See pages - of paper/HDR_COVER_page.pdf

###### Contents

  1. Introduction
  2. 1 Hot atomic vapor
    1. 1.1 Atomic ensembles
      1. 1.1.1 Light–matter interaction
      2. 1.1.2 Dicke states and collective excitation
      3. 1.1.3 What is a dense atomic medium ?
      4. 1.1.4 Doppler broadening
    2. 1.2 Non-linear optics in dense atomic media
      1. 1.2.1 2–level atoms
      2. 1.2.2 Electromagnetically induced transparency
      3. 1.2.3 Four-wave-mixing
      4. 1.2.4 Slow and fast light
  3. 2 Multimode optical memory
    1. 2.1 Gradient Echo Memory
      1. 2.1.1 Storage time and decoherence
      2. 2.1.2 GEM polaritons
      3. 2.1.3 GEM in rubidium vapor
      4. 2.1.4 Press releases
  4. Published version
  5. 2.2 GEM in a cold atom ensemble
  6. Published version
  7. 3 Quantum optics in fast-light media
    1. 3.1 Imaging with the noise of light
  8. Published version
  9. 3.2 Quantum noise and fast-light medium
  10. Published version
  11. 4 Hydrodynamics of light
    1. 4.1 What is a fluid of light ?
      1. 4.1.1 Hydrodynamic formulation of the non-linear Schrödinger equation
    2. 4.2 Overview of exciton-polaritons in a microcavity
      1. 4.2.1 Driven-dissipative Gross-Pitaevskii equation
    3. 4.3 Superfluidity
    4. 4.4 Injection of angular momentum in superfluids
  12. Published version
  13. 4.5 Fluid of light in the propagating geometry
    1. 4.5.1 Bogoliubov dispersion relation
    2. 4.5.2 Non-linear propagation in a $\chi^{(3)}$ medium
  14. Published version
  15. 5 Outlooks and future projects
    1. 5.1 Fluid of light in the propagating geometry
      1. 5.1.1 Shockwaves dynamics
      2. 5.1.2 Superfluid flow around a defect
      3. 5.1.3 Optomechanical signature of superfluidity
      4. 5.1.4 Analogue gravity
      5. 5.1.5 Disorder and interactions
    2. 5.2 Quantum simulation with photons
      1. 5.2.1 From ultracold atoms to quantum fluids of light
      2. 5.2.2 Double-$\Lambda$ configuration
      3. 5.2.3 Quantum optics
    3. 5.3 Nano-optics

## Introduction

Looking back over almost a decade of research after I graduated in 2010, I
realized that what fascinates me the most in physics are the analogies between
apparently unrelated phenomena. Connecting microscopic descriptions and
macroscopic quantities as does thermodynamics, or the general framework of
spin 1/2 particle from NMR to quantum information… These are just two examples
but one can find this everywhere in modern physics.

I cover in this work the links between what is known today as quantum
technologies and the more fundamental topic of quantum fluids. I have always
envisioned myself as a fundamental physicist, remotely interested in real-life
applications. However, it is funny to note that part of what I believed to be
fundamental science 10 years ago (quantum memories, squeezing, entanglement…)
is today entering in the phase of technological development and applications
with the major effort of the EU towards quantum technologies. This is exactly
what should motivate basic science: sometimes it does bring applications to
make our life easier and a better society and sometimes it just provides
knowledge itself without any immediate application (which also pushes toward a
better society by making it smarter). I am curious to see what quantum fluids
will bring in 10 years. This work summarizes my (modest) contribution to this
field.

The main goal of this manuscript is to provide the tools to connect non-linear
and quantum optics to quantum fluids of light. Because the physics of matter
quantum fluids and fluids of light has long been the territory of condensed
matter physicists, it is uncommon to find textbooks which draw the analogies
with quantum optics. I have taken the reverse trajectory, being trained as a
quantum optician and moving progressively to quantum gases and quantum fluids
of light. Naturally, I try to use the concepts and resources developed by the
quantum optics community to improve experiment about fluids of light, but I
also reverse the approach and ask a very simple question: what does the
concept of fluid of light bring to our understanding of non-linear and quantum
optics ? It can be rephrased as an operational question: which new effects can
we predict (and possibly observe) using the photon fluid formalism ?

This is actually a very exciting time, as we have recently demonstrated the
validity of this approach with an experiment about the dispersion relation in
a fluid of light [1]. In this experiment, described in details in the chapter
4, we have shown that light in a non-linear medium follows the Bogoliubov
dispersion: a constant group velocity at small wavevectors and a linear
increase with $k$ at large wavevectors. If you trust the formal analogy
between the non-linear Schrödinger equation and the paraxial propagation of
light in a non-linear medium, this result is not surprising. However, what is
amazing is that, this description allowed us to design a non-linear optics
experiment and helped us to understand it. This experiment can also be thought
as a correction to the Snell-Descartes law in a non-linear medium. In this
analogy,the group velocity is directly linked to the position of the beam in
the transverse plane at the output of the medium. A linear increase with $k$
at large wavevectors translates to the standard refraction law: increasing the
incidence angle will increase the distance from the optical axis after the
medium. However, at small angle the constant (non-zero) group velocity tells
us that, whatever the angle of incidence, the beam will exit the medium at the
same position ! The refraction law is independent of the incident angle.
Moreover, another idea can be extracted from quantum gases formalism: the
group velocity depends on the medium density (at small k). Therefore the
refraction law does not depend on the incident angle but on the light
intensity ! Can we do something with this ? I don’t know. Maybe we can use
this novel understanding to image through non-linear medium. Or maybe we just
understand a bit more of non-linear optics now.

This time is a very exciting one also because now we know that this approach
works and we have tens of ideas for novel non-linear optics experiments
testing the cancellation of drag force due to superfluidity, observing the
Hawking radiation, or the Zel’dhovich effect… The next challenge is to bring
this description to the field of quantum optics, where quantum noise and
entanglement are of primary importance. What is the hydrodynamic analogue of
squeezing or of an homodyne detection ? Understanding the effects of
interactions in complex quantum systems beyond the mean-field paradigm
constitutes a fundamental problem in physics. This manuscript just briefly
ventures in this territory… But this will be, for sure, the next direction of
my scientific career.

In fact I did not venture in this territory, but this manuscript intends to
provide the tools to do so in the future. Chapter 1 is a brief summary of what
are the tools needed for quantum optics in a warm atomic medium: the two-level
atoms model, electromagnetically induced transparency, four-wave-mixing,
cooperative effects and decoherence. In chapter 2, I describe one type of
optical quantum memory based on the gradient echo memory protocol. I present
two implementations: in a warm vapor and in high optical depth cold atomic
cloud and cover how these two implementations are complementary with their
specific strengths and weaknesses. In chapter 3, I move to quantum optics with
the study of imaging using the noise properties of light and propagation of
quantum noise in a fast light medium. I complete the description of my
previous works in chapter 4 including more recent experiments about fluid of
light in an exciton-polariton microcavity and in an atomic vapor. The final
chapter is devoted to describe several outlooks and collaborative projects I
have recently initiated.

##  Chapter 1 Hot atomic vapor

###  1.1 Atomic ensembles

When learning about light-atom interaction, various approaches can be
discussed. Light can be described as a classical electromagnetic field or as
quantum elementary excitations: the photons. Similarly an atom can be seen as
a classical oscillating dipole or treated using quantum mechanics. The
description of the interaction can then take any form mixing these 4 different
perspectives. While for most experiments a classical approach is sufficient,
it is sometimes needed to invoke $\hbar$ and its friends to explain a specific
behaviour. In this work, I will try, as much as possible, to avoid an
artificial distinction between quantum and classical phenomena, as it does not
bring much to the understanding of the effects. Sometimes, a purely classical
description is even more intuitive.

In this first section, I remind readers of the basic tools to appreciate
atomic ensemble physics, without entering into the details and complexities of
genuine alkali atomic structures. I describe the differences in light-matter
interaction between the case of one single atom and an atomic ensemble, and
explain what I call a dense atomic medium. A short discussion will illustrate
the relationship between collective excitations and Dicke states [2]. This
work is mainly focused on warm atomic ensembles (with the notable exception of
section 2.2) and therefore a discussion on the role of Doppler broadening and
the assumptions linked to it will conclude this part.

####  1.1.1 Light–matter interaction

Let us start with the simplest case: one atom interacting with light in free
space. Two processes, time-reversals of each other, can happen: absorption and
stimulated emission111Obviously, a third process, called spontaneous emission,
is also possible, but for this specific discussion we do not need it.. An
important question is: what is the shadow of an atom ? Or more precisely what
is the dipole cross-section $\sigma_{0}$ of one atom interacting with light in
free space ? It is interesting to note that the answer to this question does
not depend on the description of the atom as a classical oscillating dipole or
a two-level atom. With $\lambda$ being the light wavelength, the resonant
cross section for a two-level atom222More generally, the cross section for an
atom with $F_{g}$ and $F_{e}$ being the total angular momenta for the ground
and the excited states respectively, has the form
$\sigma=\frac{2F_{e}+1}{2F_{g}+1}\frac{\lambda^{2}}{2\pi}$. can be written as:
[3]:

$\sigma_{0}=\frac{3\lambda^{2}}{2\pi}.$ (1.1)

From this relation, we can immediately conclude that it is a difficult task to
make light interact with a single atom. Indeed, focusing light in free-space
is limited by diffraction to a spot of area $\sim\pi\lambda^{2}/4$. Therefore,
even a fully optimized optical system, will not reach a focusing area of
$\sigma_{0}$. What is surprising here, is that it does not depend on the
choice of your favorite atom.

Nevertheless various tricks can be tested to improve this coupling. A common
criteria to discuss the interaction strength is called cooperativity $C$ and
is given by the ratio between the dipole cross section and the mode area of
the light:

$C=\frac{\sigma_{0}}{\text{Area}_{\text{mode}}}.$ (1.2)

The cooperativity gives the ratio between the photons into the targeted mode
and those emitted to other modes.

A simple way of increasing $C$ is to put your single atom in an optical
cavity. By doing so you will approximately multiply C by the number of round
trips inside the cavity. The cooperativity will be modified to:

$C=\frac{\sigma_{0}}{\text{Area}_{\text{mode}}}\frac{1}{T},$ (1.3)

with $T$ being the output mirror transmission. A second approach would be to
reduce the area of the mode below diffraction limit. It is obviously
impossible with a propagating field but it has been observed that using the
evanescent field near a nano-structure can indeed provide a mode area well
below $\pi\lambda^{2}/4$. In this work we will use another way of improving
coupling: by adding more than one atom we can increase the cooperativity by
the number of atoms $N$ as

$C=\frac{\sigma_{0}}{\text{Area}_{\text{mode}}}N.$ (1.4)

The cooperativity is therefore equal to the inverse of the number of atoms
needed to observe non-linear effects. Unfortunately, adding more atoms comes
with a long list of associated problems that we will discuss throughout this
work.

####  1.1.2 Dicke states and collective excitation

Getting a large cooperativity means having a large probability for a photon to
be absorbed by the atomic medium. However, we find here an important
conceptual question: if a photon is absorbed by the atomic ensemble, in what
direction should I expect the photon to be re-emitted ? If there is an optical
cavity around the atomic ensemble, it is natural to expect the photon to be
more likely emitted in the cavity mode, since it is most strongly coupled to
the atoms. The coupling efficiency $\beta$ is the ratio between the decay rate
in the cavity mode to the total decay rate and is given by:

$C=\frac{\beta}{1-\beta}.$ (1.5)

But, what if we do not place the atoms in a cavity ? Why shouldn’t we expect
the recovered field in any arbitrary direction ? This would make the job of an
experimental physicist way trickier…

##### Dicke states

We can get a physical intuition of why light is mainly re-emitted in the
forward direction by introducing Dicke states. In 1954, Robert Dicke predicted
in Ref. [2] that the behaviour of a cloud of excited atoms would change
dramatically above a density of 1 atom per $\lambda^{3}$. Indeed, when the
inter-atomic distance becomes smaller than the wavelength of the emitted
photons, it becomes impossible to distinguish which atoms are responsible for
the emission of individual photons. This indistinguishability lead to a
spontaneous phase-locking of all atomic dipoles everywhere in the medium, and
therefore a short and directional burst of light is emitted in the forward
direction. The anisotropic nature of the emission can be understood simply as
a constructive interference due to the alignment of atomic dipoles thanks to
the dipole-dipole interaction [4, 5].

##### Collective excitation

When the atomic density is lower than $\lambda^{-3}$, the dipole-dipole
interaction becomes negligible but, fortunately, directional emission can
still occur. The mechanism involved here is similar to the Dicke prediction.
When one photon interacts with the atomic ensemble, the created excitation is
delocalized in the entire cloud. Every atom participates in the absorption
process and each of them retains the phase of the incoming field in the
coherence between the ground and the excited states. If the Fock state with
only one photon is sent into the medium, the collective state of the atomic
cloud $|\underline{e}\rangle$ will be written as a coherent superposition of
all possible combinations of a single atom excited $|e\rangle$ and all the
other ones in the ground state $|g\rangle$:

$|\underline{e}\rangle=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}|g_{1},g_{2},...,g_{i-1},e_{i},g_{i+1},...,g_{N}\rangle.$
(1.6)

Once again the collective enhancement comes from a constructive interference
effect [6]. During the scattering process, an atom at position $\vec{r}_{i}$
in the cloud interacts with an incoming photon of wavevector $\vec{k}$ and
therefore acquires (in the rotating frame) a static phase term
$e^{i\vec{k}\vec{r}_{i}}$ in the collective state superposition
$|\underline{e}\rangle$. After the scattering333We assume here an elastic
Rayleigh scattering process and therefore the norm of $\vec{k}^{\prime}$ is
fixed, only its direction is a free parameter. of a photon in the direction of
a wavevector $\vec{k}^{\prime}$, the atomic ensemble is back in the collective
ground state $|\underline{g}\rangle$ but with an additional phase. The cloud
state is:

$\left(\frac{1}{\sqrt{N}}\sum_{i=1}^{N}e^{i(\vec{k}-\vec{k}^{\prime}).\vec{r}_{i}}\right)|\underline{g}\rangle.$
(1.7)

The global factor before $|\underline{g}\rangle$ gives the square root of the
probability of this process. To benefit from the $\frac{1}{N}$ enhancement the
phase term must be equal to zero and therefore we must have
$\vec{k}=\vec{k}^{\prime}$. The probability of scattering is then maximum in
the forward direction with the same wavevector $\vec{k}$ as the incident
photon.

##### Decoherence

With these two configurations (Dicke states and collective enhancement) we
have understood qualitatively why emission will be in the forward direction
after scattering through atomic vapor. Obviously there are many restrictions
to these simple explanations, because the phase coherence is not always
conserved. These troublemakers are grouped under the term: decoherence.

One main reason for the decoherence will come from the fact that atoms are
moving (and moving quite fast in a hot atomic vapor). Taking this into
account, the position of re-emission for atom $i$ will not be $\vec{r}_{i}$
and cancellation of the phase $e^{i(\vec{k}-\vec{k}^{\prime}).\vec{r}_{i}}$
will not be perfect anymore. This will be discussed in more detail in
paragraph 1.1.4.

Another drawback with real atoms is that they often have more than one excited
state with slightly different energies. If light couples to these states (even
with different coupling constants), the atomic ensemble state will acquire a
temporal phase of the order of $e^{i\Delta Et/\hbar}$ with $\Delta E$ being
the characteristic energy difference between excited states. We will come back
to this point when we will discuss 2-level atoms and rubidium D-lines (see
paragraph 1.2.1).

####  1.1.3 What is a dense atomic medium ?

As an associate professor, I am member of a CNU (Conseil National des
Universités) thematic section called dilute media and optics. Then why do I
entitle this work ”Quantum Optics in dense atomic medium”. What does dense
mean in this context ? I do not mean that I have been working with dense media
in condensed matter sense (with the notable exception of section 4.4) but
rather with an optically dense cloud. In the continuation I describe the
relationship between temperature and atomic density in a warm vapor and the
method to provide a good measure of the optical density: the optical depth.

