Photon-BEC in an Optical Microcavity

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 in the energetic ground state level below a critical temperature Tcrit. This phenomenom has been demonstrated in several dierent physical systems as, for instance, dilute ultracold Bose 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 radiation is 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 temperature 3 4 T is lowered corresponding to a vanishing chemical potential. Recent experiments( )( ) with a dye-lled optical micro resonator, performed by the group of Martin Weitz at the University of Bonn, achieved thermalization of photons in a number-conserving way. The curvature of the micro resonator provides two important ingredients which are prereq- uisites for BEC: a conning potential and a non-vanishing eective photon mass. This experiment gives new opportunities for creating coherent light. In contrast to the fty-year old laser, which operates far from thermal equilibrium, the photon BEC gains coherence by an equilibrium phase transition. Before this experiment one had several verications that massive particles behave like waves for example the interference of fullerene(5). But now the quantized electromagnetic waves in the cavity are given an eective mass, so here it is the other way around waves behave like massive particles. In the next sections I report about the details of the experimental setup and then I investigate theoretically those `particles` of light (6). First I give a short overview of the experimental setup and some important experimental facts. In the second part I am focusing on the calculation of the eective action, then I deduce the critical 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 eective 2-dimensional Bose gas with a given temperature. Thus we can calculate all physical observables quantum mechanical exact and do not get any divergencies from the semiclassical approach and even nite size corrections are already included.

1.2 Experimental setup

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

2 Figure 1.1: a. Schematic spectrum of cavity modes with absorption coecient α(ν) and uorescence strength f(ν) b. Dispersion relation of photons in the cavity (solid line) with xed longitudinal mode (q = 7) and the free photon dispersion (dashed line) c. Schematic experimental 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 grow. Thus we conclude that the energy E = c|~k| of the photons to maintain dndye ~ in this region of the cavity is higher than in the center. The two mirrors are spaced D = 1.56 µm away from each other which is exactly 3.5 optical wavelengths. Indeed , if we look at gure 1.1a one can clearly see at quantum number q = 7, which corresponds a wave length λ = 585 nm, that there is a great overlap between the absorption coecient α(ν) and the uorescence strength f(ν) of the dye. This modies spontaneous emissions such that the emission of longitudinal mode with quantum number q = 7 dominates over all other emission processes. Due to the extremely short distances of the mirrors there 14 is an eective longitudinal low frequency cut-o ωcut = ckz = 2π · 5.1 · 10 Hz where 2π λ kz = and D = 7 · , because no larger wavelength ts into the microcavity and ndyeD 2 of course the speed of light in vacuum c is modied when entering the dye solution by the refraction index . Furthermore the energy of the longitudinal mode ndye = 1.33 E = ~ωcut is far above thermal energy at room-temperature . kz 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 in the lowest available energy level inside the cavity are suppressed by the factor ∝ ~ωcut exp[−80]. The cavity photon number nph is almost not altered by the temperature T of the surrounding dye solution. This means that the thermalization process conserves the 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 can be seen from the gure 1.a the re-emission in the longitudinal mode kz dominates over all other modes. There is the largest overlap between emission coecient and uorescence strength. This means that most of the (q = 7)-photons absorbed by the dye will be re-emitted inside the cavity whereas for all other longitudinal modes there is a disbalance between absorption and emission. The kz-mode is frozen out and the kr-modes can thermalise 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) transmitted through one cavity mirror, as measured with a spectrometer for dif- ferent pump powers b. Images of the spatial radiation distribution below criticality (upper panel) and above criticality (lower panel).

The BEC of the photons inside the dye-lled cavity has been proven experimentally by investigating both the spatial and temporal coherence. One has measured spectral distributions (see gure 1.2a) which show a classical thermal distribution at room temperature and an increased intensity for λ = 585 nm. The latter corresponds to the frozen longitudinal wave vector which yields 2π . So far this is a proof for BEC in the frequency kz λ = 3.5·k domain. The experimental setup also allowsz to check the spatial domain. Hence another evidence 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 but above criticality one nds a rather sharp yellow spot sitting in the center of the cavity in the minimum of the harmonic potential, which is induced by the curvature of the cavity mirrors.

1.4 Hamiltonian for photons in an optical micro cavity

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

~ p 2 2 (1.1) E = ~c|k| = ~c kz + kr ,

4 2π where kz = is the longitudinal wave vector component with the mirror distance ndyed(r,D) D depending on the xed quantum number q = 7. In general the longitudinal wave vector component depends on the distance d(r) between the two curved mirrors with curvature . From a few geometrical considerations for this biconvex cavity one gets that R = 1 m √ d(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 temperature thermalised tranversal modes which are whereas ∝ (kBT )/(~c) 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 out the so 2 2 yields, kz(r) x = (kr) /kz

2 ~ p 2 2 (kr) (1.2) E = ~c|k| = ~c kz + kr ≈ ~ckz(r) + ~c . 2kz(r) Furthermore we have r R in our setup, i.e. r ∝ µm R = 1 m and the Taylor expansion in kz at kz(r = 0) gives

2π 1 1 2 3 (1.3) kz ≈ + 2 r + O(r ) . ndyed(r) D RD Inserting (1.3) into (1.2) then yields

(~k )2 ( k )2 1 r 2 ~ r 2 2 (1.4) E ≈ ~ckz(r) + ~c ≈ mphc + + mphω r 2kz(r) 2mph 2 with eective photon mass ~ωcut ~kz(0) −36 which is about the mph = c2 = c = 6.7 · 10 kg magnitude 1010 smaller than the usual atomic masses. Furthermore this approximation gives an eective trap frequency √c 10 which about a factor 8 ω = DR = 2π · 4.1 · 10 Hz 10 higher than the usual atomic BEC trap frequencies.2 Thus the system in an optical microcavity is equivalent to a non-relativistic 2-dimensional Bose gas. The dispersion relation is quadratic for small transversal wave numbers as is illustrated in the dispersion relation in gure 1.1 b. Note that for kr = 0 we still have an `eective rest mass` intimidly related to the frozen kz-mode, which appears in zeroth order 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 temperature

Tcrit(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 number for the onset of Bose-Einstein condensation Ncrit(T ). The comfort of this situation is that we do not have to extract the temperature by inverting the equation of state for the particle number.

2.2 Eective action

From the dispersion (1.4) we read o that the system is equivalent to an harmonic oscillator in 2 dimensions. In the following I give a general treatment for D dimensional isotropic harmonic oscillators:

2 2 ~p (~kr) 1 2 2 2 h0 = + V (~x) = + mphω ~x + mphc . (2.1) 2mph 2mph 2 then the full Hamiltonian in second quantization with interaction looks like this

Z 2 Z g Hˆ = dDxψˆ†(~x) − ~ 4 − µ + V (~x) ψˆ(~x) + ψˆ†(~x)ψˆ†(~x)ψˆ(~x)ψˆ(~x) (2.2) 2m 2 where the canonical commutation relations read h i ψˆ†(~x),ψˆ(~x0) = δ(~x − ~x0) (2.3) h i h i ψˆ(~x),ψˆ(~x0) = ψˆ†(~x),ψˆ†(~x0) = 0 (2.4)

Now I am using the Bogoliubov description for the eld operators ψˆ†(~x), ψˆ(~x) because in our case we can consider the ground state ∗ to be macroscopically occupied. To this ψ0, ψ0

6 end we set D E D E ∗ ˆ† ˆ (2.5) ψ0 = ψ (~x) , ψ0 = ψ(~x) , h i where the expectation value is dened as 1 −βHˆ with the partition func- < • >= Z tr e • h −βHˆ i tion Z = tr e and β = 1/(kBT ). The decompostion of the eld operators looks as follows: D E ˆ† ∗ ˆ† ˆ† ˆ† ˆ† (2.6) ψ (~x) = ψ0(~x) + δψ (~x) , δψ (~x) = ψ (~x) − ψ (~x) ˆ ˆ ˆ ˆ D ˆ E 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 δψˆ†(~x), δψˆ(~x) vanish by construction. Plugging (2.6) and (2.7) into (2.2) yields:

