## Suppression of radiation damping in electromagnetic waveguide, signature of quantum decoherence in the field bath

##### Abstract

Recent development of spectral analysis of the Liouville-von Neumann equation
has revealed the fact that irreversibility is a rigorous dynamical property of
Poincaré non-integrable systems with an infinite degrees of freedom interacting
among each other through resonance coupling. In the present work we discuss
this role of resonance in some examples of matter-field coupling systems for both
classical and quantum mechanics: the one is a classical motion of a charged particle
in electromagnetic waveguide, and the other is the decoherence problem of quantum
matter-field interacting systems.
In the first part of this dissertation, we study an accelerated motion of a
charged classical dipole molecule with frequency ω1 inside the rectangular waveguide.
If the particle is in free space, it is well known that its accelerated motion will
eventually stop by radiating the field through the resonance interaction. This result
is the so-called radiation damping. For the case in the waveguide, there are two
possible situations, due to the existence of the cut-off frequency ωc of the waveguide.
Under the cut-off frequency electromagnetic wave cannot propagate inside the
waveguide. The stability of the dipole depends on the relation between ω1 and ωc.
For ω1 < ωc, the dipole cannot resonate with the field. This corresponds to the
Poincaré integrable system. For this case the dipole keeps its accelerated motion
without emitting the radiating field. Therefore the radiation damping of the dipole
molecules is suppressed inside the waveguide under the absence of resonance interaction.
The motion of this steady state somewhat resembles a quantum ground state.
We show that this steady state is dressed by electromagnetic field. The overlap
of the dressing field leads to a force analogous to van der Waals force in quantum
mechanics. The critical frequency determined by ω1 = ωc gives a critical size of the
waveguide. For heavy molecules, such as HCl, this is of order 10−5m. We show that
the size of the dressing field is the same order of the size of the waveguide. Hence
we have a macroscopic size of the dressing in the waveguide.
For ω1 > ωc, the dipole can resonate with the field, and the system becomes
non-integrable in the sense of Poincaré. As a result, the accelerated motion eventually
stop by emitting the resonance field. This corresponds to the problem of
classical radiation damping. We show that there is non-negligible deviation of exponential
decay in a short time scale of the order t ∼ 1/ω1. This corresponds to the
quantum Zeno effect, well known in quantum unstable systems. After this period,
the dipole decays exponentially in time by emitting the resonance field. We found
by choosing ωc very close to ω1, we can increase the decay rate 105
times faster than
the case where the dipole is in the free space, at the same time the emitted field
travel 10−4
times slower than the speed of light. This is again a consequence of the
existence of the cut-off frequency in the waveguide. Indeed, the cut-off frequency
leads to a non-linear dispersion relation for the electromagnetic field. To some extent,
the electromagnetic field is sticky inside the waveguide. Due to the large decay
rate and slower speed of light, the size of the wavepacket emitted by the dipole is
significantly small (about 10cm for HCl). This is even smaller than the quantum
case in free space, where the wavepacket of the field emitted by the decay of electron
in hydrogen atom is about 1m.
In the second part of this dissertation, we study a quantum matter-field
coupled system. We focus our attention on the problem of quantum decoherence in
a system of a particle coupled with a field, the Hamiltonian of which has a similar
structure to the problem of classical radiation damping mentioned above.
We apply the complex spectral representation of the Liouville-von Neumann
operator that gives a rigorous approach to the irreversible processes. We focus
our attention on the time evolution of the field, which is commonly neglected in
phenomenological approaches to the decoherence problem. We found a signature of
decoherence in the field which has a characteristic time dependence proportional to
t that comes from the secular effect between the particle and the field through the
resonance interaction that breaks time-symmetry.