Glassy Dynamics in a Disordered Heisenberg Quantum Spin System

Understanding the dynamics of strongly interacting disordered quantum systems is one of the most challenging problems in modern science, due to features such as the breakdown of thermalization and the emergence of glassy phases of matter. We report on the observation of anomalous relaxation dynamics in an isolated XXZ quantum spin system realized by an ultracold gas of atoms initially prepared in a superposition of two different Rydberg states. The total magnetization is found to exhibit subexponential relaxation analogous to classical glassy dynamics, but in the quantum case this relaxation originates from the buildup of nonclassical correlations. In both experiment and semiclassical simulations, we find the evolution toward a randomized state is independent of the strength of disorder up to a critical value. This hints toward a unifying description of relaxation dynamics in disordered isolated quantum systems, analogous to the generalization of statistical mechanics to out-of-equilibrium scenarios in classical spin glasses.

Popular Summary

Glasses have the microscopic structure of a fluid, however, they flow—or relax—extremely slowly, even slowing down over long timescales. In classical glasses, this behavior is governed by thermal fluctuations acting on the complex network of interacting atoms. Our work surprisingly reveals that such relaxation behavior is also found in isolated quantum systems, where the dynamics is dominated solely by quantum effects such as interference and entanglement.

Isolated quantum systems prepared out of thermal equilibrium are generally believed to relax to thermal equilibrium, however, little is known about how this actually proceeds. While various intriguing phenomena beyond equilibrium have been observed, a comprehensive understanding is impeded by the difficulty of describing quantum many-body dynamics theoretically.

We experimentally realize a prototypical disordered and well-isolated quantum spin system using a gas of highly excited atoms that interact with each other via dipolar forces. By observing the decay of the global spin polarization, we find relaxation behavior of the same nature as in classical glasses. In addition, we can tune the degree of disorder in the system and find that the decay law is largely independent of the degree of disorder, which might indicate an underlying, yet unexplained, universality.

Our discovery is likely to inspire further theoretical investigations aimed at a deeper understanding of the dynamics of disordered isolated quantum systems far from equilibrium. In fact, the apparent universality of the relaxation, in conjunction with the similarity to classical glasses, might hint at the existence of an effective overarching theoretical framework describing slow relaxation dynamics beyond thermal equilibrium.

Figure 1

Relaxation dynamics in a disordered quantum spin system. (a) Protocol for initialization and readout of the many-body spin system composed of Rydberg atoms. Spin states $|\uparrow ⟩$ and $|\downarrow ⟩$ correspond to two different Rydberg states. (b) Exact simulation of 12 spins interacting via a Heisenberg XXZ Hamiltonian. The plot shows the magnetization for a single realization (gray curve) and the disorder average over 1000 realizations (black curve) which relaxes as function of time given in units of the median of the mean-field interaction strengths $J_{\mathrm{mf}}$. The dashed line indicates the mean-field prediction that does not relax. The microscopic expectation values of $⟨S_x^{(i)}⟩$, $⟨S_y^{(i)}⟩$, and $⟨S_z^{(i)}⟩$ for each spin are plotted at three different time steps on the Bloch sphere. The reduction of the expectation values (magnetization) is a consequence of the spreading of entanglement (visualized by the blue bonds between spins).

Figure 2

Buildup of entanglement quantified by the time evolution of the second order Rényi entropy of the few-particle simulation from Fig. 1. The plot shows the ensemble averaged entropy for a single realization (gray curve) and the disorder average over 1000 realizations (black curve). Since the full system remains pure, the Rényi entropy is a measure of entanglement that increases on similar timescales to the maximal value of ${\mathcal{S}}_i^{(2)}=1$ as the magnetization relaxes to zero.

Figure 3

Implementation of the XXZ spin-$1/2$ model in Rydberg gases. (a) Experimental procedure. Left: A laser pulse at 780 nm (red) and 480 nm (blue) excites a controlled fraction of the ${}^{87}\mathrm{Rb}$ atoms to the $|\downarrow ⟩=|48S⟩$ Rydberg state. Middle: A microwave field couples the $|\downarrow ⟩$ state to the $|\uparrow ⟩=|49S⟩$ state to perform a Ramsey experiment. Right: A blue 480 nm laser depopulates the $|\downarrow ⟩$ Rydberg state, before the $|\uparrow ⟩$ state is ionized by an electric field. The ions are detected by a multichannel plate. (b) Ramsey fringes showing a high degree of phase coherence for a detuning of $\mathrm{\Delta }/2\pi =0.47\mathrm{MHz}$ for a peak spin density of ${\rho }_S^0=6.0(15)\times {10}^7{\mathrm{cm}}^{-3}$. The solid line shows discrete truncated Wigner approximation simulations for that density.

Figure 4

Many-body relaxation dynamics of a Heisenberg XXZ Rydberg spin system. The dots show the temporal evolution of the two magnetization components $⟨S_x⟩$ and $⟨S_y⟩$ from a tomographic spin readout. Error bars are determined from 120 realizations of the experiment at a peak spin density of ${\rho }_S^0=1.2(3)\times {10}^9{\mathrm{cm}}^{-3}$. The observed dynamics are clearly inconsistent with the mean-field prediction including imperfect initial state preparation (dotted line). The solid lines are DTWA predictions without free parameters. The shaded area indicates the systematic uncertainty of the measured density. The dashed line depicts a fit of the data with a stretched exponential function yielding a stretching exponent of $\beta =0.32(2)$.

