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Physical Review A

Probing the topological phase transition in the Su-Schrieffer-Heeger Hamiltonian using Rydberg-atom synthetic dimensions

Y. Lu, C. Wang, S. K. Kanungo, F. B. Dunning, T. C. Killian

Published Aug 15, 2024• 11 citations
Publisher

Abstract

We experimentally investigate the Su-Schrieffer-Heeger (SSH) Hamiltonian using Rydberg-atom synthetic dimensions constructed, in a single atom, from a ladder of six neighboring $n\phantom{\rule{0.222222em}{0ex}}^{3}S_{1}$ Rydberg states in which adjacent states are coupled with two-photon transitions using microwave fields. Alternating strong or weak tunneling rates, controlled by adjusting the microwave amplitudes, are varied to explore the topological phase transition inherent in the SSH Hamiltonian as a function of the ratio of the tunneling rates. For each ratio, quench dynamics experiments, in which the system is initially prepared in one of the bulk Rydberg states and then subjected to the microwave fields, are performed to measure the population evolution of the system. From the dynamics measurements, we extract the mean chiral displacement and verify that its long-time average value converges towards the system winding number. The loss of nontrivial topology is also examined by probing the energy spectrum of the system in steady state and observing the disappearance of the zero-energy edge states. The results show that even a system with as few as six levels can demonstrate the essential characteristics of the SSH Hamiltonian.

