Many-body tunneling dynamics of Bose-Einstein condensates and vortex states in two spatial dimensions

Raphael Beinke, Shachar Klaiman, Lorenz S. Cederbaum, Alexej I. Streltsov, and Ofir E. Alon

Phys. Rev. A 92, 043627 – Published 27 October 2015

Abstract

In this work, we study the out-of-equilibrium many-body tunneling dynamics of a Bose-Einstein condensate in a two-dimensional radial double well. We investigate the impact of interparticle repulsion and compare the influence of angular momentum on the many-body tunneling dynamics. Accurate many-body dynamics are obtained by solving the full many-body Schrödinger equation. We demonstrate that macroscopic vortex states of definite total angular momentum indeed tunnel and that, even in the regime of weak repulsions, a many-body treatment is necessary to capture the correct tunneling dynamics. As a general rule, many-body effects set in at weaker interactions when the tunneling system carries angular momentum.

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Received 13 August 2015

DOI:https://doi.org/10.1103/PhysRevA.92.043627

Authors & Affiliations

Raphael Beinke^1,*, Shachar Klaiman¹, Lorenz S. Cederbaum¹, Alexej I. Streltsov¹, and Ofir E. Alon²

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

^*raphael.beinke@pci.uni-heidelberg.de

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Vol. 92, Iss. 4 — October 2015

