Antonine Rochet - Laboratoire Kastler Brossel

An Introduction to Quantum Fluid of Light
Realized by : Antonine Rochet
Pierre and Marie Curie University - Paris VI, Master 2 LuMI
Supervised by : Quentin Glorieux
Kastler Brossel Laboratory (LKB) - ENS - UMPC - CNRS
February 8, 2016
An Introduction to Quantum Fluid of Light
Rochet Antonine
Contents
1
2
Quantum Fluid : from particle to photon
1.1 Quantum fluid of particles . . . . . . . . . . . . . . .
1.2 Fluid of light : mathematical description . . . . . . .
1.2.1 Propagation of an electric field in a nonlinear
1.2.2 Gross-Piteavskii equation . . . . . . . . . . .
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medium . . .
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2
2
3
3
4
Fluid of Light in the LKB Group
2.1 Fluid of polaritons . . . . . . . .
2.1.1 Experimental setup . . .
2.1.2 Experimental results . . .
2.2 Fluid of light in an atomic vapor
2.2.1 The laser detuning . . . .
2.2.2 The density . . . . . . .
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I would like to thank Quentin Glorieux for his precious help for this project through his availability,
instructive approach and his good mood. A thought to Alberto Bramati for his contribution to my
understanding. It was pleasant!
2
An Introduction to Quantum Fluid of Light
Rochet Antonine
Introduction
In 1900 Max Planck discovered the quantification of the electromagnetic interactions through the study
of the black body radiation. The thermodynamical study of this radiation leads to consider the system
as an ideal gas of photons without any interaction in thermodynamical equilibrium [1]. Later, this
system has been identified to a massless Bose gas of non-interacting particles. Recently, it has been
realized that under suitable circumstances photons can acquire an effective mass and will behave as a
quantum fluid of light with photon-photon interactions. Our purpose, on a first part is to understand
what leads to name this system a quantum fluid of light through a theoretical approach.
Today, worldwide, around thirty quantum fluid of light experiments exist. At the Kastler Brossel
Laboratory, a team of the Quantum Optics Group formed by Alberto Bramati, Quentin Glorieux and
Elisabeth Giacobino has been working on a particular quantum fluid since 2007 : polaritons fluid.
In a second part, we will show an eloquent result of their experiment highlighting the superfluidy of
polaritons. One of the group perspective is to use an atomic gas as medium to generate a quantum
fluid of light, that will constitute our last part.
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An Introduction to Quantum Fluid of Light
1
Rochet Antonine
Quantum Fluid : from particle to photon
In classical physics, a liquid is defined by the interactions existing between its entities (van der Waals
or dipolar interactions) contrasting with a perfect gas, where particles does not interact at all. In
quantum mecanics, comparisons are made between those two macroscopic phases and some quantum
systems description. We will focus on quantum fluids, quantum counter part of liquid systems and
more specifically on quantum fluids of light. The purpose of this first part is to discover what is hidden
behind this non-intuitive notion. To begin, we will give an overview on what is a quantum fluid of
massive particles and in a second part we will make a link with fluid of photons.
1.1
Quantum fluid of particles
In quantum physics, we consider the wave-like behavior of particles. Every object can be characterize
by its wavelength (with the momentum p) :
λ=
h
.
p
In a gas at thermodynamical equilibrium at a temperature T , every
q particle has an energy on the
2
p2
order of E = 2m = kB T . That corresponds to a wavelength λdB = mk~B T called thermal de Broglie
wavelength.
Figure 1: Schematic representation of the transition between a classical gas and a Bose Einstein
condensate taken from [2]
.
If this wavelength is cloth to the average distance between particles d, particles wave functions interact
with each other ((c) on figure 1), quantum effects emerge on the system and we have a quantum fluid.
