Conditions in favour of a realistic multilevelled description of a decohering quantum system
are examined. In this regard the first crucial observation is that the thermal effects,
contrary to the conventional belief, play a minor role at low temperatures in the
decoherence properties. The system–environment coupling and the environmental
energy spectrum dominantly affect the decoherence. In particular, zero temperature
quantum fluctuations or non-equilibrium sources can be present and influential
on the decoherence rates in a wide energy range allowed by the spectrum
of the environment. A crucial observation against the validity of the two-level
approximation is that the decoherence rates are found to be dominated not by the
long time resonant but the short time off-resonant processes. This observation
is demonstrated in two stages. Firstly, our zero temperature numerical results
reveal that the calculated short time decoherence rates are Gaussian-like (the
time dependence of the density matrix is led by the second time derivative at
t = 0). Exact analytical results are also permitted in the short time limit, which, consistent with
our numerical results, reveal that this specific Gaussian-like behaviour is a property of the
non-Markovian correlations in the environment. These Gaussian-like rates have no dependence
on any spectral parameter (position and the width of the spectrum) except, in totality, the
spectral area itself. The dependence on the spectral area is a power law. Furthermore, the
Gaussian-like character at short times is independent of the number of levels
(N), but the numerical value of the decoherence rates is a monotonic function of
N. In this context, we demonstrate that leakage, as a characteristic multilevel effect, is
dominated by the non-resonant processes.
The long time behaviour of decoherence is also examined. Since our spectral model allows
Markovian environmental correlations at long times, the decoherence rates in this regime
become exponential independently from the number of levels. The latter and the coupling
strengths play the major role in the quantitative values of the rates and the rates are
independent of the other spectral parameters.
The validity of the presented results is restricted only by their reliance on the
Born–Oppenheimer approximation. This approximation is strongly dependent on the
external observational time and its reliability depends on an additional timescale. In the
rest of the work, the crossover between the short and the long time behaviour of
the density matrix of the multilevelled system is examined using an intuitive
argument. It is shown that the Born approximation weakens as the resonant
couplings become more effective at long times. This implies that, in calculations made
with this approximation in the long time regime, a need for a justification arises
for the reliability of the results. This justification is made for the present work.