ˆ ˆ ˆ H = H0 + H1 (2.8) where the free Hamiltonian reads Z ˆ ∗ ˆ† H0 = ψ0(~x)[h0(~x) − µ] ψ0(~x) + δψ (~x)[h0(~x) − µ] ψ0(~x) ∗ ˆ ˆ† ˆ (2.9) + ψ0(~x)[h0(~x) − µ] δψ(~x) + δψ (~x)[h0(~x) − µ] δψ(~x) and the interaction-part of the Hamiltonian is given by g Z Hˆ = |ψ (~x)|4 + 2 |ψ (~x)|2 ψ∗(~x)δψˆ(~x) + (ψ∗(~x))2δψˆ†(~x)δψˆ†(~x) 1 2 0 0 0 0 2 ˆ† 2 ˆ† ˆ + 2 |ψ0(~x)| ψ0(~x)δψ (~x) + 4 |ψ0(~x)| δψ (~x)δψ(~x) ∗ ˆ† ˆ ˆ 2 ˆ† ˆ† + 2ψ0(~x)δψ (~x)δψ(~x)δψ(~x) + (ψ0(~x)) δψ (~x)δψ (~x) ˆ† ˆ† ˆ ˆ† ˆ† ˆ ˆ + 2ψ0(~x)δψ (~x)δψ (~x)δψ(~x) + δψ (~x)δψ (~x)δψ(~x)δψ(~x) (2.10) h i The partition function is −βHˆ and the eective action is ∗ 1 . Z = tr e Γeﬀ [ψ0,ψ0] = − β ln(Z) The expressions (2.9),(2.10) are pretty long and will become extremely complicated once you insert this in the exponential for the partition function. The problem here is we have terms which are not only quadratic or quartic in the uctuation operators δψˆ ,δψˆ† 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 solution with a pertubative approach in g. But before we do this we already know from quantum statistics the real physical elds ∗ extremize the eective action which then is ψ0 , ψ0 Γeﬀ identical to the free energy(8). In performing this extremalisation we get rst

7 " # δΓ[ψ∗,ψ ] δ 1 1 δHˆ 0 0 −βHˆ (2.11) ∗ = − ∗ ln(Z) = tr e ∗ δψ0 δψ0 β Z δψ0 D E g n D E = h[h − µ] ψ i + [h − µ] δψˆ + 2 |ψ |2 ψ + 2 |ψ |2 δψˆ 0 0 0 2 0 0 0 D E D E D E D Eo ∗ ˆ ˆ 2 ˆ† ˆ† ˆ ˆ† ˆ ˆ ! (2.12) + 2ψ0δψδψ + 2(ψ0) δψ + 4ψ0δψ δψ + 2δψ δψδψ = 0.

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

n 2 ∗ D ˆ Ê D ˆ† Ê D ˆ† ˆ Êo (h0 − µ)ψ0 = −g |ψ0| ψ0 + ψ0 δψδψ + 2ψ0 δψ δψ + δψ δψδψ =: gCδψˆ† (2.13) and, of course, an analogous expression for the complex conjugate. We are now able to ˆ ˆ ˆ† replace terms of H0 which are linear in δψ ,δψ by terms which are linear in g. Taking this into account we nd the following eective potential:

1 Z g g Γ = − ln tr exp −β dDxψ∗(h − µ)ψ + δψˆ† · C + C · δψˆ + δψˆ†(h − µ)δψˆ + Hˆ eﬀ β 0 0 2 δψˆ† 2 δψˆ 0 1 (2.14)

Linearisation in g of the trace yields . 1 h i g g Γ = − ln tr exp −βHˆ 1 + δψˆ† · C + C · δψˆ + Hˆ (2.15) eﬀ β 0 2 δψˆ† 2 δψˆ 1 with the Hamiltonian Z ˆ D ∗ ˆ† ˆ (2.16) H0 = d xψ0(h0 − µ)ψ + δψ (h0 − µ)δψ.

The trace is performed over the Fock-base . Since we have already explic- h{nk}|k∈ \0 itly treaten the ground state, the uctuation operatorN has to exclude the ground state according to ˆ P∞ . The unperturbed Hamiltonian ˆ (2.16) is now δψ(~x) = k6=0 ϕk(x)ˆak H0 diagonalised in the Fock base. Since we perform the trace we only need the diagonal elements of the perturbed part which means:

1 ˆ D E(0) tr e−βH0 O(δψˆ†,δψ,ˆ (δψˆ)3) = O(δψˆ†,δψ,ˆ (δψˆ)3) = 0 Z0

(0) ˆ with h•i = 1 tr e−βH0 • . Z0 Thus the eective action (2.15) reduces to

8 1 Z Γ = − ln tr exp −β dDxψ∗(h − µ)ψ + δψˆ†(h − µ)δψˆ (2.17) eﬀ β 0 0 0 βg Z × 1 − dDx |ψ (~x)|4 + 4 |ψ (~x)|2 δψˆ†(~x)δψˆ(~x) + δψˆ†(~x)δψˆ†(~x)δψˆ(~x)δψˆ(~x) 2 0 0 which nally leads to

1 1 βg Z D E(0) Γ = − ln Z − ln 1 − dDx |ψ (~x)|4 + 4 |ψ (~x)|2 δψˆ†(~x)δψˆ(~x) (2.18) eﬀ β 0 β 2 0 0 D E(0) + δψˆ†(~x)δψˆ†(~x)δψˆ(~x)δψˆ(~x) .

