Experimental platforms based on trapped ions, cold molecules, and Rydberg atoms have made possible the investigation of highly nonlocal spin-$1/2$ Hamiltonians with long-range couplings. Here, we study the effects of such nonlocal couplings in the long-range spin-$1/2$ XXZ Heisenberg Hamiltonian. We calculate explicitly the two-spin energy spectrum, which describes all possible energetic configurations of two spins pointing in a specific direction embedded in a background of spins with opposite orientation. For fast decay of the spin-spin couplings, we find that the two-spin energy spectrum is characterized by well-defined discrete values, corresponding to bound states, separated by a set of continuum states describing the scattering region. In the deep long-range regime instead, the bound states disappear as they get incorporated by the scattering region. The presence of two-spin bound states results to be crucial to determine both two- and many-spin dynamics. On one hand, radically different two-spin spreadings can be observed by tuning the decay of the spin couplings. On the other hand, two-spin bound states enable the dynamical stabilization of effective antiferromagnetic states in the presence of ferromagnetic couplings. Finally, we propose a novel scheme based on a trapped-ion quantum simulator to experimentally realize the long-range XXZ model and to study its out-of-equilibrium properties.
Experimental platforms based on trapped ions, cold molecules, and Rydberg atoms have made possible the investigation of highly nonlocal spin-1/2 Hamiltonians with long-range couplings. Here, we study the effects of such nonlocal couplings in the long-range spin-1/2 XXZ Heisenberg Hamiltonian. We calculate explicitly the two-spin energy spectrum, which describes all possible energetic configurations of two spins pointing in a specific direction embedded in a background of spins with opposite orientation. For fast decay of the spin-spin couplings, we find that the two-spin energy spectrum is characterized by well-defined discrete values, corresponding to bound states, separated by a set of continuum states describing the scattering region. In the deep long-range regime instead, the bound states disappear as they get incorporated by the scattering region. The presence of two-spin bound states results to be crucial to determine both two- and many-spin dynamics. On one hand, radically different two-spin spreadings can be observed by tuning the decay of the spin couplings. On the other hand, two-spin bound states enable the dynamical stabilization of effective antiferromagnetic states in the presence of ferromagnetic couplings. Finally, we propose a novel scheme based on a trapped-ion quantum simulator to experimentally realize the long-range XXZ model and to study its out-of-equilibrium properties.
Quantum-optical techniques allow for generating controllable spin-spin interactions between ions, making trapped ions an ideal quantum simulator of Heisenberg chains. A single parameter, the detuning of the Raman coupling, allows to switch between ferromagnetic and antiferromagnetic chains, and to modify the range of the interactions. On the antiferromagnetic side, the system can be tuned from an extreme long-range limit, in which any pair of ions interacts with almost equal strength, to interactions with a decay. By exact diagonalization, we study how a system of up to 20 ions behaves upon tuning the interactions. We find that it undergoes a transition from a dimerized state with extremely short-ranged correlations towards a state with quasi long-range order, that is, algebraically decaying correlations. The dynamical evolution of the system after a local quench is shown to strongly vary in the two regimes: While in the dimerized limit, the excitation remains localized for long times, propagating spinons characterize the dynamics of the quasi-long-range ordered system. Taking a look onto the ferromagnetic side of the system, we demonstrate the feasibility of witnessing non-locality of quantum correlations by measuring two-particle correlators.
Quantum-optical techniques allow for generating controllable spin-spin interactions between ions, making trapped ions an ideal quantum simulator of Heisenberg chains. A single parameter, the detuning of the Raman coupling, allows to switch between ferromagnetic and antiferromagnetic chains, and to modify the range of the interactions. On the antiferromagnetic side, the system can be tuned from an extreme long-range limit, in which any pair of ions interacts with almost equal strength, to interactions with a decay. By exact diagonalization, we study how a system of up to 20 ions behaves upon tuning the interactions. We find that it undergoes a transition from a dimerized state with extremely short-ranged correlations towards a state with quasi long-range order, that is, algebraically decaying correlations. The dynamical evolution of the system after a local quench is shown to strongly vary in the two regimes: While in the dimerized limit, the excitation remains localized for long times, propagating spinons characterize the dynamics of the quasi-long-range ordered system. Taking a look onto the ferromagnetic side of the system, we demonstrate the feasibility of witnessing non-locality of quantum correlations by measuring two-particle correlators.
