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Stabilizing spin coherence through environmental entanglement in strongly dissipative quantum systems

Soumya Bera, Serge Florens, Harold U. Baranger, Nicolas Roch, Ahsan Nazir, and Alex W. Chin
Phys. Rev. B 89, 121108(R) – Published 18 March 2014
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Abstract

The key feature of a quantum spin coupled to a harmonic bath—a model dissipative quantum system—is competition between oscillator potential energy and spin tunneling rate. We show that these opposing tendencies cause environmental entanglement through superpositions of adiabatic and antiadiabatic oscillator states, which then stabilizes the spin coherence against strong dissipation. This insight motivates a fast-converging variational coherent-state expansion for the many-body ground state of the spin-boson model, which we substantiate via numerical quantum tomography.

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  • Received 20 July 2013
  • Revised 26 February 2014

DOI:https://doi.org/10.1103/PhysRevB.89.121108

©2014 American Physical Society

Authors & Affiliations

Soumya Bera1, Serge Florens1, Harold U. Baranger2, Nicolas Roch3,1, Ahsan Nazir4,5, and Alex W. Chin6

  • 1Institut Néel, CNRS and Université Grenoble Alpes, F-38042 Grenoble, France
  • 2Department of Physics, Duke University, Durham, North Carolina 27708, USA
  • 3Laboratoire Pierre Aigrain, École Normale Supérieure, CNRS (UMR 8551), Université Pierre et Marie Curie, Université Denis Diderot, 24 rue Lhomond, 75231 Paris Cedex 05, France
  • 4Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
  • 5Photon Science Institute, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
  • 6Theory of Condensed Matter Group, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom

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Vol. 89, Iss. 12 — 15 March 2014

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  • Figure 1
    Figure 1

    Origins of polaron and antipolaron displacements in environmental wave functions. Black dashed lines are the spin-dependent potential energies of a single harmonic oscillator in the absence of spin tunneling (Δ=0), while blue (red) curves sketch the single-mode wave functions of the oscillator (in real space X) on the 〈σz〉=1(−1) potential surfaces. (a) High-frequency modes (ω≫Δ) tend to rest in the bottom of their (spin-dependent) potential due to the large energy of other displacements. This leads to the formation of adiabatic polarons. (b) and (c) Low-frequency modes (ω≪Δ) have shallow potentials with well-separated minima. The interminima wave-function overlap is reduced while the potential cost of other displacements becomes less prohibitive. The oscillators can either climb up the potential landscape, gaining some tunneling energy [panel (b)], or become superposed with oppositely displaced states—“antipolarons”—so that tunneling and potential energy are both optimal [panel (c)]. Superposition of such polaron and antipolaron states generates multimode entanglement in the complete environmental wave function.

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  • Figure 2
    Figure 2

    Upper panel: Ground-state coherence −〈σx〉 as a function of dissipation strength α computed with the NRG (circles) for Δ/ωc=0.01 and compared to the results of the expansion Eq. (3) with N=1,2,3,4 coherent states. The inclusion of an antipolaronic component to the wave function (N≥2) has a drastic effect on the spin coherence. Lower panel: Displacements determined at order N=1 (dashed line) and N=4 (solid lines), showing the emergence of three antipolaron states at low energies, which merge smoothly onto the polaron state at high energy (adiabatic regime) [35]. Parameters are α=0.5 and Δ/ωc=0.01.

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  • Figure 3
    Figure 3

    Left: Spin-diagonal Wigner distribution, as obtained from the NRG, computed for a high-energy mode. Polaron formation leads to a classical shift, X=fkcl, that is fully captured by the Silbey-Harris state (single coherent state). Right: The spin-off-diagonal Wigner function is badly approximated by the single polaron N=1 state, as antipolaron contributions dominate in this quantity. A single antipolaron (N=2) quickly restores the correct magnitude, and reveals a pair of shifted Gaussians, as expected (see text). (Parameters here are α=0.8 and Δ/ωc=0.01.)

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