Resonant dynamics of spin cluster in a periodically driven one-dimensional Rydberg lattice

TL;DR

This work investigates the resonant dynamics of spin clusters in a periodically driven one-dimensional Rydberg lattice. By deriving an effective domain-wall Hamiltonian and identifying a resonant driving frequency $F$ set by the Rydberg interaction, it shows that at resonance the spin cluster expands ballistically with a reduced spreading rate relative to facilitation, while near resonance the dynamics exhibit Bloch-like oscillations and a coexistence of expansion and confinement. The results reveal rich dynamical regimes arising from the interplay of long-range interactions and time-dependent driving, offering new avenues for controlled quantum state manipulation in programmable quantum simulators. The findings are relevant for experimental realization in Rydberg platforms and may generalize to other long-range interacting systems and related spin-chain models.

Abstract

Rydberg lattice under facilitation conditions can feature kinetic constraints, leading to ballistic and nonergodic behavior at different detuning intensities. Here, we demonstrate that a resonant driving field can achieve effects similar to those under facilitation conditions. We focus on the relaxation dynamics of spin clusters in a periodically driven Rydberg spin lattice. Through an effective Hamiltonian for the domain walls of the spin cluster, it is shown that when the driving frequency is resonant with the Rydberg interaction, the spin cluster exhibits ballistic expansion with half the spreading rate compared to the case of facilitation conditions. However, near the resonant point, the spin cluster displays confinement behavior of the Bloch-like oscillations. These results demonstrate the rich dynamic behaviors in the driven Rydberg spin lattices and may find applications in quantum state manipulation.

Figures (6)

• Figure 1: (a) Schematic illustration of the 1D Rydberg lattice under periodic driving field $\mathit{\Omega} \left( t \right)=\mathit{\Omega}_{0}\cos\left(\omega t\right)$. There are long-range interactions $V_{i,j}=\mathcal{V}_{0}/\left| i-j \right|^{6}$ between atoms in the Rydberg excited states $\left| \uparrow \right>$, and $\left| \downarrow \right>$ represents the ground state of an atom. The bottom panels present the numerical results of time evolutions of the Rydberg density $\left< \hat{n}_{j} \right>$ for different initial excitations and driving frequencies of driving fields. (a) Time evolutions of Rydberg density for simple spin cluster initial state under the resonant frequency $\omega=5.0867$, which is resonant with the Rydberg interaction and the system exhibits expansion dynamics of spin cluster. (b) Time evolutions of Rydberg density for a Gaussian distribution of spin cluster as initial state near the resonant frequency with $\omega=5.2867$. In contrast, the system exhibits oscillation dynamics in this case. The size of the system is set as $N=20$, and the strength of the field and interaction are taken as $\mathit{\Omega}_{0}=1$ and $\mathcal{V}_{0}=5$, respectively.
• Figure 2: Numerical results of time average overlap $\overline{O(t)}$ between the evolved states computed from the Rydberg spin Hamiltonian and the effective Hamiltonian for different driven frequency $\omega$. The initial states are all taken as the spin cluster states in Eq. (\ref{['state_j1j2']}) with the positions of domain walls $j_{1}=8$ and $j_{2}=11$. The size of the system is set as $N=18$, and the strength of the field and interaction are taken as $\mathit{\Omega}_{0}=1$ and $\mathcal{V}_{0}=5$, respectively. The results indicate that the effective Hamiltonian works well in the high frequency drive region.
• Figure 3: Time evolutions of Rydberg density $\left< \hat{n}_{j} \right>$ and density variance $\delta \sigma(t)=\sigma(t)-\sigma(0)$ for the system under the resonant frequency of the driving field with different initial states. The frequency of the driving field is set as $\omega=5.0867$, which is resonant with the Rydberg interaction. The top panels present time evolution of Rydberg density $\left< \hat{n}_{j} \right>$ for the initial excitations of (a1) simple spin cluster $\left| \psi \left( j_{1},j_{2} \right) \right>$ in Eq. (\ref{['state_j1j2']}) with the positions of domain walls $j_{1}=46$ and $j_{2}=55$; the Gaussian distribution of the spin cluster in Eq. (\ref{['state_Gau']}) with different central momenta (b1) $k_{0}=0$, (c1) $k_{0}=\pi/2$ and (d1) $k_{0}=\pi$. The width, central position of the Gaussian distribution and size of the spin cluster are taken as $d=4$, $c_{0}=50.5$ and $r_{0}=10$, respectively. The bottom panels (a2)-(d2) show the density variance $\delta \sigma(t)$ on double-logarithmic scales (black dots) corresponding to the Rydberg density in (a1)-(d1), and the solid lines represent the linear fittings for the data. The size of the system is set as $N=100$, and the strength of the field and interaction are taken as $\mathit{\Omega}_{0}=1$ and $\mathcal{V}_{0}=5$, respectively. In the cases of (a)-(c), it can be observed that the exponent $\beta$ of density variances decrease due to the collision of the domain walls. While the case in (c) with central momentum $k_{0}=\pi$, the spin cluster exhibits ballistic expansion with exponent $\beta=2$.
• Figure 4: Total Rydberg density $\mathcal{N}(t)$ as a function of time for the resonant cases in Fig. \ref{['fig_resonant']}(a1)-(d1). The frequency of the driving field is set as the resonant value $\omega=5.0867$. The size of the system is set as $N=100$, and the strength of the field and interaction are taken as $\mathit{\Omega}_{0}=1$ and $\mathcal{V}_{0}=5$, respectively. Combining with the results in Fig. \ref{['fig_resonant']}, one can see that the total Rydberg density increase due to the collision of domain walls, and the total Rydberg density is conserved for the case with central momentum $k_{0}=\pi$.
• Figure 5: Time evolution of Rydberg density $\left< \hat{n}_{j} \right>$, density variance $\delta \sigma(t)=\sigma(t)-\sigma(0)$ and autocorrelation function $\mathcal{A}(t)$ for the system near the resonant frequency of the driving field with different initial states. The frequency of the driving field is set as $\omega=5.1367$. The top panels (a1)-(d1) present time evolutions of Rydberg density $\left< \hat{n}_{j} \right>$ for the initial excitations taken as the same as those in Fig. \ref{['fig_resonant']}(a1)-(d1). The middle panels (a2)-(d2) show the density variance $\delta \sigma(t)$ corresponding to the Rydberg density in (a1)-(d1). The bottom panels show the autocorrelation function $\mathcal{A}(t)$ between the initial states and evolved states corresponding to the cases in (a1)-(d1). The size of the system is set as $N=100$, and the strength of the field and interaction are taken as $\mathit{\Omega}_{0}=1$ and $\mathcal{V}_{0}=5$, respectively. The periodic boundary condition is taken for the system. For the cases in (a)-(c), the systems exhibit confinement dynamics of spin cluster without revival due to the near resonant condition and the collision of spin cluster domain walls, while for the Gaussian distribution with $k_{0}=\pi$ in (d), the spin cluster exhibits oscillation dynamics with high-amplitude revival due to the absent of the collision.