Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Exponential Decay With a Memory

R. T. Robiscoe, John C. Hermanson

Published Oct 1, 1972• 8 citations
Publisher

Abstract

We examine the exponential decay law for an excited quantum system in the case where the state amplitude has a “memory,” that is, where the rate at which the amplitude is decaying at a particular time depends to some extent on values of the amplitude at previous times. For a particular choice of memory, characterized by a memory time τ, the amplitude obeys the differential equation of a damped harmonic oscillator. When τ is greater than a certain critical memory time τc, comparable to the natural lifetime of the state, the amplitude becomes an exponentially damped oscillation in time. When τ is less than τc, the decay is nearly exponential with a decay rate somewhat greater than that calculated from standard theory. By comparing theory with experiment, we can place an upper limit on the memory time for the radiative decay of atoms.

DOI
10.1119/1.1986867

Related papers

Nonclassical face of quantum decay processes

K. Urbanowski

A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).

Journal of Physics Conference Series•Aug 1, 2017•0 citations

Radiative Decay of Coupled Atomic States

Peter R. Fontana, Donald J. Lynch

The radiative decay of an atom with two excited states coupled by an external perturbation is investigated. The differential equations of motion are Fourier transformed and the probability amplitudes are obtained by contour integration. The real parts of the poles in the complex plane are the perturbed energies of the excited states, and the imaginary parts yield the decay characteristics. The decay probabilities of the excited states contain three different decay terms; two exponential decays and one modulated exponential decay. The probabilities of the final states give the frequency distribution of the emitted photons as a function of time. In an Appendix, the Heitler-Ma formalism is used to eliminate the final states of the system, and the resulting equations which contain damping terms are compared with the phenomenological method.

Physical Review A•Aug 1, 1970•28 citations

Decay of a metastable state: A variational approximation

E. Freidkin, Peter S. Riseborough, Peter Hänggi A

We consider the decay of a metastable state, coupled to a thermal reservoir, by quantum tunnelling. We calculate the leading exponential term in the W.K.B. expansion of the decay rate through a variational approximation. The approach is simple, quick and versatile, when compared with the alternative method which involves the numerical solution of non-linear, non-local, integro-differential equations. We have demonstrated the versatility of the variational approach by applying it to a few types of potentials, most of which have not been previously studied. In the few cases where numerical results are available, the variational approximation gives excellent agreement. In the cases where no numerical results are available for comparison, the variational approximation reproduces trends that are either physically reasonable or that may be inferred by interpolation between exactly soluble limits. We also address the question: What are the effects of frequency dependent friction, or a finite noise correlation time, on the quantum tunnellling rate? This question is one that has been recently broached in the discussion of experiments on highly damped r.f. SQUIDs and Josephson junctions. We find that the effect of the memory time is to reduce the lifetime of the metastable state, towards its undamped value. Again, we find that the variational approximation reproduces the exact variation of the action to a surprisingly good degree of accuracy.

Physica A Statistical Mechanics and its Applications•Apr 1, 1987•4 citations

The true quantum face of the “exponential” decay: Unstable systems in rest and in motion

K. Urbanowski

Results of theoretical studies and numerical calculations presented in the literature suggest that the survival probability P0(t) has the exponential form starting from times much smaller than the lifetime τ up to times t ⪢τ and that P0(t) exhibits inverse power–law behavior at the late time region for times longer than the so–called crossover time T ⪢ τ (The crossover time T is the time when the late time deviations of P0(t) from the exponential form begin to dominate). More detailed analysis of the problem shows that in fact the survival probability P0(t) can not take the pure exponential form at any time interval including times smaller than the lifetime τ or of the order of τ and it has has an oscillating form. We also study the survival probability of moving relativistic unstable particles with definite momentum . These studies show that late time deviations of the survival probability of these particles from the exponential–like form of the decay law, that is the transition times region between exponential–like and non-exponential form of the survival probability, should occur much earlier than it follows from the classical standard considerations.

EPJ Web of Conferences•Jan 1, 2017•0 citations

The Decay of Resonance Radiation by Spontaneous Emission

Emil J. Hellund

A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.

