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#### The onset time of Fermi's golden rule

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##### Citation

Debierre, V., & Lassalle, E. (2021). The onset time of Fermi's golden rule.* Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,* *477*(2248):
20200777. doi:10.1098/rspa.2020.0777.

Cite as: https://hdl.handle.net/21.11116/0000-000A-4B0E-4

##### Abstract

Fermi's golden rule describes the decay dynamics of unstable quantum

systems coupled to a reservoir, and predicts a linear decay in time.

Although it arises at relatively short times, the Fermi regime does not

take hold in the earliest stages of the quantum dynamics. The standard

criterion in the literature for the onset time of the Fermi regime is

t(F)similar to 1/Delta omega, with Delta omega the frequency interval

around the resonant transition frequency omega (0) of the system, over

which the coupling to the reservoir does not vary appreciably. In this

work, this criterion is shown to be inappropriate in general for

broadband reservoirs, where the reservoir coupling spectrum takes the

form R (omega)proportional to omega (eta), and for which it is found

that for eta >1, the onset time of the Fermi regime is given by

t(F)proportional to(omega (X)/omega (0))(eta -1)x1/omega (0) where omega

(X) is the high-frequency cutoff of the reservoir. Therefore, the onset

of the Fermi regime can take place at times orders of magnitude larger

than those predicted by the standard criterion. This phenomenon is shown

to be related to the excitation of the off-resonant frequencies of the

reservoir at short times. For broadband reservoirs with eta <= 1, and

for narrowband reservoirs, it is shown that the standard criterion is

correct.

systems coupled to a reservoir, and predicts a linear decay in time.

Although it arises at relatively short times, the Fermi regime does not

take hold in the earliest stages of the quantum dynamics. The standard

criterion in the literature for the onset time of the Fermi regime is

t(F)similar to 1/Delta omega, with Delta omega the frequency interval

around the resonant transition frequency omega (0) of the system, over

which the coupling to the reservoir does not vary appreciably. In this

work, this criterion is shown to be inappropriate in general for

broadband reservoirs, where the reservoir coupling spectrum takes the

form R (omega)proportional to omega (eta), and for which it is found

that for eta >1, the onset time of the Fermi regime is given by

t(F)proportional to(omega (X)/omega (0))(eta -1)x1/omega (0) where omega

(X) is the high-frequency cutoff of the reservoir. Therefore, the onset

of the Fermi regime can take place at times orders of magnitude larger

than those predicted by the standard criterion. This phenomenon is shown

to be related to the excitation of the off-resonant frequencies of the

reservoir at short times. For broadband reservoirs with eta <= 1, and

for narrowband reservoirs, it is shown that the standard criterion is

correct.