# Short Bragg pulse spectroscopy for a paraxial fluids of light

Clara Piekarski  Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-
PSL Research University, Collège de France, Paris 75005, France  Wei Liu
Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research
University, Collège de France, Paris 75005, France  Jeff Steinhauer
Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research
University, Collège de France, Paris 75005, France  Department of Physics,
Technion—Israel Institute of Technology, Technion City, Haifa 32000, Israel
Elisabeth Giacobino  Laboratoire Kastler Brossel, Sorbonne Université, CNRS,
ENS-PSL Research University, Collège de France, Paris 75005, France  Alberto
Bramati  Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL
Research University, Collège de France, Paris 75005, France  Quentin Glorieux
<EMAIL_ADDRESS>Laboratoire Kastler Brossel, Sorbonne Université,
CNRS, ENS-PSL Research University, Collège de France, Paris 75005, France

###### Abstract

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

††preprint: APS/123-QED

Fluids of light in the paraxial configuration have emerged as an original
approach to study degenerate Bose gases [1]. Several important results have
recently established this platform as a potential analogue quantum simulator,
including the demonstrations of superfluidity of light [2, 3], the observation
of the Berezinskii–Kosterlitz–Thouless transition [4] and pre-condensation
[5], the evidence of photon droplets [6] and the creation of analogue rotating
black hole geometries [7, 8]. Paraxial fluids of light rely on the direct
mathematical analogy that can be drawn between the Gross-Pitaevskii equation
describing the mean field evolution of a Bose-Einstein condensate (BEC) and
the non-linear Maxwell equation describing the propagation of light within a
$\chi^{(3)}$ non-linear medium [9, 1]. The growing interest about this
platform comes from the various advantages that makes paraxial fluids of light
a complementary system to atomic BEC. First, optical detection techniques are
highly sensitive (single-photon counting, homodyne detection) and allows for
measuring with high precision the density distribution in position and
momentum space as well as the phase. Second, since there is no gravity force
acting on a photon fluid, there is no need for a trapping potential and
homogeneous density can be easily achieved. Moreover, an external control
potential can be applied either during the entire evolution [3, 10], or only
for a short period of time as in this work.

An essential characterization tool for atomic BEC is coherent Bragg
spectroscopy [11]. The original implementation was based on two-photon Bragg
scattering and allowed for the measurement of the true momentum distribution,
which was significantly narrower than that observed by time of flight [12].
For atomic BEC, it has been realized very early that this technique would also
allow for measuring the dispersion relation [13], which describes how each
frequency component of a wavepacket evolves and the dynamic structure factor,
which is the Fourier transform of the density correlation function [14] and is
essential to the description of many-body systems [15, 16].

Several variants of this method have been developed for exciton-polaritons
[17] and for atomic BEC, including momentum-resolved spectroscopy [18], multi-
branch spectroscopy [19] and tomographic imaging [20]. Most of these
techniques rely on measuring the energy of the condensate’s linear excitations
known as Bogoliubov quasi-particles. Interestingly, in paraxial fluids of
light, the dispersion relation has been recently obtained not using Bragg
spectroscopy but by measuring the group velocity of two counter-propagating
wavepackets in the transverse plane [2]. However, this experiment, as well as
a similar implementation in a thermo-optic medium [21], have a resolution for
weak non-linearities (small sound velocities). Indeed, in this regime, the two
wavepackets propagate too slowly to separate and due to interferences no
analytical dispersion relation can be derived [22].

In this Letter, we implement an optical analogue of Bragg spectroscopy to
measure the phonon dispersion and the static structure factor in a paraxial
fluid of light. We show that short Bragg pulses used for phase imprinting
technique in atomic BEC can be achieved in fluids of light using wavefront
shaping with a spatial light modulator. This technique not only allows for
measuring the dispersion relation with a better resolution at small sound
velocities but it also gives access to the excitation strength for phonons:
the static structure factor. We found that the static structure factor is
significantly reduced from that of free particles, revealing indirectly the
quantum depletion, consisting of pair-correlated particles, in a paraxial
fluid of light [13].

This paper is organized as follows. We first introduce the formalism of a
paraxial fluid of light and the short Bragg pulse technique which inspired our
approach. We describe numerically and experimentally the optical
implementation of Bragg spectroscopy. We then measure the dispersion relation
and evaluate the maximum resolution. Finally, we present a measurement of the
zero temperature static structure factor in agreement with the Feynman
relation for an homogeneous Bose gas [23, 15].