Figure 5

Rescaled magnetization dynamics for different densities and disorder strengths. (a) 2D representation of a spin system with different densities and thus strengths of disorder, characterized by the mean number of spins per blockade radius ${(a_0/R_{\mathrm{bl}})}^{-2}$. The bar denotes the mean $x$ and standard deviation $\sigma$ of the interparticle distances ${\min }_{i\ne j}{r}_{ij}$. (b) Data points represent measurements of the averaged magnetization $⟨S_x⟩$ for spin densities ${\rho }_S^0=0.43(11)\times {10}^9{\mathrm{cm}}^{-3}$ (blue), $0.8(2)\times {10}^9{\mathrm{cm}}^{-3}$ (green), $1.2(3)\times {10}^9{\mathrm{cm}}^{-3}$ (orange and red, corresponding to two different datasets). By rescaling time with the effective interaction strength $C_6/a_0^6$, the data collapse on a single curve described by a stretched exponential function with stretching exponent $\beta =0.32(2)$ (dashed line). The inset shows $\beta$ as a function of the corresponding ratio ${(a_0/R_{\mathrm{bl}})}^{-3}$ for the experimental data and DTWA calculations (solid gray line).

Figure 6

Numerical simulation of the dynamics for a uniform density distribution using the DTWA. (a) The simulated dynamics before rescaling. After the quadratic onset, the magnetization decay is fitted well by the stretched exponential function (dashed lines). (b) The fitted decay rate ${\gamma }_J$ (dashed black line) agrees well with the median of the mean-field interaction strengths $J_{\mathrm{mf}}$ (solid gray line), which is the typical energy scale of the system. In the weakly interacting limit, $J_{\mathrm{mf}}$ scales linearly with $C_6/a^6$ (dotted gray line). The colored dots denote the decay rate derived from the four simulations depicted in (a). (c) For small (${\rho }_{\mathrm{S}}=1.25\times 10^8{\mathrm{cm}}^{-3}$, blue line and blue dot in the inset) and intermediate densities (${\rho }_{\mathrm{S}}=3.51\times 10^8{\mathrm{cm}}^{-3}$, green line and green dot in inset and ${\rho }_{\mathrm{S}}=8.73\times 10^8{\mathrm{cm}}^{-3}$, yellow line and yellow dot in the inset), the numerical data collapses on one curve after rescaling time with $C_6/{\tilde{a}}^6$, where $\tilde{a}$ plays the role of an effective distance that takes into account the roles of disorder and power law interactions. For densities as large as ${\rho }_{\mathrm{S}}=2.11\times {10}^9{\mathrm{cm}}^{-3}$ (dotted red line and red dot in the inset), the spatial order introduced by the blockade is so large that the dynamics does not follow the universal behavior observed for smaller densities. Inset: disorder averaged stretching exponent $\beta$ (solid line) as a function of the order in the system expressed by ${(\tilde{a}/R_{\mathrm{bl}})}^{-3}$. Below a critical value of ${(\tilde{a}/R_{\mathrm{bl}})}^{-3}\lesssim 0.7$ the exponent becomes constant. The shaded area indicates statistical uncertainty. The dots denote the exponent $\beta$ resulting from the single disorder realizations of (a). The dashed line is obtained from the fluctuator model [see Eq. (6) in Sec. 6].

Figure 7

Normalized probability distribution of nearest-neighbor interactions $g(J)$ for the densities given in Fig. 6 (blue line, ${\rho }_S=1.25\times {10}^8{\mathrm{cm}}^{-3}$; yellow line, ${\rho }_S=8.73\times {10}^8{\mathrm{cm}}^{-3}$; red line, ${\rho }_S=2.11\times {10}^9{\mathrm{cm}}^{-3}$). The dashed line depicts a distribution corresponding to randomly placed spins. The blockade effect induces correlations in the system and hence the probability distribution at high densities becomes more peaked. This results in a decreased disorder compared to the completely random case. The inset shows the same data with linear axis to illustrate the narrowing of the distribution functions.

Figure 8

Reconstruction of the magnetization. Left: measurements of the total number of spins $M_{\uparrow +\downarrow }$. Right: measurement of the population in the $|\uparrow ⟩$ state $M_{\uparrow }(\varphi )$ after a readout pulse of phase $\varphi$. Both are affected by the population in auxiliary states $N_a$. The measurement $M_{\uparrow }$ is fitted by a sinusoidal function (orange line), from which we extract the mean value ${\overline{M}}_{\uparrow }$. The amplitude $A$ of the fit, normalized by the total number of spin $N_{\uparrow +\downarrow }$, indicates the magnetization in the $xy$ plane.

Figure 9

Comparison between DTWA and MACE. Both moving average cluster expansion (MACE) and DTWA perform well to describe the experimental data at late times. At intermediate times (between 1 and 3 $\mu \mathrm{s}$), MACE predicts faster depolarization dynamics.

Figure 10

Scaling of the disorder strength as a function of density. The Kullback-Leibler divergence ${\mathcal{D}}_{\mathrm{KL}}$ increases monotonically with $(a/R_{\mathrm{bl}}{)}^{-3}$. This quantifies the additional correlations induced by the Rydberg blockade effect. The dots indicate the Kullback-Leibler divergence corresponding to the specific simulations presented in Fig. 6.