Short Bragg Pulse Spectroscopy for a Paraxial Fluids of Light

Short Bragg pulse spectroscopy for a paraxial ﬂuids of light

Clara Piekarski,1 Wei Liu,1 Jeff Steinhauer,1, 2 Elisabeth Giacobino,1 Alberto Bramati,1 and Quentin Glorieux1, ∗ 1Laboratoire Kastler Brossel, Sorbonne Université,CNRS, ENS-PSL Research University, Collègede France, Paris 75005, France 2Department of Physics, Technion—Israel Institute of Technology, Technion City, Haifa 32000, Israel (Dated: November 26, 2020) We implement Bragg spectroscopy in a paraxial fluid of light. Analogues of short Bragg pulses are imprinted on a photon fluid by wavefront shaping using a spatial light modulator. We measure the dispersion relation and evidence a parabolic single-particle regime as well as a linear phonon regime even for very weakly interacting photons and low sound velocity. Finally, we report a measurement of the static structure factor, S(k), and we demonstrate the presence of pair-correlated excitations, revealing indirectly the quantum depletion in a paraxial fluid of light

Fluids of light in the paraxial configuration have Several variants of this method have been developed emerged as an original approach to study degenerate for exciton-polaritons [17] and for atomic BEC, including Bose gases [1]. Several important results have recently momentum-resolved spectroscopy [18], multi-branch established this platform as a potential analogue spectroscopy [19] and tomographic imaging [20]. Most quantum simulator, including the demonstrations of of these techniques rely on measuring the energy of superfluidity of light [2, 3], the observation of the the condensate’s linear excitations known as Bogoliubov Berezinskii–Kosterlitz–Thouless transition [4] and pre- quasi-particles. Interestingly, in paraxial fluids of light, condensation [5], the evidence of photon droplets [6] and the dispersion relation has been recently obtained not the creation of analogue rotating black hole geometries [7, using Bragg spectroscopy but by measuring the group 8]. Paraxial fluids of light rely on the direct mathematical velocity of two counter-propagating wavepackets in the analogy that can be drawn between the Gross-Pitaevskii transverse plane [2]. However, this experiment, as well as equation describing the mean field evolution of a Bose- a similar implementation in a thermo-optic medium [21], Einstein condensate (BEC) and the non-linear Maxwell have a resolution for weak non-linearities (small sound equation describing the propagation of light within a χ(3) velocities). Indeed, in this regime, the two wavepackets non-linear medium [1, 9]. The growing interest about propagate too slowly to separate and due to interferences this platform comes from the various advantages that no analytical dispersion relation can be derived [22]. makes paraxial fluids of light a complementary system to atomic BEC. First, optical detection techniques In this Letter, we implement an optical analogue of are highly sensitive (single-photon counting, homodyne Bragg spectroscopy to measure the phonon dispersion detection) and allows for measuring with high precision and the static structure factor in a paraxial fluid of the density distribution in position and momentum space light. We show that short Bragg pulses used for phase as well as the phase. Second, since there is no gravity imprinting technique in atomic BEC can be achieved in force acting on a photon fluid, there is no need for fluids of light using wavefront shaping with a spatial a trapping potential and homogeneous density can be light modulator. This technique not only allows for easily achieved. Moreover, an external control potential measuring the dispersion relation with a better resolution can be applied either during the entire evolution [3, 10], at small sound velocities but it also gives access to the or only for a short period of time as in this work. excitation strength for phonons: the static structure An essential characterization tool for atomic BEC factor. We found that the static structure factor is is coherent Bragg spectroscopy [11]. The original significantly reduced from that of free particles, revealing implementation was based on two-photon Bragg indirectly the quantum depletion, consisting of pair- scattering and allowed for the measurement of the correlated particles, in a paraxial fluid of light [13]. true momentum distribution, which was significantly This paper is organized as follows. We first introduce narrower than that observed by time of flight [12]. the formalism of a paraxial fluid of light and the short For atomic BEC, it has been realized very early Bragg pulse technique which inspired our approach. that this technique would also allow for measuring arXiv:2011.12935v1 [cond-mat.quant-gas] 25 Nov 2020 We describe numerically and experimentally the optical the dispersion relation [13], which describes how each implementation of Bragg spectroscopy. We then measure frequency component of a wavepacket evolves and the the dispersion relation and evaluate the maximum dynamic structure factor, which is the Fourier transform resolution. Finally, we present a measurement of the zero of the density correlation function [14] and is essential to temperature static structure factor in agreement with the the description of many-body systems [15, 16]. Feynman relation for an homogeneous Bose gas [15, 23]. In a