##### Atomic density

A strong advantage of atomic vapors is that their density can be tuned at will
by simply changing their temperature. As alkali vapors are not perfect gases,
there is a correction to the Boyle law given by C. Alcock, V. Itkin and M.
Horrigan in Ref. [7]. The vapor pressure $p$ in Pascal is given by:

$p(\text{Pa})=10^{A+\frac{B}{T}},$ (1.8)

with $A=9.318$ and $B-4040$ K. The atomic density $n_{at}$ is then given by
dividing the vapor pressure by $k_{B}$ the Boltzmann constant times the
temperature $T$:

$n_{at}=\frac{p(\text{Pa})}{k_{B}T}.$ (1.9)

Figure 1.1: Atomic density of rubidium as function of temperature in Celsius.
Right scale is given in unit of $\lambda^{3}$ for the D2 line.

##### Optical depth

Taking into account the density, light propagating in an atomic medium will
have a linear absorption $\alpha$ given by:

$\alpha=n_{at}\sigma_{0}.$ (1.10)

The optical depth is simply defined as the product of $\alpha$ by $L$, the
length of the atomic medium. Based on this definition we can directly write
the Beer law of absorption for a beam of initial intensity $I_{0}$:

$I=I_{0}\exp{[-\alpha L]}.$ (1.11)

For the D2 line of rubidium $\sigma_{0}=1.25\times 10^{-9}$ cm-2. We can note
that the value is slightly different from Eq. 1.1 due to the multiplicity of
atomic states.

##### Saturation intensity

This derivation is only valid on resonance and with a weak electromagnetic
field. In fact, saturation can modify the behaviour of the medium. To quantify
this effect, we introduce the saturation intensity $I_{sat}$ which is defined
as the value of intensity for which the cross section is reduced by half
compared to the low intensity case. This leads to the redefinition of the
atomic cross section $\sigma$ as [8]:

$\sigma=\frac{\sigma_{0}}{1+I/I_{sat}}.$ (1.12)

We can refine even more this model by adding a detuning
$\Delta=\omega_{L}-\omega_{0}$ between the excitation (laser) field frequency
and the atomic transition. In this case we have:

$\sigma(\Delta)=\frac{\sigma_{0}}{1+4(\Delta/\Gamma)^{2}+I/I_{sat}}.$ (1.13)

The saturation intensity is defined for a resonant excitation. However, in
practice, it is often useful to know if the atomic medium is saturated or not
at a large detuning from the resonance. A simple way to obtain a quick insight
on this question is defining an off-resonance saturation intensity
$I_{sat}(\Delta)$. It is given by [8]:

$I_{sat}(\Delta)=\left[1+4\frac{\Delta^{2}}{\Gamma^{2}}\right]I_{sat}^{0},$
(1.14)

with $I_{sat}^{0}$ the resonant saturation intensity. We have then
reformulated Eq. 1.12 with $I_{sat}(\Delta)$ instead of $I_{sat}$. We can
remember that, when $I=I_{sat}(\Delta)$, the cross section is reduced by half
compared to low intensity.

####  1.1.4 Doppler broadening

Unfortunately, in an atomic cloud, every atom does not contribute equally to
the light-matter interaction. This is a consequence of the Doppler effect.
Indeed, atoms moving at the velocity $\vec{v}$ will result to a frequency
shift $\sim\vec{k}\vec{v}$. In a warm atomic vapor this effect is crucial to
understand the dynamics of the system, as Doppler broadening can be much
greater than the natural linewidth $\Gamma$. For an atom of a mass $m$ at
temperature $T$ the Doppler linewidth $\Gamma_{D}$ is given by:

$\Gamma_{D}=\sqrt{\frac{k_{B}T}{m\lambda^{2}}}.$ (1.15)

At 100∘C, this gives $\Gamma_{D}\approx 250$ MHz for Rb D2 line, compared to
$\Gamma\approx 6$ MHz.

An important consequence is that the atomic response, derived in the next
section, will have to be integrated over the atomic velocity distribution to
obtain quantitative predictions. We will not include this calculation here, as
it adds unnecessary complexity but no immediate intelligibility. We refer the
interested reader to [9, 10, 11]

Figure 1.2: Doppler linewidth of rubidium D2 line as function of temperature.

###  1.2 Non-linear optics in dense atomic media

> There is no two-level atom and rubidium is not one of them
>
> William D. Phillips

This famous quote from Bill Phillips reminds us, that how useful can the two-
level atom model be, it still remains simplistic compared to the complexity of
alkali atomic structures. In this section, I briefly recall three important
features of this model: linear absorption $\text{Im}(\chi^{(1)})$, linear
phase shift $\text{Re}(\chi^{(1)})$ and non-linearity $\chi^{(3)}$ near a two-
level atomic resonance. This well known derivation is an important concept
settling the background of this work. Along this section I will highlight the
theoretical tools one by one to progressively cover more and more complex
situations. Specifically, we are in this work focusing on non-linear effects
with large intensities. This means not only a large $\text{Re}(\chi^{(3)})$
but also a large $\text{Re}(\chi^{(3)})\times I$.

Therefore I introduce:

  * •

the saturation of non-linearity at larger intensity ($\chi^{(5)}$ correction
term);

  * •

the concept of electromagnetically induced transparency (EIT) and possible
extension when the probe beam is not perturbative anymore;

  * •

the process of four-wave-mixing.

####  1.2.1 2–level atoms

Let us consider the interaction of a monochromatic electric field with a
system of $N$ two-level atoms. This interaction process can be described by
the optical Bloch equation [12]:

$\frac{d\hat{\rho}}{dt}=-i\frac{i}{\hbar}\left[\hat{H},\hat{\rho}\right]-\hat{\Gamma}\hat{\rho},$
(1.16)

where $\hat{\rho}$ is the density matrix of the atomic system, $\Gamma$ is the
decay rate of the excited state, $\hat{H}=\hat{H}_{0}+\hat{V}$ is the
Hamiltonian of the system with the non-perturbative part $\hat{H}_{0}$ and the
interaction $\hat{V}$ which can be written as
$\hat{V}=-\mathbf{d}\cdot\mathbf{E}=-\hbar\Omega/2$ in the dipole
approximation. $\Omega$ is the Rabi frequency.

We can denote the ground and the excited state of the atom as $|g\rangle$ and
$|e\rangle$ respectively with the resonant transition frequency $\omega_{eg}$.
With these notations we can rewrite the Bloch equation (1.16) for the slowly
varying amplitudes $\sigma_{ij}(t)$ of the density matrix elements
$\rho_{ij}(t)=\sigma_{ij}(t)\exp(-i\omega_{ij}t)$ in the following form:

$\displaystyle\dot{\rho}_{gg}$ $\displaystyle=$ $\displaystyle
i\frac{\Omega}{2}\left(\sigma_{ge}-\sigma_{eg}\right)+\Gamma\rho_{ee}$
$\displaystyle\dot{\rho}_{ee}$ $\displaystyle=$
$\displaystyle-i\frac{\Omega}{2}\left(\sigma_{ge}-\sigma_{eg}\right)-\Gamma\rho_{ee}$
$\displaystyle\dot{\sigma}_{ge}$ $\displaystyle=$
$\displaystyle-i(\Delta-i\Gamma/2)\sigma_{ge}-i\frac{\Omega}{2}\left(\rho_{ee}-\rho_{gg}\right),$
(1.17)

where $\Omega$ is the Rabi frequency of the probe field and
$\Delta=\omega-\omega_{eg}$ is the laser detuning from the excited state. To
obtain this system of equations we have applied the Rotating Wave
Approximation (RWA). This approximation allows to eliminate the fast decaying
terms and to rewrite the Bloch equation for slow-varying amplitudes [12]. The
elements $\rho_{gg}$ and $\rho_{ee}$ correspond to population of the ground
and the excited states respectively, while the elements
$\sigma_{eg}=\sigma_{ge}^{*}$ correspond to the atomic coherence. We can
rewrite the system of equations (1.17) as:

$\displaystyle\dot{\rho}_{ee}-\dot{\rho}_{gg}$ $\displaystyle=$
$\displaystyle-i\Omega\left(\sigma_{ge}-\sigma_{eg}\right)-\Gamma\left(\rho_{ee}-\rho_{gg}+1\right)$
$\displaystyle\dot{\sigma}_{ge}$ $\displaystyle=$
$\displaystyle-i(\Delta-i\Gamma/2)\sigma_{ge}-i\frac{\Omega}{2}\left(\rho_{ee}-\rho_{gg}\right),$
(1.18)

taking into account the condition that $\rho_{gg}+\rho_{ee}=1$. Because the
amplitudes $\rho_{gg}$, $\rho_{ee}$ and $\sigma_{eg}$ are slow-varying we can
assume that $\dot{\rho}_{gg}=\dot{\rho}_{ee}=\dot{\sigma}_{ge}=0$ and the
solution of (1.18) can be found in the following form:

$\displaystyle\sigma_{ge}$ $\displaystyle=$
$\displaystyle-\frac{\Omega/2\left(\rho_{ee}-\rho_{gg}\right)}{\Delta-i\Gamma/2},$
$\displaystyle\rho_{ee}-\rho_{gg}$ $\displaystyle=$
$\displaystyle-\frac{\Delta^{2}+\Gamma^{2}/4}{\Delta^{2}+\Gamma^{2}/4+\Omega^{2}/2}.$
(1.19)

The response of the medium on the interaction with light can be described in
terms of the atomic polarization $\mathbf{P}$. The vector of polarization
relates to the electric field with a proportional coefficient:

$\mathbf{P}=\varepsilon_{0}\chi\mathbf{E},$ (1.20)

where $\chi$ is the atomic susceptibility.

In general, if we neglect frequency conversion processes, the polarization
$\mathbf{P}$ can be written as an expansion in Taylor series in the electric
field as:

$P=\varepsilon_{0}\chi^{(1)}E+\varepsilon_{0}\chi^{(2)}|E|^{2}+\varepsilon_{0}\chi^{(3)}|E|^{2}\cdot
E+\varepsilon_{0}\chi^{(4)}|E|^{4}+\varepsilon_{0}\chi^{(5)}|E|^{4}\cdot
E+...$ (1.21)

Here $\chi^{(1)}$ is known as a linear susceptibility, higher order terms are
known as a second-order, a third-order, or n-order susceptibilities. In
general, for anisotropic materials the susceptibility is a $(n-1)$-order rank
tensor. In our consideration we expand the Taylor series up to the rank $5$ to
take into account the nonlinear response of the atomic medium. For a centro-
symmetric medium, the even terms vanish and the polarization of the system can
be written as [12]:

$P=\varepsilon_{0}\chi^{(1)}E+\varepsilon_{0}\chi^{(3)}|E|^{2}\cdot
E+\varepsilon_{0}\chi^{(5)}|E|^{4}\cdot E+...$ (1.22)

The polarization can be found in terms of the density matrix elements:

$P=N\ \mu_{eg}\sigma_{ge},$ (1.23)

where $\mu_{eg}$ is the dipole moment of the transition. From this expression
we can find a full polarization of the atomic system:

$P=-\frac{N|\mu_{eg}|^{2}(\Delta+i\Gamma/2)}{\hbar(\Delta^{2}+\Gamma^{2}/4+\Omega^{2}/2)}E=-\frac{4N|\mu_{eg}|^{2}}{\hbar\Gamma^{2}}\cdot\frac{\Delta+i\Gamma/2}{1+4\Delta^{2}/\Gamma^{2}+2\Omega^{2}/\Gamma^{2}}E.$
(1.24)

We can write $P$ as function of the saturation intensity $I_{\text{sat}}$
using:

$\frac{I}{I_{\text{sat}}}=2\left[\frac{\Omega}{\Gamma}\right]^{2}\text{ with
}I=\frac{1}{2}n_{0}\varepsilon_{0}cE^{2}.$ (1.25)

One obtains:

$P=-\frac{4N|\mu_{eg}|^{2}}{\hbar\Gamma^{2}}\cdot\frac{\Delta+i\Gamma/2}{1+4\Delta^{2}/\Gamma^{2}+I/I_{\text{sat}}}E.$
(1.26)

We should note that the real part of the susceptibility corresponds to the
refractive index of the medium, while the imaginary part gives information
about the absorption.

Next we calculate the zeroth, first and second contributions to the
polarization of a collection of two-level atoms. By performing a power series
expansion of Eq. (1.26) in the quantity $I/I_{\text{sat}}$:

$P\approx-\frac{4N|\mu_{eg}|^{2}}{\hbar\Gamma^{2}}\frac{\Delta+i\Gamma/2}{(1+4\Delta^{2}/\Gamma^{2})}E\cdot\left[1-\frac{I/I_{\text{sat}}}{1+4\Delta^{2}/\Gamma^{2}}+\frac{I^{2}/I_{\text{sat}}^{2}}{(1+4\Delta^{2}/\Gamma^{2})^{2}}\right].$
(1.27)

We now equate this expression with Eq. 1.22 to find the three first orders of
the atomic polarization:

$\displaystyle\varepsilon_{0}\chi^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{4N|\mu_{eg}|^{2}}{\hbar\Gamma^{2}}\cdot\frac{\Delta+i\Gamma/2}{(1+4\Delta^{2}/\Gamma^{2})}$
$\displaystyle\varepsilon_{0}\chi^{(3)}|E|^{2}$ $\displaystyle=$
$\displaystyle\
\frac{4N|\mu_{eg}|^{2}}{\hbar\Gamma^{2}}\cdot\frac{\Delta+i\Gamma/2}{(1+4\Delta^{2}/\Gamma^{2})}\cdot\frac{I/I_{\text{sat}}}{1+4\Delta^{2}/\Gamma^{2}}$
$\displaystyle\varepsilon_{0}\chi^{(5)}|E|^{4}$ $\displaystyle=$
$\displaystyle-\frac{4N|\mu_{eg}|^{2}}{\hbar\Gamma^{2}}\cdot\frac{\Delta+i\Gamma/2}{(1+4\Delta^{2}/\Gamma^{2})}\cdot\frac{I^{2}/I_{\text{sat}}^{2}}{(1+4\Delta^{2}/\Gamma^{2})^{2}}.$
(1.28)

We introduce the usual444The coefficients $1,3$ and $10$ are used because we
are only concern with the non-linear effects conserving the input frequency.
For example the $\chi^{(3)}$ term can lead to several frequency conversion,
and we only keep triplets like: $(+\omega,+\omega,-\omega)$,
$(+\omega,-\omega,+\omega)$ and $(-\omega,+\omega,+\omega)$. This coefficient
is given by the binomial coefficient $\tbinom{3}{2}$. power series expansion:

$\chi_{\text{eff}}=\chi^{(1)}+3\chi^{(3)}|E|^{2}+10\chi^{(5)}|E|^{4}.$ (1.29)

This notation allows us to write the effective refractive index
$n_{\text{eff}}$ as:

$n_{\text{eff}}^{2}=1+\chi_{\text{eff}}.$ (1.30)

We use the standard definition [12] of the non-linear indices ($n_{2}$ and
$n_{3}$) : $n_{\text{eff}}=n_{0}+n_{2}I+n_{3}I^{2}$ and we can expand
$n_{\text{eff}}^{2}$ in:

$n_{\text{eff}}^{2}=n_{0}^{2}+2n_{0}n_{2}I+(2n_{0}n_{3}+n_{2}^{2})I^{2}.$
(1.31)

We can then connect the indices to the expression of $\text{Re}(\chi)$. We
show the scaling of $\text{Re}(\chi)$ in Figure 1.3.

