Decoherence-free evolution of time-dependent superposition states of two-level systems and thermal effects

In this paper we detail some results advanced in a recent letter [Prado et al., Phys. Rev. Lett. 102, 073008 (2009).] showing how to engineer reservoirs for two-level systems at absolute zero by means of a time-dependent master equation leading to a nonstationary superposition equilibrium state. We also present a general recipe showing how to build nonadiabatic coherent evolutions of a fermionic system interacting with a bosonic mode and investigate the influence of thermal reservoirs at finite temperature on the fidelity of the protected superposition state. Our analytical results are supported by numerical analysis of the full Hamiltonian model.

Figure 1

Evolution of the numerically computed fidelity for both cases of absolute zero (solid line) and finite temperature, with the average number of thermal photons $\stackrel{̲}{n}=0.01$ (dashed line) and $\stackrel{̲}{n}=0.1$ (dotted line). The constant value $\mathcal{F}=0.5$ indicated by the dash-dotted line refers to the case where the coupling $g$ between the atomic system and the cavity mode is null.

Figure 2

Fidelity of the protected state $|+(t)\rangle$ undergoing adiabatic and nonadiabatic evolutions at zero temperature. We consider the values ${\Delta }_c/{\Gamma }_{\mathrm{eng}}\simeq 0.1$ (solid line), ${\Delta }_c/{\Gamma }_{\mathrm{eng}}\simeq 1$ (dashed line), and finally, ${\Delta }_c/{\Gamma }_{\mathrm{eng}}\simeq 3$ (dotted line).

Figure 3

Fidelity of the protected state $|+(t)\rangle$ undergoing nonadiabatic evolutions, with ${\Delta }_c=3\times {10}^2\gamma$ (${\Delta }_c/{\Gamma }_{\mathrm{eng}}\simeq 3$), for absolute zero (solid line) and finite temperature, with the average number of thermal photons $\stackrel{̲}{n}=0.01$ (dashed line) and $\stackrel{̲}{n}=0.1$ (dotted line).

Figure 4

Fidelity of the protected state $|+(t)\rangle$ undergoing nonadiabatic evolutions, versus the average thermal photons $\stackrel{̲}{n}$ characterizing the finite temperatures with $g=10\gamma$ (solid line), $20\gamma$ (dashed line), $50\gamma$ (dotted line), and ${10}^2\gamma$ (dash-dotted line).