In a third-order nonlinear Kerr medium, the evolution of the electric field is
given by the nonlinear Schrödinger equation (NLSE), written within the
paraxial and slowly-varying-envelope approximation as:

$i\dfrac{\partial E}{\partial
z}=\left(-\dfrac{1}{2k_{0}}\nabla_{\perp}^{2}-k_{0}n_{2}|E|^{2}+i\dfrac{\alpha}{2}\right)E\text{,}$
(1)

where $k_{0}$ is the wavevector, $\alpha$ is the linear absorption
coefficient, $n_{2}$ is the nonlinear index . The subscript $\perp$ refers to
the transverse $(x,y)$ plane. We define $\Delta n=n_{2}|E|^{2}$ as the
nonlinear refractive index.
What is remarkable about this equation is that it is similar to the Gross-
Pitaevskii equation (GPE), which describes the evolution the wavefunction
$\Psi$ of a weakly-interacting Bose-Einstein condensate:

$i\hbar\dfrac{\partial\Psi(\textbf{r},t)}{\partial
t}=\left(-\dfrac{\hbar^{2}}{2m}\nabla^{2}+\mathcal{V}(\textbf{r})+g|\Psi(\textbf{r},t)|^{2}\right)\Psi(\textbf{r},t)\text{
,}$ (2)

where $\mathcal{V}$ is the trapping potential, m is the bosonic mass and $g$
is the interaction parameter, which is positive for a stable condensate.
A major difference between the equations (1) and (2) is that in the NLSE, the
time-derivative is replaced by a spatial derivative in the direction of
propagation $z$. To map the NLSE onto the GPE, we define an effective time
$\tau=z/c$. This space-time mapping means that each transverse plane inside
the non-linear medium is formally analogous to a 2D Bose gas of photons after
the corresponding effective time of evolution $\tau$. Since the $z$ dimension
acts as an effective time dimension, this configuration is referred as 2D+1
geometry.
The comparison between the NLSE and the GPE yields expressions for the
effective photon mass $\overline{m}=\hbar k/c$ and for the interaction term
$\overline{g}=-\hbar\Delta n$. In our case, the stability condition
$\overline{g}>0$ corresponds to $\Delta n<0$ (i.e. self defocusing regime).
One can notice that there is no trapping potential term in Eq. (1), as fluids
of light do not need to be held in a trap.

Figure 1: Principle of the experiment and simplified setup. A 780 nm laser
beam is elongated in the $x$-direction and sent on a spatial light modulator
(SLM). The SLM displays a vertical grating, which imprints a sinusoidal phase
modulation of wavevector $k_{x}$. The SLM plane is imaged at the input of the
6.8 cm rubidium vapor cell. This phase modulation creates two counter-
propagating left (L) and right (R) phonons at +$k_{x}$ and -$k_{x}$
represented in green and orange. Initially in phase opposition (constant input
density), the phonons constructively interfere after some effective time
$\tau=z/c$, giving a maximum of density contrast. The output plane of the non-
linear medium (a rubidium vapor cell) is imaged on a camera to study the
fringes contrast at z=L.

In a weakly-interacting BEC the excitation spectrum is given by the Bogoliubov
dispersion relation $\Omega_{B}(\textbf{k}_{x})$ [24], which can be rewritten
for a photon fluid as [1]:

$\Omega_{B}(\textbf{k}_{x})=\sqrt{(\dfrac{k_{x}^{2}}{2k_{0}})^{2}+\Delta
nk_{x}^{2}}\text{ ,}$ (3)

Within space-time mapping, $\Omega_{B}$ has units of inverse of a length. Eq.
(3) shows two regimes of dispersion, whose transition happens around
$k_{x}=k_{0}\sqrt{|\Delta n|}=1/\xi$ with $\xi$ is the healing length. For
$k_{x}\xi<1$, the dispersion is linear and Bogoliubov excitations present a
phonon-like behavior: $\Omega_{B}\approx k_{x}\Delta n$. For $k_{x}\xi>1/$,
the dispersion becomes quadratic: $\Omega_{B}\approx k_{x}^{2}/2k_{0}$. In
this regime, excitations behave like massive free particles.