$\displaystyle n_{0}$ $\displaystyle=$
$\displaystyle\sqrt{1+Re\left[\chi^{(1)}\right]},$ $\displaystyle n_{2}$
$\displaystyle=$
$\displaystyle\frac{3Re\left[\chi^{(3)}\right]}{n_{0}^{2}\varepsilon_{0}c},$
$\displaystyle n_{3}$ $\displaystyle=$
$\displaystyle\frac{20Re\left[\chi^{(5)}\right]}{n_{0}^{3}\varepsilon_{0}^{2}c^{2}}-\frac{n_{2}^{2}}{2n_{0}}.$
(1.32)

Figure 1.3: $\text{Re}(\chi^{(1)})$ in black ,$\text{Re}(\chi^{(3)})$ in
dashed green,$\text{Re}(\chi^{(5)})$ in dashed red

##### What to optimize ?

When looking at the previous derivation of the non-linear susceptibility, we
see that 3 different quantities can be optimized depending on the goal of your
experiments.

  * •

If you want to observe non-linear effects at the single photon level, for
example to create a photonic transistor, you want to optimize the value of
Re$\left[\chi^{(3)}\right]$ in the limit $I\ll I_{sat}$;

  * •

If you want to observe dispersive effects as slow and fast light propagation,
you will want to optimize the value of derivative of Re$\left[\chi\right]$ as
function of the frequency. This is the configuration we will study in chapter
3;

  * •

If you are interested in a large value for $\Delta n=n_{2}\times I$, you have
access to 2 knobs: increasing $n_{2}$ or increasing $I$. However you must stay
within the limit of $I<I_{sat}$ otherwise the Taylor expansion does not hold
anymore. That is why we have done the calculation to the next order
$I/I_{sat}$. This is the configuration we will study in chapter 4.

##### Scaling for effective 2-level atoms.

We briefly review the dispersive limit. When the laser is detuned far enough
from the atomic transition, we have a simplification of the atomic response.

First, the approximation of the 2-level atoms becomes more precise because the
contribution from all the levels averages to an effective contribution. This
is the case when the detuning $\Delta$ is much larger than the hyperfine
splitting energy scale (typically around $\sim 500$ MHz for Rb).
The second consequence is that the absorptive part of the susceptibility
Im$(\chi)$ becomes negligible with respect to the dispersive part Re$(\chi)$.
In this limit, the medium is virtually transparent but there is still a non-
negligible phase shift (a linear and a non-linear one). At $\Delta\gg\Gamma$,
we can simplify Eq. 1.28 to obtain for the linear part of the absorption:

$\text{absorption: Im}(\chi^{(1)})\propto\frac{N}{\Delta^{2}},$ (1.33)

and for the non-linear dispersive part:

$\text{phase shift: Re}(\chi^{(3)})\propto\frac{N}{\Delta^{3}}.$ (1.34)

The non-linear absorption is neglected as it scales with
$\frac{1}{\Delta^{4}}$, as well as the linear phase shift which just results
in a redefinition of the phase reference.

In the far detuned limit, an intuitive idea to improve the non-linear phase
shift is to simply increase the atomic density, by rising the cloud
temperature as described in section 1.1.3. This is indeed true as
Re$(\chi^{(3)})$ scales with $N$. However, this is often critical to conserve
a large (fixed) transmission while increasing the non-linear phase shift. This
condition of fixed transmission means that $\frac{N}{\Delta^{2}}$ is a
constant. In other words, we can rewrite the phase shift Re$(\chi^{(3)})$ as
this constant times $\frac{1}{\Delta}$.

We see that the intuitive vision is no longer valid if we want to keep a fixed
transmission: to maximize the phase shift at a given transmission it is
therefore favorable to reduce $\Delta$, which in consequence leads to a lower
temperature (to keep $\frac{N}{\Delta^{2}}$ constant). This is obviously
limited by the initial hypothesis of far detuned laser ($\Delta\gg\Gamma$).

In order to verify this model we have measured the non-linear phase shift for
various temperatures and detuning and this is reported in chapter 4. In the
next paragraph, I explain how to conduct this measurement.

##### Measurement of the non-linear phase shift for effective 2-level atoms.

Figure 1.4: Typical non-linear phase shift measurement for effective 2-level
atomic cloud. Detuning $\delta$ is given to D1 Rb85 line. Cell temperature is
130C. For larger intensity a clear deviation is observed due to $n_{3}$ term
(not shown). Insets are examples of obtained far field measurements.  Figure
1.5: Comparison between 2-level atoms model and experimental measurement of
$n_{2}$. Blue dots are experimental data presented in Fig. 1.4, and black line
is the model of Eq. 1.32. The model is scaled by the atomic density which is a
fitting parameter.

The typical method to measure the non-linear phase shift of a sample is to
realize a z-scan experiment [13, 14]. However this technique works better with
a thin layer of material. For thick samples, we can use a technique
demonstrated in Ref. [15]. This allows to measure the accumulated phase
$\Delta\phi$ along the propagation. The phase accumulated can be written as:

$\Delta\phi(r)=k\int_{z_{0}}^{z_{0}+L}n_{2}I(r,z)dz,$ (1.35)

with $z_{0}$ is the coordinate of the front of the sample and $L$ is its
length. The intensity profile of the beam can then be replaced by the Gaussian
profile of a TEM(0,0) beam at the input plane. We obtain:

$\Delta\phi(r)=k\int_{z_{0}}^{z_{0}+L}n_{2}\Delta\phi(r)\frac{w_{0}^{2}}{w(z)^{2}}e^{-2r^{2}/w(z)^{2}}dz,$
(1.36)

with $w_{0}$ is the waist radius,
$w(z)=w_{0}\sqrt{1+\left(\frac{z}{z_{R}}\right)^{2}}$ and $z_{R}=\frac{\pi
w_{0}^{2}}{\lambda}$. It is then possible to derive the far field diffraction
pattern in the Fraunhofer limit (see Ref. [15] for details). From Eq. 1.36, it
is clear that the phase shift in the center : $\Delta\phi(r=0)$ is directly
proportional to $n_{2}$ and the central intensity $I(0,0)$. We will therefore
observe a switch from a bright spot to dark spot in the center of the
diffraction pattern when the phase is modified by $\pi$ and back to a bright
spot again for a phase change of $2\pi$. By simply counting the number of
rings $N_{Rings}$ that appear while slowly increasing $I(0,0)$, we can
estimate the non-linear phase shift accumulated along $z$. In the limit of
long Rayleigh length ($z\ll z_{R}$) we can approximate $\Delta n$ to:

$\Delta n=n_{2}I\simeq\frac{\lambda}{L}N_{Rings}.$ (1.37)

In Figure 1.4, I present an experimental characterization of $n_{2}$ for
rubidium 85. From this figure, we have extracted the value of $n_{2}$ as
function of the detuning from the atomic transition. To validate the 2-level
atoms model, I plot in Figure 1.5, the results of the numerical model (Eq.
1.32) after integration over the Doppler profile and compare it to the
experimental data. We see that for large detuning the model is in excellent
agreement. However as we get closer to the resonance the contribution of the
$\chi^{(5)}$ term start to be not negligible anymore and $n_{2}$ is reduced555
$\chi^{(3)}$ and $\chi^{(5)}$ are of opposite signs. compared to the value
predicted by $\chi^{(3)}$. This is the main limitation to obtain a larger
$\Delta n=n_{2}I$ in experiments.

To get a better understanding of this effect we have compared
$\chi^{(5)}I^{2}$ and $\chi^{(3)}I$ for the two-level model. We see in Fig.
1.6, that the contribution of $|\chi^{(5)}I^{2}|$ is huge when we get closer
to resonance but at detuning larger than 2.5 GHz, it can be safely neglected
at this intensity.

Figure 1.6: Comparison between $|\chi^{(5)}I^{2}|$ and $|\chi^{(3)}I|$. Here
$I=2.10^{5}$ W/m2 (which is still much smaller than the effective saturation
intensity at this detuning).

####  1.2.2 Electromagnetically induced transparency

In the previous section we have discussed non-linearity in a 2-level atoms
model. If we add one more level, more complex schemes using atomic coherences
can be exploited. Here, I briefly describe the interaction of the system of
$3$-level atoms with two electromagnetic fields in order to study the effect
of electromagnetically induced transparency (EIT) [16, 17].

The interaction process can be described by the evolution of the optical Bloch
equations, as it was done in the section 1.2.1. The three levels are noted:
$|g\rangle$ for ground, $|e\rangle$ for excited, $|s\rangle$ for supplementary
third level. The optical Bloch equation (1.16) can be rewritten for slowly-
varying amplitudes in the case of a $3$-level atom as following [9, 12]:

$\displaystyle\dot{\rho}_{gg}$ $\displaystyle=$ $\displaystyle
i\frac{\Omega_{p}}{2}(\sigma_{ge}-\sigma_{eg})+\Gamma_{eg}\rho_{ee}$
$\displaystyle\dot{\rho}_{ss}$ $\displaystyle=$ $\displaystyle
i\frac{\Omega_{c}}{2}(\sigma_{se}-\sigma_{es})+\Gamma_{es}\rho_{ee}$
$\displaystyle\dot{\rho}_{ee}$ $\displaystyle=$
$\displaystyle-i\frac{\Omega_{p}}{2}(\sigma_{ge}-\sigma_{eg})-i\frac{\Omega_{c}}{2}(\sigma_{se}-\sigma_{es})-\Gamma\rho_{ee}$
(1.38) $\displaystyle\dot{\sigma}_{ge}$ $\displaystyle=$
$\displaystyle-i(\Delta_{p}-i\Gamma/2)\sigma_{ge}-i\frac{\Omega_{p}}{2}(\rho_{ee}-\rho_{gg})+i\frac{\Omega_{c}}{2}\sigma_{gs}$
$\displaystyle\dot{\sigma}_{se}$ $\displaystyle=$
$\displaystyle-i(\Delta_{c}-i\Gamma/2)\sigma_{se}-i\frac{\Omega_{c}}{2}(\rho_{ee}-\rho_{ss})+i\frac{\Omega_{p}}{2}\sigma_{sg}$
$\displaystyle\dot{\sigma}_{gs}$ $\displaystyle=$
$\displaystyle-i(\Delta_{p}-\Delta_{c}-i\gamma_{0})\sigma_{gs}-i\frac{\Omega_{p}}{2}\sigma_{es}+i\frac{\Omega_{c}}{2}\sigma_{ge}.$

Here we characterize two electromagnetic fields: the probe field with the Rabi
frequency $\Omega_{p}$ interacts between the initially populated ground state
$|g\rangle$ and the excited state $|e\rangle$, while the control field with
the Rabi frequency $\Omega_{c}$ couples the excited state $|e\rangle$ with the
initially empty second ground state $|s\rangle$. The probe and the control
fields are detuned from the corresponding atomic resonances with detunings
$\Delta_{p}=\omega_{p}-\omega_{eg}$ and $\Delta_{c}=\omega_{c}-\omega_{es}$
respectively. Decay rates $\Gamma_{eg}$ and $\Gamma_{es}$ can be found with
the Clebsch-Gordan coefficients of the corresponding transitions, and they
satisfy to the condition $\Gamma_{eg}+\Gamma_{es}=\Gamma$. The decay rate
$\gamma_{0}$ corresponds to the decay rate of the ground states coherence
between $|g\rangle$ and $|s\rangle$.

In the RWA we assume $\rho_{ii}$ and $\sigma_{ij}$ as slowly varying
amplitudes. With these conditions the system (1.2.2) can be solved in the
steady-state regime when

$\dot{\rho}_{gg}=\dot{\rho}_{ee}=\dot{\rho}_{ss}=\dot{\sigma}_{ge}=\dot{\sigma}_{se}=\dot{\sigma}_{gs}=0.$

In the following, we are interested in the nonlinear components of the atomic
susceptibility $\chi$, which is the proportionality coefficient between the
atomic polarization and the electric field, see Eq. 1.20. We solve the system
(1.2.2) numerically. The polarization induced by the probe field can be found
in terms of the coherence at the corresponding atomic transition
$\sigma_{ge}$, in the same way how it was done in Eq. 1.23.

We can extract quantities similar to 2-level atoms:
$\chi^{(1)},\chi^{(3)},\chi^{(5)}$ from a linear expansion of $\chi$ obtained
numerically. The code to implement these simulations in Mathematica is
available here.

####  1.2.3 Four-wave-mixing

Adding one more level, one can add even more complexity (and also more fun) in
the light-matter interactions. I describe here, a typical configuration called
double-$\Lambda$, where two intense pumps, one probe beam and one conjugate
beam interact (see Figure 1.7). A lot of details about this configuration can
be found in [18, 19, 20, 21, 22, 23, 24].

Figure 1.7: Levels scheme for four-wave-mixing in the double-$\Lambda$
configuration.

Here, I will just give an intuitive description of a few phenomenon. One
important point to understand, is there is only one pump laser in real
experimental configuration. So if it is detuned by $\Delta$ from the
transition $|1\rangle\rightarrow|3\rangle$ it has to be detuned by
$\Delta+\omega_{0}$ from the transition $|2\rangle\rightarrow|4\rangle$. If
$\Delta>0$ and $\Delta\gg\Gamma$, it directly implies a steady state for the
population with a large amount of the atoms in state $|2\rangle$.
An interesting approach, to understand the quantum correlations which appears
between the probe and conjugate is to think of four-wave-mixing as a DLCZ
memory protocol [25]. Indeed, one starts with all the population in
$|2\rangle$. Sometimes (not often because of the large detuning) a
$|2\rangle\rightarrow|4\rangle$ pump photon will write his phase in the atomic
coherence and induce the emission of an anti-Stokes (conjugate) photon. In the
DLCZ [25] language, when this anti-Stokes (conjugate) photon is detected it
implies that the memory has been loaded. After a given time, the memory can be
read (efficiently due to small detuning) by a pump photon on the
$|1\rangle\rightarrow|3\rangle$ transition. This process is accompanied by the
emission of a probe photon, in a coherent manner (as describe in section
1.1.2).

In chapter 3, I use this technique to generate entangled pulses of light
between the probe and the conjugate [20]. I also use another property of this
system: to provide small group velocity and anomalous dispersion, in order to
observe slow and fast light [26].

####  1.2.4 Slow and fast light

We begin by applying the curl operator to the Maxwell equations to obtain the
Helmholtz equation

$(\nabla^{2}E+k^{2})\mathbf{E}=0,$ (1.39)

where we have $k=\mu_{0}\epsilon_{0}\epsilon_{r}\omega^{2}$. As usual we
assume the beam to propagate along the $+z$ direction and we ignore
polarization. Solutions of the Helmholtz equation that fulfills these
conditions are:

$E(z,t)=E_{0}(z)e^{i(kz-\omega t)}.$ (1.40)

The phase velocity is defined as the velocity at which the phase of this
solution moves:

$v_{\phi}=\frac{\omega}{k}$ (1.41)

By replacing $k$ with its definition, it can be rewritten with the speed of
light in vacuum $c$ and the index of refraction $n$:

$v_{\phi}=\frac{\omega}{\sqrt{\mu_{0}\epsilon_{0}\epsilon_{r}(\omega)\omega^{2}}}=\frac{c}{n(\omega)}.$
(1.42)

This is a well known result but it hides in the dependency on $\omega$ of
$\epsilon_{r}(\omega)$ that different frequency will propagate at different
velocities. It has no consequence for monochromatic waves, but it implies that
a pulse will distort while propagating in a dispersive media. Slow and fast
light terminology comes from this effect: a light pulse (basically a wave-
packet) that propagates slower than $c$ will be qualified as slow-light and
reciprocally if it does propagate faster than $c$ it will be qualified as
fast-light [27, 28]. Let us precise this terminology.

