Nonescape Probabilities Research Articles

Mean lifetime of a false vacuum in terms of the Krylov-Fock nonescape probability

The Krylov-Fock expression of non-decay (or survival) probability, which allows to evaluate the deviations from the exponential decay law (nowadays well established experimentally), is more informative as it readily provides the distribution function for the lifetime as a random quantity. Guided by the well established formalism for describing nuclear alpha decay, we use this distribution function to figure out the mean value of lifetime and its fluctuation rate. This theoretical framework is of considerable interest inasmuch as it allows an experimental verification. Next, we apply the Krylov-Fock approach to the decay of a metastable state at a finite temperature in the framework of thermo-field dynamics. In contrast to the existing formalism, this approach shows the interference effect between the tunnelings from different metastable states as well as between the tunneling and the barrier hopping. This effect looks quite natural in the framework of consistent quantum mechanical description as a manifestation of the "double-slit experiment". In the end we discuss the field theory applications of the results obtained.

Narrow Escape of Interacting Diffusing Particles.

The narrow escape problem deals with the calculation of the mean escape time (MET) of a Brownian particle from a bounded domain through a small hole on the domain's boundary. Here we develop a formalism which allows us to evaluate the nonescape probability of a gas of diffusing particles that may interact with each other. In some cases the nonescape probability allows us to evaluate the MET of the first particle. The formalism is based on the fluctuating hydrodynamics and the recently developed macroscopic fluctuation theory. We also uncover an unexpected connection between the narrow escape of interacting particles and thermal runaway in chemical reactors.

Buildup of symmetrization entanglement for the nonescape probability of two identical particles

We study the nonescape probability for tunneling decay of two identical noninteracting particles which are entangled due to symmetrization. As a consequence, the two-particle nonescape probability, which gives the probability that the particles remain within the potential region, may be written as the sum of two terms: a direct term and an exchange term which is added or subtracted according to symmetrization. At the initial time the direct term decays exponentially in the usual way, but the exchange term vanishes due to the orthonormality of the initial states. As time evolves the exchange term builds up until it becomes comparable to the direct term at time ${t}_{0}$. Thereafter, the symmetric and antisymmetric nonescape probabilities decay in a distinctive way. The buildup time typically lasts several lifetimes and implies that during the relevant exponentially decaying regime the probability of finding the particles within the interaction region is insensitive to the symmetry of the decaying particles. That is, the particles decay as if they were distinguishable. This might be experimentally verified in the tunneling decay of atoms in ultracold gases.

Exact analytical non-Hermitian formulation of the time evolution of decay of one and two identical quantum particles

An analytical solution to the time evolution of decay of one and two identical noninteracting particles is presented using the formalism of resonant states. It is shown that the time-dependent wave function and hence the survival and nonescape probabilities for the initial state of a single particle and entangled symmetric and antisymmetric initial states of two identical particles evolve in a distinctive form along the exponential and long-time nonexponential decaying regimes. In particular, for the last regime, they exhibit different inverse power of time behaviors.

Escape dynamics of a Bose-Hubbard dimer out of a trap

We consider a potential scattering of Bose-Hubbard dimer in 1D optical lattice. A numerical approach based on effective non-Hermitian Hamiltonian has been developed for solving the scattering problem. It allows to compute the tunneling and dissociation probabilities for arbitrary shape of the potential barrier and arbitrary kinetic energy of the dimer. The developed approach has been used to address the problem of two-particle decay out of a trap. In particular, it is shown that the presence of dissociation channels significantly decreases non-escape probability due to single-particle escape to those channels.

Anomalous Zeno effect for sharply localized atomic states

We analyze the nonquadratic in time Zeno effect which arises when a few-atom state initially trapped between two high laser-induced barriers is briefly released to free evolution. We identify the Zeno time, analyze the energy distributions of those atoms which have escaped and those that remained inside the trap, and obtain a simple relation between the survival and nonescape probabilities. The relevant time scales are such that the effect would be observable for the atomic species used in current laser experiments.

Fidelity of fermionic-atom number states subjected to tunneling decay

Atom-number states are a valuable resource for ultracold chemistry, atom interferometry and quantum information processing. Recent experiments have achieved their deterministic preparation in trapped few-fermion systems. We analyze the tunneling decay of these states, both in terms of the survival probability, and the non-escape probability, which can be extracted from measurements of the full counting statistics. At short times, the survival probability exhibits deviations from the exponential law. The decay is governed by the multi-particle Zeno time which exhibits a signature of quantum statistics and contact interactions. The subsequent exponential regime governs most of the dynamics, and we provide accurate analytical expressions for the associated decay rates. Both dynamical regimes are illustrated in a realistic model. Finally, a global picture of multi-particle quantum decay is presented.

