Reduced density matrices and coherence of trapped interacting bosons

The first- and second-order correlation functions of trapped interacting Bose-Einstein condensates are investigated numerically on a many-body level from first principles. Correlations in real space and momentum space are treated. The coherence properties are analyzed. The results are obtained by solving the many-body Schrödinger equation. It is shown in an example how many-body effects can be induced by the trap geometry. A generic fragmentation scenario of a condensate is considered. The correlation functions are discussed along a pathway from a single condensate to a fragmented condensate. It is shown that strong correlations can arise from the geometry of the trap, even at weak interaction strengths. The natural orbitals and natural geminals of the system are obtained and discussed. It is shown how the fragmentation of the condensate can be understood in terms of its natural geminals. The many-body results are compared to those of mean-field theory. The best solution within mean-field theory is obtained. The limits in which mean-field theories are valid are determined. In these limits the behavior of the correlation functions is explained within an analytical model.

Figure 1

(Color online) Energy per particle, natural orbital, and natural geminal occupations of the ground state of $N=1000$ bosons at ${\lambda }_0=0.01$ in a harmonic trap with a central barrier. Shown is the dependence on the barrier height. Top: energy per particle $E∕N$ of the many-body solution. Inset: energy difference per particle between the best mean-field and the GP solution, $(E_{BMF}-E_{GP})∕N$ (triangles), and between the many-body and GP solution, $(E_{MCTDHB}-E_{GP})∕N$ (circles). Middle: the eigenvalues $n_1^{\left(1\right)}$ and $n_2^{\left(1\right)}$ of the first-order RDM ${\rho }^{\left(1\right)}(x_1\mid x_1^{\prime })$. The ground state fragments with increasing barrier height. Bottom: the eigenvalues $n_1^{\left(2\right)}$, $n_2^{\left(2\right)}$, and $n_3^{\left(2\right)}$ of the second-order RDM ${\rho }^{\left(2\right)}(x_1,x_2\mid x_1^{\prime },x_2^{\prime })$. The dashed lines in the middle and bottom panel indicate upper and lower bounds on the largest eigenvalue of the first- and second-order RDMs. See text for details. The quantities shown are dimensionless.

Figure 2

(Color online) Natural orbitals and geminals at different barrier heights. First column: the natural orbitals ${\alpha }_1^{\left(1\right)}\left(x_1\right)$ (dashed red line) and ${\alpha }_2^{\left(1\right)}\left(x_1\right)$ (solid blue line) of the many-body solution at different barrier heights $A=0,13,19,24$, from top to bottom. The trapping potential is shown as a dashed-dotted black line in the first column. The state of the system changes from condensed to fragmented between $A=13$ and $A=24$. Second to fourth columns: natural geminals ${\alpha }_1^{\left(2\right)}(x_1,x_2)$, ${\alpha }_2^{\left(2\right)}(x_1,x_2)$, and ${\alpha }_3^{\left(2\right)}(x_1,x_2)$ from left to right at the same barrier heights as above. While the natural orbitals remain qualitatively unchanged during the fragmentation transition, the natural geminals take on their final shapes only when the system becomes fully fragmented. At low barrier heights only one natural geminal is occupied. At high barriers three natural geminals are occupied, see Fig. 1. The total energy is minimized by the occupation of a natural geminal that contributes practically nothing to the interaction energy. See text for more details. The quantities shown are dimensionless.

Figure 3

(Color online) Density distribution, momentum distribution and first-order coherence. The first two columns show the one-particle density ${\rho }^{\left(1\right)}(x_1\mid x_1)∕N$ and the one-particle momentum distribution ${\rho }^{\left(1\right)}(k_1\mid k_1)∕N$ of the many-body solution (solid blue lines) and of the BMF solution (dashed red line), respectively. From top to bottom the height of the central barrier is $A=0,13,19,24$. The BMF result agrees well with the many-body result for a large range of barrier heights. Only at $A=19$, in the many-body regime, deviations are visible in the momentum distribution. See text for details. The third column shows the absolute value squared of the normalized first-order correlation function ${\mid g^{\left(1\right)}(x_1^{\prime },x_1)\mid }^2$ at the same barrier heights. An initially coherent condensate splits into two separate condensates which are no longer mutually coherent. Only the coherence within each of the wells is preserved. The quantities shown are dimensionless.

Figure 4

(Color online) Density distribution, momentum distribution, and first-order coherence obtained by using the GP equation for high barriers. The first two columns show the GP one-particle density ${\rho }^{\left(1\right)}(x_1\mid x_1)∕N$ (left) and the GP one-particle momentum distribution ${\rho }^{\left(1\right)}(k_1\mid k_1)∕N$ (middle) at barrier heights $A=19$ and $A=24$ (solid green lines). In the first column the trapping potential is also shown (dashed-dotted black line). The GP equation models the density well, but fails at the computation of the momentum distribution, compare with Fig. 3. The third column shows the absolute value squared of the normalized first-order correlation function ${\mid g^{\left(1\right)}(x_1^{\prime },x_1)\mid }^2$ computed with the GP equation at the same barrier heights. The normalized first-order correlation function is incorrectly described by the solution of the GP equation. The quantities shown are dimensionless.

Figure 5

(Color online) Two-particle density and two-particle momentum distribution at different barrier heights. The first two columns (from left to right) show the two-particle density ${\rho }^{\left(2\right)}(x_1,x_2\mid x_1,x_2)∕N(N-1)$ of the many-body (left) and BMF (right) wave function for the barrier heights $A=0,13,19,24$, from top to bottom. At low barrier heights $(A=0,13)$ the system is condensed, and the two-particle density factorizes into a product of the one-particle densities. At higher barriers $(A=19,24)$, the system fragments and the two-particle density does not factorize into a product of the one-particle densities. The fragmentation transition is not visible in the two-particle density. The results of the many-body and BMF wave function cannot be distinguished at any barrier height. The third and fourth column show the two-particle momentum distribution ${\rho }^{\left(2\right)}(k_1,k_2\mid k_1,k_2)∕N(N-1)$ of the many-body (left) and BMF (right) solution at the same barrier heights as above. The transition from a condensed state to a fragmented state is clearly visible. At $A=19$ the BMF solution does not reproduce the many-body results. The system is in a true many-body state, inaccessible to mean-field methods. At higher barriers $A⩾24$ the system fully fragments, and a mean-field description is applicable again. The quantities shown are dimensionless.

Figure 6

(Color online) Second-order coherence at different barrier heights. The first two columns (from left to right) show the diagonal of the normalized second-order correlation function in real space $g^{\left(2\right)}(x_1,x_2,x_1,x_2)$ of the many-body (left) and BMF (right) solution at barrier heights $A=0,13,19,24$, from top to bottom. $g^{\left(2\right)}(x_1,x_2,x_1,x_2)$ is very close to one at all barrier heights. Note the scale. The system seems to be second-order coherent and the results of the many-body and BMF solution agree well with each other. The third and fourth column depict the diagonal of the normalized second-order correlation function in momentum space $g^{\left(2\right)}(k_1,k_2,k_1,k_2)$ of the many-body (left) and BMF (right) solution at the same barrier heights. The fragmentation transition is clearly visible between $A=13$ and $A=24$. At $A=19$ there are strong many-body correlations between the momenta (local maxima in black color) and $g^{\left(2\right)}(k_1,k_2,k_1,k_2)$ exhibits a complicated pattern, see text for more details. A mean-field description fails here. In the limit of high barriers, $A⩾24$, the correlations of the many-body state become somewhat weaker and become again describable by those of the BMF solution. The quantities shown are dimensionless.