Quantum Field Theory: Green's Functions and Polaritons, Lecture notes of Physics

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Polariton condensation: A

Green’s Function approach

Jonathan Keeling

MathNanoSci Intensive Programme, L’Aquila 2010.

List of Figures

1.1 Schematic electronic spectrum & coupled quantum wells.... 7 1.2 Schematic semiconductor microcavity.............. 11 1.3 Schematic polariton spectrum................... 12

2.1 Disorder localised exciton states................. 18

3.1 Diagrammatic expansion of photon Green’s functions...... 25 3.2 Normal and anomalous self energies............... 26 3.3 Spectrum of normal modes in the condensed system...... 28 3.4 Spectral weight of for broadened case.............. 29

4.1 Cartoon of system coupled to pumping and decay baths..... 31 4.2 Keldysh time path contour..................... 34

5.1 Occupation of pumping baths................... 40 5.2 Critical temperature vs density.................. 43 5.3 Critical κ, γ for clean system.................... 44 5.4 Critical κ, γ for disordered system................. 44

6.1 Inverse Green’s functions and spectral weight.......... 50 6.2 Distribution set by competition of baths............. 51 6.3 Zeros of <, =

[

(DR)−^1

]

from diagrammatic approach....... 53 6.4 Zeros of <, =

[

(DR)−^1

]

from Maxwell-Bloch equations...... 54

iv

Introduction

These lectures discuss the use of Green’s functions in understanding con- densation in non-equilibrium systems. The lectures thus have two aims, conceptual and technical. The conceptual aim of these lectures is to introduce the idea of quantum condensation in general, and of condensation of quasiparticles in semicon- ductors in particular. The quasiparticles that will be studied in detail are exciton-polaritons; but some historic overview including exciton conden- sation will also be provided. Excitons are bound electron hole pairs in semiconductors, and exciton-polaritons are superpositions of excitons and photons (resulting from repeated interconversion of photons to excitons and vice versa). These quasiparticle condensates are of interest for several rea- sons: firstly, their light effective mass potentially allows condensation at high temperatures, up to room temperature; secondly, particularly for po- laritons, they are non-equilibrium, and thus provide a bridge to lasing. As such, one conceptual aim of these lectures is to show the link between lasing and condensation illustrated for polaritons. I will also try to highlight the similarity between different kinds of quantum condensates, particularly dis- cussing the relation between the theory used here for polariton condensates to the theory of a weakly interacting dilute Bose gas (WIDBG). As well as the conceptual aims, these lectures also illustrate an appli- cation of Green’s function approaches to condensates. There are thus also technical aims as follows: firstly, to show how Green’s functions may be used to determine the fluctuation spectrum, and physical response func- tions (such as luminescence intensity) of a multi-component condensate; secondly to introduce non-equilibrium Green’s functions as a way to dis- cuss systems driven out of equilibrium, and to show how these reduce to simpler descriptions in various limits. These lectures will assume some familiarity with second quantised no- tation, and do not provide a complete overview of Green’s functions (for which, see one of the references listed below). However, they will provide a brief summary, intended to act either as a reminder for those already familiar, or as an enumeration of the important results required for those unfamiliar. The lectures divide into a shorter first section, discussing fluctuations in an equilibrium system, and a longer second section discussing the non- equilibrium system. The aims in both are to introduce the apparatus (Green’s functions, non-equilibrium approaches) that are required to ad- dress the problem, while simultaneously discussing the results that can be

v

Chapter 1

Introduction to condensation

and polaritons

This section of notes provides an introduction to condensation in general, and to condensation of excitons and polaritons in particular. In order to provide a general framework, we will first discuss condensation of a gas of weakly interacting Bosons; such a model is certainly appropriate to de- scribe experiments on cold dilute atomic gases, and can be appropriate to a description of polaritons at low enough densities. However, to under- stand polariton condensation across a range of densities, we will in the next chapter introduce a somewhat less standard model of polariton condensates which takes into account saturation of excitons at high densities.

