PhD thesis in English

3. Rotating ideal BECever, it does depend on β c , as well as on the energy spectrum of the system.If the system is close to a d-dimensional harmonic oscillator, which is the casefor the potential (3.4) with the small anharmonicity relevant for the experiment, forlarge values of J we have approximately∞∑∞∑j=J+1 n=1e −jβc(En−E0) ≈ d e−(J+1)βcω, (3.18)1 − e−βcω where ω denotes an effective harmonic frequency and d is the dimensionality ofthe corresponding system. In our case, we apply the semi-classical correction onlyto the x − y part of the potential and for this reason d = 2. For the case of alarge anharmonicity, the effective frequency ω would depend on κ, representing theharmonic expansion of the potential around its minimum. With such an estimate,Eq. (3.16) reduces tod∆β c ≈ −1 − e × e −(J+1)βcω. (3.19)−βcω ∂N/∂β c + (J + 1) e −(J+1)βcω dω1−e −βcωThe term (J+1) e −(J+1)βcω in the denominator of the second factor can be neglectedfor large enough values of the cutoff J, yielding as a simplified version of the aboveexpression:d e −(J+1)βcω∆β c ≈ −∂N/∂β c (1 − e −βcω ) . (3.20)In order to use the derived estimates for ∆β c , apparently one would alreadyhave to know the sought-after value of β c as well as the difficult derivative ∂N/∂β c .However, in practical applications this obstacle can be circumvented as follows.The expressions (3.19) and (3.20) can be used for fitting the numerical data forβ c (J) = β c −∆β c , as is illustrated in Fig. 3.7. In this standard approach, all unknownvalues are fit parameters, obtained numerically by the least-square method. Notethat not only β c is obtained by such a fitting procedure, but also other parameters,such as ∂N/∂β c , or the effective harmonic frequency ω. The important point here isto capture the correct J-dependence, while all other parameters do not depend onit, so that they can be extracted by fitting. For example, in Fig. 3.7 we have used72