Quantum and classical fidelity for singular perturbations of the inverted and harmonic oscillator
Let us consider the quantum/versus classical dynamics for Hamiltonians of the form (0.1) H g ϵ : = P 2 2 + ϵ Q 2 2 + g 2 Q 2 , where ϵ = ± 1 , g is a real constant. We shall in particular study the quantum fidelity (Q.F.) between H g ϵ and H 0 ϵ defined as (0.2) F Q ϵ ( t , g ) : = 〈 exp ( − i t H 0...
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Published in | Journal of mathematical analysis and applications Vol. 326; no. 2; pp. 908 - 928 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
San Diego, CA
Elsevier Inc
15.02.2007
Elsevier |
Subjects | |
Online Access | Get full text |
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Summary: | Let us consider the quantum/versus classical dynamics for Hamiltonians of the form
(0.1)
H
g
ϵ
:
=
P
2
2
+
ϵ
Q
2
2
+
g
2
Q
2
,
where
ϵ
=
±
1
,
g is a real constant. We shall in particular study the quantum fidelity (Q.F.) between
H
g
ϵ
and
H
0
ϵ
defined as
(0.2)
F
Q
ϵ
(
t
,
g
)
:
=
〈
exp
(
−
i
t
H
0
ϵ
)
ψ
,
exp
(
−
i
t
H
g
ϵ
)
ψ
〉
for 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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ISSN: | 0022-247X 1096-0813 |
DOI: | 10.1016/j.jmaa.2006.03.044 |