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arXiv:cond-mat/0612306

Two-level systems coupled to an oscillator: Excitation transfer and energy exchange

Peter L. Hagelstein, Irfan U. Chaudhary

Published Dec 12, 2006• 4 citations
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Abstract

We consider models in which two sets of matched two-level systems are coupled to a common oscillator in the case where the oscillator energy is small relative to the two-level transition energies. Since the two sets of two-level systems are coupled indirectly through the oscillator, excitation transfer from one set of two-level systems to the other is possible. In addition, the excitation energy from the two-level systems may be exchanged with the oscillator coherently, even though the oscillator energy may be orders of magnitude smaller than the two-level system transition energy. In the lossless case, we demonstrate these effects numerically, and also use an approximate diagonalization to show that these effects are expected from the model Hamiltonian. We augment the model to include loss effects, and show that loss enhances the excitation transfer effect by breaking the severe cancelation between different paths that occurs in the lossless case. We describe a simple approximate model wavefunction appropriate when the loss increases rapidly with energy. Within this model approximation, we present numerical and analytical results for excitation transfer and energy transfer rates, showing that they are greatly increased. Our study of these models is motivated in part by claims of excess heat production in electrochemical experiments in heavy water. We examine the question of whether the rates associated with this kind of model are sufficiently large to be relevant to the experimental claims. We find that consistency is possible given recent experimental results showing strong screening effects in low energy deuteron-deuteron fusion experiments in metals.

DOI
10.48550/arxiv.cond-mat/0612306

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We consider models in which two sets of matched two-level systems are coupled to a common oscillator in the case where the oscillator energy is small relative to the two-level transition energies. Since the two sets of two-level systems are coupled indirectly through the oscillator, excitation transfer from one set of two-level systems to the other is possible. In addition, the excitation energy from the two-level systems may be exchanged with the oscillator coherently, even though the oscillator energy may be orders of magnitude smaller than the two-level system transition energy. In the lossless case, we demonstrate these effects numerically, and also use an approximate diagonalization to show that these effects are expected from the model Hamiltonian. We augment the model to include loss effects, and show that loss enhances the excitation transfer effect by breaking the severe cancelation between different paths that occurs in the lossless case. We describe a simple approximate model wavefunction appropriate when the loss increases rapidly with energy. Within this model approximation, we present numerical and analytical results for excitation transfer and energy transfer rates, showing that they are greatly increased. Our study of these models is motivated in part by claims of excess heat production in electrochemical experiments in heavy water. We examine the question of whether the rates associated with this kind of model are sufficiently large to be relevant to the experimental claims. We find that consistency is possible given recent experimental results showing strong screening effects in low energy deuteron-deuteron fusion experiments in metals.
We consider a generalization of the spin-boson model in which two different two-level systems are coupled to an oscillator, under conditions where the oscillator energy is much less than the two-level system energies, and where the oscillator is highly excited. We find that the two-level system transition energy is shifted, producing a Bloch-Siegert shift in each two-level system similar to what would be obtained if the other were absent. At resonances associated with energy exchange between a two-level system and the oscillator, the level splitting is about the same as would be obtained in the spin-boson model at a Bloch-Siegert resonance. However, there occur resonances associated with the transfer of excitation between one two-level system and the other, an effect not present in the spin-boson model. We use a unitary transformation leading to a rotated system in which terms responsible for the shift and splittings can be identified. The level splittings at the anticrossings associated with both energy exchange and excitation transfer resonances are accounted for with simple two-state models and degenerate perturbation theory using operators that appear in the rotated Hamiltonian.
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.
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.
The time-dependent behavior of a two-level system interacting with a quantum oscillator system is analyzed in the case of a coupling larger than both the energy separation between the two levels and the energy of the quantum oscillator ($\ensuremath{\Omega}<\ensuremath{\omega}<\ensuremath{\lambda}$, where $\ensuremath{\Omega}$ is the frequency of the transition between the two levels, $\ensuremath{\omega}$ is the frequency of the oscillator, and $\ensuremath{\lambda}$ is the coupling between the two-level system and the oscillator). Our calculations show that the amplitude of the expectation value of the oscillator coordinate decreases as the two-level system undergoes the transition from one level to the other, while the transfer probability between the levels is staircaselike. This behavior is explained by the interplay between the adiabatic and nonadiabatic regimes encountered during the dynamics with the system acting as a quantum counterpart of the Landau-Zener model. The transition between the two levels occurs as long as the expectation value of the oscillator coordinate is driven close to zero. On the contrary, if the initial conditions are set such that the expectation values of the oscillator coordinate are far from zero, the system will remain locked on one level.
The stochastic Liouville equation and exact quantum mechanical simulations have been used to test the validity of rate law predictions for the vibrational energy relaxation in a three-level system. The model system consists of a low lying ground state which is radiatively coupled to only one of three closely spaced excited states. All three excited states are strongly coupled to baths of known statistical properties (either Gaussian or Poisson). We follow the energy exchange between the excited states after the light is turned off. Two different types of bath correlation have been considered: (A) The couplings between the different levels are the same and (B) the couplings are uncorrelated with each other. We find that relaxation differs for these two cases, in contradication to the prediction of the simple rate law description. We also present the first quantitative study of the interference between different pathways that can occur in such a three-level system. We find that the interference has a considerable effect on the decay of the middle level for the same bath (case A), even in the Markovian limit, while for independent baths (case B) interference becomes important only in the non-Markovian limit. These results are entirely different from the prediction of the conventional rate laws. We also discuss the effect of the excitation on the subsequent relaxation and the coherence transfer processes that lead to an oscillatory decay in the non-Markovian limit.
We present expressions for rates of radiationless transitions in two-level systems with arbitrary coupling to a set of damped vibrational modes; calculations are limited to transitions slower than vibrational relaxation. Explicit results are discussed within the harmonic approximation and Condon approximation; systematic methods are introduced for treating non-Condon and anharmonic effects. Results relevant to experiment are (1) significant deviations from the energy-gap law when a small number of vibrational modes is coupled to the transition, (2) dependence of transition rate on homogeneous and inhomogeneous vibrational linewidths, (3) nonexponential decay for transitions with vibrational frequency shifts, and (4) persistence of electronic coherence for long times either at low temperature or when frequency shifts are small. Our approach differs from previous work in the partitioning of the Hamiltonian and in the methods of evaluating operator averages; these differences are discussed in relation to the adiabatic approximation and the irreversibility of the transition.