third-order nonlinear Kerr medium, the evolution of the electric field is given by the nonlinear Schrödinger ∗ [email protected] equation (NLSE), written within the paraxial and slowly- 2 varying-envelope approximation as: imprinted non-linear medium SLM phase ∂E 1 2 2 α i = − ∇⊥ − k0n2|E| + i E, (1) ∂z 2k0 2 output intensity where k0 is the wavevector, α is the linear absorption coefficient, n2 is the nonlinear index . The subscript ⊥ input intensity L & R moving refers to the transverse (x, y) plane. We define ∆n = phonons 2 n2|E| as the nonlinear refractive index. What is remarkable about this equation is that it is similar to the Gross-Pitaevskii equation (GPE), which FIG. 1. Principle of the experiment and simplified setup. A describes the evolution the wavefunction Ψ of a weakly- 780 nm laser beam is elongated in the x-direction and sent on a spatial light modulator (SLM). The SLM displays a interacting Bose-Einstein condensate: vertical grating, which imprints a sinusoidal phase modulation 2 of wavevector kx. The SLM plane is imaged at the input of the ∂Ψ(r, t) ~ 2 2 i~ = − ∇ + V(r) + g|Ψ(r, t)| Ψ(r, t), 6.8 cm rubidium vapor cell. This phase modulation creates ∂t 2m two counter-propagating left (L) and right (R) phonons at (2) +kx and -kx represented in green and orange. Initially where V is the trapping potential, m is the bosonic mass in phase opposition (constant input density), the phonons and g is the interaction parameter, which is positive for constructively interfere after some effective time τ = z/c, a stable condensate. giving a maximum of density contrast. The output plane A major difference between the equations (1) and (2) is of the non-linear medium (a rubidium vapor cell) is imaged that in the NLSE, the time-derivative is replaced by a on a camera to study the fringes contrast at z=L. spatial derivative in the direction of propagation z. To map the NLSE onto the GPE, we define an effective time τ = z/c. This space-time mapping means that frequency content, which ensures the creation of counter- each transverse plane inside the non-linear medium is propagating phonons at wave-vectors +kx and −kx, formally analogous to a 2D Bose gas of photons after which corresponds to a standing wave in the BEC density. The density perturbation after a time t, defined by the corresponding effective time of evolution τ. Since 2 2 the z dimension acts as an effective time dimension, this δn(t) = |ψ(t)| − |ψ(t = 0)| , is given by (4): configuration is referred as 2D+1 geometry. δn(t) = US (k)cos(k.r)sin(ωt) , (4) The comparison between the NLSE and the GPE yields 0 expressions for the effective photon mass m = ~k/c and where U is a constant quantifying the excitation strength for the interaction term g = −~∆n. In our case, the and S0(k) is the zero-temperature static structure factor. stability condition g > 0 corresponds to ∆n < 0 (i.e. In [25], the authors measured the Fourier transform ρk of self defocusing regime). One can notice that there is no δn(t), (which also oscillates at ω(kx)) at different times. trapping potential term in Eq. (1), as fluids of light do They extracted the Bogolioubov pulsation from the zero- not need to be held in a trap. crossings of ρk, and the zero-temperature static structure In a weakly-interacting BEC the excitation spectrum factor from the extrema. is given by the Bogoliubov dispersion relation ΩB(kx) To design an analogue technique for paraxial fluids of [24], which can be rewritten for a photon fluid as [1]: light, we had to make two major modifications. First, s we only have access to one value of t which is given, 2 kx 2 2 in our analogy, by the length of the non-linear medium. ΩB(kx) = ( ) + ∆nkx , (3) 2k0 Therefore, instead of probing the density perturbation as function of time, we probe it as function of k at fixed Within space-time mapping, ΩB has units of inverse of a effective time τ = L/c From Eq. (4), it is clear that we length. Eq. (3) shows two regimes of dispersion, whose can still obtain the dispersion relation from the minima p transition happens around kx = k0 |∆n| = 1/ξ with of δn and the structure factor from the maxima of δn. ξ is the healing length. For kxξ < 1, the dispersion is We explain latter in the text how the constant factor U linear and Bogoliubov excitations present a phonon-like can be canceled with a normalization procedure using a behavior: ΩB ≈ kx∆n. For kxξ > 1/, the dispersion non-interacting gas. 2 becomes quadratic: ΩB ≈ kx/2k0. In this regime, The second important change is on the creation of excitations behave like massive free particles. the phase modulation. Since we use a fluid of light, we Bragg spectroscopy in atomic BEC relies on counting can directly imprint a phase on the laser beam with a the number of scattered atoms as a function of the spatial light modulator (SLM) and image it on the input frequency difference between two Bragg beams. A plane of the medium. By imposing a sinusoidal phase variant of this configuration has been presented in [25] pattern on the SLM with a given wavelength and a given and relies on short Bragg pulses at two symmetrically depth, we create two left and right propagating phonons tilted angles to imprint a phase pattern on a BEC (see Fig. 1) with the exact same characteristics as in at time t = 0. The short pulse results in a broad short Bragg pulse spectroscopy. This is in fact a general 3