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

D E(0) D E(0)2 δψˆ†(~x)δψˆ†(~x)δψˆ(~x)δψˆ(~x) = 2 δψˆ†(~x)δψˆ(~x) . (2.19)

The two eld operator expectation value is related to the Green function and the propagator in the following way:

D E(0) 1 ˆ† ˆ (0) X ∗ (2.20) ψ (~x)ψ(~x) = G (~x; ~x) = ϕ~m(~x) ϕ~m(~x) exp (β [E~m − µ]) − 1 ~m ∞ X = exp (βµl)(~x,β~l|~x,0) (2.21) l=1 This is a general result from quantum statistics, particularly in our case we have to take explicit care of the mean eld decomposition. The δψˆ†(~x), δψˆ(~x) are eld operators where the ground state is excluded, so get of (2.20),(2.21):

D E(0) 1 ˆ† ˆ X ∗ (2.22) δψ (~x)δψ(~x) = ϕ~m(~x) ϕ~m(~x) exp (β [E~m − µ]) − 1 ~m6=~0 ∞ X (2.23) = exp (βµl) [(~x,β~l|~x,0) − |ϕ~0(~x)| exp (−βE~0l)] l=1

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

9 written as: Z ∞ h g 2i 1 Y X Γ = dDxψ∗(~x) h − µ + |ψ (~x)| ψ (~x) − ln exp −β(E − µ)n eﬀ 0 0 2 0 0 β ~k ~k ~k6=~0 n~k=0 Z ∞ D 2 X ∗ 1 + 2g d x |ψ0(~x)| ϕ~m(~x)ϕ~m(~x) exp β (E~m − µ) − 1 ~m6=~0 Z ∞ ∞ D X X ∗ 1 ∗ 1 + g d x ϕ~m(~x)ϕ~m(~x) ϕ~n(~x)ϕ~n(~x) exp β (E~m − µ) − 1 exp β (E~n − µ) − 1 ~n6=~0 ~m6=~0 (2.24)

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

∞ ∞ X xl X 1 ln(1 − x) = − , qm = (2.25) l 1 − q l=1 m=0 for all so we should ensure that this is our case. We know 1 is a |q| < 1 n~k = exp[β(E~ −µ)]−1 non-negative number, thus we must demand that and get k µ < E~k

Z ∞ (0) 1 Y X Γ = dDxψ∗(~x)[h − µ] ψ (~x) − ln exp −β(E − µ)n (2.26) eﬀ 0 0 0 β ~k ~k ~k6=~0 n~k=0 D ∞ Z 1 X Y X = dDxψ∗(~x)[h − µ] ψ (~x) − ln exp (−β(E − µ)n ) (2.27) 0 0 0 β ki ki ~k6=~0 i=1 nki =0 where the i is now the spatial component label. We can have dierent trap frequencies ωi and furthermore dierent quantum numbers ki which nally results in dierent energies per direction 1 . Note that the ground state energy reads Eki = ~ωi(ki + 2 )

10 ~ PD . Evaluating the geometrical series of the in (2.27) yields E0 = 2 i=1 ωi nki

Z 1 D 1 (0) D ∗ X Y (2.28) Γeﬀ = d xψ0(~x)[h0 − µ] ψ0(~x) − ln . β 1 − exp (−β(Eki − µ)) ~k6=~0 i=1 Inserting the Taylor series of the logarithm (2.25) we get

Z D ∞ 1 X X X exp (−β(Ek − µ)) l = dDxψ∗(~x)[h − µ] ψ (~x) − i (2.29) 0 0 0 β l ~k6=~0 i=1 l=1 Z D ∞ 1 X X X exp (−β(Ek − µ)) l = dDxψ∗(~x)[h − µ] ψ (~x) − i . (2.30) 0 0 0 β l i=1 l=1 ki,~k6=~0

