Universal and measurable entanglement in the spin-boson model

Ru-Fen Liu, Chia-Chu Chen

Cuixia Wang, Ding Xiong, Guoqiang Huang +1 more

Angela Kopp, Karyn Le Hur

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 $E(\ensuremath{\alpha},\ensuremath{\Delta},h)$, where $\ensuremath{\alpha}$ is the dissipation strength, $\ensuremath{\Delta}$ is the tunneling amplitude between qubit states, and $h$ is the level asymmetry. For $1\ensuremath{-}\ensuremath{\alpha}\ensuremath{\gg}\ensuremath{\Delta}/{\ensuremath{\omega}}_{c}$ and $(\ensuremath{\Delta},h)\ensuremath{\ll}{\ensuremath{\omega}}_{c}$, we show that the Kondo energy scale ${T}_{K}$ controls the entanglement between the qubit and the bosonic environment (${\ensuremath{\omega}}_{c}$ is a high-energy cutoff). For $h\ensuremath{\ll}{T}_{K}$, the disentanglement proceeds as $(h/{T}_{K}{)}^{2}$; for $h\ensuremath{\gg}{T}_{K}$, $E$ vanishes as $({T}_{K}/h{)}^{2\ensuremath{-}2\ensuremath{\alpha}}$, up to a logarithmic correction. For a given $h$, the maximum entanglement occurs at a value of $\ensuremath{\alpha}$ which lies in the crossover regime $h\ensuremath{\sim}{T}_{K}$. We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise.

Kurt Fischer

We introduce a boson-impurity model, and prove that it is in the same universality class as the one-dimensional Ising model with algebraic decaying interaction. This boson-impurity model can be derived from the spin-boson model. The spin-boson model itself is therefore in the same universality class as well. Because the boson-impurity model contains only virtual spin-fluctuations, these are already enough to drive the quantum phase transition of the sub-ohmic spin-boson model.

Jeffrey Scott Pratt

Karyn Le Hur

The concept of entanglement entropy appears in multiple contexts, from black hole physics to quantum information theory, where it measures the entanglement of quantum states. We investigate the entanglement entropy in a simple model, the spin-boson model, which describes a qubit (two-level system) interacting with a collection of harmonic oscillators that models the environment responsible for decoherence and dissipation. The entanglement entropy allows to make a precise unification between entanglement of the spin with its environment, decoherence, and quantum phase transitions. We derive exact analytical results which are confirmed by Numerical Renormalization Group arguments both for an ohmic and a subohmic bosonic bath. Those demonstrate that the entanglement entropy obeys universal scalings. We make comparisons with entanglement properties in the quantum Ising model and in the Dicke model. We also emphasize the possibility of measuring this entanglement entropy using charge qubits subject to electromagnetic noise; such measurements would provide an empirical proof of the existence of entanglement entropy.