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

Contents

1 Quantum Fluid : from particle to photon 2

1.1 Quantum fluid of particles . . . 2

1.2 Fluid of light : mathematical description . . . 3

1.2.1 Propagation of an electric field in a nonlinear medium . . . 3

1.2.2 Gross-Piteavskii equation . . . 4

2 Fluid of Light in the LKB Group 6 2.1 Fluid of polaritons . . . 6

2.1.1 Experimental setup . . . 6

2.1.2 Experimental results . . . 8

2.2 Fluid of light in an atomic vapor . . . 10

2.2.1 The laser detuning . . . 10

2.2.2 The density . . . 10

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!

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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 aquantum fluid of lightthrough 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.

1 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 momentump) :

λ= h p.

In a gas at thermodynamical equilibrium at a temperature T , every particle has an energy on the order ofE = _2m^p2 =kBT. That corresponds to a wavelength λ_dB =

q

~²

mkBT 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 particlesd, particles wave functions interact with each other ((c) on figure1), 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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a Bose Einstein condensate of Rubidium atoms (m = 10⁵m_e) 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 de- scribe 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 motionD=ε0E+P.

- the induction directly proportional to the magnetic fieldB=µ0H.

According to Maxwell’s equations ∇ ⊗E =−^∂B_∂t and ∇ ⊗H= ^∂D_∂t and by a well known process [4], we find the electric field propagation equation :

∆²E− 1 c²

∂²E

∂t² =µ₀∂²P

∂t² .

If we consider a nonlinear media : the polarization is the sum of a linear polarization proportional to the electric fieldP_l=ε₀χEadded to a nonlinear termP_nl =ε₀χ⁽²⁾E²+ε₀χ⁽³⁾E³. The first order of the electric susceptibilityχis linked to the relative linear dielectric constantby= 1 +χ. We introduce a spatial modulation of the relative dielectric constantδ(r⊥, z), supposed to be slowly varying in space (we will come back on its meaning latter).

Considering a propagation alongz axis, we separate transversal and longitudinal dynamic withr_⊥ = (x, y) ,k_⊥= (kx, ky) and kz =k0e_z= ^ω_c⁰√

ez. We obtain:

∂²E

∂z² +∇²_⊥E−(+δ) c²

∂²E

∂t² =µ₀∂²P

∂t² .

The paraxial approximation consists to consider |k⊥| |k₀| so the electric field can be written E(r⊥, z, t) = ξ(r⊥, z)e^{i(k0z−ω0t)} with ξ(r⊥, z) a slowly varying envelope on the transversal direction.

We use the same expression for the nonlinear polarizationP_{N L}(r⊥, z, t) =p_nl(r⊥, z)e^{i(k0z−ω0t)} and we obtain :

∂²_z ξ

k₀² + 2i∂z ξ k0

+∇²_⊥ ξ k₀² +δ

ξ=− 1 ε0pnl.

We can neglected the second derivative on z considering that ^|∂2zξ|

k²₀ ^|∂_k^zξ|

0 ∼ ^|∇2⊥ξ|

k₀² . We have a general expression :

i∂ξ

∂z =− 1

2k₀∇²_⊥ξ− k0

2 (δ ξ+ 1

ε₀p_nl).

Let us consider a medium that is centrosymmetric : its geometry is such thatχ⁽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) =ε0χ⁽³⁾E³(r⊥, z, t)

P_nl(r⊥, z, t) =ε₀χ⁽³⁾(³₄|ξ(r_⊥, z)|²E(r⊥, z, t) +¹₄cos(3(k₀z−ω₀t))ξ³(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∂ξ

∂z =− 1 2k0

∇²_⊥ξ−k₀

2 (δ+χ⁽³⁾|ξ|²)ξ

We define a potentialV(r⊥, z) = ^−k₂⁰δ(r⊥, z) and a constantg= ^−χ₂⁽³⁾k0 obtaining the form : i∂ξ

∂z =− 1

2k₀∇²_⊥ξ⁽ⁱ⁾−V(r⊥, z)ξ⁽ⁱⁱ⁾+g|ξ|²ξ⁽ⁱⁱⁱ⁾ (1) There are three terms :

(i)equivalent to a kinetic energy of the transversal photons with an effective mass m_{ef 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 whereχ₍₃₎ <0.

