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Progress of Theoretical Physics

Theoretical Study of Quantum Coherence in Spin-Boson System

Hiroshi Kawamoto

Published Dec 1, 1989• 2 citations
Publisher

Abstract

We study a two-state system which is coupled to a bosonic environment with the Ohmic spectral density. In this paper, two methods are used. One is the quantum Langevin equation and the other the reduced density matrix. Through these investigations the time reversal symmetry is emphasized because the spectral density of the environment gives rise to a long-ranged collision effect on the spin part. By using the second method, we find, at T=0, (1) a purely oscillatory motion for α≪ αc (α is a dimentionless dissipation coefficient and αc ≃0.27); (2) a divergent frequency of the oscillation for α= αc; (3) a localization for α> αc. On the other hand, a simple decoupling of the Langevin equation merely gives the lowest order calculation in α. It is notable that the approximation used here is reliable regardless of a range of α. Concerning this a detailed illustration is shown.

DOI
10.1143/ptp.82.1044

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We study a two-state system which is coupled to a bosonic environment with the Ohmic spectral density. In this paper, two methods are used. One is the quantum Langevin equation and the other the reduced density matrix. Through these investigations the time reversal symmetry is emphasized because the spectral density of the environment gives rise to a long-ranged collision effect on the spin part. By using the second method, we find, at T=0, (1) a purely oscillatory motion for α≪ αc (α is a dimentionless dissipation coefficient and αc ≃0.27); (2) a divergent frequency of the oscillation for α= αc; (3) a localization for α> αc. On the other hand, a simple decoupling of the Langevin equation merely gives the lowest order calculation in α. It is notable that the approximation used here is reliable regardless of a range of α. Concerning this a detailed illustration is shown.
Within the lowest-order Born approximation, we present an exact calculation of the time dynamics of the spin-boson model in the Ohmic regime. We observe non-Markovian effects at zero temperature that scale with the system-bath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by $\sqrt{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ is the coupling strength to the environment. We also explore the calculation of the next-order Born approximation: we show that, at the expense of very large computational complexity, interesting physical quantities can be rigorously computed at fourth order using computer algebra, presented completely in an accompanying MATHEMATICA file. We compute the $O(\ensuremath{\alpha})$ corrections to the long time behavior of the system density matrix; the result is identical to the reduced density matrix of the equilibrium state to the same order in $\ensuremath{\alpha}$. All these calculations indicate precision experimental tests that could confirm or refute the validity of the spin-boson model in a variety of systems.
An investigation as to the possibility of the quantum phase coherence in a circuit containing the Josephson junction is made. In the study the primary part of the system is assumed to be a two-state system, and so the quantum oscillation between two states is expected. For an analysis of the influence exerted by an environment two types of environments are considered. One type has the Lorentzian spectral density and the other has the Ohmic one. In both types a total loss of the phase coherence is found when a strength of a coupling between the primary part of the system and the environment, α ≡φ2 / \hbarRN(φ ≃φ0), satisfies a condition √\mathstrutα ≫Δω0 τc. Here \hbar Δω0 is a level splitting and τc a relaxation time of the environments. Furthermore, we find that the total loss of the phase coherence leads to a quantum tunneling governed by a Poisson process with a tunnelin rate Ω2 / Γ (Ω is the renormalized natural frequency and Γ the damping constant). It must be emphasized that the Ohmic spectral density of the environments gives rise to an instability in the primary part when the coupling strength α (√α ≫ Δω0 τc) exceeds some critical value.
We summarize in this work the relations of the spin-boson Hamiltonian, which was recently studied in connection with the phenomenon of quantum coherence in the presence of dissipation, to three different fermionic Hamiltonians. These relations were obtained through well-known equivalences between Fermi and Bose operators in one dimension. The fermionic Hamiltonians correspond to (a) a two-level system coupled linearly with a fermionic bath, (b) the resonant-level model, and (c) the anisotropic Kondo model. For the first Hamiltonian we reobtain and discuss the relationship between the dimensionless dissipation coefficient \ensuremath{\alpha} and the phase shift of the fermions. The resonant-level model allows us to study the properties of the two-level system for values of \ensuremath{\alpha} around (1/2). At \ensuremath{\alpha}=(1/2) the model reduces to the Toulouse limit, where the Hamiltonian is exactly soluble. A pure exponential decay is obtained for the relaxation of P(t)=〈${\ensuremath{\sigma}}_{z}$(t)〉 for t>0, given that for t<0 the system is known to be localized in one of the two states. The comparison with the Kondo model gives the long-time-limit behavior of the system for (1/2)1 and provides a connection between universal numbers obtained previously for the spin-boson Hamiltonian and the Kondo model.
The loss of coherence in a general open quantum system interacting with a bosonic environment is analyzed. The reservoir is initially in a thermal state. The reduced dynamics is described by a non-Markovian time-local master equation. We consider spectral densities that are sub- or super-Ohmic at low frequencies and arbitrarily shaped at high frequencies. In the super-Ohmic regime, for noninteger frequency powers larger than 2, long time survival of coherence appears. In the latter regime, at vanishing temperature, the asymptotic amount of surviving coherence is stabilized to its initial value, up to a phase factor, by properly increasing the bandwidth and decreasing the low-frequency profile of the spectral density. For noninteger positive frequency powers less than 2, stretched exponential-like decoherence is found over long times. The relaxations to the asymptotic configurations become arbitrarily slow by approaching the frequency power 2 of the super-Ohmic regime. The same dependence on temperature, spectral density, and scale frequency appears for purity and concurrence of two qubits and coherence of a qubit.
The dynamics of the spin-boson Hamiltonian is considered in the stochastic approximation. The Hamiltonian describes a two-level system coupled to an environment, and is widely used in physics, chemistry, and the theory of quantum measurement. We demonstrate that the method of stochastic approximation, a general method for consideration of dynamics of an arbitrary system interacting with environment, is powerful enough to reproduce qualitatively the striking results by Leggett et al. [Rev. Mod. Phys. 59, 1 (1987)] found earlier for this model system. The results include an exact expression of the dynamics in terms of the spectral density, and show the appearance of two interesting regimes for the system, i.e., pure oscillating and pure damping ones. Correlators describing the environment are also computed.
Within the lowest-order Born approximation, we present an exact calculation of the time dynamics of the spin-boson model in the Ohmic regime. We observe non-Markovian effects at zero temperature that scale with the system-bath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by α\sqrtαα​, where ααα is the coupling strength to the environment. These calculations indicate precision experimental tests that could confirm or refute the validity of the spin-boson model in a variety of systems.