Dynamics of a Single Spin-1/2 Coupled to x- and y-Spin Baths: Algorithm and Results

Novotny, M. A.; Guerra, Marta L.; Raedt, Hans De; Michielsen, K.; Jin, Fengping

doi:10.1016/j.phpro.2012.05.015

Cited by 4 publications

(4 citation statements)

References 21 publications

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“…We leave as future work the coupling between a system S composed of spin-1/2 objects (qubits) and an environment E composed of harmonic oscillators. In particular, we have recently been able to build on exact calculations of a single spin coupled to specific types of spin environment [33] to devise an algorithm that does not have computer memory constraints limited by the size of D E [34,35]. We are working to extend this algorithm to other types of environment and for more than one spin in the system S.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Quantum decoherence scaling with bath size: Importance of dynamics, connectivity, and randomness

et al. 2013

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We consider the decoherence of a quantum system S coupled to a quantum environment E. For states chosen uniformly at random from the unit hypersphere in the Hilbert space of the closed system S + E we derive a scaling relationship for the sum of the off-diagonal elements of the reduced density matrix of S as a function of the size D E of the Hilbert space of E. This sum decreases as 1/ √ D E as long as D E 1. We test this scaling prediction by performing large-scale simulations which solve the time-dependent Schrödinger equation for a ring of spin-1/2 particles, four of them belonging to S and the others to E, and for this ring with small world bonds added in E and/or between S and E. The spin-1/2 particles experience nearest-neighbor interactions that are identical for the interactions within S and random for the interactions within E and between S and E, or that are all identical. Provided that the time evolution drives the whole system from the initial state toward a scaling state, a state which has similar properties as states belonging to the class of quantum states for which we derived the scaling relationship, the scaling prediction holds. We examine various interaction parameters and initial states for our model system to find whether or not the time evolution reaches the class of states that have the scaling property. For the homogeneous ring we find that the evolution for select initial states does not reach these scaling states. This conclusion is not modified if we add some homogeneous random connections. For a ring we find that some randomness in the interaction parameters is required so that most initial configurations are driven toward the scaling state. Furthermore, if the amount of randomness is small the time required to reach the scaling states may be very large. For the case of all random interactions in E the ring is driven toward the scaling state. Adding small world bonds between S and E with random interaction strengths may decrease the time required to reach the scaling state or may prevent the scaling state from being reached. For the latter case we show that increasing the complexity of the environment by adding extra connections within the environment suffices to observe the predicted scaling behavior.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Quantum decoherence scaling with bath size: Importance of dynamics, connectivity, and randomness

et al. 2013

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“…This contrasts with the exponential dependence in t 2 and N for only an x-bath. A physically interesting question that can be studied with the expanded algorithm is under which circumstances the approach to the asymptotic value for the quantum purity and von Neumann entropy is power law or is exponential for different combinations of x-, y-, and z-baths [18]. For y-baths one could use…”

Section: Conclusion and Discussionmentioning

confidence: 99%

An Efficient Algorithm for Simulating the Real-Time Quantum Dynamics of a Single Spin-1/2 Coupled to Specific Spin-1/2 Baths

Novotny

Guerra

Raedt

et al. 2012

J. Phys.: Conf. Ser.

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An efficient algorithm for the computation of the real-time dependence of a single quantum spin-1/2 coupled to a specific set of quantum spin-1/2 baths is presented. The specific spin baths have couplings only with the spin operators S x between bath spins and the central spin. We calculate spin expectation values, the quantum purity, the von Neumann entropy, and the off-diagonal components of the reduced density matrix for the central spin once the bath spins have been traced out. The algorithm does not require the storage of any vector larger than of size 2, even though the size of the Hilbert space is 2 N+1 , where N is the number of bath spins. Results are presented for the central spin connected to different sizes and types of spin baths, and for different initial states for the central spin and for the bath spins. Results are also compared to those for more general baths.

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“…Note that such a symmetry is also obeyed in the case that there is no interaction between the environment spins, e.g. for an environment Hamiltonian [52,53]. In this particular case, it is only required that H S is an even function and H SE an odd function under reversal of all spin components of the system spins.…”

Section: E Verification By Spin Hamiltoniansmentioning

confidence: 99%

Quantum decoherence and thermalization at finite temperature within the canonical-thermal-state ensemble

Novotny

Jin

Yuan³

et al. 2016

Phys. Rev. A

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We study measures of decoherence and thermalization of a quantum system S in the presence of a quantum environment (bath) E. The whole system is prepared in a canonical thermal state at a finite temperature. Applying perturbation theory with respect to the system-environment coupling strength, we find that under common Hamiltonian symmetries, up to first order in the coupling strength it is sufficient to consider the uncoupled system to predict decoherence and thermalization measures of S. This decoupling allows closed form expressions for perturbative expansions for the measures of decoherence and thermalization in terms of the free energies of S and of E. Numerical results for both coupled and decoupled systems with up to 40 quantum spins validate these findings.PACS numbers: 03.65. Yz, 75.10.Jm, 75.10.Nr, 05.45.Pq Decoherence and thermalization are two basic concepts in quantum statistical physics [1]. Decoherence renders a quantum system classical due to the loss of phase coherence of the components of a system in a quantum superposition via interaction with an environment (or bath). Thermalization drives the system to a stationary state, the (micro) canonical ensemble via energy exchange with a thermal bath. As the evolution of a quantum system is governed by the time-dependent Schrödinger equation, it is natural to raise the question how classicality could emerge from a pure quantum state. This question is becoming more important technologically, for example in designing quantum computers [2] where decoherence effects are a major impediment in engineering implementations.Various theoretical and numerical studies have been performed, trying to answer this fundamental question, e.g., the microcanonical thermalization of an isolated quantum system [3][4][5][6], canonical thermalization of a system coupled to a (much) larger environment [3,[7][8][9][10][11][12][13][14][15][16][17], and of two identical quantum systems at different temperatures [18,19]. In our earlier work [20], we found that at infinite temperature the degree of decoherence of a system S scales with the dimension of the environment E if the state of the whole system S + E is randomly chosen from the Hilbert space of the whole system. We showed that in the thermodynamic limit, the system S decoheres thoroughly.In this Letter, we investigate for the first time quantitatively how classicality emerges from a pure quantum state when S + E is of a finite size and at a finite temperature. We assume that the whole system S + E is in a canonical thermal state, a pure state at finite inverse temperature β [21][22][23], and investigate measures of the decoherence and the thermalization of S. This canonical thermal state could be the result of a thermalization process of the whole system S + E coupled to a large heat bath, which we do not consider any further. The state of the system S is described by the reduced density matrix. The degree of decoherence of the system S is measured in terms of σ , defined below in terms of measures of the offdiagonal c...

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Dynamics of a Single Spin-1/2 Coupled to x- and y-Spin Baths: Algorithm and Results

Cited by 4 publications

References 21 publications

Quantum decoherence scaling with bath size: Importance of dynamics, connectivity, and randomness

Quantum decoherence scaling with bath size: Importance of dynamics, connectivity, and randomness

An Efficient Algorithm for Simulating the Real-Time Quantum Dynamics of a Single Spin-1/2 Coupled to Specific Spin-1/2 Baths

Quantum decoherence and thermalization at finite temperature within the canonical-thermal-state ensemble

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