Coherent Atom Transport via Enhanced Shortcuts to Adiabaticity: Double-Well Optical Lattice

Theoretical studies of coherent atom transport have as yet mainly been restricted to one-dimensional model systems with harmonic trapping potentials. Here we investigate this important phenomenon—a prerequisite for a variety of quantum-technology applications based on cold neutral atoms—under much more complex physical circumstances. More specifically yet, we study fast atomic transport in a moving double-well optical lattice, whose three-dimensional (anharmonic) potential is nonseparable in the $x$-$y$ plane. We first propose specific configurations of acousto-optic modulators that give rise to the moving-lattice effect in an arbitrary direction in this plane. We then determine moving-lattice trajectories that enable single-atom transport using two classes of quantum-control methods: shortcuts to adiabaticity (STA), here utilized in the form of inverse engineering based on a quadratic-in-momentum dynamical invariant of Lewis-Riesenfeld type, and their recently proposed modification termed enhanced STA (eSTA). Subsequently, we quantify the resulting single-atom dynamics by numerically solving the relevant time-dependent Schrödinger equations and compare the efficiency of STA- and eSTA-based transport by evaluating the respective fidelities. We show that—except for the regime of shallow lattices—the eSTA method consistently outperforms its STA counterpart. This study has direct implications for neutral-atom quantum computing based on collisional entangling two-qubit gates and quantum sensing of constant homogeneous forces via guided-atom interferometry.

Figure 1

Density plot of the optical potential of a DWOL in the $x$-$y$ plane. Two different choices for the unit cell of this lattice are indicated with the dashed black and white lines. The point around which the harmonic approximation of the full DWOL potential is developed is denoted by “X.”

Figure 2

Illustration of the interwell barrier in a DWOL for different values of angles $\beta$ and $\theta$. (a) Double well without a tilt between the two individual wells, (b) double well with a tilt between the two wells, and (c) single-well configuration. The dashed and dash-dot lines respectively represent the ground-state wave functions of individual wells in (a) and (b). In the single-well configuration [(c)] they represent the wave functions corresponding to the two lowest eigenstates (the ground state and the first excited state, respectively).

Figure 3

Schematic of a setup that enables a 2D lattice formed from a folded retroreflected beam. The two AOMs tune the frequency up and down, resulting in a moving DWOL. Two counterpropagating beams in the $z$ direction, not shown here, form an optical lattice.

Figure 4

Schematic of a setup that enables a 2D lattice formed by four different beams. The two AOMs for one pair of counterpropagating beams tune the frequency up and down, resulting in a moving DWOL within the $x$-$y$ plane. They are controlled separately through the DDS system. An additional pair of counterpropagating laser beams, not shown here, forms an optical lattice in the $z$ direction.

Figure 5

Path of a DWOL potential minimum as a function of time, obtained using the STA approach, for transport times $t_f$ comparable to (a), and an order of magnitude longer than (b) the internal timescale $T_x$. The chosen parameter values are the following: lattice depth $U_{d,0}=1500E_R$, transverse waists $w_{0,x}=w_{0,y}=4.2\times {10}^3{l}_x$, $\beta =3\pi /20$, $\theta =pi/2$, ${\varphi }_z=\pi /2$, and the transport distance is set to $d_x=5\pi /k_L\approx 160l_x$.

Figure 6

Path of a DWOL potential minimum as a function of time, obtained using the eSTA approach, for transport times $t_f$ that are comparable to (a) and an order of magnitude longer than (b) the internal timescale $T_x$. The parameters chosen are the following: lattice depth $U_{d,0}=1500E_R$, transverse waists $w_{0,x}=w_{0,y}=4.2\times {10}^3{l}_x$, $\beta =3\pi /20$, relevant angles $\theta =\pi /2$, ${\varphi }_z=\pi /2$, and the transport distance is set to $d_x=5\pi /k_L\approx 160l_x$.

Figure 7

The squared modulus $|{\psi }_0(x,y,z=0){|}^2$ of the ground-state wave function (i.e., the ground-state probability density) in the $x$-$y$ plane for $U_{d,0}=300E_R$. The transverse beam waists are set to $w_{x/y,0}=4.2\times {10}^3{l}_x$. At the same time, the relevant angles are chosen such that $\beta =3\pi /20$ and $\theta =\varphi =\pi /2$.

