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Revista de la Sociedad Colombiana de Física

Dinámica no Markoviana de un Espin Central en una Red de Espines Interactuantes

Rafael Gutiérrez, J. Rodríguez Garzón

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

In this work we study the dynamics of a central spin coupled to a bath of N spin- 1 2 , arranged in a spin star configuration. The coupling between the central spin with each spin of the bath is Heisenberg X-Y type. Considering the bath as a collective environment spin, we find the exact dynamics for the thermodynamic limit of the bath particles, N �¨ ��. The results show colapses and revivals for populations. This is considered a signal of non-Markovian dynamics.

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In this work we study the dynamics of a central spin coupled to a bath of N spin- 1 2 , arranged in a spin star configuration. The coupling between the central spin with each spin of the bath is Heisenberg X-Y type. Considering the bath as a collective environment spin, we find the exact dynamics for the thermodynamic limit of the bath particles, N �¨ ��. The results show colapses and revivals for populations. This is considered a signal of non-Markovian dynamics.
The reduced dynamics of a central spin coupled to a bath of NNN spin-12\frac{1}{2}21​ particles arranged in a spin star configuration is investigated. The exact time evolution of the reduced density operator is derived, and an analytical solution is obtained in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ of an infinite number of bath spins, where the model shows complete relaxation and partial decoherence. It is demonstrated that the dynamics of the central spin cannot be treated within the Born-Markov approximation. The Nakajima-Zwanzig and the time-convolutionless projection operator technique are applied to the spin star system. The performance of the corresponding perturbation expansions of the non-Markovian equations of motion is examined through a comparison with the exact solution.
The dynamics of a central spin-1/2 in presence of a local magnetic field and a bath of N spin-1/2 particles is studied in the thermodynamic limit. The interaction between the spins is Heisenberg XY type and the bath is considered to be a perfect thermal reservoir. In this case, the evolution of the populations of the reduced density matrix is obtained for different temperatures. A Born approximation is made but not a Markov approximation resulting a non-Markovian dynamics. The measure of the entanglement between the spin and the bath is obtained by means of the von Neumann entropy. For low temperatures, results show that there are oscillations of populations and of the von Neumann entropy as measure of entanglement, indicating that the central spin becomes disentangled of the spins of the bath with characteristic time periods in which it is possible to extract or recuperate information. In the regime of high temperatures, the evolution shows a final maximum entangled state with entropy S = ln 2 as it is expected for a two level system. I
The dynamics of a central spin-1/2 in presence of a local magnetic field and a bath of N spin-1/2 particles is studied in the thermodynamic limit. The interaction between the spins is Heisenberg XY type and the bath is considered to be a perfect thermal reservoir. In this case, the evolution of the populations of the reduced density matrix are obtained for different temperatures. A Born approximation is made but not a Markov approximation resulting a non-Markovian dynamics. The measure of the way that the system mixes is obtained by means of the von Neumann entropy. For low temperatures, results show that there are oscillations of populations and of the von Neumann entropy, indicating that the central spin becomes a pure state with characteristic time periods in which it is possible to extract or recuperate information. In the regime of high temperatures, the evolution shows a final maximum mixed state with entropy S=ln 2 as it is expected for a two level system.
A model of an arbitrary spin coupled to a bath of spins 12\frac{1}{2}21​ in a star configuration is considered. The exact reduced dynamics of the central spin is found for the case of noncorrelated initial conditions of the system and the bath. The exact solution is used to test two approximation techniques, namely, the Nakajima-Zwanzig projection operator technique and the time-convolutionless projection operator technique corresponding to the second order of the coupling constant. Two types of projection operators are used for deriving the master equations, and the results are compared with the exact solution for a central spin equal to one. It is shown that the approximate master equations reproduce the exact dynamics on time scales 1/(AN)1/(A\sqrt{N})1/(AN​), where AAA is the coupling constant and NNN is the number of spins in the bath.
We discuss two central spin models coupled, respectively, to a quantum Heisenberg XY spin star environment [1] and to an antiferromagnetic environment [2]. In the first model [1], the exact quantum dynamics of the reduced density matrix of two coupled spin qubits in a quantum Heisenberg XY spin star environment in the thermodynamic limit at arbitrarily finite temperatures is obtained using a novel operator technique. In this approach, the transformed Hamiltonian becomes effectively Jaynes-Cumming like and thus the analysis is also relevant to cavity quantum electrodynamics. This special operator technique is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. To study their entanglement evolution, the concurrence of the reduced density matrix of the two coupled central spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled central spins, the thermal temperature of the spin environment and the interaction between the constituents of the spin environment. We also investigate the effect of detuning which in our model can be controlled by the strength of a locally applied external magnetic field. It is found that the detuning has a significant effect on the entanglement generation between the two spin qubits.
We provide a thorough analysis of the entanglement dynamics of an interacting two-qubit system in the non-Markovian regime. In such a regime, the time scale on which the reservoir degrees of freedom evolve is either of the same order of magnitude or less than that on which the system evolves. We used an exact numerical method; the Quasi-Adiabatic Path Integral (QUAPI) technique, to describe the corresponding qubit dissipative dynamics in such a non-Markovian regime. We computed the time evolution of the density operator for the quibits, from which we quantified the coherences and population dynamics, as well as the qubit-bath coupling effects. Using negativity as a metric, we calculated the dynamics of non-local quantum correlations (entanglement), and indentified a non-Markovian quantum phenomena in terms of early stage disentanglement, and the collapse and revival of entanglement.