Escape behavior of a quantum particle in a loop coupled to a lead

We consider a one-dimensional loop of circumference $L$ crossed by a constant magnetic flux $\ensuremath{\Phi}$ and connected to an infinite lead with coupling parameter $\ensuremath{\varepsilon}$. Assuming that the initial state ${\ensuremath{\psi}}_{0}$ of the particle is confined inside the loop and evolves freely, we analyze the time evolution of the nonescape probability $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)$, which is the probability that the particle will still be inside the loop at some later time $t$. In appropriate units, we found that $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)={P}_{\ensuremath{\infty}}({\ensuremath{\psi}}_{0},\ensuremath{\Phi})+{\ensuremath{\sum}}_{k=1}^{\ensuremath{\infty}}{C}_{k}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})/{t}^{k}$. The constant ${P}_{\ensuremath{\infty}}({\ensuremath{\psi}}_{0},\ensuremath{\Phi})$ is independent of $L$ and $\ensuremath{\varepsilon}$, and vanishes if ${\ensuremath{\psi}}_{0}$ has no bound state components or if $|\mathrm{cos}(\ensuremath{\Phi})|\ensuremath{\ne}1$. The coefficients ${C}_{1}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})$ and ${C}_{3}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})$ depend on the initial state ${\ensuremath{\psi}}_{0}$ of the particle, but only the momentum $k=\ensuremath{\Phi}/L$ is involved. There are initial states ${\ensuremath{\psi}}_{0}$ for which $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)\ensuremath{\sim}{C}_{\ensuremath{\delta}}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})/{t}^{\ensuremath{\delta}}$, as $t\ensuremath{\rightarrow}\ensuremath{\infty}$, where $\ensuremath{\delta}=1$ if $\mathrm{cos}(\ensuremath{\Phi})=1$ and $\ensuremath{\delta}=3$ if $\mathrm{cos}(\ensuremath{\Phi})\ensuremath{\ne}1$. Thus, by submitting the loop to an external magnetic flux, one may induce a radical change in the asymptotic decay rate of $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)$. Interestingly, if $\mathrm{cos}(\ensuremath{\Phi})=1$, then ${C}_{1}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})$ decreases with $\ensuremath{\varepsilon}$ (i.e., the particle escapes faster in the long run) while in the case $\mathrm{cos}(\ensuremath{\Phi})\ensuremath{\ne}1$, the coefficient ${C}_{3}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})$ increases with $\ensuremath{\varepsilon}$ (i.e., the particle escapes slower in the long run). Assuming the particle to be initially in a bound state of the loop with $\ensuremath{\Phi}=0$, we compute explicit relations and present some numerical results showing a global picture in time of $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)$. Finally, by using the pseudospectral method, we consider the interacting case with soft-core Coulomb potentials.

Time evolution of decay of two identical quantum particles

An analytical solution for the time evolution of decay of two identical non interacting quantum particles seated initially within a potential of finite range is derived using the formalism of resonant states. It is shown that the wave function, and hence also the survival and nonescape probabilities, for factorized symmetric and entangled symmetric/antisymmetric initial states evolve in a distinctive form along the exponentially decaying and nonexponential regimes. Our findings show the influence of the Pauli exclusion principle on decay. We exemplify our results by solving exactly the s-wave delta shell potential model.

Enhancement of non-escape probability in open chaotic system due to quantum localization induced by periodic orbits

Abstract We study semiclassically the quantum non-escape probability of a wavepacket initially localized in spatial region Ω , in an open system with fully chaotic classical limit. We show that at time t , the classically diffusive exponential decay is suppressed in leading order of ℏ by the quantum localization due to periodic orbits of period 2 t that are passing through Ω . Each contributing periodic orbit is weighted by the probability density of the initial wave packet along the periodic orbit and a prefactor which counts the stability of the periodic orbit. We argue that the periodic orbits correction will give non-trivial fluctuations superimposed on the monotonic exponential decay at small time and gives arise to anomalous slow decay at large time.

Free initial wave packets and the long-time behavior of the survival and nonescape probabilities

The behavior of both the survival $S(t)$ and the nonescape $P(t)$ probabilities at long times for the one-dimensional free-particle system is shown to be closely connected to that of the initial wave packet at small momentum. We prove that both $S(t)$ and $P(t)$ asymptotically exhibit the same power-law decrease at long times, when the initial wave packet in momentum representation behaves as ${k}^{m}$ with $m=0$ or $1$ at small momentum. On the other hand, if the integer m becomes greater than $1,$ $S(t)$ and $P(t)$ decrease in different power laws at long times.

Comment on "Resonant spectra and the time evolution of the survival and nonescape probabilities".

In [G. Garcia-Calderon, J. L. Mateos, and M. Moshinsky, Phys. Rev. Lett. 74, 337 (1995)], the time evolution of the quantum decay of a state initially located within an interaction region of finite range was investigated. In particular, it was shown that the survival and nonescape probabilities behave differently at very large times. The purpose of this Comment is to show that they have the same asymptotic behavior.

