Quantum random walks using quantum accelerator modes

We discuss the use of high-order quantum accelerator modes to achieve an atom optical realization of a biased quantum random walk. We first discuss how one can create coexistent quantum accelerator modes, and hence how momentum transfer that depends on the atoms’ internal state can be achieved. When combined with microwave driving of the transition between the states, a different type of atomic beam splitter results. This permits the realization of a biased quantum random walk through quantum accelerator modes.

Figure 1

(Color online) Numerical simulation for the quantum accelerator modes that are produced with $g=9.81{\mathrm{ms}}^{-2}$, $T=132.0\mu \mathrm{s}$. In column 1 (a, c, e), ${\varphi }_d=0.8\pi$ and the (5, 2) quantum accelerator mode is produced; in column 2 (b, d, f), ${\varphi }_d=2.4\pi$ and the (3, 1) quantum accelerator mode results. (a) and (b) show the momentum variation with the number of kicks for a single initial plane wave with $\beta =0$, (c) and (d) show the evolution of an atomic cloud with initial temperature $5\mu \mathrm{K}$, and (e) and (f) show stroboscopic Poincaré sections determined by Eq. (2), with $T=132.0\mu \mathrm{s}$ $(ϵ=-0.135)$. The colorbar indicates the population, in arbitrary units.

Figure 2

(Color online) Variation of the percentage of atoms in the (5, 2) and (3, 1) quantum accelerator modes as ${\varphi }_d$ changes. The atomic ensemble is prepared at $5\mu \mathrm{K}$, and $T=132.0\mu \mathrm{s}$. The (5, 2) mode (squares) appears with lower kicking strength, and (3, 1) mode (circles) appears when the kicking strength increases. The error bar illustrates the typical spread of population in a specific quantum accelerator mode obtained from the simulation.

Figure 3

(Color online) Numerical simulation for the quantum accelerator modes that are produced with $g=20.10{\mathrm{ms}}^{-2}$, $T=137.0\mu \mathrm{s}$. In column 1 (a, c, e), ${\varphi }_d=0.8\pi$ and the (5, $-4$) quantum accelerator mode is produced; in column 2 (b, d, e), ${\varphi }_d=2.4\pi$ and the (1, $-1$) (b, d, f) quantum accelerator mode results. (a) and (b) show the momentum variation with the number of kicks for a single initial plane wave with $\beta =0$, (c) and (d) show the evolution of an atomic cloud with initial temperature $5\mu \mathrm{K}$, and (e) and (f) show stroboscopic Poincaré sections determined by Eq. (2), with $T=137.0\mu \mathrm{s}(ϵ=0.336)$. The colorbar indicates the population, in arbitrary units.

Figure 4

(Color online)(a) Momentum variation with number of kicks, for $\beta =0$, ${\varphi }_d=2.2\pi$ for state $\mid a⟩$ and ${\varphi }_d=0.7\pi$ for state $\mid b⟩$, with $T=132.0\mu \mathrm{s}$ and $g=9.81{\mathrm{ms}}^{-2}$. A state-flipping microwave pulse is applied after the 25th kick. The zoom-in around the switch point is shown in (b).

Figure 5

(Color online) (a) Phase map of quantum accelerator modes, with $\beta =0.1$, $T=131.0\mu \mathrm{s}$, $g=7.26{\mathrm{ms}}^{-2}$, ${\varphi }_d=3.8\pi$ for state $\mid a⟩$ (red dots, ) and ${\varphi }_d=0.6\pi$ for state $\mid b⟩$ (blue dots). (b) Momentum variation with number of kicks; a state-flipping $\pi$ pulse occurs after the 8th kick.

Figure 6

(Color online) The momentum distribution (in arbitrary units) of the biased quantum random walk with a Hadamard coin after 50 steps, starting in state $\mid a⟩\otimes \mid 0⟩$ for (a) and (c), and state $\mid b⟩\otimes \mid 0⟩$ for (b) and (d). The parameter $\gamma$, defined in Eq. (7), is 0 for (a), (b) and 0.5 for (c), (d). The momentum increase is 0.25 units per step to the negative direction and 0.1 units per step to the positive direction.