Universal and measurable entanglement entropy in the spin-boson model

Abstract

We study the entanglement between a qubit and its environment from the spin-boson model with Ohmic dissipation. Through a mapping to the anisotropic Kondo model, we derive the entropy of entanglement of the spin $E(α,Δ,h)$, where $α$ is the dissipation strength, $Δ$ is the tunneling amplitude between qubit states, and $h$ is the level asymmetry. For $1-α\gg Δ/ω_c$ and $(Δ,h) \ll ω_c$, we show that the Kondo energy scale $T_K$ controls the entanglement between the qubit and the bosonic environment ($ω_c$ is a high-energy cutoff). For $h\ll T_K$, the disentanglement proceeds as $(h/T_K)^2$; for $h\gg T_K$, $E$ vanishes as $(T_K/h)^{2-2α}$, up to a logarithmic correction. For a given $h$, the maximum entanglement occurs at a value of $α$ which lies in the crossover regime $h\sim T_K$. We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise.