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arXiv:cond-mat/0612095 · Physical Review Letters

Universal and Measurable Entanglement Entropy in the Spin-Boson Model

Angela Kopp, Karyn Le Hur

Published May 31, 2007• 62 citations
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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 $E(\ensuremath{\alpha},\ensuremath{\Delta},h)$, 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.

DOI
10.1103/physrevlett.98.220401

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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(\ensuremathα,\ensuremathΔ,h)E(\ensuremath{\alpha},\ensuremath{\Delta},h)E(\ensuremathα,\ensuremathΔ,h), where \ensuremathα\ensuremath{\alpha}\ensuremathα is the dissipation strength, \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is the tunneling amplitude between qubit states, and hhh is the level asymmetry. For 1\ensuremath−\ensuremathα\ensuremath≫\ensuremathΔ/\ensuremathωc1\ensuremath{-}\ensuremath{\alpha}\ensuremath{\gg}\ensuremath{\Delta}/{\ensuremath{\omega}}_{c}1\ensuremath−\ensuremathα\ensuremath≫\ensuremathΔ/\ensuremathωc​ and (\ensuremathΔ,h)\ensuremath≪\ensuremathωc(\ensuremath{\Delta},h)\ensuremath{\ll}{\ensuremath{\omega}}_{c}(\ensuremathΔ,h)\ensuremath≪\ensuremathωc​, we show that the Kondo energy scale TK{T}_{K}TK​ controls the entanglement between the qubit and the bosonic environment (\ensuremathωc{\ensuremath{\omega}}_{c}\ensuremathωc​ is a high-energy cutoff). For h\ensuremath≪TKh\ensuremath{\ll}{T}_{K}h\ensuremath≪TK​, the disentanglement proceeds as (h/TK)2(h/{T}_{K}{)}^{2}(h/TK​)2; for h\ensuremath≫TKh\ensuremath{\gg}{T}_{K}h\ensuremath≫TK​, EEE vanishes as (TK/h)2\ensuremath−2\ensuremathα({T}_{K}/h{)}^{2\ensuremath{-}2\ensuremath{\alpha}}(TK​/h)2\ensuremath−2\ensuremathα, up to a logarithmic correction. For a given hhh, the maximum entanglement occurs at a value of \ensuremathα\ensuremath{\alpha}\ensuremathα which lies in the crossover regime h\ensuremath∼TKh\ensuremath{\sim}{T}_{K}h\ensuremath∼TK​. 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.
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, $\ensuremath{\alpha}={\ensuremath{\alpha}}_{M}<1.$
Scaling and universality in the Ohmic two-state system is investigated by exploiting the equivalence of this model to the anisotropic Kondo model. For the Ohmic two-state system, we find universal scaling functions for the specific heat ${C}_{\ensuremath{\alpha}}(T)$, static susceptibility ${\ensuremath{\chi}}_{\ensuremath{\alpha}}(T)$, and spin relaxation function ${S}_{\ensuremath{\alpha}}(\ensuremath{\omega})$, depending on the reduced temperature $T/{\ensuremath{\Delta}}_{r}$ (frequency $\ensuremath{\omega}/{\ensuremath{\Delta}}_{r}$), with ${\ensuremath{\Delta}}_{r}$ the renormalized tunneling frequency, and uniquely specified by the dissipation strength $\ensuremath{\alpha}$ ( $0<\ensuremath{\alpha}<1$). The scaling functions can be used to extract $\ensuremath{\alpha}$ and ${\ensuremath{\Delta}}_{r}$ in experimental realizations.
We study the von Neumann block entropy in the Kondo necklace model for different anisotropies $\ensuremath{\eta}$ in the XY\mathit{XY}XY interaction between conduction spins using the density matrix renormalization group method. It was found that the block entropy presents a maximum for each $\ensuremath{\eta}$ considered, and, comparing it with the results of the quantum criticality of the model based on the behavior of the energy gap, we observe that the maximum block entropy occurs at the quantum critical point between an antiferromagnetic and a Kondo singlet state, so this measure of entanglement is useful for giving information about where a quantum phase transition occurs in this model. We observe that the block entropy also presents a maximum at the quantum critical points that are obtained when an anisotropy $\ensuremath{\Delta}$ is included in the Kondo exchange between localized and conduction spins; when $\ensuremath{\Delta}$ diminishes for a fixed value of $\ensuremath{\eta}$, the critical point increases, favoring the antiferromagnetic phase.
The ground state and the spectral structure of lower-lying excited states of a dissipative two-level system coupled to a sub-Ohmic bath (s = 1/2) with nonzero bias have been studied using the unitary transformation method. By calculating the ground-state entanglement entropy, the ground-state average of σ(z)(G), and the static susceptibility of the two-level system, we explore the nature of the transition (crossover) between the delocalized and localized state of the two-level system. Furthermore, we calculate the time-dependent expectation (σ(z)(t)) and the time evolution of the entanglement entropy to show that, when the system undergoes a transition (crossover) from the delocalized to the localized state, the time evolution of the two-level system changes from coherent to decoherent dynamics.
The dynamics of a central spin-1/2 in presence of a local magnetic field and a bath of N spin-1/2 particles is studied in the thermodynamic limit. The interaction between the spins is Heisenberg XY type and the bath is considered to be a perfect thermal reservoir. In this case, the evolution of the populations of the reduced density matrix is obtained for different temperatures. A Born approximation is made but not a Markov approximation resulting a non-Markovian dynamics. The measure of the entanglement between the spin and the bath is obtained by means of the von Neumann entropy. For low temperatures, results show that there are oscillations of populations and of the von Neumann entropy as measure of entanglement, indicating that the central spin becomes disentangled of the spins of the bath with characteristic time periods in which it is possible to extract or recuperate information. In the regime of high temperatures, the evolution shows a final maximum entangled state with entropy S = ln 2 as it is expected for a two level system. I