Consistent perturbative treatment of the subohmic spin-boson model yielding arbitrarily small $T_2/T_1$ decoherence time ratios

We present a perturbative treatment of the subohmic spin-boson model which remedies a crucial flaw in previous treatments. The problem is traced back to the incorrect application of a Markov type approximation to specific terms in the temporal evolution of the reduced density matrix. The modified solution is consistent both with numerical simulations and the exact solution obtained when the bath-coupling spin-space direction is parallel to the qubit energy-basis spin. We therefore demonstrate that the subohmic spin-boson model is capable of describing arbitrarily small ratios of the $T_2$ and $T_1$ decoherence times, associated with the decay of the off-diagonal and diagonal reduced density-matrix elements, respectively. An analytical formula for $T_2/T_1$ at the absolute zero of temperature is provided in the limit of a subohmic bath with vanishing spectral power law exponent. Small ratios closely mimic the experimental results for solid state (flux) qubits, which are subject predominantly to low-frequency electromagnetic noise, and we suggest a reanalysis of the corresponding experimental data in terms of a nonanalytic decay of off-diagonal coherence.

Figure 1

The decay factor $exp[-K(t\left)\right]$ in the ohmic and subohmic regime, $0\le n\le 1$; $\eta =5\times {10}^{-3}$ and $\theta ={60}^{\circ }$; the ultraviolet cutoff ${\omega }_c=100{\omega }_s$. For comparison, the dominant decay factor $exp[-t/2T_1]$ is shown by the dotted line.

Figure 2

Decoherence time ratio $\frac{T_2}{T_1}$ for $n=0$ as a function of coupling angle $\theta$. The analytical result [Eq. (16)] is shown by the red line and the numerical results by solving Eq. (4) are indicated by dots. The parameters $\eta$ and ${\omega }_c$ are the same as in Fig. 1.