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Figure 1
Schematic plot of the system's setup. Shown is a cut through the potential $V (r)$ (shaded gray area) of the 2D radial double well, as well as typical density cuts of the ground-state GP orbitals $ϕ_{Λ}^{IN}$ and $ϕ_{Λ}^{OUT}$ for both $L = 0$ and $L = N$ . The number of bosons is $N = 100$ . The IN and OUT regions of the trap are separated by the potential barrier at $R_{B}$ (vertical dotted black line). For $L = 0$ , the density maximum in the IN region is located exactly in the trap's center, i.e., at $r = 0$ (dotted red curve). On the other hand, for $L = N$ , one can clearly see the node of $ϕ_{Λ}^{IN}$ (dashed red). In the OUT region, $ϕ_{Λ}^{OUT}$ for $L = N$ (blue, double dotted) is pushed further to the crater's wall than for $L = 0$ (solid blue). Horizontal lines denote typical (angular-momentum-dependent) values of the ground-state energies at the mean-field level for the corresponding orbitals, i.e., $E_{Λ}^{IN}$ (solid red) for $ϕ_{Λ}^{IN}$ and $E_{Λ}^{OUT}$ (solid blue) for $ϕ_{Λ}^{OUT}$ . See text for more details. All quantities are dimensionless.
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Figure 2
(a) Dependence of the ground-state energy on the position of the radial barrier $R_{B}$ . The angular momentum is $L = 0$ , the number of bosons is $N = 100$ , and the number of orbitals used is $M = 1$ . $E_{0}^{IN}$ (dashed red curve) and $E_{0}^{OUT}$ (solid green) denote, respectively, the ground-state energies for the IN and OUT regions for $Λ = 0, E_{2}^{IN}$ (dotted blue) and $E_{2}^{OUT}$ (magenta, double dotted) for $Λ = 2$ . The black vertical lines at $R_{1} = 3.271$ and $R_{2} = 3.412$ denote the crossing points for $Λ = 0$ and $Λ = 2$ , respectively. Increasing the repulsion strength shifts the crossing point to larger radii. (b) Same as in (a) but for $L = N$ . The corresponding crossing points are $R_{3} = 4.089 (Λ = 0)$ and $R_{4} = 4.093 (Λ = 0.2)$ . The angular momentum $L$ affects the position of the crossing points comparatively stronger than the repulsion strength $Λ$ . Note that the angular momentum is macroscopic; all particles carry it. See text for more details. All quantities are dimensionless.
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Figure 3
(a) Single-particle tunneling dynamics for different barrier positions $R_{B}$ . The orbital angular momentum is $l = 0$ . The dynamics are started from the OUT region. Close to the crossing point $R_{1} = 3.271$ , the amplitude of $P_{OUT} (t)$ is maximal and almost the whole density tunnels (red solid curve). The period is in a very good agreement with the predicted value $τ_{1} = 110.231$ . Leaving the crossing point, both the amplitude and period become smaller ( $R_{B} = 3.1$ , dotted blue; $R_{B} = 3.5$ , dashed black). (b) Dependence of the amplitude $A$ and period $τ$ (plotted on the same axis) on $R_{B}$ for the same parameters as in (a). The indices IN and OUT denote the two different initial conditions considered. Close to the crossing point $R_{1}$ , both the amplitude and period become maximal and give the same result for the two initial conditions. See text for more details. All quantities are dimensionless.
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Figure 4
Mean-field $(M = 1)$ and many-body $(M = 4)$ tunneling dynamics of the interacting system at the crossing point $R_{2} = 3.412$ . The angular momentum is $L = 0$ , the number of particles is $N = 100$ , and the interaction strength is $Λ = 2$ . The bosons are initially prepared in the OUT region. Whereas the GP theory predicts unperturbed oscillations of $P_{OUT} (t)$ (dotted gray curve), the amplitude is damped at the many-body level (solid red) and saturates after approximately 14 cycles. Only the first two natural orbitals are significantly occupied (solid green and blue), the other two (solid magenta and light blue; curves lie atop of each other) carry just a small fraction of the particles. However, the frequencies of the GP and many-body density oscillations are essentially the same. (Inset) Same simulation but for strong repulsion $(Λ = 6)$ at the respective crossing point $R_{B} = 3.608$ . Many fewer particles tunnel back and forth; the dynamics obtained from the GP and mctdhb(4) descriptions start to deviate from each other already during the first cycle, both in their frequencies and amplitudes. After four cycles, the four natural orbitals are already macroscopically occupied (not shown). Note the different time scales. See text for more details. All quantities are dimensionless.
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Figure 5
Real, imaginary, and absolute value of the GP, mean-field orbital (top panels) and of the first two natural orbitals $α_{1}$ and $α_{2}$ of the many-body simulation ( $M = 4$ , middle and bottom panels) at $t = 18.4 τ_{1}$ . The other parameters of the system are the same as in the main figure of Fig. 4. Whereas the mean-field orbital is localized in the external rim, $α_{1}$ and $α_{2}$ from the many-body computation are delocalized, covering both the IN and OUT regions. See text for more details. All quantities are dimensionless.
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Figure 6
Dependence of the amplitude $A$ and period $τ$ (plotted on the same axis) on $R_{B}$ in the noninteracting system, i.e., the single-particle case. The orbital angular momentum is $l = 1$ . The indices IN and OUT denote the two different initial conditions considered. Close to the crossing point $R_{3} = 4.089$ , the period and amplitude of the density oscillations are maximal and the two initial conditions give equivalent results for the tunneling dynamics. Leaving the crossing point, both the amplitude and period become smaller. See text for more details. All quantities are dimensionless.
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Figure 7
Many-body $(M = 4)$ tunneling dynamics of a vortex state at the crossing point $R_{4} = 4.093$ . The number of particles is $N = 100$ , the total angular momentum is $L = N$ , and the interaction strength is $Λ = 0.2$ . The bosons are released from the OUT region. The damped oscillation of $P_{OUT} (t)$ (solid red) is enveloped by the natural occupations of $α_{1}$ and $α_{2}$ (solid green, blue). The remaining natural orbitals do not become significantly occupied (solid magenta, light blue; curves lie atop of each other). (Upper inset) Dynamics of $P_{OUT} (t)$ for the GP, mean-field ( $M = 1$ , dotted gray) and many-body ( $M = 4$ , solid red) descriptions between $t = 150 τ_{1}$ and $t = 155 τ_{1}$ . The system's parameters are the same as in the main figure. The mean-field description does not account for the density collapse; the vortex state is already fragmented. However, the tunneling frequencies of the GP and mctdhb(4) results are essentially the same. (Lower inset) Same simulation for strong repulsion, $Λ = 2$ , at the respective crossing point $R_{B} = 4.123$ . The other parameters of the system are the same as in the main figure. The dynamics obtained from the mean-field (dotted gray) and many-body descriptions (solid red) start to deviate from each other already after a few cycles, both in frequency and amplitude. After 10 cycles, the four natural orbitals are already macroscopically occupied (not shown). See text for more details. All quantities are dimensionless.
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Figure 8
Real, imaginary, and absolute value of the GP, mean-field orbital (top panels) and of the first two natural orbitals $α_{1}$ and $α_{2}$ of the many-body simulation ( $M = 4$ , middle and bottom panels) at $t = 183.3 τ_{2}$ . The system's parameters are the same as in the main figure of Fig. 7. Whereas the mean-field orbital is localized in the external rim, the natural orbitals in the many-body computation are delocalized, covering both the IN and OUT regions. $α_{1}$ only has a node in the trap's center, whereas $α_{2}$ has a second, ring-shaped node at the barrier's position $R_{4}$ . See for comparison Fig. 5 and the text for more details. All quantities are dimensionless.
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Figure 9
(a) Many-body tunneling dynamics for $N = 10$ particles. The number of orbitals is $M = 10$ , the angular momentum is $L = 0$ , and the interaction strength is $Λ = 2$ . The bosons are released from the IN region. The dashed red curve denotes the time evolution of $P_{IN} (t)$ . Only the first four natural orbitals become macroscopically occupied [solid curves from top to bottom: $n_{1} (t)$ in green, $n_{2} (t)$ in blue, $n_{3} (t)$ in magenta, and $n_{4} (t)$ in light blue (atop of the $n_{3} (t)$ curve)]. The $n_{5} (t)$ and $n_{6} (t)$ natural occupations (in light green and yellow; bottommost solid curves lying atop each other) carry only 1% of the particles. (Inset) The remaining natural occupation numbers [solid curves from top to bottom: $n_{7} (t)$ in light blue, $n_{8} (t)$ in yellow, $n_{9} (t)$ in black, and $n_{10} (t)$ in brown], all staying below 0.15%. The fact that only the first four natural orbitals are significantly occupied justifies the description of the tunneling dynamics with $M = 4$ time-adaptive orbitals for $Λ = 2$ . (b) The same simulation as in (a) but releasing the particles from the OUT region, which gives similar but not identical results. Most important, only the first four natural orbitals become macroscopically occupied. All quantities are dimensionless.
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