Spectral properties of a periodically kicked quantum Hamiltonian

Combescure, Monique

doi:10.1007/bf01025846

Cited by 63 publications

(66 citation statements)

References 13 publications

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“…This observation is linked to the conjecture put forward by Combescure [15] and partially addressed by Bourget [7]. After the completion of this work, we became aware of a recent paper by Bourget [8] which successfully resolves the issues with Milek and Seba's work by building on the earlier work in [7].…”

Section: Introductionmentioning

confidence: 74%

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On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

McCaw

McKellar

2005

Journal of Mathematical Physics

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By a straightforward generalisation, we extend the work of Combescure [J. Stat. Phys. 59, 679 (1990)] from rank-1 to rank-N perturbations. The requirement for the Floquet operator to be pure point is established and compared to that in Combescure. The result matches that in McCaw and McKellar [J. Math. Phys. 46, 032108 (2005)]. The method here is an alternative to that work. We show that if the condition for the Floquet operator to be pure point is relaxed, then in the case of the δ-kicked Harmonic oscillator, a singularly continuous component of the Floquet operator spectrum exists. We also provide an in-depth discussion of the conjecture presented in the work of Combescure of the case where the unperturbed Hamiltonian is more general. We link the physics conjecture directly to a number-theoretic conjecture of Vinogradov [The Method of Trigonometrical Sums in the Theory of Numbers (Interscience, London, 1954)] and show that a solution of Vinogradov's conjecture solves the physics conjecture. The result is extended to the rank-N case. The relationship between our work and the work of Bourget [J. Math. Anal. Appl. 276, 28 (2002); 301, 65 (2005)] on the physics conjecture is discussed.

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Section: Introductionmentioning

confidence: 74%

“…This result is the essence of Lemma 1 in [15]. When H 0 is pure point, it is a trivial calculation to show that …”

Section: Rank-n Generalisation Of Combescure's First Theoremmentioning

confidence: 81%

On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

McCaw

McKellar

2005

Journal of Mathematical Physics

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“…(2) with a normalized vector |v [15]. SinceV satisfiesV 2 =V , the quantum map (2) has a periodicity about λ.…”

Section: Quantum Map Under a Rank-1 Perturbationmentioning

confidence: 99%

Cheon’s anholonomies in Floquet operators

Miyamoto

Tanaka

2007

Phys. Rev. A

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Anholonomies in the parametric dependences of the eigenvalues and the eigenvectors of Floquet operators that describe unit time evolutions of periodically driven systems, e.g., kicked rotors, are studied. First, an example of the anholonomies induced by a periodically pulsed rank-1 perturbation is given. As a function of the strength of the perturbation, the perturbed Floquet operator of the quantum map and its spectrum are shown to have a period. However, we show examples where each eigenvalue does not obey the periodicity of the perturbed Floquet operator and exhibits an anholonomy. Furthermore, this induces another anholonomy in the eigenspaces, i.e., the directions of the eigenvectors, of the Floquet operator. These two anholonomies are previously observed in a family of Hamiltonians [T. Cheon, Phys. Lett. A 248, 285 (1998)] and are different from the phase anholonomy known as geometric phases. Second, the stability of Cheon's anholonomies in periodically driven systems is established by a geometrical analysis of the family of Floquet operators. Accordingly, Cheon's anholonomies are expected to be abundant in systems whose time evolutions are described by Floquet operators. As an application, a design principle for quantum state manipulations along adiabatic passages is explained.

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“…Unfortunately the quantum dynamics in the time-dependent case proved itself to be rather difficult to analyze in its full generality and complexity. The systems which allow for at least partially analytical treatment and whose dynamics has been perhaps best studied from various points of view are either driven harmonic oscillators [4,17,10,15] or periodically kicked quantum Hamiltonians [11,12,5,7,8,25]. On a more general level, it is widely believed that there exist close links between long time behavior of a quantum system and its spectral properties.…”

Section: Introductionmentioning

confidence: 99%

On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

2007

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We consider quantum Hamiltonians of the form H(t) = H + V (t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E n ∼ n α , with 0 < α < 1. In particular, the gaps between successive eigenvalues decay as n α−1 . V (t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate V (t) m,n ≤ ε |m − n| −p max{m, n} −2γ for m = n where ε > 0, p ≥ 1 and γ = (1−α)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and ε is small enough. More precisely, for any initial condition Ψ ∈ Dom(H 1/2 ), the diffusion of energy is bounded from above as H Ψ (t) = O(t σ ) where σ = α/(2⌈p − 1⌉γ − 1 2 ). As an application we consider the Hamiltonian H(t) = |p| α + εv(θ, t) on L 2 (S 1 , dθ) which was discussed earlier in the literature by Howland.

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Spectral properties of a periodically kicked quantum Hamiltonian

Cited by 63 publications

References 13 publications

On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

Cheon’s anholonomies in Floquet operators

On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

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