arXiv:0810.2532 · Journal of Physics A Mathematical and Theoretical

An exactly solvable model for the dynamics of two spin-\frac{1}{2} particles embedded in separate spin star environments

Yamen Hamdouni

Published Jul 13, 2009• 14 citations

Abstract

Exact analytical results for the dynamics of two interacting qubits, each of which is embedded in its own spin star bath, are presented. The time evolution of the concurrence and the purity of the two-qubit system are investigated for finite and infinite numbers of environmental spins. The effect of qubit–qubit interactions on the steady state of the central system is investigated.

DOI: 10.1088/1751-8113/42/31/315301

Related papers

Exact dynamics of a two-qubit system in a spin star environment

Yamen Hamdouni, M. Fannes, Francesco Petruccione

Physical Review BJun 19, 200682 citations

Quantum dynamics of double-qubits in a spin star lattice with an XY interaction

Jun Jing, Zhi-Guo Lü

arXiv:quant-ph/0612115Dec 14, 20060 citations

Dynamics of quantum coherence in a spin-star system: Bipartite initial state and coherence distribution

Chandrashekar Radhakrishnan, Zhiguo Lü, Jun Jing +1 more

Physical Review AOct 31, 201927 citations

Dynamics of two central spins immersed in spin baths

Devvrat Tiwari, Shounak Datta, Samyadeb Bhattacharya +1 more

arXiv:2205.04135Physical Review ASep 30, 202210 citations

Non-Markovian dynamics in a spin star system: Exact solution and approximation techniques

Heinz‐Peter Breuer, Daniel Klaus Burgarth, Francesco Petruccione

arXiv:quant-ph/0401051Physical Review BJul 30, 2004256 citations

Dynamics of the homogeneous two-qubit X X Z central spin model with the spin bath prepared in superpositions of symmetric Dicke states

Zejiang Li, Pei Yang, Wen‐Long You +1 more

arXiv:2006.00550Physical Review ASep 16, 202011 citations

arXiv:0810.2532 · Journal of Physics A Mathematical and Theoretical

An exactly solvable model for the dynamics of two spin-\frac{1}{2} particles embedded in separate spin star environments

Yamen Hamdouni

Published Jul 13, 2009• 14 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8113/42/31/315301

Related papers

Exact dynamics of a two-qubit system in a spin star environment

Yamen Hamdouni, M. Fannes, Francesco Petruccione

We derive the exact reduced dynamics of a central two-qubit system in a spin star configuration. The exact evolution of the reduced system density matrix is obtained and we compute the limit of an infinite number of environment spins. Initially pure states of the central system evolve into mixed ones and we determine the decoherence-free states of the model. The long-time behavior is studied, partial decoherence is shown to be a result of the coupling of the qubits to the environment, and entanglement evolution of the central system is investigated.We derive the exact reduced dynamics of a central two-qubit system in a spin star configuration. The exact evolution of the reduced system density matrix is obtained and we compute the limit of an infinite number of environment spins. Initially pure states of the central system evolve into mixed ones and we determine the decoherence-free states of the model. The long-time behavior is studied, partial decoherence is shown to be a result of the coupling of the qubits to the environment, and entanglement evolution of the central system is investigated.

Physical Review BJun 19, 200682 citations

Quantum dynamics of double-qubits in a spin star lattice with an XY interaction

Jun Jing, Zhi-Guo Lü

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.

arXiv:quant-ph/0612115Dec 14, 20060 citations

Dynamics of quantum coherence in a spin-star system: Bipartite initial state and coherence distribution

