The quantum fidelity for the time-periodic singular harmonic oscillator

Combescure, Monique

doi:10.1063/1.2178153

Cited by 33 publications

(40 citation statements)

References 42 publications

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“…Moreover the response of a quantum system to an external perturbation when the size δ of the perturbation increases can manifest intriguing properties such as recurrences or decay in time of the so-called Loschmidt Echo (or "quantum fidelity") [7], [8]. By Loschmidt Echo we mean the following: starting from a quantum HamiltonianĤ in L 2 (R d ), whose classical counterpart H has a chaotic dynamics, and adding to it a " perturbation"Ĥ δ =Ĥ + δV , then we compare the evolutions in time U(t) := e −itĤ/ , U δ (t) := e −itĤ δ / of initial quantum wavepackets ϕ sufficiently well localized around some point z in phasespace; more precisely the overlap between the two evolutions, or rather its square absolute value, is:…”

Section: Introductionmentioning

confidence: 99%

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

Combescure

Robert

2006

Ann. Henri Poincaré

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The notion of Loschmidt echo (also called "quantum fidelity") has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of "quantum chaos", quantum computation and decoherence. In this paper, we study this quantity semiclassically (as → 0), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called "Ehrenfest time" C| log |, we are able to estimate semiclassically the Loschmidt Echo as a function of t (time), (Planck constant), and δ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the "return probability" (again on reference states being the coherent states) by the same method, as a function of t and .

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Section: Introductionmentioning

confidence: 99%

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

Combescure

Robert

2006

Ann. Henri Poincaré

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“…Then we have proven the general result (see [5]): Proposition 2.1. Let u , θ be the functions defined above.…”

Section: Quantum Fidelity For a Suitable Class Of Reference States (Pmentioning

confidence: 94%

Quantum and classical fidelity for singular perturbations of the inverted and harmonic oscillator

Combescure¹,

Combescure

2007

Journal of Mathematical Analysis and Applications

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Let us consider the quantum/versus classical dynamics for Hamiltonians of the formwhere = ±1, g is a real constant. We shall in particular study the quantum fidelity (Q.F.) between H g and H 0 defined asfor some reference state ψ in the domain of the relevant operators. We shall also propose a definition of the classical fidelity (C.F.), already present in the literature [G. Benenti, G. Casati, G. Veble, On the stability of classical chaotic motion under systems' perturbations, Phys. Rev. E 67 (2003) 055202(R); B. Eckhardt, Echoes in classical dynamical systems, J. Phys. A: Math. Gen. 36 (2003) 371-380; T. Prosen, M. Znidaric, Stability of quantum motion and correlation decay, J. Math. Phys. A: Math. Gen. 35 (2002) 1455-1481; G. Veble, T. Prosen, Faster than Lyapunov decays of classical Loschmidt Echo, Phys. Rev. Lett. 92 (2003) 034101] and compare it with the behavior of the quantum fidelity, as time evolves, and as the coupling constant g is varied.

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“…This is to say that estimate (10) holds true for the time evolution governed by the Hamiltonian H + V (t) as well.…”

Section: Afterwards One Can Apply Theorem 5 To the Hamiltonian H +V 1mentioning

confidence: 99%

“…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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The quantum fidelity for the time-periodic singular harmonic oscillator

Cited by 33 publications

References 42 publications

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

Quantum and classical fidelity for singular perturbations of the inverted and harmonic oscillator

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

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