Interacting quantum spin models are remarkably useful for describing different types of physical, chemical, and biological systems.Significant understanding of their equilibrium properties has been achieved to date, especially for the case of spin models with short-range couplings.However, progress toward the development of a comparable understanding in long-range interacting models, in particular out-of-equilibrium, remains limited.In a recent work, we proposed a semiclassical numerical method to study spin models, the discrete truncated Wigner approximation (DTWA), and demonstrated its capability to correctly capture the dynamics of one-and two-point correlations in one-dimensional (1D) systems.Here we go one step forward and use the DTWA method to study the dynamics of correlations in two-dimensional (2D) systems with many spins and different types of long-range couplings, in regimes where other numerical methods are generally unreliable.We compute spatial and time-dependent correlations for spin-couplings that decay with distance as a power-law and determine the velocity at which correlations propagate through the system.Sharp changes in the behavior of those velocities are found as a function of the power-law decay exponent.Our predictions are relevant for a broad range of systems including solid state materials, atom-photon systems and ultracold gases of polar molecules, trapped ions, Rydberg, and magnetic atoms.We validate the DTWA predictions for small 2D systems and 1D systems, but ultimately, in the spirt of quantum simulation, experiments will be needed to confirm our predictions for large 2D systems.
Chains of coupled two-level atoms behave as one-dimensional quantum spin systems, exhibiting free magnons and magnon bound states. While these excitations are well studied for closed systems, little consideration has been given to how they are altered by the presence of an environment. This will be especially important in systems that exhibit nonlocal dissipation, e.g., systems in which the magnons decay due to optical emission. In this paper, we consider free magnon excitations and two-magnon bound states in an XXZ chain with nonlocal dissipation. We prove that while the energy of the bound state can lie outside the two-magnon continuum of energies, the decay rate of the bound state has to always lie within the two-magnon continuum of decay rates. We then derive analytically the bound-state solutions for a system where both the XY interaction and nonlocal dissipation are nearest neighbor or next-nearest neighbor, finding that the inclusion of nonlocal dissipation allows more freedom in engineering the energy and decay rate dispersions for the bound states. Finally, we numerically study a model of an experimental setup that should allow the realization of dissipative bound states by using Rydberg-dressed atoms coupled to a photonic crystal waveguide. We demonstrate that this model can exhibit many key features of our simpler models.
The quantum simulation of complex many body systems holds promise for understanding the origin of emergent properties of strongly correlated systems, such as high-Tc superconductors and spin liquids. Cold atomic systems provide an almost ideal platform for quantum simulation due to their excellent quantum coherence, initialization and readout properties, and their ability to support several forms of interactions. In this thesis, I present experiments on the quantum simulation of long range Ising models in the presence of transverse magnetic fields with a chain of up to sixteen ultracold 171-Yb+ ions trapped in a linear radio frequency Paul trap. Two hyperfine levels in each of the 171-Yb+ ions serve as the spin-1/2 systems. We detect the spin states of the individual ions by observing state-dependent fluorescence with single site resolution, and can directly measure any possible spin correlation function. The spin-spin interactions are engineered by applying dipole forces from precisely tuned lasers whose beatnotes induce stimulated Raman transitions that couple virtually to collective phonon modes of the ion motion. The Ising couplings are controlled, both in sign and strength with respect to the effective transverse field, and adiabatically manipulated to study various aspects of this spin model, such as the emergence of a quantum phase transition in the ground state and spin frustration due to competing antiferromagnetic interactions. Spin frustration often gives rise to a massive degeneracy in the ground state, which can lead to entanglement in the spin system. We detect and characterize this frustration induced entanglement in a system of three spins, demonstrating the first direct experimental connection between frustration and entanglement. With larger numbers of spins we also vary the range of the antiferromagnetic couplings through appropriate laser tunings and observe that longer range interactions reduce the excitation energy and thereby frustrate the ground state order. This system can potentially be scaled up to study a wide range of fully connected spin networks with a few dozens of spins, where the underlying theory becomes intractable on a classical computer.
Motivated directly by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as $1/{r}^{\ensuremath{\alpha}}$, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct and well-studied phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (nonfrustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for $\ensuremath{\alpha}\ensuremath{\lesssim}3$, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a U(1) continuous symmetry, and introduce a possibly exotic tricritical point with no direct parallel in short-range interacting spin chains. Importantly, we show that the main signatures of all five phases found could be observed experimentally in the near future.