Abstract Many of the recently developed quantum systems that are considered promising candidates for the underlying technology of quantum computing enjoy long-range interaction, which is even tunable in some of them. Most of these systems can be described using the Heisenberg spin model to represent their interaction with the same system or a different one in hybrid structures. We consider here a finite two-dimensional spin system with a varying spin-spin long-range interaction under the effect of an external uniform magnetic field. We investigate the dynamics of the system and its asymptotic behavior at different degrees of anisotropy and interaction range under coupling to a thermal dissipative environment, starting from different initial states. We show that the system spin state and bipartite entanglement evolve in time to reach asymptotic values that are enhanced significantly by the degree of anisotropy and interaction range, whereas the thermal dissipative environment degrades the entanglement asymptotic equilibrium value as the temperature increases. Interestingly, while the robustness of the spin system against the environment’s thermal dissipative effects increases with the interaction range and degree of anisotropy, the entanglement between the far sites on the lattice shows its highest resilience in the partially anisotropic system, which might be attributed to a critical behavior taking place in the system. While the system’s early dynamics varies significantly depending on the initial state, the asymptotic equilibrium value is found to be completely independent.
Abstract Many of the recently developed quantum systems that are considered promising candidates for the underlying technology of quantum computing enjoy long-range interaction, which is even tunable in some of them. Most of these systems can be described using the Heisenberg spin model to represent their interaction with the same system or a different one in hybrid structures. We consider here a finite two-dimensional spin system with a varying spin-spin long-range interaction under the effect of an external uniform magnetic field. We investigate the dynamics of the system and its asymptotic behavior at different degrees of anisotropy and interaction range under coupling to a thermal dissipative environment, starting from different initial states. We show that the system spin state and bipartite entanglement evolve in time to reach asymptotic values that are enhanced significantly by the degree of anisotropy and interaction range, whereas the thermal dissipative environment degrades the entanglement asymptotic equilibrium value as the temperature increases. Interestingly, while the robustness of the spin system against the environment’s thermal dissipative effects increases with the interaction range and degree of anisotropy, the entanglement between the far sites on the lattice shows its highest resilience in the partially anisotropic system, which might be attributed to a critical behavior taking place in the system. While the system’s early dynamics varies significantly depending on the initial state, the asymptotic equilibrium value is found to be completely independent.
Recently new novel magnetic phases were shown to exist in the asymptotic steady states of spin systems coupled to dissipative environments at zero temperature. Tuning the different system parameters led to quantum phase transitions among those states. We study, here, a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian Lindblad environment at finite temperature. We show how applying an inhomogeneous magnetic field to the system at different degrees of anisotropy may significantly affect the spin states, and the entanglement properties and distribution among the spins in the asymptotic steady state of the system. In particular, applying an inhomogeneous field with an inward (growing) gradient toward the central spin is found to considerably enhance the nearest neighbor entanglement and its robustness against the thermal dissipative decay effect in the completely anisotropic (Ising) system, whereas the beyond nearest neighbor ones vanish entirely. The spins of the system in this case reach different steady states depending on their positions in the lattice. However, the inhomogeneity of the field shows no effect on the entanglement in the completely isotropic (XXX) system, which vanishes asymptotically under any system configuration and the spins relax to a separable (disentangled) steady state with all the spins reaching a common spin state. Interestingly, applying the same field to a partially anisotropic (XYZ) system does not just enhance the nearest neighbor entanglements and their thermal robustness but all the long-range ones as well, while the spins relax asymptotically to very distinguished spin states, which is a sign of a critical behavior taking place at this combination of system anisotropy and field inhomogeneity.
We consider a finite two-dimensional Heisenberg triangular spin lattice at different degrees of anisotropy coupled to a dissipative Lindblad environment obeying the Born-Markovian constrain at finite temperature. We show how applying an inhomogeneous magnetic field to the system may significantly affect the entanglement distribution and properties among the spins in the asymptotic steady state of the system. Particularly, applying an inhomogeneous field with an inward (growing) gradient toward the central spin is found to considerably enhance the nearest neighbor entanglement and its robustness to the thermal dissipative decay effect in the completely anisotropic (Ising) system, whereas all the beyond nearest neighbor entanglements vanish entirely. Applying the same field to a partially anisotropic (XYZ) system, does not only enhance the nearest neighbor entanglements and their robustness but also all the beyond nearest neighbor ones. Nevertheless, the inhomogeneity of the field shows no effect on the asymptotic behavior of the entanglement in the isotropic (XXX) system, which vanishes under any system configuration. Moreover, the same inhomogeneous field exhibits the most influential impact, compared with the other ones, on the the spin dynamics as well. Although in the isotropic system the spins relax to a separable (disentangled) steady state with all the spins reaching a common spin state regardless of the field inhomogeneity, the spins in the steady state of the completely anisotropic system reach different distinguished spin states depending on their positions in the lattice. However, in the XYZ system, though the anisotropy is lower, the spin states become even more distinguished, accompanying the long range quantum correlation across the system, which is a sign of a critical behavior taking place at this combination of system anisotropy and field inhomogeneity.
