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Journal of Physics B Atomic Molecular and Optical Physics

Time-dependent tunnelling via path integrals. Connection to results of the quantum mechanics of decaying states

Theodosios Geo Douvropoulos, Cleanthes A. Nicolaides

Published Oct 24, 2002• 10 citations
Publisher

Abstract

The probability, P ( t ), of the irreversible dissipation into a continuous spectrum of an initially ( t = 0) localized (Ψ o ) nonstationary state acquires, as time increases, 'memory' due to the lower energy bound of the spectrum, and eventually follows a nonexponential decay (NED). Regardless of the degree of dependence on energy, the magnitude of this deviation from exponential decay depends on the degree of proximity to threshold, and on whether the theory employs a real energy distribution, one form of which is g ( E ) ≡ ⟨Ψ o |δ( H − E )|Ψ o ⟩, or a complex energy distribution, G ( E ) ≡ ⟨Ψ o |( H − E + i0)|Ψ o ⟩. It is the latter that is physically consistent, since it originates from the singularity at t = 0, which breaks the S -matrix unitarity, in accordance with the non-Hermitian character of decaying states. In order to test the quantum mechanical theory, we carried out semiclassical path integral calculations of the P ( t ) for an isolated narrow tunnelling state, whereby the truncated Fourier transform of a semiclassical Green function, G sc ( E ), is obtained. The results are in agreement with the analytic results of quantum mechanics when energy and time asymmetry are taken into account. It is shown that the analytic structure of G sc ( E ) is [ D regular + D pole ], where D pole is a finite sum over complex poles, which are the complex eigenvalues, W n , that the potential can support. The W n are given by E n + Δ n − (i/2)Γ n , where E n are the real eigenvalues of the corresponding bound potential, Γ n are the energy widths and Δ n are the energy shifts, both expressed in terms of computable semiclassical quantities. The spherical harmonic oscillator (SHO) with and without angular momentum, and unstable ground states of diatomic molecules, are treated as particular cases. The exact spectrum of the SHO is recovered only when the Kramers–Langer semiclassical expression for the centrifugal potential is used, thereby bypassing the difficulty of the singularity at r = 0. The spectrum from the use of the quantal form l ( l + 1) reduces to that of l ( l + 1/2) 2 in the limit of large l , i.e., for orbits far from r = 0. Using previously computed energies and widths for the vibrational levels of He 2 2+ 1σ g 2 1 Σ g + , the application of two formulae for P ( t ), one derived from a Lorentzian real energy distribution and the other from the corresponding complex energy distribution, shows that, for the lowest level, in the former case NED starts after about 193 lifetimes, and in the latter after about 102 lifetimes. The fact that this difference is large should have consequences for the deeper understanding of irreversibility at the quantum level.

DOI
10.1088/0953-4075/35/21/310

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For the understanding of irreversibility at the quantum level, the formation and decay of transient (unstable) states play a fundamental role. If the system is treated within Hermitian quantum mechanics, the resulting energy distribution of the resonance state, whose Fourier transform yields the time-dependent probability of decay, $P(t),$ is real. The physical constraint of the lower bound in the energy spectrum introduces ``memory,'' and causes nonexponential decay (NED) to set in after $t\ensuremath{\gg}\ensuremath{\tau},$ where \ensuremath{\tau} is the lifetime defined by exponential decay. The closer to threshold the decaying state is, the earlier NED appears. Apart from the constraint of $E>~0,$ the constraint of $t>~0$ must be accounted for at the same time. It results from the discontinuity at $t=0$ of the solution of the time-dependent Schr\"odinger equation, which breaks the unitarity associated with the S matrix and gives rise to a complex energy distribution, as a manifestation of the non-Hermitian property of the decaying states. For a narrow isolated resonance state, for which the self-energy is essentially energy-independent, analytic results for ${P}_{\mathrm{NED}}(t)$ obtained from semiclassical path-integral calculations agree with the quantum-mechanical ones when both physical constraints $E>0$ and $t>0$ are taken into account. As an example of the difference in the magnitude of the ${P}_{\mathrm{NED}}(t)$ when using a real and a complex energy distribution, application is made to the decay of the unstable ${\mathrm{He}}_{2}^{2+}{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ ground molecular state.

