Photon-BEC in an optical

microcavity

a research training report by Tobias Rexin

Supervisor: Priv.-Doz. Dr. Axel Pelster

July 11, 2011

1 Introduction

1.1 Bose-Einstein-condensation of photons in optical

microcavity

Bose-Einstein condensation (BEC) is the macroscopic accumulation of bosonic particles inthe energetic ground state level below a critical temperature Tcrit. This phenomenom hasbeen demonstrated in several di�erent physical systems as, for instance, dilute ultracoldBose gases such as sodium(1) or exitons in solid state matter(2), but for one of the most ob-vious Bose gases, namely blackbody radiation, it is yet unobserved. Blackbody radiationis electromagnetic radiation which is in thermal equilibrium with the cavity walls. In-stead of undergoing BEC the photons disappear in the cavity walls when the temperatureT is lowered corresponding to a vanishing chemical potential. Recent experiments(3)(4)

with a dye-�lled optical micro resonator, performed by the group of Martin Weitz at theUniversity of Bonn, achieved thermalization of photons in a number-conserving way. Thecurvature of the micro resonator provides two important ingredients which are prereq-uisites for BEC: a con�ning potential and a non-vanishing e�ective photon mass. Thisexperiment gives new opportunities for creating coherent light. In contrast to the �fty-yearold laser, which operates far from thermal equilibrium, the photon BEC gains coherenceby an equilibrium phase transition.Before this experiment one had several veri�cations that massive particles behave likewaves for example the interference of fullerene(5). But now the quantized electromagneticwaves in the cavity are given an e�ective mass, so here it is the other way around wavesbehave like massive particles. In the next sections I report about the details of the exper-imental setup and then I investigate theoretically those `particles` of light (6). First I givea short overview of the experimental setup and some important experimental facts. Inthe second part I am focusing on the calculation of the e�ective action, then I deduce thecritical particle number Ncrit for both the non-interactioning and the interacting Bose gas.The interesting feature here is that we have to deal with an e�ective 2-dimensional Bosegas with a given temperature. Thus we can calculate all physical observables quantummechanical exact and do not get any divergencies from the semiclassical approach andeven �nite size corrections are already included.

1.2 Experimental setup

Photons are trapped in a curved optical micro resonator, where the curvature of themirrors induce an e�ective harmonic trapping potential (see �gure 1.1c). Outside the

2

Figure 1.1: a. Schematic spectrum of cavity modes with absorption coe�cientα(ν) and �uorescence strength f(ν) b. Dispersion relation of pho-tons in the cavity (solid line) with �xed longitudinal mode (q = 7)and the free photon dispersion (dashed line) c. Schematic experi-mental setup with trapping potential imposed by the curvd mirrors.

center of the mirrors the distance d becomes shorter and the allowed wave vectors |~k| =2π

dndyegrow. Thus we conclude that the energy E = ~c|~k| of the photons to maintain

in this region of the cavity is higher than in the center. The two mirrors are spacedD = 1.56 µm away from each other which is exactly 3.5 optical wavelengths. Indeed , ifwe look at �gure 1.1a one can clearly see at quantum number q = 7, which corresponds awave length λ = 585 nm, that there is a great overlap between the absorption coe�cientα(ν) and the �uorescence strength f(ν) of the dye. This modi�es spontaneous emissionssuch that the emission of longitudinal mode with quantum number q = 7 dominates overall other emission processes. Due to the extremely short distances of the mirrors thereis an e�ective longitudinal low frequency cut-o� ωcut = ckz = 2π · 5.1 · 1014 Hz wherekz = 2π

ndyeDand D = 7 · λ

2, because no larger wavelength �ts into the microcavity and

of course the speed of light in vacuum c is modi�ed when entering the dye solution by therefraction index ndye = 1.33. Furthermore the energy E = ~ωcut of the longitudinal modekz is far above thermal energy at room-temperature exp [−~ωcut/(kBT )] ≈ exp[−80].The photon statistics, e.g. the photon number nph of a classical blackbody radiator,is determined by T 4 due to the Stefan-Boltzmann law whereas thermal excitations inthe lowest available energy level ∝ ~ωcut inside the cavity are suppressed by the factorexp[−80]. The cavity photon number nph is almost not altered by the temperature Tof the surrounding dye solution. This means that the thermalization process conservesthe average photon number. Thermal equilibrium can be achieved by absorption and re-emission processes in the dye solution which is acting as a heat bath for photons. As canbe seen from the �gure 1.a the re-emission in the longitudinal mode kz dominates over allother modes. There is the largest overlap between emission coe�cient and �uorescencestrength. This means that most of the (q = 7)-photons absorbed by the dye will bere-emitted inside the cavity whereas for all other longitudinal modes there is a disbalancebetween absorption and emission. The kz-mode is frozen out and the kr-modes canthermalise due to the rovibrational energy levels of the dye solution.

3

1.3 Proving BEC and experimental results

Figure 1.2: a. Spectral intensity distributions (connected circles) transmittedthrough one cavity mirror, as measured with a spectrometer for dif-ferent pump powers b. Images of the spatial radiation distributionbelow criticality (upper panel) and above criticality (lower panel).

