Overlap of exact and Gross-Pitaevskii wave functions in Bose-Einstein condensates of dilute gases

It has been proven theoretically for bosons with two-body repulsive interaction potentials in the dilute limit that the Gross-Pitaevskii equation provides the exact energy and density per particle as does the basic many-particle Schrödinger equation [E. H. Lieb and R. Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. Here, we investigate the overlap of the Gross-Pitaevskii and exact ground-state wave functions. It is found that this overlap is always smaller than unity and may even vanish despite the fact that both wave functions provide the same energy and density per particle. Consequences are discussed.

Figure 1

Numerical error of the MCTDHB method with two orbitals for the exact energy per particle (solid) and overlap of the many-body and mean-field wave functions (dashed line) as a function of the number of bosons. The calculations are performed for the HIM model discussed in the text. The value of $\mathrm{\Lambda }=100$ is used in the calculation and the exact overlap in the dilute limit is 0.704.

Figure 2

The overlap of the GP and the numerically computed many-body wave functions in the case of 1D and 2D double-well trap potentials. Upper panel: The two trap potentials used in the calculations (in the 2D example $\frac{1}{2}{y}^2$ has to be added). Lower panel: The overlap as a function of the interaction parameter for three different boson numbers $N$. Note that for each value of $\mathrm{\Lambda }$ the points computed for different values of $N$ fall on top of each other. The 1D and 2D results are connected by solid and dashed lines, respectively.

Figure 3

The impact of boson correlation on the boson number fluctuation in one of the wells of the double-well one-dimensional trap potentials shown in the upper panel of Fig. 2. Upper panel: The expansion coefficients $C_n$ of the many-body wave function in Eq. (8). Note that the overlap $S\left(N\right)$ depicted in Fig. 2 is provided by the first coefficient $C_0$. Lower panel: The boson number fluctuations relative to the number $N/2$ of bosons in the left well as a function of $\mathrm{\Lambda }$ for the two double-well potentials. Note that in the GP theory this relative boson number fluctuation is always 0.5. In all calculations $N={10}^7$.

Figure 4

The difference between the many-body and GP energies per particle as a function of the number of bosons in the BEC. Note that as the number of bosons increases the energy tends further to the GP result. The results shown were calculated for the potential given in the manuscript in Fig. 1 and an interaction constant of $\mathrm{\Lambda }=0.5$. Solid and empty markers correspond to the 1D and 2D examples, respectively.

Figure 5

The maximum of the absolute value of the difference between the many-body and GP boson density per particle as a function of the number of bosons in the BEC. The small deviation between the many-body and GP densities essentially shows that we have reached the dilute limit. The results shown were calculated for the potential given in the manuscript in Fig. 1 and an interaction constant of $\mathrm{\Lambda }=0.5$. Solid and empty markers correspond to the 1D and 2D examples, respectively.

Figure 6

The overlap between the many-body and GP wave functions as a function of the number of bosons in the BEC. The results shown were calculated for the potential given in the manuscript in Fig. 1 where the solid line refers to the 1D potential and the dashed line refers to the 2D potential. The depicted results are for an interaction constant of $\mathrm{\Lambda }=0.1$.