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

∞ " Z ~ω1 D ∗ 1 X exp (βµl) exp −β 2 l = d xψ0(~x)[h0 − µ] ψ0(~x) − + ... β l 1 − exp(−β ω1l) l=1 ~ !# exp −β ~ωD l PD ω + 2 − exp −β i=1 ~ i l (2.31) 1 − exp(−β~ωDl) 2

In the isotropic case in D-dimensions, where all trap frequencies wi are all equal, we nd Z (0) D ∗ Γeﬀ = d xψ0(~x)[h0 − µ] ψ0(~x)

∞ D 1 X exp (βµl) exp −β~ω l 1 − 2 − 1 . (2.32) β l D l=1 [1 − exp(−β~ωl)] Thus the equation of state reads

Z ∞ ∂Γeﬀ 2 X D 1 N = − = dDx |ψ (~x)| + exp (βµl) exp −β ω l − 1 ∂µ 0 ~ 2 D l=1 [1 − exp(−β~ωl)] (2.33)

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

11 the nal result for the particle number:

∞ D X 1 N µ = ω = N = 2 − 1 . (2.34) crit ~ 2 crit D l=1 [1 − exp(−β~ωl)] Note that this series converges quite fast and is easy to evaluate numerically since we have treaten the ground state explicitly. Additionally there is no dimensional problem, it converges in every dimension D. Plugging in the numbers for Troom = 300 K, D = 2 and ω = 2π · 4.1 · 1010Hz (see section 1.3) we get

∞ ( ) X 1 Ncrit = 2 − 1 , (2.35) 1.054571628·10−34·2π·4.1·1010 2 l=1 1 − exp(− 1.3806504·10−23·3·102 l) which yields N ≈ 78000. crit 9 If ~ω 1) a semiclassical approximation( ) can be performed. The semiclassical result kB T is also included in the quantum mechanical exact calculation. A Taylor expansion in ~ω k T of (2.34) and the inclusion of the ground state yields B

ζ(D) ζ(2) Ncrit,semi = 2 = 2 ≈ 76500. (2.36) (β~ω)D (β~ω)2 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 to 10 add them in the semiclassical case( ) with γ being the Euler-Mascheroni constant.

1 ζ(2) − ln(β~ω) + γ − 2 Ncrit,semi+ﬁnites = 2 + ≈ 78000 (2.37) (β~ω)2 (β~ω) Up to few particles the semiclassical correction (2.37) gives almost the same result as the quantum mechanical exact result (2.35). The reason for this is the ratio ~ω = 0.006 for kBT 2 10 the photon BEC with Troom ∝ 10 K and ω ∝ 10 Hz. Note that for the usual atomic BEC we have −7 and 2 which gives a ratio of ~ω . TBEC,Atom ∝ 10 K ω ∝ 10 Hz k T = 0.048 Thus the semiclassical error for the photon BEC is supposed to be small andB even with the rst semiclassical correction one obtains nearly the exact result. If we compare this 4 with the experimental result of Ncrit = (6.3 ± 2.4) · 10 ,the theoretical prediction is within the error bar. In the next section we investigate the impact of interaction on the critical particle number.

2.4 Interacting photon gas

Now we go a step further and investigate the interacting Bose gas where g 6= 0. In section 2.2 we derived the free energy by extremizing the eective action Γeﬀ with respect to the

12 elds ∗. Afterwards we argued that the interaction parameter is suppossed to be ψ0 , ψ0 g small in this way we could obtain the `eective linearised` free energy Γeﬀ in (2.24) without getting any trouble with operator valued exponential from the grand canonical partition function. I only refer to the D dimensional isotropic harmonic oscillator for the sake of simplicity, but it is also possible to generalize it and treat the anisotropic oscillator with dierent ωi. Furthermore one has to pay attention that the propagator (section 3) and some sums are now independent products. First I want to take care of the expressions excluding the ground state and (~k = ~m = ~n 6= ~0).

