Time-Resolved Observation of Thermalization in an Isolated Quantum System

We use trapped atomic ions forming a hybrid Coulomb crystal and exploit its phonons to study an isolated quantum system composed of a single spin coupled to an engineered bosonic environment. We increase the complexity of the system by adding ions and controlling coherent couplings and, thereby, we observe the emergence of thermalization: Time averages of spin observables approach microcanonical averages while related fluctuations decay. Our platform features precise control of system size, coupling strength, and isolation from the external world to explore the dynamics of equilibration and thermalization.

Figure 1

Complexity of the Hamiltonian studied numerically. Parameters are ${\omega }_1/(2\pi )=0.7\mathrm{MHz}$, ${\overline{n}}_{j=1,\dots ,N}=1$. (a) Dimension of truncated Hilbert space $\dim [{\mathcal{H}}_{\text{trunc}}(N)]$ for corresponding fractions of initial-state population $\mathrm{tr}{\rho }_{\text{trunc}}(0)$ lying within ${\mathcal{H}}_{\text{trunc}}$ (solid lines). For $N=3$, for example, 85% lie within $\dim ({\mathcal{H}}_{\text{trunc}})\approx 2^{10}$ (circle). We derive $D_{\mathrm{eff}}(N,\mathrm{\Omega },{\omega }_z)$ up to $\dim ({\mathcal{H}}_{\text{trunc}})=2^{16}$ (dashed line). (b) Choosing $\mathrm{\Omega }$ and varying ${\omega }_z$ (dashed line), we can tune the spin-phonon coupling into resonance with different modes (sketched at the bottom) and boost the system size. Note that the actual number of participating states is much larger than the normalized quantity $D_{\mathrm{eff}}$; see Eq. (5). (c) For fixed $\mathrm{\Omega }(N)$ [cf. dashed line in (b)], $D_{\mathrm{eff}}({\omega }_z)$ increases significantly with $N$. This enables the systematic investigation of equilibration and thermalization depending on the system size. Error bars show systematic numerical uncertainties [36].

Figure 2

Measured unitary time evolution $⟨{\sigma }_z(t)⟩$. Experimental results (black dots, error bars: 1 s.d.) for $N=1,\dots ,5$ compared to full ED (solid lines). We exclude numerical results for $N=5$ due to their large systematic uncertainties. Oscillations (time fluctuations) of high amplitude during the transient duration $t/{\tau }_S\lesssim 1$ are driven by the evolution of $\rho (0)$ towards the ground state of $H$. (a) For ${\omega }_z=0$ and increasing $N$, excitation is coherently exchanged with a growing number of modes resulting in revivals at ${\tau }_{\mathrm{rev}}$ (shaded areas). (b) For ${\omega }_z\approx \mathrm{\Omega }(N)$, expectation values fluctuate around a negative offset. Revivals and this nontrivial bias emphasize the coherence of the dynamics. (c) Histograms of experimental measurements sample the probability distribution which underlies $⟨{\sigma }_z(t)⟩$. Here, we show these for $t\in [{\tau }_S,13{\tau }_S]$, ${\omega }_z\approx \mathrm{\Omega }(N)$, and $N=1$, 5 to exemplify the quantities ${\mu }_{\exp }(⟨{\sigma }_z⟩)$ and ${\delta }_{\exp }(⟨{\sigma }_z⟩)$.

Figure 3

Time averages and mean amplitudes of time fluctuations of $⟨{\sigma }_z(t)⟩$. These are calculated from experimental traces (black dots, error bars: 1 s.d., derived from the underlying probability distribution of $⟨{\sigma }_z(t)⟩$ [36]) for varying ${\omega }_z$ and $N=1,\dots ,5$ and comparison to full ED (solid lines). (a) Increasing ${\omega }_z$ shifts the ground state of $H$, adjusts spin-mode couplings, and varies $D_{\mathrm{eff}}$. Even for small systems, we find agreement of time averages with microcanonical averages, ${\mu }_{\exp }(⟨{\sigma }_z⟩)\approx {\mu }_{\text{micro}}(⟨{\sigma }_z⟩)$ (dashed lines, shaded areas indicate systematic uncertainties). As $D_{\mathrm{eff}}$ rapidly increases with $N$, time averages follow microcanonical averages for a larger range of ${\omega }_z$. (b) ${\delta }_{\exp }(⟨{\sigma }_z⟩)$ gradually decreases with $N$, and correlated resonances in ${\mu }_{\exp }(⟨{\sigma }_z⟩)$ and ${\delta }_{\exp }(⟨{\sigma }_z⟩)$ fade away, indicating that we tune our system from microscopic to mesoscopic size.

Figure 4

Scaling of mean amplitudes of time fluctuations with $N$ and $D_{\mathrm{eff}}$. (a) We plot ${\delta }_{\exp }(⟨{\sigma }_z⟩)$ (error bars: 1 s.d.) as a function of $N$. The spread for $N\le 2$ highlights finite-size effects, and we show an average value for each $N$ (large symbols, error bars: 1 s.d.). For $N=1,\dots ,4$, we observe a decay that ceases for $N=5$. (b) ${\delta }_{\exp }(⟨{\sigma }_z⟩)$ as a function of calculated $D_{\mathrm{eff}}$ (error bars: systematic uncertainties), which captures the dependence of the effective system size on all experimental parameters. We compare to a scaling ${\delta }_{\infty }\propto 1/\sqrt{D_{\mathrm{eff}}}$ (solid line), motivated for our system, and our measurements agree for $D_{\mathrm{eff}}\lesssim 25$. Further increasing $D_{\mathrm{eff}}$, the system needs longer durations to resolve decreasing energy differences in the environment, unveiling the importance of time scales.