Decay process accelerated by tunneling in its very early stage

We examine a fast decay process that arises in the transition period between the Gaussian and exponential decay processes in quantum decay systems. It is usually expected that the decay is decelerated by a confinement potential barrier. However, we find a case where the decay in the transition period is accelerated by tunneling through a confinement potential barrier. We show that the acceleration gives rise to an appreciable effect on the time evolution of the nonescape probability of the decay system.

Analytical approach to the wave function of a decaying quantum system

Recently we outlined an analytical method of solving the time-dependent Schr\"odinger equation for a model which simulates a decaying quantum system such as an $\ensuremath{\alpha}$-decaying nucleus. A particle in the model is initially confined around the origin and leaks out, tunneling through a potential barrier. The solution can be expressed as a linear combination of the Moshinsky functions, each of which is associated with a pole of the scattering $\mathbf{S}$ matrix of the model. In this paper we give a full account of the method with a few explicit examples. We examine deviations from the exponential decay law at very large times. We comment on a recent controversy regarding the t dependence of the survival and nonescape probabilities when t is very large.

Comment on “Resonant Spectra and the Time Evolution of the Survival and Nonescape Probabilities”

A Comment on the Letter by G. Garcia-Calderon, J. L. Mateos, and M. Moshinsky, Phys. Rev. Lett. 74, 337 (1995). The authors of the Letter offer a Reply.Received 2 December 1996DOI:https://doi.org/10.1103/PhysRevLett.80.4353©1998 American Physical Society

García-Calderón, Mateos, and Moshinsky Reply:

In our paper [Phys. Rev. Lett. 74, 337 (1995)], we derived an exact expression for the survival and nonescape probabilities as an expansion in terms of resonant states. It was shown that these quantities exhibit at long times a different behavior. Although both decay as a power law, they have different exponents. In this paper we show that, contrary to the claim in the Comment of R. M. Cavalcanti (quant-ph/9704023), the nonescape probability decay for long times as an inverse power law.

Survival and Nonescape Probabilities for Resonant and Nonresonant Decay

In this paper we study the time evolution of the decay process for a particle confined initially in a finite region of space, extending our analysis given recently (Phys. Rev. Lett.74, 337 (1995)). For this purpose, we solve exactly the time-dependent Schrödinger equation for a finite-range potential. We calculate and compare two quantities: (i) the survival probabilityS(t), i.e., the probability that the particle is in the initial state after a timet; and (ii) the nonescape probabilityP(t), i.e., the probability that the particle remains confined inside the potential region after a timet. We analyze in detail the resonant and nonresonant decay. In the former case, after a very short timeS(t) andP(t) decay exponentially, but for very long times they decay as a power law, albeit with different exponents. For the nonresonant case we obtain that both quantities differ initially. However, independently of the resonant and nonresonant character of the initial state we always find a transition to the ground state of the system which indicates a process of “loss of memory” in the decay.

Resonant spectra and the time evolution of the survival and nonescape probabilities.

The time evolution of quantum decay of an arbitrary state initially located within an interaction region of finite range is investigated. We compare the survival [ital S]([ital t]) and nonescape [ital P]([ital t]) probabilities. Our approach considers the full resonant spectra of the system using a novel representation of the time-dependent Green function. It is shown that, for an initial state near a resonance, [ital S]([ital t]) and [ital P]([ital t]) exhibit at long times a different behavior; not only the onset to a power law decay occurs at a different time, but also instead of the well known [ital S]([ital t])[similar to][ital t][sup [minus]3] we obtain [ital P]([ital t])[similar to][ital t][sup [minus]1].

Greuling-Goertzel Calculations in Heavy Water

The Greuling-Goertzel approximation is applied to the calculation of neutron slowing down in heavy water in the B/sub 1/ and P/sub 1/ approximations. Numerical calculations of the flux and nonescape probability assuming constant cross sections indicate the inconsistent Greuling-Goertzel B/sub 1/ approximation to be the most favorable. (auth)

On the Validity of the Second Fundamental Theorem for Small Reactors

The second fundamental theorem of reactor theory gives a general expression for the nonescape probability. To check the validity of this expression for small sizes, first-flight nonescape probabilities are calculated for neutrons which have originated from either a persisting or a uniform stationary distribution in slabs of halfwidths ranging from 0.1 to 10 mean free paths. Exact values computed directly from the integral formulation are compared with the approximate values obtained by expanding the distributions in eigen solutions of the wave equation and applying the general theorems assuming that the linear extrapolation of the final flux vanishes on the extrapolation surface. It is found that the nonescape probabilities given by the fundamental theorem remain quite accurate even when the size of the reactor is decreased to the order of the mean free path. For a slab which is only two mean free paths wides the fractional difference from the exact value is 1.5 per cent for the persisting distribution and 2.5 per cent for the uniform distribution. (auth)