The BEC of the photons inside the dye-�lled cavity has been proven experimentally byinvestigating both the spatial and temporal coherence. One has measured spectral distri-butions (see �gure 1.2a) which show a classical thermal distribution at room temperatureand an increased intensity for λ = 585 nm. The latter corresponds to the frozen longitudi-nal wave vector kz which yields λ = 2π

3.5·kz . So far this is a proof for BEC in the frequencydomain. The experimental setup also allows to check the spatial domain. Hence anotherevidence for the achieved BEC is the in-situ spot (see �gure 1.2b) captured by the camera(see �gure 1.1c). Below criticality there is only a thermal 'cloud' of the cavity photons butabove criticality one �nds a rather sharp yellow spot sitting in the center of the cavity inthe minimum of the harmonic potential, which is induced by the curvature of the cavitymirrors.

1.4 Hamiltonian for photons in an optical micro cavity

We start with the relativistic energy-wave vector relation for photons where we explicitlydecompose |~k| into its longitudinal component kz and the radial symmetric componentkr:

E = ~c|~k| = ~c√kz

2 + kr2, (1.1)

4

where kz = 2πndyed(r,D)

is the longitudinal wave vector component with the mirror distance

D depending on the �xed quantum number q = 7. In general the longitudinal wave vectorcomponent depends on the distance d(r) between the two curved mirrors with curvatureR = 1 m. From a few geometrical considerations for this biconvex cavity one gets thatd(r) = D − 2(R −

√R2 − r2) where r is now the radius to the optical axis, so that for

r = 0 we once again get the mirror distance D = 1.56 µm in the center of the cavity.First we can assume kr � kz which is reasonable because kr corresponds to the room tem-perature thermalised tranversal modes which are∝ (kBT )/(~c) whereas kz ∝ 80(kBT )/(~c)due to the small resonator distance. The known binomic formula for small x : (

√1 + x ≈

1 + 1/2 ·x) is applied to the square-root of the energy where we have explicitly pulled outthe kz(r) so x = (kr)

2/k2z yields,

E = ~c|~k| = ~c√kz

2 + kr2 ≈ ~ckz(r) + ~c

(kr)2

2kz(r). (1.2)

Furthermore we have r � R in our setup, i.e. r ∝ µm � R = 1 m and the Taylorexpansion in kz at kz(r = 0) gives

kz ≈2π

ndyed(r)

[1

D+

1

RD2r2 +O(r3)

]. (1.3)

Inserting (1.3) into (1.2) then yields

E ≈ ~ckz(r) + ~c(~kr)

2

2kz(r)≈ mphc

2 +(~kr)2

2mph

+1

2mphω

2r2 (1.4)

with e�ective photon mass mph = ~ωcut

c2= ~kz(0)

c= 6.7 · 10−36 kg which is about the

magnitude 1010 smaller than the usual atomic masses. Furthermore this approximationgives an e�ective trap frequency ω = c√

DR2

= 2π · 4.1 · 1010 Hz which about a factor 108

higher than the usual atomic BEC trap frequencies.Thus the system in an optical microcavity is equivalent to a non-relativistic 2-dimensionalBose gas. The dispersion relation is quadratic for small transversal wave numbers as isillustrated in the dispersion relation in �gure 1.1 b. Note that for kr = 0 we still havean `e�ective rest mass` intimidly related to the frozen kz-mode, which appears in zerothorder of the kz(r)-expansion.

5

2 Theoretical description of

Photon-BEC

2.1 Ultracold vs room temperature Bose gas

The determination of the phase boundary between the gas and the BEC phase is of funda-mental interest. In the case of ultracold Bose gases one considers the critical temperatureTcrit(N) as a function of the particle number N , this time we have a given temperature T ,namely the room temperature T = 300 K, and then calculate the critical particle numberfor the onset of Bose-Einstein condensation Ncrit(T ). The comfort of this situation isthat we do not have to extract the temperature by inverting the equation of state for theparticle number.

2.2 E�ective action

From the dispersion (1.4) we read o� that the system is equivalent to an harmonic os-cillator in 2 dimensions. In the following I give a general treatment for D dimensionalisotropic harmonic oscillators:

h0 =~p2

2mph

+ V (~x) =(~kr)2

2mph

+1

2mphω

2~x2 +mphc2. (2.1)

then the full Hamiltonian in second quantization with interaction looks like this

H =

∫dDxψ†(~x)

[− ~2

2m4− µ+ V (~x)

]ψ(~x) +

∫g

2ψ†(~x)ψ†(~x)ψ(~x)ψ(~x) (2.2)

where the canonical commutation relations read[ψ†(~x),ψ(~x′)

]= δ(~x− ~x′) (2.3)[

ψ(~x),ψ(~x′)]

=[ψ†(~x),ψ†(~x′)

]= 0 (2.4)

Now I am using the Bogoliubov description for the �eld operators ψ†(~x), ψ(~x) because inour case we can consider the ground state ψ0, ψ

∗0 to be macroscopically occupied. To this

6

end we set

ψ∗0 =⟨ψ†(~x)

⟩, ψ0 =

⟨ψ(~x)

