The coupling of a qubit to a macroscopic reservoir plays a fundamental role in understanding the complex transition from the quantum to the classical world. Considering a harmonic environment, we use both intuitive arguments and numerical many-body quantum tomography to study the structure of the complete wavefunction arising in the strong-coupling regime, reached for intense qubit-environment interaction. The resulting strongly-correlated many-body ground state is built from quantum superpositions of adiabatic (polaron-like) and non-adiabatic (antipolaron-like) contributions from the bath of quantum oscillators. The emerging Schrödinger cat environmental wavefunctions can be described quantitatively via simple variational coherent states. In contrast to qubit-environment entanglement, we show that non-classicality and entanglement among the modes in the reservoir are crucial for the stabilization of qubit superpositions in regimes where standard theories predict an effectively classical spin.
The quantitative description of the quantum entanglement between a qubit and its environment is considered. Specifically, for the ground state of the spin-boson model, the entropy of entanglement of the spin is calculated as a function of $\ensuremath{\alpha},$ the strength of the ohmic coupling to the environment, and $\ensuremath{\varepsilon},$ the level asymmetry. This is done by a numerical renormalization group treatment of the related anisotropic Kondo model. For $\ensuremath{\varepsilon}=0,$ the entanglement increases monotonically with $\ensuremath{\alpha},$ until it becomes maximal for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}{1}^{\ensuremath{-}}.$ For fixed $\ensuremath{\varepsilon}>0,$ the entanglement is a maximum as a function of $\ensuremath{\alpha}$ for a value, $\ensuremath{\alpha}={\ensuremath{\alpha}}_{M}<1.$
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
TK 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/TK)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.
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.
Based on the quantum coherence theory, we employ the [Formula: see text]-norm measure to explore the steady-state coherence (SSC) in the spin-boson model. In this model, the environment is a non-Markovian bosonic bath with Ohmic-like spectral density. More generally, the interaction coupling between the qubit and the environment is a linear superposition of pure dephasing and pure damping coupling. Governed by the non-Markovian dynamics, some compact expressions of the SSC have been obtained at both zero temperature and high temperature. It shows that the hybrid coupling significantly affects the SSC. Moreover, a comprehensive analysis has been conducted to investigate the effects of various crucial factors, including the temperature of the bosonic bath, the qubit’s tunneling and the Ohmicity parameter. This analysis provides an effective approach for maintaining a relatively high SSC by manipulating system parameters.