Quantum walk of two interacting bosons

We study the effect of interactions on the propagation of quantum correlations in the bosonic two-body quantum walk. The combined effect of interactions and Hanbury Brown–Twiss interference results in unique spatial correlations which depend on the strength of the interaction, but not on its sign. We experimentally measure the weak interaction limit of these effects using light propagating in a highly nonlinear photonic lattices. Finally, we propose an experimental approach to observe the strong interaction limit using few atoms in optical lattices.

Figure 1

Two-particle quantum walk. (a) An illustration of two identical atoms initially placed on two sites of an optical lattice, and allowed to tunnel to neighboring sites and interfere. (b) The evolution of the particle density plotted against position (in lattice sites) for the case of a single-atom quantum walk. (c) The evolution of the density for a two-atom quantum walk. This density is only weakly affected by interactions, while the two-particle correlations are strongly modified.

Figure 2

Two-particle correlations for interacting quantum walkers. Left column: Correlations after propagation time $T=4$ in units of $1/J$, where initially the two particles are placed at the same site, $|{\psi }_{\text{initial}}\rangle ={\left(a_0^{†}\right)}^2|0\rangle$. The correlations are plotted against the positions of the two particles $q$ and $r$, in lattice sites away from the origin. (a) For zero interactions, the particles show no interference in the correlations [10]. (b)–(f) As the interaction is increased, the correlations show the formation of bound pairs, while the density distribution (shown on the right-hand side of each plot) becomes increasingly localized. Right column: Similar results for an initial condition in which the two particles are placed at adjacent sites $|{\psi }_{\text{initial}}\rangle =a_1^{†}{a}_0^{†}|0\rangle$. (g) At $U=0$ the correlation shows spatial bunching. (h)–(j) The correlations change as the interaction $\left|U\right|$ is increased, while the density is only weakly affected. (k) At strong interactions the correlation is transformed to spatial antibunching, similar to the correlation that would be exhibited by two noninteracting fermions initially placed in the same configuration (l). All results are identical for both attractive and repulsive interactions.

Figure 3

Spectrum and two-particle eigenmodes of Eq. (1) for $U=8.$ The spectrum is separated into two minibands. The higher band consists of bound pair states in which the two particles only occupy the same sites (nonzero values mainly on the diagonal $r=q$, top inset), while the lower band consists of states in which the two particles have small probability to occupy the same site (lower inset). The gap is proportional to the interaction strength $U$. Attractive interactions $U<0$ yield an identical yet inverted spectrum. The position axis in the insets is given in lattice sites.

Figure 4

Experimental measurements of the fluctuation in intensity correlations for nonlinear thermal light. (a) The fluctuations in the intensity correlations in the linear case, when two thermal beams are injected into two adjacent sites, corresponding to $|{\psi }_{\text{initial}}\rangle =a_1^{†}{a}_0^{†}|0\rangle$. The results show spatial bunching [10]. (b) The predictions of the quantum theory for two noninteracting particles initially placed at the same locations. (c) Experimental results for nonlinear thermal waves. (d) The predictions of the quantum theory for two interacting particles. Note the similarity between the classical results and the quantum prediction in both cases. Positions are given in lattice sites.

Figure 5

Simulations comparing the fluctuations for the quantum (top panel) and the classical (bottom) cases, with increasing $\left|U\right|$ or nonlinear coefficient $\left|\gamma \right|$, correspondingly, for $|{\psi }_{\text{initial}}\rangle =a_1^{†}{a}_0^{†}|0\rangle$. The two sets of results look similar for weak interactions (a)–(d). However, beyond $\left|U\right|=1.5$ (e)–(h) the two results diverge—the quantum correlations transform to fermioniclike correlations, while the classical correlations become increasingly localized. In all cases the positions are given in lattice sites.