The average distance d is directly linked to the particles density n by d = n−1/3 . Two parameters
determine the thermal wavelength : the temperature of the system and the particles mass. For example,
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An Introduction to Quantum Fluid of Light
Rochet Antonine
a Bose Einstein condensate of Rubidium atoms (m = 105 me ) can form at low temperature (1µK)
whereas a liquid Helium (composed by two protons and two electrons) is formed at 2,2 K [3]. This
quantum fluid phase is characterized by a bimodal distribution of particles. On one hand, there is
a massive occupation of the fundamental state. On the other hand, some of the thermal states are
occupied following a Maxwell-Boltzmann distribution.
This condensate state is observed with bosons (particles with integer spin) because they can be in the
same quantum state. Here we have a first similitude with a photon, belonging to the boson family.
A phase transition occurs in an atomic gas because of interactions between particles whereas photons
"don’t see" each other. We have to pay attention to similitudes through the mathematical description.
1.2
Fluid of light : mathematical description
Interactions between light and matter constitute a great field of study. First, we are going to describe the interactions between an electric field and a nonlinear medium and identify our result to
the Shrödinger formalism. Secondly, we will discuss the similarities with a Bose Einstein condensate
description.
1.2.1
Propagation of an electric field in a nonlinear medium
We consider a non magnetic media (without free charge and current) with :
- the electric motion D = ε0 E + P.
- the induction directly proportional to the magnetic field B = µ0 H.
According to Maxwell’s equations ∇ ⊗ E = − ∂B
∂t and ∇ ⊗ H =
we find the electric field propagation equation :
∆2 E −
∂D
∂t
and by a well known process [4],
1 ∂2E
∂2P
=
µ
.
0
c2 ∂t2
∂t2
If we consider a nonlinear media : the polarization is the sum of a linear polarization proportional to
the electric field Pl = ε0 χE added to a nonlinear term Pnl = ε0 χ(2) E2 +ε0 χ(3) E3 . The first order of the
electric susceptibility χ is linked to the relative linear dielectric constant by = 1+χ. We introduce a
spatial modulation of the relative dielectric constant δ(r⊥ , z), supposed to be slowly varying in space
(we will come back on its meaning latter).
Considering a propagation along z axis, we separate transversal and longitudinal dynamic with r⊥ =
√
(x, y) , k⊥ = (kx , ky ) and kz = k0 ez = ωc0 ez . We obtain:
∂2E
( + δ) ∂ 2 E
∂2P
2
+
∇
E
−
=
µ
.
0
⊥
∂z 2
c2
∂t2
∂t2
The paraxial approximation consists to consider |k⊥ | |k0 | so the electric field can be written
E(r⊥ , z, t) = ξ(r⊥ , z)ei(k0 z−ω0 t) with ξ(r⊥ , z) a slowly varying envelope on the transversal direction.
We use the same expression for the nonlinear polarization PN L (r⊥ , z, t) = pnl (r⊥ , z)ei(k0 z−ω0 t) and we
obtain :
∂z2 ξ
∂z ξ ∇2⊥ ξ δ
1
+
pnl .
+
2i
+ ξ=−
2
2
k0
ε0 k0
k0
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An Introduction to Quantum Fluid of Light
Rochet Antonine
We can neglected the second derivative on z considering that
expression :
i
|∂z2 ξ|
k02
|∂z ξ|
k0
∼
|∇2⊥ ξ|
.
k02
We have a general
k0 δ
∂ξ
1 2
1
∇ ξ− ( ξ+
pnl ).
=−
∂z
2k0 ⊥
2 ε0
Let us consider a medium that is centrosymmetric : its geometry is such that χ(2) = 0. We can develop
the nonlinear polarization using a real electric field expression E(r⊥ , z, t) + E(r⊥ , z, t)∗ :
Pnl (r⊥ , z, t) = ε0 χ(3) E 3 (r⊥ , z, t)
Pnl (r⊥ , z, t) = ε0 χ(3) ( 43 |ξ(r⊥ , z)|2 E(r⊥ , z, t) + 14 cos(3(k0 z − ω0 t))ξ 3 (r⊥ , z))).
