Embedding Quantum Many-Body Scars into Decoherence-Free Subspaces

Quantum many-body scars are nonthermal excited eigenstates of nonintegrable Hamiltonians, which could support coherent revival dynamics from special initial states when scars form an equally spaced tower in the energy spectrum. For open quantum systems, engineering many-body scarred dynamics by a controlled coupling to the environment remains largely unexplored. Here, we provide a general framework to exactly embed quantum many-body scars into the decoherence-free subspaces of Lindblad master equations. The dissipative scarred dynamics manifest persistent periodic oscillations for generic initial states, and can be practically utilized to prepare scar states with potential quantum metrology applications. We construct the Liouvillian dissipators with the local projectors that annihilate the whole scar towers, and utilize the Hamiltonian part to rotate the undesired states out of the null space of dissipators. We demonstrate our protocol through several typical models hosting many-body scar towers and propose an experimental scheme to observe the dissipative scarred dynamics based on digital quantum simulations and resetting ancilla qubits.

Figure 1

Schematic illustration of the protocol for embedding quantum many-body scars into decoherence-free subspaces of Lindblad master equations. (a) The equally spaced many-body scar tower $\{|S_n⟩\}$ (red lines) is embedded onto the imaginary axis of the Liouvillian spectrum as nondecaying eigenmodes in the form of $\{|S_n⟩⟨S_m|\}$ (red crosses). (b) The dissipators drive the system into their common null space (sometimes equals the scar subspace) and the Hamiltonian part of the Liouvillian rotates the undesired states out of the null space to make them decay away.

Figure 2

Numerical results for the Liouvillian spectrum and dissipative scarred dynamics. (a) Liouvillian spectrum of the toy model hosting Dicke states as scars. Scarred eigenmodes are equidistantly embedded on the imaginary axis (the red dotted line) in the form of $\{|S_n⟩⟨S_m|\}$. $L=6$, $\mathrm{\Omega }=2\pi$, $\gamma =1$, $V_{j,j+1}={\sigma }_j^x$. (b) Total spin-$z$ dynamics for the toy model, starting from three different initial states. ${\theta }_j\in [0,\pi ]$ are some random rotation angles. ${\rho }_R$ is a random physical density matrix. (c) Spectrum of the non-Hermitian AKLT Hamiltonian with or without the three-local projectors. $L=8$, $\gamma =2$. Liouvillian dynamics of the quantum jump rate (d) and the scar subspace overlap (e) for the domain-wall preserving model, starting from two initial states. $L=8$, $\mathrm{\Delta }=0.5$, $J=1$, $\gamma =1$, $V_{j,j+1}={\sigma }_j^x{\sigma }_{j+1}^x$.

Figure 3

(a) An illustration of the experimental scheme to implement the dissipative scarred dynamics. (b) Observable dynamics simulated by the quantum trajectory method with the initial state $\exp (i{\mathrm{\Sigma }}_j{\theta }_j{\sigma }_j^x)\{{\prod }_{j=2}^{L-1}[1+(-1{)}^j{P}_{j-1}^0{\sigma }_j^{+}{P}_{j+1}^0]|\downarrow \cdots \downarrow ⟩\}$ [14], ${\theta }_j\in [0,0.2\pi ]$ (top panel), and $|\uparrow \uparrow \uparrow \uparrow \downarrow \downarrow \downarrow \downarrow ⟩$ (bottom panel; trajectories omitted due to the plot range). For both panels we take 1000 trajectories. $\delta t=0.1$, $L_{j=1,L}={\sigma }_j^{-}$, other $L_{j\ne 1,L}={\sigma }_j^{-}{\sigma }_{j+1}^{-}$, $L=8$, $\mathrm{\Delta }=0.5$, $J=1$, $\gamma =1$.