Quantum information processing in solid states: A critique of two-level approximation

We examine the effect of multilevels on decoherence and dephasing properties of a quantum system consisting of a non-ideal two level subspace, identified as the qubit and a finite set of higher energy levels above this qubit subspace. The whole system is under interaction with an environmental bath through a Caldeira-Leggett type coupling. The model that we use is an rf-SQUID under macroscopic quantum coherence and coupled inductively to a flux noise characterized by an environmental spectrum. The model interaction can generate dipole couplings which can be appreciable for a number of high levels. The decoherence properties of the qubit subspace is examined numerically using the master equation formalism of the system’s reduced density matrix. We numerically examine the relaxation and dephasing times as the environmental frequency spectrum, and the multilevel system parameters are varied at zero temperature. We observe that, these time scales receive contribution from all available energies in the noise spectrum (even well above the system’s energy scales) stressing the dominant role played by the non-resonant (virtual) transitions. The relaxation and dephasing times calculated, strongly depend on the number of levels within the range of levels for which appreciable couplings are produced. Under the influence of these effects, we remark that the validity of the two level approximation is restricted not by the temperature but by these dipole couplings as well as the availability of the environmental modes at low temperatures.