Chandrashekar Radhakrishnan, Zhiguo Lü, Jun Jing +1 more

We investigate the transient dynamics of quantum coherence for a system of two central spins in a spin-star environment by employing a numerical procedure based on a Laguerre polynomial expansion scheme. The dynamics of the total, local, and global coherence are calculated for different values of the anisotropy parameter, the system-bath interaction strengths, and temperature for different initial bipartite states. Significant dynamical features of quantum coherence are found as follows: (i) an $X$ state can only have global coherence; (ii) a state with only initial local coherence gains global coherence during the course of evolution by the induced correlations between the two-qubit system and the common bath; (iii) an incoherent state gains coherence by interacting with an external bath. We find there are two primary ways to gain coherence for an incoherent state: one is by interacting with the external quantum bath and the other is through interconversion of other quantum properties such as purity into coherence. Finally, we demonstrate that our results for the system in an infinite bath also hold qualitatively when the system is in contact with a finite bath.We investigate the transient dynamics of quantum coherence for a system of two central spins in a spin-star environment by employing a numerical procedure based on a Laguerre polynomial expansion scheme. The dynamics of the total, local, and global coherence are calculated for different values of the anisotropy parameter, the system-bath interaction strengths, and temperature for different initial bipartite states. Significant dynamical features of quantum coherence are found as follows: (i) an $X$ state can only have global coherence; (ii) a state with only initial local coherence gains global coherence during the course of evolution by the induced correlations between the two-qubit system and the common bath; (iii) an incoherent state gains coherence by interacting with an external bath. We find there are two primary ways to gain coherence for an incoherent state: one is by interacting with the external quantum bath and the other is through interconversion of other quantum properties such as purity into coherence. Finally, we demonstrate that our results for the system in an infinite bath also hold qualitatively when the system is in contact with a finite bath.

Physical Review AOct 31, 201927 citations

Dynamics of two central spins immersed in spin baths

Devvrat Tiwari, Shounak Datta, Samyadeb Bhattacharya +1 more

In this article we derive the exact dynamics of a two-qubit (spin $1/2$ ) system interacting centrally with separate spin baths composed of qubits in a thermal state. Furthermore, each spin of the bath is coupled to every other spin of the same bath. The corresponding dynamical map is constructed. It is used to analyze the non-Markovian nature of the two qubit central spin dynamics. We further observe the evolution of quantum correlations like entanglement and discord under the influence of the environmental interaction. Moreover, we demonstrate the comparison between this exact two-qubit dynamics and the locally acting central spin model in a spin bath. This work is a stepping stone towards the realization of non-Markovian heat engines and other quantum thermal devices.In this article we derive the exact dynamics of a two-qubit (spin $1/2$) system interacting centrally with separate spin baths composed of qubits in a thermal state. Furthermore, each spin of the bath is coupled to every other spin of the same bath. The corresponding dynamical map is constructed. It is used to analyze the non-Markovian nature of the two qubit central spin dynamics. We further observe the evolution of quantum correlations like entanglement and discord under the influence of the environmental interaction. Moreover, we demonstrate the comparison between this exact two-qubit dynamics and the locally acting central spin model in a spin bath. This work is a stepping stone towards the realization of non-Markovian heat engines and other quantum thermal devices.

arXiv:2205.04135Physical Review ASep 30, 202210 citations

Non-Markovian dynamics in a spin star system: Exact solution and approximation techniques

Heinz‐Peter Breuer, Daniel Klaus Burgarth, Francesco Petruccione

The reduced dynamics of a central spin coupled to a bath of $N$ spin- $\frac{1}{2}$ 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 reduced dynamics of a central spin coupled to a bath of $N$ spin-$\frac{1}{2}$ 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.

arXiv:quant-ph/0401051Physical Review BJul 30, 2004256 citations

Dynamics of the homogeneous two-qubit X X Z central spin model with the spin bath prepared in superpositions of symmetric Dicke states

