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arXiv:1406.4983 · Physical Review B

Generalized multipolaron expansion for the spin-boson model: Environmental entanglement and the biased two-state system

Soumya Bera, Ahsan Nazir, Alex W. Chin, Harold U. Baranger, Serge Florens

Published Aug 7, 2014• 77 citations
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Abstract

We develop a systematic variational coherent-state expansion for the many-body ground state of the spin-boson model, in which a quantum two-level system is coupled to a continuum of harmonic oscillators. Energetic constraints at the heart of this technique are rationalized in terms of polarons (displacements of the bath states in agreement with classical expectations) and antipolarons (counterdisplacements due to quantum tunneling effects). We present a comprehensive study of the ground-state two-level system population and coherence as a function of tunneling amplitude, dissipation strength, and bias (akin to asymmetry of the double-well potential defining the two-state system). The entanglement among the different environmental modes is investigated by looking at spectroscopic signatures of the bipartite entanglement entropy between a given environmental mode and all the other modes. We observe a drastic change in behavior of this entropy for increasing dissipation, indicative of the entangled nature of the environmental states. In addition, the entropy spreads over a large energy range at strong dissipation, a testimony to the wide entanglement window characterizing the underlying Kondo state. Finally, comparisons to accurate numerical renormalization-group calculations and to the exact Bethe ansatz solution of the model demonstrate the rapid convergence of our variationally optimized multipolaron expansion, suggesting that it should also be a useful tool for dissipative models of greater complexity, as relevant for numerous systems of interest in quantum physics and chemistry.

DOI
10.1103/physrevb.90.075110

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We develop a systematic variational coherent-state expansion for the many-body ground state of the spin-boson model, in which a quantum two-level system is coupled to a continuum of harmonic oscillators. Energetic constraints at the heart of this technique are rationalized in terms of polarons (displacements of the bath states in agreement with classical expectations) and antipolarons (counterdisplacements due to quantum tunneling effects). We present a comprehensive study of the ground-state two-level system population and coherence as a function of tunneling amplitude, dissipation strength, and bias (akin to asymmetry of the double-well potential defining the two-state system). The entanglement among the different environmental modes is investigated by looking at spectroscopic signatures of the bipartite entanglement entropy between a given environmental mode and all the other modes. We observe a drastic change in behavior of this entropy for increasing dissipation, indicative of the entangled nature of the environmental states. In addition, the entropy spreads over a large energy range at strong dissipation, a testimony to the wide entanglement window characterizing the underlying Kondo state. Finally, comparisons to accurate numerical renormalization-group calculations and to the exact Bethe ansatz solution of the model demonstrate the rapid convergence of our variationally optimized multipolaron expansion, suggesting that it should also be a useful tool for dissipative models of greater complexity, as relevant for numerous systems of interest in quantum physics and chemistry.
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 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.
We propose a ground-state ansatz for the Ohmic spin-boson model that improves upon the variational treatment of Silbey and Harris for biased systems in the scaling limit. In particular, it correctly captures the smooth crossover behavior expected for the ground-state magnetization when moving between the delocalized and localized regimes of the model, a feature that the variational treatment is unable to properly reproduce, while it also provides a lower ground-state energy estimate in the crossover region. We further demonstrate the validity of our intuitive ground-state by showing that it leads to predictions in excellent agreement with those derived from a nonperturbative Bethe-ansatz technique. Finally, recasting our ansatz in the form of a generalized polaron transformation, we are able to explore the dissipative two-state dynamics beyond weak system-environment coupling within an efficient time-local master equation formalism.
The spin-boson (SB) model is a standard prototype for quantum dissipation, which we generalize in this work, to explore the dissipative effects on a one-dimensional spin-orbit (SO) coupled particle in the presence of a sub-ohmic bath. We analyze this model by extending the well-known variational polaron approach, revealing a localization transition accompanied by an intriguing change in the spectrum, for which the doubly degenerate minima evolves to a single minimum at zero momentum as the system-bath coupling increases. For translational invariant system with conserved momentum, a continuous magnetization transition occurs, whereas the ground state changes discontinuously. We further investigate the transition of the ground state in the presence of harmonic confinement, which effectively models a quantum dot-like nanostructure under the influence of the environment. In both the scenarios, the entanglement entropy of the spin-sector can serve as a marker for these transitions. Interestingly, for the trapped system, a cat-like superposition state corresponds to maximum entanglement entropy below the transition, highlighting the relevance of the present model for studying the effect of decoherence on intra-particle entanglement in the context of quantum information processing.
We investigate the effects of initial states on the dynamics of a quantum spin-boson model using an efficient variational polaron master equation approach. The transformation of a factorized system-environment initial state into the variational polaron frame introduces inhomogeneous terms in the master equation, essential for accurately describing non-Markovian dynamics beyond the weak coupling regime. To efficiently account for these effects, we formulate the dynamics in an extended auxiliary Liouville space, converting the resulting integro-differential equations into coupled ordinary differential equations. We further introduce a simplified variational polaron technique for computing the time evolution of expectation values of general observables, including those that do not commute with the polaron transformation, overcoming a major limitation of previous approaches and enabling the unified treatment of coherence and population dynamics. The effects of transformed initial states generated from arbitrary factorized initial states are systematically characterized through expectation values and trace-distance analysis, providing insight into the associated bath-relaxation dynamics over a broad range of tunneling amplitudes, temperatures, and coupling strengths. Comparison with the weak-coupling master equation reveals a delay phenomenon associated with multi-bath-quanta processes, pronounced at intermediate coupling strengths and tunneling amplitudes. We further show that coherent superposition initial states induce stronger nonequilibrium effects than localized states, with inhomogeneous terms playing a crucial role in the short-time dynamics. The proposed formalism applies to both undriven and resonantly driven spin-boson models, providing a computationally efficient framework for open quantum-system dynamics with arbitrary factorized initial states and general physical observables.