Detailed Balance of Thermalization Dynamics in Rydberg-Atom Quantum Simulators

Dynamics of large complex systems, such as relaxation towards equilibrium in classical statistical mechanics, often obeys a master equation that captures essential information from the complexities. Here, we find that thermalization of an isolated many-body quantum state can be described by a master equation. We observe sudden quench dynamics of quantum Ising-like models implemented in our quantum simulator, defect-free single-atom tweezers in conjunction with Rydberg-atom interaction. Saturation of their local observables, a thermalization signature, obeys a master equation experimentally constructed by monitoring the occupation probabilities of prequench states and imposing the principle of the detailed balance. Our experiment agrees with theories and demonstrates the detailed balance in a thermalization dynamics that does not require coupling to baths or postulated randomness.

Figure 1

Setup. (a) The wave front of the trap laser is modulated by a liquid-crystal spatial light modulator (SLM) and imaged by a telescope (L1, L2) and an objective lens (L3). The fluorescence image of the trapped atoms is captured by a camera and analyzed to feedback the SLM for array compactification by atom shuttling [15, 16, 17, 18]. Then, 480 and 780 nm lasers excite the array to a $67S$ Rydberg state, forming a tunable Ising-like spin-$1/2$ chain. (b) Zigzag chain fluorescence images (Gaussian filtered for clarity). Time dependence of Rydberg fraction after the quench for (c) the linear chain of $N=10$ and $\theta =180°$ and (d) the zigzag chain of $N=20$ and $\theta =60°$. The experimental data (circles) are compared with the computation (solid lines) and the result (dashed) of the master equation constructed based on the experimental data. All error bars, $\pm$ standard error of the mean. (c),(d)(right) Chain configurations with blockade radius of $r_B={(|C_6|/2\pi \mathrm{\Omega })}^{1/6}=6.5\mu \mathrm{m}$ and lattice spacing of $d=4.0\pm 0.2\mu \mathrm{m}$.

Figure 2

Detailed balance in the linear chain of $N=10$. (a) The measured $P_n(t)$ (color bars) and its standard deviation ($\pm \sigma$, overlaid black lines). At $t>2\mu \mathrm{s}$, the measured data (red dots) and the theoretical predictions (black line) of the steady-state values $P_n^{\mathrm{eq}}$ are shown. The red dashed line evolves along the tallest bar, showing the coherent oscillation of $P_n$. (b) The ratio ${\mathrm{\Gamma }}_{n\to n-1}/{\mathrm{\Gamma }}_{n-1\to n}$ (red dots) of transition rates are obtained (with error bar $\pm \sigma$) from $P_n^{\mathrm{eq}}$ and compared with the theoretical prediction (black dots). (c) Graph for the thermalization dynamics. Its nodes (circles) represent low-energy eigenstates of the prequench Hamiltonian $H_0$, classified by the number $n$ of spin-up atoms in the states. ${\nu }_n$ is the number of the eigenstates having $n$ spin-up atoms. Each red link connecting two nodes indicates transitions between the corresponding states by $H_I$.

Figure 3

Diffusion over prequench states. Time average ($t\ge t_{\text{relax}}$) of the measured probability $|C_m{|}^2$ of occupying the $m$th eigenstates of the prequench Hamiltonian $H_0$ for the linear chain of $N=10$. The position of the $D$th eigenstate is indicated by the dashed line. (Inset) The same for the $\theta =60°$ zigzag chain of $N=10$.

Figure 4

Thermalization value. Time average ${\overline{f}}_R$ of the measured Rydberg fraction as a function of system size $N$ for the linear chain (red circles) and the zigzag chain of $\theta =60°$ (blue diamonds). The error bars of $\pm \sigma$ are shown. For comparison, the MPS result (solid lines) and the ETH prediction (at $N=19$ for the linear chain and at $N=21$ for the zigzag chain; color stars) are shown. Note that the data points of the shortest chains ($N=2$, $\theta =180°$) and ($N=3$, $\theta =60°$) have smaller error bars than the other points, because they are obtained from a data set for which a defect-free chain is formed from a chain of smaller initial size.