We introduce a general variational framework to address the tunneling of hot Fermi systems. We use the representation of the trace of the imaginary time τ = i t propagator as a functional integral type of a sum over complete sets of states at intermediate propagation slices. We assume that these states are τ -dependent and generated by an arbitrary trial Hamiltonian H 0 ( τ ) . We then use the convexity inequality to derive H 0 ( τ ) controlled variational bound for a trial action functional. This functional has a general structure consisting of two parts - statistically weighted quantum penetrability and dynamical tunneling entropy. We examine how this structure incorporates the basic physics of tunneling of hot Fermi systems. Using the variational inequality one can optimise the dynamical parameters controlling the action functional for any choice of the trial problem. As an application we take H 0 ( τ ) to describe imaginary time dynamics of non interacting Bogoliubov-de Gennes (BdG) quasiparticles . Optimising its dynamical parameters we extend the tunneling theory of hot Fermi systems to the Hartree-Fock-Bogoliubov (HFB) frame and derive the corresponding generalisation of imaginary time temperature dependent BdG mean field equations. As in the trial action the prominent feature of these equations is an inseparable interplay between quantum dynamical and entropic statistical effects. In the zero temperature limit these equations describe the “false ground state” tunneling decay of superfluid Fermi systems (spontaneous fission in nuclear physics). With increasing excitation energy (effective temperature) the decay process is gradually evolving from pure quantum tunneling to statistical “bottle neck” escape mechanism. Correspondingly the statistically weighted dynamical penetrability part in the action gradually decreases while the tunneling entropy increases with increasing effective temperature.
We introduce a general variational framework to address the tunneling of hot Fermi systems. We use the representation of the trace of the imaginary time τ = i t propagator as a functional integral type of a sum over complete sets of states at intermediate propagation slices. We assume that these states are τ -dependent and generated by an arbitrary trial Hamiltonian H 0 ( τ ) . We then use the convexity inequality to derive H 0 ( τ ) controlled variational bound for a trial action functional. This functional has a general structure consisting of two parts - statistically weighted quantum penetrability and dynamical tunneling entropy. We examine how this structure incorporates the basic physics of tunneling of hot Fermi systems. Using the variational inequality one can optimise the dynamical parameters controlling the action functional for any choice of the trial problem. As an application we take H 0 ( τ ) to describe imaginary time dynamics of non interacting Bogoliubov-de Gennes (BdG) quasiparticles . Optimising its dynamical parameters we extend the tunneling theory of hot Fermi systems to the Hartree-Fock-Bogoliubov (HFB) frame and derive the corresponding generalisation of imaginary time temperature dependent BdG mean field equations. As in the trial action the prominent feature of these equations is an inseparable interplay between quantum dynamical and entropic statistical effects. In the zero temperature limit these equations describe the “false ground state” tunneling decay of superfluid Fermi systems (spontaneous fission in nuclear physics). With increasing excitation energy (effective temperature) the decay process is gradually evolving from pure quantum tunneling to statistical “bottle neck” escape mechanism. Correspondingly the statistically weighted dynamical penetrability part in the action gradually decreases while the tunneling entropy increases with increasing effective temperature.
We use mean-field theory to investigate the dynamics of superfluid fermions. This thesis includes our two works. The first one is to study Josephson oscillations and self-trapping of superfluid fermions in a double-well potential with time-dependent Bogoliubov-de Gennes equations. We investigate the behaviour of a two-component Fermi superfluid. We numerically solve the time-dependent Bogoliubov-de Gennes equations and characterize the regimes of Josephson oscillations and self-trapping for different potential barriers and initial conditions. In the weak link limit the results agree with a two-mode model where the relative population and the phase difference between the two wells obey coupled nonlinear Josephson equations. A more complex dynamics is predicted for large amplitude oscillations and large tunneling. The second one is to calculate the dynamic structure factor of unitary fermions. We have studied the dynamic structure factor of unitary fermions both at zero and finite temperature using the Bogoliubov-de Gennes theory and also Superfluid Local Density Approximation. We have derived the expression of the linear response function and the dynamic structure factor in the random phase approximation. At zero temperature, the SLDA+RPA formalism indeed provides a better accuracy at low momentum transfer and also its static structure factor is closer to quantum Monte Carlo value than that in BdG+RPA; however SLDA seems to give worse results for the molecular excitations at large momentum transfer. We have discussed the role of temperature and the comparison between SLDA and BdG, as well as with experimental data. The analysis is still at a preliminary level, but it suggests that mean-field theories can indeed be used to extract quantitative information about the order parameter and the excitations of the system by two-photon Bragg scattering experiments.
