Quantum dynamics of the dissipative two-state system coupled with a sub-Ohmic bath

Abstract

The decoherence of a two-state system coupled with a sub-Ohmic bath is investigated theoretically by means of the perturbation approach based on a unitary transformation. It is shown that the decoherence depends strongly and sensitively on the structure of environment. Nonadiabatic effect is treated through the introduction of a function ${\xi }_k$ which depends on the boson frequency and renormalized tunneling. The results are as follows:(1) the non-equilibrium correlation function $P(t)$, the dynamical susceptibility ${\chi }^{\prime \prime }(\omega )$ and the equilibrium correlation function $C(t)$ are analytically obtained for $s\le 1$; (2) the phase diagram of thermodynamic transition shows the delocalized-localized transition point ${\alpha }_l$ which agrees with exact results and numerical data from the Numerical Renormalization Group; (3) the dynamical transition point ${\alpha }_c$ between coherent and incoherent phase is explicitly given for the first time. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for $\alpha >{\alpha }_c$, the coherent dynamics disappear); (4) the Shiba's relation and sum rule are exactly satisfied when $\alpha \le {\alpha }_c$; (5) an underdamping-overdamping transition point ${\alpha }_c^{\ast }$ exists in the function $S(\omega )$. Consequently, the dynamical phase diagrams in both ohmic and sub-Ohmic case are mapped out. For $\Delta \ll {\omega }_c$, the critical couplings (${\alpha }_l,{\alpha }_c$ and ${\alpha }_c^{\ast }$) are proportional to ${\Delta }^{1-s}$.

26pages,13figures,submitted to Phys. Rev. B

References in corpus (1)

• Numerical Renormalization Group for Bosonic Systems and Application to the Subohmic Spin-Boson Model

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