Entangled particles in a dynamically controlled trap

Thomas Stefan Häberle and Matthias Freyberger

Phys. Rev. A 89, 052332 – Published 29 May 2014

Abstract

We control the entanglement of motion between two colliding particles by dynamically varying the frequency of their common trap. In particular, we restrict the possible frequencies to a narrow interval and limit the average energy of the particles to a maximal value. In order to solve this restricted optimization problem we propose a modified gradient method. It turns out that the motional entanglement generated by collisions can be considerably improved.

Received 12 November 2013

DOI:https://doi.org/10.1103/PhysRevA.89.052332

Authors & Affiliations

Thomas Stefan Häberle and Matthias Freyberger

Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, D-89069 Ulm, Germany

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Issue

Vol. 89, Iss. 5 — May 2014

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Images

Figure 1
(a) Optimized static trap frequency $u_{S}$ shown as a function of scaled evolution time $ω_{0} T$ for the interaction strength $g = \frac{3}{4} ɛ_{0} l_{0}$ [see Eq. (2)] and the initial particle distance $R = 5 l_{0}$ [see Eq. (4)]. Note that the values of $u_{S}$ are restricted to the interval $u_{S} \in [0.9, 1.1]$ . In addition, the maximum allowed energy has been set to $E_{max} = 25 ɛ_{0}$ . We clearly see that the trap frequency $u_{S}$ strongly depends on the time $T$ , for which we like to achieve maximized entanglement between the particles. As an example we demonstrate in panel (b) the time evolution of the von Neumann entropy $S$ until the final time $T = 8 π / ω_{0}$ calculated for the corresponding optimized static trap frequency. The visible steps in $S$ are the signature of colliding particles in the center of the trap. Initially each collision adds entanglement to the particles. However, already the seventh collision begins to disentangle them. These steps in the von Neumann entropy are synchronous to peaks of the average interaction energy $E_{int} (t)$ , Eq. (17), shown in panel (c). This further visualizes that motional entanglement changes at times, for which the particles have a high probability to be in the center of the trap.
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Figure 2
Controlled (red-colored solid line) and optimized uncontrolled (blue-colored dashed line) entanglement measured by the von Neumann entropy $S$ as a function of scaled final time $ω_{0} T$ . In panel (a) the basic parameters have been chosen as in Fig. 1. However, in panel (b) we vary the trap frequency $ω (t)$ by $\pm 20 %$ instead of $\pm 10 %$ as in panel (a). In both cases the maximally allowed energy value has again been set to $E_{max} = 25 ɛ_{0}$ . It is obvious that a dynamic trap can significantly improve entanglement, in particular for longer times. Hence the control of a classical parameter will lead to much stronger nonclassical correlations in the motion of two particles.
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Figure 3
(a) Control function $φ (t)$ for maximal entanglement at the final time $T = 8 π / ω_{0}$ . All parameters have been chosen as in Fig. 1 and the maximally allowed energy value has again been set to $E_{max} = 25 ɛ_{0}$ . This control function leads to the scaled trap frequency $u (t)$ [Eq. (7)] shown in panel (b). The control restriction creates a relatively simple and almost rectangular signal rocking the trap. The corresponding time evolution of entanglement $S$ is depicted in panel (c) as a red-colored solid curve. We compare it here to the evolution in the optimized static trap (blue-colored dashed curve) already shown in Fig. 1. The time-dependent steepness of the trap washes out the distinctive step-like behavior in the entanglement but leads to a considerable improvement at final time $T = 8 π / ω_{0}$ . Finally, we see in panel (d) that the distinctive peak-like structure of the interaction energy $E_{int}$ in the static case [blue-colored dashed curve, see also Fig. 1] has also been smeared out by the controlled trap (red-colored solid curve). This exemplifies that, under control, the particles stay longer in the interaction region.
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Figure 4
We show the average energy $E (t)$ [Eq. (8)] in a controlled (red-colored solid line) and optimized static (blue-colored dashed line) trap as a function of scaled time $ω_{0} t$ . We clearly see that $E (t)$ always stays well below $E_{max} = 25 ɛ_{0}$ . Note that there is a wide time window in which the entanglement has been improved in a controlled trap even if the average energy is lower than the total energy in the optimized static trap.
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