Out-of-equilibrium steady states of a locally driven lossy qubit array

We find a rich variety of counterintuitive features in the steady states of a qubit array coupled to a dissipative source and sink at two arbitrary sites, using a master equation approach. We show there are setups where increasing the pump and loss rates establishes long-range coherence. At sufficiently strong dissipation, the source or sink effectively generates correlation between its neighboring sites, leading to a striking density-wave order for a class of “resonant” geometries. This effect can be used more widely to engineer nonequilibrium phases. We show the steady states are generically distinct for hard-core bosons and free fermions, and differ significantly from the ones found before in special cases. They are explained by generally applicable ansatzes for the long-time dynamics at weak and strong dissipation. Our findings are relevant for existing photonic setups.

Figure 1

Contrasting steady-state correlations $\langle {\widehat{b}}_i^{†}{\widehat{b}}_j\rangle$ of hard-core bosons on a 1D lattice subject to incoherent pump with rate ${\gamma }_{+}$ and loss with rate ${\gamma }_{-}$ (a) at opposite ends, with ${\gamma }_{+}={\gamma }_{-}=10J$, and (b) at the center, with ${\gamma }_{-}=4{\gamma }_{+}$, where $J$ is the tunneling.

Figure 2

Steady-state correlations $\langle {\widehat{b}}_i^{†}{\widehat{b}}_j\rangle$ for pump and loss at neighboring sites $p=L/2, q=L/2+1$ for ${\gamma }_{+}={\gamma }_{-}:=\gamma$ and $L=8$ at (a) weak dissipation, $\gamma =0.6J$, and (b) strong dissipation, $\gamma =8J$. (c) Coherence between reflection-symmetric sites as a function of the pump/loss rate. (d) Correlation length $\xi$ obtained by fitting the coherences to an exponential.

Figure 3

Steady-state density $n_i$ of free fermions and hard-core bosons in the weak-dissipation limit with ${\gamma }_{+}={\gamma }_{-}$ for (a) $L=8, p=2, q=6$, from exact diagonalization and using a product-of-modes ansatz [Eqs. (10, 11)], and (b) $L=23, p=1, q=12$, using the ansatz. (c) Occupation of single-particle modes and (d) density correlation between sites $k$ and $\tilde{k}:=L+1-k$ for the setup in (b), where $\langle {\widehat{n}}_i,{\widehat{n}}_j\rangle :=\langle {\widehat{n}}_i{\widehat{n}}_j\rangle -n_i{n}_j$.

Figure 4

(a) Resonant modes in adjacent segments for a specific pump-loss setup in the Zeno limit, ${\gamma }_{\pm }\gg J$. (b) Steady-state correlation between free-fermion modes in segments $\nu =1,2,3$, showing coherence among resonant modes. (c) Density of free fermions and hard-core bosons from exact diagonalization (${\gamma }_{+}={\gamma }_{-}$).

Figure 5

(a) Lowest-energy resonant mode in a class of setups parametrized by integers $l=3$ and $r=2$. (b) Steady-state density of free fermions and hard-core bosons with ${\gamma }_{+}={\gamma }_{-}$ in the Zeno limit from exact diagonalization and our ansatz [Eq. (13)]. (c)–(d) Density-density correlations of the fermions and bosons, respectively. (e)–(f) Scaling of the density bump $n_{q+1}$ with system size, $q$ being the loss site, with the same color convention as in (b).