Fig. 1 a) and b) show the spatio-temporal over-density di-agrams. We present the corresponding density profiles atdifferent times in figure 1 c) and d). The 1D density dataare averaged over the y direction for | y | < . τ . In the 2D geometry, interestingly, the steepening FIG. 2. Pressure analysis: a),c): 1D & 2D over-pressureprofiles evaluated at different effective times τ . Each follow-ing profile shifted vertically by 2 for better visibility. b),d)show the 1D and 2D spatio-temporal diagrams of the over-pressure evolution, respectively. The dotted black lines showthe trajectory of expansion at the speed of sound accord-ing to the parabolic equation with the prefactor given by k/L = 107 mm − in both geometries. The blue dotted linesshow the same trajectories shifted horizontally by 250 µm and200 µm in 1D and 2D cases, respectively. It corresponds toexternal undisturbed area used for the measurement of thedifferential pressure. Dashed green rectangles around τ = 40show the presence of a second shock due to an increasing dif-ferential pressure. of the shock front is less pronounced. Moreover, a densitymuch lower than the background density is observed inthe center of the 2D profiles for long time τ >
20, whichis not the case in 1D. This negative differential densityhas a direct consequence on the differential pressure cal-culated using Eq. (17).
STATIC PRESSURE
To isolate the effect of the perturbation on the staticpressure, we compute the over-pressure from images ofthe background with and without the bump taken atsame effective times τ ( P , ∆), using Eq. (16) and (14).The over-pressure as a function of time τ is shown inFig. (2) b) and d) and profiles averaged along y in the 1Dcase and radially in the 2D case are presented in Fig. (2)a) and c) for various times.The trajectory of a density pulse spreading with no
1D differential pressure2D differential pressure Fit
FIG. 3. Differential pressure calculated from Eq. (17) for the1D (circular dots) and the 2D cases (square dots). The uncer-tainty bars correspond to the statistical analysis of multipleimages. The pressure is normalized as described in the maintext. Blue line is the ambient pressure outside of the shock.Black dashed line is the Friedlander model for a blast wavedescribed in Eq. (18) with P s = 1 and t ∗ = 20. dispersion at the speed of sound can be expressed as fol-lows: r = c s ( τ ) × ( L/c ). The coefficient can be calcu-lated using the time dependence of the sound velocity: c s = c (cid:112) τ / ( kL ) obtained from Eqs. (6) and (8). It di-rectly leads to τ = kr /L and knowing that: L = 75 mmand k = 8 × mm − , one gets: τ = 107 × r . Thecoefficient does not depend on the dimensionality of thesystem.In the pressure maps (Fig. (2) b) and d)), we haveadded a black dashed line following this trend: τ =107 × x (1D) and τ = 107 × r (2D). As expected, thistrajectory follows closely the shock front in the 1D ge-ometry. The differential pressure is defined as the pres-sure difference between inside and outside of the shockas expressed in Eq. (17). The undisturbed pressure asfunction of time is evaluated along the same trend line τ = 107 × ( r ext − r ) , translated r = 250 µ m in 1D and r = 200 µ m in 2D, which corresponds to ∼ . x = 0) is presented in Fig. 3. 1D (red circles)and 2D (gray triangles) geometries are compared from τ = 0 to τ = 45. An important difference can be seenbetween the two geometries: in the 2D situation the dif-ferential pressure becomes negative at τ = 20 as it goesto zero in the 1D case. The observation of the negativepressure is the typical signature of a blast wind. This measurement reveals the dramatic impact of the geome-try on blast wind in a fluid of light and exemplifies theanalogy with classical hydrodynamics. To quantify thisanalogy, we use the Friedlander waveform model whichis known to describe the dynamics of physical quantitiesin a free-field (i.e. in a open 3-dimensional space) blastwave [2]. In this model the differential pressure followsan exponential decay of the form:∆ ˜ P = P s e − τ/t ∗ (1 − τ /t ∗ ) , (18)where P s and t ∗ are two parameters which correspondsrespectively to the peak differential pressure immediatelybehind the shock and to the time when the differentialpressure becomes negative. The period when the hydro-static pressure is above the ambient value is known as thepositive phase, and the period when the properties arebelow the ambient value is the negative phase. We use P s = 1 (since the differential pressure is normalized) and t ∗ = 20 and plot the corresponding model with a blackdashed line in Fig. 3. An intriguing feature can also beseen in the 2D time evolution at τ = 40. Close to theminimum of the negative phase, a second peak of differ-ential pressure is observed (the single point at τ = 40Fig. 3 is the average of several realizations with errorsbars indicating the standard deviation of the measure-ment) in our optical analogue which is reminiscent of thesecond shock observed in classical explosion. In classicalblast wave dynamics, this second shock is believed to be aconsequence of the expansion and subsequent implosionof the detonation products and source materials. Ourresults suggest that this second shock might be of moregeneral nature than currently thought. VELOCITY