Bragg spectroscopy in atomic BEC relies on counting the number of scattered
atoms as a function of the frequency difference between two Bragg beams. A
variant of this configuration has been presented in [25] and relies on short
Bragg pulses at two symmetrically tilted angles to imprint a phase pattern on
a BEC at time $t=0$. The short pulse results in a broad frequency content,
which ensures the creation of counter-propagating phonons at wave-vectors
$+k_{x}$ and $-k_{x}$, which corresponds to a standing wave in the BEC
density. The density perturbation after a time $t$, defined by $\delta
n(t)=|\psi(t)|^{2}-|\psi(t=0)|^{2}$, is given by (4):

$\delta n(t)=US_{0}(k)\text{cos}(\textbf{k.r})\text{sin}(\omega t)\text{ ,}$
(4)

where U is a constant quantifying the excitation strength and $S_{0}(k)$ is
the zero-temperature static structure factor. In [25], the authors measured
the Fourier transform $\rho_{k}$ of $\delta n(t)$, (which also oscillates at
$\omega(k_{x})$) at different times. They extracted the Bogolioubov pulsation
from the zero-crossings of $\rho_{k}$, and the zero-temperature static
structure factor from the extrema.

To design an analogue technique for paraxial fluids of light, we had to make
two major modifications. First, we only have access to one value of $t$ which
is given, in our analogy, by the length of the non-linear medium. Therefore,
instead of probing the density perturbation as function of time, we probe it
as function of $k$ at fixed effective time $\tau=L/c$ From Eq. (4), it is
clear that we can still obtain the dispersion relation from the minima of
$\delta n$ and the structure factor from the maxima of $\delta n$. We explain
latter in the text how the constant factor $U$ can be canceled with a
normalization procedure using a non-interacting gas.

The second important change is on the creation of the phase modulation. Since
we use a fluid of light, we can directly imprint a phase on the laser beam
with a spatial light modulator (SLM) and image it on the input plane of the
medium. By imposing a sinusoidal phase pattern on the SLM with a given
wavelength and a given depth, we create two left and right propagating phonons
(see Fig. 1) with the exact same characteristics as in short Bragg pulse
spectroscopy. This is in fact a general strength of paraxial fluids of light,
since any phase modulation (analogous to any short external potential) can be
applied on the initial state of our system.

Figure 2: (a) and (b) show the simulated density modulation for medium of 7.5
cm and a transverse wave-vector of $k_{x}$=3.6$\times$ 104 m-1, with
$\Delta$n=0 (a) and $\Delta$n=2.1$\times$10-5 (b). (c) and (d) show
experimental images of the density obtained respectively for $k_{x}$=27.8 mm-1
(minimum of contrast) and $k_{x}$=43.2 mm-1 (maximum of contrast) in the non-
interacting case ($\Delta n$=0). These images are taken with a large
modulation depth for illustration. (e) gives a inset of (d).

Propagation of the density perturbation $\delta n(z)$, calculated for a 1D+1
fluid of light with sinusoidal phase modulation at $k_{x}$, is shown in Fig. 2
(a) and 2 (b). Interference fringes along the transverse axis $x$ with wave-
vector $k_{x}$ and fringes along the propagation axis $z$ (i.e. effective time
$\tau$) with frequency $\omega(k_{x})$ can be observed. Fig. 2(a) presents the
simulated pattern in the non-interacting case while Fig 2(b) includes
repulsive interactions given by $|\Delta$n$|$=2.1$\times$10-5. The reduction
of fringe period along $z$ reflects the difference between the free-particle
dispersion (a) and the Bogoliubov dispersion (b) due to the interaction
energy. As mentioned earlier, we cannot directly observe the time evolution in
our paraxial fluid of light, since we can only see it at the effective times
$\tau$=0 and $\tau$=L/c. However, we can measure the density contrast at the
end of the medium ($z=L$) and identify the contrast minima as function of
$k_{x}$. For a given modulation $k_{x}$, if we define the contrast at $t=L/c$
as $C=(n_{max}-n_{min})/(n_{max}+n_{min})$, we can show using Eq. (4) that
$C=US_{0}(k)\text{sin}(\omega t)$. Minima occurs when
$\omega(k_{x})=p\dfrac{\pi}{L},$ where p is an integer. The knowledge of the
successive minimum locations in $k_{x}$ will therefore give the dispersion
relation, since the corresponding value of $\omega$ is known.

Figure 3: (a) Contrast results (smoothed and normalized) for the reference
far-off resonance case (grey) with $\Delta$n=0 and for a close-to-resonance
set (red) with an input power of 90mW, a cell temperature of 128∘C and an
absorption of 50%. The shift of the minima of contrast toward the smaller
$k_{x}$ for the -1.5 GHz set, showed by the grey arrow, is an evidence of the
nonlinear effect taking place. (b) Dispersion $\omega(k_{x})$ extracted from
the minima of contrast of the reference far-off-resonance set, and for three
data sets with decreasing detunings $\Delta$=-2.5 GHz, $\Delta$=-2 GHz and
$\Delta$=-1.5 GHz with a laser power increased to 105 mW. The dots show the
extracted experimental values $\omega(k_{x})=p\pi/L$ for the pth minimum and
the full lines the fits of the Bogolioubov dispersion. The resulting values of
$\Delta n$ are indicated in the legend.

