Optimal control of atom transport for quantum gates in optical lattices

By means of optimal control techniques we model and optimize the manipulation of the external quantum state (center-of-mass motion) of atoms trapped in adjustable optical potentials. We consider in detail the cases of both noninteracting and interacting atoms moving between neighboring sites in a lattice of a double-well optical potentials. Such a lattice can perform interaction-mediated entanglement of atom pairs and can realize two-qubit quantum gates. The optimized control sequences for the optical potential allow transport faster and with significantly larger fidelity than is possible with processes based on adiabatic transport.

Figure 1

(Color online) (a) Initial configuration: the potential (solid lines) is in the $\lambda /2$ configuration and the single-particle wave functions are localized in the left (dashed) or right (dotted) well. (b)–(c) Intermediate snapshots obtained by lowering the central barrier and tilting the potential. (d) End of the process: the particle initially in the right well ends in the ground state of the single well, while the particle initially in the left well ends in the first excited state.

Figure 2

Two possible sequences (a) (left) and (b) (right) employed to shift the atoms from a double- to a single-well configuration are shown in the left and right part of the panel. For each sequence we show the time dependence of $V_0$, $\beta$, $\theta$ for a sequence duration $T$. For sequence (b) we show the time dependence of $\theta$ for two settings of $\text{EOM}\theta$: $-0.42\pi$ (solid) and $-0.48\pi$ (dashed).

Figure 3

Instantaneous spectrum of the 1D Hamiltonian for sequences (a) and (b) for $\text{EOM}\theta$: $-0.42\pi$.

Figure 4

Population of the first three eigenstates of the Hamiltonian, ground (top), first excited (middle), second excited (bottom), at the end of sequence a as a function of the sequence duration $T$. The experimental data (symbols) are compared to the four sets of numerical data (lines) obtained for $\theta /\pi =-0.454+\Delta {\theta }_a/\pi$, with $\Delta {\theta }_a/\pi =0$ (dot-dash), $-0.02$ (dash), $-0.03$ (solid), and $-0.04$ (dot), while $-0.454$ is the nominal value of $\theta /\pi$ expected from the calibrations.

Figure 5

Population (overlap absolute squared) of the first three eigenstates of the Hamiltonian at the end of sequence (b) as a function of ${\theta }_b$. The duration of the sequence is fixed to $T=0.5\text{ms}$. The experimental data (symbols) are in good agreement with the numerical data (lines). In this graph the $x$ axis for the experimental has been shifted by an offset of $-0.015$ with respect to the initial calibration.

Figure 6

Infidelities $\left(1-f_n^{\alpha }\right)$ for the atom initially in the left ($\alpha =L,n=1$, squares) and in the right well ($\alpha =R,n=0$, plus) as a function of the optimization step.

Figure 7

Initial (dotted) and optimized waveforms (solid) $\beta \left(t\right)$ and $\theta \left(t\right)$ as a function of time for a sequence of $T=0.15\text{ms}$.

Figure 8

(Color online) Comparison between the evolution of the atoms with and without optimal control. Top (left to right): nonoptimized case, absolute square value of the wave functions as a function of time (atoms initially in the left and right well respectively); 1D trapping potential as a function of time; projections $p_n\left(t\right)$ at time $t$ of the state initially in the left well onto the instantaneous eigenstates $|{\varphi }_n\left(t\right)⟩$ with $n=0$ (blue solid), $n=1$ (red dashed), $n=2$ (green dotted), $n=3$ (magenta dot-dashed). Bottom: analogous plots for the optimized case.

Figure 9

(Color online) Absolute square values of the relevant symmetric wave functions in the coordinates of the two atoms: (a) Initial wave function in the state $|{\tilde{\Psi }}_{\text{in}}⟩$ with one atom in the left well and one atom in the right well; (b) wave function of the target state $|{\tilde{\Phi }}_{\text{tg}}⟩$. (c) evolved wave function using the nonoptimized sequences of Fig. 2b giving a fidelity $F_{\text{int}}=0.22$ (for $T=150\mu \text{s}$); (d) evolved wave function using the optimized sequences of Fig. 7 giving a fidelity $F_{\text{int}}=0.93$.

Figure 10

The normalized Fourier transform magnitudes $\left|\tilde{\beta }\left(f\right)\right|$ (solid) and $\left|\tilde{\theta }\left(f\right)\right|$ (dashed) of the optimized control sequences $\beta \left(t\right)$ and $\theta \left(t\right)$ shown in Fig. 7. The spectra are normalized to the value at the fundamental frequency $1/T=6.67\text{kHz}$.