For blast waves, there are no simple thermodynamicrelationships between the physical properties of the fluidat a fixed point [30]. This means that the temporal evo-lution of the static pressure measured at a fixed point isnot sufficient to calculate the temporal evolution of thevelocity or the dynamic pressure from that single mea-surement. To fully describe the physical properties ofa fluid in a blast wave it is necessary to independentlymeasure at least three of the physical properties, such as,the static pressure, the density and the fluid velocity orthe dynamic pressure. In the last section of this work, wereport the measurement of last two physical properties,which are vector quantities.The fluid velocity is calculated from its phase (seeEq. (11)) which is measured using off-axis interferomet-ric imaging. The off-axis configuration consists in thetilted recombination of the signal beam with the refer-ence beam on the camera plane. This results in the setof linear fringes evolving along the relative tilt directionand locally deformed (stretched or compressed) accord-ing to the beams relative curvature. Using a collimatedGaussian beam as the reference, the measured curvature
FIG. 4. Fluid velocities from the off-axis interferometry. a),b)Space-time evolution of the Mach number with respect tothe background’s local speed of sound, in the 1D and 2Dgeometry, respectively. The dotted black line in a) showsthe calculated trajectory of expansion at the speed of sound(see main text). c),d) show the background’s ˜ v (blue) andtotal ˜ v (red) Mach number profiles, at different times, forthe 1D (x coordinate) and 2D geometry (radial coordinate),respectively. Each following profile shifted vertically (spacingof 1) for visibility. is the one of the signal beam. The acquired interfero-gramm carries the information on the beam phase viaits amplitude modulated term. This term shows spatialperiodicity and in the Fourier space it translates to twopeaks shifted by a distance proportional to the off-axistilt angle, symmetric with respect to the origin. By nu-merically calculating the spatial spectrum and filteringone of these peaks, the inverse Fourier transform gives thebeam complex envelope with a spatial resolution boundby the fringe wavelength. The measured phase is un-wrapped and the contribution due to the relative tilt isremoved by subtracting the phase ramp. The resultingphase is averaged and numerically differentiated to getthe velocity map.Using this procedure, the off-axis interferograms of thebackground fluid and of the background fluid with theperturbation are analyzed to give access to v ( r, τ ) and v ( r, τ ), respectively. The difference of these quantitiesgives the perturbation velocity v ( r, τ ). The non-zerovelocity v of the background fluid arises from its finitesize causing its expansion due to a non-zero pressure gra-dient. The knowledge of v is essential to calculate theeffective interaction g and therefore the time τ and thesound velocity. Indeed, φ = τ ˜ ρ can be accessed byintegrating v over the transverse coordinate and usingthe fact that φ (˜ r → ∞ , τ ) →
0. Knowing τ , the sound velocity is c s ( r ⊥ , τ ) = c (cid:112) τ ˜ ρ ( r ⊥ , τ ) / ( k L ).The velocity maps normalized by the local sound ve-locity (in Mach units) are presented in figure 4 a) andb) for the 1D and 2D configurations, respectively. Sincevelocity is a vector quantity, negative values correspondto a propagation along − x direction. Figure 4 c) and d)show the corresponding profiles obtained for three spe-cific times τ = 2; 23 and 45. The maximal speed of soundat these times is 0.18, 0.62 and 0.86 percent of the speedof light in vacuum. Positive outward velocity, as well aszero velocity at the center is observed at all times bothin the 1D and 2D cases. Whereas it is intuitively ex-pected in the 1D geometry with the differential pressurenever dropping to negative values, it also holds in the 2Dcase in which a negative phase for the differential pres-sure exists. A possible explanation lies in the fact thatwhen the negative phase is reached for the differentialpressure, the perturbation has already expanded enoughsuch that the net resulting force is smaller due to a largerdistance. It is also worth noting that the velocity is atleast 2 times larger in the 1D geometry than in 2D, asseen by comparison of the y-axis scales in Figure 4 c) andd). Additionally, clear steepening of the velocity profilesis observed in the 1D case reaching a Mach number of 1at the steepest position. a) b)c) d) Dynamic pressure (1D) Dynamic pressure (2D)
FIG. 5. Dynamic pressure analysis. a) and b) show thespatio-temporal evolution maps of the dynamic pressure pro-files, for the 1D (the x component) and 2D geometry (theradial component), respectively. Below, the c) and d) pan-els show various superimposed dynamic pressure profiles atdifferent times, in 1D and 2D geometry, respectively.
