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 Universit´e,CNRS, ENS-PSL Research University, Coll`egede 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 Schr¨odinger ∗ [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 = − r? − k0n2jEj + 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 n2jEj 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~ = − r + V(r) + gjΨ(r; t)j 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) = j (t)j − j (t = 0)j , is given by (4): configuration is referred as 2D+1 geometry. δn(t) = US (k)cos(k.r)sin(!t) , (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 !(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 j∆nj = 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.