Zejiang Li, Pei Yang, Wen‐Long You +1 more

We obtain exact dynamics of a two-qubit central spin model (CSM) consisting of two interacting qubits homogeneously coupled to a spin bath via the $XXZ$ -type coupling, with the bath initially prepared in linear superpositions of the symmetric Dicke states. Using the interaction picture Hamiltonian with respect to the non-spin-flipping part of the model, we derive a sequence of equations of motion within each magnetization sector satisfied by the probability amplitudes of the time-evolved state. These equations of motion admit analytical solutions for the single-qubit CSM in which one of the two central qubits decouples from the rest of the system. Based on this, we provide a quantitative interpretation to the observed collapse-revival phenomena in the single-qubit Rabi oscillations when the bath is prepared in the spin coherent state. We then study the disentanglement and coherence dynamics of two initially entangled noninteracting qubits when the two qubits interact with individual baths or with a common bath. For individual baths, the coherent dynamics is found to be positively correlated to the single-qubit purity dynamics, and entanglement sudden disappearance and revivals are observed in both cases. The entanglement creation of two initially separable qubits coupled to a common bath is also studied and collapse and revival behaviors in the entanglement dynamics are observed. Choosing the equally weighted state and the $W$ -class states as the bath initial states, we finally study the dynamics of entanglement between two individual bath spins and demonstrate the entanglement sharing mechanism in such a system.We obtain exact dynamics of a two-qubit central spin model (CSM) consisting of two interacting qubits homogeneously coupled to a spin bath via the $XXZ$-type coupling, with the bath initially prepared in linear superpositions of the symmetric Dicke states. Using the interaction picture Hamiltonian with respect to the non-spin-flipping part of the model, we derive a sequence of equations of motion within each magnetization sector satisfied by the probability amplitudes of the time-evolved state. These equations of motion admit analytical solutions for the single-qubit CSM in which one of the two central qubits decouples from the rest of the system. Based on this, we provide a quantitative interpretation to the observed collapse-revival phenomena in the single-qubit Rabi oscillations when the bath is prepared in the spin coherent state. We then study the disentanglement and coherence dynamics of two initially entangled noninteracting qubits when the two qubits interact with individual baths or with a common bath. For individual baths, the coherent dynamics is found to be positively correlated to the single-qubit purity dynamics, and entanglement sudden disappearance and revivals are observed in both cases. The entanglement creation of two initially separable qubits coupled to a common bath is also studied and collapse and revival behaviors in the entanglement dynamics are observed. Choosing the equally weighted state and the $W$-class states as the bath initial states, we finally study the dynamics of entanglement between two individual bath spins and demonstrate the entanglement sharing mechanism in such a system.

arXiv:2006.00550Physical Review ASep 16, 202011 citations

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.

X

state can only have global coherence; (ii) a state with only initial local coherence gains global coherence during the course of evolution by the induced correlations between the two-qubit system and the common bath; (iii) an incoherent state gains coherence by interacting with an external bath. We find there are two primary ways to gain coherence for an incoherent state: one is by interacting with the external quantum bath and the other is through interconversion of other quantum properties such as purity into coherence. Finally, we demonstrate that our results for the system in an infinite bath also hold qualitatively when the system is in contact with a finite bath.

In this article we derive the exact dynamics of a two-qubit (spin

1/2

) system interacting centrally with separate spin baths composed of qubits in a thermal state. Furthermore, each spin of the bath is coupled to every other spin of the same bath. The corresponding dynamical map is constructed. It is used to analyze the non-Markovian nature of the two qubit central spin dynamics. We further observe the evolution of quantum correlations like entanglement and discord under the influence of the environmental interaction. Moreover, we demonstrate the comparison between this exact two-qubit dynamics and the locally acting central spin model in a spin bath. This work is a stepping stone towards the realization of non-Markovian heat engines and other quantum thermal devices.

The reduced dynamics of a central spin coupled to a bath of

N

spin-

\frac{1}{2}

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.

We obtain exact dynamics of a two-qubit central spin model (CSM) consisting of two interacting qubits homogeneously coupled to a spin bath via the

XXZ

-type coupling, with the bath initially prepared in linear superpositions of the symmetric Dicke states. Using the interaction picture Hamiltonian with respect to the non-spin-flipping part of the model, we derive a sequence of equations of motion within each magnetization sector satisfied by the probability amplitudes of the time-evolved state. These equations of motion admit analytical solutions for the single-qubit CSM in which one of the two central qubits decouples from the rest of the system. Based on this, we provide a quantitative interpretation to the observed collapse-revival phenomena in the single-qubit Rabi oscillations when the bath is prepared in the spin coherent state. We then study the disentanglement and coherence dynamics of two initially entangled noninteracting qubits when the two qubits interact with individual baths or with a common bath. For individual baths, the coherent dynamics is found to be positively correlated to the single-qubit purity dynamics, and entanglement sudden disappearance and revivals are observed in both cases. The entanglement creation of two initially separable qubits coupled to a common bath is also studied and collapse and revival behaviors in the entanglement dynamics are observed. Choosing the equally weighted state and the

W

-class states as the bath initial states, we finally study the dynamics of entanglement between two individual bath spins and demonstrate the entanglement sharing mechanism in such a system.