2 (a) (b) (c) 1 (a) 0.5 -0.8 -0.4 0 0.4 0.8 -0.8 -0.4 0 0.4 0.8 1 0 0 1.01 0.8 0.4 0 2 2 2 (d) 0.3 1 0.6

4 1.0 0 4 -2 -1 0 1 2 0.4 0.2 (e) 6 6 0.2 0.1 0.1 0 (b) L 0.99 -0.2 -0.1 0 0.1 0.2 0 0 0 10 20 30 40 50 60 70 80 0 20 40 60 80

FIG. 2. (a) and (b) show the simulated density modulation for medium of 7.5 cm and a transverse wave-vector of kx=3.6× FIG. 3. (a) Contrast results (smoothed and normalized) for 104 m−1, with ∆n=0 (a) and ∆n=2.1×10−5 (b). (c) and (d) the reference far-off resonance case (grey) with ∆n=0 and show experimental images of the density obtained respectively for a close-to-resonance set (red) with an input power of −1 −1 ◦ for kx=27.8 mm (minimum of contrast) and kx=43.2 mm 90mW, a cell temperature of 128 C and an absorption of (maximum of contrast) in the non-interacting case (∆n=0). 50%. The shift of the minima of contrast toward the smaller These images are taken with a large modulation depth for kx for the -1.5 GHz set, showed by the grey arrow, is an illustration. (e) gives a inset of (d). evidence of the nonlinear effect taking place. (b) Dispersion ω(kx) extracted from the minima of contrast of the reference far-off-resonance set, and for three data sets with decreasing strength of paraxial fluids of light, since any phase detunings ∆=-2.5 GHz, ∆=-2 GHz and ∆=-1.5 GHz with a laser power increased to 105 mW. The dots show the extracted modulation (analogous to any short external potential) th experimental values ω(kx) = pπ/L for the p minimum and can be applied on the initial state of our system. the full lines the fits of the Bogolioubov dispersion. The Propagation of the density perturbation δn(z), resulting values of ∆n are indicated in the legend. calculated for a 1D+1 fluid of light with sinusoidal phase modulation at kx, is shown in Fig. 2 (a) and 2 (b). Interference fringes along the transverse axis x with perturbative approximation. For a given kx, the phase wave-vector kx and fringes along the propagation axis applied on the SLM is φ(x) = 0.1 cos(kx.x). In order z (i.e. effective time τ) with frequency ω(kx) can be to eliminate the unmodulated reflection on the SLM, we observed. Fig. 2(a) presents the simulated pattern in superpose a vertical grating to the horizontal sinusoidal the non-interacting case while Fig 2(b) includes repulsive −5 one and select only the first vertical order in the Fourier interactions given by |∆n|=2.1×10 . The reduction of plane. The SLM is imaged at the entrance of the non- fringe period along z reflects the difference between the linear medium with a demagnification factor of three to free-particle dispersion (a) and the Bogoliubov dispersion increase its resolution. The beam waists at the medium (b) due to the interaction energy. As mentioned earlier, entrance are wx=0.15 mm, wy=1.5 mm. We consider we cannot directly observe the time evolution in our a local density approximation in order to compare the paraxial fluid of light, since we can only see it at experimental data with the 1-dimensional simulations. the effective times τ=0 and τ=L/c. However, we can We verified that the fringe wavelengths were in agreement measure the density contrast at the end of the medium down to 1% with the k imposed on the SLM. (z = L) and identify the contrast minima as function of x As a non-linear medium we used a 6.8 cm rubidium k . For a given modulation k , if we define the contrast x x cell containing a natural mixture of 28% of 87Rb and at t = L/c as C = (n − n )/(n + n ), max min max min 72% of 85Rb. The interaction parameter ∆n is tuned by we can show using Eq. (4) that C = US0(k)sin(ωt). π adjusting the cell temperature, hence the atomic density Minima occurs when ω(k ) = p , where p is an integer. x L in the cell [26, 27], the laser detuning and the laser The knowledge of the successive minimum locations in power. One should be careful that the linear absorption kx will therefore give the dispersion relation, since the α increases as well when going closer to resonance. We corresponding value of ω is known. estimate a maximum level of absorption to not perturb To implement this procedure experimentally, we use the model to be α = 13 cm−1 (60% transmission), and a 780 nm diode-laser for which we can finely adjust make sure to work under this value. The output plane the detuning with respect to the D2 resonance line of of the cell is then imaged on a camera. Typical beam 87Rb. The beam is elongated with two cylindrical lenses picture obtained for the non-interacting case (∆n=0) are −1 to cover the entire surface of the SLM along the x shown on Fig. 2(c) and 2(d) for kx=27.8 mm (contrast −1 direction. A sinusoidal phase modulation along x is minimum) and kx=43.2 mm (contrast maximum). imprinted using a SLM and is kept sufficiently weak This non-interacting case is obtained experimentally by that the resulting intensity modulation does not locally setting a large detuning (∆=-6 GHz) from the 87Rb modify the nonlinear index ∆n. The phase modulation F = 2 to F 0 transition. depth is limited to 10%, to remain within the Bogoliubov We then took 475 images of the cell output with 4

−1 −1 modulations ranging from kx=5 mm to kx=100 mm 1.2 −1 with a step dkx=0.2 mm . To measure the density modulation δn, we normalize the images by a reference 1 taken without phase modulation and then select a central 0.8 window of the fluid (see Fig.2(e)). After integration of the intensity along the y (vertical) axis, we calculate the 0.6 contrast which is directly proportional to δn(τ = L/c). In figure 3(a), we plot the normalized contrast as a 0.4 function of kx and highlight the minima positions for a 0.2 non-interacting and a weakly interacting (∆=-1.5 GHz) 0 fluid of light. For the latter case, we see a clear shift of 0 10 20 30 40 50 60 70 80 90 the contrast minima toward smaller values of kx, which indicates the presence of interaction. In Fig. 3(b), we plot the k positions of the contrast x FIG. 4. Zero temperature static structure factor minima (marked in Fig. 3(a) and shifted vertically of π/L measurement. The experimental conditions are the same as as explained previously) using four different detunings green curve of figure 3, but at a temperature 20 degrees lower (∆=-2.5 GHz, ∆=-2 GHz and ∆=-1.5 GHz with a laser (108◦C), leading to ∆n = 2 × 10−6. The solid red line is the power of 105 mW). The fits using Eq. (3) to extract the model given by 6. Dashed line at S(kx) = 1 is the structure value of ∆n which quantify the interactions are given factor of a non-interacting gas (coherent state). in solid lines. The dots are the experimental points and the full lines are fits to the Bogolioubov relation. The far-off resonance set matches exactly the quadratic measure U which actually depends on k [24]. U is dispersion ω = k2/2k , which corresponds to ∆n = 0 x x 0 determined by the phase imprinting efficiency of the with an uncertainty of 2 × 10−7 for this set. Then, for SLM, the modulation transfer function of the optical the three weakly interacting sets with detunings ∆=- system and the depth of phase modulation. These 2.5 GHz (green), ∆=-2 GHz (yellow), ∆=-1.5 GHz (red), three parameters are not modified while changing ∆n, the fits of the Bogolioubov dispersion relations give the therefore the S(k ) for the interacting case is obtained respective values of ∆n = 5 × 10−6, ∆n = 7 × 10−6 and x by dividing the contrast maxima values for ∆n 6= 0 by ∆n = 10 × 10−6. the calibrated value of U. The results are presented in We can evaluate the sensitivity of this method for Fig. 4 for a weakly interacting fluid (∆n = 2 × 10−6) as resolving weak interaction (small |∆n|). The energy well as the Feynman relation [15, 16, 23]: offset accumulated in the linear part of the dispersion translates into an energy shift at large kx. The dispersion 2 curve with ∆n 6=0 is vertically shifted relative to the non- kx/2k0 S(kx) = . (6) interacting one (∆n = 0). We can calculate this shift at ΩB(kx) high kx as: s Our experimental data (dots) show a good agreement 2 2 2 kx 2 kx with the Feynman relation (solid line) with no free ΩB(kx)−ωlin(kx) = kx|∆n| + ( ) − ≈ k0|∆n| . 