Dynamics of the Two-State System with Ohmic Dissipation

The authors study a symmetric two-state system with "bare" tunneling frequency ₀, dimensionless Ohmic dissipation coefficient, and heat-bath cutoff ₂. Defining ₑ₀ ({{₀}{₂}) }^{ (1-) } for 1, they find to lowest order in {ₑ}{₂}, kT{₂} (a) for all ₑ, incoherent relaxation at a rate ({{₀}^2}{₂}) ({}2) () (+1{2) } ({{₂}) }^2-1; (b) for T=0, 12<<1, incoherent relaxation at a rate ₑ; and (c) for T=0, 0<<12, damped oscillations with frequency ₑ and Q factor 12cot ({2) } (1-) plus a power-law background.