Local-Field Corrections as a Regularization Method for the Spin-Boson Model

The decoherence rate of a ‘central spin’ in a bosonic bath of magnetic fluctuations is computed using the spin-boson model. The magnetic fluctuations are treated in a fully quantum mechanical way by using the macroscopic quantum electrodynamics formalism and are expressed in terms of the classical electromagnetic Green’s function of the system. The resulting frequency integral formally diverges but it can be regularized by applying real-cavity, local-field corrections to the location of the ‘central spin’. This results in a cut-off function in terms of the magnetic permeability of the background material that leads to convergence at both high and low frequencies. This cut-off function appears naturally from the formalism and thus removes the need to rely on ad-hoc arguments to justify the form of the cut-off function. Furthermore, the magnetic permeability and the nature of interactions in quantum electrodynamics illuminate the connection between the two main models of ‘central spin’ decoherence, the spin-boson model and the spin-bath model, demonstrating how the two very different models are able to correctly model the same underlying physics.

The coefficients G G G λ (r, r , ω) are given by G G G e (r, r , ω) = i ω 2 c 2 h πε 0 Imε(r , ω)G G G(r, r , ω) , with the backward arrow referring to the fact that the operator acts on the right hand variable (here r ). The function, G G G(r, r , ω), is the electromagnetic Green's function, which is the solution to the Helmholtz equation for a point source Here, ε ε ε(r, ω) and µ µ µ(r, ω) are the electric permittivity and magnetic permeability tensors respectively.
The field expectation value can be computed as follows where we have assumed that the bosonic operators evolve freely and noted that the expectation value of the other bilinear combinations of the bosonic operators vanish upon taking expectation values. Using the remaining non-vanishing thermal expectation values where n th (ω) is the thermal photon number at temperature, T , (k B is Boltzmann's constant) and the integral relation for the Green's function and hence The time integrals give Thus Noting that one arrives at the expression in Eq. 22 in the main text.

Free Space
The free space Greens function reads We are interested in theẑẑ component of the curl of the Green's function Evaluating this expression leads tô Hence which is Eq. 24 in the main text.

Homogeneous Media
The homogeneous media Greens function reads Unfortunately, theẑẑ component of the curl of the Green's function for coincident limits diverges unless the imaginary part of the refractive index, n(ω), vanishes where upon it equalŝ

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One can regularize the expression for the coherence by performing local field corrections on the spin. We model the embedding of the spin into the homogeneous medium by applying a real-cavity model, where we assume the spin lies at the centre of a spherical cavity. The Green's function for the field inside a spherical cavity reads where R T E and R T M are the reflection coefficients for the T E and T M polarized waves respectively and the M M M mln (k, r) and N N N mln (k, r) dyads are given by where j l (x) are spherical Bessel functions of the first kind and P m l (x) are the associated Legendre polynomials. One can compute the curl of the individual dyads using then by taking r, r → 0 one finds that the only contribution is from l = 1 and m = 0. Hence, the T M mode vanishes and only the T E mode contributes. Thus, the Green's function reduces tô The reflection of the T E modes at the cavity interface can be described in terms of the Mie scattering coefficient where z 0 = ωR c /c and z = n(ω)ωR c /c, with R c the radius of the cavity, and j 1 (z) and h 1 (z) are, respectively, spherical Bessel and Hankel functions of the first kind for l = 1,