Quantum Fluids of Light

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Quantum Fluids of Light

Titus Franz 12.05.2015

Abstract

Bose-Einstein condensation (BEC) is an exciting quantum phase of matter without classical analogue. This report presents how to achieve a BEC of photons using a cavity-embedded quantum well. In addition, it is shown how the propagation of light in nonlinear media like atomic vapours can simulate in a relatively simple way the Gross- Pitaevski equation which describes BEC of interacting bosons. Finally, superuidity and the generation of vortices, two prominent examples of the exotic behaviour of Bose-Einstein condensates, are portrayed.

1 Bose-Einstein condensation of non-interacting particles

1.1 Introduction

Right from the beginning of the theoretical development of quantum mechanics physi- cists realized the signicance and diversity of new quantum phases. Already in 1924 to 1925 Satyendra Nath Bose and Albert Einstein discussed the statistic of Bosons leading to the theory of Bose-Einstein condensates (BEC) [12], an entirely novel phase of matter. Bosons, i.e. particles with integer spin, may occupy the same single-particle state, as they don't obey the Pauli-principle. Therefore, it is possible, that at nite temperature the ground state is macroscopically populated. This is a new state of matter with exotic behaviour like superconductivity without classical analogue [3], [18], [5], [10].

1.2 The grand canonical ensemble

Neglecting the interaction between the bosons, the system may be represented in the grand canonical ensemble with xed temperature T and chemical potential µ. The system is described by the particle number equation

N =X

k

nk=X

k

1

e^β(^k^−µ)−1 (1)

wherenkis the number of particles with wave vectork,β= _k¹

BT is the inverse temperature and _k = ^¯^h_2m²^k² is the free boson dispersion. Note thatµ <0, otherwisen_k <0, which is unphysical.

In the thermodynamical limit, the sum is replaced by an integralP

k→ lim

V→∞

V (2π)^d

R∞ 0 d^dk. For a critical density n= ^N_V > nc where µ= 0, the particle number equation has no longer a solution and the single particle mode n0 withk= 0has to be singled out:

n=n0+ Z

k6=0

d^dk (2π)^d

1

e^β(^k^−µ)−1. (2)

At this point all additional bosons accumulate in the ground state and n0 becomes macroscopic, the system is described by a Bose-Einstein-Condensate (BEC).

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1.3 BEC of photons

BEC has already been observed in various physical systems like atomic gases or solid- state quasiparticles. However, black-body radiation, the maybe most prominent example of Bose gases, does not show this behaviour.

This is explained by the vanishing chemical potential. The photon number is not conserved when the cavity is cooled down, i.e. the photons are absorbed in the cavity walls instead of being accumulated in the ground state mode [14].

2 Microcavity exciton-polaritons

2.1 Introduction

One way to surpass the previously described problems and to observe Bose-Einstein condensation of photons are so called cavity-conned quantum wells. The cavity mirrors of those systems provide a conning potential and an eective mass to the photons which are now equivalent to massive bosons. The semiconductor helps to achieve ther- malisation by phonon-exciton interactions [18].

2.2 Photons in a microcavity

The Fabry-Perot cavity used in this experiment is made out of two parallel Bragg mirrors separated byλ_c. Each Bragg mirror consists of alternating layers of thickness ei =_4n^λ⁰

i, whereλ0is the wavelength of maximal reection andni is the optical index of layer i ∈ {a, b} (see gure 1). This cavity is excited with light incident under an angleθwith respect to the normal, such that only the z-component of the wave vector k^γ = (k^γ_k, k^γ_z)satises the resonance condition

k^γ_z = 2πnc

λ₀ (3)

Under these conditions for small incident angleθthis leads to an energy of the cavity which approximates to a parabolic dispersion

Eγ(k_k^γ =¯hc n_c

s 2πnc

λ₀ ²

+ k^γ_k2

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≈hc λ0

+¯h²k²_k

2m^∗_γ (5)

with an eective mass termm^∗_γ = ⁿ_λ²^c^h

0c.

The photon lifetime of the microcavity is given by τc =Lef f

nc

π F

c (6)

with an eective cavity length L_{ef f} =λ_c

1 + 2_nⁿ^aⁿ^b

b−na

and a cavity nesseF.

