Ground state and dynamics of the biased dissipative two-state system: Beyond variational polaron theory

Ahsan Nazir, Dara P. S. McCutcheon, and Alex W. Chin

Phys. Rev. B 85, 224301 – Published 11 June 2012

Abstract

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.

Received 2 February 2012

DOI:https://doi.org/10.1103/PhysRevB.85.224301

Authors & Affiliations

Ahsan Nazir^1,*, Dara P. S. McCutcheon^1,2,3, and Alex W. Chin⁴

¹Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
²Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
³London Centre for Nanotechnology, University College London, United Kingdom
⁴Theory of Condensed Matter Group, University of Cambridge, J J Thomson Avenue, Cambridge, CB3 0HE, United Kingdom

^*a.nazir@imperial.ac.uk

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Vol. 85, Iss. 22 — 1 June 2012

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Images

Figure 1
(a) Magnetization $M$ as a function of system-bath coupling $α$ for the Ohmic spin-boson ground state. Results from the Silbey-Harris variational treatment are shown as open circles, while the solid curves are plotted using the ansatz presented in this work [see Eqs. (12) and (14)]. Part (b) shows the difference in the ground-state energies predicted by the Silbey-Harris theory and the present ansatz (both of which are negative). In both plots $Δ = 10^{- 2} ω_{c}$ , with the arrows indicating values of $ε = 0.5 Δ$ (blue), $ε = 0.1 Δ$ (red), $ε = 10^{- 2} Δ$ (green), $ε = 10^{- 4} Δ$ (orange), and $ε = 10^{- 8} Δ$ (purple) in decreasing order.Reuse & Permissions
Figure 2
Magnetization $M$ as a function of system-bath coupling $α$ calculated from the present ansatz (solid curves) and calculated using the exact scaling-limit expressions obtained via the Bethe ansatz (crosses), presented in Ref. 61. The arrows indicate decreasing values of $ε / Δ$ , as in Fig. 1. The two plots correspond to different values of $Δ / ω_{c}$ , as indicated.Reuse & Permissions
Figure 3
Ground-state spin coherence $〈 σ_{x} 〉$ as a function of system-bath coupling $α$ calculated from the present ansatz (solid curves) and calculated using the exact scaling-limit expressions obtained via the Bethe ansatz (crosses), presented in Ref. 61. The arrows indicate the bias values $ε = 0.5 Δ$ (blue), $ε = 0.1 Δ$ (red), and $ε = 10^{- 2} Δ$ (green), in decreasing order. The two plots correspond to different values of $Δ / ω_{c}$ , as indicated.Reuse & Permissions
Figure 4
Ground-state energy $λ$ as a function of system-bath coupling $α$ calculated from the present ansatz (solid curves) and calculated using expressions obtained via the Bethe ansatz (crosses), presented in Ref. 61. The arrows indicate the bias values $ε = 0.5 Δ$ (blue), $ε = 0.1 Δ$ (red), and $ε = 10^{- 2} Δ$ (green), in decreasing order. The two plots correspond to different values of $Δ / ω_{c}$ , as indicated.Reuse & Permissions
Figure 5
Magnetization dynamics ${〈 σ_{z} 〉}_{t}$ computed within the secular approximation as a function of (a) $Δ_{r} t$ for the present theory and (b) $Δ_{SH} t$ for Silbey-Harris theory. Solid curves: the arrows indicate the coupling strengths $α = 0.1$ (blue), $α = 0.25$ (red), $α = 0.4$ (green), $α = 0.5$ (orange), and $α = 0.6$ (purple, shown for our ansatz only) in increasing order. The dotted curves correspond to the exact (scaling-limit) solution at $α = 0.5$ , extracted from a mapping to the Toulouse Hamiltonian and computed from the formula given in Ref. 2. The dashed curves show the same exact solution, but now plotted in units of $T_{k} t$ , where $T_{k} = Δ^{2} / ω_{c}$ is the Kondo scale for the Toulouse problem.[2] Other parameters: $Δ = 10^{- 2} ω_{c}$ , $ε = 10^{- 2} Δ$ , and $T = 0$ .Reuse & Permissions

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