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 fieldPl=ε0χEadded to a nonlinear termPnl =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 =k0ez= ωc0√
The paraxial approximation consists to consider |k⊥| |k0| so the electric field can be written E(r⊥, z, t) = ξ(r⊥, z)ei(k0z−ω0t) with ξ(r⊥, z) a slowly varying envelope on the transversal direction.
We can neglected the second derivative on z considering that |∂2zξ|
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)
Pnl(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
(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 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
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, ..., ~rN, t) = φ(~r1, t)...φ(~rN, 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 massmexc∼me. 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⊥=kk.
.
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 mp ∼ 10−5me 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]
.
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:
We can dissociate two regimes depending on the incident angle of the laser beam :
-k⊥2mcs/~corresponding to a large momentum excitation withWBog(k⊥)∼ k2m2⊥. We recognize a parabolic dispersion similar to a single particle, a photon.
- k⊥ 2mcs/~corresponding to a small momentum excitation withWBog(k⊥)∼ kk⊥
0ζ. We have a sonic dispersion (phonon) : WBog(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 thancs (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 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]