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Incident light λ

| {z }

| {z } | {z }

optical cavity of length λc=m_2n^λ_c layera

with thickness ea=m_4n^λ_a

layerb with thickness

eb=m_4n^λ_b

θ

Figure 1: Microcavity of length λ₀ consisting of two Bragg mirrors. Both mirrors are made of alternaying layers of thicknesse_a ande_b. The cavity is excited with light of wavelengthλand with wave vector k^γ = (k^γ_k,k^γ_z)under an angleθ. (Source: Own representation based on [24].)

2.3 Excitons in a seminconductor quantum well

In order to increase the coupling of the excitons in a semiconductor to light, the excitons are conned in a 2D quantum well (see gure 2. As a result, along the z direction the excitonic eigenenergies are quantized, while in the (x,y)-plane the excitons show a parabolic dispersion relation. Taking into account, that the ground state energy is well separated from higher lying states, the energy spectrum may be written as

E_exc^2D(k^exc_k ) =Eexc+

¯ hk_k^exc2

2m^∗_exc (7)

where Eexc is the energy of the bottom of the exciton band, k_k^exc is the exciton wave vector in the (x,y)-plane and m^∗_excis the exciton eective mass.

2.4 Strong coupling between photons and excitons

A strong coupling between photons and excitons is obtained by placing one or more semiconductor quantum wells (see 2.3) in a Fabry-Perot microcavity (see 2.2). This is depicted in gure 3 In dierence to bulk semiconductors only the parallel component of the momentum is conserved when conned excitons couple to light [13]. In addition, the z component of the electromagnetic eld is quantized. This allows to excite a single exciton by choosing the photon angle of incidence:

k^exc=k^γ = Enc

¯

hc sinθ^c, (8)

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Figure 2: A GaAs-InGaAs quantum well as it is used in the experiment. The crystal is grown in z direction and the thickness of the InGaAs layer is 8nm. (Right) The energy diagram as a function of the z-position. The excitons are trapped in the InGaAs-zone (red). (Source: [4])

where E is the photon energy andθ^c is photon angle of diraction inside the cavity.

Neglecting the exciton-exciton energies this system of photons and excitons may be described by the Hamiltonian

Hˆ =X

k

Eexc(k)ˆb^†_kˆbk+Eγ(k)ˆa^†_kˆak+¯hΩR

2

ˆ

a^†_kˆbk+ ˆb^†_kˆak

, (9)

where ˆb_k,ˆb^†_k and ˆa_k, ˆa^†_k are the destruction and annihilation operators of an exciton respectively a photon at wave vector k = |k_k|. ΩR is the the Rabi frequency and corresponds to the frequency of the photon-exciton conversion. Note that the excitons are fermionic pairs and therefore don't underlie denite statistics. However, at low exciton density they may be described as Bosons [19].

The Hamiltonian above can be diagonalized by the unitary transformation ˆ

ρ^(LP)_k ˆ ρ^{(U P}_k ⁾

!

=

−C_k X_k X_k C_k

ˆ a_k ˆbk

. (10)

The eigenstatesρˆ^(LP)_k andρˆ^{(U P}_k ⁾of the Hamiltonian are called upper respectively lower polaritons, their eigenenergies are given by:

E_{(U P)}=1 2

Eexc(k) +Eγ(k) + q

δ_k²+ (¯hΩR)²

, (11)

E_(LP₎= 1 2

E_exc(k) +E_γ(k)−q

δ²_k+ (¯hΩ_R)²

. (12)

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InGaAs GaAs GaAs

Incident light λ

| {z }

| {z } | {z }

optical cavity of length λc=m_2n^λ_c layera

with thickness ea=m_4n^λ_a

layerb with thickness

eb=m_4n^λ_b

θ

| {z } | {z }

Bragg mirror

| {z }

Bragg mirror quantum well

Polariton

Figure 3: The quantum well from gure 2 is embedded in the microcavity from gure 1. The microcavity of lengthλ0consists of two Bragg mirrors. Both mirrors are made of alternating layers of thickness ea and eb. The cavity is excited with light of wavelength λ and with wave vector k^γ = (k^γ_k,k^γ_z) under an angle θ. The quantum well is made of a layer of InGaAs between layers of GaAs. (Source: Own representation based on [4]).

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2.5 The interaction Hamiltonian and Gross-Pitaevski Equation

The polariton-polariton interaction is due to exciton-exciton and exciton-photon interactions [9]. The latter arise from Coulomb electron-hole interactions and are described by an eective Hamiltonian proportional toˆb^†_k+qˆbk⁰−qˆbkˆbk⁰. The former is due to sat- uration of the coupling between photons and the pair of electrons forming the exciton.