Figure 8

Illustration of the motion of a single-atom wave packet located in the left well of a double well at $t=0$ and transported to the distance $d_x=30.2l_x$ at $t=t_f$. The snapshots of the wave-packet motion, which corresponds to an eSTA-based moving-DWOL trajectory, are shown for (a) $t=0.2t_f$, (b) $t=0.4t_f$, and (c) $t=0.8t_f$. The values of the system parameters $(U_{d,0},w_{0,x},w_{0,y},\beta ,\varphi ,\theta )$ are the same as in Fig. 7.

Figure 9

The dependence of the atom-transport fidelity on the transport time $t_f$ for a lattice depth $U_{d,0}$ of (a) $50E_R$, (b) $100E_R$, and (c) $150E_R$. The transport distance $d_x$ along the $x$ axis is set to $158l_x$, while the transverse beam waists are $w_{x/y,0}=4.2\times {10}^3{l}_x$. The relevant angles are chosen such that $\beta =3\pi /20$ and $\theta =\varphi =\pi /2$.

Figure 10

The dependence of the atom-transport fidelity on the transport time $t_f$ for a lattice depth $U_{d,0}$ of (a) $250E_R$, (b) $400E_R$, and (c) $500E_R$. The transport distance $d_x$ along the $x$ axis is set to $158l_x$. For the Gaussian beam along the $z$ axis, the transverse beam waists are set to $w_{x/y,0}=4.2\times {10}^3{l}_x$. The relevant angles are chosen such that $\beta =3\pi /20$ and $\theta =\varphi =\pi /2$.

Figure 11

The dependence of the atom-transport fidelity on the transport time $t_f$ for a lattice depth $U_{d,0}$ of (a) $300E_R$, (b) $400E_R$, and (c) $600E_R$. The transport distance $d_y$ along the $y$ axis is set to $158l_y$. For the Gaussian beam along the $z$ axis, the transverse beam waists are set to $w_{x/y,0}=4.2\times {10}^3{l}_x$. The relevant angles are chosen such that $\beta =3\pi /20$ and $\theta =\varphi =\pi /2$.

Figure 12

The dependence of the atom-transport fidelity on the transport time $t_f$ for a potential strength $U_{d,0}$ of (a) $300E_R$, (b) $400E_R$, and (c) $600E_R$. The transport distance $d_r$ along the diagonal is set to $448l_r$. For the Gaussian beam along the $z$ axis, the transverse beam waists are set to $w_{x/y,0}=4.2\times {10}^3{l}_x$. The relevant angles are chosen such that $\beta =3\pi /20$ and $\theta =\varphi =\pi /2$.

Figure 13

The dependence of the atom-transport fidelity on the transport time $t_f$ for a lattice depth $U_{d,0}$ of $1500E_R$. The transport distance $d_x$ along the $x$ axis is set to (a) $127l_x$, (b) $158l_x$, and (c) $191l_x$. For the Gaussian beam along the $z$ axis, the transverse beam waists are set to $w_{x/y,0}=4.2\times {10}^3{l}_x$. The relevant angles are chosen such that $\beta =3\pi /20$ and $\theta =\varphi =\pi /2$.

Figure 14

The dependence of the atom-transport fidelity on the transport time $t_f$ for a potential strength $U_{d,0}$ of $1500E_R$. The transport distance $d_y$ along the $y$ axis is set to (a) $127l_y$, (b) $158l_y$, and (c) $191l_y$. For the Gaussian beam along the $z$ axis, the transverse beam waists are set to $w_{x/y,0}=4.2\times {10}^3{l}_x$. The relevant angles are chosen such that $\beta =3\pi /20$ and $\theta =\varphi =\pi /2$.

Figure 15

The dependence of the atom-transport fidelity on the transport time $t_f$ for a lattice depth $U_{d,0}$ of $1500E_R$. The transport distance $d_r$ along the diagonal is set to (a) $127l_r$, (b) $158l_r$, and (c) $191l_r$. For the Gaussian beam along the $z$ axis, the transverse beam waists are set to $w_{x/y,0}=4.2\times {10}^3{l}_x$. The relevant angles are chosen such that $\beta =3\pi /20$ and $\theta =\varphi =\pi /2$.