There are two terms : one relative to the third harmonic generation and the other to an optical Kerr
effect. We will focus on the last one, neglecting the other. We finally obtain, for the transverse field :
i
We define a potential V (r⊥ , z) =
i
∂ξ
k0
1 2
∇⊥ ξ − (δ + χ(3) |ξ|2 )ξ
=−
∂z
2k0
2
−k0
2 δ(r⊥ , z)
and a constant g =
−χ(3)
2 k0
obtaining the form :
∂ξ
1 2 (i)
=−
∇ ξ − V (r⊥ , z)ξ (ii) + g|ξ|2 ξ (iii)
∂z
2k0 ⊥
(1)
There are three terms :
(i) equivalent to a kinetic energy of the transversal photons with an effective mass mef f ∝ k0
(ii) a potential-like form which is attractive or repulsive according to a higher or a lower refractive
index with δn(r⊥ , z) ∝ δ(r⊥ ,z)
(iii) a non linear term relative to a field interaction with itself. At this moment we can introduce an
effective photon-photon interaction. If χ(3) > 0 (or g < 0), the interaction is attractive and on the
contrary, if χ(3) < 0 (or g > 0), the interaction become repulsive. In what is next, we will consider
only the case where χ(3) < 0.
What is important to notice here is that we have a Schrödinger-like equation (with ~ = 1) describing the transverse field spatial evolution in a nonlinear centrosymmetric medium in the paraxial
approximation.
1.2.2
Gross-Piteavskii equation
For a Bose-Einstein condensate, the Hamiltonian describing a N bosons system with a mass m is :
Ĥ =
X
1≤j≤N
(−
~2 2
1X
∇~rj + V (~rj )) +
U (|~rj − ~rl |)
2m
2
j6=l
The first sum is relative to the kinetic energy of each particle and their interaction with a confinement potential necessary to trap and cool bosons. The second sum describes boson-boson interactions. We make the hypothesis that at T = 0 each particle is in the same state : Ψ(~r1 , ..., ~rN , t) =
φ(~r1 , t)...φ(~rN , t). After long calculations emerge the Gross-Piteaveskii Equation (GPE) for a macroscopic wave function describing a dilute Bose Einstein condensate [5] :
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An Introduction to Quantum Fluid of Light
i~
Rochet Antonine
∂φ
~2 2
=−
∇ φ − V (r⊥ , t)φ + g|φ|2 φ
∂t
2m ⊥
A visual comparison with the Schrödinger-like equation highlight a resemblance :
i
∂ξ
1 2
∇ ξ − V (r⊥ , z)ξ + g|ξ|2 ξ
=−
∂z
2k0 ⊥
Two differences must be highlighted :
- the order parameter in GPE is a wave function so the term |φ|2 is a boson density. In the
Schrödinger-like equation, |ξ|2 is proportional to a transverse field intensity or equivalently to the
photon density.
- the GPE is relative to a temporal evolution of the wave function whereas the Schrödinger equation
describes a spatial evolution. On one hand, we have the description of the transversal field profile on
a plan z = z0 at different values t. On the other hand, we have at a time t0 the description at different
values z of the transversal field profile. For example, it can describe the temporal evolution of the
bosons density in a 2D Bose-Einstein condensate.
Photons behavior in a non linear medium is the analog of a massive particle one in a condensate
phase if we assign them an effective mass proportional to the wave vector k0 . Consequently, we call this
collective state induced by a nonlinear medium a quantum fluid of light. This theoretical prediction
made in the 90’s of a fluid of photon existence has been experimentally observed. We are going to
focus on an experiment from the LKB Quantum Optics Group concerning polaritons fluid.