This leads to an anharmonic term proportional toˆa^†_k+qˆb^†_k0−qˆb_kˆb_k0. The inverse of the unitary transformation (11) is applied to the interaction Hamiltonians and all terms containingρˆ^{(U P)}are discarded as in the experiment only the low lying polaritons are excited with a spectrally narrow laser. The resulting Hamiltonian is given by [8,9,23]:

Hˆ = ˆHlin+ ˆHint=X

k



Ekρˆ^†_kρˆk+1 2

X

k⁰,q

V_k,k^pol−pol0,q ρˆ^†_k−qρˆ^†_k0+qρˆkρˆk⁰



. (13) By taking the Fourier-Back-Transform of the annihilation operator

ˆ ρk= 1

V Z

drΨ(r)eˆ ⁻ⁱ^k·r/¯^h (14) and the approximation that the potential only depends on the transferred momentum q [11]

V_k,k^pol−pol0,q =V_q^pol−pol= Z

drV^pol−pol(r)e⁻ⁱ^k·r/¯^h, (15) we nd the interaction Hamiltonian of a Bose-gas:

Hˆ = Z

dr ¯h²

2m∇Ψˆ^†∇Ψ + ˆˆ Ψ^†Vext(r) ˆΨ

+1 2

Z Z

drdr⁰

Ψˆ^†(r) ˆΨ^†(r⁰)V^pol−pol(r−r⁰) ˆΨ(r⁰) ˆΨ(r)

. (16)

From this Hamiltonian the dynamic of the system is derived using the Heisenberg equation:

i¯h∂Ψ(r, t)ˆ

∂t =h

Ψ(r, t),ˆ Hˆi

=

−¯h²∇²

2m +Vext(r, t) + Z

dr⁰Ψˆ^†(r⁰, t)V^pol−pol(r⁰−r) ˆΨ(r⁰, t)

Ψ(r, t).ˆ (17) The idea of deriving the Gross-Pitaevski equation is similar to the Bogoliubov ansatz that the ground state of the Bose gas remains macroscopically populated even with interactions. Indeed, this is fullled and a Bose-Einstein condensation has been observed for polaritons above some critical excitation power threshold (see gure 4) [17].

We write therefore:

Ψ =ˆ a₀ρˆ₀+X

i6=0

a_iρˆ₀≈a₀ρˆ₀≈Ψ₀. (18)

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Figure 4: Observation for Bose-Einstein condensation for polaritons in the same ex- perimental set-up as in gure 3. The left panels correspond to an excitation spectrum below the excitation power threshold Pthr, the middle cone is at P_thr, the right one above P_thr. a: Far-eld image of the emission cone of

±23. The vertical axis shows emission intensity. Above the threshold, a sharp peak forms in the centre of the cone, i.e. all polaritons are in the low- est momentum statek_k= 0. The ground state is macroscopically populated which justies the approximation in equation 18. b: The same data as ina, but resolved in energy. (Source: [17])

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In the last step the Bogoliubov approximation was realized, i.e. the annihilation/creation operator was replaced by its mean value. The Gross-Pitaevski equation follows now by taking a contact-point interaction between the particlesV^pol−pol(r⁰−r) =gδ(r⁰−r):

d

dtΨ0(r, t) =

−¯h²∇²

2m +Vext(r, t) +g|Ψ0(r, t)|²

Ψ0(r, t). (19)

2.6 The driven-dissipative Gross-Pitaevski equation

One way of injecting polaritons in the microcavity is the "(quasi-)resonant injection":

The microcavity is constantly pumped with a continuous laser of frequency ω_L. The special case of∆ =Elas−ELP = 0where the energy of the laserElas= ¯hωLequals the polariton energyELP, is called resonant pumping. In this case the system is described by the driven-dissipative Gross-Pitaevski equation:

d

dtΨ0(r, t) =

−¯h²∇²

2m +Vext(r, t) +g|Ψ0(r, t)|²−i¯hγ 2

Ψ0(r, t) +FL(r, t). (20) Here a pump term FL(r, t) and a relaxation term proportional to the decay of the lower polaritons γ_k^LP ≡γ are added to the equation.

3 Quantum uids of light in atomic vapour

3.1 Introduction

As we have seen in the last chapter, a BEC of light can be achieved in microcavities. This system can eectively be described using the Gross-Pitaevski-equation (19).

However, the timescale of the system is too fast to observe the out-of-equilibrium dynamics. In the following chapter a dierent system of light will be shown, which can be used to simulate the time-dependent Gross-Pitaevski equation. The deduction of this hydrodynamical description follows the approach of [6].

