PhD thesis in English

3. Rotating ideal BECN - N 03.0·10 5 J=302.5·10 5J=10002.0·10 51.5·10 53.00·10 51.0·10 52.95·10 50.5·10 52.90·10 580 90 100 110 120 130 14000 20 40 60 80 100 120 140E max / − hω ⊥Figure 3.4: Number of thermally excited atoms N − N 0 calculated as a function ofthe maximal available two-dimensional energy eigenvalue E max at T = 63.30 nK.The results are given for two different values of the cumulant cutoff J for a criticallyrotating condensate, with the same parameters as in Fig. 3.3. The horizontal linecorresponds to the number of atoms N = 3 · 10 5 in the experiment [12].ergy eigenvalue. A semiclassical correction to this value, can be calculated accordingto Ref. [65] as∫∆Z 1 (β, E max ) =d⃗r d⃗p(2π) 3 e−βH(⃗r,⃗p) Θ(H(⃗r, ⃗p) − E max ) , (3.13)where H(⃗r, ⃗p) as before represents the classical Hamiltonian of the system, while Θdenotes the Heaviside step-function.For the trap potential (3.4), in z-direction we have a pure harmonic potential,which can be treated exactly. Therefore, we focus only on the two-dimensionalproblem in the x − y plane. In this case, the semiclassical correction for the singleparticlepartition function (3.13) can be expressed in terms of the complementaryerror function:∆Z (2)1 (β, E max) = 1 { e−βE max2β κ√ π+κβ e β(1−η2 ) 24κ[−(1 − η 2 ) √ (1 − η 2 ) 2 + 4κE max]× Erfc(√βE max + β(1 − η2 ) 24κ)},(3.14)68