Synthetic dimensions are powerful tools for quantum simulation and computation. They are realized by harnessing internal or external degrees of freedom of an atom or molecule, which can mimic the motion of an electron in a real-space lattice potential. Such degrees of freedom are highly tunable and can be engineered to create configurations difficult to access or realize in real space. Some of the exciting possibilities include realizing higher dimensions systems[1, 2, 3], nontrivial real space[4, 5] and band structure[6, 7] topologies and artificial gauge fields[8, 9]. Experiments have utilized various degrees of freedom to create synthetic dimensions, such as motional[10, 11], spin[8, 12, 13, 14] and rotational[15] levels of atoms and molecules, and frequency modes, spatial modes, and arrival times in photonic systems[16]. Atomic synthetic dimensions have demonstrated artificial gauge fields, spin-orbit coupling, chiral edge states using Raman-coupled ground magnetic sublevels[8, 12, 17] of atoms, and phenomena such as Anderson localization using two-photon Bragg transitions by coupling free-particle momentum states[18]. Here, we harness the Rydberg levels of 84Sr to realize a synthetic lattice for studying quantum matter. Resonant millimeter-wave (mm-wave) radiation coupling Rydberg levels |i⟩ and |j⟩ with amplitude Ωij (Rabi frequency) are described by the same Hamiltonian as a particle tunneling between lattice sites |i⟩ and |j⟩ with tunneling amplitude Jij = Ωij /2. The mathematical equivalence to particles moving in a real-space lattice enables Rydberg levels to function as a synthetic spatial dimension. Rydberg-atom synthetic dimensions offer control over connectivity, tunneling rates and on-site potentials, which allows for the creation of a broad range of synthetic dimensional system. The capabilities of such a system are demonstrated by realizing the famous Su-Schrieffer-Heeger (SSH) model[19] and studying its topologically protected edge states(TPS).
Synthetic dimensions are powerful tools for quantum simulation and computation. They are realized by harnessing internal or external degrees of freedom of an atom or molecule, which can mimic the motion of an electron in a real-space lattice potential. Such degrees of freedom are highly tunable and can be engineered to create configurations difficult to access or realize in real space. Some of the exciting possibilities include realizing higher dimensions systems[1, 2, 3], nontrivial real space[4, 5] and band structure[6, 7] topologies and artificial gauge fields[8, 9]. Experiments have utilized various degrees of freedom to create synthetic dimensions, such as motional[10, 11], spin[8, 12, 13, 14] and rotational[15] levels of atoms and molecules, and frequency modes, spatial modes, and arrival times in photonic systems[16]. Atomic synthetic dimensions have demonstrated artificial gauge fields, spin-orbit coupling, chiral edge states using Raman-coupled ground magnetic sublevels[8, 12, 17] of atoms, and phenomena such as Anderson localization using two-photon Bragg transitions by coupling free-particle momentum states[18]. Here, we harness the Rydberg levels of 84Sr to realize a synthetic lattice for studying quantum matter. Resonant millimeter-wave (mm-wave) radiation coupling Rydberg levels |i⟩ and |j⟩ with amplitude Ωij (Rabi frequency) are described by the same Hamiltonian as a particle tunneling between lattice sites |i⟩ and |j⟩ with tunneling amplitude Jij = Ωij /2. The mathematical equivalence to particles moving in a real-space lattice enables Rydberg levels to function as a synthetic spatial dimension. Rydberg-atom synthetic dimensions offer control over connectivity, tunneling rates and on-site potentials, which allows for the creation of a broad range of synthetic dimensional system. The capabilities of such a system are demonstrated by realizing the famous Su-Schrieffer-Heeger (SSH) model[19] and studying its topologically protected edge states(TPS).
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.
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 The concept of synthetic dimensions works particularly well in atomic physics, quantum optics, and photonics, where the internal degrees of freedom (Zeeman sublevels of the ground state, metastable excited states, or motional states for atoms, and angular momentum states or transverse modes for photons) provide the synthetic space. In this Perspective article we report on recent progress on studies of synthetic dimensions, mostly, but not only, based on the research realized around the Barcelona groups (ICFO, UAB), Donostia (DIPC), Poznan (UAM), Kraków (UJ), and Allahabad (HRI). We describe our attempts to design quantum simulators with synthetic dimensions, to mimic curved spaces, artificial gauge fields, lattice gauge theories, twistronics, quantum random walks, and more.
"As the progress of science pushes the frontiers of studying nature to both the extremely large scale and the incredibly small scale, experiments can become increasingly difficult to perform. Therefore, a scale model experiment for these kinds of physics can be useful as long as it is relatively easy to access and control. This is the idea of quantum simulation, where one system exactly mimics another when they are subjected to equivalent physical scenarios. Topological systems, those with a property that is stabilized by their specific configuration, are often quite challenging to experimentally probe. Therefore, test bed systems have been widely and successfully applied to studying topology in an effort to advance the field, which promises enticing applications such as robust quantum technology. Our particular scheme uses a novel type of ""synthetic"" dimension for atoms of a Bose–Einstein condensate, which we call a momentum-space lattice, to emulate one-dimensional lattice models. The idea of a synthetic dimension or synthetic lattices is a useful construct to describe how the particle dynamics can evolve in degrees of freedom that are not directly related to transport in real space. This can relate to population dynamics in an internal degree of freedom, such as spin, or, in our case, in the dynamics of free-particle-like atomic momentum states. We expose a Bose–Einstein condensate to two laser fields that are capable of driving transitions between different momentum states of the condensate atoms through a two-photon Bragg diffraction process. Neighboring states in the free-particle dispersion relation are characterized by a unique momenta and energy spacing such that we can independently control every aspect of their coupling. In this thesis, we report on the design and construction of an apparatus for the production of 87Rb Bose–Einstein condensates as well as the partial construction of an apparatus for the production of 39K Bose–Einstein condensates. We also discuss the experimental realization of, and theoretical background for, our momentum-space lattice. Using the momentum-space lattice, we perform experiments on not only topological systems, but also on driven artificial spins featuring chaotic dynamics, as well as faster-than-adiabatic quantum state control and preparation. Our measurements of topological systems are based in the 1D Su–Schrieffer–Heeger model, where we report on the direct observation of the localized, solitonic edge state created by the topology inherent in the system. Through the addition of exactly tunable tunneling disorder, we observe the destruction of the band topology due to disorder at a random-singlet transition. Furthermore, we drive a trivial system into a topological phase through the addition of disorder, observing a counter-intuitive phenomenon known as the topological Anderson insulator for the first time. We go on to report on our driven, tunable spin experiment, where we probe effective spin ""squeezing"" (in the absence of any entanglement in our noninteracting system), the out-of-time-ordered correlation function, and chaos. And finally, we use our momentum-space lattice to demonstrate a technique which allows a system to be transformed faster than the adiabatic limit with no diabatic transitions in a highly non-trivial regime. Due to the versatility of the momentum-space lattice, the results detailed here have a broad impact. Our observations of topological phenomena in elementary and canonical 1D systems are some of the first experimental verifications of long-standing predictions. Further, our studies of the interplay between classical chaos and quantum systems may help shed light on this complex and rich problem. And from a more ""applied science"" perspective, our demonstrations of generalized counter-diabatic techniques are widely beneficial and applicable to many systems where adiabaticity is challenged."
Rydberg-atom synthetic dimensions in the form of a lattice of 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.
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.