To implement this procedure experimentally, we use a 780 nm diode-laser for
which we can finely adjust the detuning with respect to the D2 resonance line
of 87Rb. The beam is elongated with two cylindrical lenses to cover the entire
surface of the SLM along the $x$ direction. A sinusoidal phase modulation
along $x$ is imprinted using a SLM and is kept sufficiently weak that the
resulting intensity modulation does not locally modify the nonlinear index
$\Delta n$. The phase modulation depth is limited to 10%, to remain within the
Bogoliubov perturbative approximation. For a given $k_{x}$, the phase applied
on the SLM is $\phi(x)=0.1\cos(k_{x}.x)$. In order to eliminate the
unmodulated reflection on the SLM, we superpose a vertical grating to the
horizontal sinusoidal one and select only the first vertical order in the
Fourier plane. The SLM is imaged at the entrance of the non-linear medium with
a demagnification factor of three to increase its resolution. The beam waists
at the medium entrance are $w_{x}$=0.15 mm, $w_{y}$=1.5 mm. We consider a
local density approximation in order to compare the experimental data with the
1-dimensional simulations. We verified that the fringe wavelengths were in
agreement down to 1% with the $k_{x}$ imposed on the SLM.

As a non-linear medium we used a 6.8 cm rubidium cell containing a natural
mixture of 28% of 87Rb and 72% of 85Rb. The interaction parameter $\Delta n$
is tuned by adjusting the cell temperature, hence the atomic density in the
cell [26, 27], the laser detuning and the laser power. One should be careful
that the linear absorption $\alpha$ increases as well when going closer to
resonance. We estimate a maximum level of absorption to not perturb the model
to be $\alpha=13~{}\text{cm}^{-1}$ (60% transmission), and make sure to work
under this value. The output plane of the cell is then imaged on a camera.
Typical beam picture obtained for the non-interacting case ($\Delta n$=0) are
shown on Fig. 2(c) and 2(d) for $k_{x}$=27.8 mm-1 (contrast minimum) and
$k_{x}$=43.2 mm-1 (contrast maximum). This non-interacting case is obtained
experimentally by setting a large detuning ($\Delta$=-6 GHz) from the 87Rb
$F=2$ to $F^{\prime}$ transition.

We then took 475 images of the cell output with modulations ranging from
$k_{x}$=5 mm-1 to $k_{x}$=100 mm-1 with a step $dk_{x}$=0.2 mm-1. To measure
the density modulation $\delta n$, we normalize the images by a reference
taken without phase modulation and then select a central window of the fluid
(see Fig.2(e)). After integration of the intensity along the $y$ (vertical)
axis, we calculate the contrast which is directly proportional to $\delta
n(\tau=L/c)$. In figure 3(a), we plot the normalized contrast as a function of
$k_{x}$ and highlight the minima positions for a non-interacting and a weakly
interacting ($\Delta$=-1.5 GHz) fluid of light. For the latter case, we see a
clear shift of the contrast minima toward smaller values of $k_{x}$, which
indicates the presence of interaction.

In Fig. 3(b), we plot the $k_{x}$ positions of the contrast minima (marked in
Fig. 3(a) and shifted vertically of $\pi/L$ as explained previously) using
four different detunings ($\Delta$=-2.5 GHz, $\Delta$=-2 GHz and $\Delta$=-1.5
GHz with a laser power of 105 mW). The fits using Eq. (3) to extract the value
of $\Delta n$ which quantify the interactions are given 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
dispersion $\omega=k_{x}^{2}/2k_{0}$, which corresponds to $\Delta n=0$ with
an uncertainty of $2\times 10^{-7}$ for this set. Then, for the three weakly
interacting sets with detunings $\Delta$=-2.5 GHz (green), $\Delta$=-2 GHz
(yellow), $\Delta$=-1.5 GHz (red), the fits of the Bogolioubov dispersion
relations give the respective values of $\Delta n=5\times 10^{-6}$, $\Delta
n=7\times 10^{-6}$ and $\Delta n=10\times 10^{-6}$.