1.1 Condensation of a Bose gas

The basic idea of condensation refers to the macroscopic occupation of a single quantum mechanical mode. A straightforward conversion of this statement into an explicit formula is the requirement that the one particle density matrix:

ρ(r, r′) = 〈ψ†(r)ψ(r′)〉 =

j

nj φ∗ j (r)φj (r′) (1.1)

should have at least one eigenvalue, nj which scales as the number of par- ticles in the system[13] (where normalisation

dr|φ|^2 = 1 is assumed). i.e. for N particles, such that Tr(ρ) = N , one requires limN →∞ nj /N to be finite. This criterion means that there is a macroscopic number of parti- cles in a single mode. In the case of a translationally invariant system, the single particle wavefunctions φj (r) should take the form of plane waves φj (r) = exp(ik · r)/

V , and so the existence of a macroscopic eigenvalue means that

lim |r−r′|→∞

ρ(r, r′) =

N

V

will be finite. This latter criterion is referred to as off diagonal long range order.

1

CHAPTER 1. INTRODUCTION TO CONDENSATION AND

POLARITONS

Non-interacting Bose gas

The simplest example in which such behaviour is seen is a gas of non- interacting bosons in three dimensions. From the statistical mechanics of a Bose gas, the number of particles in a volume V is given by:

N =

V

(2π)^3

d^3 knB (k) = V m^3 /^2 √ 2 π^2 ¯h^3

0

dnB ()

d (1.2)

Using the form of the Bose distribution, and writing z = eβμ, this can be rewritten as:

n =

N

V

(mkB T )^3 /^2 √ 2 π^2 ¯h^3

π 2

I 3 / 2 (z), Ip(z) =

Γ(p)

0

dx zxp−^1 ex^ − z

dx. (1.3) If μ → 0 , z → 1, then the occupation of the lowest energy state will diverge; in this limit there is a macroscopic eigenvalue of the density matrix, corre- sponding to the k = 0 eigenstate. One may show that I 3 / 2 (1) = 2.612, and so this macroscopic occupation arises at a non-zero temperature:

TBEC =

2 π¯h^2 mkB

( (^) n

2. 612

While the non-interacting gas shows condensation, the behaviour of the interacting gas is more interesting, as it is only in this case that one sees features such as superfluidity (arising from the modified excitation spectrum). Furthermore, interactions are crucial in resolving what happens when there are two almost degenerate single particle states. Therefore, we turn next to the weakly interacting Bose gas.

Weakly interacting dilute Bose gas

The WIDBG model can be written as:

H − μN =

k

(k − μ)ψ† kψk +

k,k′,q

U

2 V

ψ† k+qψ† k′−qψk′ ψk (1.5)

d^3 r

[

ψ†

¯h^2 ∇^2 2 m − μ

ψ +

U

ψ†ψ†ψψ

]

where ψ(r) =

k ψke ik·r/√V. In the interacting case, we may start to describe the phase transition to a condensed state somewhat differently, by considering the conditions required in order that a condensed state should be the state that minimises the free energy of the system. If we consider a state in which all N particles are in the same single particle wavefunction φ(r), then the free energy of this state, arising from the above Hamiltonian, is of the form:

F = N

d^3 r

[

¯h^2 2 m |∇φ|^2 − μ|φ|^2 +

U (N − 1)

|φ|^4

]

Minimising this with respect to φ leads to a self consistent equation for the single particle wavefunction φ, taking into account the interactions between

CHAPTER 1. INTRODUCTION TO CONDENSATION AND

POLARITONS

disturbing, but there are a number of explanations. Firstly, in order that a state should be able to show interference effects, it must have a sufficiently well defined phase, and since phase and number are conjugate, thus an indefinite number of particles is necessary to show coherent interference. Secondly, when we consider fluctuations below, it will be seen that one can find states with fixed total numbers of particles, but indeterminate division of particles between the k = 0 mode and non-zero k modes and these states arise by considering fluctuations on top of the coherent state.

Fluctuations

We next consider the effective Hamiltonian for fluctuations on top of the macroscopically occupied state. This has several purposes: it allows one to find the excitation spectrum, which controls how the condensate responds to certain probes; it allows one to understand the depletion of condensate density at finite temperature, as excitations are populated instead of the ground state; and it also allows one to improve upon the ground state ansatz assumed above, in which all particles are in the same state. Working in momentum space, we will consider fluctuations on top of the coherent state |Ψ〉 = exp(λψ† 0 )|Ω〉, where λ =

N =

V μ/U as discussed above. This means one may replace the operator ψk by ψk → λδk + δψk, in terms of the finite momentum fluctuations ψk. Substituting this into the WIDBG Hamiltonian then yields:

Hfluct =

k

(k − μ)ψ k†ψk +

U

λ^2 V

4 ψ k†ψk + ψ† kψ−†k + ψkψ−k

where we have used H → H = H −μN , and assumed that ψ 0 is real. Then, substituting the value of λ^2 , we get:

Hfluct =

k

(k + μ)ψ k†ψk + μ 2

ψ† kψ†−k + ψkψ−k

k

ψ† k ψ−k

) (^ 

k +^ μ^ μ μ k + μ

ψk ψ−†k

(k + μ) 2

Here, the last line involved changing the order of two operators, so the final term comes from the commutator of [ψ−k, ψ†−k]. One now uses a Bogoliubov transformation to diagonalise the matrix, noting that since this will mix ψ†−k and ψk, it is not a unitary transformation, but rather:

( ψk ψ−†k

cosh(θ) sinh(θ) sinh(θ) cosh(θ)

φk φ†−k

after which the matrix in the Hamiltonian becomes: ( (k + μ) cosh(2θ) + μ sinh(2θ) (k + μ) sinh(2θ) + μ cosh(2θ) (k + μ) sinh(2θ) + μ cosh(2θ) (k + μ) cosh(2θ) + μ sinh(2θ)

1.1. CONDENSATION OF A BOSE GAS 5

Hence, taking sinh(2θ) = −μ/ξ, cosh(2θ) = ( + μ)/ξ, and ξ^2 = ( + μ)^2 − μ^2 = ( + 2μ), one finally gets:

Hfluct =

k

ξkφ† kφk + ξk − k − μ 2

From this form, we may note several points:

• The mean-field ansatz was not the ground state, since in the absence of populating the finite k modes, the energy of Hfluct would be zero, whereas Eq. (1.18) implies it may be smaller, since k + μ > ξk.
• Regardless of the temperature^1 , the energy of an excitation with momentum k is given by ξk, hence the spectrum of fluctuations is straightforward.
• At a finite temperature, there will be a thermal occupation of the fluctuations, hence the total occupation is N = Ncond + Nfluct. The population of the fluctuations Nfluct is given by: ∑

k

〈ψ† kψk〉 =

k

cosh^2 (θ)〈φ† kφk〉 + sinh^2 (θ)〈φ−kφ†−k〉

k

k + μ ξk

nB (ξk) + (k + μ − ξk) 2 ξk

Thus, with increasing temperature, for a fixed N , the value of |λ|^2 will fall. This reproduces the standard form for the critical temperature for Bose-condensation, but starting from the non-condensed side.^2

Nature of the ground state

Let us briefly note the existence in Eq. (2.22) of a contribution that exists even at T = 0. This is another reflection of the statement that the empty state is not the ground state; turning this around, we may note that the ground state of the fluctuation Hamiltonian, |Ω〉 obeys:

0 = φk|Ω〉 =

cosh(θk)ψk − sinh(θk)ψ†−k

which is satisfied by:

|Ω〉 =

exp

tanh(θk)ψ k†ψ−†k

(^1) Up to the fact that the condensate density will vary with temperature (^2) To find the total density, one may use that N = −dF/dμ. Evaluating this for H = H 0 + Hfluct gives:

N = |lambda|^2 −

X k

〈ψ† k ψk 〉 + 〈ψk ψ−k 〉 + 〈ψ† k ψ†−k 〉 2

!

. (1.20)

The negative sign in this is at first surprising, but can be understood as meaning that, in the presence of fluctuations, Ncond = N − Nfluct 6 = | lambda|^2 , since | lambda|^2 was only the mean field estimate of the condensate density.

1.2. EXCITONS AND POLARITONS 7

generally convenient to work in terms of density and phase fluctuations, i.e. ψ =

ρ + πeiφ, where ρ is the mean field density, and π, φ the fluctuations. In such a language, it is the contribution of phase fluctuations that destroys long range order, and leads to power law decay of correlations. Although the non-interacting gas does not show any transition, just as in the three dimensional case, the transition temperature for the weakly interacting case can be roughly estimated from the non-interacting result, such that one has kB Tc ' 2 π¯h^2 n/m.

1.2 Excitons and polaritons

Because the transition temperature for a Bose gas depends inversely on its mass, it is favourable for condensation to consider particles with rela- tively light masses. One way to produce particles with particularly light masses is to consider quasiparticles inside semiconductors. Since a bosonic quasiparticle is required, one of the simplest quasiparticles one might con- sider here is the exciton, a bound pair of an electron and hole. An even lighter quasiparticle can be produced by considering a mixture of excitons and photons, i.e. exciton-polaritons. The following section will introduce excitons, exciton condensation, and microcavity exciton-polaritons.