A wavepacket has the general form:

$E(z,t)=\int E_{0}(k)e^{i(kz-\omega(k)t)}\text{dk},$ (1.43)

where $E_{0}(k)$ is the Fourier transform of $E_{0}(z)$ at $t=0$. If the
spectrum $E_{0}(k)$ is sufficiently narrowband (i.e. the pulse is not too
short), we can call the central value $k_{c}$ and Taylor expand $\omega$
around $k_{c}$:

$\omega(k)\approx\omega_{c}+\left.\frac{\partial\omega}{\partial
k}\right\rvert_{k=k_{c}}(k-k_{c}).$ (1.44)

Using this approximation we can write:

$E(z,t)\approx E\left(z-t\left.\frac{\partial\omega}{\partial
k}\right\rvert_{k_{c}},0\right)\times
e^{i(k_{c}\left.\frac{\partial\omega}{\partial
k}\right\rvert_{k_{c}}-\omega_{c})t}.$ (1.45)

In this simple results we can see that the pulse will propagate largely
undistorted (up to an overall factor phase and as long as Eq. 1.44 is a good
approximation) at the group velocity given by:

$v_{g}=\left.\frac{\partial\omega}{\partial k}\right\rvert_{k_{c}}.$ (1.46)

As derived previously about 2-level atoms, it is often more convenient to
express the group velocity as a function of the variation of $n$ with
frequency. We have

$\displaystyle\frac{\partial\omega}{\partial k}$ $\displaystyle=$
$\displaystyle\frac{c}{n}-\frac{ck}{n^{2}}\frac{\partial n}{\partial k}$
(1.47) $\displaystyle=$
$\displaystyle\frac{c}{n}-\frac{\omega}{n}\frac{\partial\omega}{\partial
k}\frac{\partial n}{\partial\omega}.$

We can then write the group velocity:

$v_{g}=\frac{c}{n(\omega_{c})+\omega_{c}\left.\frac{\partial
n}{\partial\omega}\right\rvert_{\omega_{c}}}.$ (1.48)

This equation gives us direct access of the group velocity if we know
$n(\omega_{c})$ (which is the case now that we master optical Bloch
equations). The group velocity is then given by the speed of light in vacuum
divided by a term that includes both the index of refraction at the carrier
frequency and the derivative of the index around the carrier frequency. The
denominator is commonly called the group index $n_{g}$ and it can take values
larger or smaller than unity [29].

In the vast majority of dielectric media, far away from resonance,
$\left.\frac{\partial n}{\partial\omega}\right\rvert_{\omega_{c}}$ is usually
positive and $n_{g}>1$. However, it is possible (using EIT for example) to
obtain $\left.\frac{\partial n}{\partial\omega}\right\rvert_{\omega_{c}}<0$.
This type of medium is said to have anomalous dispersion. For sufficiently
large negative value, it is possible to reach $n_{g}<1$: a negative group
velocity. I report on the use of this type of medium in chapter 3.

Figure 1.8: Real part of the the linear refractive index for the EIT
configuration described in section 1.2.2. We can see that around
$\omega-\omega_{c}=0$ the slope of $\frac{\partial n}{\partial\omega}$ is
large and negative. This is a perfect place of observe a negative group
velocity.

##  Chapter 2 Multimode optical memory

List of publications related to this chapter :

  * •

Temporally multiplexed storage of images in a gradient echo memory.
Q. Glorieux, J. B. Clark, A. M. Marino, Z. Zhou, and P. D. Lett.
Optics Express 20, 12350 (2012)

  * •

Gradient echo memory in an ultra-high optical depth cold atomic ensemble.
B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K.
Lam, N. P. Robins, and B. C. Buchler.
New Journal of Physics 15, 085027 (2013)

  * •

Spatially addressable readout and erasure of an image in a gradient echo
memory.
J. B. Clark, Q. Glorieux and P. D. Lett.
New Journal of Physics 15, 035005 (2013)
Not included.

  * •

An ultra-high optical depth cold atomic ensemble for quantum memories.
B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K.
Lam, N. P. Robins, and B. C. Buchler.
Journal of Physics, 467, 012009 (2013)
Not included.

###  2.1 Gradient Echo Memory

Light in vacuum propagates at $c$. This statement is not only a fundamental
principle of physics it is also the foundation of the definition of the meter
in the international system of units. Delaying [30], storing [31] or advancing
[32] light are therefore only possible in a medium (i.e. not in vacuum). In
the next two chapters, I will show how to play with the speed of light in a
optically dense atomic medium [33].

The main motivation to delay or store photons in matter is the need to
synchronize light-based communication protocols. Quantum technologies promise
an intrinsically secure network of long distance quantum communication.
However for a realistic implementation, the quantum internet will need quantum
repeaters in order to compensate losses in long distance channels [34]. The
core element of a quantum repeater is a quantum memory [35] which can store
and release photons, coherently and on demand. This chapter is covering this
topic with the presentation of an important memory protocol: the gradient echo
memory and two implementations of this protocol one in a warm vapor [36] and
the second in a cold atomic cloud [37].

If you are somewhat familiar with MRI, understanding qualitatively gradient
echo memory (GEM) is straightforward. Imagine you want to encode quantum
information in a light. Various approaches are available (from polarization
encoding, time-bin, orbital angular momentum…) but for qubit encoding (i.e. an
Hilbert space of dimension 2), you always can map your information into a spin
1/2 system and represent it on the Bloch sphere. Let’s say your information is
of the generic form
$|\psi\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle+e^{i\phi}|1\rangle\right)$,
this means on the equator of the Bloch sphere with a latitude $\phi$ as shown
on Fig. 2.1 i). In GEM, during the storage process one applies a spatially
dependant energy shift in order to enlarge the transition [38, 39]. This
technique enables the storage of photons with a broad spectrum (i.e. short in
time).

Figure 2.1: GEM protocol. i) Bloch sphere with the state initialized on the
equator. ii) and iii) are views from top of the equatorial plan. ii) Dephasing
with the gradient in one direction. iii) Rephasing with the gradient in the
other direction. The dephasing is shown in the rotating frame.

Because $|\psi\rangle$ is not the energy ground state, the state will start to
rotate in the equatorial plane of the Bloch sphere. However, due to the
spatially dependant energy shift, each frequency component will rotate at a
different speed and dephase as shown in figure 2.1 ii) The trick used in
gradient echo techniques is to reverse the time evolution of the dephasing
process. By simply inverting the energy gradient after the evolution time $T$,
we can see on figure 2.1 iii) that the frequency components will start to
precess in the opposite direction with opposite speed111This is true if not
only the energy gradient is switched but also the sign of the energy. If only
the energy gradient changes sign, the components will still precess in the
same direction but the slowest components will become the fastest.. At a time
$2T$, all frequency components will be in phase again and the collective
enhancement described in chapter 1 can occur.

Various approaches have been tested to produce a spatially varying energy
shift. The two most successful techniques are using an AC-Stark shift in rare-
earth crystal and using a Zeeman shift on atomic vapors by applying a magnetic
field gradient. In this chapter I describe the Zeeman shift implementation in
warm and cold atomic ensembles [40, 36, 37].

####  2.1.1 Storage time and decoherence

I should now describe how to convert coherent light excitation into a matter
excitation. In the two-level-atom configuration, a quantum field
$\hat{\mathcal{E}}(z,t)$ (carrying the information) is sent into the atomic
vapor with ground state $|g\rangle$ and excited state $|e\rangle$.

Here, we are focusing on the propagation of a light pulse into a medium,
therefore the time derivative terms in the Maxwell-Bloch equations have to be
conserved, but the transverse gradient is neglected222The opposite
approximation is done in the Chapter 4, where the time evolution is not
considered but the diffraction in the transverse plan is.. Using the rotating
wave approximation described in chapter 1, and in the presence of a spatially
varying magnetic field, the evolution equations are given by:

$\displaystyle\frac{\partial}{\partial t}\hat{\sigma}_{ge}(z,t)$
$\displaystyle=$
$\displaystyle[-\Gamma+i\eta(t)z]\hat{\sigma}_{ge}(z,t)-ig\hat{\mathcal{E}}(z,t)$
$\displaystyle\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
z}\right)\hat{\mathcal{E}}(z,t)$ $\displaystyle=$ $\displaystyle
igN\hat{\sigma}_{ge}(z,t).$ (2.1)

We have introduced the term $i\eta(t)z$ which represents the energy shift form
the spatially varying magnetic field. $\eta(t)$ is the gradient slope that can
be inverted (or modified in a more complex manner) at the desired time. From
this coupled equations it is straightforward to point out the problem of the
two-level-atom configuration. The information is transferred from the field to
the atomic coherence $\hat{\sigma}_{ge}(z,t)$. However this coherence will
decay with the decay rate of the excited state $\Gamma$. This is a terrible
problem for a quantum memory ! The good news is: this problem can be solved
using a third level.

Figure 2.2: Adiabatic elimination notation summary.

The solution comes from mapping the photonic excitation into the ground state
coherence of a three-level atom as discussed in section 1.2.2. Using a two-
photon detuning equal to zero and a large one photon detuning compared to
$\Gamma$, it is possible to apply the adiabatic elimination of the excited
state and rewrite the evolution equation for the ground state coherence in a
way similar to equations 2.1.1:

$\displaystyle\frac{\partial}{\partial t}\hat{\sigma}_{gs}(z,t)$
$\displaystyle=$
$\displaystyle[-\gamma+i\eta(t)z-i\frac{\Omega^{2}}{\Delta}]\hat{\sigma}_{gs}(z,t)-i\frac{g\Omega}{\Delta}\hat{\mathcal{E}}(z,t)$
$\displaystyle\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
z}\right)\hat{\mathcal{E}}(z,t)$ $\displaystyle=$ $\displaystyle
i\frac{g^{2}N}{\Delta}\hat{\mathcal{E}}(z,t)+i\frac{gN\Omega}{\Delta}\hat{\sigma}_{gs}(z,t).$
(2.2)

We apply a transformation to this equations by going into the rotating frame
at frequency $\frac{g^{2}N}{\Delta}$. Moreover we neglect the ac-Stark shift
induced by the control field and we finally obtain [40, 41]:

$\displaystyle\frac{\partial}{\partial t}\hat{\sigma}_{gs}$ $\displaystyle=$
$\displaystyle[-\gamma+i\eta(t)z]\hat{\sigma}_{gs}-i\frac{g\Omega}{\Delta}\hat{\mathcal{E}}$
$\displaystyle\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
z}\right)\hat{\mathcal{E}}$ $\displaystyle=$ $\displaystyle i\frac{g\Omega
N}{\Delta}\hat{\sigma}_{gs}.$ (2.3)

We found an equation similar to Eq. 2.1.1 but where the decoherence $\Gamma$
has been replaced by $\gamma$ which is the long-lived coherence between the
two ground states. This is a major improvement as the ground state coherent is
immune from spontaneous emission, and therefore much longer memory time can be
envisioned (and demonstrated actually [42]). Another important improvement is
obtained by going to three levels and modifying the coupling $g$ to
$\frac{g\Omega}{\Delta}$, which can be modified and tuned with the Rabi
frequency of the control field.

A final transformation [26] is made to the equations 2.1.1 by changing $z$ to
the retarded frame $z^{\prime}=z+ct$:

$\displaystyle\frac{\partial}{\partial t}\hat{\sigma}_{gs}$ $\displaystyle=$
$\displaystyle[-\gamma+i\eta(t)z]\hat{\sigma}_{gs}-i\frac{g\Omega}{\Delta}\hat{\mathcal{E}}$
$\displaystyle\frac{\partial}{\partial z^{\prime}}\hat{\mathcal{E}}$
$\displaystyle=$ $\displaystyle iN\frac{g\Omega}{\Delta}\hat{\sigma}_{gs}.$
(2.4)

####  2.1.2 GEM polaritons

Equations 2.1.1 can be solved numerically but new insights emerge when writing
them in the picture of normal modes called GEM-polaritons [39, 43]. The
polariton picture for light-matter interaction in a 3-level system has been
pioneered by Lukin and Fleischhauer with the introduction of the dark-state
polariton [6]. This pseudo particle is a mixed excitation of light and matter
(atomic coherence) where the relative weight can be manipulated with the
external control field. This is a boson-like particle which propagates with no
dispersion and a tunable group velocity. It is possible to consider a similar
approach for GEM. Neglecting the decoherence and taking the spatial Fourier
transform of Eq. 2.1.1, we can write [43]:

$\displaystyle\frac{\partial}{\partial t}\hat{\sigma}_{gs}(t,k)$
$\displaystyle=$ $\displaystyle-\eta\frac{\partial}{\partial
k}\hat{\sigma}_{gs}(t,k)+i\frac{g\Omega}{\Delta}\hat{\mathcal{E}}(t,k)$
$\displaystyle k\hat{\mathcal{E}}(t,k)$ $\displaystyle=$ $\displaystyle
N\frac{g\Omega}{\Delta}\hat{\sigma}_{gs}(t,k).$ (2.5)

In analogy to [6], we consider the polariton-like operator in k-space:

$\hat{\psi}(k,t)=k\hat{\mathcal{E}}(t,k)+N\frac{g\Omega}{\Delta}\hat{\sigma}_{gs}(t,k).$
(2.6)

This particle will follow an equation of motion:

$\left(\frac{\partial}{\partial t}+\eta(t)\frac{\partial}{\partial
k}-iN\frac{g^{2}\Omega^{2}}{\Delta^{2}k}\right)\hat{\psi}(k,t)=0$ (2.7)

This equation indicates that $|\psi(k,t)|$ propagates un-distorted along the
$k$-axis with a speed defined by the gradient $\eta$. We are here in front of
a strange polariton.

The first difference with dark-state-polariton is that it does propagate in
the $k$-plane instead of the normal space.

The second difference, maybe more fundamental is that the weight of one of the
two components depends on $k$. If the polariton is normalized (and it has to
be, in order to fulfill the bosonic commutation relation), lower spatial
frequencies (short $k$) will be more an atomic excitation and higher spatial
frequencies will be more a field excitation. We see here the limit of the
analogy with the dark-state polariton. The GEM polariton picture is mainly
useful to understand that the velocity of the pseudo particle is given by
$\eta(t)$ and therefore will change sign when $\eta(t)$ will be flipped. We
have here another interpretation of the rephasing process explained with the
Bloch sphere, earlier in the chapter.

####  2.1.3 GEM in rubidium vapor

Figure 2.3: a) Orthogonal linearly polarized Raman beams used in the $\Lambda$
GEM experiments presented in this thesis. When propagating along the direction
of the magnetic field, a linear polarization can be viewed as a superposition
of $\sigma+$ and $\sigma-$ polarizations. We therefore simultaneously drive
two Raman transitions in any given echo experiment. b) An unshifted hyperfine
structure for the D1 transition in 85Rb. c) A weak bias magnetic field to
Zeeman shift the hyperfine structure of the ground state. In the weak field
limit, the $m_{F}$ levels are linearly and oppositely shifted in the F = 2 and
F = 3 manifolds. For the GEM protocol, we couple $m_{F}=2$ and $m_{F}=0$
magnetic sublevels. d) Complete depiction of the two Raman transitions. Figure
and caption are extracted from the J. Clark thesis [26], whom I supervised at
NIST.

I give here a short description of the implementation of GEM in a hot Rubidium
85 vapor. More details are given in the article: Temporally multiplexed
storage of images in a gradient echo memory, attached to this work [36]. The
first task is to isolate a pair of ground states that can be individually
addressed by light fields and whose energy splitting can be tuned by a linear
Zeeman shift. This is done by applying a bias magnetic field (about 50 G) to
lift the degeneracy of the $m_{F}$ sub-levels for F=2 and F=3 ground states
(see figure 2.3).
The second step is to apply a gradient magnetic field along the atomic vapor
(a 20 cm long cell) to control the inhomogeneous broadening of the two-photon
Raman transition. The amplitude of this gradient field must be smaller than
the bias field and large enough to accept the full bandwidth of the light
pulse we expect to store. Typically, in rubidium a rule of thumb is a Zeeman
shift of 1 MHz per Gauss. So a gradient with 1 Gauss amplitude already allows
for the storage of 1 microsecond Gaussian pulses.
An important part of the development of such setup consists in the precise
control of the gradient and the ability to reverse it quickly. Indeed in the
echo description we have done so far, the inversion of the gradient is
supposed to be instantaneous. It is clear that switching a large current in an
inductor instantaneously is a tricky issue. More details on the methods used
are given in the article Gradient echo memory in an ultra-high optical depth
cold atomic ensemble attached to this work [37].