⟩, (2.5)

where the expectation value is de�ned as < • >= 1Z

tr[e−βH •

]with the partition func-

tion Z = tr[e−βH

]and β = 1/(kBT ). The decompostion of the �eld operators looks as

follows:

ψ†(~x) = ψ∗0(~x) + δψ†(~x) , δψ†(~x) = ψ†(~x)−⟨ψ†(~x)

⟩(2.6)

ψ(~x) = ψ0(~x) + δψ(~x) , δψ(~x) = ψ(~x)−⟨ψ(~x)

⟩. (2.7)

Formula (2.5) already implies that the expectation values of the �uctuations δψ†(~x), δψ(~x)vanish by construction.Plugging (2.6) and (2.7) into (2.2) yields:

H = H0 + H1 (2.8)

where the free Hamiltonian reads

H0 =

∫ψ∗0(~x) [h0(~x)− µ]ψ0(~x) + δψ†(~x) [h0(~x)− µ]ψ0(~x)

+ ψ∗0(~x) [h0(~x)− µ] δψ(~x) + δψ†(~x) [h0(~x)− µ] δψ(~x) (2.9)

and the interaction-part of the Hamiltonian is given by

H1 =g

2

∫|ψ0(~x)|4 + 2 |ψ0(~x)|2 ψ∗0(~x)δψ(~x) + (ψ∗0(~x))2δψ†(~x)δψ†(~x)

+ 2 |ψ0(~x)|2 ψ0(~x)δψ†(~x) + 4 |ψ0(~x)|2 δψ†(~x)δψ(~x)

+ 2ψ∗0(~x)δψ†(~x)δψ(~x)δψ(~x) + (ψ0(~x))2δψ†(~x)δψ†(~x)

+ 2ψ0(~x)δψ†(~x)δψ†(~x)δψ(~x) + δψ†(~x)δψ†(~x)δψ(~x)δψ(~x) (2.10)

The partition function is Z = tr[e−βH

]and the e�ective action is Γeff [ψ∗0,ψ0] = − 1

βln(Z).

The expressions (2.9),(2.10) are pretty long and will become extremely complicated onceyou insert this in the exponential for the partition function. The problem here is wehave terms which are not only quadratic or quartic in the �uctuation operators δψ ,δψ†

so we cannot close the algebra (Zassenhausen formula(7) for exponentials of operators).Since the interaction parameter g is supposed to be small, we can try to �nd a solutionwith a pertubative approach in g. But before we do this we already know from quantumstatistics the real physical �elds ψ∗0 , ψ0 extremize the e�ective action Γeff which then is

identical to the free energy(8). In performing this extremalisation we get �rst

7

δΓ[ψ∗0,ψ0]

δψ∗0= − δ

δψ∗0

1

βln(Z) =

1

Ztr

[e−βH

δH

δψ∗0

](2.11)

= 〈[h0 − µ]ψ0〉+⟨

[h0 − µ] δψ⟩

+g

2

{2⟨|ψ0|2 ψ0

⟩+⟨

2 |ψ0|2 δψ⟩

+⟨

2ψ∗0δψδψ⟩

+⟨

2(ψ0)2δψ†⟩

+⟨

4ψ0δψ†δψ⟩

+⟨

2δψ†δψδψ⟩}

!= 0. (2.12)

Note that there is no problem by interchanging the derivative of the Hamiltonian with itsexponential, as long as we do this under the trace. Hence we obtain from (2.6),(2.7),(2.12)that the real physical �elds ful�ll the following condition:

(h0 − µ)ψ0 = −g{[|ψ0|2 ψ0

]+ ψ∗0

⟨δψδψ

⟩+ 2ψ0

⟨δψ†δψ

⟩+⟨δψ†δψδψ

⟩}=: gCδψ†

(2.13)

and, of course, an analogous expression for the complex conjugate. We are now able toreplace terms of H0 which are linear in δψ ,δψ† by terms which are linear in g. Takingthis into account we �nd the following e�ective potential:

Γeff = − 1

βln tr

(exp

[−β∫dDxψ∗0(h0 − µ)ψ +

g

2δψ† · Cδψ† +

g

2Cδψ · δψ + δψ†(h0 − µ)δψ + H1

])(2.14)

Linearisation in g of the trace yields .

Γeff = − 1

βln tr

(exp

[−βH0

] (1 +

g

2δψ† · Cδψ† +

g

2Cδψ · δψ + H1

))(2.15)

with the Hamiltonian

H0 =

∫dDxψ∗0(h0 − µ)ψ + δψ†(h0 − µ)δψ. (2.16)

The trace is performed over the Fock-base 〈{nk}|k∈N\0. Since we have already explic-itly treaten the ground state, the �uctuation operator has to exclude the ground stateaccording to δψ(~x) =

∑∞k 6=0 ϕk(x)ak. The unperturbed Hamiltonian H0 (2.16) is now

diagonalised in the Fock base. Since we perform the trace we only need the diagonalelements of the perturbed part which means:

1

Z0

tr(e−βH0O(δψ†,δψ,(δψ)3)

)=⟨O(δψ†,δψ,(δψ)3)

⟩(0)

= 0

with 〈•〉(0) = 1Z0

tr(e−βH0•

).

