Consistent perturbative treatment of the subohmic spin-boson model yielding arbitrarily smallT2/T1decoherence time ratios

Abstract

We present a perturbative treatment of the subohmic spin-boson model which remedies a crucial flaw in previous treatments. The problem is traced back to the incorrect application of a Markov type approximation to specific terms in the temporal evolution of the reduced density matrix. The modified solution is consistent both with numerical simulations and the exact solution obtained when the bath-coupling spin-space direction is parallel to the qubit energy-basis spin. We therefore demonstrate that the subohmic spin-boson model is capable of describing arbitrarily small ratios of the T2 and T1 decoherence times, associated to the decay of the off-diagonal and diagonal reduced density-matrix elements, respectively. An analytical formula for T2/T1 at the absolute zero of temperature is provided in the limit of a subohmic bath with vanishing spectral power law exponent. Small ratios closely mimic the experimental results for solid state (flux) qubits, which are subject predominantly to low-frequency electromagnetic noise, and we suggest a reanalysis of the corresponding experimental data in terms of a nonanalytic decay of off-diagonal coherence.Since the advent of quantum information technology, the problem of two-state systems aka qubits, immersed in environmental (bath) degrees of freedom has increasingly gained importance, cf., e.g., 1-3 . The spin-boson model, succinctly describing such a physical situation, namely an open two-state quantum system interacting with a bath, accounts for two-state energy relaxation and dephasing, and therefore has been studied extensively 4,5 . It was, for example, used to describe physical contexts as diverse as flux qubits implemented using superconducting quantum interference devices (SQUIDs) 6-10 , electron transfer in biomolecules 11 , and phonon coupling in atomic tunneling 12 . The spin-boson model introduces a continuum of independent simple harmonic oscillators with a given spectral weight distribution as the environment, assuming them to couple with the qubit linearly. The Hamiltonian of the entire (closed) system therefore reads (setting = 1)Here, − → σ = (σ x , σ y , σ z ) are the usual Pauli matrices andb † k ,b k are creation and annihilation operators of harmonic oscillators in the (infinitely extended) bath, labelled by the quantum number(s) k, which can stand, e.g., for the momentum of the bath excitations. The coupling direction − → n is parametrized by two angles θ,ϕ as − → n = (sin θ cos ϕ, sin θ sin ϕ, cos θ).The bath is conventionally characterized by the quantity J(ω) ≡ k λ 2 k δ(ω − ω k ), which determines the dynamics of the spin-boson model. This spectral density of the bath is effectively a density of states summation weighed by the coupling strength squared λ 2 k , hence the name. Usually it is assumed that J(ω) is of a power law form up to a cutoff, J(ω) = ηω, where ω c is the (physical) cutoff frequency of the bath, assumed to be much larger than the system frequency spacing ω s . The strength of coupling is parametrized by the dimensionless η. Baths h...

Exact Open Quantum System Dynamics Using the Hierarchy of Pure States (HOPS)

“…3) which is consistent with other results [37,38]. The usefulness of the consistent perturbative treatment proposed in [28] is confirmed by finding very good agreement with our exact numerical results using HOPS (see Fig. 5).…”

“…As expected very good agreement was found in the Ohmic case (s=1) for various bias values and temperatures reaching from T =0 to T =10∆. However in the sub-Ohmic case, for non-zero temperature and non-zero bias, deviations occur which are due to the failure of the master equation with time independent rates and which can be cured by introducing time dependent rates [28].…”

“…Otherwise L 0 is identical zero which suppresses the contribution of J(0). It has been pointed out [28] that these problems arise from taking the limit t → ∞ in (17). Keeping the actual Γ(t, ω) yields time dependent coefficients for the master equation, restoring a growing but finite value for the ω=0 contribution (see Fig.…”

“…In a more careful perturbative treatment (Ref. [28]) this contribution is replaced by the time dependent rate Re(Γ(t, ω = 0)). For all parameters very good agreement between HOPS and this extended master equation can be seen in Fig.…”

“…To benchmark HOPS we solve the spin-boson model. In the weak coupling limit and for a fast decaying BCF the dynamics gained from HOPS is compared with calcu-lations employing the quantum optical master equation and its correct extension to a sub-Ohmic environment (as explained in [28]). Furthermore, when the spin is influenced mainly by classical noise induced by a high temperature environment, HOPS is compared with HEOM utilizing a MT decomposition and eHEOM.…”