What is important to notice here is that we have a Schrödinger-like equation (with ~ = 1) de- scribing 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 massm is : Hˆ = X

1≤j≤N

(−~² 2m∇_~²_r

j+V(~rj)) + 1 2

X

j6=l

U(|~rj−~rl|)

The first sum is relative to the kinetic energy of each particle and their interaction with a confine- ment potential necessary to trap and cool bosons. The second sum describes boson-boson interac- tions. We make the hypothesis that at T = 0 each particle is in the same state : Ψ(~r₁, ..., ~r_N, t) = φ(~r₁, t)...φ(~r_N, t). After long calculations emerge the Gross-Piteaveskii Equation (GPE) for a macro- scopic wave function describing a dilute Bose Einstein condensate [5] :

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i~∂φ

∂t =−~²

2m∇²_⊥φ−V(r⊥, t)φ+g|φ|²φ

A visual comparison with the Schrödinger-like equation highlight a resemblance : i∂ξ

∂z =− 1 2k0

∇²_⊥ξ−V(r⊥, z)ξ+g|ξ|²ξ

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 planz=z0 at different valuest. On the other hand, we have at a timet0 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 vectork0. 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.

2 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 termg|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 Fluid of polaritons

2.1.1 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 massm_exc∼m_e. In the LKB experiment, the semi-conductor is itself in a micro-cavity (figure2). 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 (figure3). 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] withkz =k0

andk⊥=k_k.

.

To create a polariton condensate, all polaritons have to go in the lower branch aroundk⊥= 0 through a thermal process : a non-resonant laser excite polaritons, they relax in the upper branch, emitted a

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Figure 3: Dispersion of the upper polariton (UP) and lower polariton (LP) at resonance with a con- tribution scale of the excitonic or photonic dispersion. Image taken in [2]

.

photon atk⊥ 6= 0and 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 statek⊥ = 0 in coexistence with a thermal distribution of the excited states. It is interesting to notice that the effective mass of a polariton is m_p ∼ 10⁻⁵m_e and for T ∼20K,λ_dB = 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]

.

2.1.2 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 tok⊥). 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] :

WBog(k⊥) = s

k²_⊥ 2m(k_⊥²

2m + 2gn). (2)

In (2),g is the constant of interaction between polaritons andnthe polaritons density. We define the Bogoliubov speed of soundcs =p

gn/m:

WBog(k⊥) = s

k_⊥² 2m(k_⊥²

2 + 2mcs).

We can dissociate two regimes depending on the incident angle of the laser beam :

-k⊥2mc_s/~corresponding to a large momentum excitation withW_Bog(k⊥)∼ ^k_2m^2⊥. We recognize a parabolic dispersion similar to a single particle, a photon.

- k⊥ 2mcs/~corresponding to a small momentum excitation withWBog(k⊥)∼ _k^k⊥

0ζ. We have a sonic dispersion (phonon) : W_Bog(k⊥)∼csk⊥.

In figure5we can see experimental results of the LKB group (2009) compared with simulated results.

The experiment occurs atk⊥ 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 wherek⊥2mcs/~. In the center, this is a transitional regime where the fluid speed is still higher thanc_s (supersonic fluid). Finally, on the right the fluid speed is slower than c_s(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 photon- photon 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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(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]

2.2 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 term between photons is g = ^−χ₂⁽³⁾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 coefficientχ⁽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+n2I with n0 n2I andI =|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 withn2 <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 hereα(ω) the absorption coefficient and n(ω) = n2(ω)[9]

.

According to Kramer Krönig relations linking absorption and refraction, we have absorption and re- fraction profiles depending on the laser frequencyω(figure6). To choose a detuning value (δ =ω−ω0) two things are taken into account. First, a weak absorption is required because to probe what hap- pened 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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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 .

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 semicon- ductor 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] :

W_Bog(k⊥) = s

k_⊥² 2k₀(k²_⊥

2k₀ −k₀χ⁽³⁾|ξ|²

). (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

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[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).

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