DOI
10.1103/physreva.110.023318

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We experimentally investigate the Su-Schrieffer-Heeger (SSH) Hamiltonian using Rydberg-atom synthetic dimensions constructed, in a single atom, from a ladder of six neighboring n3S1n\phantom{\rule{0.222222em}{0ex}}^{3}S_{1}n3S1​ Rydberg states in which adjacent states are coupled with two-photon transitions using microwave fields. Alternating strong or weak tunneling rates, controlled by adjusting the microwave amplitudes, are varied to explore the topological phase transition inherent in the SSH Hamiltonian as a function of the ratio of the tunneling rates. For each ratio, quench dynamics experiments, in which the system is initially prepared in one of the bulk Rydberg states and then subjected to the microwave fields, are performed to measure the population evolution of the system. From the dynamics measurements, we extract the mean chiral displacement and verify that its long-time average value converges towards the system winding number. The loss of nontrivial topology is also examined by probing the energy spectrum of the system in steady state and observing the disappearance of the zero-energy edge states. The results show that even a system with as few as six levels can demonstrate the essential characteristics of the SSH Hamiltonian.
We simulate the the Su-Schrieffer-Heeger (SSH) model using Rydberg-atom synthetic dimensions constructed, in a single atom, from a ladder of six neighboring n 3S1n\:^3S_1n3S1​ Rydberg states in which adjacent states are coupled with two-photon transitions using microwave fields. Alternating strong/weak tunneling rates, controlled by adjusting the microwave amplitudes, are varied to map out the topological phase transition as a function of the ratio of the tunneling rates. For each ratio, quench dynamics experiments, in which the system is initially prepared in one of the bulk Rydberg states and then subjected to the microwave fields, are performed to measure the population evolution of the system. From the dynamics measurements, we extract the mean chiral displacement and verify that its long-time average value converges towards the system winding number. The topological phase transition is also examined by probing the energy spectrum of the system in steady state and observing the disappearance of the zero-energy edge states. The results show that even a system with as few as six levels can demonstrate the essential characteristics of the SSH Hamiltonian.
Rydberg-atom synthetic dimensions in the form of a lattice of n3S1^3S_13S1​ levels, 58≤n≤6358\leq n \leq 6358≤n≤63, coupled through two-photon microwave excitation are used to examine dynamics within the single-particle Su-Schrieffer-Heeger (SSH) Hamiltonian. This paradigmatic model of topological matter describes a particle hopping on a one-dimensional lattice with staggered hopping rates. Tunneling rates between lattice sites and on-site potentials are set by the microwave amplitudes and detuning, respectively. An atom is first excited to a Rydberg state that lies within the lattice and then subject to the microwave dressing fields. After some time, the dressing fields are turned off and the evolution of the population distribution in the different final lattice sites monitored using field ionization. The measurements show the existence of long-lived symmetry-protected edge states and reveal the existence of direct long-distance tunneling between the edge states. The results are in good agreement with model calculations and further demonstrate the potential of Rydberg-atom synthetic dimensions to simulate and faithfully reproduce complex Hamiltonians.
Rydberg-atom synthetic dimensions in the form of a lattice of n3S1n^{3}S_{1}n3S1​ levels, $58\ensuremath{\le}n\ensuremath{\le}63$, coupled through two-photon microwave excitation are used to examine dynamics within the single-particle Su-Schrieffer-Heeger Hamiltonian. This paradigmatic model of topological matter describes a particle hopping on a one-dimensional lattice with staggered hopping rates. Tunneling rates between lattice sites and on-site potentials are set by the microwave amplitudes and detuning, respectively. An atom is first excited to a Rydberg state that lies within the lattice and then subject to the microwave dressing fields. After some time, the dressing fields are turned off and the evolution of the population distribution in the different final lattice sites monitored using field ionization. The measurements show the existence of long-lived symmetry-protected edge states and reveal the existence of direct long-distance tunneling between the edge states. The results are in good agreement with model calculations and further demonstrate the potential of Rydberg-atom synthetic dimensions to simulate and faithfully reproduce complex Hamiltonians.
We demonstrate a platform for synthetic dimensions based on coupled Rydberg levels in ultracold atoms, and we implement the single-particle Su-Schrieffer-Heeger (SSH) Hamiltonian. Rydberg levels are interpreted as synthetic lattice sites, with tunneling introduced through resonant millimeter-wave couplings. Tunneling amplitudes are controlled through the millimeter-wave amplitudes, and on-site potentials are controlled through detunings of the millimeter waves from resonance. Using alternating weak and strong tunneling with weak tunneling to edge lattice sites, we attain a configuration with symmetry-protected topological edge states. The band structure is probed through optical excitation to the Rydberg levels from the ground state, which reveals topological edge states at zero energy. We verify that edge-state energies are robust to perturbation of tunneling-rates, which preserves chiral symmetry, but can be shifted by the introduction of on-site potentials.
Abstract A discrete degree of freedom can be engineered to match the Hamiltonian of particles moving in a real-space lattice potential. Such synthetic dimensions are powerful tools for quantum simulation because of the control they offer and the ability to create configurations difficult to access in real space. Here, in an ultracold 84 Sr atom, we demonstrate a synthetic-dimension based on Rydberg levels coupled with millimeter waves. Tunneling amplitudes between synthetic lattice sites and on-site potentials are set by the millimeter-wave amplitudes and detunings respectively. Alternating weak and strong tunneling in a one-dimensional configuration realizes the single-particle Su-Schrieffer-Heeger (SSH) Hamiltonian, a paradigmatic model of topological matter. Band structure is probed through optical excitation from the ground state to Rydberg levels, revealing symmetry-protected topological edge states at zero energy. Edge-state energies are robust to perturbations of tunneling-rates that preserve chiral symmetry, but can be shifted by the introduction of on-site potentials.
Engineering synthetic dimensions, where the physics of additional spatial dimensions is simulated within the internal states of a quantum system, allows the realisation of phenomena not otherwise accessible in experiments. Ultracold ground-state polar molecules are an ideal platform to encode synthetic dimensions, offering access to large Hilbert spaces of long-lived internal states associated with the rotational and hyperfine degrees of freedom, that can be coupled together with microwave fields to simulate tunnelling. Here, to benchmark the advantages of ultracold molecules, we encode a 1D synthetic lattice in the rotational states of ultracold RbCs molecules and use it to investigate the well-known Su-Schrieffer-Heeger (SSH) model, a minimal model displaying topological properties. To probe the system, we perform spectroscopy using an auxiliary rotational state and study the time dynamics after deterministic state preparation. We demonstrate long coherence times, typically ~500 times the lattice tunnelling period, even for a synthetic lattice using 8 rotational states. Observations of dynamics at long times with full site-resolved readout of the synthetic dimension allow us to test the effects of chiral and non-chiral perturbations on the topologically protected edge states. Our work lays the foundation for further quantum simulations using the rich internal structure of molecules, including dipolar string phases in interacting samples of molecules, and adiabatic state preparation of many-body Hamiltonians.