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Journal of Condensed Matter Nuclear Science

Thoughts on a Model of Vysotskii and Vysotskyy

Peter L. Hagelstein

Published May 6, 2024• 0 citations
Publisher

Abstract

Motivated by the recent proposal of Vysotstkii and Vysotskyy of the possibility of correlated coherent state evolution of an ion wave function in a low density magnetized plasma leading to low energy nuclear reactions, we have considered the construction of a dynamical squeezed state wave packet solution for a particle with a dynamic charge in a constant magnetic field. For a z- directed magnetic field, the dynamics simplify if the wave packet is constructed to have the same dynamics in x and y. From the solution developed, evolution equations are obtained for the first and second moments. In the case of the first moments, the dynamics that result are Newton’s laws for a particle in a magnetic field. The second moments evolve according to what would be expected for a squeezed state solution, except that the breathing frequency is reduced by a factor of two. The uncertainty product for the squeezed state is a Schrodinger-Robertson relation. The amount of squeezing required for the fluctuation energy to be sufficient to result in nuclear reactions is extremely large, and such a wave packet would be quite fragile in the sense that collisions with other particles would prevent organized “breathing” mode evolution except at very low plasma density. Following the proposal of Vysotskii and Vysotskyy, we constructed generalization of the squeezed state wave packet in which an excited state SHO eigenfunction is squeezed, and find that the evolution of the squeezing parameters is similar as for a squeezed state solution. The uncertainty principle is of the form of the Schrodinger-Robertson relation, but with a prefactor that substantially increases the uncertainty. An issue that was identified in the proposed scheme is localization associated with electron ionization and recombination that has the potential to spoil the delocalization and phase of the wave packet. It is noted that in a minor modification of the scheme non-classical energy exchange may be possible for charged particles traveling through a dynamic and spatially inhomogeneous magnetic field, as long as the changes in the field as seen by the particle are non-adiabatic relative to the squeezing dynamics.

