Quantum Walk in Degenerate Spin Environments

We study the propagation of a hole in degenerate (paramagnetic) spin environments. This canonical problem has important connections to a number of physical systems, and is perfectly suited for experimental realization with ultracold atoms in an optical lattice. At the short-to-intermediate time scale that we can access using a stochastic-series-type numeric scheme, the propagation turns out to be distinctly nondiffusive with the probability distribution featuring minima in both space and time due to quantum interference, yet the motion is not ballistic, except at the beginning. We discuss possible scenarios for long-term evolution that could be explored with an unprecedented degree of detail in experiments with single-atom resolved imaging.

Figure 1

Hole propagation in a spin environment. (a) The hole starts at the lower left corner. As it moves, it leaves behind a trace of altered spins. In (b),(c), the final destination is the upper right corner, yet the paths taken differ (purple dashed lines), and so do the resulting spin states. In (d), we show a self-retracing trajectory described as a necklace of zero-area loops; all such trajectories interfere because at the end the hole returns to the origin, and the spin environment is inherently preserved.

Figure 2

Probability of finding the hole at the origin as a function of time (in units of inverse hopping, $1/J$). The panels (a) and (b) show the same data for different time intervals. The curves correspond to different spin values $N=2$, 3,4, and $\infty$, the Brinkman and Rice self-retracing approximation, and ballistic propagation (see legend). All curves, except that for ballistic motion, follow each other closely up to the first minimum at $t\approx 0.9$, but beyond that point, the details of the spin environment become important to the hole propagation. The BR approximation is very accurate up to the first minimum, but quickly fails at longer $t$.

Figure 3

Spatial probability distributions $\rho (\mathbf{r},t)$ [Eq. (5)]. Images (a)–(c) correspond to $N=\infty$ at $t=0.9$, $t=1.375$ and $t=1.875$, while (d) shows the case of ballistic motion in a polarized lattice at $t=2$. In (a), $\rho (0)$ is close to zero due to destructive interference, changing to maximum in (b) only to become a local minimum again in (c) at $t=1.875$, which also corresponds to a shallow local minimum in the survival probability, see Fig. 2. In the ballistic case, strong destructive interference is possible at arbitrary long times.

Figure 4

(a) Logarithmic plot of the mean-square displacement as a function of time. The lines correspond to cubic polynomial fits to data. (b) MSD exponent $\alpha$. At small $t$ it is close to 2, as expected for coherent propagation in a perfect lattice. Within the time scale of these simulations, $\alpha$ does not converge to a constant value, so the precise nature of the propagation at large $t$ (specifically if the motion is diffusive) remains an open question.

Figure 5

Hole propagation in a dynamically passive environment amounts to a quantum mechanical realization of the $N$-puzzle game [29], where a particular order is sought with as few tile movements as possible. While the classical problem is NP hard, in the quantum mechanical system, all possible trajectories are realized in superposition.