Spin-1 quantum walks

We study the quantum walks of two interacting spin-1 bosons. We derive an exact solution for the time-dependent wave function, which describes the two-particle dynamics governed by the one-dimensional spin-1 Bose-Hubbard model. We show that propagation dynamics in real space and mixing dynamics in spin space are correlated via the spin-dependent interaction in this system. The spin-mixing dynamics has two characteristic frequencies in the limit of large spin-dependent interactions. One of the characteristic frequencies is determined by the energy difference between two bound states, and the other frequency relates to the cotunneling process of a pair of spin-1 bosons. Furthermore, we numerically analyze the growth of the spin correlations in quantum walks. We find that long-range spin correlations emerge showing a clear dependence on the sign of the spin-dependent interaction and the initial state.

Figure 1

Spin-mixing dynamics in the single-site system $\left(J=0\right)$. Solid and dashed lines represent the populations as a function of time in the spin state $\alpha =0$ and the sum of the populations in spin states $\alpha =\pm 1$, respectively.

Figure 2

Energy spectra for $U_2/U_0=0.3$ and $U_0/J=10$ as a function of the center-of-mass quasimomentum $K$. Thick solid and dashed lines correspond to the energy bands for bound states with $F=2$ and $F=0$, respectively. The bundle of thin solid lines represents the scattering continuum. The energy is defined in units of $J$.

Figure 3

Normalized number of bound states and scattering states in each space spanned by Eqs. (8) and (9) as a function of the spin-dependent interaction. Thick and thin lines represent the normalized number of atoms in bound states and scattering states, respectively. Solid and dashed lines correspond to the indices $\lambda =U_2$ and $\lambda =-2U_2$, respectively. We assume a condition where two $\alpha =0$ atoms occupy the same site and we choose $U_0/J=1$.

Figure 4

Interaction dependence of $X_B,X_S$, and $X_{BS}$ at $t=0$. (a) $U_2/U_0=0.3$. Solid and dashed lines represent $X_B$ and $X_S$, respectively. (b) Interference term $X_{BS}\left(0\right)$ for three $U_2/U_0$ values. Solid, dot-dashed, and dashed lines correspond to $U_2/U_0=0.3$, 1, and 5, respectively.

Figure 5

Time dependence of the interference term $X_{BS}$. We choose three sets of interactions $U_0$ and $U_2$ in the vicinity of the maximum point of $X_{BS}\left(0\right)$ in Fig. 4. Solid, dot-dashed, and dashed lines correspond to $(U_0/J,U_2/U_0)=(1,0.3),(2,1)$, and $(0.5,5)$, respectively.

Figure 6

Spin-mixing dynamics in a 1D optical lattice for four $U_2/J$ values under the condition $U_2/U_0=0.3$: (a) $U_2/J=6$, (b) $U_2/J=3$, (c) $U_2/J=1.5$, and (d) $U_2/J=0.3$. Solid and dashed lines represent populations in spin state $\alpha =0$ and the sum of the populations in spin states $\alpha =\pm 1$, respectively.

Figure 7

(a) Time dependence of the bound-state term $X_B\left(t\right)$. Interactions are $U_0/J=10$ and $U_2/U_0=0.3$. Envelope functions plotted by dashed lines correspond to $\pm {\mathcal{J}}_0\left({\epsilon }^{\prime }t\right)$. (b) Three characteristic frequencies of spin-mixing dynamics as a function of $U_0$: ${\omega }_{\mathrm{low}}$ (circle), ${\omega }_{-}\left(\epsilon \right)$ (dashed line), and ${\omega }_{-}\left({\epsilon }^{\prime }\right)$ (solid line).

Figure 8

Two-particle spin correlations at $t=5/J$ calculated with $U_0/J=2$ and $\theta =0$. The vertical and horizontal axes represent the lattice site indices. (a), (b) Spin correlations starting from the initial state $|{\mathrm{\Psi }}_{0,0}^0\left(0\right)\rangle$ where two spin-1 bosons are placed at the origin. (c), (d) Spin correlations for the initial state $|{\mathrm{\Psi }}_{0,1}^0\left(0\right)\rangle$ where two bosons are placed at the origin and at the first site. We assume a positive $U_2/U_0$ in (a) and (c) and a negative $U_2/U_0$ in (b) and (d).