Time Evolution of Bath Properties in Spin-Boson Dynamics.

The dynamical behaviors of a two-level system (TLS) coupled to a harmonic dissipative bath has been studied extensively using a variety of analytical and numerical methods. The focus of the vast majority of these studies has been on the properties of the TLS, averaged with respect to the bath degrees of freedom. In this work, we use real-time path integral methods to probe the behavior of select bath degrees of freedom during the dynamics of a symmetric two-level system (TLS) coupled to a dissipative bath by calculating system-bath densities (SBD) and coordinate expectation values. Overall, the SBD motion on each diabatic state is simpler than the motion of the total density. In the weak coupling regime, which characterizes the parameters of oscillators that comprise such a bath, the SBD on each TLS state remains primarily compact and Gaussian-like, such that its peak is well characterized by the mode expectation value. In the absence of a dissipative environment, nonadiabatic density depletion leads to spikes in coordinate expectation values. The evolution of the SBD peak trajectory for two discrete modes exhibits Lissajous patterns with frequency-dependent shapes that strongly resemble classical trajectory motion on a torus. These patterns become more complex when the coupling of the mode to the TLS is increased outside of this regime, leading to persistent small amplitude oscillations in the TLS populations characterized by a very slow decay and SBD trajectories that exhibit behaviors reminiscent of chaotic classical systems. Indirect coupling to a dissipative bath has a stabilizing effect on the dynamics, eliminating spikes, synchronizing the SBD motion on the two diabatic states and regularizing the SBD trajectory to simple rectangular Lissajous-like shapes with a slowly shrinking boundary, regardless of the mode frequencies.