Thus the e�ective action (2.15) reduces to

8

Γeff = − 1

βln tr

[exp

(−β∫dDxψ∗0(h0 − µ)ψ + δψ†(h0 − µ)δψ

)(2.17)

×(

1− βg

2

∫dDx |ψ0(~x)|4 + 4 |ψ0(~x)|2 δψ†(~x)δψ(~x) + δψ†(~x)δψ†(~x)δψ(~x)δψ(~x)

)]which �nally leads to

Γeff = − 1

βln Z0 −

1

βln

(1− βg

2

∫dDx |ψ0(~x)|4 + 4 |ψ0(~x)|2

⟨δψ†(~x)δψ(~x)

⟩(0)

(2.18)

+⟨δψ†(~x)δψ†(~x)δψ(~x)δψ(~x)

⟩(0)).

In the next step we apply the Wick-rule in order to calculate the expectation value of thefour �eld operators:⟨

δψ†(~x)δψ†(~x)δψ(~x)δψ(~x)⟩(0)

= 2⟨δψ†(~x)δψ(~x)

⟩(0)2

. (2.19)

The two �eld operator expectation value is related to the Green function and the propa-gator in the following way:⟨

ψ†(~x)ψ(~x)⟩(0)

= G(0)(~x; ~x) =∑~m

ϕ∗~m(~x)1

exp (β [E~m − µ])− 1ϕ~m(~x) (2.20)

=∞∑l=1

exp (βµl) (~x,β~l|~x,0) (2.21)

This is a general result from quantum statistics, particularly in our case we have to takeexplicit care of the mean �eld decomposition. The δψ†(~x), δψ(~x) are �eld operators wherethe ground state is excluded, so get of (2.20),(2.21):

⟨δψ†(~x)δψ(~x)

⟩(0)

=∑~m 6=~0

ϕ∗~m(~x)1

exp (β [E~m − µ])− 1ϕ~m(~x) (2.22)

=∞∑l=1

exp (βµl) [(~x,β~l|~x,0)− |ϕ~0(~x)| exp (−βE~0l)] (2.23)

The free energy Γeff with explicit ground state treatment and linearisation in g can be

9

written as:

Γeff =

∫dDxψ∗0(~x)

[h0 − µ+

g

2|ψ0(~x)|2

]ψ0(~x)− 1

βln∏~k 6=~0

∞∑n~k=0

exp(−β(E~k − µ)n~k

)+ 2g

∫dDx |ψ0(~x)|2

∞∑~m 6=~0

ϕ∗~m(~x)ϕ~m(~x)1

exp β (E~m − µ)− 1

+ g

∫dDx

∞∑~n6=~0

∞∑~m 6=~0

ϕ∗~m(~x)ϕ~m(~x)1

exp β (E~m − µ)− 1ϕ∗~n(~x)ϕ~n(~x)

1

exp β (E~n − µ)− 1

(2.24)

2.3 Ncrit for non-interacting photon gas

In the previous section we have derived the e�ective action in �rst order of g. This will bethe starting point for calculating the critical particle number Ncrit for the non-interactingphoton gas, so we set the interacion parameter g = 0. In order to obtain the criticalparticle number we have to take the derivative of Γeff (2.24) with respect to µ, but beforewe do this we can further simplify the non-interacting e�ective potential. Using thegeometric series in n~k and the series expansion for ln(1− x) we end up with:

ln(1− x) = −∞∑l=1

xl

l,

∞∑m=0

qm =1

1− q(2.25)

for all |q| < 1 so we should ensure that this is our case. We know n~k = 1

exp[β(E~k−µ)]−1

is a

non-negative number, thus we must demand that µ < E~k and get

Γ(0)eff =

∫dDxψ∗0(~x) [h0 − µ]ψ0(~x)− 1

βln∏~k 6=~0

∞∑n~k=0

exp(−β(E~k − µ)n~k

)(2.26)

=

∫dDxψ∗0(~x) [h0 − µ]ψ0(~x)− 1

β

∑~k 6=~0

lnD∏i=1

∞∑nki

=0

exp (−β(Eki − µ)nki) (2.27)

where the i is now the spatial component label. We can have di�erent trap frequen-cies ωi and furthermore di�erent quantum numbers ki which �nally results in di�er-ent energies per direction Eki = ~ωi(ki + 1

2). Note that the ground state energy reads

10

E0 = ~2

(∑Di=1 ωi

). Evaluating the geometrical series of the nki in (2.27) yields

Γ(0)eff =

∫dDxψ∗0(~x) [h0 − µ]ψ0(~x)− 1

β

∑~k 6=~0

lnD∏i=1

1

1− exp (−β(Eki − µ)). (2.28)

Inserting the Taylor series of the logarithm (2.25) we get

=

∫dDxψ∗0(~x) [h0 − µ]ψ0(~x)− 1

β

∑~k 6=~0

D∑i=1

∞∑l=1

exp (−β(Eki − µ)) l

l(2.29)

=

∫dDxψ∗0(~x) [h0 − µ]ψ0(~x)− 1

β

D∑i=1

∞∑l=1

∑ki,~k 6=~0

exp (−β(Eki − µ)) l

l. (2.30)

Evaluating the geometrical series of the ki we �nally end up with

=

∫dDxψ∗0(~x) [h0 − µ]ψ0(~x)− 1

β

∞∑l=1

exp (βµl)

l

[exp

(−β ~ω1

2l)

1− exp(−β~ω1l)+ ...