DOI
10.70923/001c.124969

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Motivated by the recent proposal of Vysotstkii and Vysotskyy of the possibility of correlated coherent state evolution of an ion wave function in a low density magnetized plasma leading to low energy nuclear reactions, we have considered the construction of a dynamical squeezed state wave packet solution for a particle with a dynamic charge in a constant magnetic field. For a z- directed magnetic field, the dynamics simplify if the wave packet is constructed to have the same dynamics in x and y. From the solution developed, evolution equations are obtained for the first and second moments. In the case of the first moments, the dynamics that result are Newton’s laws for a particle in a magnetic field. The second moments evolve according to what would be expected for a squeezed state solution, except that the breathing frequency is reduced by a factor of two. The uncertainty product for the squeezed state is a Schrodinger-Robertson relation. The amount of squeezing required for the fluctuation energy to be sufficient to result in nuclear reactions is extremely large, and such a wave packet would be quite fragile in the sense that collisions with other particles would prevent organized “breathing” mode evolution except at very low plasma density. Following the proposal of Vysotskii and Vysotskyy, we constructed generalization of the squeezed state wave packet in which an excited state SHO eigenfunction is squeezed, and find that the evolution of the squeezing parameters is similar as for a squeezed state solution. The uncertainty principle is of the form of the Schrodinger-Robertson relation, but with a prefactor that substantially increases the uncertainty. An issue that was identified in the proposed scheme is localization associated with electron ionization and recombination that has the potential to spoil the delocalization and phase of the wave packet. It is noted that in a minor modification of the scheme non-classical energy exchange may be possible for charged particles traveling through a dynamic and spatially inhomogeneous magnetic field, as long as the changes in the field as seen by the particle are non-adiabatic relative to the squeezing dynamics.
Abstract Features of the Robertson–Schrödinger coordinate–momentum and energy–time uncertainty relations and connection between them have been considered. A method has been proposed to determine the duration of giant energy fluctuations of particles in a correlated coherent state. This method makes it possible to justify both a huge (many orders of magnitude) increase in the probability of the tunnel effect with the subsequent low-energy nuclear reaction and the automatic selection of low-energy reaction channels involving charged particles and the exclusion of the production of radioactive daughter isotopes. It has been shown that the same mechanism of formation of correlated coherent states explains a very significant suppression of gamma radiation observed in such reactions stimulated by the virtual energy as compared to similar reactions proceeding at a high “real” energy of particles.
For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as 〈qi〉〈{q}_{i}〉〈qi​〉, 〈pi〉〈{p}_{i}〉〈pi​〉 and 〈qipj〉〈{q}_{i}{p}_{j}〉〈qi​pj​〉, 〈pipj〉〈{p}_{i}{p}_{j}〉〈pi​pj​〉, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.
A method for the formation of correlated coherent states of low-energy particles in a parabolic potential well owing to the full-scale low-frequency modulation ω( t ) = ω 0 sinΩ t of the parameters of this well has been considered. It has been shown that such a modulation in the absence of a stochastic force acting on a particle results in the fast formation of correlated coherent states and in an increase in the correlation coefficient and transparency of the potential barrier to the limiting values | r ( t )| max → 1 and D → 1. The presence of the stochastic force significantly affects the evolution of correlated coherent states, decreasing the rate of an increase in the correlation coefficient | r ( t )| max (at Ω ≤ 10 −4 ω 0 ) and limiting it at the level | r (∞)| max < 1 (at Ω = (0.001–0.1)ω 0 ); | r (∞)| max increases with a decrease in the frequency of modulation and decreases with an increase in the intensity of the stochastic force. It has been shown that, at a realistic relation between the parameters, low-frequency modulation can ensure such | r | max value that the transparency of the potential barrier for low-energy particles increases by a factor of 10 50 –10 100 or larger. The mechanism of the formation of correlated coherent states for charged particles in a gas or a low-pressure plasma placed in a low-frequency magnetic field has been considered. We have determined the relation between the magnetic field strength and modulation frequency, as well as the relation between the temperature and density of the gas (plasma), at which the method under consideration can be used to optimize nuclear reactions at low energies.
We introduce excited coherent states, β,α;n≔a†nβ,α, where n is an integer and states β,α denote the coherent states of a charged particle in a uniform magnetic field. States β,α minimize the Schrödinger-Robertson uncertainty relation while having the nonclassical properties. It has been shown that the resolution of identity condition is realized with respect to an appropriate measure on the complex plane. Some of the nonclassical features such as sub-Poissonian statistics and quadrature squeezing of these states are investigated. Our results are compared with similar Agarwal’s type photon added coherent states (PACSs) and it is shown that, while photon-counting statistics of β,α,n are the same as PACSs, their squeezing properties are different. It is also shown that for large values of β, while they are squeezed, they minimize the uncertainty condition. Additionally, it has been demonstrated that by changing the magnitude of the external magnetic field, Bext, the squeezing effect is transferred from one component to another. Finally, a new scheme is proposed to generate states β,α;n in cavities.
Abstract The quasi-classical dynamics of a charged particle in a weak magnetic field in the presence of dissipative losses caused by ionization of the medium is studied. The approximate approach proposed here is a generalization of the Caldirola—Kanai method for quantizing the translational motion of particles in dissipative media. It is shown that a weak curvature of a classical trajectory by the magnetic field is accompanied by an isotropic increase of uncertainty of the particle coordinates to the some maximum value at the moment the localized probability density wave packet stops. The limitation of the increase of coordinate uncertainty is due to irreversible ionization losses.
We investigate the dynamics of a charged particle interacting with a multimode quantized electromagnetic field and obtain an analytic solution for the full electron--field system. This framework enables the calculation of position expectation values and uncertainties for arbitrary wave packets and field states, allowing us to identify quantum corrections to the corresponding classical motion. While the corrections to the position expectation value are weak and largely insensitive to the quantum state of the field, the wave packet broadening exhibits a pronounced dependence on the field state. In particular, the quantum uncertainty of the radiation is directly imprinted onto the spatial uncertainty of the particle. We illustrate these effects for Gaussian wave packets interacting with coherent, Fock, and squeezed states, including bright squeezed vacuum. The interaction with a finite-duration laser pulse is also analyzed as a multimode example. Our results provide a transparent analytic route toward understanding how quantum fluctuations of light influence electron dynamics in strong-field settings.