Uncertainty product of an out-of-equilibrium many-particle system

Shachar Klaiman, Alexej I. Streltsov, and Ofir E. Alon

Phys. Rev. A 93, 023605 – Published 1 February 2016

Abstract

The variance of a many-particle operator is a very sensitive probe of many-body correlations. In contrast to the expectation value of such operators which depends on the condensation fraction of the Bose-Einstein condensate (BEC), their variance depends on the absolute number of depleted particles. Consequentially, the many-body correlations incorporated within the variance can be observed even in the limit of an infinite number of particles (keeping the interaction parameter, i.e., the product of the number of particles times the scattering length, constant) when the system becomes 100% condensed. In the present work we show both analytically and numerically the effect of many-body correlations on the variance and uncertainty product for an out-of-equilibrium BEC. Our work demonstrates the need to go beyond a mean-field description of the condensate in order to correctly describe these quantities even when only a single particle is depleted and the system is by definition 100% condensed.

Received 2 September 2015

DOI:https://doi.org/10.1103/PhysRevA.93.023605

Physics Subject Headings (PhySH)

Bose-Einstein condensates

Mean field theory

Atomic, Molecular & Optical

Authors & Affiliations

Shachar Klaiman^1,*, Alexej I. Streltsov^1,†, and Ofir E. Alon^2,‡

¹Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany
²Department of Physics, University of Haifa at Oranim, Tivon 36006, Israel

^*shachar.klaiman@pci.uni-heidelberg.de
^†alexej.streltsov@pci.uni-heidelberg.de
^‡ofir@research.haifa.ac.il

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Vol. 93, Iss. 2 — February 2016