The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.
We present a new variational method for investigating the ground state and out of equilibrium dynamics of quantum many-body bosonic and fermionic systems. Our approach is based on constructing variational wavefunctions which extend Gaussian states by including generalized canonical transformations between the fields. The key advantage of such states compared to simple Gaussian states is presence of non-factorizable correlations and the possibility of describing states with strong entanglement between particles. In contrast to the commonly used canonical transformations, such as the polaron or Lang-Firsov transformations, we allow parameters of the transformations to be time dependent, which extends their regions of applicability. We derive equations of motion for the parameters characterizing the states both in real and imaginary time using the differential structure of the variational manifold. The ground state can be found by following the imaginary time evolution until it converges to a steady state. Collective excitations in the system can be obtained by linearizing the real-time equations of motion in the vicinity of the imaginary time steady-state solution. Our formalism allows us not only to determine the energy spectrum of quasiparticles and their lifetime, but to obtain the complete spectral functions and to explore far out of equilibrium dynamics such as coherent evolution following a quantum quench. We illustrate and benchmark this framework with several examples: a single polaron in the Holstein and Su-Schrieer-Heeger models, non-equilibrium dynamics in the spin-boson and Kondo models, the superconducting to charge density wave phase transitions in the Holstein model.
Quantum tunneling remains unexplored in many regimes of many-body quantum physics, including the effect of quantum phase transitions on tunneling dynamics. In general, the quantum phase is a statement about the ground state and has no relation to far-from-equilibrium dynamics. Although tunneling is a highly dynamical process involving many excited states, we find that the quantum phase of the Bose-Hubbard model determines phase-dependent tunneling outcomes for the quantum tunneling escape, or quasi-bound problem. Superfluid and Mott insulator correlations lead to a new quantum tunneling rate, the quantum fluctuation rate. This rate shows surprising and highly dynamical features, such as oscillatory interference between trapped and escaped atoms and a completely different macroscopic quantum tunneling behavior for superfluid and Mott insulator phases. In the superfluid phase we find that escape dynamics are wave-like and coherent, leading to interference patterns in the density with a rapid decay process which is non-exponential. Quantum entropy production peaks when about half the atoms have escaped. In the Mott phase, despite stronger repulsive interactions, tunneling is significantly slowed by the presence of a Mott gap, creating an effective extra barrier to overcome. Only one atom can tunnel at a time, yet the decay process is nearly linear, completely defying the single-particle exponential model. Moreover, quantum entropy peaks when only about one quarter of the atoms have escaped. These and many other such effects go beyond the usual notions of single-particle quantum tunneling, quantum statistical effects on tunneling, and well-known semi-classical approaches from WKB to instanton theory. These results thus open up a new regime of exploration of far-from-equilibrium dynamics for quantum simulators and quantum dynamics.
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
The decay of Bose-Einstein condensates from a metastable ground state into collapse due to macroscopic quantum tunneling is investigated using a semiclassical approximation to Feynman’s path integral formalism. Applying a variational ansatz of a single Gaussian to the wave function determined by the Gross-Pitaevskii equation, a special choice of the Gaussian width parameters yields a mean-field energy functional in Hamiltonian form. The temporal spatial extension of the condensate then is described in the picture of a particle moving in an external potential. In this picture the decay of the condensate wave function is investigated using the bounce trajectory method, accounting for the action of the tunneling orbit in imaginary time and quadratic fluctuations around it, the latter described by the Gelfand-Yaglom differential equation. Additionally, the tunneling formalism is used for describing the condensate wave function by a trial function of a superposition of several Gaussians. Using this ansatz a Hamiltonian form of the mean-field energy does not exist and the formalism of describing the tunneling process has to be extended. The bounce trajectory is described in the parameter space of the Gaussian width parameters and the equations of motion for the imaginary time evolution are obtained by applying a time-dependent variational principle. The action of the bounce trajectory is investigated as a function of the number of Gaussians taken into account and the contribution of the fluctuations is evaluated using the monodromy matrix.