arXiv:1302.3511 · Physical Review A

Analytical study of quadratic and nonquadratic short-time behavior of quantum decay

Sergio Cordero, Gastón García‐Calderón

Published Dec 26, 2012• 16 citations

Abstract

The short-time behavior of quantum decay of an unstable state initially located within an interaction region of finite range is investigated using a resonant expansion of the survival amplitude. It is shown that in general the short-time behavior of the survival probability $S(t)$ has a dependence on the initial state and may behave as either $S(t)=1\ensuremath{-}O({t}^{3/2})$ or $1\ensuremath{-}O({t}^{2})$. These cases are illustrated by solvable models. The experiment reported by Wilkinson et al. [Nature (London) 387, 575 (1997)] does not distinguish between the above short-time behaviors.

DOI: 10.1103/physreva.86.062116

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arXiv:1302.3511 · Physical Review A

Analytical study of quadratic and nonquadratic short-time behavior of quantum decay

Sergio Cordero, Gastón García‐Calderón

Published Dec 26, 2012• 16 citations

arXiv HTML PDF

Abstract

S(t)

has a dependence on the initial state and may behave as either

S(t)=1\ensuremath{-}O({t}^{3/2})

1\ensuremath{-}O({t}^{2})

. These cases are illustrated by solvable models. The experiment reported by Wilkinson et al. [Nature (London) 387, 575 (1997)] does not distinguish between the above short-time behaviors.

DOI: 10.1103/physreva.86.062116

Related papers

Resonant Spectra and the Time Evolution of the Survival and Nonescape Probabilities

Gastón García‐Calderón, J. L. Mateos, M. Moshińsky

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 $S(t)$ and nonescape $P(t)$ 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, $S(t)$ and $P(t)$ 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 $S(t)\ensuremath{\sim}{t}^{\ensuremath{-}3}$ we obtain $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}$.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 $S(t)$ and nonescape $P(t)$ 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, $S(t)$ and $P(t)$ 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 $S(t)\ensuremath{\sim}{t}^{\ensuremath{-}3}$ we obtain $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}$.

Physical Review LettersJan 16, 199587 citations

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The characteristic short-time behavior of the survival probability of a decaying state, which underlies the quantum Zeno effect (QZE), is shown to follow quite simply from the Schr\"odinger equation even when the spectrum of the Hamiltonian has no finite lower bound, provided the mean energy of the initial state is finite. The failure of the Breit-Wigner form of energy distribution to predict QZE is traced to the nonexistence of a well-defined mean energy. On the basis of a suitably modified distribution, it is explicitly demonstrated that the existence of a finite lower bound to the energy distribution only affects the asymptotic decay law. For short times, the survival probability depends quadratically on time if the variance of the distribution is also finite, not otherwise. For a sufficiently well-behaved energy distribution, the asymptotic decay must be as ${t}^{{\ensuremath{-}}^{n}}$ if the spectrum has a finite lower bound, or faster than any inverse power of t if there is no lower bound.The characteristic short-time behavior of the survival probability of a decaying state, which underlies the quantum Zeno effect (QZE), is shown to follow quite simply from the Schr\"odinger equation even when the spectrum of the Hamiltonian has no finite lower bound, provided the mean energy of the initial state is finite. The failure of the Breit-Wigner form of energy distribution to predict QZE is traced to the nonexistence of a well-defined mean energy. On the basis of a suitably modified distribution, it is explicitly demonstrated that the existence of a finite lower bound to the energy distribution only affects the asymptotic decay law. For short times, the survival probability depends quadratically on time if the variance of the distribution is also finite, not otherwise. For a sufficiently well-behaved energy distribution, the asymptotic decay must be as ${t}^{{\ensuremath{-}}^{n}}$ if the spectrum has a finite lower bound, or faster than any inverse power of t if there is no lower bound.

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Long time deviations from the exponential decay law: possible effects in particle physics and cosmology

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An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. We find that the instantaneous energy of the unstable state for a large class of models of unstable states tends to the minimal energy of the system ${\cal E}_{min}$ as $t\rightarrow\infty$ which is much smaller than the energy of this state for $t$ of the order of the lifetime of the considered state. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for $t$ from the exponential time region. Taking into account results obtained for a model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as $σ$ meson as well as having astrophysical and cosmological consequences.An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. We find that the instantaneous energy of the unstable state for a large class of models of unstable states tends to the minimal energy of the system ${\cal E}_{min}$ as $t\rightarrow\infty$ which is much smaller than the energy of this state for $t$ of the order of the lifetime of the considered state. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for $t$ from the exponential time region. Taking into account results obtained for a model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as $σ$ meson as well as having astrophysical and cosmological consequences.

arXiv:1007.1742Jul 10, 20100 citations

S(t)

has a dependence on the initial state and may behave as either

S(t)=1\ensuremath{-}O({t}^{3/2})

1\ensuremath{-}O({t}^{2})

. These cases are illustrated by solvable models. The experiment reported by Wilkinson et al. [Nature (London) 387, 575 (1997)] does not distinguish between the above short-time behaviors.

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

S(t)

and nonescape

P(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,

S(t)

and

P(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 $S(t)\ensuremath{\sim}{t}^{\ensuremath{-}3}$ we obtain $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}$.

We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude

a(t)

of the state considered is found. Our attention is focused on the late time properties of

a(t)

and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as

t

goes to the infinity more slowly than any exponential function of

t

. This effect can be described using a time-dependent decay rate

γ(t)

and then the Khalfin result means that this

γ(t)

is not a constant but at late times it tends to zero as

t

goes to the infinity. It appears that the energy

E(t)

of the unstable state behaves similarly: It tends to the minimal energy

E_{min}

of the system as

t \to \infty

. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of

E(t)

and

γ(t)

. We discuss also possible implications of such a late time asymptotic properties of

E(t)

and

γ(t)

and cases where these properties may manifest themselves.

N

, the exponent becomes quadratic in

N

in the fermionic case. The same results apply to strongly interacting many-body systems related by the generalized Bose-Fermi duality. The faster fermionic decay can be traced back to the effective hard-core interactions between particles, which are as well the decaying products, and exhibit spatial antibunching which hinders the reconstruction of the initial unstable state. The results are illustrated with a paradigmatic model of quantum decay from a trap allowing leaks by tunneling, whose dynamics is described exactly by means of an expansion in resonant states.

{\cal E}_{min}

t\rightarrow\infty

which is much smaller than the energy of this state for

t

of the order of the lifetime of the considered state. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for

t

from the exponential time region. Taking into account results obtained for a model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as

σ

meson as well as having astrophysical and cosmological consequences.