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International Journal of Quantum Chemistry•Nov 2, 2006•2 citations
The probability, P ( t ), of the irreversible dissipation into a continuous spectrum of an initially ( t = 0) localized (Ψ o ) nonstationary state acquires, as time increases, 'memory' due to the lower energy bound of the spectrum, and eventually follows a nonexponential decay (NED). Regardless of the degree of dependence on energy, the magnitude of this deviation from exponential decay depends on the degree of proximity to threshold, and on whether the theory employs a real energy distribution, one form of which is g ( E ) ≡ ⟨Ψ o |δ( H − E )|Ψ o ⟩, or a complex energy distribution, G ( E ) ≡ ⟨Ψ o |( H − E + i0)|Ψ o ⟩. It is the latter that is physically consistent, since it originates from the singularity at t = 0, which breaks the S -matrix unitarity, in accordance with the non-Hermitian character of decaying states. In order to test the quantum mechanical theory, we carried out semiclassical path integral calculations of the P ( t ) for an isolated narrow tunnelling state, whereby the truncated Fourier transform of a semiclassical Green function, G sc ( E ), is obtained. The results are in agreement with the analytic results of quantum mechanics when energy and time asymmetry are taken into account. It is shown that the analytic structure of G sc ( E ) is [ D regular + D pole ], where D pole is a finite sum over complex poles, which are the complex eigenvalues, W n , that the potential can support. The W n are given by E n + Δ n − (i/2)Γ n , where E n are the real eigenvalues of the corresponding bound potential, Γ n are the energy widths and Δ n are the energy shifts, both expressed in terms of computable semiclassical quantities. The spherical harmonic oscillator (SHO) with and without angular momentum, and unstable ground states of diatomic molecules, are treated as particular cases. The exact spectrum of the SHO is recovered only when the Kramers–Langer semiclassical expression for the centrifugal potential is used, thereby bypassing the difficulty of the singularity at r = 0. The spectrum from the use of the quantal form l ( l + 1) reduces to that of l ( l + 1/2) 2 in the limit of large l , i.e., for orbits far from r = 0. Using previously computed energies and widths for the vibrational levels of He 2 2+ 1σ g 2 1 Σ g + , the application of two formulae for P ( t ), one derived from a Lorentzian real energy distribution and the other from the corresponding complex energy distribution, shows that, for the lowest level, in the former case NED starts after about 193 lifetimes, and in the latter after about 102 lifetimes. The fact that this difference is large should have consequences for the deeper understanding of irreversibility at the quantum level.
For the understanding of irreversibility at the quantum level, the formation and decay of transient (unstable) states play a fundamental role. If the system is treated within Hermitian quantum mechanics, the resulting energy distribution of the resonance state, whose Fourier transform yields the time-dependent probability of decay, P(t),P(t),P(t), is real. The physical constraint of the lower bound in the energy spectrum introduces ``memory,'' and causes nonexponential decay (NED) to set in after $t\ensuremath{\gg}\ensuremath{\tau},$ where \ensuremath{\tau} is the lifetime defined by exponential decay. The closer to threshold the decaying state is, the earlier NED appears. Apart from the constraint of E> 0,E>~0,E> 0, the constraint of t> 0t>~0t> 0 must be accounted for at the same time. It results from the discontinuity at t=0t=0t=0 of the solution of the time-dependent Schr\"odinger equation, which breaks the unitarity associated with the S matrix and gives rise to a complex energy distribution, as a manifestation of the non-Hermitian property of the decaying states. For a narrow isolated resonance state, for which the self-energy is essentially energy-independent, analytic results for PNED(t){P}_{\mathrm{NED}}(t)PNED​(t) obtained from semiclassical path-integral calculations agree with the quantum-mechanical ones when both physical constraints E>0E>0E>0 and t>0t>0t>0 are taken into account. As an example of the difference in the magnitude of the PNED(t){P}_{\mathrm{NED}}(t)PNED​(t) when using a real and a complex energy distribution, application is made to the decay of the unstable ${\mathrm{He}}_{2}^{2+}{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ ground molecular state.
It is shown how the semiclassical theory of path integrals can be implemented in a practical manner for the analysis of a potential that combines the two-state system of a double well potential (DWP) with decay into a continuous spectrum. This potential may correspond to a variety of physical situations in physics and chemistry. The structure of the formalism and of the results is such that it allows computation not only for analytic but also for numerically given potentials. The central theme is the determination of the energy-dependent Green’s function, which is shown to consist of a regular part and a part containing simple and double complex poles. These poles represent the position of the energy levels, as well as the energy widths and shifts due to the interaction with the continuous spectrum. When applied to the bound DWP without tunneling, the theory is shown to reduce in certain limits to known results from the Jeffreys–Wentzel–Kiamers–Bhrillouin approximation. If the system is taken to be prepared in the first well, the interactions with the remaining of the potential lead to two types of transition rates. One represents the transient motion toward a virtual equilibrium state of the DWP. It emerges as a positive imaginary part of the self-energy. The other represents the decay into the continuum and emerges as a negative imaginary part of the pole. Comparison of the two mechanisms of nonstationarity is made for different magnitudes of the second barrier relative to the first one. Since the system decays to the continuum while oscillating, the theory obtains a correction to the frequency of oscillation in the DWP due to the interaction with the continuum. This phenomenon is observable in real two-state systems, if an external perturbation which affects mainly one state converts it into a resonance state.
