Variance as a sensitive probe of correlations

Bose-Einstein condensates made of ultracold trapped bosonic atoms have become a central venue in which interacting many-body quantum systems are studied. The ground state of a trapped Bose-Einstein condensate has been proven to be 100% condensed in the limit of infinite particle number and constant interaction parameter [Lieb and Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. The meaning of this result is that properties of the condensate, noticeably its energy and density, converge to those obtained by minimizing the Gross-Pitaevskii energy functional. This naturally raises the question whether correlations are of any importance in this limit. Here, we demonstrate both analytically and numerically that even in the infinite particle limit many-body correlations can lead to a modification of the variance of any operator compared to that expected from the Gross-Pitaevskii result. The deviation of the variance stems from its explicit dependence on terms of the reduced two-body density matrix which otherwise do not contribute to the energy and density in this limit. This makes the variance a sensitive probe of many-body correlations even when the energy and density of the system have already converged to the Gross-Pitaevskii result. We use the many-particle position and momentum operators to exemplify this persistence of correlations. Implications of this many-body effect are discussed.

Figure 1

Variance of the many-particle position operator, $\widehat{X}={\sum }_{j=1}^N{\widehat{x}}_j$, of a weakly interacting Bose-Einstein condensate held in a symmetric trap for different barrier heights. Results for $N=1000$ (in green), $10000$ (in blue), $100000$ (in magenta), and $1000000$ (in red) bosons are shown. The interaction parameter is $\mathrm{\Lambda }={\lambda }_0(N-1)=0.1$. (a) Shown is the variance $\frac{1}{N}{\mathrm{\Delta }}_{\widehat{X}}^2$ (full curves) and the Gross-Pitaevskii variance ${\mathrm{\Delta }}_{\widehat{x},GP}^2$ (in black; dashed curve). Large differences arise from certain barrier height, indicating that the correlations term ${\mathrm{\Delta }}_{\widehat{x},\mathrm{correlations}}^2$ becomes dominant. All four curves for the different numbers of particles lie atop each other. (b) The many-body energy and (c) the depletion are seen to approach and coincide with the Gross-Pitaevskii results, as is expected from the literature. Note the small values on the $y$ axes of panels (b) and (c). In contrast, the variance converges to a value different from the Gross-Pitaevskii results for not too shallow barriers. See the text for further discussion. The quantities shown are dimensionless.

Figure 2

Variance of the center-of-mass position and momentum operators and their uncertainty product. An illustrative numerical example with one-dimensional bosons. The one-body Hamiltonian is $\widehat{h}\left(x\right)=-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}+\frac{x^2}{2}$, and the particle-particle interaction is contact, $\widehat{W}(x_1-x_2)=\delta (x_1-x_2)$. Shown are results for the interaction parameters $\mathrm{\Lambda }={\lambda }_0(N-1)=1$, 10, and 100 as a function of the number of particles $N$. Gross-Pitaevskii results (in color, dotted and dashed curves) and analytical, many-body results (in black, full curve) are compared. (a) The vanishing of ${\mathrm{\Delta }}_{{\widehat{X}}_{\mathrm{c}.\mathrm{m}.}}^2$ and (b) the divergence of ${\mathrm{\Delta }}_{{\widehat{P}}_{\mathrm{c}.\mathrm{m}.}}^2$ with increasing $N$, see Eq. (19), are shown. (c) The position-momentum uncertainty product ${\mathrm{\Delta }}_{{\widehat{X}}_{\mathrm{c}.\mathrm{m}.}}^2{\mathrm{\Delta }}_{{\widehat{P}}_{\mathrm{c}.\mathrm{m}.}}^2$ computed at the Gross-Pitaevskii level is seen, with increasing $\mathrm{\Lambda }$, to deviate from the analytical, many-body value of $\frac{1}{4}$. The variance and uncertainty product constitute a sensitive probe of correlations enduring the infinite-particle limit. See the text for more details. The quantities shown are dimensionless.

Figure 3

Density of a Bose-Einstein condensate held in a symmetric trap. The barrier height is (a) $A=3$ and (b) $A=12$. The number of bosons is $N=100000$ and the interaction parameter is $\mathrm{\Lambda }={\lambda }_0(N-1)=0.1$. The density (in red; full curve) coincide with the Gross-Pitaevskii limit (in black; dashed curve), as is expected from the literature, whereas the respective variances deviate from each other, as seen in Fig. 1. See the text for more details. The quantities shown are dimensionless.

Figure 4

Same as Fig. 1 but for an asymmetric trap. Results for $N=100000$ and the two interaction parameters $\mathrm{\Lambda }={\lambda }_0(N-1)=0.5$ and 1 are shown. Note the small amount of depletion. The variance behaves qualitatively the same as in the symmetric trap, compare to Fig. 1, and deviates from the Gross-Pitaevskii limit for not too shallow barriers. The quantities shown are dimensionless.

Figure 5

Same as Fig. 3 but for an asymmetric trap. The number of bosons is $N=100000$ and the interaction parameter is $\mathrm{\Lambda }={\lambda }_0(N-1)=0.5$. The density and the Gross-Pitaevskii result coincide. Compare to the respective variances in Fig. 4. The quantities shown are dimensionless.

Figure 6

(a) Many-particle radial variance $\frac{1}{N}({\mathrm{\Delta }}_{\widehat{X}}^2+{\mathrm{\Delta }}_{\widehat{Y}}^2+{\mathrm{\Delta }}_{\widehat{Z}}^2)$ (in red; full curve) of a three-dimensional Bose-Einstein condensate made of $N=100$ atoms in a cubic-shaped unharmonic trap as a function of the interaction parameter $\mathrm{\Lambda }={\lambda }_0(N-1)$. The Gross-Pitaevskii radial variance ${\mathrm{\Delta }}_{\widehat{x},GP}^2+{\mathrm{\Delta }}_{\widehat{y},GP}^2+{\mathrm{\Delta }}_{\widehat{z},GP}^2$ is shown in black (dashed curve). (b) The difference between the many-body and Gross-Pitaevskii energy (in blue; full curve) and the depletion (in magenta; dashed curve). (c) The many-body density and (d) the Gross-Pitaevskii density for $\mathrm{\Lambda }=100$. Note the depletion is less than $1\%$. Shown are contours for $\frac{\rho \left(\mathbf{r}\right)}{N}=0.02$ and $0.001$. The many-body density coincide with the Gross-Pitaevskii density, whereas the respective variances clearly deviate from each other. See text for further details. The quantities shown are dimensionless.

Figure 7

Variance of $\widehat{X}$ of a Bose-Einstein condensate held in the symmetric trap of Fig. 1 for different barrier heights. Shown is $\frac{1}{N}{\mathrm{\Delta }}_{\widehat{X}}^2$ for $N=10$ and 100 bosons for increasing number $M$ of time-dependent orbitals. The curves sit atop each other, and the variance is seen to converge. The quantities shown are dimensionless.