3.2 Propagation of light in a non-linear medium

The Kerrχ⁽³⁾non-linearity of an optical medium can be seen as an eective interaction between photons. A monochromatic beam of frequency ω₀ propagating in z-direction is described by the non-linear wave equation

0 =∂_z²E(r, z) +∇²_⊥E(r, t) +ω²₀ c²

+δ(r, z) +χ³⁾|E(r, z)|²

E(r, z). (21) HereE(r, z)is the amplitude of the electromagnetic eld at position(r, z) = (x, y, z), is the dielectric constant and δ(r, z)its variation in space which is assumed to be small. Under this condition the paraxial approximation is justied and the second

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derivative of E(r, z) = E(r, z)e^−ik⁰^z van be neglected. Under this approximation equation 21 takes the form of:

i∂zE(r, z) =−∇²_⊥E(r, t)−k²₀ 2

δ(r, z) +χ³⁾|E(r, z)|²

E(r, z). (22) This equation is equivalent to the Gross-Pitaevski equation in the following sense:

The time evolution in the Gross-Pitaevski equation is replaced by the propagation in space (z-direction). Therefore, a space-time mapping is needed to compare both systems.

The spatial variation of the dielectric constant correlates to the potential termVext(r, t) ˆ=^−k⁰^δ(r,z)₂ and the Kerr nonlinearity χ⁽³⁾ gives rise to the photon-photon interaction constant

g=−^χ⁽³⁾₂^k⁰.

4 Superuidity

4.1 Introduction

In the previous chapter we have seen that both systems, the exciton-polaritons in a quantum well and the uid of light in a nonlinear medium are described by the same mathematical equation and therefore equivalent. Therefore, the following discussion of properties like the excitation spectrum, superuidity or the existence of vortices is valid for both systems. For simplicity the notation introduced in section 2 for the microcavity is used as it resembles the standard notation of the Gross-Pitaevski equation for Bose-gases.

4.2 Landau criterion for superuidity

Superuidity was rst observed for Helium 4 below the critical temperature of Tλ = 2.17K by P. Kapitza, F.F. Allen and D. Misseneren in 1937 [1, 16]. The theoretical explanation was found 4 years later by Landau who gave simple criterion for which quantum mechanics prohibits any dissipation of the moving uid [20].

Landau considered a uid at low temperature moving with constant velocity v in a tube. In the frame of reference moving with the uid the total energy for an excitation with momentum p is given by E₀+(p). Here E₀ is the non-excited uid energy and (p is the excitation energy. Under a Galilean transformation the energy and momentum in the referential of the tube are given by

E⁰=E₀+(p)−p·v+1

2Mv². (23)

In order dissipate its energy the change of energy in the rest frame needs to be negative.

Therefore we nd the condition for nite viscosity:

∆E =(p)−p·v<0 (24)

⇔v > v_c=(p)

p . (25)

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If the velocity is small enough that no excitation is possible, the system is superuid.

This leads to the Landau criterion for superuidity:

v < vc= lim

p→0

(p)

p . (26)

4.3 Excitation spectrum of an interacting BEC

In the last chapter we have seen that it is sucient to calculate the excitation spectrum in order to determine whether the system is in the superuid or supersonic phase. The perturbed wave function is obtained from the static wave functionφ0by a Bogoliubov transformation:

Ψ(r, t) =h

φ0(r) +Ae^{i(l·r−ωt)}B^∗e^{−i(l·r−ωt)}i

e⁻ⁱ^µt^h^¯. (27) Here Aand B^∗ are the small perturbation amplitudes. This ansatz is injected in the Gross-Pitaevski equation and the terms evolving withe^±ωt are isolated. The obtained set of coupled equation is solved and the excitation spectrum is found:

¯

hω_±=± s

¯h²k² 2m

²

+¯h²k²

m gn₀. (28)

For small k this equation is linear and the speed of sound is given by cs= ∂ω

∂k _k=0=

rgn0

m . (29)

If the velocity v = ^hk^¯_m⁰ of the condensate is slower than the speed of sound c_s, the condensate is according to the Landau criterion (equation 26) in the superuid regime.

On the other hand for high velocitiesv > c_sthe condensate is able to scatter elastically, this regime is called "Cerenkov-regime". This behaviour was observed experimentally, for a wave vector gure 5.