Finally, an important reason why GEM is a powerful tool for storing and
retrieving light is that it is intrinsically temporally multimode: one can
store multiple pulses at different times inside the memory and retrieve them
independently [44, 36, 45]. But it is also spatially multimode, so images can
be stored in the transverse plane (only limited by decoherence due to atomic
diffusion). We have leveraged these two advantages to demonstrate the first
storage of a ”movie” inside an atomic memory [36]. Results are presented in
the next pages.

####  2.1.4 Press releases

These results led to multiple press articles covering this topic. I reproduce
here some excerpts.

The sequence of images that constitute Hollywood movies can be stored handily
on solid-state media such as magnetic tape or compact diskettes. At the Joint
Quantum Institute images can be stored in something as insubstantial as a gas
of rubidium atoms. No one expects a vapor to compete with a solid in terms of
density of storage. But when the ”images” being stored are part of a quantum
movie –the coherent sequential input to or output from a quantum computer
–then the pinpoint control possible with vapor will be essential. Phys.org

One of the enabling technologies for a quantum internet is the ability to
store and retrieve quantum information in a reliable and repeatable way. One
of the more promising ways to do this involves photons and clouds of rubidium
gas. Rubidium atoms have an interesting property in that a magnetic field
causes their electronic energy levels to split, creating a multitude of new
levels. MIT Technology Review

Having stored one image (the letter N), the JQI physicists then stored a
second image, the letter T, before reading both letters back in quick
succession. The two ”frames” of this movie, about a microsecond apart, were
played back successfully every time, although typically only about 8 percent
of the original light was redeemed, a percentage that will improve with
practice. According to Paul Lett, one of the great challenges in storing
images this way is to keep the atoms embodying the image from diffusing away.
The longer the storage time (measured so far to be about 20 microseconds) the
more diffusion occurs. The result is a fuzzy image. Science Daily.

## Article 1: Temporally multiplexed storage of images in a gradient echo
memory

### Preprint

The preprint version of this article is accessible on  arXiv:1205.1495.

### Published version

Temporally multiplexed storage of images in a gradient echo memory.
Q. Glorieux, J. B. Clark, A. M. Marino, Z. Zhou, and P. D. Lett.
Optics Express 20, 12350 (2012)

###  2.2 GEM in a cold atom ensemble

Shifting from warm atomic vapor (with the simplicity of high and tunable
density) to laser-cooled atoms trapped in a magneto-optical trap (MOT) has
many advantages. The first of them being the reduction of decoherence due to
atomic motion. However, in order to obtain the large optical depth required
for an efficient gradient echo memory, a substantial optimization work has
been needed. In the paper attached to this work, I highlight the various
techniques I developed and used during my stay at the Australian National
University under the supervision of Ping Koy Lam.

For quantum memory applications we aim to reach low temperatures and the very
large optical depths (OD) required for high efficiency. Because OD is related
to the integrated absorption of photons through a sample, there are a number
of ways to increase the OD. These include: increase the atom number (for
instance by increasing the length of the cloud) or increase the density. Low
temperature is also important as the thermal diffusion of atoms is a limiting
factor for the memory lifetime.

In the attached paper, I present a MOT that achieves a peak OD of over 1000 at
a temperature of 200 $\mu$K. We obtain this result by combining three
techniques:

  * •

The optimization of the static loading of the MOT through geometry. The
optimal shape for our atomic ensemble is a cylinder along the direction of the
memory beams to allow for maximum absorption of the probe. This can be
achieved by using rectangular, rather than circular, quadrupole coils or using
a 2D MOT configuration.

  * •

The use of a spatial dark spot. The density in the trapped atomic state is
limited by re-absorption of fluorescence photons within the MOT (leading to an
effective outwards radiation pressure). By placing a dark spot of
approximately 7.5 mm in diameter in the repump, atoms at the centre of the MOT
are quickly pumped into the lower ground state and become immune to this
unwanted effect, allowing for a higher density of atoms in the centre of the
trap, as first demonstrated in [46].

  * •

The use of optical de-pumping, followed by a compression phase with a temporal
dark spot [47].

With the setup built using these techniques, we have implemented the gradient
echo memory scheme on the D1 line of Rubidium. The results shown here
demonstrate a memory efficiency of 80 $\pm$ 2% and a coherence time up to 195
$\mu$s. This coherence time is a factor of eight greater than GEM experiments
done in atomic vapor (described in the previous section).

## Article 2: Gradient echo memory in an ultra-high optical depth cold atomic
ensemble

### Preprint

The preprint version of this article is accessible on  arXiv:1211.7171.

### Published version

Gradient echo memory in an ultra-high optical depth cold atomic ensemble.
B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K.
Lam, N. P. Robins, and B. C. Buchler.
New Journal of Physics 15, 085027 (2013)

##  Chapter 3 Quantum optics in anomalous dispersion media

List of publications related to this chapter :

  * •

Imaging using quantum noise properties of light.
J. B. Clark, Z. Zhou, Q. Glorieux, A. M. Marino and P. D. Lett.
Optics Express 20, 17050 (2012)

  * •

Quantum mutual information of an entangled state propagating through a fast-
light medium.
J. B. Clark, R. T. Glasser, Q. Glorieux, U. Vogl, T. Li, K. M. Jones, and P.
D. Lett. Nature Photonics 14, 123024 (2014)

  * •

Generation of pulsed bipartite entanglement using four-wave mixing.
Q. Glorieux, J. B. Clark, N. V. Corzo, and P. D. Lett.
New Journal of Physics 14, 123024 (2012). Not included.

  * •

Extracting spatial information from noise measurements of multi-spatial-mode
quantum states.
A. M. Marino, J. B. Clark, Q. Glorieux and P. D. Lett.
European Physical Journal D 66, 288 (2012). Not included.

  * •

Rotation of the noise ellipse for squeezed vacuum light generated via four-
wave mixing.
N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, R. T. Glasser and P. D.
Lett. Physical Review A 88, 043836 (2013). Not included.

  * •

Experimental characterization of Gaussian quantum discord generated by four-
wave mixing.
U. Vogl, R. T. Glasser, Q. Glorieux, J. B. Clark, N. V. Corzo, and P. D. Lett.
New Journal of Physics 87, 010101 (2013). Not included.

  * •

Advanced quantum noise correlations.
U. Vogl, R. T. Glasser, J. B. Clark, Q. Glorieux, T. Li, N. V. Corzo, and P.
D. Lett. New Journal of Physics 16, 013011 (2014). Not included.

### Introduction

A large part of my work at NIST-JQI has been devoted to the study and use of
noise properties of entangled beams generated using four-wave-mixing in warm
atomic vapor. It is not possible to describe here all these experiments but I
will highlight two results that are valuable in relation to the other works
presented in this manuscript.

We have seen in the previous chapter that GEM has the ability to store
multiple spatial modes in the transverse direction. In the first experiment
that I describe in this chapter, we used the multimode properties of to
quantum noise [48] to be able to identify an object with a higher precision
than would normally be achieved with a typical laser.

The second result I will comment concerns the study of the opposite process of
a quantum memory: advancing quantum information [32]. We shown that under
certain conditions, an atomic vapor can exhibit an anomalous dispersion and
give rise to a group velocity larger than $c$. In the context of quantum
communication, these results can be shocking as it is clear that no signal can
travel faster than light [49]. However, we demonstrated the role of quantum
noise to erase all information while velocity becomes larger than $c$.

###  3.1 Imaging with the noise of light

Imagine that you have an object (an intensity or a phase mask) that you want
to image but you are also limited in the number of photons you can send onto
this object. We propose two techniques (one classical using thermal noise and
one quantum using quantum correlations) and compare the uncertainty in shape
estimation between both. To simplify the comparison we restricted the shape
estimation to a 1D problem.
The resource needed for this experiment is a pair of squeezed vacuum beams
generated using four-wave-mixing in a atomic vapor. These beams are too weak
to be measured directly with an amplified photodiode and need to be homodyned
with local oscillators to be detected. Taken independently both beams (also
known as probe and conjugate) exhibit an excess noise compared to a coherent
state. They are indeed thermal states. However, when the intensity difference
or the phase sum is analyzed, we observed a noise below the shot noise
indicating quantum correlations.
The parameter we are trying to estimate is the angle of a bow-tie mask (1D
problem) placed on the conjugate path. To do so the local oscillator (LO) is
designed with a bow-tie shape of same size and same center as the mask, using
a spatial light modulator. However the orientation of the mask is unknown.

The classical technique consists in overlapping the bow-tie shaped LO with the
conjugate beam (after the mask) and measure the detected noise. The LO is then
rotated and when the noise is maximized, the search is finished and the LO
angle is an estimation of the mask angle. Indeed the fraction of the conjugate
beam which passes through the mask contains spatial modes with extra-thermal
noise compared to vacuum. In the limiting case of zero overlap between the LO
and the mask, the LO will just beat with vacuum and detect shot noise because
no light coming from the conjugate beams interfere with it.

The quantum technique makes use of the second part of the entangled pair (the
probe beam), which surprisingly does not have to interact directly with the
mask. For this measurement, we simply added a second LO with the same angle to
beat with the probe beam and then measured the quadrature that minimized the
detected noise. This time, the search is terminated when the quadrature noise
is minimal. In the paper Imaging using the quantum noise properties of light,
we have demonstrated that the precision in the estimation of the mask angle is
enhanced by a factor 6 with the quantum technique compared to the classical
one, even though the second beam does not interact directly with the mask.

The second experiment described in this paper is a direct application of this
principle to the field of pattern recognition. We start by defining a pattern
set (the alphabet letters). We are able to imprint this set on the local
oscillator with a spatial light modulator as shown in figure 3.1. An unknown
mask from the set is placed in the path of the conjugate beam and we compare
the two techniques to see if pattern recognition is enhanced using quantum
correlations.

On the presented results (mask of the letter Z) by taking into account the
uncertainty on the noise measurements it is not possible to conclude on the
pattern in the classical case because more than one pattern noise measurement
lie within one standard deviation. On the quantum case however, the letter Z
is identified with more than 7 standard deviation certainty.

Figure 3.1: Example of probe and conjugate LO shape created using a spatial
light modulator. The intense dot in the middle is the (filtered) pump beam.
Each letter is an independent image and they are collated for presentation.

## Article 3: Imaging using quantum noise properties of light

### Preprint

The preprint version of this article is accessible on arXiv:1207.1713.

### Published version

Imaging using quantum noise properties of light.
J. B. Clark, Z. Zhou, Q. Glorieux, A. M. Marino and P. D. Lett.
Optics Express 20, 17050 (2012)

###  3.2 How quantum noise affects the propagation of information.

We have already discussed various approaches to delay light and quantum
information carried by entanglement. Although entanglement cannot be used to
signal superluminally [50], it is thought to be an essential resource in
quantum information science [51, 52]. It is then of fundamental interest to
study what happens to entanglement when part of an entangled system propagates
through a fast light medium. Much work has been done to understand fast-light
phenomena associated with anomalous dispersion, which gave rise to group
velocities that are greater than the speed of light in vacuum, as described in
chapter 1 [53]. For classical pulses propagating without the presence of
noise, it has been well established theoretically that the ”pulse front”
propagates through a linear, causal medium at the speed of light in vacuum
[54]. It is often argued that this part of the pulse carries the entirety of
the pulse’s classical information content since the remainder of the signal
can in principle be inferred by measuring the pulse height and its
derivatives, just after the point of non-analyticity has passed [55, 56].
Experimentally, in the inevitable presence of quantum noise, pulse fronts may
not convey the full story of what is readily observed in the laboratory.
In the paper Quantum mutual information of an entangled state propagating
through a fast-light medium, we discuss in details the detrimental role of
quantum noise. Instead of using a pulse with a given shape, in this work we
encode the information in correlated quantum noise, and therefore the
definition of a ”pulse front” becomes impossible. The fluctuations of the
probe and conjugate electric fields are not externally imposed, and they
present no obvious pulse fronts or non-analytic features to point to as
defining the signal velocity. As such, classically-rooted approaches to
defining the signal or information content of the individual modes are not
readily applicable to this system. To quantify the quantity of information
usable in the entangled beams we use the criteria of inseparability (Fig. 3.2
for details). As a reminder, note that an inseparability lower than 2
guarantees the presence of entanglement.

The entangled beams are generated using four-wave-mixing in a hot atomic
vapor, and one of the two beams then propagates through a tunable fast-light
medium (with group velocity larger than $c$). This fast light medium can be
tuned to slow-light by changing the experimental parameters and this allows
for direct comparison. Due to the Kramers-Kronig relation, a group velocity
larger than $c$, is systematically associated with gain or loss through the
medium. The main point of our study is to understand the role of this gain or
loss on the quantum information encoded in our beams.

The first step consists in measuring the arrival time of the cross-correlation
maximum (or minimum) when the conjugate beam propagates in vacuum (actually in
air) and compare it to the case of propagation in a fast-light medium. This
approach is analogous to the classical technique of measuring the arrival time
of a light pulse maximum. After accumulating enough statistics, we have
observed an advancement of the normalized correlation peak by $3.7$ ns
$\pm~{}0.1$ ns. Compared to the 300 ns of correlation time, this corresponds
to a un-ambiguous relative advancement of more than 1%.

Figure 3.2: Determination of inseparability $\mathcal{I}$ using a spectrum
analyzer. Part a) corresponds to the reference case while b) corresponds to
data taken with fast light. The minimum noise power of the homodyne difference
signal corresponds to squeezing of the $\hat{X}$ quadrature difference (i.e.
$\langle\hat{X}_{-}^{2}\rangle$) while the minimum noise power of the homodyne
sum gives the squeezing of phase sum $\langle\hat{Y}_{+}^{2}\rangle$. With the
fast-light cell inserted into the path of the conjugate, we were able to see
$\mathcal{I}<2$. The figure shows how the inseparability can be calculated
(from the data obtained from the spectrum analyzers) by adding together the
values of the appropriate minima and averaging. Adapted from [26]

However this advancement is coming with a reduction of the quadrature
squeezing from $-3$ dB below the shot noise in the case of free-space to
$-2.3$ dB in the case of fast-light media (Fig. 3.2). Obviously this reduction
is not observed when comparing the cross-correlation maximum or normalized.
Therefore, in a second set of measurements, we report the inseparability as
function of time. Similarly we see an advancement of the inseparability peak
in the case of the fast-light medium. This was expected as the inseparability
is computed from normalized cross-correlation maximum and minimum (see figure
3.2 for details).

More insights can be found in the behaviour of the leading edge of
inseparability. Indeed, the inseparability can be used to define the mutual
information (using the covariance matrix) and therefore, a larger value of
inseparability means more mutual information shared between the two sides of
the entangled pair. If at some moment in time, the inseparability has a larger
value than later it means that the amount of information is larger. What we
observe in the experiment is that the noise added by the fast light medium,
consistently reduces the value of inseparability of the leading edge in order
that it never outperforms the case of free-space. In the opposite case of slow
light, we also have shown clear evidence of delaying not only the maximum of
inseparability but also the trailing edge.

It is interesting to contrast this asymmetry in the fast- and slow-light
behaviour of the mutual information with the results of previous experiments
studying the velocity of classical information propagating through dispersive
media. In refs [55] and [57], new information associated with a ’non-analytic’
point in the field was found to propagate at $c$ in the presence of slow- and
fast-light media alike. Because of copyright only the first page of the paper
is reproduced here. For full access please check on our webpage :
www.optiquequantique.fr or Nature Photonics 8, 515 (2014).

## Article 4: Quantum mutual information of an entangled state propagating
through a fast-light medium

### Preprint

The preprint version of this article is accessible on arXiv:1405.7726.

