Probing the sign of the Hubbard interaction by two-particle quantum walks

We address the discrimination between attractive and repulsive interaction in systems made of two identical bosons propagating on a one-dimensional lattice, and suggest a probing scheme exploiting the dynamical properties of the corresponding two-particle quantum walks. In particular, we show that the sign of the interaction leaves a clear signature in the dynamics of the two walkers, which is governed by the Hubbard model, and in their quantum correlations, thus permitting one to discriminate between the two cases. We also prove that these features are strictly connected to the band structure of the Hubbard Hamiltonian.

Figure 1

(Left) Band structure (spectrum) of ${\mathcal{H}}_N\left(V\right)$ and $-{\mathcal{H}}_N(-V)$ for $V=8$ and $N=5$. Energies are given in units of $J$, wave vectors in units of $2\pi /N$. (Right) Deviation $D_V\left(N\right)$ between the spectra of ${\mathcal{H}}_N\left(V\right)$ and ${\mathcal{H}}_N(-V)$ for $V=8$ at different values of $N$. The two spectra are exactly opposite $[D_V\left(N\right)=0]$ only for even $N$. The red line is a phenomenological fit for odd $N$ values.

Figure 2

Band structure $\omega \left(K\right)$ of a chain with $N=30$ sites at $V/J=+8$ (left, red) and $V/J=-8$ (right, blue).

Figure 3

(Upper panel) Evolution of two-sites correlations ${\tilde{\mathrm{\Gamma }}}_{i,j}$ in states $\left|{\mathrm{\Psi }}_1\right.〉,\left|{\mathrm{\Psi }}_2\right.〉$, and $\left|{\mathrm{\Psi }}_3\right.〉$ under ${\mathcal{H}}_{+}$ and ${\mathcal{H}}_{-}$, with $V=\pm 8$ and $J=1$. (Lower panel) The same for states $\left|{\mathrm{\Psi }}_4\right.〉$ and $\left|{\mathrm{\Psi }}_5\right.〉$.

Figure 4

Time evolution of entanglement of particles under ${\mathcal{H}}_{\pm }$. (Upper left) States $\left|{\mathrm{\Psi }}_1\right.〉,\left|{\mathrm{\Psi }}_2\right.〉$, and $V=\pm 8$. (Upper right) States $\left|{\mathrm{\Psi }}_4\right.〉,\left|{\mathrm{\Psi }}_5\right.〉$ under ${\mathcal{H}}_{\pm }, V=\pm 8$. (Lower left) State $\left|{\mathrm{\Psi }}_5\right.〉$ and different values of $V$. (Lower right) State $\left|{\mathrm{\Psi }}_6\right.〉$ and different values of $V$.

Figure 5

Evolution of two-sites correlations ${\tilde{\mathrm{\Gamma }}}_{i,j}$ in state $\left|{\mathrm{\Psi }}_5\right.〉$ under ${\mathcal{H}}_{+}$ and ${\mathcal{H}}_{-}$ for different values of $V$ and for $J=1$. Differences between evolutions are more apparent at low potential energies (i.e., $V\sim J$).

Figure 6

(a) Band structures for ${\mathcal{H}}_{+}$ (red crosses) and ${\mathcal{H}}_{-}$ (blue dots) at $V=\pm 8$. The main subband and the miniband are clearly visible, as well as the symmetry of the two spectra. (b) Evolution of the eigenvalues ${\omega }_i^{+}$ at increasing $V$: for high values of the potential energy, there is a clear separation between the miniband (which becomes almost flat) and the main subband, while at low $V$ the two subbands are entwined. The gray vertical dashed line indicates $V=8$ (a). (c) Radial wave function for the first eigenstate of ${\mathcal{H}}_{+}$ (upper, red) and ${\mathcal{H}}_{-}$(lower, blue).

Figure 7

Eigenvalues of ${\mathcal{H}}_{+}$ with relative multiplicity (degeneracy), ordered by decreasing energy.

Figure 8

Projections of number states ${\left|1,1\right.〉}_s, {\left|1,2\right.〉}_s, {\left|1,3\right.〉}_s$, and ${\left|1,4\right.〉}_s$ over the eigenstates of ${\mathcal{H}}_{+}$ (first row, red) and ${\mathcal{H}}_{-}$ (second row, blue) for a potential energy of $\left|V\right|=8$. Notice that, as expected, ${\left|1,4\right.〉}_s$ behaves like ${\left|1,2\right.〉}_s$, and their projections are also identical in modulus (since the two states are translationally equivalent).

Figure 9

Total difference $\mathrm{\Delta }\left(V\right)$ among the projections of a state $\left|\mathrm{\Psi }\right.〉$ over the eigenstates of $\mathcal{H}$ for positive and negative $V$.