Probing Quantum Thermalization of a Disordered Dipolar Spin Ensemble with Discrete Time-Crystalline Order

We investigate thermalization dynamics of a driven dipolar many-body quantum system through the stability of discrete time crystalline order. Using periodic driving of electronic spin impurities in diamond, we realize different types of interactions between spins and demonstrate experimentally that the interplay of disorder, driving, and interactions leads to several qualitatively distinct regimes of thermalization. For short driving periods, the observed dynamics are well described by an effective Hamiltonian which sensitively depends on interaction details. For long driving periods, the system becomes susceptible to energy exchange with the driving field and eventually enters a universal thermalizing regime, where the dynamics can be described by interaction-induced dephasing of individual spins. Our analysis reveals important differences between thermalization of long-range Ising and other dipolar spin models.

Figure 1

Experimental system and observation of DTC order. (a) Periodically driven, interacting NV centers. During each Floquet period $T$ (dotted box), NV centers interact for time ${\tau }_1$, then experience pulsed microwave rotations of duration ${\tau }_2$, at resonant frequencies ${\omega }_1$ or ${\omega }_2$. After $n$ Floquet cycles, the population difference between $|0⟩$ and $|-1⟩$ is measured. (b) Distinct Floquet time evolutions realized: ${\mathbb{Z}}_2$ Ising, with only Ising interactions, and ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_3$, with both Ising and spin-exchange interactions. (c)–(f) Representative time traces of the normalized spin polarization $P(nT)$ and Fourier spectra $|S(\nu ){|}^2$ of the ${\mathbb{Z}}_3$ DTC order at a perturbation $\epsilon /\pi =0.06$ with $T=70\mathrm{ns}$ (c),(d) and 130 ns (e),(f). In (c),(e), blue, gray, and green points correspond to $P(t)$ at $t\equiv T,2T,3T$ (mod $3T$), respectively.

Figure 2

Phase diagram of DTC order for short drive periods. (a)–(c) Phase diagrams in semi-log scale. The phase boundary (markers) is identified as a crystalline fraction of 10%. The dotted line indicates the linear phase boundary predicted by a self-consistent mean-field analysis [52]. Shaded areas denote a universal dephasing regime corresponding to Markovian thermalization where $T>T^{*}$ (see Fig. 4). In (a), the dashed line represents the theoretical prediction from Ref. [25]. Errorbars denote 95% confidence intervals of the phase boundary [52]. (d) Short-$T$ phase diagram in linear scale [markers as in (a)–(c)]. (e) Bloch sphere illustrating the screening effect of spin-exchange interactions. $h_z$ and $h_y$ are mean fields arising from Ising and spin-exchange interactions, respectively, and $\epsilon /T$ is the perturbing field. The black arrow corresponds to the mean field solution $|{\psi }_{\mathrm{MF}}⟩$.

Figure 3

Long interaction time behavior of DTC order. (a),(b) Temporal decay of ${\mathbb{Z}}_3$ DTC order $|S(\nu =1/3){|}^2$, for different $\epsilon$, as a function of sweeping window position $n_{\text{sweep}}$. Dashed lines denote the noise floor. (c) Late-time decay rate $\mathrm{\Gamma }$ as a function of $\epsilon$, with phenomenological quadratic fit. Each data point results from an average over simple exponential fits of $|S(\nu =1/3){|}^2$ starting from $n_{\text{sweep}}=15–20$ to exclude initial transients resulting from a short-time, non-universal dephasing. Error bars denote the statistical error of the fit results. The arrow indicates the mean-field phase boundary.

Figure 4

Universal thermalization dynamics for different Floquet Hamiltonians. (a) Exponents of the stretched exponential fits versus $T$. Data points denote the average over $\beta$ values extracted at different $\epsilon$, error bars are the standard deviation of the mean. Lines denote fits to extract the saturation timescale $T^{*}$ (arrows), identified where $\overline{\beta }=0.9$. (b) Late-time decay rate $\mathrm{\Gamma }$ as a function of $\epsilon$ (after a global offset ${\mathrm{\Gamma }}_0$ has been subtracted, see [52]) at $T=3.5$, 3.5, 2.3, and $0.3\mu \mathrm{s}$ for the ${\mathbb{Z}}_2$-Ising, ${\mathbb{Z}}_2$, ${\mathbb{Z}}_3$, and two-group ${\mathbb{Z}}_3$ cases, respectively [markers as in (a)]. Solid line indicates a dephasing model fit, predicting $\mathrm{\Gamma }={\epsilon }^2/2+{\mathrm{\Gamma }}_0$. Error bars as in Fig. 3. Cross markers show numerical results of an infinite-range, interacting spin system with both Ising and spin-exchange interactions [52]. Error bars in the numerics represent the standard deviation of decay rate distributions for different realizations.