(2.23) Z h g i Γ = dDxψ∗(~x) h − µ + |ψ (~x)|2 ψ (~x) eﬀ 0 0 2 0 0 h i   ∞ exp (βµl) exp −β PD ~ωi l 1 X i 2 X − exp (−βE l) − 1 β l  ~m  l=1 ~k=0   Z ∞ ∞ D 2 X X ∗ + 2g d x |ψ0(~x)| exp βµl ·  ϕ~m(~x) exp (−βE~ml) ϕ~m(~x) l=1 ~m6=~0 ∞ ∞ Z X X + g dDx exp βµ(l + j) l=1 j=1

 ∞ ∞  X ∗ X ∗ (2.38) ×  ϕ~m(~x) exp (−βE~ml) ϕ~m(~x) ϕ~n(~x) exp (−βE~nj) ϕ~n(~x) ~m6=~0 ~n6=~0 Using (2.23) and taking care of the ground state we get Z h g i Γ = dDxψ∗(~x) h − µ + |ψ (~x)|2 ψ (~x) eﬀ 0 0 2 0 0 h i   ∞ exp (βµl) exp −β PD ~ωi l 1 X i 2 X − exp (−βE l) − 1 β l  ~m  l=1 ~k=0 Z ∞ D 2 X 2 + 2g d x |ψ0(x)| exp βµl · (~x,l~β|~x,0) − |ϕ0(~x)| exp (−βE0l) l=1 Z ∞ ∞ D X X 2 + g d x exp βµ(l + j) · (~x,l~β|~x,0) − |ϕ0(~x)| exp (−βE0l) l=1 j=1 2 (2.39) × (~x,j~β|~x,0) − |ϕ0(~x)| exp (−βE0j) .

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 furthermore the ground state is shifted. Expanding both the ground state wave function which is our order parameter ψ0 and the chemical potential µ in orders of g

(0) (1) (0) (1) (2.40) ψ0 = ψ0 + ψ0 + ... µ = µ + µ + ... Furthermore, we still have

∗ δΓeﬀ [ψ0,ψ0] ! (2.41) ∗ = 0 δψ0 Inserting (2.39) in (2.41) yields

∗ δΓeﬀ [ψ0,ψ0] 2 ∗ = { h0 − µ + g |ψ0(x)| δψ0 ∞ ) X 2 ! (2.42) +2g exp βµl · (x,l~β|x,0) − |ϕ0(x)| exp (−βE0l) ψ0 = 0 l=1

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

(0) (0) ! (0) (2.43) h0 − µ ψ0 = 0 → µ = E0 for the condensate phase, where (0) . Correspondingly, we obtain in rst order in ψ0 6= 0 g

2 (0) (1) (0) (1) h0 − µ ψ0 = − g ψ0 − µ ∞ ) X (0) 2 (0) (2.44) +2g exp(βµ l) · (~x,l~β|~x,0) − |ϕ0| exp (−βE0l) ψ0 . l=1

Here we have the problem to determine both (1) and (1) from equation (2.44). Multi- ψ0 µ

14 plying (2.44) with ∗(0) and intergrating yields ψ0

Z Z 2 ∗(0) (0) (1) ∗(0) (0) (1) ψ0 h0 − µ ψ0 = − ψ0 g ψ0 (x) − µ ∞ ) X (0) 2 (0) +2g exp(βµ l) · (x,l~β|x,0) − |ϕ0(x)| exp (−βE0l) ψ0 l=1 (2.45)

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

Z Z 2 D ∗(0) (1) (0) D ∗(0) (0) d xψ0 (~x)µ ψ0 (~x) = d xψ0 (~x) g ψ0 (~x) ∞ ) X (0) 2 (0) +2g exp(βµ l) · (~x,l~β|~x,0) − |ϕ0(~x)| exp (−βE0l) ψ0 (~x). l=1 (2.46)

2 √ With the norm for the order parameter R (0) , note that (0) ψ0 (x) = N0 ψ0 = N0 · ϕ0 where ϕ0 is ground state eigen function of the harmonic oscillator

D mphω 4 h mphω 2i ϕ0 = exp − ~x (2.47) π~ 2~ we get

Z 2 (1) g ∗(0) (0) µ = ψ0 ψ0 (x) N0 ∞ ) X (0) 2 (0) (2.48) +2g exp(βµ l) · (~x,l~β|~x,0) − |ϕ0(~x)| exp (−βE0l) ψ0 l=1 Z 4 D (0) = gN0 d x ϕ0 (~x) ∞ Z 2 D X (0) 2 (0) (2.49) + 2g d x exp(βµ l) · (~x,l~β|~x,0) − |ϕ0(~x)| exp (−βE0l) ϕ0 (~x) l=1 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.