We consider a finite one-dimensional Heisenberg XYZ spin chain under the influence of a dissipative Lindblad environment obeying the Born-Markovian constraint in presence of an external magnetic field with closed and open boundary conditions. We present an exact numerical solution for the Lindblad master equation of the system in the Liouville space. The dynamics and asymptotic behavior of the nearest-neighbor and beyond-nearest-neighbor pairwise entanglements in the system are investigated under the effect of spatial anisotropy, temperature, system size, and different initial states. The entanglements in the free spin system exhibit nonuniform oscillatory behavior that varies significantly depending on the system size, anisotropy, and initial state. The xy spatial anisotropy dictates the asymptotic behavior of the different entanglements in the system under the influence of the environment regardless of the initial state. Higher anisotropy yields higher steady-state value of the nearest-neighbor entanglement whereas a complete isotropy wipes it out. The longer range entanglements respond differently to the anisotropy variation. The anisotropy in the z direction may enhance the entanglements depending on the interplay with the magnetic field applied in the same direction. As the temperature is raised, the steady state of the short-range entanglements is found to be robust within very small nonzero temperature range that depends critically on the spatial anisotropy. Moreover, the end to end entanglement transfer time and speed through the open boundary chain vary considerably based on the degree of anisotropy and temperature of the environment.
Abstract We consider one dimensional Ising spin system in a transverse uniform time-dependent magnetic field. The asymptotic behavior of the bipartite entanglements between the terminal spin and each one of the other spins along the chain is investigated and compared at different spin-spin interaction ranges, from nearest neighbor to infinite long range, under the separate action of two different magnetic fields, constant and time-varying. We find that each of the nearest neighbor and next to nearest neighbor bipartite entanglements reach an asymptotic final state that is independent of the initial condition or the variation in the interaction range showing perfect ergodic behavior at quite short interaction ranges. However, the nearest neighbor entanglement maintains this behavior at a slightly longer ranges. The other bipartite entanglements assume a zero value within these interaction ranges. At intermediate short and long interaction ranges, the asymptotic states of all entanglements become strongly dependent on the initial state and the interaction range, deviating from the ergodic behavior observed before. The maximum asymptotic entanglement attainable between a pair of spins takes place at a long interaction range value that increases with the distance between the spins. At the infinite long range interaction, the dynamics of all bipartite entanglements coincide. great care should be taken in constructing both.
We consider a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian environment at finite temperature in the presence of an external uniform magnetic field. We show how inserting a magnetic impurity in the spin system can be used to effectively control the dynamics and asymptotic state of the system. A strong impurity, at a border or central site, was found to enhance its entanglement with the other spins and their thermal robustness to the dissipative environment reaching an asymptotic state that is independent of the system initial state. However, it reduces the entanglement among the other spins in the lattice and their thermal robustness and may diminish them depending on its strength and the environment temperature. Moreover, the effect of the impurity increases significantly as the degree of anisotropy of the spin system increases. Therefore, the entanglement distribution over the different sites of the lattice can be modulated by tuning the impurity strength, system anisotropy and environment temperature. The impurity can be used as a switch that turns the entanglement on among specific spins and off among others simultaneously.
We consider an infinite one-dimensional anisotropic XY spin chain with a nearest-neighbor time-dependent Heisenberg coupling J(t) between the spins in presence of a time-dependent magnetic field h(t). We discuss a general solution for the system and present an exact solution for particular choice of J and h of practical interest. We investigate the dynamics of entanglement for different degrees of anisotropy of the system and at both zero and finite temperatures. We find that the time evolution of entanglement in the system shows nonergodic and critical behavior at zero and finite temperatures and different degrees of anisotropy. The asymptotic behavior of entanglement at the infinite time limit at zero temperature and constant J and h depends only the parameter $\ensuremath{\lambda}=J/h$ rather than the individual values of J and h for all degrees of anisotropy but changes for nonzero temperature. Furthermore, the asymptotic behavior is very sensitive to the initial values of J and h and for particular choices we may create finite asymptotic entanglement regardless of the final values of J and h. The persistence of quantum effects in the system as it evolves and as the temperature is raised is studied by monitoring the entanglement. We find that the quantum effects dominate within certain regions of the kT-$\ensuremath{\lambda}$ space that vary significantly depending on the degree of the anisotropy of the system. Particularly, the quantum effects in the Ising model case persist in the vicinity of both its critical phase transition point and zero temperature as it evolves in time. Moreover, the interplay between the different system parameters to tune and control the entanglement evolution is explored.