DYNAMIC PRESSURE
Alternatively, we can measure the dynamic pressureto compute a third thermodynamic quantity: the totalpressure. The dynamic pressure is also a vector quantityand can be obtained from a phase measurement similar tofluid velocity using Eq. (15). The dynamic pressure mapsare presented in Figs. 5 a) and b). Once again Figs. 5c) and d) show dynamic pressure profiles for three se-lected times. In 1D, the dynamic pressure forms a steepoverpressure characteristic of the shock front which in-creases as function of time. In the 2D geometry, on thecontrary the dynamic pressure reaches a plateau at theshock front without forming a steep overpressure peak.This behavior is in agreement with the velocity distribu-tions presented previously.
CONCLUSION
Relying on detailed measurements of all thermo-dynamic quantities in a fluid of light blast wave, wehave demonstrated for the first time the occurence of ablast wave in a fluid of light. We compare 1D and 2Dgeometry and report the observation of a negative phase during the blast only for the 2-dimensional case. Thedifferential pressure in the 2D geometry is compared tothe classical hydrodynamics of Friedlander blast-waveand we see a very good agreement with this model. Ve-locity maps and dynamic pressure are finally presentedto complete the study. Our work opens the way toprecise engineering of a fluid of light density and velocitydistribution which will prove to be a valuable tool todesign new experiments studying superfluid turbulence[31] or analogue gravity where an excitation of a fluid oflight changes from a subsonic to a supersonic region.
ACKNOWLEDGMENTS
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Experimental details
The scheme of the experimental setup is shown on Figure S1. Toptica DLCpro 780 withTA was used for all measurements. The laser frequency was tuned around 780 nm and measured with a MogWaveMultimeter LambdaMeter and calibrated with Saturable absorption spectroscopy (SAS). The laser beam was modecleaned with a single mode fiber and then split into the Background, Bump and the Reference arms. The respectiveintensity ratio was fixed by the angles of the Half-Wave-Plates (HWP), placed before the Polarizing Beam Splitters(PBS), in agreement with experimental requirements: ˜ ρ ( r = 0 , τ = 0) ≈ µm diameter pinhole centered at the overlap area anda photodiode. The relative phase needs to be locked in order to minimize permanently this signal and make itinsensitive to perturbations such as air currents. Therefore the photodiode signal was transformed into an error signalof a piezoelectric mirror mount controlling the relative phase. The error signal generation from the photodiode signalwas realized with the PyRPL software running on a Red Pitaya FPGA [29]. The modulation frequency was around2-3 kHz. FIG. S1. Schematic visualization of the experimental setup. Diode laser frequency calibration was performed with SaturableAbsorption Spectroscopy (SAS), and during the experiment the frequency measurement was performed with a MogWaveLambdameter. The laser was mode cleaned with a polarization maintaining single mode siber (PMSMF), before being splitinto the Background, Bump and the Reference. The Background-Bump interference arm complementary to the Rb vapor cellwas cropped with a 200 µm diameter pinhole (Ph) to measure the the power of the overlap area on a photodiode. This signalwas minimized by controlling the relative phase via piezoelectric motion of a mirror mount to have permanently constructiveinterference on the vapor cell arm. The error signal was generated from the photodiode signal with the PyRPL lockbox software. Vapor Temperature
One of the useful knobs to control the light-matter interaction in hot vapor cells is the atomic density. The latteris directly linked to the vapor pressure via the ideal gas law (neglecting the atom-atom interactions). It equals theRb vapor’s saturation pressure at thermal liquid-gas equilibrium and can be increased by several orders of magnitudewhen heating the cell from 50 ° C to 150 ° C. Keeping the vapor temperature constant during the experiment istherefore necessary to control the atomic susceptibility. In our experiment, several electric resistors were woundaround the cell and connected in parallel to a DC power supply to heat up the cell. The vapor temperature wasaccessed by measuring the transmission spectrum around the Rb D2 line in the weak beam limit. The frequencycalibration was performed via Saturable absorption spectroscopy, as shown on Figure S1. The experimental spectrumwas fitted with the linear susceptibility model developped in [32] taking into account all hyperfine transitions of bothisotopes and the collisional self-broadening due to resonant dipole-dipole interactions [33], with the atomic densityand the number fraction of Rb isotope as free parameters. The temperature was measured before and after eachexperiment to prevent any temperature drift.
