Optimal quantum control of Bose-Einstein condensates in magnetic microtraps

Transport of Bose-Einstein condensates in magnetic microtraps, controllable by external parameters such as wire currents or radio-frequency fields, is studied within the framework of optimal control theory (OCT). We derive from the Gross-Pitaevskii equation the optimality system for the OCT fields that allow efficient channeling of the condensate between given initial and desired states. For a variety of magnetic confinement potentials we study transport and wave-function splitting of the condensate, and demonstrate that OCT drastically outperforms simpler schemes for the time variation of the microtrap control parameters.

Figure 1

(Color online) Results of our simulation for the single-well potential (7) and $x_0=5$. (a) Cost function $J(\psi ,\lambda )$ for different transfer times $T$ and for a linear time variation of $\lambda$ (solid line) and optimized variations (symbols). The dotted line shows results of a simulation where an additional $x^4$ term is added to the potential (see text). The inset reports the optimized control fields $\lambda \left(t\right)$ for transfer times of 3 (solid line), 6 (dashed line), and 9 (dotted line). (b) Time evolution of potential and wave function $\mid \psi (x,t)\mid$ for linear $\lambda$ and $T=9$. (c)–(e) Density plot of $\mid \psi (x,t)\mid$ for linear $\lambda$ as a function of time and position, and for transfer times of $T=\left(\mathrm{c}\right)$ 9, (d) 6, and (e) 3. On the bottom of each panel we show the desired wave function (dash-dotted line) and $\mid \psi (x,T)\mid$ (solid line). The solid lines in the density plots represent the equipotential lines of the confinement potential. $\left({\mathrm{c}}^{\prime }\right)–\left({\mathrm{e}}^{\prime }\right)$ Same as (c)–(e) but for optimized control fields.

Figure 2

(Color online) Time-integrated Wigner function $w(x,p)$ for the single-well transfer processes shown in Fig. 1. The solid lines show the equipotential lines for the Wigner functions of the initial and final wave functions ${\psi }_0$ and $\psi \left(T\right)$, respectively.

Figure 3

(Color online) Same as Fig. 1 but for the double-well potential (9). The inset in (a) shows the optimized $\lambda \left(t\right)$ for $T=6$ (solid line) and 9 (dashed line). (b) Time evolution of $\mid \psi (x,t)\mid$ and $V(x,\lambda \left(t\right))$ for $T=9$. (c), (d) Density plot of $\mid \psi (x,t)\mid$ for $T=\left(\mathrm{c}\right)$ 9 and (d) 6. $\left({\mathrm{b}}^{\prime }\right)–\left({\mathrm{d}}^{\prime }\right)$ Same as (b)–(d) but for optimized control fields.

Figure 4

(Color online) Absolute value of Wigner function $\mid w(x,p;T)\mid$ at the end of the transfer process [Figs. 3b, 3c, 3d] for linear $\lambda$ variation and for transfer times of $T=\left(\mathrm{a}\right)$ 9 and (b) 6. $\left({\mathrm{a}}^{\prime }\right)$, $\left({\mathrm{b}}^{\prime }\right)$ Same as (a), (b) but for optimized control fields. The interference pattern around the origin is a measure of the coherence properties of the split condensate.

Figure 5

(Color online) Results for the magnetic microtrap of Hänsel et al. [13] and for ${}^{87}\mathrm{Rb}$ atoms confined in the $m_F=2$ state. (a) Cost function for linear $t∕T$ (solid line), square-root $\sqrt{t∕T}$ (dashed line), and optimized (symbols) variation of $\lambda \left(t\right)$. The optimized $J(\psi ,\lambda )$ is magnified in the left inset. The right inset reports the optimal $\lambda \left(t\right)$ for $T=8$ (solid line) and $15\mathrm{ms}$ (dashed line). (b) Contour plot of magnetic confinement potential as a function of $\lambda$. (c), (d) Wave-function evolution for linear variation of $\lambda$ and transfer times of $T=\left(\mathrm{c}\right)$ 15 and (d) $8\mathrm{ms}$. $\left({\mathrm{c}}^{\prime }\right),\left({\mathrm{d}}^{\prime }\right)$ Same as (c), (d) but for optimized control.

Figure 6

(Color online) Results for the magnetic microtrap of Lesanovsky et al. [21] and for ${}^{87}\mathrm{Rb}$ atoms confined in the $m_F=2$ state. (a) Magnetic confinement potential for different rf field strengths. (b) Linear (dashed line) and optimized control parameter $\lambda \left(t\right)$ as obtained from the solutions of the one-dimensional (solid line) and two-dimensional (dotted line, indistinguishable from solid line) Schrödinger equation. (c) Density plot of wave-function evolution $\int \mid \psi (x,y;t)\mid dy$ for linear $\lambda$ variation. (d) Same as (c) but for optimized $\lambda \left(t\right)$. (e) Density plot of wave-function evolution $\int \mid \psi (x,y;t)\mid dx$ for optimized $\lambda \left(t\right)$.

Figure 7

(Color online) Results of our simulations performed with the nonlinear Gross-Pitaevskii equation for the double-well potential (9). (a) Ground-state wave function ${\psi }_0$ and effective potential $V_0+\kappa {\mid {\psi }_0\mid }^2$ for $\lambda =0$ (single well) and different values of the nonlinearity parameter $\kappa$. (b) Density plot of the cost function $J(\psi ,\lambda )$ as a function of transfer time $T$ and nonlinearity parameter $\kappa$, and for a linear $\lambda$ variation. The contour at $\kappa =0$ corresponds to the solid line shown in Fig. 3a. (c) Cost function as a function of $\kappa$ and for transfer time $T=8$. The solid line and symbols correspond to the linear and optimized variation of $\lambda$, respectively. $\left(\mathrm{d}\right),\left({\mathrm{d}}^{\prime }\right)$ Density plot of $\mid \psi (x,t)\mid$ for $\kappa =20$ and $T=8$, and for a (d) linear and $\left({\mathrm{d}}^{\prime }\right)$ optimized $\lambda$ variation.