Quantum walks and quantum simulations with Bloch-oscillating spinor atoms

We propose a scheme for the realization of a quantum walker and a quantum simulator for the Dirac equation with ultracold spinor atoms in driven optical lattices. A precise control of the dynamics of the atomic matter wave can be realized using time-dependent external forces. If the force depends on the spin state of the atoms, the dynamics will entangle the inner and outer degrees of freedom, which offers unique opportunities for quantum information and quantum simulation. Here we introduce a method to realize a quantum walker based on the state-dependent transport of spinor atoms and a coherent driving of the internal state. In the limit of weak driving the dynamics are equivalent to that of a Dirac particle in $1+1$ dimensions. Thus it becomes possible to simulate relativistic effects such as Zitterbewegung and Klein tunneling.

Figure 1

Dynamics of ultracold Cesium atoms in a driven optical lattice. (a) A static external field leads to Bloch oscillations. (b) A sinusoidal driving of the field leads to directed transport. We assumed an optical lattice with a wave length $\lambda =1064\hspace{0.5em}\mathrm{nm}$, a depth of $V_0=E_R$ and a maximum field strength of $F_0/m=0.42\hspace{0.5em}\mathrm{m}/{\mathrm{s}}^2$. The time is given in units of the Bloch period $T_B=8.44\hspace{0.5em}\mathrm{ms}$ and the period of the external driving $T_{\mathrm{drive}}=10\hspace{0.5em}\mathrm{ms}$, respectively.

Figure 2

Quantum walk of ultracold Cs atoms in a driven magnetic gradient field. (a) The coin operations: Rabi frequency of the microwave $\pi /2$ pulses inducing transitions between the two hyperfine levels. (b) The shift operation: A driven magnetic gradient field induces state-dependent transport. (c) The resulting dynamics of the total atomic density $|{\psi }_{\uparrow }(x,t)|{}^2+|{\psi }_{\downarrow }(x,t)|{}^2$. Parameters are given in the text.

Figure 3

Characterization of a Bloch quantum walk. (a) Increase of the standard deviation $\Delta x=\sqrt{\langle x^2\rangle -\langle x\rangle {}^2}$. (b) The total atomic population $|{\psi }_{\uparrow }(x,t)|{}^2+|{\psi }_{\downarrow }(x,t)|{}^2$ after 15 driving periods. The parameters are given in the text.

Figure 4

Zitterbewegung of spinor atoms in a driven magnetic gradient field. (a) Evolution of the position expectation value for different values of the rotation angle $\theta$. (b) and (c) Evolution of the probability densities $|{\psi }_{\uparrow }(x,t)|{}^2$ and $|{\psi }_{\downarrow }(x,t)|{}^2$ for $\theta =0.1\pi$. The remaining parameters are given in the text.

Figure 5

Dynamics of spinor atoms in the Dirac regime with an additional potential step. (a)–(c) Evolution of the atomic density (left) and the position expectation value $\langle x(t)\rangle$ (right) for different heights of the potential step: (a) $V_{\mathrm{step}}/E_R=0$, (b) 0.015, and (c) 0.1. (d) Position expectation value $\langle x(t)\rangle$ after a fixed propagation time $t_{\mathrm{final}}=150\hspace{0.5em}\mathrm{ms}$ as a function of the potential step $V_{\mathrm{s}}$. Results of a full quantum simulation (solid blue line) are compared to the effective discrete Dirac approximation (open circles). The remaining parameters are the same as in Fig. 4 with $\theta =0.1\pi$.

Figure 6

Quantum walk in the presence of decoherence. (a) Increase of the standard deviation $\Delta x(t)$ for different values of the dephasing rate $\kappa$. (b) The standard deviation in a loglog plot showing a different scaling $\Delta x(t)~t^{\alpha }$ in the quantum regime ($\kappa =0$) and in the classical regime ($\kappa =100\hspace{0.5em}{\mathrm{s}}^{-1}$). The dashed black line is a linear fit with the result $\alpha =0.94$ and $\alpha =0.59$, respectively. (c) The standard deviation $\Delta x(t_{\mathrm{final}})$ after a fixed time $t_{\mathrm{final}}=150\hspace{0.5em}\mathrm{ms}$ as a function of $\kappa$.

Figure 7

Atomic Zitterbewegung in the presence of decoherence. Shown is the position expectation value $\langle x(t)\rangle$ for different values of the dephasing rate $\kappa$. Parameters are the same as in Fig. 4 with $\theta =0.2\pi$.