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arXiv:2102.07474 · Physical Review Research

Dynamically assisted tunneling in the impulse regime

Christian Kohlfürst, Friedemann Queißer, Ralf Schützhold

Published Aug 13, 2021• 29 citations
arXivHTMLPDF

Abstract

We study the enhancement of tunneling through a potential barrier $V(x)$ by a time-dependent electric field with special emphasis on pulse-shaped vector potentials such as ${A}_{x}(t)={A}_{0}/{cosh}^{2}(\ensuremath{\omega}t)$. In addition to the known effects of preacceleration and potential deformation already present in the adiabatic regime, as well as energy mixing in analogy to the Franz-Keldysh effect in the nonadiabatic (impulse) regime, the pulse ${A}_{x}(t)$ can enhance tunneling by ``pushing'' part of the wave function out of the rear end of the barrier. Aside from the natural applications in condensed matter and atomic physics, these findings could be relevant for nuclear fusion, where pulses ${A}_{x}(t)$ with $\ensuremath{\omega}=1\phantom{\rule{0.28em}{0ex}}\mathrm{keV}$ and peak field strengths of ${10}^{16}\phantom{\rule{0.28em}{0ex}}\mathrm{V}/\mathrm{m}$ might enhance tunneling rates significantly.

DOI
10.1103/physrevresearch.3.033153

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Time as a Path to Thermonuclear Fusion

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Abstract We develop a new theoretical framework for quantum tunneling based on the Temporal Theory of the Universe (TTU) and the Temporal Wave Principle (TWP), in which the physical field of time τ(x, t, Θ) modifies the effective tunneling dynamics. In this approach, quantum particles are treated as τ-excitations with internal hyper-temporal oscillations, while spatial gradients of the temporal field (∇τ) directly influence both the effective potential barrier and the particle’s effective mass. We derive the Temporal Tunneling Equation (TTE) by incorporating τ-dependent corrections into the WKB action. The key result is that the tunneling probability Γ acquires an exponential enhancement factor of the form Γ ≈ Γ₀ exp[ Λ (∇τ)² ], where Λ is an explicit coefficient determined by TTU parameters (α, κ, m_τ).Thus, stronger τ-gradients reduce the Euclidean action, lower the effective barrier height, and increase the tunneling rate. This mechanism is universal and does not rely on thermal activation, zero-point vibration models, or exotic interactions. Importantly, the predicted enhancement can be tested experimentally using STM tunneling, Josephson junctions, THz-modulated barriers, cold atoms, and high-Q resonators. The results demonstrate that temporal geometry plays an active role in low-energy quantum phenomena, and that controlled τ-gradients offer a new pathway to engineer tunneling processes in condensed matter and atomic systems. Keywords Temporal field τ; TTU; TWP; quantum tunneling; WKB; hyper-time Θ; effective mass; temporal gradients; Schrödinger emergence; barrier suppression; tunneling enhancement; δτ-waves; condensed matter; STM; Josephson junctions. Table of Contents Time as a Path to Thermonuclear Fusion 1. Introduction 1.1. The problem of thermonuclear fusion1.2. Limitations of temperature-driven tunneling1.3. Temporal field τ as a physical substrate1.4. From TTU to temporal enhancement of tunneling1.5. Purpose and contributions of this work 2. Temporal Field and Gradient Dynamics 2.1. Definition of τ(x, t, Θ)2.2. Temporal gradients and temporal curvature2.3. Effective potential modification: U_eff = U₀ − B(∇τ)²2.4. Effective mass modification: m_eff = m₀ − C(∇τ)²2.5. Validity of the gradient expansion2.6. Physical interpretation of temporal geometry 3. Temporal Tunneling Equation (TTE) 3.1. Standard WKB exponent and its limitations3.2. Temporal correction to the Euclidean action3.3. Derivation of δS_E ∝ (∇τ)²3.4. Final formula: Γ = Γ₀ exp[ Λ (∇τ)² ]3.5. Mechanisms of barrier reduction3.6. Universality of τ-enhanced tunneling 4. Temporal Mechanisms in Thermonuclear Fusion 4.1. Coulomb barrier suppression via temporal curvature4.2. Mass reduction of reacting nuclei4.3. Temporal thinning of the barrier4.4. Effective Gamow factor under τ-dynamics4.5. Conditions for fusion at reduced temperatures 5. Stellar Fusion as a Temporal Phenomenon 5.1. The paradox of insufficient stellar temperatures5.2. Required tunneling vs observed luminosities5.3. δτ-backgrounds inside stellar plasma5.4. τ-gradient catalysis in solar fusion5.5. Resolution of the long-standing stellar tunneling problem 6. Toward Temporal Fusion Reactors (TFR) 6.1. Engineered ∇τ fields in laboratory plasmas6.2. τ–EM hybrid modes as fusion catalysts6.3. Temporal plasma compression and χ-nonlinearity6.4. Design principles for TFR architectures6.5. Scaling laws and efficiency predictions 7. Experimental Signatures and Tests 7.1. Fusion rate anomalies at constant temperature7.2. EM-frequency dependence predicted by TTE7.3. Temporal refractive effects and τ-sidebands7.4. Plasma transparency shifts7.5. Solid-state and tabletop tests of τ-effects 8. Discussion 8.1. Difference from thermal catalysis8.2. Difference from zero-point vibration models8.3. Why standard QM underestimates tunneling8.4. How τ-gradients reshape nuclear barriers8.5. Implications for cosmology and plasma physics 9. Conclusion • Time is a physical driver of tunneling• TTE provides a quantitative mechanism• Fusion becomes geometry-controlled• Temporal fields offer a new path to thermonuclear energy• Predictions are testable in current devices Reference Appendices Appendix A. Full WKB Derivation of the TTEAppendix B. Θ-Compactification and the Topological Origin of ħAppendix C. Laboratory Temporal Field BackgroundsAppendix D. Emergent Schrödinger Equation from τ(x,t,Θ)Appendix E. δτ-Waves and Universal τ-DispersionAppendix F. Temporal Correction to the Euclidean ActionAppendix G. Material-Dependent τ-ResponsesAppendix H. Temporal Plasma PhysicsAppendix I. Cosmological δτ-FluctuationsAppendix J. Temporal Fusion Reactors (TFR)Appendix K. Particles as τ-VorticesAppendix L. Chronon Modes and f-Mode SpectroscopyAppendix M. Experimental Bounds on α, κ, χAppendix N. Word-Friendly Formula Reference