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Figure 1
Time-dependent center-of-mass position-momentum uncertainty product $Δ_{{\hat{X}}_{c.m.}}^{2} (t) Δ_{{\hat{P}}_{c.m.}}^{2} (t)$ of a BEC released from a harmonic trap. An illustrative numerical example with one-dimensional bosons. The one-body Hamiltonian is $- \frac{1}{2} \frac{\partial^{2}}{\partial x^{2}} + \frac{x^{2}}{2}$ , and the interboson interaction is contact, $λ_{0} \hat{W} (x_{1} - x_{2}) = λ_{0} δ (x_{1} - x_{2})$ . Shown and compared as a function of time $t$ are the Gross-Pitaevskii results for the interaction parameters $Λ = λ_{0} (N - 1) = 1$ (red dashed curve), $Λ = 10$ (green dash-dotted curve), and $Λ = 100$ (blue dash-double-dotted curve) and the analytical, many-body result $\frac{1}{4} (1 + t^{2})$ valid $\forall Λ$ [Eq. (19)] (black solid curve). The position-momentum uncertainty product computed at the Gross-Pitaevskii level differs from the analytical, many-body result. The difference increases upon increasing $Λ$ , meaning that the pace of growth of the uncertainty at the mean-field level depends on the interaction parameter $Λ$ (the $y$ axis is plotted in logarithmic scale). The many-body uncertainty product grows as $t^{2}$ , and the mean-field uncertainty product is seen to grow in a similar manner in time (see Ref. [26] for the mean-field analysis of the expansion). The uncertainty product constitutes a macroscopic probe of the time-dependent correlations of a BEC, even when the system becomes 100% condensed in the limit of an infinite number of particles. See the text for more details. The quantities shown are dimensionless.
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Figure 2
(a) Time-dependent survival probability in the left well, $P_{L} (t)$ , of a BEC in a one-dimensional bosonic Josephson junction. The interboson interaction is contact, $λ_{0} \hat{W} (x_{1} - x_{2}) = λ_{0} δ (x_{1} - x_{2})$ . The interaction parameter is $Λ = λ_{0} (N - 1) = 0.01$ . Shown are the many-body results for $N = 1 000, 10 000$ , and $100 000$ bosons (in color, solid curves) using $M = 2$ time-dependent orbitals, and the Gross-Pitaevskii, mean-field result (in black, dashed curve). All four curves, the three many-body results and the mean-field one, lie on top of each other. Full tunneling of the density back and forth between the left and right wells in seen. The inset shows the smoothness of the survival probability at its extrema. The panels to the right depict snapshots of the time-dependent density (with the double-well potential in the background) at three different times, when the density is (b) localized in the left well, (c) delocalized over both wells, (d) or localized in the right well. The many-body (in color, solid curves) and mean-field (in black, dashed curve) results lie on top of each other. That the many-body and mean-field results for the time-dependent density and survival probability coincide indicates that the system is condensed [also see Fig. 3]. See the text for more details. The quantities shown are dimensionless.
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Figure 3
(a) Time-dependent center-of-mass position-momentum uncertainty product $Δ_{{\hat{X}}_{c.m.}}^{2} (t) Δ_{{\hat{P}}_{c.m.}}^{2} (t)$ of a BEC in the one-dimensional bosonic Josephson junction of Fig. 2. Shown are many-body results using $M = 2$ time-dependent orbitals. The interaction parameter is $Λ = λ_{0} (N - 1) = 0.01$ . The system is weakly interacting. The time-dependent uncertainty product computed at the many-body level is seen to grow in time in an oscillatory manner. The period of oscillations is essentially $\frac{1}{2} t_{Rabi}$ . For comparison and later analysis, the Gross-Pitaevskii dynamics only oscillates and is thus seen to deviate from the many-body results as time progresses. The results for $N = 1 000$ (red), $10 000$ (green), and $100 000$ (light blue) bosons lie on top of each other and suggest convergence of the many-body uncertainty product with increasing number of bosons and at constant interaction parameter in the infinite-particle limit. (b) The time-dependent depletion decreases with $N$ and at constant $Λ$ (note the small values on the $y$ axis, which is in logarithmic scale). Side by side, the curves appear to be shifted vertically from each other, suggesting that the total time-dependent number of particles residing in the higher natural orbitals is constant. The uncertainty product is hereby demonstrated to be a property of a BEC which depends on the total number of depleted particles and not on the depleted fraction which becomes 0% in the infinite-particle limit. See the text for more details. The quantities shown are dimensionless.
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Figure 4
Time-dependent center-of-mass position-momentum uncertainty product $Δ_{{\hat{X}}_{c.m.}}^{2} (t) Δ_{{\hat{P}}_{c.m.}}^{2} (t)$ of a BEC in the one-dimensional bosonic Josephson junction of Fig. 2: analysis of its growth in time. Results for $N = 1000$ bosons for the interaction parameters $Λ = λ_{0} (N - 1) = 0.005$ (red, small-amplitude oscillating curve), $0.01$ (green, middle-amplitude oscillating curve), and $0.02$ (blue, large-amplitude oscillating curve) are depicted. Shown are many-body results using $M = 2$ time-dependent orbitals. The black, monotonously increasing curves are least-squares fits of the maxima of the uncertainty product to a polynomial of the form $a (Λ) + b (Λ) t^{2}$ . The numerical results suggest that the envelope of the time-dependent uncertainty product grows as $Λ^{2} t^{2}$ , to leading orders in time and the interaction parameter. See the text for more details. The quantities shown are dimensionless.
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Figure 5
Time-dependent center-of-mass position-momentum uncertainty product $Δ_{{\hat{X}}_{c.m.}}^{2} (t) Δ_{{\hat{P}}_{c.m.}}^{2} (t)$ of a BEC in the one-dimensional bosonic Josephson junction of Fig. 2: convergence of the out-of-equilibrium many-body dynamics. Results for $N = 100$ bosons for an increasing number $M$ of the time-dependent orbitals are shown. The curves lie on top of each other, and the uncertainty product is seen to converge with increasing $M$ . The quantities shown are dimensionless.
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