The survival amplitude G(t)G(t)G(t) of a nonstationary state decaying into a purely continuous spectrum is treated in terms of an integral transform of an energy distribution with $\ensuremath{\infty}>E\ensuremath{\geqslant}0$. We examine three such distributions. Two are real functions, the Lorentzian gL(E){g}^{L}(E)gL(E) and a modified Lorentzian G(E)=gL(E)E1∕2G(E)={g}^{L}(E){E}^{1∕2}G(E)=gL(E)E1∕2, and one is the complex version of gL(E),gcL(E){g}^{L}(E),{g}_{c}^{L}(E)gL(E),gcL​(E). Real distributions are associated with Hermitian treatments while complex ones result from non-Hermitian treatments. The difference between the two has repercussions on the G(t)G(t)G(t) for nonexponential decay (NED) and on the understanding of irreversible decay at the quantum level. For all three distributions, we derive analytically amplitudes (propagators) for NED and then show that these satisfy differential equations, from which additional insight into the decay process for very long and very short times can be obtained. By making analogy with the classical Langevin equation, the terms of the differential equation that are derived when the simpler gL(E){g}^{L}(E)gL(E) and gcL(E){g}_{c}^{L}(E)gcL​(E) are employed, are interpreted using concepts such as friction and fluctuation. On the other hand, when gL(E){g}^{L}(E)gL(E) is multiplied by an energy-dependent factor, as in G(E)G(E)G(E), the results are, as expected, more complicated and the interpretability of the differential equation satisfied by the NED propagator loses clarity.
A semiclassical theory of multidimensional tunneling is formulated to calculate the tunneling wave function, energy splitting in a double well and decay rate constant from a metastable state. First, the tunneling wave function is calculated by analytic continuation of a quantized torus prepared in analytic forms using either the Chapman–Garrett–Miller method or the Birkhoff–Gustavson normal form method. For a weakly nonintegrable system, tunnelings are confirmed to be classified into two qualitatively different domains; pure tunneling in the I region and mixed tunneling in the C region. Semiclassical wave functions agree with quantum mechanical ones within a few percent both in classically allowed and tunneling regions. Breakdown of this simple picture is exemplified for cases of relatively strong couplings. Second, expressions of the tunneling energy splitting in a double well potential and the decay rate from a metastable state are derived. The wave function near a well formulated above is connected with the semiclassical Green’s function in the deep tunneling region. The latter is expressed by complex trajectories which start from the complex quantized torus. A preliminary numerical comparison with the quantum mechanical value is also done for the energy splitting.
We adapt the semiclassical technique, as used in the context of instanton transitions in quantum field theory, to the description of tunneling transmissions at finite energies through potential barriers by complex quantum mechanical systems. Even for systems initially in their ground state, not generally describable in semiclassical terms, the transmission probability has a semiclassical (exponential) form. The calculation of the tunneling exponent uses analytic continuation of degrees of freedom into a complex phase space as well as analytic continuation of the classical equations of motion into the complex time plane. We test this semiclassical technique by comparing its results with those of a computational investigation of the full quantum mechanical system, finding excellent agreement.
Abstract A recently published methodology based on the semiclassical path integral theory was applied in a double well structure and gave the analytic form of the system's Green's function. This type of potential can describe the ammonia molecule as far as the motion of the nitrogen atom perpendicular to the hydrogen plane is discussed. Because of the fact that a double well describes a bound system and correspondingly stationary states (constructed by the symmetric and antisymmetric superposition of the eigenstates of the two unperturbed wells), it was expected that the energy spectrum would be real, in a form of doublets due to the splitting effect that takes place. However, the result was a pair of complex poles, which had a clearly positive imaginary part for each member. The present work explains the role of the imaginary parts of the complex poles as the decay rate of quantities defined as the energy indeterminacies, which are directly related to the fact that energy is not well determined in a classically forbidden region of motion. These quantities come as a function of (dκ)/dE, which is the derivative of the classical action inside the potential barrier, with respect to energy. The major contribution comes from the turning points, and then the imaginary parts are responsible, not only for the conservation of energy, but for the correct sign of time as well. In this way, a different approach for the tunneling process is adopted, in which the entry or exit of the particle from the potential barrier takes place inside a neighborhood of the turning point, as though the latter was broadened and fluctuating. The magnitude of the previously mentioned decay rate is equal to ω/π, where ω is the frequency of the classical oscillations inside one well. In contrast, the inversion frequency is generated by the part of the complex pole that is unrelated to (dκ)/dE and is much smaller in magnitude than the classical frequency, since it is given as ω/π exp(−κ). In this way, the period of the energy fluctuations is much smaller than the internal period of the system produced by the oscillating communication of the two classically allowed regions of motion. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007