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Perturbative method for generalized spectral decompositions

Laura, Roberto, Castagnino, Mario, R. Id Betán

Published Jan 1, 1999• 16 citations
OpenAlex

Abstract

Imposing analytic properties to states and observables we construct a perturbative method to obtain a generalized biorthogonal system of eigenvalues and eigenvectors for quantum unstable systems. A decay process can be described using this generalized spectral decomposition, and the final generalized state is obtained.

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Comment on "Nonexponential Decay Via Tunneling in Tight-Binding Lattices and the Optical Zeno Effect"

Elena Rufeil Fiori, Horacio M. Pastawski

S. Longhi [1] studied the survival probability P(t) of an unstable state coupled to a tight-binding lattice finding an exact analytical solution that describes the nonexponential decay. When the first coupling is smaller than the others, he shows that P(t) has a natural decomposition into two terms; one is the exponential decay, consistent with the Gamow's approach, and the other is the correction to this decay. The first purpose of this Comment is to show that the condition imposed on the system; the site energy of the first site is set in the center of the band, limits the generality of the results. A decomposition based on the spectral properties of the system removes this restriction [2]. Other point of this Comment is that, for the weak coupling limit, the author stated that a Zeno effect occurs for t less than t*, where t* is the smallest root of the equation g_eff(t*)=g_0, and for the strong coupling limit an anti-Zeno effect occurs for t close to t**, where t** is the first peak of g_eff. By using a spectral formulation we show that, the anti-Zeno effect not only requires a strong coupling limit, also is necessary to choose the resonance energy and width such that a destructive interference can occur. We present this interference and give an analytical expression and physically interpretation for t* and t**.

arXiv:0710.1086•Oct 4, 2007•1 citation
We present a general formalism for performing a time-dependent Bogoliubov analysis of a dynamically unstable Bose-Einstein condensate, which extends the quasiparticle projection method of Morgan et al. [Phys. Rev. A 57, 3818 (1998)] to cases with a complex spectrum. By introducing the proper left eigenvectors associated with each regime, we construct a biorthogonal basis. While the usual Bogoliubov normalization ⟨u∣u⟩−⟨v∣v⟩=1\langle u | u \rangle - \langle v | v \rangle = 1⟨u∣u⟩−⟨v∣v⟩=1 may not hold in this basis, it still allows for a complete mode decomposition and an accurate reconstruction of arbitrary perturbations over time. This approach extends the applicability of the Bogoliubov framework beyond the stable regime, providing a consistent analysis of the time evolution of unstable condensates. As a proof of concept, we apply the method to a one-dimensional condensate with attractive interactions, which is dynamically unstable and evolves into nonstationary localized structures seeded by small perturbations. Overall, the present method provides a complete and robust mode expansion that remains meaningful beyond the linear regime and useful for characterizing the macroscopic development of instabilities.
S. Longhi [1] studied the survival probability P(t) of an unstable state coupled to a tight-binding lattice finding an exact analytical solution that describes the nonexponential decay. When the first coupling is smaller than the others, he shows that P(t) has a natural decomposition into two terms; one is the exponential decay, consistent with the Gamow's approach, and the other is the correction to this decay. The first purpose of this Comment is to show that the condition imposed on the system; the site energy of the first site is set in the center of the band, limits the generality of the results. A decomposition based on the spectral properties of the system removes this restriction [2]. Other point of this Comment is that, for the weak coupling limit, the author stated that a Zeno effect occurs for t less than t*, where t* is the smallest root of the equation g_eff(t*)=g_0, and for the strong coupling limit an anti-Zeno effect occurs for t close to t**, where t** is the first peak of g_eff. By using a spectral formulation we show that, the anti-Zeno effect not only requires a strong coupling limit, also is necessary to choose the resonance energy and width such that a destructive interference can occur. We present this interference and give an analytical expression and physically interpretation for t* and t**.