Hilbert space fragmentation in driven-dephasing Rydberg atom array

TL;DR

We address Hilbert space fragmentation in a driven-dephasing chain of strongly interacting Rydberg atoms. An effective dephasing PXP model is derived via a unitary rotation in the strong-blockade limit, producing ${H}_{\text{PXP}}$ and a Liouvillian ${\mathcal{L}}(\rho)=-i[H_{\text{PXP}},\rho]+\mathcal{D}(\rho)$ that reproduces the observed metastable dynamics. The fragmentation is unraveled by a commutant algebra using a consecutive double excitations addressing operator (CDEA), yielding a fragmented Hilbert space with a number of subspaces $f(L)$ that satisfies $f(L)=f(L-1)+f(L-2)+f(L-4)$ and grows as $f(L)\sim A \alpha^L$ with $\alpha\approx1.754877$. Metastable plateaus and their scaling with dephasing follow from a mean-field rate $\gamma_{\text{mf}}= \frac{\Omega^2 \gamma}{V^2}$, and plateau lifetimes scale as $\tau \sim 1/\gamma_{\text{mf}}$. These results illuminate dissipative constrained dynamics and offer a route to control Hilbert space fragmentation in Rydberg simulators by tuning interactions and dephasing.

Abstract

We investigate the onset and mechanism of Hilbert space fragmentation (HSF) in a chain of strongly interacting Rydberg atoms subject to local dephasing. It is found that the emergence of multiple long-lived metastable states is fundamentally tied to HSF of the driven-dephasing Rydberg atom system. We demonstrate that the manifesting HSF is captured by a dephasing PXP model that supports multiple degenerate zero modes. These modes form disconnected, block-diagonal subspaces of maximally mixed states, which consist of many-body spin states sharing the same symmetry. A key result is the identification of the underlying symmetry in the HSF, where conserved quantities in each subspace are defined by the consecutive double excitation addressing operator. Moreover, we show explicitly that the number of the fragmented Hilbert space grows exponentially with the chain length, following a modified Fibonacci sequence. Our work provides insights into many-body dynamics under dynamical constraints and opens avenues for controlling and manipulating HSF in Rydberg atom systems.

Figures (6)

• Figure 1: (a) Atom array setting. Atoms are resonantly excited from the ground state $|0\rangle$ to electronically excited Rydberg state $|1\rangle$ by a laser field (Rabi frequency $\Omega$). In Rydberg state, atoms experience a dephasing with rate $\gamma$. The van der Waals interaction between Rydberg atoms is dominated by the NN interaction $V$ between the neighboring atoms. (b) Dynamics of the atom strongly depends on initial states. Typical observable exhibits metastable evolution (solid) in the intermediate timescale before thermalization. The metastable dynamics is effectively described by the dephasing PXP model (dashed). (c) The degenerate zero modes of the dephasing PXP model form distinctive diagonal blocks in the Hilbert space. Dynamical evolution in one block (arrow) is not affected by other blocks. A frozen state (dot-dashed) is an isolated basis and invariant with time. See text for details.
• Figure 2: Metastable dynamics and fragmentation. (a) Magnetization $M$ of the Rydberg atom chain. Metastable dynamics takes place from $\Omega t=0$ to $\Omega t\lesssim 100$ (dashed curves) if the initial state is a frozen state. When started with other basis states (solid), the metastability occurs in interval $1\lesssim \Omega t\lesssim 100$. The dependence on the initial state and clustering of $M$ indicate the onset of fragmentation. The dynamics thermalizes to the infinite temperature state $\rho_{\infty}$ in the long time limit. (b) Purity of the atom chain. $\rho_{\text{o}}$ is largely pure when starting from the frozen state (dashed). Other basis (solid) partially retain coherence in the metastable state. (c) Complex spectrum of Liouvillian ${\tilde{\mathcal{L}}}$. There are 20 nearly degenerate modes (arrow) around the stationary mode ($\lambda_j= 0$). (d) Real parts $\text{Re}(\lambda_j)$ of the quasi-degenerate (square) and degenerate (circle) modes. The inset shows scaling of the gap (minimum of $\text{Re}(\lambda_j)$) with $L$. Increasing $L$, the gap decreases and approaches to the mean field result $\gamma_{\text{mf}}$ (dashed). (e) Magnetization of the dPXP model. Due to the fragmentation, $M$ does not change with time after the initial transient period (solid). In the frozen state, $M$ is a constant (dashed). (f) Purity of the dPXP model. The steady $P$ is determined by dimensions of the respective subspace, as marked in the figure. In (b), the purity in the metastable state is close to the one of the dPXP model. (g) Complex spectra of the Liouvillian $\mathcal{L}(\rho)$. There are 21 degenerate zero modes, whose real parts are shown in panel (d). (h) Fragmented Hilbert space of the dPXP model. The index $j$ is the decimal number converted from the binary representation of the basis. Basis states of each subspace are given in Table \ref{['table:HSF']}. We have considered $V=50\Omega$ in panels (a)-(d), and $L=6$, $\gamma=2\Omega$ in all the panels.
• Figure 3: (a) Dynamics of $\langle\mathcal{A}\rangle_{\text{o}}$. In the metastable regime ($\Omega t\le 100$), the CDEA operator is almost constant. Parameters are same with ones in Fig. \ref{['fig2']}. (b) Number $f(L)$ of the zero modes with respect to chain length $L$. As shown by the solid line, $f(L)$ increases with $L$ exponentially. For a given $L$, $f(L)$ is larger than the Fibonacci sequence (square), but smaller than the Hilbert space dimension $D=2^L$ (triangle).
• Figure S1: Dynamics of the total magnetization and purity of the Rydberg atom chain with $L=7,\ 8,\ 9$. In all panels, we choose $V=50\Omega$ and $\gamma=2\Omega$.
• Figure S2: (a) Decay time $\tau$ versus dephasing strength $\gamma$. The black dashed line stands for Eq. (\ref{['overall_trans']}). The orange points are taken from the spectra gap of the Liouvillian $\mathcal{L}(\rho_{\text{o}})$. The square points are obtained from the time when the dynamics of total magnetization reaches $1/e$ of the steady state computed with the dPXP model. In both cases, the initial state is chosen as $|000000\rangle$. (b) Decay time $\tau$ versus chain length $L$. Data points are the Liouvillian gap of the dPXP model, i.e. the smallest value along the real axis near the zero mode. Other parameters are $V=50\Omega$ and $\gamma=2\Omega$ in panel (b).