Z D (1) D mphω h mphω 2i µ = gN0 d x exp −2 ~x π~ ~ ∞ X + 2g exp(βµ(0)l) l=1 ( D Z m ω D 1 2 m ω lβ ω × dDx ph exp −~x2 ph tanh ~ + 1 π~ 2 sinh(lβ~ω) ~ 2 Z D D D mphω h mphω 2i − exp −β~ω l d x exp −2 ~x (2.50) 2 π~ ~ These integrals are of Gaussian type. With (2.41) and a few substitutions one nally gets    D ∞ mphω 2 X D 1 = g N0 + 2 exp β~ω l D − 1 2π~ 2 lβ~ω 2 l=1 sinh(lβ~ω) tanh 2 + 1 (2.51)

D " ∞ !# 2 mphω X 1 (2.52) = g N0 + 2 D − 1 2π 2 ~ l=1 (1 − exp (−β~ωl)) Now we get µ up to rst order in g quantum mechanical excact

µ = µ(0) + µ(1)

D " ∞ !# 2 D mphω X 1 (2.53) = ~ω + g N0 + 2 D − 1 . 2 2π 2 ~ l=1 [1 − exp (−β~ωl)] A semiclassical derivation would lead to the following result:

D D 2 2 D mphω mph D (2.54) µ = ~ω + g N0 + g 2ζ 2 2π~ 2π~2β 2 Here we see once again why it is quite smart to perform the quantum mechanical excact calculation. In D = 2, which is our case, we would get semiclassically the divergent Riemann 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 we have to derive the full eective action Γeﬀ (2.39) with respect to µ: ∂Γ Z N = − eﬀ = dDx |ψ (~x)|2 ∂µ 0 ∞   X D X + exp (βµl) exp −β ω l exp −βE l − 1 ~ 2  ~k  l=1 ~k=0 Z ∞ D 2 X 2 − 2gβ d x |ψ0(~x)| l exp(βµl) · (~x,l~β|~x,0) − |ϕ0(~x)| exp (−βE0l) l=1 Z ∞ ∞ D X X 2 − gβ d x (l + j) exp[βµ(l + j)] · (~x,l~β|~x,0) − |ϕ0(~x)| exp (−βE0l) · l=1 j=1 2 (2.55) (~x,j~β|~x,0) − |ϕ0(~x)| exp (−βE0j) .

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

∞ X D 1 N = N + exp βµ(0)l (1 + gβlµ(1)) exp −β ω l − 1 0 ~ 2 D l=1 (1 − exp(−β~ωl))   D ∞ mphω 2 X (0) 1 − gβ2N0 l · exp βµ l D − 1 2π~ lβ~ω lβ~ω lβ~ω 2 l=1 2 sinh 2 sinh 2 + cosh 2  D ∞ ∞  mphω 2 X X (0)  1 − gβ (l + j) exp βµ (l + j) D 2π 2 ~ j=1 l=1  lβ~ω jβ~ω (j+l)β~ω  8 sinh 2 sinh 2 sinh 2 1 − exp (−βE0l) D jβ~ω jβ~ω jβ~ω 2 2 sinh 2 sinh 2 + cosh 2 1 − exp (−βE0j) D lβ~ω lβ~ω lβ~ω 2 2 sinh 2 sinh 2 + cosh 2

+ exp (−βE0(l + j))} . (2.56)

17 (0) (1) With µ = E0 and µ from (2.53) and some hyperbolic addition theorems we get ∞ X (1) 1 N = N0 + 2 (1 + gβlµ ) D − 1 l=1 (1 − exp (−lβ~ω)) D ( ∞ " # mphω 2 X 1 − gβ N0 l D − 1 2π 2 ~ l=1 (1 − exp (−lβ~ω)) ∞ ∞ " X X 1 +2 (l + j) D 2 j=1 l=1 {(1 − exp (−lβ~ω)) (1 − exp (−jβ~ω)) (1 − exp (−(l + j)β~ω))} #) 1 1 (2.57) − D − D + 1 . (1 − exp (lβ~ω)) 2 (1 − exp (lβ~ω)) 2

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

Ncrit(T,ω,g) = ∞ D " ∞ !#! X mphω 2 X 1 1 2 1 + gβl 2 D − 1 D − 1 2π 2 (1 − exp (−lβ ω)) l=1 ~ j=1 (1 − exp (−β~ωj)) ~ D ( ∞ ∞ " mphω 2 X X 1 1 − 2gβ (l + j) − D − D + 1 2π 2 2 ~ j=1 l=1 (1 − exp (lβ~ω)) (1 − exp (lβ~ω)) #) 1 (2.58) D {(1 − exp (−lβ~ω)) (1 − exp (−jβ~ω)) (1 − exp (−(l + j)β~ω))} 2

