Dynamics of two center spins in a Quantum Heisenberg XY system

Jun Jing, Zhi-Guo Lü

Published Dec 1, 2006• 0 citations

Abstract

The dynamics of two coupled spins of 1/2 coupled to spin-bath of a quantum Heisenberg XY type \\cite{Breuer, Yuan} is studied. The center pair of spins served as an quantum open subsystem were initially prepared in a Bell state or a product state and the bath consisted of $N$ ($N\\to\\infty$ as the thermodynamic limit) spins-1/2 is in a thermal state at different temperatures from the beginning. Transformed by the Holstein-Primakoff operator, the model will be treated effectively as two spin qubits embedded in a single mode cavity. Then the von-Neumann entropy, z-component summation and the concurrence of the center spins can be determined by a novel polynomial scheme for the time-evolution of quantum systems. It is found that (i) with increasing temperature, the bath plays a more strong destroy effect on the coherence or entanglement of the subsystem; (ii) the larger the coupling strength between the subsystem spins, the less the variation of the initial state; (iii) the stronger the interaction between the subsystem and the bath, the faster the variation of the initial state.

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Dynamics of two center spins in a Quantum Heisenberg XY system

Jun Jing, Zhi-Guo Lü

Published Dec 1, 2006• 0 citations

OpenAlex

Abstract

The dynamics of two coupled spins of 1/2 coupled to spin-bath of a quantum Heisenberg XY type \cite{Breuer, Yuan} is studied. The center pair of spins served as an quantum open subsystem were initially prepared in a Bell state or a product state and the bath consisted of

N

(

N\\to\\infty

as the thermodynamic limit) spins-1/2 is in a thermal state at different temperatures from the beginning. Transformed by the Holstein-Primakoff operator, the model will be treated effectively as two spin qubits embedded in a single mode cavity. Then the von-Neumann entropy, z-component summation and the concurrence of the center spins can be determined by a novel polynomial scheme for the time-evolution of quantum systems. It is found that (i) with increasing temperature, the bath plays a more strong destroy effect on the coherence or entanglement of the subsystem; (ii) the larger the coupling strength between the subsystem spins, the less the variation of the initial state; (iii) the stronger the interaction between the subsystem and the bath, the faster the variation of the initial state.

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The dynamics of two coupled spins-1/2 interacting with a spin-bath via the quantum Heisenberg XY coupling is studied. The pair of central spins served as a quantum open subsystem are initially prepared in two types of states: the product states and the Bell states. The bath, which consists of $N$ (in the thermodynamic limit $N\to\infty$ ) mutually coupled spins-1/2, is in a thermal state at the beginning. By the Holstein-Primakoff transformation, the model can be treated effectively as two spin qubits embedded in a single mode cavity. The time-evolution of the purity, z-component summation and the concurrence of the central spins can be determined by a Laguerre polynomial scheme. It is found that (i) at a low temperature, the uncoupled subsystem in a product state can be entangled due to the interaction with bath, which is tested by the Peres-Horodecki separability; however, at a high temperature, the bath produces a stronger destroy effect on the purity and entanglement of the subsystem; (ii) when the coupling strength between the two central spins is large, they are protected strongly against the bath; (iii) when the interaction between the subsystem and the bath is strong, the collapse of the two spin qubits from their initial entangled state is fast.The dynamics of two coupled spins-1/2 interacting with a spin-bath via the quantum Heisenberg XY coupling is studied. The pair of central spins served as a quantum open subsystem are initially prepared in two types of states: the product states and the Bell states. The bath, which consists of $N$ (in the thermodynamic limit $N\to\infty$) mutually coupled spins-1/2, is in a thermal state at the beginning. By the Holstein-Primakoff transformation, the model can be treated effectively as two spin qubits embedded in a single mode cavity. The time-evolution of the purity, z-component summation and the concurrence of the central spins can be determined by a Laguerre polynomial scheme. It is found that (i) at a low temperature, the uncoupled subsystem in a product state can be entangled due to the interaction with bath, which is tested by the Peres-Horodecki separability; however, at a high temperature, the bath produces a stronger destroy effect on the purity and entanglement of the subsystem; (ii) when the coupling strength between the two central spins is large, they are protected strongly against the bath; (iii) when the interaction between the subsystem and the bath is strong, the collapse of the two spin qubits from their initial entangled state is fast.

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J. Rodríguez Garzón, Rafael Gutiérrez

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. IThe 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

Aug 4, 20130 citations

N

(

N\\to\\infty

N

(in the thermodynamic limit

N\to\infty

) mutually coupled spins-1/2, is in a thermal state at the beginning. By the Holstein-Primakoff transformation, the model can be treated effectively as two spin qubits embedded in a single mode cavity. The time-evolution of the purity, z-component summation and the concurrence of the central spins can be determined by a Laguerre polynomial scheme. It is found that (i) at a low temperature, the uncoupled subsystem in a product state can be entangled due to the interaction with bath, which is tested by the Peres-Horodecki separability; however, at a high temperature, the bath produces a stronger destroy effect on the purity and entanglement of the subsystem; (ii) when the coupling strength between the two central spins is large, they are protected strongly against the bath; (iii) when the interaction between the subsystem and the bath is strong, the collapse of the two spin qubits from their initial entangled state is fast.

The dynamics of two

1∕2

-spin qubits under the influence of a quantum Heisenberg

XY

-type spin-bath is studied. After the Holstein-Primakoff transformation, numerical polynomial scheme is used to give the time-evolution calculation of the center qubits initially prepared in a product state or a Bell state. Then the concurrence of the two qubits, the

z

-component moment of either of the subsystem spins, and the fidelity of the subsystem are shown; they exhibit sensitive dependence on the anisotropic parameter, the temperature, the coupling strength, and the initial state. It is found that (i) the larger the anisotropic parameter $\ensuremath{\gamma}$, the bigger the probability of maintaining the initial state of the two qubits; (ii) with increasing temperature

T

, the bath plays a stronger destroy effect on the dynamics of the subsystem, so does the interaction

{g}_{0}

between the subsystem and the bath; and (iii) the time evolution of the subsystem is dependent on the initial state. The revival of the concurrence does not always mean the restoration of the state. Further, the dynamical properties of the subsystem should be judged by the combination of concurrence and fidelity.

The dynamics of entanglement and fidelity for a subsystem of two separate spin-

1∕2

qubits prepared in Bell states is investigated. One of the subsystem qubit labeled

A

is under the influence of a Heisenberg

XY

spin bath, while another one labeled

B

is uncoupled with that. We discuss two cases: (i) the number of bath spins is infinite, $N\ensuremath{\rightarrow}\ensuremath{\infty}$, and (ii)

N

is finite,

N=40

. In both cases, the bath is initially prepared in a thermal equilibrium state. It is shown that the time dependence of the concurrence and the fidelity of the two subsystem qubits can be controlled by tuning the parameters of the spin bath, such as the anisotropic parameter, the temperature, and the coupling strength with qubit

A

. It is interesting to find that the dynamics of the concurrence is independent of four different initial Bell states and that of the fidelity is divided into two groups.

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 an 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.