Hidden Phase of the Spin-Boson Model

A quantum two-level system immersed in a sub-Ohmic bath experiences enhanced low-frequency quantum statistical fluctuations which render the nonequilibrium quantum dynamics highly non-Markovian. Upon using the numerically exact time-evolving matrix product operator approach, we investigate the phase diagram of the polarization dynamics. In addition to the known phases of damped coherent oscillatory dynamics and overdamped decay, we identify a new third region in the phase diagram for strong coupling showing an aperiodic behavior. We determine the corresponding phase boundaries. The dynamics of the quantum two-state system herein is not coherent by itself but slaved to the oscillatory bath dynamics.

Figure 1

Polarization $P(t)$ for $T=0$, $s=0.7$, and different coupling strengths $\alpha$. The arrow intersects the lines in ascending order of coupling strengths. Local minima (maxima) are marked by a red cross (green diamond).

Figure 2

Polarization $P(t)$ for $T=0$, $\alpha =0.8$, and different spectral exponents $s$. The arrow intersects the lines in ascending order of spectral exponents. Local minima are marked with a red cross.

Figure 3

Polarization $P(t)$ for $T=0$, $s=0.3$, and different coupling strengths $\alpha$. The arrow intersects the lines in ascending order of coupling strengths. Local minima (maxima) are marked by a red cross (green diamond).

Figure 4

Phase diagram of the (sub-)Ohmic spin-boson model at $T=0$ and ${\omega }_c=10\mathrm{\Omega }$. The symbols represent coupling strengths $\alpha (s)$ at which a transition occurs and are linearly interpolated for better visibility. Blue cross: the first local minimum vanishes when increasing coupling out of the coherent to the incoherent domain. Orange diamond: a local minimum appears for $\mathrm{\Omega }t<8$ when increasing coupling strength out of the incoherent domain. Green circle: the first local maximum vanishes when going from the coherent to the pseudocoherent domain.

Figure 5

Polarization $1-P(t)$ for $T=0$, $\alpha =0.8$, $s=0.3$, and different bath cutoff frequencies ${\omega }_c$. The arrow intersects the lines in ascending order of cutoff frequencies. Local maxima are marked with a green diamond. Note that the ${\omega }_c=10\mathrm{\Omega }$ line corresponds to the $s=0.3$ line in Fig. 2.

Figure 6

Polarization $P(t)$ for $T=0$, $s=0.45$, and different coupling strengths $\alpha$. The arrow intersects the lines in ascending order of coupling strengths. Local minima (maxima) are marked with a red cross (green diamond).