(11) For known interaction parameter g one can determine Ncrit. Checking the literature 2 one nds ~ where is extracted from a Gross-Pitaevski equation t of the exper- g = m g˜ g˜ 4 ph imental data( ). 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˜) = ∞ X 1 2 2 − 1 l=1 (1 − exp (−lβ~ω)) ( ∞ ∞ β ω X X 1 1 + 2˜g ~ 2j − 1 − 1 2π (1 − exp (−β ωj)) 2 j=1 l=1 ~ (1 − exp (−lβ~ω)) 1 −(l + j) {(1 − exp (−lβ~ω)) (1 − exp (−jβ~ω)) (1 − exp (−(l + j)β~ω))} 1 1 − − + 1 . (2.59) (1 − exp (lβ~ω)) (1 − exp (lβ~ω)) These double sums can be calculated numerically. However it turns out that the double sum converges very slowly. The numerical evaluation by Mathematica may contain numerical uncertainties. Nevertheless the result is:

Ncrit(T,ω,g˜) = 78030 + 511 − 295 ≈ 78250 (2.60) | {z } interaction contribution

Thus we conclude that, due to the small interaction parameter, the shift of Ncrit is very small. 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 [x,p] = I (3.1) i~ and the harmonic oscillator Hamiltonian in 1-D with ω being the oscillator frequency

p2 mω2x2 H = + (3.2) 2m 2 One easily obtains the equations of motion in the Heisenberg picture

dp 1 = [p,H] = −mω2x (3.3) dt i~ dx 1 p = [x,H] = (3.4) dt i~ m Taking the derivative of (3.4) we get with help of (3.3)

d2x = −ω2x (3.5) dt2 This equation no longer evolves operator products so we can solve it directly p x = x cos[ω(t − t )] + 0 sin[ω(t − t )]. (3.6) 0 0 mω 0 The propagator is dened as

U(x,t; x0,t0) = (x,t|x0,t0) = hx,t|x0,t0i . (3.7) Now we calculate the complex conjugate as follows

∗ ∗ (3.8) (A∂x0 + Bx0)U = xU with the abreviations ~ and . A = imω sin[ω(t − t0)] B = cos[ω(t − t0)] ∗ Solving this dierential equation for x0 and U yields

∗ 1 ∂U x − Bx0 (3.9) ∗ = U ∂x0 A

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

1 B U ∗ = N ∗(t,t ) exp − x2 − xx + f(x) . (3.10) 0 A 2 0 0

In the last step I put the x dependent integration constant f(x) into the exponential, such ∗ that N is independent from x and x0. We are left with the yet unknown functions N and 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) 1 B 1 B N ∗(t,t ) exp − x2 − xx + f(x) = N(t,t ) exp − x2 − xx + f(x ) 0 A 2 0 0 0 A 2 0 0 (3.12)

Since N is independent of x and x0 the exponentials should be the same B B f(x ) − x2 = f(x) − x2 (3.13) 0 2 0 2 B as this holds for all x and x , we conclude f(x) − x2 = g(t,t ) (3.14) 0 2 0

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

2 2 imω [(x + x0) cos[ω(t − t0)] − 2xx0] U(x,t; x0,t0) = N(t,t0) exp . (3.15) 2~ sin[ω(t − t0)] The completeness relation determines N Z Z ∗ dx0U(x1,t; x0,t0)U (x2,t; x0,t0) = dx0 hx1,t|x0,t0i hx0,t0|x2,ti = hx1,t|x2,ti = δ(x1 − x2) (3.16)

Remembering the Fourier identity Z dk exp(ikz) = 2πδ(z) (3.17) we get from (3.15) and (3.16)

Z 2π sin[ω(t − t )] dx U(x ,t; x ,t )U ∗(x ,t; x ,t ) = |N(t,t )|2 ~ 0 δ(x − x ). (3.18) 0 1 0 0 2 0 0 0 mω 1 2

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

r 2 2 mω imω [(x + x0) cos[ω(t − t0)] − 2xx0] U(x,t; x0,t0) = exp . 2πi~ sin[ω(t − t0)] 2~ sin[ω(t − t0)] (3.19)

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

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