Multiparticle Quantum Walks and Fisher Information in One-Dimensional Lattices

Recent experiments on quantum walks (QWs) demonstrated a full control over the statistics-dependent walks of single particles and two particles in one-dimensional lattices. However, little is known about the general characterization of QWs at the many-body level. Here, we rigorously study QWs, Bloch oscillations, and the quantum Fisher information for three indistinguishable bosons and fermions in one-dimensional lattices using a time-evolving block decimation algorithm and many-body perturbation theory. We show that such strongly correlated QWs not only give rise to statistics-and-interaction-dependent ballistic transports of scattering states and of two- and three-body bound states but also allow a quantum enhanced precision measurement of the gravitational force. In contrast to the QWs of the fermions, the QWs of three bosons exhibit strongly correlated Bloch oscillations, which present a surprising time scaling $t^3$ of the Fisher information below a characteristic time $t_0$ and saturate to the fundamental limit of $t^2$ for $t>t_0$.

Figure 1

The spectra of three fermions (a),(b) and three bosons (d),(e) for $L=61$ and different values of interaction strength. Each point represents an eigenenergy $E$ for a given total momentum $K$, and the red and blue dots correspond to the energies of three- and two-body bound states, respectively. (c) and (f) show the spectra of the 3BSs of fermions and bosons, respectively. The solid lines denote the perturbation results of Eqs. (2) and (3), which agree well with the ED calculation (symbols). All spectra in either (c) or (f) are shifted by a constant.

Figure 2

Time evolutions of density distribution for three fermion (a)–(d) and three boson (e)–(h) systems with $L=101$ and different values of attractions; see text. Results (a)–(d) match the analytic results for the XXZ Heisenberg model [44].

Figure 3

Density-density correlation functions $C_{i,j}$ for both the fermionic (upper panel) and bosonic (lower panel) systems with a size $L=101$ at the time $t=22$. Corresponding profiles $C_{i=0,j}$ around $j=0$ are shown in subsets.

Figure 4

Left (a)–(c) and right (d)–(f) panels show the time evolutions of density distributions for three bosons and three fermions with $L=101$ and $F=0.1$, respectively. Multiple fractional Bloch oscillations are observed. The corresponding frequencies are shown in the bottom row.

Figure 5

The time-dependent quantum FI $\mathcal{F}/4t^2$ vs time $t$ for single-particle and multiparticle (co)-BOs. Parameters: $L=101$, $V=-10$, and $F=0.1$. The FI is proportional to $t^3$ for $t<t_0$, while $\mathcal{F}\sim t^2$ for $t>t_0$.