Generation of robust entangled states in a non-Hermitian periodically driven two-band Bose-Hubbard system

A many-body Wannier-Stark system coupled to an effective reservoir is studied within a non-Hermitian approach in the presence of a periodic driving. We show how the interplay of dissipation and driving dynamically induces a subspace of states which are very robust against dissipation. We numerically probe the structure of these asymptotic states and their robustness to imperfections in the initial-state preparation and to the size of the system. Moreover, the asymptotic states are found to be strongly entangled making them interesting for further applications.

Figure 1

Scheme of single-particle Wannier-Stark ladder. The bold lines correspond to the two-lowest (quasi) bound states (marked by $a$ and $b$) considered in this paper. The structure can be understood as the remainders of the Bloch bands after turning on the external field. The wave packet (red) represents a Wannier-Stark state that leaks to the continuum of higher bands. This process is marked by the arrow going to the left in the upper panel.

Figure 2

Energy diagram and allowed couplings in the many-body system for $N/L=2/2$, i.e., for two particles in two lattice sites. The color of the arrow defines the process, i.e., induced by hopping (blue), by the force (red and yellow), or by the interactions (green).

Figure 3

(a) Long-time average of the population inversion $W\left(t\right)$ as a function of the amplitude $A$ and frequency $\omega$ of the drive $F\left(t\right)$. The initial condition is the Mott state $|11,00\rangle$. Here ${\omega }_0=\mathrm{\Delta }-U$ and ${\gamma }_b=0.08J_{\alpha }$, with $J_{\alpha }\equiv \left|J_{a,b}\right|$. (b) Decay rates computed from the quasienergy spectrum of ${\widehat{H}}^{\prime }$ for different system sizes $N/L$ vs the dissipation strengths (see legends). The inset shows the projections of the Fock states onto the Floquet eigenstates for $N/L=2/2$. The most stable state corresponds to a linear combination of only two Fock states belonging to the lowest resonant manifold. The parameters taken from Ref. [16] are $\mathrm{\Delta }=0.1$, $J_a=-0.006$, $J_b=0.006$, $U_a=U_b=U_x=0.034$, $c_0=-0.098$, and $c_{\pm }=0.035$.

Figure 4

Panels (a) and (b) show $W\left(t\right)$ for $N/L=2/2$ and the final-time normalized distribution $p\left(\right\{|D_{\left\{n\right\}}^t{|}^2\left\}\right)$ for the asymptotic state vs the dissipation strength. The initial condition is the Mott-insulator state defined in the main text. Panels (c) and (d) show $p\left(\right\{|D_{\left\{n\right\}}^t{|}^2\left\}\right)$ for $N/L=4/4$ and $N/L=6/6$. Initial conditions: (black-filled circles) $|{\psi }_0^1\rangle$, (blue-squares) $\frac{1}{\sqrt{2}}(|{\psi }_0^2\rangle +|{\psi }_0^3\rangle )$ and (red-filled triangles) $\frac{1}{\sqrt{2}}|{\psi }_0^1\rangle +\frac{1}{2}(|{\psi }_0^2\rangle +|{\psi }_0^3\rangle )$ defined in the main text. The green asterisks in panel (c) correspond to an average over 100 initial random states of the type $\sum D_{\left\{n\right\}}|n_1^a{n}_2^a\cdots ,00\cdots \rangle$. Note that the states of interest satisfy ${\sum }_l({\widehat{n}}_l^b-{\widehat{n}}_l^a)=0$ and $n_l^a+n_l^b=1$. These conditions imply a band-exchange symmetry of the populations discussed at the end of Sec. 3. The parameters are the same as for Fig. 3 with $|c_{\mu }A/J_a|\simeq 10$ and $\omega ={\omega }_0$.

Figure 5

Entanglement negativity for $N/L=2/2$ and $N/L=4/4$ for increasing dissipation ${\gamma }_b/J_{\alpha }$. The initial state is a nonentangled Mott-insulator state. The parameters are the same as in Figs. 3 and 4.