Universal and measurable entanglement in the spin-boson model

We study the entanglement between a qubit and its environment by calculating the von Neumann entropy of the spin in the delocalized phase of the spin-boson model. Using a well-known mapping between the spin-boson model with Ohmic dissipation and the anisotropic Kondo model, we obtain exact results for the entanglement entropy $E$ at arbitrary dissipation strength $\alpha$ and level asymmetry $h$. We show that the Kondo energy scale $T_K$ controls the entanglement between the qubit and the bosonic environment. For $h \ll T_K$, we find that $E=E(h=0)-\frac{2e^{b/(2-2\alpha)} \Gamma[1+1/(2-2\alpha)]}{\pi \ln 2 \Gamma[1+\alpha/(2-2\alpha)]} (\frac{h}{T_K})^2$, where $b=\alpha \ln \alpha + (1-\alpha) \ln (1-\alpha)$. The universal $(h/T_K)^2$ scaling reflects the Fermi liquid nature of the Kondo ground state. In the limit $h \gg T_K$, E vanishes as $(T_K/h)^{2-2\alpha}$, up to a logarithmic correction. We thoroughly explore the phase space $(\alpha, h)$; for a given $h$, the maximal entanglement occurs in the crossover regime $h \sim T_K$. We also emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise.