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Journal of Condensed Matter Nuclear Science

Local Approximation for the Lossy Spin–boson Model

Peter L. Hagelstein, Irfan U. Chaudhary

Published Jun 1, 2011• 24 citations
Publisher

Abstract

We are interested in the problem of coherent energy exchange in the spin-boson model augmented with loss, in the limit of infinite loss. This model is interesting because it predicts rapid coherent energy exchange under conditions when the transition energy of the two-level systems is much greater than the characteristic energy of the oscillator. Here we introduce the local approximation, in which we assume that the matrix elements change little when an energy exchange event occurs. This approximation results in a periodic system when a resonance condition is satisfied, which is much easier to solve than the original problem. We consider a model problem where we analyze the self-energy and indirect coupling matrix elements numerically and in the local approximation, with good agreement. Systematic results from the local approximation are obtained for the self-energy and indirect coupling matrix elements in models where 3, 5, 7, 9, and 11 quanta are exchanged.

DOI
10.70923/001c.72154

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We are interested in the problem of coherent energy exchange in the spin-boson model augmented with loss, in the limit of infinite loss. This model is interesting because it predicts rapid coherent energy exchange under conditions when the transition energy of the two-level systems is much greater than the characteristic energy of the oscillator. Here we introduce the local approximation, in which we assume that the matrix elements change little when an energy exchange event occurs. This approximation results in a periodic system when a resonance condition is satisfied, which is much easier to solve than the original problem. We consider a model problem where we analyze the self-energy and indirect coupling matrix elements numerically and in the local approximation, with good agreement. Systematic results from the local approximation are obtained for the self-energy and indirect coupling matrix elements in models where 3, 5, 7, 9, and 11 quanta are exchanged.
In the Fleischmann–Pons experiment, energy is produced without commensurate energetic reaction products. To account for this we have proposed new models in which coherent energy exchange occurs between two-level systems with a large energy quantum, and an oscillator with a much smaller characteristic energy. In earlier work we demonstrated using perturbation theory that the lossy spin-boson model is capable of efficient coherent energy exchange when many oscillator quanta are exchanged for a two-level system quanta on resonance. Here we introduce a formulation that isolates nearly degenerate states in the lossy spin-boson model, and allows the computation of the self-energy and indirect coupling matrix elements. From calculations of simple lossy spin boson models we establish a scaling law and define the dimensionless coupling constant g for the system. Direct calculations using this approach is not possible for large models in which a great many oscillator quanta are exchanged, but the formulation provides a foundation for understanding this kind of model, and will allow us to develop more powerful approximate methods.
The lossy spin-boson model is of interest to us since it predicts efficient energy exchange between identical two-level systems and an oscillator when the transition energy is a large (odd) multiple of the the oscillator energy. This model is not so easy to solve directly in the strong coupling regime, so we have developed approximate versions of the model that are easier to analyze. Here we introduce the pulse and amplitude approximation which compares very well with exact numerical solutions when the coupling is strong, and when the characteristic oscillator energy is much less than the transition energy. We examine discrete and continuum versions of the approximation, and find that they give good results for the solutions and for the self-energy. We show that the indirect coupling matrix element can be estimated from differences in the eigenvalue of the pulse optimization constraint for solutions with different phases.
In the Fleischmann–Pons experiment, excess heat is thought to have a nuclear origin due to the amount of energy produced, yet there are no commensurate energetic particles. This has motivated us over the years to focus attention on models in which a large quantum is fractionated into a great many small quanta. We have found that many two-level systems with a large transition energy are able to exchange energy coherently with an oscillator with a much smaller characteristic energy as long as decay channels are present in the vicinity of the two-level transition energy. In previous work we analyzed this basic model, and obtained estimates for the coherent energy exchange rate in the strong coupling limit. In this work we consider a version of this model where the upper states of the two-level systems are unstable. In this case, there is no coherent energy exchange, but instead we find a dynamical polarization effect which we have analyzed. We extend the model to the case of three-level systems, and generalize the result to apply to general N -level systems. Coherent energy exchange is possible within the context of a donor and receiver model, where the receiver transitions have unstable upper states. We give results for the donor dynamics in this case. This model provides a foundation for a new kind of model that we put forth recently for which the predictions appear to be closely connected to experiment.
Energy in the Fleischmann–Pons experiment is produced without commensurate energetic particles, and 4He is seen in amounts proportional to the energy produced (with a ratio of energy to number near 24 MeV). Correspondingly we focus on the D2/4He transition as a two-level system coupled to an oscillator, in order to make a connection with the lossy spin–boson model considered in previous work. Because of the strong Coulomb repulsion between the deuterons, the associated coupling matrix element is very small, and there is no possibility of converting the transition energy to oscillator quanta within a simple lossy spin–boson model. This motivates us to generalize to more complicated model that includes a set of (donor) two-level systems for the D2 / 4 He transition, and a set of (receiver) two-level systems that are strongly coupled to the oscillator. We analyze the resulting model in the limit that the receiver system is very strongly coupled. Within this formulation, the associated dynamics can be interpreted in terms of a transition from D2 to 4He with direct conversion (and fractionation) of the large energy quantum to the oscillator, once the coupling with the receiver system is sufficiently strong.
Excess heat in the Fleischmann–Pons experiment is observed without commensurate energetic particles, which is inconsistent with known nuclear reactions. Any proposed model must address this at the outset. At present, we recognize are two general approaches to the problem: mechanisms which transfer the reaction energy directly to a condensed matter mode, which requires the fractionation of a large quantum into many small quanta; mechanisms in which the large energy quantum is converted to kinetic energy over a very large number of neighboring nuclei. We have focused on the first approach, and we have found a model that seems to accomplish this. Here we introduce the model, which we call the “lossy spin-boson model”. We find that energy exchange in the lossless spin-boson model is hindered due to destructive interference effects. Augmenting the model with loss removes the destructive interference, and we use perturbation theory on a specific example for illustration. Feshbach projection operators and the Brillouin–Wigner formalism which we adopt to describe loss in the model are reviewed in Appendix A.
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.