Excitons

Excitons result from Coulomb interactions between excited electrons, and holes in a semiconductor. They require an almost filled valence band, with a small number of unoccupied states to give valence band holes, and a partly occupied conduction band. In some indirect gap semi-metals, this configuration may exist in the ground state, however we will discuss here direct band gap semiconductors, where such a situation can only be cre- ated by optical excitation, transferring electrons between the valence and conduction band (see Fig. 1.1).

Energy

Momentum

Holes

z

holes

electrons

Energy

Figure 1.1: Left: Optically excited semiconductor, with an al- most filled valence band, with a small number of valence band holes and conduction band electrons created. Right: Coupled quantum wells, with applied perpendicular electric field, so that electrons and holes are spatially separated into two quantum wells.

The behaviour of the system can be described in terms of a Hamiltonian

CHAPTER 1. INTRODUCTION TO CONDENSATION AND

POLARITONS

for electrons with k near the band gap:

H =

k

c(k)a† ckack + v(k)a† vkavk

q

V (^) qee ρeqρe −q + V (^) qhh ρhq ρh −q − 2 V (^) qeh ρeqρh −q

where ρeq =

k a

† ck+qack and^ ρ hq = ∑ k avk−qa

† vk, and the Coulomb inter- action is V (q) = e^24 π/q^2. For small values of k, the conduction and valence band electron energies can be expanded assuming parabolic bands, c(k) = ¯h^2 k^2 / 2 mc, v(k) = −Eg − ¯h^2 k^2 / 2 |mv|. This Hamiltonian, written in terms of creation operators for conduction and valence band electrons can be rewritten via h†−k = avk, ck = ack as a Hamiltonian for valence band holes and conduction band electrons; this has the result that the kinetic energy term then describes positive mass holes as well as positive mass conduction band electrons. At low densities of electrons and holes, one may consider the single electron-hole pair problem, which leads to the two particle Hamiltonian:

H = Eg − ¯h^2 ∇^2 e 2 mc

¯h^2 ∇^2 h 2 mv

e^2 4 π|re − rh|

This has Hydrogen like solutions, parametrised by the exciton Rydberg Ryex = μe^4 / 2 ^2 ¯h^2 = (μ/m∗^2 )Ry, and the exciton Bohr radius aex = ¯h^2 /μe^2 = (m∗/μ)a 0 , where μ is the reduced electron-hole mass, and m∗ the free electron mass. Compare to Hydrogen, the most significant effect is that  =  0 r with r ' 11 for typical semiconductors; this leads to a much increased Bohr radius, and reduced binding energy. As such, it is quite feasible to approach exciton densities where excitons are separated by a distance comparable to their Bohr radius. As one moves away from low densities, the problem of the electron- hole ground state becomes more complicated, since the high density of electronic excitation can screen the Coulomb interaction inside each exciton, leading to a possible transition to a Mott insulator phase, or some other non excitonic phase at high densities. In fact, the full phase diagram of the electron-hole system as a function of density and electron/hole mass ratio remains not fully determined. Two further complications regarding the exciton system should be men- tioned:

• The first is that the excitons can recombine, i.e. since the conduction band electrons could be formed by optical excitation, there is a non- zero probability that electrons in the conduction band can transfer back to the conduction band, emitting a photon. The rate of re- combination can be calculated in terms of the light-matter coupling, which one may write as Hint = −eA(R)(pe/mc − ph/mh), depends on the properties of the semiconductor, via the transition matrix ele- ment μcv = 〈ck|p/m∗|v − k〉, in terms of matrix elements of the single electron states.

CHAPTER 1. INTRODUCTION TO CONDENSATION AND

POLARITONS

This then prompts one to write the exciton condensate wavefunction as:

|Ψex. cond.〉 = N exp

k

λka† ckavk

k

uk + vka† ckavk

The second expression follows from the first by replacing the exponential of the sum by a product of exponentials, and then by noting that since a, a†^ are Fermionic operators, the Taylor expansion of the exponential should stop after the first term. We have then introduced uk, vk via λk = vk/uk, |uk|^2 + |vk|^2 = 1, with the latter criterion ensuring normalisation. For low densities, the form vk ' λk =