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Dynamic Resonant Tunneling

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The physics of dynamic resonant tunneling is investigated.First, the resonant tunneling effect through an opaque barrier via a delta-function well is illustrated.Then, it is shown that, even in the adiabatic regime, where the dynamics can be governed by an analytic solution, the particle can be activated to higher energies.If the well varies quickly enough that the particle cannot escape from the well during the energetic elevation, the activation can be enhanced, as was anticipated by Azbel.However, and this is the main result of this work, the quasi-bound state of the well can even "reduce" the activation.In fact, because the resonant energy of the well matches twice the incoming particle's energy, and if the contribution to the wave function from both parts destructively interferes, then the particle cannot dwell in the well and activation is suppressed.This effect can be utilized in frequency-controlled transistors, and it is even speculated that it may explain the reason that humans can distinguish between tens of thousands of different odors with merely few hundreds of odor receptors.Lastly, the short time dynamics of a very fast perturbative well is also discussed.

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By developing the preceding work on the fast forward of transient phenomena of quantum tunneling by Khujakulov and Nakamura [Phys. Rev. A 93, 022101 (2016)], we propose a scheme of the exact fast forward of adiabatic control of stationary tunneling states with use of the electromagnetic field. The idea allows the acceleration of both the amplitude and phase of wave functions throughout the fast-forward time range. The scheme realizes the fast-forward observation of the transport coefficients under the adiabatically changing barrier with the fixed energy of an incoming particle. As typical examples, we choose systems with (1) Eckart's potential with tunable asymmetry and (2) double $\ensuremath{\delta}$-function barriers under tunable relative height. We elucidate the driving electric field to guarantee the stationary tunneling state during a rapid change of the barrier and evaluate both the electric-field-induced temporary deviation of transport coefficients from their stationary values and the modulation of the phase of complex scattering coefficients.

