Non-Hermitian Skin Effect and Chiral Damping in Open Quantum Systems

One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namely, that the eigenstates are exponentially localized at the boundary of the system. For open quantum systems, a short-time evolution can often be well described by the effective non-Hermitian Hamiltonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvillian superoperators generate time evolution. In this Letter, we find that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It is shown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such that the damping in a class of open quantum systems is algebraic under periodic boundary conditions but exponential under open boundary conditions. Moreover, the non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a sharp wave front. These phenomena are beyond the effective non-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged open quantum dynamics.

Figure 1

(a) SSH model with staggered hopping $t_1$ and $t_2$, with the ovals indicating the unit cells. The Bloch Hamiltonian is $h(k)=(t_1+t_2\cos k){\sigma }_x+t_2\sin k{\sigma }_y$. The fermion loss and gain are described by the dissipators $L^l$ and $L^g$ [Eq. (2)] in the master equation framework. (b) A different realization of the same model. The hopping Hamiltonian $h(k)=(t_1+t_2\cos k){\sigma }_x+t_2\sin k{\sigma }_z$, and the dissipators are $L_x^l=\sqrt{{\gamma }_l}{c}_{xA}$ and $L_x^g=\sqrt{{\gamma }_g}{c}_{xA}^{†}$, (b) is equivalent to (a) via a basis change ${\sigma }_y\leftrightarrow {\sigma }_z$. Because gain and loss is on site, (b) is more feasible experimentally.

Figure 2

(a) The damping of the fermion number towards the steady state of a periodic boundary chain with length $L=150$ (unit cell). The damping is algebraic for cases $A$, $B$ with $t_1\le t_2$, while exponential for $C$, $D$ with $t_1>t_2$. The initial state is the completely filled state ${\prod }_{x,s}{c}_{xs}^{†}|0⟩$. (b) The eigenvalues of the damping matrix $X$. Blue: periodic boundary; Red: open boundary. The Liouvillian gap of the periodic-boundary chain vanishes for $A$ and $B$, while it is nonzero for $C$ and $D$. For the open-boundary chain, the Liouvillian gap is nonzero in all four cases. This drastic spectral distinction between open and periodic boundary comes from the NHSE (see text).

Figure 3

(a) The particle number damping of a periodic boundary chain (solid curve) and open-boundary chains for several chain length $L$. The long-time damping of a periodic chain follows a power law, while the open boundary chain follows an exponential law after an initial power law stage. (b) The site-resolved damping. The left end ($x=1$) enters the exponential stage from the very beginning, followed sequentially by other sites. For both (a) and (b), the initial state is the completely filled state ${\prod }_{x,s}{c}_{xs}^{†}|0⟩$, therefore, $\mathrm{\Delta }(0)$ the identity matrix $I_{2L\times 2L}$. $t_1=t_2=1,{\gamma }_g={\gamma }_l=0.2$.

Figure 4

Time evolution of ${\tilde{n}}_x(t)=n_x(t)-n_x(\infty )$, which shows damping of particle number $n_x(t)$ towards the steady state. (a) ${\tilde{n}}_x(t)$ of the main model with dissipators given by Eq. (2) (referred to as model I). Left: periodic boundary; Right: open boundary. The chiral damping is clearly seen in the open boundary case. The dark region corresponds to the exponential damping stage seen in Fig. 3. (b) ${\tilde{n}}_x(t)$ of model II, whose damping matrix $X$ [Eq. (15)] has no NHSE. The Liouvillian gap is nonzero and the same for periodic and open boundary chains. Common parameters: $t_1=t_2=1;{\gamma }_g={\gamma }_l=0.2$.