Collective cluster nucleation dynamics in 2D Ising quantum magnets

Abstract

Strongly interacting many-body systems often show collective properties that are non-trivially related to the microscopic degrees of freedom. Collectivity is responsible for intriguing ground state properties, for example, in superconductors. However, collective effects may also govern the non-equilibrium response of quantum systems, not only in condensed matter physics but also in quantum field theories modeling the properties of our universe. Understanding emergent collective dynamics from first principles, in particular in non-perturbative regimes, is therefore one of the central challenges in quantum many-body physics. Here we report on the observation of collective cluster nucleation in 2D quantum Ising systems realized in an atomic Rydberg array. We observe a confined regime in which the steady-state cluster size is energy-dependent and a deconfined regime characterized by kinetically constrained dynamics of cluster nucleation. Our results mark a qualitative leap for quantum simulations with Rydberg arrays and shed light on highly collective non-equilibrium processes in one of the most important textbook models of condensed matter physics with relevance from quantum magnets and the kinetics of glass formers to cosmology.

Figures (7)

• Figure 1: Illustration of the many-body spectrum in a 2D Ising model.(a) Sketch of the quench response of the square-lattice Ising model with nearest-neighbor interactions for an initial state with all spins in the $\ket|\downarrow>$-state (light circles) as a function of $h_z$ and $\Omega$. For quenches to weak transverse field $\Omega$, the response is sharp and concentrated around discrete many-body resonances in $h_z$. At $h_z=h_z^S=-4J$, single spin flips are resonant, while confined clusters of increasing size (dark red circles with cluster sizes indicated by the numbers) become resonant at larger $h_z$ up to the deconfinement point at $h_z=h_z^A=-2J$ (marked with the letter A). Here, the dynamics are characterized by avalanche-like growth of clusters. With increasing $\Omega$, the resonances broaden (indicated by the gray shading), shift, and the cluster sizes mix due to quantum fluctuations (light red circles). (b) Confined clusters behave as collective objects with characteristic shapes determined by energetic constraints. In the classical ($\Omega \rightarrow 0$) limit, the excitation energy of a cluster is determined by the number of unsatisfied bonds (black lines) along its perimeter, each increasing the energy by $J$. The collective resonance is located at the total cluster energy divided by the number of spins in the cluster, as indicated below the cluster sketches. The square-shaped cluster at the lower right is located at an energy of $h_z=-2J$, where the addition of further spins is resonant such that the cluster size is unconfined. (c) Typical experimental snapshot after a quench to the deconfinement resonance at $h_z = -2J$. Four clusters of Rydberg atoms, that is, connected sites of flipped spins identified by missing atoms, are highlighted by the linked black squares.
• Figure 2: Spectral response.(a) Two-dimensional histogram of the cluster size distribution versus longitudinal field after $\qty{2}{\micro \second}$ evolution time with $\Omega_\text{max}=2\pi \times \qty{2.24}{\mega \hertz}$. To highlight the tails of the distribution, we use a linear color scale up to 10 counts and a logarithmic scale above. The inset shows normalized cuts along the vertical axis for 2-, 3-, 8-, and 15-atom clusters as indicated by the colored arrows. (b) Resonance positions extracted from Gaussian fits to the data shown in the inset of (a) versus cluster size (dots) together with the classical expectation (solid line). The colored dots correspond to the cuts shown in (a). (c) Amplitude of the Gaussian fits versus cluster size. The coloring is as in (b). (d) Mean number of individual flipped spins (gray) and mean number of clusters (brownish colors) together with the mean cluster size (reddish colors) versus longitudinal field. The lighter colors for the cluster number and size indicate data taken at shorter evolution times of $\qty{0.5}{\micro \second}$ (light) and $\qty{1}{\micro \second}$ (medium-light). (e) Normalized cluster size distribution at the facilitation resonance (red) and at $\widetilde{h}_z = \pm \qty{4}{\mega \hertz}$ (brown, yellow) after $\qty{2}{\micro \second}$ evolution time.
• Figure 3: Deconfined cluster formation kinetics. All data is taken on the deconfinement resonance at $\widetilde{h}_z = \widetilde{h}_z^A = 0$ and with $\Omega_\text{max} = 2\pi\times\qty{2.0}{MHz}$. (a) Growth dynamics for the two largest clusters in the system. For each run, we identify the two largest clusters and average their sizes individually over all runs (middle red for the largest, light red for the second-largest cluster). In dark red, we show the mean size of the largest cluster conditioned to runs in which only one cluster was identified. A linear fit reveals a growth rate of $\qty{24 \pm 4}{\text{sites} \per \micro \second}$, where the uncertainty is dominated by the selection of data points included in the fit (all vs. $t>\qty{0.25}{\micro \second}$). Error bars indicate the standard error of the mean. (b) Number of cluster collisions versus time. A collision is defined as a bond on which cluster growth is blocked by a nearby cluster, that is, where we identify two adjacent $\uparrow$-spins around a $\downarrow$-spin and where both $\uparrow$-spins belong to distinct clusters (see inset). (c) Histograms showing the evolution of the distribution of cluster sizes. We bin the data in intervals of $\qty{0.15}{\micro \second}$ and show the area-normalized counts for increasing evolution times from $\qty{0.1}{\micro \second}$ to $\qty{1.45}{\micro \second}$ (dark to light). The individual histograms are offset vertically for better visibility, as indicated by the thin gray lines.
• Figure 4: Shape constraints and cluster number conservation.(a) The number of loops in the clusters versus time (red) and the number of isolated spin flips (gray) show a similar increase (decrease) rate after an initial transient behavior. (b) The light red data shows the sum of the loops and the number of isolated flipped spins. For times larger than $\qty{0.25}{\micro \second}$, this sum is approximately constant. The gray line shows the mean number of isolated spins plus the minimum number of spins in clusters needed to flip down to remove all loops. It saturates fully, as expected for a signal induced by the finite detection fidelity. The mean number of clusters is conserved, as shown by the yellow data that quickly saturates.
• Figure 5: Raman sideband cooling. In red we show the Raman spectra without sideband cooling and in blue with sideband cooling applied. (a) Radial Raman spectrum. (b) Axial Raman spectrum.