Non-linear refractive index variation measurement
The intensity dependent refractive index of our hotatomic vapor is the key parameter governing the fluid’s dynamics as it is linked to the effective evolution time:
FIG. S2. Vapor’s transmission and its intensity dependent refractive index measurement. a) and b) show the maximalrefractive index variation calculated from the off-axis interferograms of the backgroung beam with a reference, in 1D and 2Dgeometry, respectively. c) Background beam’s transmission spectrum with respect to Rb cooling transition measured atdifferent input powers. Dashed line is the theory of a linear multilevel vapor at temperature 150 ° C and 0.5 % the isotopicfraction of Rb inside the cell. Checking the ”Kerr” approximation: d) and e) show the variation of the refractive index withlaser power at fixed laser detuning in both geometries. τ = ∆ nk L and its speed of sound: c s = c √ ∆ n . In this work it was measured using the off-axis interferometry whichgives access to the transverse phase variations at the cell exit plane. The transverse phase profile of the Backgroundbeam is assumed to depend as follows on the beam’s intensity I( r ): φ th ( r , L ) = k L n I ( r )1 + I ( r ) /I s + φ (S1)Where n is the Kerr index, I s the saturation intensity of the Kerr effect and φ a constant phase. The gradientof the phase, giving access to the fluid velocity, is numerically calculated and fitted with ∇ φ th with n and I s asfree parameters. Figure S2 a) and b) show measured maximal variation of refractive index for different experimentalconfigurations of the laser detuning ∆ and power P . Each point corresponds to a processed image. c) Shows thetransmission spectra through the cell for different input powers. No saturation of the absortpion can be evidenced.The black dashed line is the theoretical calculation of the linear susceptibility used for the measurement of the vapor’stemperature. Finally, d) and e) show the variation of the refractive index with intensity. The graphs show that theresults of this work are obtained below the regime of the saturation of the Kerr effect. Background beam’s expansion
In the theoretical discussion developed in the main text and for the ∆ n measurement it is assumed that thebackground fluid beam’s density is invariant with time. The experimental data to verify this hypothesis are shown inFigure S3. No expansion in the x direction of the 1D case was observed. The expansion is most pronounced in thetransverse y direction of the 1D case. In the 2D case the background’s insignificant expansion is observed. Relevance of the Quantum Pressure
As mentioned in the main text, the Quantum pressure was neglected in the theoretical description of the experi-mental data as we are interested in the fluid’s behavior in the long wavelength limit. This term is known to have adispersive contribution to the shockwave profile which, upon steepening, becomes composed of an increased amountof various momentum components moving at different velocities. To evaluate the relevance of the Quantum Pressure
FIG. S3. Background fluid’s expansion. a) shows the Background’s expansion in the transverse y direction in the 1D geometryand b) shows the Background’s radial expansion in the 2D geometry. a) b)
FIG. S4. Quantum pressure calculated from experimental density profiles. a) in the 1D geometry and b) in the 2D geometry.Same colormap is used for both graphs. in this work we calculated it from the experimental density profiles at different evolution times for both 1D and 2Dgeometry as: ˜ P q = 12 √ ˜ ρ ˜ ∇ ⊥ (cid:112) ˜ ρ (S2)Depending on the dimensionality the Laplacian was calculated as: ˜ ∇ ⊥ = ξ ∂ /∂x in 1D or as: ˜ ∇ ⊥ = ξ [ ∂ /∂r +(1 /r ) × ∂/∂r ] in 2D using the radial symmetry. With this formulation the value of the dimensionless Quantum Pressuredirectly compares with the dimensionless density (stemming from interactions) in the right hand side of the Euler-likeMadelung equation. The result is shown on Figure S4 a) for the 1D and b) for the 2D case. The Quantum Pressureseems to be most pronounced at the vicinity of the Shock front in 1D. In 2D it seems to decay with time. In bothcases it does not exceed 0.1 for times τ >
5. This validates the theoretical approach chosen in work and consistingin neglecting the Quantum Pressure. For lower times the calculation seems inaccurate. This may be due to a largeuncertainty on the healing length ξξ