Fock space prethermalization and time-crystalline order on a quantum processor

Abstract

Periodically driven quantum many-body systems exhibit a wide variety of exotic nonequilibrium phenomena and provide a promising pathway for quantum applications. A fundamental challenge for stabilizing and harnessing these highly entangled states of matter is system heating by energy absorption from the drive. Here, we propose and demonstrate a disorder-free mechanism, dubbed Fock space prethermalization (FSP), to suppress heating. This mechanism divides the Fock-space network into linearly many sparse sub-networks, thereby prolonging the thermalization timescale even for initial states at high energy densities. Using 72 superconducting qubits, we observe an FSP-based time-crystalline order that persists over 120 cycles for generic initial Fock states. The underlying kinetic constraint of approximately conserved domain wall (DW) numbers is identified by measuring site-resolved correlators. Further, we perform finite-size scaling analysis for DW and Fock-space dynamics by varying system sizes, which reveals size-independent regimes for FSP-thermalization crossover and links the dynamical behaviors to the eigenstructure of the Floquet unitary. Our work establishes FSP as a robust mechanism for breaking ergodicity, and paves the way for exploring novel nonequilibrium quantum matter and its applications.

Figures (4)

• Figure 1: Schematic illustration of Fock space prethermalization (FSP).a-b, Fock space network in the thermal ( a) and FSP ( b) regimes. The thickness of lines denotes the relative hopping strength $|\mathcal{T}_{\boldsymbol{s}_1, \boldsymbol{s}_2}|$ of different bonds in each case, which is obtained numerically for the Floquet unitary without the perfect $\pi$-pulse, i.e., $e^{-{\rm i} J \sum_j \tilde{\sigma}^z_j(\lambda_2)\tilde{\sigma}^z_{j+1}(\lambda_2)}U_{\rm p}(\varphi_1,\lambda_1,\varphi_2)$. Bonds with strength weaker than $4\%$ of the maximum values are not plotted. Fock-space sites $\boldsymbol{s}$ are colored according to their total domain wall numbers $W(\boldsymbol{s})$. In ( a), with $\lambda_1=1.2$, qubits can flip freely, rendering a densely connected Fock-space network. In ( b), with $\lambda_1=0.1$, dominant Ising interactions enforce the approximate domain wall conservation and result in a sparse network, with four scarred states (marked by big green and purple dots) nearly disconnected. c, Kinetic constraints in FSP regime enforced by strong Ising interactions ($J\gg\lambda_1$). At leading order, hopping among inter-DW sectors is suppressed by total DW conservation, while intra-DW hopping is only allowed if it locally conserves DW numbers, i.e., a spin can only flip if its neighbors are anti-parallel. A DW is denoted by a blue dashed line. d, Illustrative numerical results of the quasienergy $\varepsilon_n$ and the averaged DW number $\langle w \rangle_n$ of each eigenstate $|\varepsilon_n\rangle$. We show $\lambda_1=0.1,0,4,1.2$ representing FSP, critical, and thermal regimes, respectively. Simulations for larger systems are presented in Supplementary Information section 2B. In all cases, we set $J=1$ for a 10-qubit ring, and fix $\lambda_1/\lambda_2=2$.
• Figure 2: FSP-induced time-crystalline wave packet dynamics.a-d, Measured dynamics of radial probability distribution $\Pi(d,nT)$ in FSP regime at $\lambda_1=2\lambda_2=0.1$ ( a, b) and thermal regime at $\lambda_1=2\lambda_2=1.2$ ( c, d). The initial state $|\boldsymbol{s}_0\rangle$ is prepared to the "1FM" pattern. e, Illustration of initial Fock states investigated in this work. $\uparrow, \downarrow$ denote $s^j = 1$ and $0$, respectively. "1FM" and "2FM" patterns differ from an overall AFM pattern by one or two mini ferromagnetic regions highlighted by the blue shades. To mitigate the unequal qubit errors in experiments, we average all results over five repeated experiments where the initial patterns are globally shifted along the physical qubit ring. The “random” state exemplifies one of five randomly sampled patterns, and the experimental results are averaged over them. The qubits satisfying DW constraints to flip are denoted by red colors. f, Measured normalized mean Hamming distance $\langle x \rangle/L$ in FSP regime ($\lambda_1=2\lambda_2=0.1$) for "1FM", "2FM", and "random" initial states, and in thermal regime at $\lambda_1=2\lambda_2=1.2$ for the "1FM" initial state. Error bars, when present, are derived from 10 random samples of $\varphi_2$ across all figures in the text.
• Figure 3: Site-resolved dynamics of equal-time correlators.a, Snapshots of the measured equal-time correlator $C_{jk}(t)$ at $\lambda_1=0.1$ for the "1FM" initial state, with the flipped qubit $Q_{36}$ highlighted in red. b, Measured dynamics of $A_j(t)$ for the "1FM" initial state. Black dashed lines represent analytical predictions for light-cone propagation speed. c, Measured snapshots of $C_{jk}(t)$ at $t=30T$ for $\lambda_1=0.2, 0.3, 0.4$, and $0.5$. Dashed lines indicate the light-cone boundary. $\lambda_1/\lambda_2$ is fixed to 2 during these experiments.
• Figure 4: Finite-size scaling of FSP-thermalization crossover.a, Measured DW probability distribution $\mathcal{D}(w,nT)$ in the Fock space prethermal ($\lambda_1=0.1$), critical ($\lambda_1=0.4$), and thermal ($\lambda_1=1.2$) regimes. b, Measured dynamics of the normalized averaged DW number $\langle w\rangle/L$ as a function of perturbation strength $\lambda_1$. c, Maximum amplitude of the Fourier spectra of DW dynamics for system sizes $L=24, 40, 56,$ and $72$. Fourier transformation is performed over the time window $t\in[2T,20T]$ (gray regime in b). d, Measured dynamics of normalized wave packet width $\Delta x/\sqrt{L}$ at $\lambda_1=0.1$ (circles), $0.4$ (triangles), and $1.2$ (squares). For each $\lambda_1$, experimental results are shown for system sizes $L=24, 40, 56,$ and $72$. e, Time-averaged $\Delta x/\sqrt{L}$ (averaged over the time window $t\in[10T,30T]$, i.e., gray regime in d) as a function of $\lambda_1$ for different system sizes $L$. Error bars in b- e stem from 10 random samples of $\varphi_2$.