+exp

(−β ~ωD

2l)

1− exp(−β~ωDl)− exp

(−β∑D

i=1 ~ωi2

l

)](2.31)

In the isotropic case in D-dimensions, where all trap frequencies wi are all equal, we �nd

Γ(0)eff =

∫dDxψ∗0(~x) [h0 − µ]ψ0(~x)

− 1

β

∞∑l=1

exp (βµl) exp(−β~ωD

2l)

l

{1

[1− exp(−β~ωl)]D− 1

}. (2.32)

Thus the equation of state reads

N = −∂Γeff

∂µ=

∫dDx |ψ0(~x)|2 +

∞∑l=1

exp (βµl) exp

(−β~ωD

2l

){1

[1− exp(−β~ωl)]D− 1

}(2.33)

Until now we did not take into account spin degrees of freedom so that we have to multiplyour result for N by the factor 2 for the two possible polarisations of photons. And since weare looking for the phase boundary between gas phase and BEC we have to set N0 = 0, µto E0 (this is once again an out�ow from the extremalisation see also (2.41)) which gives

11

the �nal result for the particle number:

N

(µcrit = ~ω

D

2

)= Ncrit = 2

∞∑l=1

{1

[1− exp(−β~ωl)]D− 1

}. (2.34)

Note that this series converges quite fast and is easy to evaluate numerically since wehave treaten the ground state explicitly. Additionally there is no dimensional problem, itconverges in every dimension D. Plugging in the numbers for Troom = 300 K, D = 2 andω = 2π · 4.1 · 1010Hz (see section 1.3) we get

Ncrit = 2∞∑l=1

{1[

1− exp(−1.054571628·10−34·2π·4.1·1010

1.3806504·10−23·3·102l)]2 − 1

}, (2.35)

which yields Ncrit ≈ 78000.If ~ω

kBT� 1) a semiclassical approximation(9) can be performed. The semiclassical result

is also included in the quantum mechanical exact calculation. A Taylor expansion in ~ωkBT

of (2.34) and the inclusion of the ground state yields

Ncrit,semi = 2ζ(D)

(β~ω)D= 2

ζ(2)

(β~ω)2≈ 76500. (2.36)

This is a deviation of about 2 % which can be explained by �nite-size corrections. Al-though they are already included in the quantum mechanical exact solution one has toadd them in the semiclassical case(10) with γ being the Euler-Mascheroni constant.

Ncrit,semi+finites = 2

[ζ(2)

(β~ω)2+− ln(β~ω) + γ − 1

2

(β~ω)

]≈ 78000 (2.37)

Up to few particles the semiclassical correction (2.37) gives almost the same result as thequantum mechanical exact result (2.35). The reason for this is the ratio ~ω

kBT= 0.006 for

the photon BEC with Troom ∝ 102 K and ω ∝ 1010 Hz. Note that for the usual atomicBEC we have TBEC,Atom ∝ 10−7 K and ω ∝ 102 Hz which gives a ratio of ~ω

kBT= 0.048.

Thus the semiclassical error for the photon BEC is supposed to be small and even withthe �rst semiclassical correction one obtains nearly the exact result. If we compare thiswith the experimental result of Ncrit = (6.3± 2.4) · 104,the theoretical prediction is withinthe error bar. In the next section we investigate the impact of interaction on the criticalparticle number.

2.4 Interacting photon gas

Now we go a step further and investigate the interacting Bose gas where g 6= 0. In section2.2 we derived the free energy by extremizing the e�ective action Γeff with respect to the

12

�elds ψ0 , ψ∗0. Afterwards we argued that the interaction parameter g is suppossed to be

small in this way we could obtain the `e�ective linearised` free energy Γeff in (2.24) withoutgetting any trouble with operator valued exponential from the grand canonical partitionfunction. I only refer to the D dimensional isotropic harmonic oscillator for the sake ofsimplicity, but it is also possible to generalize it and treat the anisotropic oscillator withdi�erent ωi. Furthermore one has to pay attention that the propagator (section 3) andsome sums are now independent products. First I want to take care of the expressionsexcluding the ground state and (~k = ~m = ~n 6= ~0).

