Kerim Savran

The off-resonant aspects of decoherence and a critique of the two-level approximation

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.

A critique of the Two Level Approximation

The conditions in favor and necessity of a realistic multileveled description of a decohering quantum system is examined. Under these conditions approximate techniques to simplify a multileveled system by its first two levels is unreliable and a realistic multilevel description in the formulation of decoherence is unavoidable. In this regard, the first crucial observation is that, the validity of the two level approximation of a multileveled system is not controlled purely by sufficiently low temperatures. That the type of system-environment coupling and the environmental spectrum have a dominant role over the temperature is demonstrated. Particularly, zero temperature quantum fluctuations induced by the Caldeira-Leggett type linear coordinate coupling can be influential in a wide energy range in the systems allowed transitions. The second crucial observation is that the decoherence times being among the system’s short time scales are found to be dominated not by the resonant but non-resonant processes.

Different behaviour of magnetic impurities in crystalline and amorphous states of superconductors

It has been observed that the effect of magnetic impurities in a superconductor is drastically different depending on whether the host superconductor is in the crystalline or the amorphous state. Based on the recent theory of Kim and Overhauser (KO), it is shown that as the system is getting disordered, the initial slope of the Tc depression is decreasing by a factor √l/ξ0, when the mean free path l becomes smaller than the BCS coherence length ξ0, which is in agreement with experimental findings. In addition, for a superconductor in a crystalline state in the presence of magnetic impurities the superconducting transition temperature Tc drops sharply from about 50% of Tc0 (for a pure system) to zero near the critical impurity concentration. This pure limit behaviour was indeed found by Roden and Zimmermeyer in crystalline Cd. Recently, Porto and Parpia have also found the same pure limit behaviour in superfluid He-3 in aerogel, which may be understood within the framework of the KO theory.