Correlations in few two-component quantum walkers on a tilted lattice

We study the effect of intercomponent interactions on the dynamical properties of quantum walkers. We consider the simplest situation of two indistinguishable noninteracting walkers on a tilted optical lattice interacting with a walker from a different component. The mediated effect of the third particle is then analyzed in the backdrop of various controlling parameters. The interaction-induced two-particle correlations are shown to be nontrivially affected by the particle statistics, choice of initial states, and tilt configurations of the lattice. Our analysis thus offers an overall picture and serves as a starting point of a study of interacting multicomponent quantum walkers.

Figure 1

Evolution of the single-particle density profile ${\gamma }_{Ai}$ for two different tilt values of the lattice and initial distance between occupied sites $d=2$. Horizontal dashed lines indicate half time of Bloch oscillation period $T_{1/2}$. As argued in the main text, these results are independent of the quantum statistics of $A$ particles. Time is measured in the natural unit of the problem, $\hslash /J$.

Figure 2

Two-particle density profile ${\mathrm{\Gamma }}_{ij}$ at the half time of a Bloch oscillation period $T_{1/2}$ with tilt value $F/J=0.2$ for a system of two noninteracting bosons (left) and fermions (right) initially prepared in the state $|{\mathrm{\Psi }}_0\rangle ={\widehat{a}}_0^{†}{\widehat{a}}_d^{†}|\mathtt{vac}\rangle$. Note that regardless of the initial distance $d$, the distributions are significantly distinct for different statistics.

Figure 3

Time dependence of mean distance $D\left(t\right)$ between noninteracting $A$ particles prepared in the state $|{\mathrm{\Psi }}_0\rangle ={\widehat{a}}_0^{†}{\widehat{a}}_d^{†}|\mathtt{vac}\rangle$ with $d=2$ in the presence of tilt strength $F/J=0.2$. Due to the bosonic enhancement, bosons spread slower than fermions and only in the limit of hard-core interactions ($U_A/J\to \infty$) are both statistics indistinguishable by this quantity. Note that for finite nonvanishing interactions between particles the Bloch oscillations are slightly slowed down and the system does not return exactly to the initial state after the Bloch period.

Figure 4

(a) Evolution of the single-particle density profiles for a system of two bosonic (left) and fermionic (right) $A$ particles initially localized at distant $d=2$ and interacting with $B$ particle ($U/J=10$) being delocalized with $\sigma =5$. (b) Corresponding two-particle density profile ${\mathrm{\Gamma }}_{ij}$ of $A$ particle and the intercomponent correlation ${\mathrm{\Xi }}_{ij}$ at the half time of a Bloch oscillation period $T_{1/2}$ for the same conditions. (c) For contrast, ${\mathrm{\Gamma }}_{ij}$ and ${\mathrm{\Xi }}_{ij}$ at $T_{1/2}$ when the $B$ particle is initially perfectly localized ($\sigma =0$). Note that for delocalized $B$ particles a specific “crosslike” structure in ${\mathrm{\Gamma }}_{ij}$ is enhanced together with the appearance of partial pairing. Compare with Fig. 5 for other initial distances $d$. In all plots tilt strength $F/J=0.2$.

Figure 5

Two-particle density profile ${\mathrm{\Gamma }}_{ij}$ at the half time of a Bloch oscillation period $T_{1/2}$ with tilt strength $F/J=0.2$ for a system of two bosons (left) and fermions (right) interacting ($U/J=10$) with the $B$ particle initially delocalized with $\sigma =5$. The initial distance between localized $A$ particles is $d=1$ and $d=3$ (top and bottom rows, respectively). Compare with Fig. 4 for other initial delocalization $\sigma$ of the $B$ particle.

Figure 6

Density distributions for the strongly interacting system ($U/J=10$) initially prepared in the state (8) with $d=2$ and $\sigma =2$ (top row), $\sigma =5$ (bottom row), and at the half time of Bloch period $T_{1/2}$ with tilt strength $F/J=0.2$. The dashed green line and red solid lines correspond to the single-particle density of $A$ particles ${\gamma }_{Ai}$ and the doublon density ${\eta }_i$, respectively.

Figure 7

Dynamics for a lattice that is tilted in opposite directions ($E_i=-F|i-i_0|$ with $F/J=0.2$), with two $A$ particles localized on both sides of the center ($d=2$), interacting with the $B$ particle centered around the middle ($U/J=10$, $\sigma =5$). Evolution of the single-particle density profiles ${\gamma }_{Ai}$ and ${\gamma }_{Bi}$ are shown in the first two rows. The next three rows respectively show the two-particle density profile ${\mathrm{\Gamma }}_{ij}$, the intercomponent correlations ${\mathrm{\Xi }}_{ij}$, and the doublon density ${\eta }_i$ (red line) with the single-particle density ${\gamma }_{Ai}$ (dashed green line) at the characteristic time $T_{1/2}$.