Γeff(2.23)=

∫dDxψ∗0(~x)

[h0 − µ+

g

2|ψ0(~x)|2

]ψ0(~x)

− 1

β

∞∑l=1

exp (βµl) exp[−β(∑D

i~ωi

2

)l]

l

∑~k=0

exp (−βE~ml)− 1

+ 2g

∫dDx |ψ0(~x)|2

∞∑l=1

exp βµl ·

∞∑~m 6=~0

ϕ∗~m(~x) exp (−βE~ml)ϕ~m(~x)

+ g

∫dDx

∞∑l=1

∞∑j=1

exp βµ(l + j)

×

∞∑~m 6=~0

ϕ∗~m(~x) exp (−βE~ml)ϕ~m(~x)∞∑~n6=~0

ϕ∗~n(~x) exp (−βE~nj)ϕ~n(~x)

(2.38)

Using (2.23) and taking care of the ground state we get

Γeff =

∫dDxψ∗0(~x)

[h0 − µ+

g

2|ψ0(~x)|2

]ψ0(~x)

− 1

β

∞∑l=1

exp (βµl) exp[−β(∑D

i~ωi

2

)l]

l

∑~k=0

exp (−βE~ml)− 1

+ 2g

∫dDx |ψ0(x)|2

∞∑l=1

exp βµl ·[(~x,l~β|~x,0)− |ϕ0(~x)|2 exp (−βE0l)

]+ g

∫dDx

∞∑l=1

∞∑j=1

exp βµ(l + j) ·[{

(~x,l~β|~x,0)− |ϕ0(~x)|2 exp (−βE0l)}

×{

(~x,j~β|~x,0)− |ϕ0(~x)|2 exp (−βE0j)}]. (2.39)

13

2.5 Shift of the chemical potential µ due to

interaction

Due to the interaction there will be a shift of the chemical potential and furthermorethe ground state is shifted. Expanding both the ground state wave function which is ourorder parameter ψ0 and the chemical potential µ in orders of g

ψ0 = ψ(0)0 + ψ

(1)0 + ... µ = µ(0) + µ(1) + ... (2.40)

Furthermore, we still have

δΓeff [ψ∗0,ψ0]

δψ∗0

!= 0 (2.41)

Inserting (2.39) in (2.41) yields

δΓeff [ψ∗0,ψ0]

δψ∗0= {

[h0 − µ+ g |ψ0(x)|2

]+2g

∞∑l=1

exp βµl ·[(x,l~β|x,0)− |ϕ0(x)|2 exp (−βE0l)

]}ψ0

!= 0 (2.42)

If we now plug in our expansion (2.40) for ψ0 and µ and sort it to orders in g we get inzeroth order in g [

h0 − µ(0)]ψ

(0)0

!= 0 → µ(0) = E0 (2.43)

for the condensate phase, where ψ(0)0 6= 0. Correspondingly, we obtain in �rst order in g

[h0 − µ(0)

]ψ

(1)0 = −

{g∣∣∣ψ(0)

0

∣∣∣2 − µ(1)

+2g∞∑l=1

exp(βµ(0)l) ·[(~x,l~β|~x,0)− |ϕ0|2 exp (−βE0l)

]}ψ

(0)0 . (2.44)

Here we have the problem to determine both ψ(1)0 and µ(1) from equation (2.44). Multi-

14

plying (2.44) with ψ∗(0)0 and intergrating yields∫

ψ∗(0)0

[h0 − µ(0)

]ψ

(1)0 = −

∫ψ∗(0)0

{g∣∣∣ψ(0)

0 (x)∣∣∣2 − µ(1)

+2g∞∑l=1

exp(βµ(0)l) ·[(x,l~β|x,0)− |ϕ0(x)|2 exp (−βE0l)

]}ψ

(0)0

(2.45)

The left-hand side of (2.45) vanishes due to (2.43). Therefore, we can deduce

∫dDxψ

∗(0)0 (~x)µ(1)ψ

(0)0 (~x) =

∫dDxψ

∗(0)0 (~x)

{g∣∣∣ψ(0)

0 (~x)∣∣∣2

+2g∞∑l=1

exp(βµ(0)l) ·[(~x,l~β|~x,0)− |ϕ0(~x)|2 exp (−βE0l)

]}ψ

(0)0 (~x).

(2.46)

With the norm for the order parameter∫ ∣∣∣ψ(0)

0 (x)∣∣∣2 = N0, note that ψ

(0)0 =

√N0 · ϕ0

where ϕ0 is ground state eigen function of the harmonic oscillator

ϕ0 =(mphω

π~

)D4

exp[−(mphω

2~

)~x2]

(2.47)

we get

µ(1) =g

N0

∫ψ∗(0)0

{∣∣∣ψ(0)0 (x)

∣∣∣2+2g

∞∑l=1

exp(βµ(0)l) ·[(~x,l~β|~x,0)− |ϕ0(~x)|2 exp (−βE0l)

]}ψ

(0)0 (2.48)

= gN0

∫dDx

∣∣∣ϕ(0)0 (~x)

∣∣∣4+ 2g

∫dDx

∞∑l=1

exp(βµ(0)l) ·[(~x,l~β|~x,0)− |ϕ0(~x)|2 exp (−βE0l)

] ∣∣∣ϕ(0)0 (~x)

∣∣∣2 (2.49)

Now we use the Wick rotated explicit form of the harmonic oscillator propagator(3.19).The Wick rotation just replaces it by τ so it rotates the real time into the imaginary time

15

axes.

