Figure 1

(Color online) Configuration with two rows of traps; the qubit is implemented “perpendicular” to the rows (dashed rectangles show which two traps form each qubit). The upper (lower) row moves left (right) with constant velocity (see arrows on right). (I) After the first step $\mid \psi ⟩=1∕\sqrt{2}(\mid -1,-⟩+\mid +1,+⟩)$. (II) and (III) The coin operation, in this case a Hadamard gate, is performed when the traps pass each other. (IV) The shift $O_{1\mathrm{D}}$ is implicit through a redefinition of the qubits. After an even (odd) number of shift operations only the even (odd) qubits (compared to the standard quantum walk definition) are defined. (V) The probability distribution after the sixth displacement operation. For the numerical simulation we used a potential that, along the line connecting the centers of two traps, reads $V\left(x\right)=\hslash {\omega }_x\min \{{\alpha }^2{(x-a)}^2,{\alpha }^2{(x+a)}^2\}$. The velocity is chosen such that during the passing of two traps a Hadamard operation is performed. The initial state is $\mid {\psi }_{\mathrm{init}}⟩=(1∕\sqrt{2})(\mid 0,-⟩+\mid 0,+⟩)$.Reuse & Permissions

Figure 2

(Color online) Configuration with one row of traps; the qubit is implemented “parallel” to the rows (gray boxes). (I) After the first step $\mid \psi ⟩=(\mid -1,+⟩+\mid +1,-⟩)∕\sqrt{2}$; (II) and (III) traps inside each qubit are approached to give the coin operation; (IV) and (V) the shift $O_{1\mathrm{D}}$ is realized through approaching traps of adjacent qubits.Reuse & Permissions

Figure 3

The probability distribution to find the atom at a specific trap site, with (I) the ground state and (II) the first excited state as the initial vibrational state, for a one-dimensional quantum walk on a finite line of 62 traps; from top to bottom distributions after $t=10$, 20, and 30 steps are shown. Parameters $V_0=200\hslash {\omega }_x$, $\alpha a_{\max }=60$, $\alpha a_{\min }=28.8$, and ${\omega }_x{t}_r=100$; ${\omega }_x{t}_{i,\pi }=20.25$ for the $\pi$ pulse and ${\omega }_x{t}_{i,\pi ∕2}=112$ for the $\pi ∕2$ pulse. (For simplicity we fixed $a_{\min }$ and then searched for the smallest $t_i$ that produces the desired operation. As tunneling already happens for $a>a_{\min }$, $t_i=0$ does not give the identity operation, and for this reason $t_{i,\pi ∕2}>t_{i,\pi }$.) (III) Like (II), but with ${\omega }_x{t}_r=200$, such that nonadiabatic excitations are suppressed.Reuse & Permissions

Figure 4

Probability distributions after $t=20$ steps for thermal Boltzmann distributions of vibrational modes; initial ground-state population (I) 50% and (II) 25%. All other parameters as in Fig. 3.Reuse & Permissions

Figure 5

The effect of shaking of the trap position on the quantum walk. Shaking is modeled by a sinusoidal variation of the trap distances around the perfect value, with frequency ${\omega }_{\text{shake}}=0.01{\omega }_x$ and amplitude $\alpha \Delta a$ (all the other parameters are as in Fig. 3). (I) The probability distributions for various values of $\alpha \Delta a$ (after tracing over the coin degree of freedom). (II) Ground-state population, (III) variance, and (IV) variational distance $\nu$ from the uniform distribution after $t=17$ steps. In (II) and (III) dashed lines and squares give the respective values for the full population, dotted lines and circles for the ground state only.Reuse & Permissions

Figure 6

(Color online) Implementation of the coin and the coin operator $C_{2\mathrm{D}}={\mathrm{H}}_{\mathrm{SD}}\otimes {\mathrm{H}}_{\mathrm{HF}}$ for the two-dimensional walk. (a) The four levels are formed as a tensor product of a delocalized qubit (left and right traps) and a hyperfine qubit (dark and gray filled circles symbolize the $\mid +⟩$ and $\mid -⟩$ hyperfine states; full and dashed lines denote the respective trapping potentials). The initial state is $\mid ++⟩$. $\left(\mathrm{a}\right)\to \left(\mathrm{b}\right)$ For ${\mathrm{H}}_{\mathrm{SD}}$ the traps are approached (for both hyperfine states). $\left(\mathrm{b}\right)\to \left(\mathrm{c}\right)$ For ${\mathrm{H}}_{\mathrm{HF}}$ a $\pi ∕2$ pulse between the two hyperfine levels is applied to all traps simultaneously. (d) The implementation of the two-dimensional walking operator $O_{2\mathrm{D}}$ as a combination of tunneling and spin dependent transport: (A) traps in horizontally adjacent traps are approached to give a $\pi$-pulse as in the one-dimensional walk; (B) the lattice is displaced in opposite vertical directions for the two hyperfine states.Reuse & Permissions

Figure 7

(Color online) Probability distributions for the two-dimensional quantum walk obtained from alternatingly applying $C_{2\mathrm{D}}$ and $O_{2\mathrm{D}}$ to the initial state $\mid {\psi }_{\mathrm{init}}⟩=\mid (0,0),++⟩$ after $t=$ (a) 1, (b) 2, and (c) 25 steps. The coordinates on $k$ and $l$ axes label the traps; thus in the $k$ direction each two traps form one site; note that the total population of each trap is shown (sum of probabilities for both hyperfine states).Reuse & Permissions

Figure 8

(Color online) Probability distribution of the two-dimensional quantum walk with the coin operator $C_{2\mathrm{D}}^{\mathrm{ent}}$ after $t=25$ steps. The initial state is $\mid {\psi }_{\mathrm{init}}⟩=\mid (0,0),++⟩$.Reuse & Permissions