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Figure 5: Observation of superuidity in a cavity-embedded quantum well. I−III (IV−VI): Near-eld images corresponding to position space (far-eld images corresponding to momentum space) of the excitation spot around a defect. Below the excitation power threshold Pthr (I and IV) the photons scatter elastically at the defect which leads to an interference pattern and a parabolic wavefront in position space and a Rayleigh-ring in momentum space. Above the thresholdPthr(IIIandVI) the polaritons are in the superuid regime. The emission pattern is strongly inuenced by the polariton- polariton interactions. The incident angle is chosen such that the wave vector k_k corresponds to a velocity below the Landau critical value, therefore no scattering can occur any more. The system shows characteristics typical for the superuid regime: In position space the interference pattern and the parabolic wavefront disappear, in momentum space the Rayleigh scattering ring is no longer observable. IIandVeventually show the onset of superuidity. (Source: [2])

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Figure 6: Observation of the Cherenkov-regime in a cavity-embedded quantum well.

I−III(IV−VI): Near-eld images corresponding to position space (far-eld images corresponding to momentum space) of the excitation spot around a defect. Below the excitation power threshold P_thr (I and IV) the photons scatter elastically at the defect which leads to an interference pattern and a parabolic wavefront in position space and a Rayleigh-ring in momentum space. Above the threshold Pthr (III and VI) the polaritons are in the superuid regime. The emission pattern is strongly inuenced by the polariton-polariton interactions. The incident angle is chosen such that the wave vector k_k corresponds to a velocity above the Landau critical value, therefore the system shows characteristics typical for the Cherenkov-regime:

In position space the wavefront becomes linear and in momentum space the scattering ring strongly deforms. IIandVshow the onset of the Cherenkov- regime.(Source: [2])

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4.4 Vortices in Bose-Einstein-Condensates

The Gross-Pitaevski equation 19 can be transformed in a hydrodynamic form using the Madelung transformation [21]

Ψ(r, t) =p

n0(r, t)e^iθ(r,t). (30) The imaginary part of the Gross-Pitaevski equation with this ansatz is given by

∂n

∂t +∇ ·(n0(r, t)v(r, t)) = 0 (31) where v(r, t) = _m^¯^h∇θ(r, t)is the velocity eld of the condensate. It follows that the condensate is irrotational as∇ ×v= _m^h^¯∇ × ∇θ= 0.

We consider now a translational invariant BEC having shaped as a disc. The wave function may therefore be written as

Ψ0(r= (r, φ)) =p

n0(r)e^.θ(φ) (32)

which leads to a velocity:

v(r) = ¯h mr

∂θ

∂φuφ= ¯h

mrluθ. (33)

In the last step it was used that the rotational invariances requires thatθis independent of the angleφand therefore constant. Further more this constant needs to be an integer because the phaseθ is only dened modulo2π. Finally, this leads to the quantization of the velocity eld around the centre:

I

C

vds= ¯h

ml. (34)

The divergence of¹_r in (33) is hidden in a vanishing densityn0(r)^r→0−→0. This can be observed in experiments, one example is shown in gure ??.

In a typical dilute BEC the energy is dominated by the kinetic energy. The kinetic energy needed to excite a vortex is proportional to l²[7]:

E_kin^l = Z Z 1

2mv²(r)n_l(r)d²r∝l².. (35) Therefore, it is more favourable to excite N vortices of chargel = 1rather than one vortex of chargel=N. One can further show that vortices of the same sign repel each other and vice versa:

Epair ∝ Z Z

(v1(r) +v2(r))²d²r (36)

⇒Epair =E_kin^l¹ +E_kin^l² +const∗l1l2. (37) This repel leads to the self-organization of the vortices to Abrikosov lattices in stirred BEC [15, 22].

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Figure 7: (Left) Schematic presentation of the pumping scheme. Four laser beams arrive on the microcavity with incidence angle θ. The wave vector k_k inside the cavity is set by the azimuthal angleφ. (Right) The experimentally obtained density (near-eld image) and phase (far-eld image) maps for dif- ferent azimuthal angle φ(from top to bottom: φ = 0,5,5,10,15,21). The increasing azimuthal angle corresponds to an increasing angular momentum which manifests itself in a higher number of vortices. Each elementary vortex consists of a singularity in the density map. Around the singularity the phase winds from 0 to 3π. According to 35 the existence of N vortices of charge l = 1is energetically more favourable than the existence of 1 vortex of charge l = N. Therefore, only vortices of charge l = 1 exist in the

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Quantum Fluids of Light