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Journal of Physics A Mathematical and General

Tunnelling in the equilibrium state of a spin-boson model

M. Fannes, Bruno Nachtergaele, André F. Verbeure

Published Apr 21, 1988• 14 citations
Publisher

Abstract

For a two-level spin-boson system describing a quantum particle in a double-well potential coupled to a quantised radiation field, the authors prove the unicity of the equilibrium state at all positive temperatures. They are also able to compute rigorously the transition probability between the thermal wavefunctions of the two wells.
DOI
10.1088/0305-4470/21/8/012

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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.
Standard minimal two-state Hamiltonian (spin-boson model) for a particle in a symmetric double-well potential, interacting with a bath, is complemented by a third particle-excited state. This state is dynamically coupled (via coupling to the bath) with the lowest two states. In addition to the standard over-barrier thermally-activated transfer 1 ↔ 3 ↔ 2 (which might be easily suppressed by choosing a sufficiently low T), the third state is shown to cause excited-state-induced subbarrier tunnelling which persists even at T = 0. This new channel shows a tendency to coherent oscillations if standard (direct-tunnelling and over-barrier-transfer) channels get suppressed. These conclusions challenge the standard theory of the asymptotic symmetry breaking.
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.
The coherent nonlinear tunneling dynamics of a boson-fermion mixture in a double-well potential is studied in this paper. Four types of phase are found for the mixture. The first one is two species localizing in different potential wells. The second one is two species coexisting in the same well. The third one is two species equally populated in two wells. The fourth one is one species equally populated in two wells while the other species is in one well. The phase transitions among these four states have been investigated. The interspecies and intraspecies interactions as well as bosonic and fermionic numbers can dramatically affect these phase transitions.