We can evaluate the sensitivity of this method for resolving weak interaction
(small $|\Delta n|$). The energy offset accumulated in the linear part of the
dispersion translates into an energy shift at large $k_{x}$. The dispersion
curve with $\Delta n\neq$0 is vertically shifted relative to the non-
interacting one ($\Delta n=0$). We can calculate this shift at high $k_{x}$
as:

$\Omega_{B}(k_{x})-\omega_{lin}(k_{x})=\sqrt{k_{x}^{2}|\Delta
n|+(\frac{k_{x}^{2}}{2k_{0}})^{2}}-\frac{k_{x}^{2}}{2k_{0}}\approx
k_{0}|\Delta n|\text{ .}$ (5)

The value of $\Delta$n can then be directly obtained by the difference with
the non-interacting reference at high $k_{x}$. We verified the validity of
this method with the 3 datasets presented on Fig. 3(b). Knowing the
uncertainty on $k_{x}$ to be $dk_{x}$=0.2 mm-1, it is possible to estimate the
smallest nonlinear index value achievable with this technique to be $\Delta
n=2\times 10^{-7}$. This is more than an order of magnitude better than
previous techniques using a group velocity measurement [2].

Figure 4: Zero temperature static structure factor measurement. The
experimental conditions are the same as green curve of figure 3, but at a
temperature 20 degrees lower (108∘C), leading to $\Delta n=2\times 10^{-6}$.
The solid red line is the model given by 6. Dashed line at $S(k_{x})=1$ is the
structure factor of a non-interacting gas (coherent state).

Finally, we present a measurement of the static structure factor $S(k_{x})$ in
a fluid of light. The structure factor is the spatial noise spectrum
(normalized to 1 for a coherent state). From (4), we explained how to isolate
$US(k_{x})$ using the the contrast maxima values. In order to remove the $U$
dependance, we measure $US(k_{x})$ for the non-interacting case. In this
regime it is known that $S(k_{x})$ must be equal to 1 at all $k_{x}$ since the
beam here is a spatially coherent state. This allows us to measure $U$ which
actually depends on $k_{x}$ [24]. $U$ is determined by the phase imprinting
efficiency of the SLM, the modulation transfer function of the optical system
and the depth of phase modulation. These three parameters are not modified
while changing $\Delta n$, therefore the $S(k_{x})$ for the interacting case
is obtained by dividing the contrast maxima values for $\Delta n\neq 0$ by the
calibrated value of $U$. The results are presented in Fig. 4 for a weakly
interacting fluid ($\Delta n=2\times 10^{-6}$) as well as the Feynman relation
[15, 16, 23]:

$S(k_{x})=\frac{k_{x}^{2}/2k_{0}}{\Omega_{B}(k_{x})}.$ (6)

Our experimental data (dots) show a good agreement with the Feynman relation
(solid line) with no free parameters, since the value of $\Delta n$ is
obtained using (5). The structure factor can be interpreted as the rate of
producing excitations at momentum $k_{x}$ in a Bose gas [14]. Fig. 4 clearly
shows that the rate of Bragg excitations is highly reduced at low $k_{x}$.
This can be explained by quantum depletion in our fluid of light [13]. Indeed,
observing $S(k_{x})<1$ at low $k_{x}$ is a consequence of the creation of
correlated pairs at $+k_{x}$ and $-k_{x}$ which minimize the total energy of
the system, known as quantum depletion [13]. In our experimental configuration
the number of points where this effect is highly visible (when $k_{x}<1/\xi$)
is limited by the cell length $L$. Indeed, if we require that the first
contrast minimum occurs at $k_{x}=1/\xi=k_{0}\sqrt{|\Delta n|}$, then the cell
length is given by $\L=\dfrac{2\pi}{\sqrt{5}k_{0}|\Delta n|}$. In order to
increase the number of point below $k_{x}=1/\xi$ further experiments could
increase the cell length but this will be ultimately limited by linear
absorption.

##  I Conclusion

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 our
implementation is robust and highly sensitive since it allows to measure the
interactions an order of magnitude weaker than previously reported.
Importantly, we present a measurement of the zero temperature static structure
factor and show a good agreement with the Feynman relation. This measurement
of the structure factor also reveals the presence of quantum depletion,
consisting of pair-correlated particles, in a paraxial fluid of light. These
results open the way to the measurement of the Tan’s contact and the
observation of beyond mean field effects in photon fluids.

###### Acknowledgements.

This work has received funding from the French ANR grants (C-FLigHT 138678 and
”Quantum Fluids of Light” ANR-16-CE30-0021) and from the European Union
Horizon 2020 Research and Innovation Program under Grant Agreement No. 820392
(PhoQuS). Q. G. and A. B. thank the Institut Universitaire de France (IUF).
The authors thank Antonine Rochet for support with 3D figure.

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