Physical Review•Mar 1, 1953•45 citations

Quantum oscillations without quantum coherence

V. V. Dobrovitski, Hans A. De Raedt, Mikhail I. Katsnelson +1 more

We study numerically the damping of quantum oscillations and the increase of entropy with time in model spin systems decohered by a spin bath. In some experimentally relevant cases, the oscillations of considerable amplitude can persist long after the entropy has saturated near its maximum, i.e. when the system has been decohered almost completely. Therefore, the pointer states of the system demonstrate non-trivial dynamics. The oscillations exhibit slow power-law decay, rather than exponential or Gaussian, and may be observable in experiments.

APS March Meeting Abstracts•Mar 1, 2002•1 citation
We examine the exponential decay law for an excited quantum system in the case where the state amplitude has a “memory,” that is, where the rate at which the amplitude is decaying at a particular time depends to some extent on values of the amplitude at previous times. For a particular choice of memory, characterized by a memory time τ, the amplitude obeys the differential equation of a damped harmonic oscillator. When τ is greater than a certain critical memory time τc, comparable to the natural lifetime of the state, the amplitude becomes an exponentially damped oscillation in time. When τ is less than τc, the decay is nearly exponential with a decay rate somewhat greater than that calculated from standard theory. By comparing theory with experiment, we can place an upper limit on the memory time for the radiative decay of atoms.
A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).
The radiative decay of an atom with two excited states coupled by an external perturbation is investigated. The differential equations of motion are Fourier transformed and the probability amplitudes are obtained by contour integration. The real parts of the poles in the complex plane are the perturbed energies of the excited states, and the imaginary parts yield the decay characteristics. The decay probabilities of the excited states contain three different decay terms; two exponential decays and one modulated exponential decay. The probabilities of the final states give the frequency distribution of the emitted photons as a function of time. In an Appendix, the Heitler-Ma formalism is used to eliminate the final states of the system, and the resulting equations which contain damping terms are compared with the phenomenological method.
We consider the decay of a metastable state, coupled to a thermal reservoir, by quantum tunnelling. We calculate the leading exponential term in the W.K.B. expansion of the decay rate through a variational approximation. The approach is simple, quick and versatile, when compared with the alternative method which involves the numerical solution of non-linear, non-local, integro-differential equations. We have demonstrated the versatility of the variational approach by applying it to a few types of potentials, most of which have not been previously studied. In the few cases where numerical results are available, the variational approximation gives excellent agreement. In the cases where no numerical results are available for comparison, the variational approximation reproduces trends that are either physically reasonable or that may be inferred by interpolation between exactly soluble limits. We also address the question: What are the effects of frequency dependent friction, or a finite noise correlation time, on the quantum tunnellling rate? This question is one that has been recently broached in the discussion of experiments on highly damped r.f. SQUIDs and Josephson junctions. We find that the effect of the memory time is to reduce the lifetime of the metastable state, towards its undamped value. Again, we find that the variational approximation reproduces the exact variation of the action to a surprisingly good degree of accuracy.
Results of theoretical studies and numerical calculations presented in the literature suggest that the survival probability P0(t) has the exponential form starting from times much smaller than the lifetime τ up to times t ⪢τ and that P0(t) exhibits inverse power–law behavior at the late time region for times longer than the so–called crossover time T ⪢ τ (The crossover time T is the time when the late time deviations of P0(t) from the exponential form begin to dominate). More detailed analysis of the problem shows that in fact the survival probability P0(t) can not take the pure exponential form at any time interval including times smaller than the lifetime τ or of the order of τ and it has has an oscillating form. We also study the survival probability of moving relativistic unstable particles with definite momentum . These studies show that late time deviations of the survival probability of these particles from the exponential–like form of the decay law, that is the transition times region between exponential–like and non-exponential form of the survival probability, should occur much earlier than it follows from the classical standard considerations.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
We study numerically the damping of quantum oscillations and the increase of entropy with time in model spin systems decohered by a spin bath. In some experimentally relevant cases, the oscillations of considerable amplitude can persist long after the entropy has saturated near its maximum, i.e. when the system has been decohered almost completely. Therefore, the pointer states of the system demonstrate non-trivial dynamics. The oscillations exhibit slow power-law decay, rather than exponential or Gaussian, and may be observable in experiments.