General properties of the evolution of unstable states at long times

Urbanowski, K.

doi:10.1140/epjd/e2009-00165-x

Cited by 65 publications

(169 citation statements)

References 22 publications

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“…and h(t) is the effective Hamiltonian governing the time evolution in the subspace of unstable states H ∥ = PH, where P = |ϕ⟩⟨ϕ| (see [31] and also [32,33] and references therein). H ⊖ H ∥ = H ⊥ ≡ QH is the subspace of decay products.…”

Section: Preliminariesmentioning

confidence: 99%

“…H ⊖ H ∥ = H ⊥ ≡ QH is the subspace of decay products. Here Q = I − P. There is the following equivalent formula for h(t) [31][32][33]:…”

Section: Preliminariesmentioning

confidence: 99%

“…are the instantaneous mass (energy) µ ϕ (t) and the instantaneous decay rate, γ ϕ (t) [31][32][33]. Here ℜ (z) and ℑ (z) denote the real and imaginary parts of z respectively.…”

Section: Preliminariesmentioning

confidence: 99%

“…The derivative of ζ(t) given by (23) equals [32,33] From the properties of the integral-exponential function E 1 (x) it follows that the equation ∂ζ(t) ∂t = 0 can be satisfied at most for some isolated values of time t. So, from the formula (24) the conclusion follows: Within the model considered there is no time interval in which ζ(t) = const, that is, there is no time interval in which the survival probability P 0 (t) has a pure exponential form. This conclusion explains the results presented in Figs (2) and (3).…”

Section: Epj Web Of Conferencesmentioning

confidence: 99%

“…Based on the relation (33), one can show that that vectors U(Λ p,µ )|µ; 0⟩ are eigenvectors for the Hamiltonian H. There is…”

Section: Moving Unstable Systems With Constant Momentummentioning

confidence: 99%

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

Urbanowski

2017

EPJ Web Conf.

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Abstract.Results of theoretical studies and numerical calculations presented in the literature suggest that the survival probability P 0 (t) has the exponential form starting from times much smaller than the lifetime τ up to times t ≫ τ, and that P 0 (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 P 0 (t) from the exponential form begin to dominate). More detailed analysis of the problem shows that in fact the survival probability P 0 (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 ⃗ p 0. 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.

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Section: Preliminariesmentioning

confidence: 99%

“…H ⊖ H ∥ = H ⊥ ≡ QH is the subspace of decay products. Here Q = I − P. There is the following equivalent formula for h(t) [31][32][33]:…”

Section: Preliminariesmentioning

confidence: 99%

“…are the instantaneous mass (energy) µ ϕ (t) and the instantaneous decay rate, γ ϕ (t) [31][32][33]. Here ℜ (z) and ℑ (z) denote the real and imaginary parts of z respectively.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Epj Web Of Conferencesmentioning

confidence: 99%

“…Based on the relation (33), one can show that that vectors U(Λ p,µ )|µ; 0⟩ are eigenvectors for the Hamiltonian H. There is…”

Section: Moving Unstable Systems With Constant Momentummentioning

confidence: 99%

See 3 more Smart Citations

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

Urbanowski

2017

EPJ Web Conf.

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Extended nonergodic states in disordered many‐body quantum systems

2017

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This work supports the existence of extended nonergodic states in the intermediate region between the chaotic (thermal) and the many-body localized phases. These states are identified through an extensive analysis of static and dynamical properties of a finite one-dimensional system with onsite random disorder. The long-time dynamics is particularly sensitive to changes in the spectrum and in the structures of the eigenstates. The study of the evolution of the survival probability, Shannon information entropy, and von Neumann entanglement entropy enables the distinction between the chaotic and the intermediate region.Comment: 10 pages, 7 figures, Contribution to the Special Issue "Many-Body Localization" in Annalen der Physi

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Nonequilibrium Quantum Dynamics of Many-Body Systems

Santos

Torres-Herrera

2017

Understanding Complex Systems

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We review our results for the dynamics of isolated many-body quantum systems described by onedimensional spin-1/2 models. We explain how the evolution of these systems depends on the initial state and the strength of the perturbation that takes them out of equilibrium; on the Hamiltonian, whether it is integrable or chaotic; and on the onset of multifractal eigenstates that occurs in the vicinity of the transition to a many-body localized phase. We unveil different behaviors at different time scales. We also discuss how information about the spectrum of a many-body quantum system can be extracted by the sole analysis of its time evolution, giving particular attention to the so-called correlation hole. This approach is useful for experiments that routinely study dynamics, but have limited or no direct access to spectroscopy, as experiments with cold atoms and trapped ions.3. The speed of the decay depends on the energy of the initial state. The decay is faster for initial states with energy close to the middle of the spectrum, where there is a large concentration of eigenstates, than for states with energies near the border of the spectrum [26-28].4. Decays faster than Gaussian occur when the energy distribution of the initial state is bimodal [28], in which case the quantum speed limit can be reached. Moving away from realistic systems, fast decays can be obtained by increasing the number of particles that interact simultaneously [26][27][28].5. After the initial fast (often Gaussian) decay, the dynamics slows down and becomes power-law. The power-law exponent depends on how the spectrum approaches its energy bounds [24,25] and on the level of delocalization of the eigenstates [31-33].6. In interacting systems with onsite disorder, the value of the power-law decay exponent detects the transition from chaos to many-body localization. This exponent coincides with the fractal dimension of the system [31][32][33] and with the slope of the logarithmic growth of the Shannon and entanglement entropies [33].7. At long times, after the power-law behavior and before saturation, the survival probability shows a dip below its infinite time average [35,36]. This is known as correlation hole and appears only in systems with level repulsion (that is, not in integrable models). The correlation hole provides a way to detect level repulsion from the dynamics, instead of having to resort to the eigenvalues. This is useful for the experiments mentioned above, which have limited access to the spectra of their systems. Since the correlation hole is a general indicator of the integrable-chaos transition, it serves also as a detector of the metal-insulator transition in interacting systems [33,35].Additional highlights of our research, which are not described in this chapter, but may be found in our references, include the following topics.1. The dynamical behavior of the Shannon entropy and entanglement entropy is equivalent [33,34]. The first is easier to compute numerically and is potentially accessible experimentally, although it is...

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General properties of the evolution of unstable states at long times

Cited by 65 publications

References 22 publications

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

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

Extended nonergodic states in disordered many‐body quantum systems

Nonequilibrium Quantum Dynamics of Many-Body Systems

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