Coherent Many-Body Spin Dynamics in a Long-Range Interacting Ising Chain

Coherent many-body quantum dynamics lies at the heart of quantum simulation and quantum computation. Both require coherent evolution in the exponentially large Hilbert space of an interacting many-body system. To date, trapped ions have defined the state of the art in terms of achievable coherence times in interacting spin chains. Here, we establish an alternative platform by reporting on the observation of coherent, fully interaction-driven quantum revivals of the magnetization in Rydberg-dressed Ising spin chains of atoms trapped in an optical lattice. We identify partial many-body revivals at up to about ten times the characteristic time scale set by the interactions. At the same time, single-site-resolved correlation measurements link the magnetization dynamics with interspin correlations appearing at different distances during the evolution. These results mark an enabling step towards the implementation of Rydberg-atom-based quantum annealers, quantum simulations of higher-dimensional complex magnetic Hamiltonians, and itinerant long-range interacting quantum matter.

Popular Summary

Quantum computers promise solutions to tasks that are intractable by classical machines, such as problems relevant to cryptography, material science, quantum chemistry, and classical optimization. Quantum magnets based on ultracold atoms trapped in optical lattices provide one way to create a special type of quantum computer known as a quantum annealer. To achieve the promised quantum speedup, a quantum annealer requires coherent and fine-tuned interactions among the spins of its atoms. An annealer can be implemented in cold atomic gases by off-resonant optical coupling to long-range interacting “Rydberg states,” states in which one electron of the atom is highly excited. Induced and controlled by laser light, such “Rydberg-dressed” interactions come with a large degree of tunability, distinguishing them from other types of engineered spin interactions in cold atomic gases. In our work, we experimentally realize quantum magnets with such optically induced long-range spin interactions.

Contrary to earlier experiments, our system features unmatched long coherence times. We probe this by tracing a process unique to the quantum world: collapse and revival of the transverse magnetization. Starting in a fully polarized state of maximal magnetization, the interaction first dephases the spins. This results in a decaying magnetization, which later revives when the spins partially resynchronize at long observation times. Local detection with a quantum gas microscope that is sensitive to single atoms allows us to shed light on the remaining decoherence processes as well as the microscopic mechanisms of the collapse and revival dynamics.

Our results mark an important step that pushes the coherence of synthetic quantum magnets toward the implementation of a Rydberg-dressed quantum annealer.

Figure 1

Schematic of the Ising spin chain and spin-selective detection. (a) Illustration of a spin chain with $N=10$ spins initialized in the fully transverse magnetized state. The dominant contributions of the Rydberg-dressed interaction between nearest and next-to-nearest spins spaced by $a_{\mathrm{lat}}$ are indicated by the dark gray and light gray arrows. The brightness of the blue color of the spins encodes the interaction strength between the exemplary selected fifth spin with the rest of the chain. The gray lines are guides to the eye to link to the interaction potential shown in (b). (b) Ab initio calculated Rydberg-dressed potential [18] for our experimental parameters $\mathrm{\Omega }/2\pi =3.57(3)\mathrm{MHz}$ and $\mathrm{\Delta }/2\pi =11.00(2)\mathrm{MHz}$, normalized to the nearest-neighbor interaction strength $|U_0|=13.1(5)\mathrm{kHz}$ (blue solid line), with the relevant potential at multiples of the lattice distance $a_{\mathrm{lat}}$ marked by blue points and gray horizontal lines. The inset shows the occurrence of frequency differences $\mathrm{\Delta }\nu$ in the many-body spectrum of the long-range interacting Ising model (gray bars) and those governing transverse magnetization dynamics (blue bars). Orange vertical lines mark the corresponding $\mathrm{\Delta }\nu$ for the Ising model with nearest-neighbor interactions only. (c) Simulated revival dynamics of the populations of spin left (red) and spin right (blue), starting from the initially fully magnetized chain in the $S^y$ direction for a defect-free chain of ten spins and long-range interactions. Clear partial revivals are observed during the evolution. The fluorescence images to the right show characteristic spin configurations for ten spins observed during the collapse and revival dynamics at times indicated by the gray lines. The spin of the atoms was detected via an in situ Stern-Gerlach sequence, which led to a spatial separation of spin-left (red) and spin-right atoms (blue). This allows for the reconstruction of the full spin and density distribution (pictograms to the right).

Figure 2

Evolution of the mean magnetization density. (a) The probability to measure atoms in the states $|\leftarrow ⟩$ ($|\to ⟩$) vs total dressing time $t$ shown as red (blue) data points. The final spin rotation before detection is indicated by the pictograms (upper and lower left corner). The upper axis is scaled in units of the inverse nearest-neighbor interaction $1/|U_0|=76(3)\mu \mathrm{s}$. The total atom number was restricted to be $N<15$ to filter out events with clear preparation errors. The solid line shows the theoretically expected dynamics, averaged over 100 initial chains randomly selected from a reference data set with an initial filling of 87(3)% and a mean atom number of 10(1.4). The shaded region marks the corresponding standard error of the mean (s.e.m.). All data points are an average over at least 50 experimental realizations (150 for $t<0.1\mathrm{ms}$). The inset shows the initial dynamics of the mean transverse magnetization density $⟨{\widehat{S}}_{\mathrm{tot}}^y⟩/N$ up to $t\approx 6/U_0$, obtained from the spin populations (green data points) with the theoretical prediction (green solid line). (b) Zoom-in on two magnetization features at $t\approx 5/U_0$ (1) and $t\approx 12/U_0$ (2), as indicated by the gray boxes in (a). Even at long times up to $12/U_0$, the nonvanishing magnetization indicates the presence of finite coherence. All error bars on the data points denote one s.e.m.

Figure 3

Local characterization of the spin dynamics. (a) Observed evolution of the connected correlations $C_d$ for spins separated by $d=a_{\mathrm{lat}}$ and (b) $d=2a_{\mathrm{lat}}$, as indicated by the pictograms. The solid line with surrounding shading shows the corresponding prediction of the same theoretical calculation as used in Fig. 2. All error bars denote one s.e.m.

Figure 4

Evolution of the spin parity in the initially unpopulated $|\to ⟩$ state. The histograms on top show the evolution of the number of detected atoms in the $|\to ⟩$ state with even numbers highlighted by the darker color. The data were binned in intervals of 0.1 ms and the bin centers are indicated. The main plot shows the extracted parity $⟨\widehat{P}⟩$ vs time, together with an exponential fit with time constant ${\tau }_P=0.13(4)\mathrm{ms}$ (solid line). All error bars denote one s.e.m.