Coherence and decoherence processes of a harmonic oscillator coupled with finite temperature field: exact eigenbasis solution of Kossakowski-Linblad's equation
Abstract
The eigenvalue problem of Kossakowski-Linblad’s kinetic equation associated
with the reduced density matrix of a harmonic oscillator interacting
with a thermal bath in equilibrium is solved. The solution gives rise to a
complete orthogonal eigenbasis endowed with Hilbert space structure that has
a weighted norm. We find that the eigenfunctions at finite temperature can
be obtained from the eigenfunction at zero temperature through a hyperbolic
rotation on the position variables. This transformation enables the extension
of the simple harmonic oscillator density matrix to that of a finite temperature.
We further investigate the decay of these extended states under our
dissipative kinetic equation. Furthermore, the Hilbert space structure enables
the proof of a H-theorem in this system. We apply the eigenbasis expansion of
an initial state to analyze decohorence as well as coherence processes. We find
that coherence process occurs at a longer time scale compared to decoherence
process. The time scales of both processes are estimated with the eigenbasis
expansion. In the same way we analyze the evolution of the coherent state.
We show that in addition to the ordinary decay time, we found another time
scale which is defined by the time when the motion of the peak of the coherent
state become comparative to the width of the coherent state. In contrast to
the ordinary decay time this new relaxation time depends on the initial value
of the momentum of the oscillator. We also find that our eigenbasis is applicable
to a class of non-linear interactions, with a slight extension of the form
of transport coefficients due to the non-linear interactions.