### Published version

Quantum mutual information of an entangled state propagating through a fast-
light medium.
J. B. Clark, R. T. Glasser, Q. Glorieux, U. Vogl, T. Li, K. M. Jones, and P.
D. Lett. Nature Photonics 14, 123024 (2014)

##  Chapter 4 Hydrodynamics of light

List of publications related to this chapter :

  * •

Injection of Orbital Angular Momentum and Storage of Quantized Vortices in
Polariton Superfluids.
T. Boulier, E. Cancellieri, N. D. Sangouard, Q. Glorieux, A. V. Kavokin, D. M.
Whittaker, E. Giacobino, and A. Bramati.
Phys. Rev. Lett. 116, 116402 (2016)

  * •

Observation of the Bogoliubov dispersion in a fluid of light.
Q. Fontaine, T. Bienaimé, S. Pigeon, E. Giacobino, A. Bramati, Q. Glorieux.
Phys. Rev. Lett., Accepted , (2018).

  * •

Coherent merging of counterpropagating exciton-polariton superfluids.
T. Boulier, S. Pigeon, E. Cancellieri, P. Robin, E. Giacobino, Q. Glorieux and
A. Bramati.
Phys. Rev. B 98, 024503 (2018).
Not included.

  * •

Lattices of quantized vortices in polariton superfluids.
T. Boulier, E. Cancellieri, N. D Sangouard, R. Hivet, Q. Glorieux, E.
Giacobino, A. Bramati.
Comptes Rendus Académie des Sciences 17, 893 (2016).
Not included.

  * •

Vortex chain in a resonantly pumped polariton superfluid.
T. Boulier, H. Tercas, DD. Solnyshkov, Q. Glorieux, E. Giacobino, G. Malpuech,
A. Bramati.
Scientific Reports 5, 9230 (2015).
Not included.

###  4.1 What is a fluid of light ?

We know, since Einstein, that vacuum photons are well described as mass-less
non-interacting particles, behaving as an ideal gas. In this chapter we
discuss experiments where photons acquire a sizeable effective mass and
tunable interactions mediated by a coupling with matter, forming a fluid of
light. This approach relies on a description of these hybrid photons in terms
of quasi-particle of light generically known as polariton [6].

####  4.1.1 Hydrodynamic formulation of the non-linear Schrödinger equation

The non-linear Schrödinger equation is used to describe a large variety of
phenomena. This equation can be written in a mathematical form, with $t$ and
$\nabla^{2}$ dimensionless:

$i\frac{\partial A}{\partial
t}=\left(-\frac{1}{2}\nabla^{2}+g|A|^{2}\right)A.$ (4.1)

$A$ is an arbitrary quantity and $g$ is the non-linear coupling coefficient.

In this manuscript, we will use this equation (with some modifications) to
study the time evolution of excitons polaritons in a microcavity (see 4.2) and
the spatial evolution of an electric field propagating in a non-linear medium
(see 4.5). In chapter 5, we will also discuss its relevance in slow light and
quantum memory experiments.

In the context of fluid of light, it is useful to use the hydrodynamic
formulation of the non-linear Schrödinger equation [58, 59, 60]. To show that
this approach is very general, I will do this derivation in the case of the
Gross-Pitaevskii equation which describes the dynamics of an atomic Bose-
Einstein condensate [61]. The Gross-Pitaevskii equation is a time dependant
non-linear Schrödinger equation including a potential term:

$i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial
t}=\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r})+g|\psi(\mathbf{r},t)|^{2}\right)\psi(\mathbf{r},t),$
(4.2)

where $\psi(\mathbf{r},t)$ is the condensate wave-function, $V(\mathbf{r})$ is
an external potential and $g$ is the non-linear coupling coefficient, which is
real. As usual the term $|\psi(\mathbf{r},t)|^{2}$ gives the local density of
the condensate. To find an hydrodynamic formulation, we need to extract a
quantity analogue to a local velocity. This can be done combining the time
evolution of $\psi$ and $\psi^{*}$ such as we find

$\frac{\partial|\psi|^{2}}{\partial
t}+\nabla\cdot\left[\frac{\hbar}{2mi}\left(\psi^{*}\nabla\psi-\psi\nabla\psi^{*}\right)\right]=0.$
(4.3)

We can note that we would have obtained the same equation in the absence of
external potential $V(\mathbf{r})$ and in the absence of the non-linear term
$g|\psi(\mathbf{r},t)|^{2}$ as both these terms are real. Indeed this
derivation has first been made by Madelung for the usual (linear) Schrödinger
equation [62, 63].

We note $\rho=|\psi(\mathbf{r},t)|^{2}$ the local density. We can see that Eq.
4.3 has the form of a continuity equation for $\rho$, and may be written as:

$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0,$ (4.4)

where we have introduced a quantity $\mathbf{v}$ analogue to a velocity:

$\mathbf{v}=\frac{\hbar}{2mi}\frac{\psi^{*}\nabla\psi-\psi\nabla\psi^{*}}{|\psi(\mathbf{r},t)|^{2}}.$
(4.5)

It is possible to obtain a simpler expression for the velocity by applying the
Madelung transformation to $\psi(\mathbf{r},t)$ and using the real part of the
wavefunction $\sqrt{\rho}$ and its phase $\phi$:

$\psi(\mathbf{r},t)=\sqrt{\rho(\mathbf{r},t)}e^{i\phi(\mathbf{r},t)}.$ (4.6)

This allows to write the velocity as:

$\mathbf{v}=\frac{\hbar}{m}\nabla\phi.$ (4.7)

For example, this derivation can be done in a similar way for the time-
evolution of an electromagnetic field in a cavity filled by a $\chi^{(3)}$
non-linear medium [64, 65, 66, 67, 68]. The equation is slightly modified (due
to cavity losses [67]) but the main idea remains: the light intensity
corresponds to the fluid density, the spatial gradient of its phase to the
fluid velocity and the collective behaviour originates from the effective
photon-photon interactions due to the non–linearity of the medium inside the
cavity.

Historically, the first connection made between superfluid hydrodynamics and
non-linear optics dates back to the 80’s. Back then, P. Coullet _et al._
related optical phase singularities to quantized vortices in superfluid [69].
A seminal attempt to experimental investigation of superfluid behaviour in a
system based on non-linear medium in a macroscopic cavity [70] has been
followed by a series of theoretical articles by R. Chiao on superfluidity of
light in an atomic medium within a Fabry–Perot cavity [71]. Surprisingly, no
experiments were reported thereafter, possibly because large non-linearities
and high-Q cavities were hardly available at the time. In parallel, non-linear
resonator filled with dye has been investigated and have allowed for the
observation of photon BEC [72].

###  4.2 General overview of exciton-polaritons in a semiconductor microcavity

Modern research on quantum fluids of light has shifted to exciton-polaritons
in micro-cavities. These semiconductor nanostructures, thanks to the progress
in nano–fabrication, offer unprecedented control of light-matter interaction.
In these systems, photons entering a Fabry–Perot cavity strongly couple to
excitonic dipolar transitions through quantum wells located at the cavity
electric-field maxima. This leads to the creation of massive interacting
bosonic quasi-particles known as exciton-polaritons.

##### Where does the photon effective mass comes from ?

From the confinement inside a Fabry-Perot cavity.

If you take an optical cavity of length $L$ and a refractive index $n_{c}$,
then the resonance condition leads to $p\lambda_{p}=2n_{c}L$, with $p$ a
positive integer and $\lambda_{p}$ the wavelength associated to the mode $p$.
The dispersion relation of light with wavevector $\mathbf{k}$ is:

$E_{c}(\mathbf{k})=\frac{\hbar c}{n_{c}}|\mathbf{k}|=\frac{\hbar
c}{n_{c}}\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}.$ (4.8)

If we assume the cavity to be along the $z$ direction, the quantification of
the modes inside the cavity leads to:

$k_{z}=p\frac{\pi}{n_{c}L}.$ (4.9)

We restrict ourselves to small angle of incidence $\theta$ on the cavity plane
and therefore $k_{z}=|\mathbf{k}|\cos(\theta)\gg k_{x},k_{y}$. We note the in-
plane wavevector $k_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}$. We obtain the general
dispersion relation for a cavity:

$\hbar\omega_{c}\simeq\frac{\hbar
ck_{z}}{n_{c}}+\frac{\hbar^{2}k_{\perp}^{2}}{2m^{*}},$ (4.10)

with the definition of an effective mass $m^{*}=n_{c}\hbar k_{z}c$. In typical
cavities we have $m^{*}\sim 5.10^{-7}m_{e}$, with $m_{e}$ the electron mass.
In this work, we will use microcavities made of GaAs and of length 2$\lambda$.
The cavity finesse is $F=3000$.

##### Where does the interactions comes from ?

From the quantum wells excitons embedded within the microcavity.

A quantum well in a semiconductor consist on thin layer (nano-scale) of small
band gap semiconductor, sandwiched between two other semiconductor with wider
band gap. In our case, we are talking about an InGaAs layer embedded in GaAs.
The material discontinuity in the growth direction leads to a confinement of
the electronic excitations inside the well region and a discretization of
their states inside the well. In semiconductor, the absorption of a photon of
adequate wavelength can promote an electron of the valence band (below the
Fermi energy) to the conduction band (above the Fermi energy). This will leave
an empty spot in the valence band that can be treated as a quasi-particle of
positive charge called hole. Both excitations having opposite electronic
charge can bound via Coulombian interaction. The energy of the electron-hole
pair being reduced, they form a pseudo-particle named exciton. The exciton is
then analogue to the 1s orbital of the hydrogen atom, with a positive charge
of large mass (the hole) and a negative charge of light mass (the electron)
bounded by the Coulombian interaction. For Wannier-Mott exciton, the exciton
size is larger than the typical crystal inter-atomic distance.

Excitons can be approximated as bosonic quasi-particles interacting mediated
hard-core interaction [73]. Defining a creation operator
$\hat{b}_{\mathbf{q}}$, for an exciton with wavevector $\mathbf{q}$. The
interaction hamiltonian is then written as:

$\hat{H}_{XX}=\frac{1}{2}\sum_{\mathbf{k},\mathbf{k^{\prime}},\mathbf{q}}V_{XX}\hat{b}^{{\dagger}}_{\mathbf{k}+\mathbf{q}}\hat{b}^{{\dagger}}_{\mathbf{k^{\prime}}-\mathbf{q}}\hat{b}_{\mathbf{k}}\hat{b}_{\mathbf{k^{\prime}}}.$
(4.11)

This expression describes a scattering process with destruction of 2 excitons
at wavevectors $\mathbf{k}$ and $\mathbf{k^{\prime}}$ and the creation of 2
excitons at wavevectors $\mathbf{k+q}$ and $\mathbf{k^{\prime}-q}$. We assume
the interaction potential $V_{XX}$ to be constant with $\mathbf{q}$ because we
are interested at small wavevectors compared to the inverse of the exciton
length scale.

I have briefly described the cavity part which gives an effective mass to the
photons and the quantum well which will provide the interactions. In the next
paragraph, I bring together (strongly) these elements and I introduce the
concept of exciton-polaritons.

##### Exciton-polaritons

The physics of coupling between these two systems is a fantastic example for
undergraduate quantum mechanics class. One starts with two bosonic particles
associated with the creation operators $\hat{a}_{\mathbf{k}}^{{\dagger}}$ and
$\hat{b}_{\mathbf{k}}^{{\dagger}}$ for cavity photons and excitons
respectively. These particles have their own eigen-energies
$E_{c}=\hbar\omega_{c}$ for the cavity photons and $E_{X}=\hbar\omega_{X}$ for
the excitons. The hamiltonian of the uncoupled system is clearly diagonal. One
adds a coherent coupling energy $\hbar\Omega_{R}$ between them and the system
is not diagonal anymore. After a simple 2-by-2 matrix diagonalization, one
obtains the two new eigen-states and eigen-energies, named upper and lower
polaritons111Obviously, for this simple treatment we have neglected the
exciton-exciton interactions described in the previous section. For a more
detailed description please refer to [74].

We will briefly derive this. In the photon-exciton basis we can write the
hamiltonian $\hat{H}$ as:

$\hat{H}=\begin{bmatrix}E_{c}&\hbar\Omega_{R}\\\ \hbar\Omega_{R}&E_{X}\\\
\end{bmatrix}.$ (4.12)

Diagonalization of this hamiltonian gives rise to two new bosonic particles
associated with their creation operators $\hat{p}_{\mathbf{k}}^{{\dagger}}$
and $\hat{u}_{\mathbf{k}}^{{\dagger}}$ for the lower and upper polaritons
respectively. As bosonic particles, these operators follow the usual
commutation relations. The new eigen-energies are :

$\displaystyle\omega_{UP}(\mathbf{k})=\frac{\omega_{C}(\mathbf{k})+\omega_{X}(\mathbf{k})}{2}+\frac{1}{2}\sqrt{[\omega_{C}(\mathbf{k})-\omega_{X}(\mathbf{k})]^{2}+4\Omega_{R}^{2}},$
(4.13)
$\displaystyle\omega_{LP}(\mathbf{k})=\frac{\omega_{C}(\mathbf{k})+\omega_{X}(\mathbf{k})}{2}-\frac{1}{2}\sqrt{[\omega_{C}(\mathbf{k})-\omega_{X}(\mathbf{k})]^{2}+4\Omega_{R}^{2}}.$

$\omega_{LP}$ and $\omega_{UP}$ are associated to the lower and upper
polaritons respectively. The eigen-energy is then the mean of the exciton
energy and photon energy, with a correction that is the root mean square of
the coupling Rabi frequency $\Omega_{R}$ and half of the cavity-exciton
detuning $\delta=\omega_{C}(\mathbf{k})-\omega_{X}(\mathbf{k})$.
The polariton eigen-states can be written using the Hopfield coefficients
[75]:

$\left\\{\begin{array}[]{lr}\hat{p}_{\mathbf{k}}=X_{\mathbf{k}}\hat{b}_{\mathbf{k}}+C_{\mathbf{k}}\hat{a}_{\mathbf{k}}\\\
\hat{u}_{\mathbf{k}}=X_{\mathbf{k}}\hat{a}_{\mathbf{k}}-C_{\mathbf{k}}\hat{b}_{\mathbf{k}}\end{array}\right.$
(4.14)

with

$\displaystyle
X_{\mathbf{k}}=\frac{1}{\sqrt{1+\left[\frac{\Omega_{R}}{\omega_{LP}(\mathbf{k})-\omega_{C}(\mathbf{k})}\right]^{2}}}$
(4.15) $\displaystyle
C_{\mathbf{k}}=\frac{-1}{\sqrt{1+\left[\frac{\omega_{LP}(\mathbf{k})-\omega_{C}(\mathbf{k})}{\Omega_{R}}\right]^{2}}}.$

To give an intuition of the behaviour of these pseudo particles we have
plotted in Figure 4.1 the dispersion relation for the upper and lower
polaritons for different detunings
$\delta=\omega_{X}(\mathbf{k})-\omega_{C}(\mathbf{k})$. This situation is
similar to what we realize experimentally [76]. The exciton mass is much
larger than the cavity photon effective mass and therefore the dispersion of
exciton is almost flat on this scale. To tune the detuning $\delta$ we can
only control the cavity energy. To do so, the cavity is slightly wedged in one
direction, providing a large choice of cavity-exciton detunings by pumping at
different positions in the sample. From this figure we can understand
intuitively the effect of detuning. For positive detuning (Fig. 4.1-a), the
lower polariton branch becomes more excitonic at $\mathbf{k}=0$ and therefore
this enhances the polariton-polariton interactions. On the opposite side, for
negative detuning (Fig. 4.1-c), the lower polariton branch becomes more
photonic. This can be used to reduce the effective mass and to get more light
outside of the system. An important characteristic of this system is that we
are only concern on the 2D evolution that in the transverse (cavity) plane.

Figure 4.1: Dispersion of upper and lower polaritons. Yellow dashed is the
bare exciton dispersion and blue dashed is the bare cavity dispersion. The
relative photonic fraction is given in the colorbar. Exciton energy: $Ex=1500$
meV, Coupling frequency: $\Omega_{R}=10$ meV, a) Blue detuned cavity $Ec=1515$
meV (Marignan), b) Cavity on-resonance $Ec=1500$ meV, Red detuned cavity
$Ec=1485$ meV.

