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arXiv:quant-ph/0405147 · Physical Review A

Qubit decoherence and non-Markovian dynamics at low temperatures via an effective spin-boson model

K. Shiokawa, B. L. Hu

Published Dec 9, 2004• 25 citations
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Abstract

Quantum Brownian oscillator model (QBM), in the Fock-space representation, can be viewed as a multilevel spin-boson model. At sufficiently low temperature, the oscillator degrees of freedom are dynamically reduced to the lowest two levels and the system behaves effectively as a two-level (E2L) spin-boson model (SBM) in this limit. We discuss the physical mechanism of level reduction and analyze the behavior of E2L-SBM from the QBM solutions. The availability of close solutions for the QBM enables us to study the non-Markovian features of decoherence and leakage in a SBM in the nonperturbative regime (e.g., without invoking the Born approximation) in better details than before. Our result captures very well the characteristic non-Markovian short time low temperature behavior common in many models.

DOI
10.1103/physreva.70.062106

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Quantum Brownian oscillator model (QBM), in the Fock-space representation, can be viewed as a multilevel spin-boson model. At sufficiently low temperature, the oscillator degrees of freedom are dynamically reduced to the lowest two levels and the system behaves effectively as a two-level (E2L) spin-boson model (SBM) in this limit. We discuss the physical mechanism of level reduction and analyze the behavior of E2L-SBM from the QBM solutions. The availability of close solutions for the QBM enables us to study the non-Markovian features of decoherence and leakage in a SBM in the nonperturbative regime (e.g., without invoking the Born approximation) in better details than before. Our result captures very well the characteristic non-Markovian short time low temperature behavior common in many models.
In quantum computation, it is of paramount importance to locate the parameter space where maximal coherence can be preserved in the qubit system. In recent years environment-induced decoherence based the quantum Brownian motion (QBM) models have been applied to two level systems (2LS) interacting with an electromagnetic field, leading to the general belief that 2LS are easily decohered. In a recent paper C. Anastopoulos and B. L. Hu [Phys. Rev. A62, (2000) 033821] derived a new exact non-Markovian master equation at zero temperature, from which they showed that this belief is actually misplaced. For a two-level atom (2LA)- electromagnetic field (EMF) system the decoherence time is rather long, comparable to the relaxation time. Theoretically this is because the dominant interaction is the hatsigmapm\\hat \\sigma_{\\pm}hatsigmapm​ type of coupling between the two levels (what constitutes the qubit) and the field, not the hatsigmaz\\hat \\sigma_zhatsigmaz​ type, which shows the QBM behavior. Depending on the coupling the field can act as a resonator (in an atom cavity) or as a bath (in QBM) and produce very different decoherent behavior in the system. This is not new to Cavity QED experimentalists: the 2LA-EMF system maintaining its coherence in sufficiently long duration is the reason why they can manipulate them so well to show interesting quantum coherence effects.
The dynamical aspects of quantum Brownian motion at low temperatures are investigated. Exact calculations of quantum entanglement among two Brownian oscillators are given without invoking the Born-Markov or Born approximation widely used for the study of open systems. Nonlocality due to finite oscillator separation is studied. Our approach is applicable for arbitrary time scale, in particular suitable to probe the short-time regime at cold temperatures where many experiments on quantum information processing are performed. We study separability criteria based on the uncertainty relation, negativity, and entanglement of formation. We found a crossover behavior in a disentanglement process between quantum- and thermal-fluctuation-dominated regimes. The fluctuation-dissipation theorem is violated for two Brownian particles interacting with a common environment. The deviation from the relation originates in the interaction mediated by the environment, which drives two particles into a steady oscillatory state, preventing the system from thermalizing. Consequently there is a residual entanglement at low temperatures for arbitrary coupling strength. This entanglement is generated from the environment nonperturbatively even when two modes are not entangled initially. When the distance between two oscillators is varied, competition between entanglement induced from the environment and modified decoherence due to finite separation leads to the characteristic distance where entanglement is minimized.
Employing a recently proposed measure for quantum non-Markovianity, we carry out a systematic study of the size of memory effects in the spin-boson model for a large region of temperature and frequency cutoff parameters. The dynamics of the open system is described utilizing a second-order time-convolutionless master equation without the Markov or rotating wave approximations. While the dynamics is found to be strongly non-Markovian for low temperatures and cutoffs, in general, we observe a special regime favoring Markovian behavior. This effect is explained as resulting from a resonance between the system's transition frequency and the frequencies of the dominant environmental modes. We further demonstrate that the corresponding Redfield equation is capable of reproducing the characteristic features of the non-Markovian quantum behavior of the model.
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The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.
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.