N/V φ 1 s(k) will apply, but as density increases, and vk → 1, the form of vk that minimise energy will start to be modified [14]. In the very high density limit, vk ' θ(k < kF ), i.e. the fermionic statistics start to play a significant role, and only near the Fermi surface does the Coulomb interaction have an effect. By minimising the energy of Eq. (1.28) with Hamiltonian (1.24), one can determine the form of uk, vk throughout the crossover from low to high density exciton condensates[15]. As for the mean field theory of the WIDBG, this approach is a good way to describe the ground state at zero temperature, but does not necessarily determine the transition temperature. For this, one must consider the pos- sibility of excitations, which deplete the condensate. Unlike the WIDBG, the exciton case contains two classes of excitations that one must consider:

• Fermionic excitations, corresponding to ionising excitons, by allowing for excitations in which uk, vk change. It is possible to straightfor- wardly incorporate these in a non-zero temperature mean field theory by calculating the partition function at finite temperature correspond- ing to these kinds of excitations, and thus find the decrease of vk, and the associated excitonic ordering at finite temperature. An analogous calculation for exciton-polaritons is discussed later. Such excitations are the dominant effect at high densities.
• At low densities, the dominant fluctuations instead concern bound excitons, but excited to finite momentum states. These excitations are exactly as for the WIDBG, and lead to a transition temperature that depends on the excitonic mass.

Exciton-Polaritons

Microcavity exciton polaritons are the result of strong coupling between photons confined in semiconductor microcavities, and excitons in QWs. A schematic of the semiconductor structures used to engineer microcavity exciton polaritons is shown in Fig. 1.2. The semiconductor microcavity is formed by two distributed Bragg reflectors, and QWs are then placed at the antinodes of the standing wave of light in the cavity. At the simplest level, the strong coupling between excitons and po- laritons can be understood in terms of a model of non-interacting bosonic

1.2. EXCITONS AND POLARITONS 11

Cavity Quantum Wells

Figure 1.2: Schematic of microcavity system. Red and white alternating layers indicate layers of alternating dielectric contrast, producing a Bragg mirror. Inside the cavity between the two mirrors, quantum wells are placed at the antinodes of the standing wave pattern of light.

modes, ψ† k creating photons, D† k creating excitons (assumed for the moment to behave as bosons, if at low enough density), where k labels the in plane momentum. One may then write

H =

ψ k† D k†

) (^ ω k ΩR/^2 ΩR/ 2 k

ψk Dk

Here, ωk is the energy of the photon mode confined in the cavity of width Lw, giving: ωk = (c/n)

k^2 + (2πN/Lw)^2 , with n is the refractive index, and N the index of the transverse mode in the cavity. For small k, the energy can be written as ωk = ω 0 + k^2 / 2 m, where m is an effective photon mass m = (n/c)(2π/Lw). In the absence of disorder, the exciton energy in the QW is εk = ε 0 + k^2 / 2 M , where M is the total exciton mass, and ε 0 = Egap − Ryex comes from the conduction-valence band gap Egap including QW confinement and the exciton binding energy (Rydberg) Ryex. For convenience, we define the bottom of the exciton band, ε 0 as the zero of energies; and denote the detuning between exciton and photon bands as δ = ω 0 − ε 0. Finally, the off diagonal term ΩR/2 describes the exciton- photon coupling, where ΩR is the Rabi frequency. Then, diagonalising the quadratic form in Eq. (1.29) gives the polariton spectrum:

ELP k ,UP=

δ +

k^2 2 M

k^2 2 m

δ +

k^2 2 M

k^2 2 m

+ Ω^2 R

This spectrum is illustrated in Fig. 1.3 It is shown there both as a function of momentum k, and also as a function of angle. The angle corresponds to the angle of emission of a photon out of the cavity; since the in-plane momentum and photon frequency are both conserved as photons escape through the Bragg mirrors, one may write (ε 0 + E kLP ) sin(θ) = ck, which (since typical values of k satisfy ω 0 , ε 0  ck) can be approximated as ω 0 sin(θ) = ck.

1.2. EXCITONS AND POLARITONS 13

Lifetime Thermalisation Linewidth Temperature Atoms 10s 10ms 2. 5 × 10 −^13 meV 10 −^8 K 10 −^9 meV Excitons 50ns 0.2ns 5 × 10 −^5 meV 1K 0.1meV Polaritons 5ps 0.5ps 0.5meV 20K 2meV Magnons 1 μs 100ns 2. 5 × 10 −^6 meV 300K 30meV Table 1.1: Characteristic timescales and energies for: particle lifetimes, times to establish a thermal distribution, linewidth due to finite lifetime, and characteristic temperatures for various can- didate condensates. Comparison of the first two describes how thermal the distribution will be; comparison of the later two de- termine the effect of finite lifetime on coherence properties.