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An Introduction to Quantum Fluid of Light
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Rochet Antonine
Fluid of Light in the LKB Group
Contrary to close particles with a mass and a charge, nearby photons "don’t see" each other. An
effective interaction, represented by the term g|ξ|2 is possible between photons. The LKB group study
polaritons fluid. We will first show an eloquent result about polaritons superfluidity then we will
discuss a group perspective : studying fluid of light generated in an atomic gas.
2.1
2.1.1
Fluid of polaritons
Experimental setup
To begin, it is important to understand what is a polariton in a semi-conductor. At low temperature,
in a semi-conductor, electrons are in the valence band. With an adequate optical excitation, electrons
acquire an energy and go to the conduction band, creating this way a hole in the valence band. This hole
is a quasi-particle carrying a positive charge and bounded with the electron by Coulomb interaction. If
the semi-conductors is confined in one direction (quantum wells), electrons and holes are "closer" and
form a hole-electron pair named an exciton with an effective mass mexc ∼ me . In the LKB experiment,
the semi-conductor is itself in a micro-cavity (figure 2). If the probability for one photon to be absorbed
or emitted by the semi-conductor is higher than the probability it leaves the cavity, then we have a
strong coupling regime between excitons and a resonant cavity mode. This strong coupling between an
exciton and a photon generate new eigenstates of the hamiltonian of the system : the upper polariton
and the lower polariton (figure 3). In dotted lines we can see the quadratic photon dispersion and the
quasi-flat exciton dispersion on the considered scale ( k⊥ < 10µm−1 ). In this case, the optical mode is
resonant with the exciton transition. If there is a blue or a red detuning, we can obtain a dispersion
relation more photonic or excitonic.
Figure 2: Schematic representation of the experimental setup of the LKB group from [2] with kz = k0
and k⊥ = kk .
.
To create a polariton condensate, all polaritons have to go in the lower branch around k⊥ = 0 through
a thermal process : a non-resonant laser excite polaritons, they relax in the upper branch, emitted a
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An Introduction to Quantum Fluid of Light
Rochet Antonine
Figure 3: Dispersion of the upper polariton (UP) and lower polariton (LP) at resonance with a contribution scale of the excitonic or photonic dispersion. Image taken in [2]
.
photon at k⊥ 6= 0 and relax again in the lower branch around k⊥ = 0. A detail not develop here but
important to underline is that, on the LKB group, they obtain polariton fluid without necessarily a
step of condensation. The process is a quite different but the resulting state is similar. On figure 4
we can see three steps of condensation with different value of the laser power P. Changing P means
changing the polaritons density. For a laser power P0 at the threshold of condensation : if P < P0
polaritons are distributed in a great number of states (thermal distribution) and when P > P0 there
is a concentration in the fundamental state k⊥ = 0 in coexistence with a thermal distribution of the
excited states. It is interesting to notice that the effective mass of a polariton is mp ∼ 10−5 me and
for T ∼ 20K, λdB = 3µ m. This temperature is several orders higher than the one of a Bose Einstein
condensate of Rubidium atoms and more accessible experimentally (cryogenic methods).
Figure 4: Polariton condensation by non resonant excitation from [6]
.
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An Introduction to Quantum Fluid of Light
2.1.2
Rochet Antonine
Experimental results
Once a polariton fluid obtained we can start a study of the system reactions to perturbations. In this
part, we will consider a specific case : the superfluid transition observed through the fluid colision on
a defect (our approach is going to be qualitative, theoretical results won’t be demonstrated).
A laser beam is sent on the semi-conductor quantum-wells in the direction of the confinement. It
targets the creation of a polaritons fluid in the transversal plan (relative to k⊥ ). This fluid owns a flow
speed controlled by the incident angle of the laser beam. How does the fluid react to the presence of
a defect (described by δ)? The Bogoliubov theory of weak perturbations, transposed to polaritons,
gives a solution to the non-linear Schrödinger equation [2] :
s
WBog (k⊥ ) =
2 k2
k⊥
( ⊥ + 2gn).