##### Strong coupling condition

In the previous paragraph, I have explained what are exciton-polaritons.
However, I have skipped one important discussion about strong coupling.
Indeed, exciton and photon have a finite lifetime in this system. Photon
lifetime $\tau_{C}$ can be obtained from the quality factor $Q$ of the cavity
as:

$\tau_{C}=\frac{Q}{\omega_{C}}.$ (4.16)

For the GaAs cavities we have used in this manuscript, the typical value of
$\tau_{C}$ is 20 ps and quality factor are in the order of $10^{4}$.
Exciton lifetime is far less under control and depends mainly of surface
irregularities inside the wells. These two quantities give rise to a
broadening of the energy by a width $\hbar$ times the inverse of the lifetime.
Typical broadening for a GaAs cavity is $\hbar\Gamma_{C}\sim 50$ $\mu$eV.

On the other hand the Rabi frequency can be estimated from the oscillator
strength of the exciton $f$, the size of the cavity $L$ and the exciton mass
$M$:

$\Omega_{r}=\sqrt{\frac{fe^{2}}{2n_{c}^{2}LM\varepsilon_{0}}}.$ (4.17)

However this expression assumes a perfect overlap of the exciton and the
photon wavefunctions. Quantum wells must be placed carefully in the anti-nodes
of the electromagnetic field in order to optimize the coupling. It is common
to increase the Rabi frequency by adding more wells as it scales with the
square root of the number of wells (because the effective oscillator strength
scales as the number of wells).

We can now give an graphical understanding of the strong coupling. The strong
coupling condition is achieved when the Rabi frequency is large enough so that
polariton states can be distinguished form the uncoupled exciton and photon
states. This condition directly depends on the exciton and cavity photon
linewidth and can be written:

$\Omega_{R}>\Gamma_{C},\Gamma_{X}.$ (4.18)

If this condition is not satisfied, there is no need to talk about polariton
states.

####  4.2.1 Driven-dissipative Gross-Pitaevskii equation

Remarkably, the time evolution of the exciton-polaritons wave-function follows
dynamics similar to the Gross-Pitaevksii equation 4.2 describing the evolution
of a dilute atomic Bose-Einstein condensate or a monochromatic light
propagating in a $\chi^{(3)}$ non-linear media as presented in section 4.1.1.
However, for exciton-polaritons, it includes additional non-equilibrium
features, streaming from their intrinsic driven-dissipative nature. The
evolution equation for the lower polaritons reads [77]:

$i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial
t}=\left(-\frac{\hbar^{2}}{2m^{*}}\nabla_{\perp}^{2}+V(\mathbf{r})-i\frac{\hbar\gamma_{LP}}{2}+g|\psi(\mathbf{r},t)|^{2}\right)\psi(\mathbf{r},t)+P(\mathbf{r},t),$
(4.19)

with $m^{*}$ the effective mass of lower polaritons.

On one side, due to the finite reflectively of the cavity’s mirrors, photons
have a finite lifetime such that eventfully, they exit the cavity. This, which
can be seen as a drawback, has proved to be essential as it allows for the
detection and measurement of polaritons (via their photonic component). This
leads to a dissipation term in the equation proportional to the polariton
linewidth: $\hbar\gamma_{LP}$. On the other hand, if photons exit the cavity
(after 20 ps typically), new particles must be added (continuously) to the
system to maintain a constant number of particles. This is done by pumping the
cavity with an external field and leads to a driving term $P(\mathbf{r},t)$ in
the equation. Finally, we should restrict the evolution to the plane of the
cavity and therefore the 3D spatial derivative is changed to 2D
$\nabla_{\perp}$.

An important feature of polariton physics is therefore that it is out of
equilibrium, as particles constantly leave the system. However, with the
seminal observation of exciton-polaritons BEC [78, 79] and the demonstration
of superfluidity through resonant Rayleigh scattering [80, 81], this system
has become a realistic tool for quantum hydrodynamics. It is worthy to notice
that the possibility to observe such fluid of light is allowed by the
conjunction of relatively strong $\chi^{(3)}$ non-linearity and a very low
effective mass. The latter is essential as it provides for a very fast
dynamics to form the fluid in a time scale smaller than the dissipation. This
allows for the formation of long-range correlations for example.

###  4.3 Superfluidity

To conclude this short introduction to light hydrodynamics, I will cover the
concept of superfluidity and apply it to light. Superfluidity is the property
of a fluid to flow without viscosity below a certain temperature. It has been
discovered in Helium 4 cooled at 2 K by Kapitza [82], Allen and Misener [83]
in 1937, about 60 years before the first observation of an atomic BEC [84].

However, in 1938 London highlights that superfluid transition temperature and
BEC temperature for an ideal gas at the same density are similar and he
proposes a link between the two phenomena [85]. Unfortunately, this
explanation is not relevant for liquid Helium 4, due to the strong
interactions which break the ideal gas model.

In 1941, Landau introduced a novel model based on two fluids: one classical
and one condensed which was much more accurate to describe the physics of
liquid Helium [86, 87]. He also proposed various criteria to define
superfludity. These criteria are still used today to assess superfluidity and
to measure the condensed fraction.

##### Rotating fluid

These criteria are based on a conceptually simple experiment. A fluid
(presumably a superfluid) is placed in a bucket and the bucket is rotated. For
a superfluid, under the critical velocity, the fluid does not move with the
bucket. Above the critical velocity, quantized vortices appear in the fluid.
This effect has been observed for the first time by Hess and Fairbank in 1967
[88]. I discuss this criteria in more details in the section 4.4, as we have
used a similar technique to store quantized vortices in a polariton
superfluid.

##### Persisting current

Another approach is to put in rotation a classical fluid and then cool it down
below the critical point. If now, one removes the rotating mechanism, a
superfluid will rotate for a virtually infinite time. This effect has been
observed in 1964 in liquid Helium [89] and in 2007 in a BEC with interactions
[90]. We can note an interesting proposal to observe a similar mechanism in
photon fluid proposed in Ref. [91].

##### Landau criteria for superfluidity

To derive these criteria we consider a uniform Bose gas with interaction (a
uniform quantum fluid) in its ground state. Imagine that we move at velocity
$\mathbf{v}$ a solid obstacle through the fluid.

We want to know at which critical velocity $v_{c}$, it starts to become
possible to create an elementary excitation in the fluid. It is more
convenient to work in the reference frame of the obstacle, as the obstacle
exerts a time-independent potential to the fluid in this frame. Let us recall
the standard Galilean transformations to write the energy $E(\mathbf{v})$ in
the frame moving at velocity $\mathbf{v}$ as function of the energy $E$ and
$\mathbf{p}$ the momentum of the fluid in one frame of reference:

$E(\mathbf{v})=E-\mathbf{p}\cdot\mathbf{v}+\frac{1}{2}M\mathbf{v}^{2},$ (4.20)

where $M$ is the total mass of the system.

If we apply it to our case, we can write the energy of the ground state as:

$E(\mathbf{v})=E_{0}+\frac{1}{2}M\mathbf{v}^{2},$ (4.21)

where $E_{0}$ is the ground state energy in the frame where the fluid is at
rest, and $\mathbf{p}=0$. We note $\epsilon_{p}$ the energy cost to add a
single excitation of momentum $\mathbf{p}$ to the system. In the original
frame the energy of the state with a single excitation is:

$E_{ex}=E_{0}+\epsilon_{p},$ (4.22)

and therefore in the moving frame:

$E_{ex}(\mathbf{v})=E_{0}+\epsilon_{p}-\mathbf{p}\cdot\mathbf{v}+\frac{1}{2}M\mathbf{v}^{2}.$
(4.23)

We can now make the difference between Eq. 4.23 and Eq. 4.21 to obtain the
energy to create an excitation in the moving frame. Thus at a velocity

$v=\frac{\epsilon_{p}}{p},$ (4.24)

it becomes possible for the obstacle to create an excitation with momentum
parallel to $\mathbf{v}$ and at higher velocities, the excitation momentum
will have an angle to $\mathbf{v}$. We can conclude with the Landau critical
velocity $v_{c}$ defined as the minimum velocity at which it is possible to
create excitation:

$v_{c}=\text{min}\left(\frac{\epsilon_{p}}{p}\right).$ (4.25)

Below the critical velocity there is no mechanism to create elementary
excitations and the liquid will exhibit superfluidity. We can note that for a
parabolic dispersion of the elementary excitations $\epsilon(p)\propto p^{2}$,
the critical velocity is $0$. On the other hand, a linear dispersion
$\epsilon(p)\propto p$ will allow for superfluidity.

##### Suppression of Rayleigh scattering

An important experiment, done in the LKB group before I joined, has
demonstrated superfluidity of polaritons using the setup described in the
section 4.2. A polariton flow is sent towards a defect and depending on
velocity and density of the fluid, elementary excitations are created or not
[81]. This experiment is really the foundation of most of the work done in
polariton hydrodynamics afterwards and I will review it briefly.

A microcavity is pumped at a small angle by a resonant laser field. The light
enters the cavity and hybridizes with excitons to create polaritons with a
momentum equal to the projection of the incident laser wavevector in the
cavity plane. Some defects can be found in microcavities due to scratches or
imperfect growth, which can be used to study the scattering behaviour of
polaritons and probe the superfluidity. In Fig. 4.2 a) we see the polariton
fluid hitting the defect and creating fringes upstream the defect. This
situation is at low driving intensity, which is equivalent to a low density of
polaritons inside the cavity. In this regime the non-linear interaction can be
neglected. Two interpretations can be given to these fringes:

  * •

We can think as opticians and interpret this pattern as an interference
pattern between incoming photons and photons scattered by the defect. The
fringe distance will change when varying the projection of the incident laser
wavevector in the cavity plane. This can be verified experimentally.

  * •

The second interpretation comes from the hydrodynamic language. When a normal
fluid propagates and hits a defect, characteristic waves appear upstream of
the defect similar to water hitting a bridge pile.

In Fig. 4.2 b), we show the k-space (the emission angle of the cavity). The
black spot at $k_{x}=0$ et $k_{y}=-0.3\mu$m-1 is the pumping laser. We can
observe a redistribution of the impulsion around a ring at constant $k$ which
is Rayleigh scattering on the defect.

To switch to the superfluid regime, we do not have to cool down the fluid as
in liquid Helium experiments, but we increase the density. This regime is
shown in Fig. 4.2 c) and d). The disappearance of interference fringes is
understood intuitively with the hydrodynamic interpretation because the fluid
of light, now flows without viscosity. Correspondingly, the Rayleigh
scattering vanishes.

The optics interpretation is more obscure as it is a cancellation of the
diffraction on the defect due the non-linear effects. We see here for the
first time, why the hydrodynamic interpretation is useful to analyze and study
optical phenomena that are difficult to predict with the non-linear optics
usual toolbox. I will follow this approach in various experiments presented in
this chapter.

Finally we can break superfluidity by increasing the velocity above the
critical speed $v_{c}$. This experiment is reported in Fig. 4.3 c) and d). We
will see in the next section that the dispersion elementary excitations in a
polariton fluid is linear at low wavevector, and this allows to define a speed
of sound $c_{s}$ which is the critical velocity. In this case, we observe an
analogue of the Cerenkov effect. Indeed the excitations created in the fluid
have a speed greater than the sound velocity in the medium, and we can observe
the characteristic cone upstream of the defect.

Figure 4.2: Comparison between a normal fluid (at low density a. and b.) and a
superfluid (at high density c. and d.). The flow speed $v$ is downwards in
real space images a) and c). Interference fringes in a) are due to scattering
on the defect. They vanish for higher intensity due to superfluid flow. In
k-space images, the Rayleigh scattering is clearly visible in b) and absent in
d) due to superfluidity. For the superfluid case $v<v_{c}$. Adapted from [81]

.

Figure 4.3: Comparison between a normal fluid (at low density a. and b. ) and
a superfluid (at high density c. and d.) above the critical speed. The flow
speed $v$ is downwards in real space images. For c) $v>v_{c}$. We can observe
the break of superfluidity with the apparition of a Cerenkov cone at the
defect position in c). The angle of the cone gives the critical velocity
(which is the speed of sound in this case). In k-space images, the Rayleigh
scattering is clearly visible in b). For the Cerenkov case d) polaritons are
scattered at impulsion larger than the $k_{i}n$ as expected by the Landau
theory. Adapted from [81].

###  4.4 Injection of angular momentum in polariton superfluids

A rich variety of photon hydrodynamical effects have been observed, from the
unperturbed superfluid stream flowing around a defect [81], to the appearance
of a Cerenkov cone in a supersonic flow [81] presented in the previous
section. Other works from the LKB group have focused on the formation of
topological excitations such as quantized vortices and dark solitons [76, 92].
In this section I give the details about recent experiments done at LKB, where
we have achieved the controlled injection of orbital angular momentum in a
polariton superfluid using two different techniques.

We have discussed in paragraph 4.3 a method proposed by Landau to probe
superfluidity using a rotating bucket. We have revisited this idea by forcing
the rotation of a polariton fluid and study the apparition of quantized
vortices. Quantized vortices are topological excitations characterized by the
vanishing of the field density at a given point (the vortex core) and the
quantized winding of the field phase from 0 to $2\pi$ around it.

##### Injection of angular momentum using a Laguerre-Gauss pump

The first method we have studied was to pump the microcavity with a Laguerre-
Gauss mode of increasing charge. We have reported the formation of a ring-
shaped array of same sign vortices [93]. As previously discussed in the linear
regime, an interference pattern is visible. This pattern has a spiral shape
and contains phase defects are visible. In the nonlinear (superfluid) regime,
the interference disappears and up to eight vortices appear, minimizing the
energy while conserving the quantized angular momentum (see figure 4.4).

Figure 4.4: Nucleation of same sign vortices. Experimental real space images
of the polariton field in the nonlinear regime. a) Polariton density and b)
phase pattern. Figure from Ref. [93]

##### Injection of angular momentum using a 4 tilted pumps

The second experiment studies the propagation of 4 superfluids created with
the same pump field at 4 different positions arranged in a square, all with an
inward flow. The pump is chosen to be resonant to allow for a fine tuning of
the polariton density without generating excitonic reservoir and,
consequently, a precise control of the non-linearities in the system.

If we send these fluids toward the center we simply observe the merging of the
4 converging fluids. However, if we tilt the in-plane flow as presented in the
figure 1 of PRL 116, 116402 (see below), we can inject an orbital angular
momentum in the system and we see the apparition of quantized vortices. We can
evaluate the injected orbital angular momentum per photon $L$ in unit of
$\hbar$ as:

$\frac{L}{\hbar}=R|\mathbf{k}|\sin\phi,$ (4.26)

with $R$ is the pump distance to the square center, $\mathbf{k}$ is the in-
plane pump wavevector (choose identical for the 4 pumps) and $\phi$ is the
tilt angle between the pump in-plane direction and the square center.

The separation between the pumps is small enough to ensure the four polariton
populations can meet, resulting in significant density at the square center.
However to ensure that in the central region polaritons are free to evolve, we
have cut the beam Gaussian tails to reduce direct illumination.

We use two different techniques for detection: the direct density measurement
and a phase measurement. The polariton phase is measured with an off-axis
interferometry setup: a beam splitter divides the real space image into two
parts, one of which is expanded to generate a flat phase reference beam, used
to make an off-axis interference pattern. With this method, the vortex
position on the image is independent of the phase of the reference beam. The
actual phase map is then numerically reconstructed with a phase retrieval
algorithm.

What is striking in this experiment is that even though the orbital angular
momentum injected is a continuous quantity, the number of topological defects
remains a quantized quantity. We have verified that the number of injected
vortices (up to 5) matches a simple model including only the tilt angle (see
figure 4 of PRL 116, 116402).

In conclusion we have observed the injection of angular momentum and the
storage of topological charges in a non-equilibrium superfluid of light [76].

## Article 5: Injection of Orbital Angular Momentum and Storage of Quantized
Vortices in Polariton Superfluids.

### Preprint

The preprint version of this article is accessible on arXiv:1509.02680.

