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Journal of Condensed Matter Nuclear Science

Quantum Tunneling in a Time-Periodic Double-Well Potential as a Driver of LENR

Vladimir I. Dubinko, Александр Сергеевич Мазманишвили, Denis V. Laptev

Published Nov 9, 2023• 5 citations
Publisher

Abstract

At sufficiently low temperatures, the reaction rates in solids are controlled by quantum rather than by thermal fluctuations. We solve the Schrödinger equation for a Gaussian wave packet in a nonstationary harmonic oscillator and derive simple analytical expressions for the increase of its mean energy and dispersion with time induced by time-periodic modulation. Applying these expressions to the modified Kramers theory, we demonstrate a strong increase of the rate of escape out of a time-periodic potential well at a constant height of the potential barrier, when the driving frequency of the well position equals its eigenfrequency, or when the driving frequency of the well width exceeds its eigenfrequency by a factor of ∼2. Physical realization of the time-periodic driving modes can be induced by localized anharmonic vibrations (LAVs), a.k.a. discrete breathers, in which the amplitude of atomic oscillations greatly exceeds the amplitude of zero-point oscillations. At low temperatures LAVs can be excited by external triggering (such as irradiation by fast particles or by a laser), which can result in strong catalytic effects due to amplification of the Kramers rate. Numerical solution of Schrodinger equation for time-periodic double well potential is obtained, which agrees qualitatively with modified Kramers rate theory. Application of the present theory to low-energy nuclear reactions is discussed.

DOI
10.70923/001c.124663

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At sufficiently low temperatures, the reaction rates in solids are controlled by quantum rather than by thermal fluctuations. We solve the Schrödinger equation for a Gaussian wave packet in a nonstationary harmonic oscillator and derive simple analytical expressions for the increase of its mean energy and dispersion with time induced by time-periodic modulation. Applying these expressions to the modified Kramers theory, we demonstrate a strong increase of the rate of escape out of a time-periodic potential well at a constant height of the potential barrier, when the driving frequency of the well position equals its eigenfrequency, or when the driving frequency of the well width exceeds its eigenfrequency by a factor of ∼2. Physical realization of the time-periodic driving modes can be induced by localized anharmonic vibrations (LAVs), a.k.a. discrete breathers, in which the amplitude of atomic oscillations greatly exceeds the amplitude of zero-point oscillations. At low temperatures LAVs can be excited by external triggering (such as irradiation by fast particles or by a laser), which can result in strong catalytic effects due to amplification of the Kramers rate. Numerical solution of Schrodinger equation for time-periodic double well potential is obtained, which agrees qualitatively with modified Kramers rate theory. Application of the present theory to low-energy nuclear reactions is discussed.
We numerically study coherent tunneling oscillations of the particles between two levels in a double-well potential in the presence of anharmonic periodic potentials. Extremely short driving pulses modify the tunneling coefficient to ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}A$, where $\ensuremath{\kappa}$ is the bare tunneling coefficient without the driving field and AAA is the pulse area of the driving wave form. The modulation amplitude of the ${\ensuremath{\kappa}}_{\mathit{eff}}$ gradually decreases as the driving wave form becomes broad and is given by ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}{J}_{0}(A)$ for the sinusoidal modulation, where J0(x){J}_{0}(x)J0​(x) denotes the ordinary Bessel function of order zero. Theoretical derivation of the effective tunneling coefficient ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}A$ is also shown for a periodic $\ensuremath{\delta}$ kick with alternating sign by means of the transfer matrix formula.
Based on the dynamical quantization method we derive a quantum phase-space non-Markovian Smoluchowski equation describing the non-inertial Brownian motion of a harmonic oscillator immersed in a generic environment. In the long-time regime we investigate the tunneling phenomenon by evaluating the quantum Kramers escape rate of a Brownian particle over a potential barrier. As far as a quantum thermal reservoir is concerned, it is found that our steady quantum Kramers rate may depend upon no friction constant in the strong friction domain. This preposterous feature may suggest the existence of non-dissipative quantum tunneling at the low-temperature range, including the zero temperature case. Lastly, we predict that our quantum escape rate non-exponentially decays on the edge of the breakdown of the dissipation-fluctuation relation.
Abstract We study the tunneling dynamics of N = 10 one-dimensional interacting bosons confined in a temporally driven double well potential that imitates a quantum seesaw and how we can manipulate these dynamics by changing the drive of the seesaw potential. We emulate the seesaw with a driven double well potential and consider two driving protocols: an harmonic constant-frequency drive and a chirped drive with linearly increasing frequency. We consider the time-dependent many-body Schrödinger equation of a repulsively interacting quasi-one-dimensional few-boson system. We solve it by using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) as implemented in the MCTDH-X software. For an harmonic drive and at small values of the driving amplitude, the dynamics of the particles become very slow rendering a stationary-like state. In a phase-space picture the population imbalance between the wells follows a trajectory which is restricted to a comparatively small region of space. For an harmonic drive at intermediate amplitudes, the dynamics become periodic in nature, implying that the bosons populate each of wells periodically. At comparatively large amplitudes of the harmonic drive, the dynamics show features of chaos in phase-space representation. For the chirped drive with a driving frequency increasing linearly in time, the imbalance of the atoms in the seesaw, however, has a temporal evolution that is faster for certain frequency ranges. The tunneling dynamics in such cases, for small amplitudes, show the appearance of quasi-periodicity with simultaneously present slow and fast oscillations. Increasing the amplitude of the chirped drive, we observe that the dynamics, although being periodic, become severely damped in their amplitude. Our study establishes that by tuning the temporal evolution of the quantum seesaw, a precise control of tunneling dynamics of the correlated bosons can be achieved. Since harmonic driving and chirp frequency modulation of the seesaw are experimentally achievable, our simulations can be experimentally realized in laboratories dealing with cold atomic gases.
A model for interstitial diffusion is studied in which the impurity atom is regarded as a Brownian particle constrained to move in a single direction and subject to a periodic potential. A quantum mechanical theory is developed for the rate at which atoms leave a given potential well. The theory makes use of an ensemble of minimum-uncertainty wave packets to describe thermal equilibrium in the well and computes the tunneling probability through the potential barrier on the basis of the Schr\"odinger-Langevin equation in order to incorporate the effect of an interacting heat bath. The result of the interactions is to reduce the departure rate at a given temperature level, but the qualitative nature of the rate expression remains unchanged and predicts Arrhenius plots with straight-line behavior at high-temperature levels and curvature due to tunneling at low-temperature levels. The dynamics of the wave packet after it leaves a well is studied by computer solution of the Schr\"odinger-Langevin equation and it is found that the persistence of motion decreases with increasing strength of interaction with the heat bath.
The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.
We make a brief review of the Kramers escape rate theory for the probabilistic motion of a particle in a potential well U(x), and under the influence of classical fluctuation forces. The Kramers theory is extended in order to take into account the action of the thermal and zero-point random electromagnetic fields on a charged particle. The result is physically relevant because we get a non null escape rate over the potential barrier at low temperatures (T -> 0). It is found that, even if the mean energy is much smaller than the barrier height, the classical particle can escape from the potential well due to the action of the zero-point fluctuating fields. These stochastic effects can be used to give a classical interpretation to some quantum tunneling phenomena. Relevant experimental data are used to illustrate the theoretical results.