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We study the enhancement of tunneling through a potential barrier V(x)V(x)V(x) by a time-dependent electric field with special emphasis on pulse-shaped vector potentials such as Ax(t)=A0/cosh2(\ensuremathωt){A}_{x}(t)={A}_{0}/{cosh}^{2}(\ensuremath{\omega}t)Ax​(t)=A0​/cosh2(\ensuremathωt). In addition to the known effects of preacceleration and potential deformation already present in the adiabatic regime, as well as energy mixing in analogy to the Franz-Keldysh effect in the nonadiabatic (impulse) regime, the pulse Ax(t){A}_{x}(t)Ax​(t) can enhance tunneling by ``pushing'' part of the wave function out of the rear end of the barrier. Aside from the natural applications in condensed matter and atomic physics, these findings could be relevant for nuclear fusion, where pulses Ax(t){A}_{x}(t)Ax​(t) with \ensuremathω=1keV\ensuremath{\omega}=1\phantom{\rule{0.28em}{0ex}}\mathrm{keV}\ensuremathω=1keV and peak field strengths of 1016V/m{10}^{16}\phantom{\rule{0.28em}{0ex}}\mathrm{V}/\mathrm{m}1016V/m might enhance tunneling rates significantly.
We study how tunneling through a potential barrier V ( x ) can be enhanced by an additional harmonically oscillating electric field E ( t ) = E 0 cos ( ω t ) or a similar oscillating force. To this end, we transform into the Kramers-Henneberger frame and calculate the coupled Floquet channels numerically. We find distinct signatures of resonances when the incident energy E equals the driving frequency ω = E , which clearly shows the breakdown of the time-averaged potential approximation in this regime. As a simple model for experimental applications (e.g., in solid-state physics), we study the rectangular potential, which can also be benchmarked with respect to analytical results. Finally, we consider the truncated Coulomb potential relevant for nuclear fusion. Published by the American Physical Society 2024
Abstract We develop a new theoretical framework for quantum tunneling based on the Temporal Theory of the Universe (TTU) and the Temporal Wave Principle (TWP), in which the physical field of time τ(x, t, Θ) modifies the effective tunneling dynamics. In this approach, quantum particles are treated as τ-excitations with internal hyper-temporal oscillations, while spatial gradients of the temporal field (∇τ) directly influence both the effective potential barrier and the particle’s effective mass. We derive the Temporal Tunneling Equation (TTE) by incorporating τ-dependent corrections into the WKB action. The key result is that the tunneling probability Γ acquires an exponential enhancement factor of the form Γ ≈ Γ₀ exp[ Λ (∇τ)² ], where Λ is an explicit coefficient determined by TTU parameters (α, κ, m_τ).Thus, stronger τ-gradients reduce the Euclidean action, lower the effective barrier height, and increase the tunneling rate. This mechanism is universal and does not rely on thermal activation, zero-point vibration models, or exotic interactions. Importantly, the predicted enhancement can be tested experimentally using STM tunneling, Josephson junctions, THz-modulated barriers, cold atoms, and high-Q resonators. The results demonstrate that temporal geometry plays an active role in low-energy quantum phenomena, and that controlled τ-gradients offer a new pathway to engineer tunneling processes in condensed matter and atomic systems. Keywords Temporal field τ; TTU; TWP; quantum tunneling; WKB; hyper-time Θ; effective mass; temporal gradients; Schrödinger emergence; barrier suppression; tunneling enhancement; δτ-waves; condensed matter; STM; Josephson junctions. Table of Contents Time as a Path to Thermonuclear Fusion 1. Introduction 1.1. The problem of thermonuclear fusion1.2. Limitations of temperature-driven tunneling1.3. Temporal field τ as a physical substrate1.4. From TTU to temporal enhancement of tunneling1.5. Purpose and contributions of this work 2. Temporal Field and Gradient Dynamics 2.1. Definition of τ(x, t, Θ)2.2. Temporal gradients and temporal curvature2.3. Effective potential modification: U_eff = U₀ − B(∇τ)²2.4. Effective mass modification: m_eff = m₀ − C(∇τ)²2.5. Validity of the gradient expansion2.6. Physical interpretation of temporal geometry 3. Temporal Tunneling Equation (TTE) 3.1. Standard WKB exponent and its limitations3.2. Temporal correction to the Euclidean action3.3. Derivation of δS_E ∝ (∇τ)²3.4. Final formula: Γ = Γ₀ exp[ Λ (∇τ)² ]3.5. Mechanisms of barrier reduction3.6. Universality of τ-enhanced tunneling 4. Temporal Mechanisms in Thermonuclear Fusion 4.1. Coulomb barrier suppression via temporal curvature4.2. Mass reduction of reacting nuclei4.3. Temporal thinning of the barrier4.4. Effective Gamow factor under τ-dynamics4.5. Conditions for fusion at reduced temperatures 5. Stellar Fusion as a Temporal Phenomenon 5.1. The paradox of insufficient stellar temperatures5.2. Required tunneling vs observed luminosities5.3. δτ-backgrounds inside stellar plasma5.4. τ-gradient catalysis in solar fusion5.5. Resolution of the long-standing stellar tunneling problem 6. Toward Temporal Fusion Reactors (TFR) 6.1. Engineered ∇τ fields in laboratory plasmas6.2. τ–EM hybrid modes as fusion catalysts6.3. Temporal plasma compression and χ-nonlinearity6.4. Design principles for TFR architectures6.5. Scaling laws and efficiency predictions 7. Experimental Signatures and Tests 7.1. Fusion rate anomalies at constant temperature7.2. EM-frequency dependence predicted by TTE7.3. Temporal refractive effects and τ-sidebands7.4. Plasma transparency shifts7.5. Solid-state and tabletop tests of τ-effects 8. Discussion 8.1. Difference from thermal catalysis8.2. Difference from zero-point vibration models8.3. Why standard QM underestimates tunneling8.4. How τ-gradients reshape nuclear barriers8.5. Implications for cosmology and plasma physics 9. Conclusion • Time is a physical driver of tunneling• TTE provides a quantitative mechanism• Fusion becomes geometry-controlled• Temporal fields offer a new path to thermonuclear energy• Predictions are testable in current devices Reference Appendices Appendix A. Full WKB Derivation of the TTEAppendix B. Θ-Compactification and the Topological Origin of ħAppendix C. Laboratory Temporal Field BackgroundsAppendix D. Emergent Schrödinger Equation from τ(x,t,Θ)Appendix E. δτ-Waves and Universal τ-DispersionAppendix F. Temporal Correction to the Euclidean ActionAppendix G. Material-Dependent τ-ResponsesAppendix H. Temporal Plasma PhysicsAppendix I. Cosmological δτ-FluctuationsAppendix J. Temporal Fusion Reactors (TFR)Appendix K. Particles as τ-VorticesAppendix L. Chronon Modes and f-Mode SpectroscopyAppendix M. Experimental Bounds on α, κ, χAppendix N. Word-Friendly Formula Reference
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].
The physics of dynamic resonant tunneling is investigated.First, the resonant tunneling effect through an opaque barrier via a delta-function well is illustrated.Then, it is shown that, even in the adiabatic regime, where the dynamics can be governed by an analytic solution, the particle can be activated to higher energies.If the well varies quickly enough that the particle cannot escape from the well during the energetic elevation, the activation can be enhanced, as was anticipated by Azbel.However, and this is the main result of this work, the quasi-bound state of the well can even "reduce" the activation.In fact, because the resonant energy of the well matches twice the incoming particle's energy, and if the contribution to the wave function from both parts destructively interferes, then the particle cannot dwell in the well and activation is suppressed.This effect can be utilized in frequency-controlled transistors, and it is even speculated that it may explain the reason that humans can distinguish between tens of thousands of different odors with merely few hundreds of odor receptors.Lastly, the short time dynamics of a very fast perturbative well is also discussed.
By developing the preceding work on the fast forward of transient phenomena of quantum tunneling by Khujakulov and Nakamura [Phys. Rev. A 93, 022101 (2016)], we propose a scheme of the exact fast forward of adiabatic control of stationary tunneling states with use of the electromagnetic field. The idea allows the acceleration of both the amplitude and phase of wave functions throughout the fast-forward time range. The scheme realizes the fast-forward observation of the transport coefficients under the adiabatically changing barrier with the fixed energy of an incoming particle. As typical examples, we choose systems with (1) Eckart's potential with tunable asymmetry and (2) double $\ensuremath{\delta}$-function barriers under tunable relative height. We elucidate the driving electric field to guarantee the stationary tunneling state during a rapid change of the barrier and evaluate both the electric-field-induced temporary deviation of transport coefficients from their stationary values and the modulation of the phase of complex scattering coefficients.