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arXiv:1812.07893 · Physical Review A

Decay rates and energies of free magnons and bound states in dissipative XXZ chains

Christopher D. Parmee, Nigel R. Cooper

Published Jun 18, 2019• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.99.063615

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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.
Experimental platforms based on trapped ions, cold molecules, and Rydberg atoms have made possible the investigation of highly nonlocal spin-1/21/21/2 Hamiltonians with long-range couplings. Here, we study the effects of such nonlocal couplings in the long-range spin-1/21/21/2 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.
We investigate collective behaviour that appears in open, many-body systems of two- and four-level atoms. Here, ``open" refers to the system interacting with an external environment that causes dissipation. We derive a general open quantum master equation to describe the system dynamics only, independent of this environment. We identify processes such as coherent exchange of virtual photons and modified decay rates caused by long-range interactions between all pairs of atoms that scale with distance by inverse power laws. We explore excitation transport within a one-dimensional chain of atoms where the atomic transition dipoles are coupled to the free radiation field. When the interatomic spacing is smaller than or comparable to the wavelength of light associated with the photon emitted from a given transition, virtual photon exchange interactions facilitate excitation transport through the chain. Atomic systems coupled to an environment display dissipative dynamics, however subradiant transport is exhibited from a variety of initial states; spontaneous emission from the chain occurs at a rate much slower than that for an individual atom. In particular, we find a region within the decay spectrum that consists entirely of subradiant states with a corresponding linear dispersion relation in the interaction energy. Identifying this subspace allows for the dispersionless transport of wave packets over long distances with near-zero decay. Moreover, the group velocity of the wave packet and direction of the transport can be controlled via an external uniform magnetic field while preserving its subradiant character. We discuss a number of experimental considerations to justify the feasibility and robustness of this protocol. Initial state preparation is outlined, utilising external laser driving to excite the system into the single excitation sector. Furthermore, we consider positional disorder by explicitly accounting for the external atomic degrees of freedom -- position and momentum -- which allows us to model the positions of all atoms by a motional state representing the occupation of a lattice well with a given width. These discussions are made in the low-temperature limit, where the atomic motion is essentially frozen, and we identify that subradiant transport is indeed robust. Finally, we explore the experimental limits of interatomic spacing and imperfect filling within an experimentally achievable optical lattice. We calculate the photon emission rate -- an experimentally measurable quantity -- and compare the emission spectra to our analysis. By limiting parameters to those achieved experimentally, we observe a reduction, yet not an absence, of collective behaviour. The simplicity and versatility of this system, together with the robustness of subradiance against disorder, makes it relevant for a range of applications such as lossless energy transport and long-time light storage. The lifetime of an atomic excitation could be increased by a factor of thousands to millions for a chain of atoms under the conditions that we explore in this thesis.
The bound state problems associated with two magnons in a linear chain of spin 1/2 with nearest and next nearest neighbor exchange integrals, J and α J , respectively, are studied. The Hamiltonian is rewritten in terms of the Fermi operators in place of spin operators, and an integral equation to determine the bound state energies is obtained. In the cases when the total wave number K of two magnons is π or π/2, analytical solutions are easily obtained. A part of the present results disagrees with Majumdar's ones. Computer studies on the bound states are performed for a finite chain composed of spins up to twenty. They are consistent with the analytical results.
In this thesis, we consider the forms of non-equilibrium phenomena that can arise in atoms or polar molecules trapped in a deep optical lattice and coupled by dipolar internal degrees of freedom. We specifically focus on only two internal states, which results in the systems studied behaving as spin−1/2-1/2−1/2 models with long-range interactions. We first study a closed system with both static and resonant near-field dipole interactions under an external drive. By studying the uniform mean-field dynamics of the system, we find the dynamics are given by Rabi oscillations, with a bifurcation in the dynamics as a function of drive strength between small scale and large scale oscillations. Analysing the stability of these oscillations to small fluctuations reveals that interactions tend to cause the oscillations to decohere. However, we find parameter regimes where coherent oscillations can persist for high enough intensity drive. We then consider the effects of an environment on the non-equilibrium dynamics of the near-field dipole model. We find that within the mean-field approximation, an environment causes the system to relax to many novel steady state spin configurations, such as spin density waves, antiferromagnetism and long-time oscillations, as well as bistabilities between these phases. To assess the validity of the mean-field approximation, we compare our mean-field results to small quantum systems. We carry out a similar analysis on a system with far-field dipole interactions, where it is necessary to introduce nonlocal dissipation, which results in several decay modes into the environment. These decay modes lead to instabilities of many of the steady state phases that occurred in the near-field dipole system, leading to the emergence of more spin density wave and oscillatory phases. Finally, we examine the dynamics on the approach to steady state in a dissipative dipolar system by studying Rydberg atoms coupled to a photonic crystal waveguide, which mediates an effective dipole-dipole interaction between the atoms. We find that if two excitations exist in the system, then bound states can form, with nonlocal dissipation resulting in a momentum dependent decay rate of the bound states and also greater freedom in engineering the bound state energy dispersion.
We study the formation of magnon-polaron excitations and the consequences of different timescales between the magnon and lattice dynamics. The spin-spin interactions along the one-dimensional lattice are ruled by a Heisenberg Hamiltonian in the anisotropic form XXZ, in which each spin exhibits a vibrational degree of freedom around its equilibrium position. By considering a magnetoelastic coupling as a linear function of the relative displacement between nearest-neighbor spins, results provide an original framework for achieving a hybridized magnon-polaron state. Such a state is characterized by high cooperation between the underlying excitations, where nondispersive profiles of the magnon wave function and the lattice deformations exhibit a bound dynamics. Traveling or stationary formation of a magnon-polaron depends on the effective magnetoelastic coupling, and the critical amount $({\ensuremath{\chi}}_{c})$ separating both regimes is shown. Different characteristic timescales of the magnon and the vibrational dynamics unveiled the threshold between such behaviors, as well as a limit value of critical magnetoelastic interaction, above which the magnon velocity no longer interferes at the critical magnetoelastic coupling capable of inducing the stationary regime.
Few-magnon bound states in quantum spin chains have been long studied and attracted much recent attention. For a higher-spin ferromagnetic XXZ chain with single-ion anisotropy, several features regarding the evolution of the low-lying two-magnon bound states with varying wave number were observed in the literature. However, most of these observations are only qualitatively understood due to the lack of analytical tools. By combining a set of exact two-magnon Bloch states and a plane-wave ansatz, we achieve a complete solution of the two-magnon problem in such a system. We identify parameter regions that support different types of two-magnon bound states, with the boundaries defined by algebraic equations. We discover a narrow region in which two single-ion bound states coexist. We show that the phase diagrams for distinct wave numbers are similar to each other, which enables us to map the evolution of the bound states to the rectilinear movement of a representative point for given parameters in a rescaled phase diagram. This dynamic picture provides quantitative interpretations of the observed features.