Non-Hermitian description of the dynamics of interchain pair tunneling

We study interchain pair tunneling dynamics based on an exact two-particle solution for a two-leg ladder. We show that the Hermitian Hamiltonian shares a common two-particle eigenstate with a corresponding non-Hermitian Hubbard Hamiltonian in which the non-Hermiticity arises from an on-site interaction of imaginary strength. Our results provide that the dynamic processes of two-particle collision and cross-leg tunneling are well described by the effective non-Hermitian Hubbard Hamiltonian based on the eigenstate equivalence. We also find that any common eigenstate is always associated with the emergence of a spectral singularity in the non-Hermitian Hubbard model. This result is valid for both Bose and Fermi systems and provides a clear physical implication of the non-Hermitian Hubbard model.

Figure 1

Schematic illustration of the concerned lattice systems. (a) Two-leg ladder for noninteracting particles. Two particles at the same site can hop simultaneously across two legs with $J$ being the interchain pair tunneling strength. (b) Two independent Hubbard chains with imaginary on-site interaction $i{U}_{\rho }$ ($\rho =A,B$).

Figure 2

Schematic illustration of the dynamics of two separable wave packets placed initially at leg $A$. (a) The two wave packets enter into leg $B$ completely associated with the probability flow from leg $A$ into leg $B$ when $J={\upsilon }_r$. The dynamic process of two such particles in leg $A$ can be described approximately through the non-Hermitian interacting system in (b) with $U_A={\upsilon }_r$ (${\upsilon }_r<0$), in which the annihilation of the two wave packets occurs through the imaginary on-site interaction.

Figure 3

The ket matrix to illustrate the connection between eigenstates of ${\overline{\mathcal{H}}}_l$ and $H$. (a) States in the $l\mathrm{th}$ row {$\left|{\overline{\chi }}_{l,l^{\prime }}\right.〉$} represent a complete set of eigenstates of ${\overline{\mathcal{H}}}_l$. The diagonal states {$\left|{\overline{\chi }}_{l,l}\right.〉$} (dashed box) are the complete set of eigenstates of $H$. (b) A block around a certain ket $\left|{\overline{\chi }}_{l_0,l_0}^{\rho }\right.〉$ satisfying Eqs. (49), (50), (52), and (53). All the rows in such a block are identical approximately. Then the diagonal states can be replaced by that in a row (green shaded).

Figure 4

Probability distribution of the evolved wave function for the initial state being two-particle incident Gaussian wave packets with $k_a=\pi /2, N_a=20, k_b=-\pi /2$, and $N_a=80$ in leg $A$ of different systems. Probability ${\left|\left.〈\mathrm{\Phi }\left(t\right)\right|n_{\rho ,j}\left|\mathrm{\Phi }\left(t\right)\right.〉\right|}^2$ ($\rho =A,B$) of the two-leg Hermitian system $H$ for (a) leg $A$ and (b) leg $B$ with a tunneling strength $J=4\left(3-2\sqrt{2}\right)$, respectively. (c) Corresponding probability distribution on leg $A$ of the non-Hermitian system $\mathcal{H}$ with imaginary on-site interaction $U_A=-4{\left(3-2\sqrt{2}\right)}^2$. (d) The red circle and blue line represent the total probability ${\sum }_j{\left|\left.〈\mathrm{\Phi }\left(t\right)\right|n_{A,j}\left|\mathrm{\Phi }\left(t\right)\right.〉\right|}^2$ of leg $A$ as functions of time $t$ for the Hermitian system $H$ and the non-Hermitian system $\mathcal{H}$, respectively. We plot ${\left|\left.〈\mathrm{\Phi }\left(t\right)\right|n_j\left|\mathrm{\Phi }\left(t\right)\right.〉\right|}^2$ at different instants $t$ in units of $1/\kappa$. One can see that when the matching condition $U_A=-J^2/{\upsilon }_r$ is satisfied, the non-Hermitian Hamiltonian can be utilized to describe the dynamics of leg $A$ of Hermitian Hamiltonian $H$, which is in accordance with the theoretical analysis in the text.