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Europhysics Letters (EPL)

Quantum Tunnelling in the Spin-Boson Model

M. Fannes, Bruno Nachtergaele, André F. Verbeure

Published Nov 1, 1987• 17 citations
Publisher

Abstract

For the quantum particle in a double-well potential, approximated by a two-level spin-boson model, we prove the unicity of the equilibrium state at all temperatures, i.e. the absence of localization. This can be understood on the basis of an exact computation of the transition probability between the thermal wave functions of the two minima.

DOI
10.1209/0295-5075/4/9/002

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A variationally optimised basis allows an accurate description of the quantum \nbehaviour of ultra-cold atoms, even in the strongly correlated regime. A rescaling \nscheme corrects discrepancies caused by using a reduced Hilbert space. This \napproach also allows the modelling of experimentally realizable double-well potentials, \nwhich still reveals the maximally-entangled states seen in fixed basis models. \nTime dynamics of these double-well systems show macroscopic tunnelling between \nwells for bosons with a sufficient interaction strength. \nThe many-body problem of interacting bosons in the highly-correlated regime \nis difficult. The number of basis states needed to describe this quantum system \naccurately quickly grows beyond computational reach. Rescaling the interaction \nstrength proves a simple and effective method of calculating exact eigenvalues in a \nreduced Hilbert space. \nBosonic systems in the double-well potential are investigated next. First, how \ndifferent eigen-states depend on the interaction strength is examined. The variationally \noptimised method has advantages over a standard fixed basis method with \nthe ability to model experimentally viable systems and explore more stronglycorrelated \nregimes. Secondly, tunnelling dynamics in the double well are studied, \nspecifically for a system where all particles initially occupy a single well. Oscillations \ncorresponding to collective tunnelling between wells are found in regimes \nwhere there are zero interactions or bosons lie in a maximally-entangled state. \nWhat governs the dynamics outside these two regimes is also considered.

•Jan 1, 2010•0 citations
For the quantum particle in a double-well potential, approximated by a two-level spin-boson model, we prove the unicity of the equilibrium state at all temperatures, i.e. the absence of localization. This can be understood on the basis of an exact computation of the transition probability between the thermal wave functions of the two minima.
We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well). We start from the full many-body Schr{ö}dinger Hamiltonian in a large-filling situation where the on-site interaction and kinetic energies are comparable. When tunneling is negligible against interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly suppressed. The modes in which the particles condense are minimizers of nonlinear Schr{ö}dinger-type functionals.
We study the transfer of heat in the nonequilibrium spin-boson model with an Ohmic dissipation. In the nonadiabatic limit we derive a formula for the thermal conductance based on a rate equation formalism at the level of the noninteracting blip approximation, valid for temperatures T>T(K), with T(K) as the Kondo temperature. We evaluate this expression analytically assuming either weak or strong couplings, and demonstrate that our results agree with exact relations. Far-from-equilibrium situations are further examined, showing a close correspondence to the linear response limit.
We present an alternative way to compute and interpret quantum tunneling in a one-dimensional double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of ${10}^{\ensuremath{-}5}$. In contrast to the classical potential, the quantum potential has a triple well if the classical wells are deep enough. Its minima are located at the positions of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instantons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semiclassical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such a case is illustrated by an example.
A variationally optimised basis allows an accurate description of the quantum \nbehaviour of ultra-cold atoms, even in the strongly correlated regime. A rescaling \nscheme corrects discrepancies caused by using a reduced Hilbert space. This \napproach also allows the modelling of experimentally realizable double-well potentials, \nwhich still reveals the maximally-entangled states seen in fixed basis models. \nTime dynamics of these double-well systems show macroscopic tunnelling between \nwells for bosons with a sufficient interaction strength. \nThe many-body problem of interacting bosons in the highly-correlated regime \nis difficult. The number of basis states needed to describe this quantum system \naccurately quickly grows beyond computational reach. Rescaling the interaction \nstrength proves a simple and effective method of calculating exact eigenvalues in a \nreduced Hilbert space. \nBosonic systems in the double-well potential are investigated next. First, how \ndifferent eigen-states depend on the interaction strength is examined. The variationally \noptimised method has advantages over a standard fixed basis method with \nthe ability to model experimentally viable systems and explore more stronglycorrelated \nregimes. Secondly, tunnelling dynamics in the double well are studied, \nspecifically for a system where all particles initially occupy a single well. Oscillations \ncorresponding to collective tunnelling between wells are found in regimes \nwhere there are zero interactions or bosons lie in a maximally-entangled state. \nWhat governs the dynamics outside these two regimes is also considered.