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Journal of Condensed Matter Nuclear Science

Lossy Spin–boson Model with an Unstable Upper State and Extension to N-level Systems

Peter L. Hagelstein, Irfan U. Chaudhary

Published Apr 1, 2013• 3 citations
Publisher

Abstract

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.

DOI
10.70923/001c.72218

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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.

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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.
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.
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.
Observations of excess heat in the absence of commensurate energetic charged particles challenges local energy and momentum conservation, a foundation of nuclear physics. We have explored models based on excitation transfer, in which global energy is conserved but local energy conservation is violated. We present recent results on both excitation transfer and anomalous energy exchange within the context of lossless spin-boson models. We introduce a rotation that allows us to isolate terms in the rotated Hamiltonian responsible for both processes. Spin-boson type models augmented with loss appear to be sufficiently strong to account for the excess heat effect.
Excess heat in the Fleischmann–Pons experiment occurs without commensurate energetic nuclear radiation, which motivated us to seek mechanisms capable of fractionating a large quantum, resulting in the lossy spin-boson model. Collimated x-ray emission in the Karabut experiment we interpreted as demonstrating the conversion of vibrational energy to nuclear excitation, as predicted from the lossy spin-boson model; this motivated us to seek physical models that could account for the effect. We found that the coupling strength associated with electron–nuclear and electron–electron coupling was too weak to fractionate keV quanta into 100 MHz vibrations, which motivated us to seek a theory in which nuclear transitions coupled directly to vibrations. These considerations led us to a lattice model with a relativistic description of compound nuclei, in which a strong phonon-nuclear interaction is present. This interaction is normally rotated out with the Foldy–Wouthuysen transformation, but in the presence of strong loss mechanisms this transformation becomes unhelpful. The model that results describes the coupling of vibrational energy to nuclear excitation, but earlier efforts to apply it to the Karabut experiment resulted in inconsistencies. This motivated us to analyze fluctuations due to electron-phonon coupling in metals, which we find here to be a weak effect for Bloch picture phonon exchange, and a somewhat stronger effect with Born–Oppenheimer phonon exchange. As no new physics were found to substantially enhance fractionation, we returned to the basic theory, and carried out a systematic analysis of the three different degrees of freedom that result from simple product approximations for the wavefunction. We find that when there are more virtual phonons than real, a significant enhancement of the coupling strength occurs, corresponding to a new anomalous regime. Conditions on the coupling matrix element and relative state occupation are derived for the anomalous regime, which favors strongly-coupled low energy transitions which constitute a large fraction of all nuclei present. The fractionation power of a sample in the anomalous regime is very much constrained, which allows us to be able to make predictions with the new model. We find that the 30.77 keV transition in 93Nb is likely to be in the anomalous regime in a Nb cathode; that the threshold fractionation energy is too low for excitation of the 30.77 keV transition; but that the threshold fractionation energy is high enough to result in the excitation of the 1565 eV transition in 201Hg, and x-ray emission in the few keV regime consistent with Karabut’s observations.
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.