µ(1) = gN0

∫dDx

(mphω

π~

)Dexp

[−2(mphω

~

)~x2]

+ 2g∞∑l=1

exp(βµ(0)l)

×

{∫dDx

(mphω

π~

)D ( 1

2 sinh(lβ~ω)

)D2

exp

[−~x2

(mphω

~

)[tanh

(lβ~ω

2

)+ 1

]]− exp

(−β~ωD

2l

)∫dDx

(mphω

π~

)Dexp

[−2(mphω

~

)~x2]}

(2.50)

These integrals are of Gaussian type. With (2.41) and a few substitutions one �nally gets

= g(mphω

2π~

)D2

N0 + 2∞∑l=1

exp

(β~ω

D

2l

)1(

sinh(lβ~ω)[tanh

(lβ~ω

2

)+ 1])D

2

− 1

(2.51)

= g(mphω

2π~

)D2

[N0 + 2

∞∑l=1

(1

(1− exp (−β~ωl))D2

− 1

)](2.52)

Now we get µ up to �rst order in g quantum mechanical excact

µ = µ(0) + µ(1)

= ~ωD

2+ g

(mphω

2π~

)D2

[N0 + 2

∞∑l=1

(1

[1− exp (−β~ωl)]D2

− 1

)]. (2.53)

A semiclassical derivation would lead to the following result:

µ = ~ωD

2+ g

(mphω

2π~

)D2N0 + g

(mph

2π~2β

)D2

2ζ

(D

2

)(2.54)

Here we see once again why it is quite smart to perform the quantum mechanical excactcalculation. In D = 2, which is our case, we would get semiclassically the divergentRiemann Zeta function ζ(1) whereas there is no divergence in (2.53).

16

2.6 Ncrit for interacting Bose gas

In order to obtain now the critical particle number Ncrit for the interacting Bose gas wehave to derive the full e�ective action Γeff (2.39) with respect to µ:

N = −∂Γeff

∂µ=

∫dDx |ψ0(~x)|2

+∞∑l=1

exp (βµl) exp

[−β~ωD

2l

]∑~k=0

exp(−βE~kl

)− 1

− 2gβ

∫dDx |ψ0(~x)|2

∞∑l=1

l exp(βµl) ·[(~x,l~β|~x,0)− |ϕ0(~x)|2 exp (−βE0l)

]− gβ

∫dDx

∞∑l=1

∞∑j=1

(l + j) exp[βµ(l + j)] ·[{

(~x,l~β|~x,0)− |ϕ0(~x)|2 exp (−βE0l)}·{

(~x,j~β|~x,0)− |ϕ0(~x)|2 exp (−βE0j)}]. (2.55)

Taking into account µ = µ(0) + µ(1) and linearising in g yields

N = N0 +∞∑l=1

exp(βµ(0)l

)(1 + gβlµ(1)) exp

(−β~ωD

2l

)[1

(1− exp(−β~ωl))D− 1

]

− gβ2N0

(mphω

2π~

)D2

∞∑l=1

l ·

exp(βµ(0)l

) 1(2 sinh

(lβ~ω

2

) [sinh

(lβ~ω

2

)+ cosh

(lβ~ω

2

)])D2

− 1

− gβ

(mphω

2π~

)D2

∞∑j=1

∞∑l=1

(l + j) exp(βµ(0)(l + j)

) 1(8 sinh

(lβ~ω

2

)sinh

(jβ~ω

2

)sinh

((j+l)β~ω

2

))D2

− exp (−βE0l)1(

2 sinh(jβ~ω

2

) [sinh

(jβ~ω

2

)+ cosh

(jβ~ω

2

)])D2

− exp (−βE0j)1(

2 sinh(lβ~ω

2

) [sinh

(lβ~ω

2

)+ cosh

(lβ~ω

2

)])D2

+ exp (−βE0(l + j))} . (2.56)

17

With µ(0) = E0 and µ(1) from (2.53) and some hyperbolic addition theorems we get

N = N0 + 2∞∑l=1

(1 + gβlµ(1))

[1

(1− exp (−lβ~ω))D− 1

]

− gβ(mphω

2π~

)D2

{N0

∞∑l=1

l

[1

(1− exp (−lβ~ω))D2

− 1

]

+2∞∑j=1

∞∑l=1

(l + j)

[1

{(1− exp (−lβ~ω)) (1− exp (−jβ~ω)) (1− exp (−(l + j)β~ω))}D2

− 1

(1− exp (lβ~ω))D2

− 1

(1− exp (lβ~ω))D2

+ 1

]}. (2.57)

Now in principle we could determine N0(N,T,ω,g) from (2.57), but since we are interestedin Ncrit we set N0 = 0 and get

Ncrit(T,ω,g) =

2∞∑l=1

(1 + gβl

(mphω

2π~

)D2

[2∞∑j=1

(1

(1− exp (−β~ωj))D2

− 1

)])[1

(1− exp (−lβ~ω))D− 1

]

− 2gβ(mphω

2π~

)D2

{∞∑j=1

∞∑l=1

(l + j)

[− 1

(1− exp (lβ~ω))D2

− 1

(1− exp (lβ~ω))D2

+ 1

1

{(1− exp (−lβ~ω)) (1− exp (−jβ~ω)) (1− exp (−(l + j)β~ω))}D2

]}(2.58)

For known interaction parameter g one can determine Ncrit. Checking the literature(11)

one �nds g = ~2mph

g where g is extracted from a Gross-Pitaevski equation �t of the exper-

imental data(4). The dimensionless interaction parameter g turns out to be g = 7 · 10−4

which is about one magnitude smaller than the usual atomic BEC interaction parameter.