2m 2m
(2)
In (2), g is the constant of interaction
between polaritons and n the polaritons density. We define the
p
Bogoliubov speed of sound cs = gn/m :
s
WBog (k⊥ ) =
2 k2
k⊥
( ⊥ + 2mcs ).
2m 2
We can dissociate two regimes depending on the incident angle of the laser beam :
- k⊥ 2mcs /~ corresponding to a large momentum excitation with WBog (k⊥ ) ∼
a parabolic dispersion similar to a single particle, a photon.
2
k⊥
2m .
- k⊥ 2mcs /~ corresponding to a small momentum excitation with WBog (k⊥ ) ∼
sonic dispersion (phonon) : WBog (k⊥ ) ∼ cs k⊥ .
We recognize
k⊥
k0 ζ .
We have a
In figure 5 we can see experimental results of the LKB group (2009) compared with simulated results.
The experiment occurs at k⊥ constant, the variable is the polaritons density. Panels I to III are close
field images (density of polaritons in the real space) and panels IV to VI are far field images (polaritons
distribution in the momentum space). From left to right, polaritons density arise. On the left, we are
in the regime where k⊥ 2mcs /~. In the center, this is a transitional regime where the fluid speed is
still higher than cs (supersonic fluid). Finally, on the right the fluid speed is slower than cs (subsonic
fluid). The fluid propagates "without seeing" the defect : the absence of friction is the signature of a
superfluid regime.
Of course, other phenomena have been observed in fluid of polaritons by the LKB group (solitons,
vortices). One of their objectives today is to use an other nonlinear medium to carry the photonphoton interaction. In alternative to a confined geometry, the team is going to use a propagative
geometry and an atomic gas of Rubidium.
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An Introduction to Quantum Fluid of Light
Rochet Antonine
(a) Experimental results
(b) theoritical results
Figure 5: Experimental images of the real- (panels I-III) and momentum- (panels IV-VI) space polariton
density at a higher wavevector. The different columns correspond to increasing polariton densities from
left to right [7]
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An Introduction to Quantum Fluid of Light
2.2
Rochet Antonine
Fluid of light in an atomic vapor
This experiment has not been realized yet and would be based on the nonlinear medium theory seen
in part 2.1 which can be applied to an electric field propagating in an atomic gas [10]. The interaction
(3)
term between photons is g = −χ2 k0 and the effective mass is proportional to k0 . Using an atomic
gas to create a fluid of light is an interesting perspective because it allows to control the non-linearity
coefficient χ(3) , that means control the strength of the photon-photon interaction. This control would
be possible mostly by varying two parameters: the laser detuning and the atomic density.
2.2.1
The laser detuning
We have seen earlier the presence of an optical Kerr effect, the refractive index is intensity dependant
: n(I) = n0 + n2 I with n0 n2 I and I = |ξ|2 the intensity. We consider the case where n2 < 0. For
a gaussian beam, the refractive index profile is equivalent to the one of a divergent lens : Kerr effect
with n2 < 0 targets a defocusing effect on the light. That is equivalent to a repulsive photon-photon
interaction with χ(3) < 0 (an attractive interaction make the system unstable but it can be studied).
Figure 6: Relation dispersion in an atomic gas with here α(ω) the absorption coefficient and n(ω) =
n2 (ω)[9]
.
According to Kramer Krönig relations linking absorption and refraction, we have absorption and refraction profiles depending on the laser frequency ω (figure 6). To choose a detuning value (δ = ω − ω0 )
two things are taken into account. First, a weak absorption is required because to probe what happened in the gas the light signal is used in transmission. Second, to arise photon-photon repulsive
interaction, n2 has to be maximum in absolute value and negative. A compromise leads to a detuning
negative (red shift) in the order of 1 GHz.