2k0 2k0 parameters, since the value of ∆n is obtained using (5) (5). The structure factor can be interpreted as the The value of ∆n can then be directly obtained by the rate of producing excitations at momentum kx in a difference with the non-interacting reference at high kx. Bose gas [14]. Fig. 4 clearly shows that the rate of We verified the validity of this method with the 3 datasets Bragg excitations is highly reduced at low kx. This presented on Fig. 3(b). Knowing the uncertainty on kx to can be explained by quantum depletion in our fluid of −1 be dkx=0.2 mm , it is possible to estimate the smallest light [13]. Indeed, observing S(kx) < 1 at low kx is a nonlinear index value achievable with this technique to be consequence of the creation of correlated pairs at +kx −7 ∆n = 2×10 . This is more than an order of magnitude and −kx which minimize the total energy of the system, better than previous techniques using a group velocity known as quantum depletion [13]. In our experimental measurement [2]. configuration the number of points where this effect is Finally, we present a measurement of the static highly visible (when kx < 1/ξ) is limited by the cell structure factor S(k ) in a fluid of light. The structure length L. Indeed, if we require that the first contrast x p factor is the spatial noise spectrum (normalized to 1 minimum occurs at kx = 1/ξ = k0 |∆n|, then the cell 2π for a coherent state). From (4), we explained how to length is given byL= √ . In order to increase isolate US(kx) using the the contrast maxima values. In 5k0|∆n| order to remove the U dependance, we measure US(kx) the number of point below kx = 1/ξ further experiments for the non-interacting case. In this regime it is known could increase the cell length but this will be ultimately that S(kx) must be equal to 1 at all kx since the beam limited by linear absorption. here is a spatially coherent state. This allows us to 5

I. CONCLUSION of the Tan’s contact and the observation of beyond mean field effects in photon fluids. In this work, we have implemented Bragg spectroscopy in a paraxial fluid of light. This technique has proved to be an essential tool to study ultracold atomic BEC and was missing in fluid of light platforms. We show that ACKNOWLEDGMENTS our implementation is robust and highly sensitive since it allows to measure the interactions an order of magnitude This work has received funding from the French weaker than previously reported. Importantly, we ANR grants (C-FLigHT 138678 and ”Quantum Fluids present a measurement of the zero temperature static of Light” ANR-16-CE30-0021) and from the European structure factor and show a good agreement with the Union Horizon 2020 Research and Innovation Program Feynman relation. This measurement of the structure under Grant Agreement No. 820392 (PhoQuS). Q. G. factor also reveals the presence of quantum depletion, and A. B. thank the Institut Universitaire de France consisting of pair-correlated particles, in a paraxial fluid (IUF). The authors thank Antonine Rochet for support of light. These results open the way to the measurement with 3D figure.

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