Dwell time and asymptotic behavior of the probability density

Muga, J. G.; Delgado, V.; Snider, R. F.

doi:10.1103/physrevb.52.16381

Cited by 35 publications

(48 citation statements)

References 8 publications

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“…This result describes the system dynamics near the EP on the longest time scales, for which the t 3 effect is quite typical in open quantum systems [6,11,30,31,[89][90][91]. Hence we see that while the EP dramatically modifies the dynamics on intermediate time scales, the long time dynamics are qualitatively the same as the ordinary case away from the EP; although the time scale T EP at which the long time effect sets in may be modified according to the proximity of the EP2A to the band edge.…”

Section: Inverse Power Law Evolution On Long Timescales Near the Ep2amentioning

confidence: 64%

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Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Garmon

Ordóñez

2017

Journal of Mathematical Physics

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It has been reported in the literature that the survival probability P (t) near an exceptional point where two eigenstates coalesce should generally exhibit an evolution P (t) ∼ t 2 e −Γt , in which Γ is the decay rate of the coalesced eigenstate; this has been verified in a microwave billiard experiment [B. Dietz, et al., Phys. Rev. E 75, 027201 (2007)]. However, the heuristic effective Hamiltonian that is usually employed to obtain this result ignores the possible influence of the continuum threshold on the dynamics. By contrast, in this work we employ an analytical approach starting from the microscopic Hamiltonian representing two simple models in order to show that the continuum threshold has a strong influence on the dynamics near exceptional points in a variety of circumstances. To report our results, we divide the exceptional points in Hermitian open quantum systems into two cases: at an EP2A two virtual bound states coalesce before forming a resonance, anti-resonance pair with complex conjugate eigenvalues, while at an EP2B two resonances coalesce before forming two different resonances. For the EP2B, which is the case studied in the microwave billiard experiment, we verify the survival probability exhibits the previously reported modified exponential decay on intermediate timescales, but this is replaced with an inverse power law on very long timescales. Meanwhile, for the EP2A the influence from the continuum threshold is so strong that the evolution is non-exponential on all timescales and the heuristic approach fails completely. When the EP2A appears very near the threshold we obtain the novel evolution P (t) ∼ 1 − C1 √ t on intermediate timescales, while further away the parabolic decay (Zeno dynamics) on short timescales is enhanced.

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Section: Inverse Power Law Evolution On Long Timescales Near the Ep2amentioning

confidence: 64%

“…The latter result for the far zone dynamics is fairly well known in the literature; for example, see Refs. [6,11,30,31,[89][90][91].…”

Section: Model I: Survival Probability Near the Ep2amentioning

confidence: 99%

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Garmon

Ordóñez

2017

Journal of Mathematical Physics

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“…A detailed analysis of this long time limit for several model potentials as well as a comparison with the free motion case (in one dimension the survival probability of a freely moving packet decays as t &1 asymptotically and in three dimensions as t &3 ) has been recently described in ref. [30]. The short time behaviour deserves careful examination, since different authors find contradictory results.…”

Section: Discussionmentioning

confidence: 99%

Survival Probability for the Yamaguchi Potential

1996

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Different decay behaviours of the survival probability are examined. Technically, the survival probability of an arbitrary state is determined by the matrix element of the resolvent of the full Hamiltonian. The analytical properties of this matrix element can be analyzed in terms of the properties of the kernel of the Lippmann Schwinger equation. For the Yamaguchi potential and several initial states, all functions are explicitly calculated. This approach allows the decomposition of the survival amplitude into a sum of decaying exponential terms and w-functions associated with the pole positions of the resolvent matrix element in the complex momentum plane. Novel decay behaviours are found for the decay of states associated with the resolvent poles and the decay of a state which is dominated by the form of the wave function rather than by the resolvent poles. Certain anomalous short time decay behaviour is also exemplified.

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“…In particular, for the most studied case of s-waves, µ = 3; the ℓ = 0 contribution would be in any case the dominant one at very large times since it is the slowest decaying term. This algebraic decay law also applies in 1D decaying systems [14]. The derivation of these integral power laws will facilitate identification of the complementary cases in which the exponents may be non-integers, so we shall devote Section II to a review of the basic elements of potential scattering theory relevant for our purposes.…”

Section: Introductionmentioning

confidence: 99%

“…It has been shown to be negligible for distant detectors in solvable models [4]. [3,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Long-time deviations from exponential decay for inverse-square potentials

2008

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Quantal systems are predicted to show a change-over from exponential decay to power law decay at very long times. Although most theoretical studies predict integer power-law exponents, recent measurements by Rothe et al. of decay luminescence of organic molecules in solution {Phys. Rev. Lett. 96 (2006) 163601} found non-integer exponents in most cases. We propose a physical mechanism, within the realm of scattering from potentials with long tails, which produces a continuous range of power law exponents. In the tractable case of the repulsive inverse square potential, we demonstrate a simple relation between the strength of the long range tail and the power law exponent. This system is amenable to experimental scrutiny.

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Dwell time and asymptotic behavior of the probability density

Cited by 35 publications

References 8 publications

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Survival Probability for the Yamaguchi Potential

Long-time deviations from exponential decay for inverse-square potentials

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