2.2.2
The density
In a Rubidium cell a little volume is maintained at colder temperature than the rest of the cell. In this
region Rubidium is in a liquid phase (classical one!). If the cell is heated, a certain proportion of this
liquid will switch to a gaseous phase. This cold area serve as a reservoir of Rubidium. Heating the cell
increase atomic density (figure 7) and this way the nonlinear effects. Raising the temperature creates
also√thermal agitation in the gas and extends the Doppler width of the absorption signal proportionally
to T . This doesn’t really affect the choice of the detuning.
A fluid of photons with a near resonance atomic vapor could form at room temperature. This
propagative geometry would be more flexible than the confined geometry seen on part 2.1. For example,
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An Introduction to Quantum Fluid of Light
Rochet Antonine
they could choose the cell shape so as to adjust the propagation length inside the medium by moving
the cell. Moreover, to generate defects, they could use an other laser beam modifying locally the
refractive index. A spacial light modulator could allow us to generate arbitrary defects.
Figure 7: Simulation of the evolution of the atomic density for Rubidium atoms
.
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An Introduction to Quantum Fluid of Light
Rochet Antonine
Conclusion
We have seen that a quantum fluid of light is the analog of a Bose-Einstein condensate with massive
particles. Different media can generate those fluids of photons, we have seen the examples of a semiconductor quantum wells in a microcavity and an atomic gas in a cell. We have shown the superfluidity of
polaritons through an experimental result in the LKB group. In an atomic vapor, it could be possible
to create a small perturbation : a second laser beam, weaker and at the same frequency than the
one used to create the fluid of light, would allow the creation of a weak modulation, thanks to the
interference pattern, in the transverse plane. The wavelength of those waves present on the top of the
fluid could be controlled by tuning the angle between the two lasers and this way reconstruct the full
dispersion relation and probe the superfluid regime according to the Bogoliubov solutions [5] :
s
WBog (k⊥ ) =
2
k2
k⊥
k0 χ(3) |ξ|2
( ⊥ −
).
2k0 2k0
(3)
As we have seen on the last part, with an atomic gas, it would be possible to tune the non-linearity
coefficient by changing the temperature of the cell or the detuning to resonance. One of the great
advantage is to control the presence of local defects and this would be a way to realize, among others,
quantum simulation of quantum transport in presence of disorder.
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References
[1] C.CARIMALO, Cours de physique statistique, Pierre et Marie Curie University (2010).
[2] S.PIGEON, Fluides quantiques et dispositifs à polaritons, PhD thesis, Diderot University - Paris
VII (2011).
[3] J.KASPRZAK, M.RICHARD, R.ANDRÉ and D.LE SI DANG, La condensation de Bose Einstein
en phase solide, Images de la physique - the scientific revue of CNRS (2007).
[4] R.W.BOYD, Nonlinear Optics (third edition), Chapter 2 : Wave equation description of nonlinear
optical interactions (2008).
[5] I.CARUSOTTO, Superfluid light in propagating geometries, Proc. R. Soc. A., 470 (2014).
[6] J.KASPRZAK, Condensation of exciton polaritons, PhD thesis, Joseph-Fourier University Grenoble I (2006).
[7] A.AMO, J.LEFRERE, S.PIGEON, C.ADRADOS, C.CIUTI, I.CARUSOTTO, R.HOUDRÉ,
E.GIACOBINO and A.BRAMATI, Superfluidity of polaritons in semiconductor microcavity, Nat.
Phys., 5:805 (2009).
[8] T.BOULIER, Réseau de vortex contrôlés et lumière non-classique avec des polaritons de microcavité, PhD thesis, Pierre et Marie Curie University - Paris VI (2014).
[9] M.JOFFRE, Cours d’optique non linéaire en régime continu et femtoseconde, École Normale
Supérieure (2014).
[10] P.BOUCHER, Superfluidity of light, Research Internship Report , Kastler Brossel Laboratory
(2015).
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