### Published version

Injection of Orbital Angular Momentum and Storage of Quantized Vortices in
Polariton Superfluids. T. Boulier, E. Cancellieri, N. D. Sangouard, Q.
Glorieux, A. V. Kavokin, D. M. Whittaker, E. Giacobino, and A. Bramati. Phys.
Rev. Lett. 116, 116402 (2016).

###  4.5 Fluid of light in the propagating geometry

As we have seen in the previous section, fluids of light allow to revisit
quantum gases experiments with the advantage of the precise control in state
preparation and detection of optical systems. So far I have presented results
about cavity-based platforms which have intrinsic losses and therefore are
restricted to driven-dissipative dynamics. In 2016, I have started a new
thematic in the LKB group to overcome this limitation with a different way of
doing quantum simulation with light, by removing the need for a Fabry-Perot
cavity. This approach brings complementary results to the exciton-polaritons
experiments and gives the possibility for our group to compare driven-
dissipative and conservative dynamics with fluids of light. Similarly to
exciton-polaritons, it relies on the formal mapping between the dynamics of
the problem we wish to simulate and the quantum system we will use to do it.

The system we propose to use is light propagating through a warm atomic vapor
and I show that this system can be theoretically described by a non-linear
Schrödinger equation similar to Eq. 4.2. To validate experimentally this
approach, I present recent results about the dispersion relation of elementary
excitations in this system and show that they follow a Bogoliubov spectrum
similar to atomic BEC or photon BEC[72] experiments.

####  4.5.1 Bogoliubov dispersion relation

Before going in the details of our system, I review briefly the physics of
elementary excitations in the Gross-Pitaevskii equation (Eq. 4.2). More
specifically, I derive the Bogoliubov dispersion relation for a uniform Bose
gas in the absence of external potential.

We start by writing the quantum state as a mean field plus a small
perturbation to linearize the Gross-Pitaevskii equation:

$\Psi(\mathbf{r},t)=[\psi_{0}(\mathbf{r},t)+\delta\psi(\mathbf{r},t)].$ (4.27)

Here $\psi_{0}(\mathbf{r},t)$ is the condensate wavefunction in the
unperturbed state. It can then be written:

$\psi_{0}=\sqrt{n(\mathbf{r})}\text{e}^{-i\mu t/\hbar},$ (4.28)

where $n(\mathbf{r}$ is the equilibrium density of particles and $\mu$ is the
chemical potential. For a uniform system $n(\mathbf{r}$ is independent of
$\mathbf{r}$ and $\mu$ is given by $\mu=|\psi_{0}|^{2}g=ng$. For
$\delta\psi(\mathbf{r},t)$, we are interested in solution of the form:

$\delta\psi(\mathbf{r},t)=\left[u(\mathbf{r})\text{e}^{-i\omega
t}-v^{*}(\mathbf{r})\text{e}^{i\omega t}\right]\text{e}^{-i\mu t/\hbar}.$
(4.29)

with $\omega$ a real quantity. Inserting this ansatz in the linearized Gross-
Pitaevskii equation we obtain two coupled equations for $u(\mathbf{r})$ and
$v(\mathbf{r})$ known as the Bogoliubov equations:

$\displaystyle\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+gn-\hbar\omega\right]u(\mathbf{r})-gn\
v(\mathbf{r})=0$ (4.30)
$\displaystyle\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+gn+\hbar\omega\right]v(\mathbf{r})-gn\
u(\mathbf{r})=0$

Because of the translational invariance in a uniform Bose gas we can take
$u(\mathbf{r})$ et $v(\mathbf{r})$ to be of the form of plane waves:

$u(\mathbf{r})=u_{q}e^{i\mathbf{q}\mathbf{r}}\text{ and
}v(\mathbf{r})=v_{q}e^{i\mathbf{q}\mathbf{r}}.$ (4.31)

By injecting these expressions in Eq. 4.30 we obtain two equations that are
consistent only if the determinant of the coefficients vanishes. Finally we
obtain:

$(\hbar\omega)^{2}=\left[\frac{\hbar^{2}q^{2}}{2m}+gn\right]^{2}-\left(gn\right)^{2}$
(4.32)

We take the positive energy solutions and we obtain the Bogoliubov dispersion
relation:

$\epsilon(q)=\hbar\omega(q)=\sqrt{\varepsilon_{0}(q)}\cdot\sqrt{\varepsilon_{0}(q)+2gn}.$
(4.33)

Here we have introduced the notation
$\varepsilon_{0}(q)=\frac{\hbar^{2}q^{2}}{2m}$ for the energy of the free-
particle without interaction.

We can see from this expression that the Bogoliubov dispersion differs from
the massive particle parabolic dispersion only at small $q$ when
$2gn\leq\varepsilon_{0}(q)$. In the limit $2gn/\varepsilon_{0}(q)\gg 1$, we
can approximate Eq. 4.33 by

$\epsilon(q)=\hbar\sqrt{\frac{gn}{m}}q.$ (4.34)

In this limit, the spectrum become linear (sound-like) and we can define a
speed of sound $c_{s}$ that scales with the square root of the density:

$c_{s}=\sqrt{\frac{gn}{m}}.$ (4.35)

If we use the Landau criteria for superfluidity (Eq. 4.25), we can see that
this dispersion allows for a minimum of the quantity $\frac{\epsilon(q)}{q}$
which is precisely equal to $c_{s}$.

Figure 4.5: Dispersion relation for massive particles (dashed green), for
sound waves (dashed red) and for Eq. 4.33 (black). $c_{s}=1$ and $\epsilon(q)$
in $\hbar$ unit.

####  4.5.2 Non-linear propagation in a $\chi^{(3)}$ medium

In the absence of sources, the propagation of light in a medium of refractive
index $n$ is given by the Helmholtz equation:

$\nabla^{2}E-\frac{n^{2}}{c^{2}}\frac{\partial^{2}E}{\partial t^{2}}=0,$
(4.36)

where $E$ is the electric field amplitude and $c$ is the speed of light in
vacuum. To take into account the propagation in a non-linear medium of we need
to a non-linear polarization term $P_{NL}$ in the form:

$\nabla^{2}E-\frac{n^{2}}{c^{2}}\frac{\partial^{2}E}{\partial
t^{2}}=\frac{1}{\varepsilon_{0}c^{2}}\frac{\partial^{2}P_{NL}}{\partial
t^{2}}.$ (4.37)

$n$ is the linear part of the refractive index.
For a $\chi^{(3)}$ non-linear medium we have
$P_{NL}=\varepsilon_{0}\chi^{(3)}E^{3}$. We assume the electric field to be
monochromatic (at frequency $\omega_{0}$) and to propagate along $z$. We
introduce $k_{0}$, the wavenumber of $E$, and we decompose the envelope
$E_{0}$ from the oscillating part:

$E(x,y,z,t)=E_{0}(x,y,z)e^{i(k_{0}z-\omega_{0}t)}.$ (4.38)

We can inject this expression of $E$ into Eq. 4.37 to get

$\nabla^{2}_{\perp}E_{0}+\frac{\partial^{2}E_{0}}{\partial
z^{2}}+2ik_{0}\frac{\partial E_{0}}{\partial
z}-\left(k_{0}^{2}-n^{2}\frac{\omega_{0}^{2}}{c^{2}}\right)E_{0}=-\chi^{(3)}|E_{0}|^{2}\frac{\omega_{0}^{2}}{c^{2}}E_{0},$
(4.39)

where $\nabla^{2}_{\perp}$ is the Laplacian in the transverse ($x,y$) plane.
We then take into account that $E_{0}$ is slowly varying along $z$, known as
the paraxial approximation, to eliminate the second derivative of $E_{0}$
along $z$. We can rewrite Eq. 4.39 with this simplification:

$\nabla^{2}_{\perp}E_{0}+2ik_{0}\frac{\partial E_{0}}{\partial
z}-\left(k_{0}^{2}-n^{2}\frac{\omega_{0}^{2}}{c^{2}}\right)E_{0}+\chi^{(3)}|E_{0}|^{2}\frac{\omega_{0}^{2}}{c^{2}}E_{0}=0.$
(4.40)

We can decompose the linear part of the refractive index as $n=n_{0}+\delta
n(\mathbf{r})$ with a mean index plus a local perturbation (supposed weak). As
$k_{0}=n_{0}\omega_{0}/c$, we can rewrite Eq. 4.40 as:

$i\frac{\partial E_{0}}{\partial
z}=-\frac{1}{2k_{0}}\nabla^{2}_{\perp}E_{0}-\frac{\delta
nk_{0}}{n_{0}}E_{0}-\frac{k_{0}\chi^{(3)}}{2n_{0}^{2}}|E_{0}|^{2}E_{0}.$
(4.41)

This equation is analogue to the Gross-Pitaevskii equation (Eq. 4.2). We will
therefore be able to use the same formalism, and in particular we can expect a
Bogoliubov dispersion relation for the small perturbation on the electric
field. We introduce the light intensity $I$ and the non-linear index $n_{2}$.
We have:

$n_{2}=\frac{3\chi^{(3)}}{2n_{0}^{2}\varepsilon_{0}c}\text{ and
}I=|E_{0}|^{2}n_{0}\varepsilon_{0}c/2.$ (4.42)

This allows to write an important quantity experimentally: the non-linear
contribution to the index of refraction: $\Delta n=n_{2}I.$

To precise the analogy with the Gross-Pitaevskii equation (GPE), I have
rewritten Eq. 4.2 and compared it to Eq. 4.40.

$\displaystyle i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial t}$
$\displaystyle=$ $\displaystyle\left(-\frac{\hbar^{2}}{2m}\nabla^{2}\ +\hskip
8.5359ptV(\mathbf{r})\hskip 8.5359pt+\ g\
|\psi(\mathbf{r},t)|^{2}\right)\psi(\mathbf{r},t)$ $\displaystyle
i\frac{\partial E_{0}(\mathbf{r_{\perp}},z)}{\partial z}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2k_{0}}\nabla^{2}_{\perp}-\frac{\delta
n(\mathbf{r_{\perp}})k_{0}}{n_{0}}-\frac{k_{0}\chi^{(3)}}{2n_{0}^{2}}|E_{0}|^{2}\right)E_{0}(\mathbf{r_{\perp}},z).$

The first remark is that the left term in GPE is a time derivative as it is a
spatial derivative along the propagation direction for Eq. 4.40. The dynamics
in Eq. 4.40 is then 2D in the transverse plane written
$\mathbf{r_{\perp}}=(x,y)$. Each slice at a given $z$ is equivalent to a time-
snapshot in the GPE. More precisely we can fix the initial state by designing
a given field at the input of the medium $z=0$ and measure the evolution after
an effective time by imaging the intensity and phase at the output of the
medium $z=L$.

We can now propose a mapping of the different quantity to directly write the
expected Bogoliubov dispersion for small perturbation in Eq. 4.40.

GPE $\displaystyle\leftrightarrow$ Non-linear Optics
$\displaystyle\psi(\mathbf{r},t)$ $\displaystyle\leftrightarrow$
$\displaystyle E_{0}(\mathbf{r_{\perp}},z)$ $\displaystyle m/\hbar$
$\displaystyle\leftrightarrow$ $\displaystyle k_{0}$ $\displaystyle
V(\mathbf{r})/\hbar$ $\displaystyle\leftrightarrow$ $\displaystyle-\delta
n(\mathbf{r_{\perp}})k_{0}/n_{0}$ $\displaystyle g/\hbar$
$\displaystyle\leftrightarrow$
$\displaystyle-\frac{k_{0}\chi^{(3)}}{2n_{0}^{2}}$

We also need to translate the frequency $\omega$ and the wavenumber $q$ of the
Bogoliubov excitations. As time is mapped to $z$, $\omega$ is mapped to a
spatial frequency in the propagation direction. It is denoted $\Omega_{B}$ to
avoid confusion. We have:

$\displaystyle\mathbf{q}$ $\displaystyle\leftrightarrow$
$\displaystyle\mathbf{k_{\perp}}$ $\displaystyle\omega(\mathbf{q})$
$\displaystyle\leftrightarrow$ $\displaystyle\Omega_{B}(\mathbf{k_{\perp}})$

We can now rewrite the Bogoliubov dispersion in the Non-linear Optics analogy:

$\epsilon(\mathbf{k_{\perp}})=\Omega_{B}(\mathbf{k_{\perp}})=\sqrt{\frac{\mathbf{k_{\perp}}^{2}}{2k_{0}}}\cdot\sqrt{\frac{\mathbf{k_{\perp}}^{2}}{2k_{0}}-\frac{k_{0}\chi^{(3)}}{n_{0}^{2}}|E_{0}|^{2}}.$
(4.43)

Similarly to Bose gases, we can define two limits depending on the value of
the wavevector $\mathbf{k_{\perp}}$ compared to the inverse of the healing
length $\xi$ defined as:

$\xi=-\frac{1}{k_{0}}\sqrt{\frac{2n_{0}^{2}}{\chi^{(3)}|E_{0}|^{2}}}.$ (4.44)

For $\mathbf{k_{\perp}}\gg 1/\xi$, we have a parabolic dispersion. For
$\mathbf{k_{\perp}}\ll 1/\xi$ we observe a sonic dispersion with a speed of
sound $c_{s}$ defined as:

$c_{s}=\sqrt{-\frac{\chi^{(3)}|E_{0}|^{2}}{2n_{0}^{2}}}.$ (4.45)

##### Measure of the Bogoliubov dispersion

So far, I have shown theoretically that we can consider the propagation in a
$\chi^{(3)}$ analogue to the dynamic of a Bose gas in 2D 222A fully quantum
theoretical framework for fluids of light in the same configuration has been
recently proposed [94].. We have recently pushed forward this effort with the
experimental verification of this proposal. Using the vocabulary of Bose
gases, in this context, offers modern perspectives on non–linear and quantum
optics experiments.

To date, experimental implementations are still in their infancy with
experiments in photo–refractive crystals [95, 96, 97], and thermo-optic
non–linear medium [98]. In a non-local $\chi^{(3)}$ thermo-optic medium
(methanol), the Bogoliubov dispersion of a fluid of light has been studied
[98] and indication of superfluid behaviour has been observed [99]. Similarly,
an iodine solution has been used to create an event horizon in light fluid
acoustic black hole analogue [100]. Even though these results highlight the
potential of the quantum fluid interpretation, the non-locality of these media
(thermo-optic non-linearity is affected by heat diffusion) is detrimental for
simulating system with local interactions. By using warm atomic vapors as
$\chi^{(3)}$ medium we have access to tunable and local interactions.
Interestingly, a few works on the elimination of diffraction in a slow light
atomic medium [101, 102], are using similar ideas without explicitly
connecting to fluid of light.

In our implementation, we use the following protocol:

  * •

We create a uniform background fluid with a pump laser propagating in a near
resonance atomic vapor.

  * •

We add a small perturbation by interfering a localize probe beam at the same
frequency at an angle with respect to the background wavevector.

  * •

We study the propagation of this perturbation by measuring the distance
propagated in the transverse plane by these narrow wavepackets at the end of
the medium.

  * •

From this measurement of the group velocity, we simply integrate with respect
to the wavevector and obtain the dispersion relation.

We have obtained a dispersion fitting a Bogoliubov spectrum confirming the
theoretical prediction. For more details the results are presented in the
article attached. I just add two comments (please read the paper first): one
to provide an intuitive explanation of this surprising effect and the second
to connect this effect to four-wave-mixing.

Anomalous refraction In our experiment [1] the refraction law is indeed
strongly modified compared to the linear case of Snell law (See figure 4.6).
It is well known that imaging inside a non-linear medium is not a recommended
task for this reason. What I want to show here, is that the quantum fluid
language provide an intuitive vision of this phenomenon. For small angles
($\mathbf{k_{\perp}}<\xi$), the probe beam light emerges always at the same
place in the output plan independently of the angle. This is strange because
changing the incident angle should normally change the refracted angle. This