18

In terms of the dimensionless interaction parameter (2.58) reads

Ncrit(T,ω,g) =

2∞∑l=1

[1

(1− exp (−lβ~ω))2 − 1

]

+ 2g

(β~ω2π

){ ∞∑j=1

∞∑l=1

2j

(1

(1− exp (−β~ωj))− 1

)[1

(1− exp (−lβ~ω))2 − 1

]−(l + j)

[1

{(1− exp (−lβ~ω)) (1− exp (−jβ~ω)) (1− exp (−(l + j)β~ω))}

− 1

(1− exp (lβ~ω))− 1

(1− exp (lβ~ω))+ 1

]}. (2.59)

These double sums can be calculated numerically. However it turns out that the dou-ble sum converges very slowly. The numerical evaluation by Mathematica may containnumerical uncertainties. Nevertheless the result is:

Ncrit(T,ω,g) = 78030 + 511− 295︸ ︷︷ ︸interaction contribution

≈ 78250 (2.60)

Thus we conclude that, due to the small interaction parameter, the shift of Ncrit is verysmall. But note all considerations done here are also valid for normal atomic BEC.

19

3 Appendix A: Propagator

Starting with the canonical commutation relation,

1

i~[x,p] = I (3.1)

and the harmonic oscillator Hamiltonian in 1-D with ω being the oscillator frequency

H =p2

2m+mω2x2

2(3.2)

One easily obtains the equations of motion in the Heisenberg picture

dp

dt=

1

i~[p,H] = −mω2x (3.3)

dx

dt=

1

i~[x,H] =

p

m(3.4)

Taking the derivative of (3.4) we get with help of (3.3)

d2x

dt2= −ω2x (3.5)

This equation no longer evolves operator products so we can solve it directly

x = x0 cos[ω(t− t0)] +p0

mωsin[ω(t− t0)]. (3.6)

The propagator is de�ned as

U(x,t;x0,t0) = (x,t|x0,t0) = 〈x,t|x0,t0〉 . (3.7)

Now we calculate the complex conjugate as follows

(A∂x0 +Bx0)U∗ = xU∗ (3.8)

with the abreviations A = ~imω

sin[ω(t− t0)] and B = cos[ω(t− t0)].Solving this di�erential equation for x0 and U∗ yields

1

U∗∂U∗

∂x0

=x−Bx0

A(3.9)

20

This equation (3.9) is solved by separation of variables. Up to a few integration constantsone obtains

U∗ = N∗(t,t0) exp

{− 1

A

[B

2x2

0 − xx0 + f(x)

]}. (3.10)

In the last step I put the x dependent integration constant f(x) into the exponential, suchthat N∗ is independent from x and x0. We are left with the yet unknown functions Nand f . Taking into account the group property and the unitarity of the propagator one�nds,

U∗(x,t;x0,t0) = U(x0,t0;x,t) (3.11)

N∗(t,t0) exp

{− 1

A

[B

2x2

0 − xx0 + f(x)

]}= N(t,t0) exp

{− 1

A

[B

2x2 − xx0 + f(x0)

]}(3.12)

Since N is independent of x and x0 the exponentials should be the same

f(x0)− B

2x2

0 = f(x)− B

2x2 (3.13)

as this holds for all x and x0, we conclude f(x)− B

2x2 = g(t,t0) (3.14)

This is already parametrized with N(t,t0) so we can set g(t,t0) = 0. The propagator hasthen the following form,

U(x,t;x0,t0) = N(t,t0) exp

{imω [(x2 + x2

0) cos[ω(t− t0)]− 2xx0]

2~ sin[ω(t− t0)]

}. (3.15)

The completeness relation determines N∫dx0U(x1,t;x0,t0)U∗(x2,t;x0,t0) =

∫dx0 〈x1,t|x0,t0〉 〈x0,t0|x2,t〉 = 〈x1,t|x2,t〉 = δ(x1 − x2)

(3.16)

Remembering the Fourier identity∫dk exp(ikz) = 2πδ(z) (3.17)

we get from (3.15) and (3.16)∫dx0U(x1,t;x0,t0)U∗(x2,t;x0,t0) = |N(t,t0)|2 2π~ sin[ω(t− t0)]

mωδ(x1 − x2). (3.18)

21

This �nally yields the form for the 1-dimensional propagator for the harmonic oscillator

U(x,t;x0,t0) =

√mω

2πi~ sin[ω(t− t0)]exp

{imω [(x2 + x2

0) cos[ω(t− t0)]− 2xx0]

2~ sin[ω(t− t0)]

}.

(3.19)

This is all shown for one dimension. If one has a D dimensional oscillator one has tomultiply the above result for each spatial dimension. Furthermore note that I used theWick-rotated version (it→ τ in my derivation)

22

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23