Longitudinal relaxation is the process by which an excited spin ensemble decays into its thermal equilibrium with the environment. In solid-state spin systems, relaxation into the phonon bath usually dominates over the coupling to the electromagnetic vacuum. In the quantum limit, the spin lifetime is determined by phononic vacuum fluctuations . However, this limit was not observed in previous studies due to thermal phonon contributions or phonon-bottleneck processes. Here we use a dispersive detection scheme based on cavity quantum electrodynamics to observe this quantum limit of spin relaxation of the negatively charged nitrogen vacancy (NV) centre in diamond. Diamond possesses high thermal conductivity even at low temperatures , which eliminates phonon-bottleneck processes. We observe exceptionally long longitudinal relaxation times T of up to 8 h. To understand the fundamental mechanism of spin-phonon coupling in this system we develop a theoretical model and calculate the relaxation time ab initio. The calculations confirm that the low phononic density of states at the NV transition frequency enables the spin polarization to survive over macroscopic timescales.

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We show that the generalized or modified WKB method can be readily applied to a mixture of long-and short-range potentials. The phase shifts of such a mixed potential are calculated to order R 2 by using the solutions of the long-range potential as the bases of the approximation and treating the differences between the long-range and mixed potential as a WKB perturbation. We demonstrate the method by a simple example using a Coulomb and inverse-square potential. Comparison of the phase shifts with the exact results shows excellent agreement. The phase shifts can then be separated into contributions due to the long-range and shortrange potentials plus a contribution due to the mixture or coupling of the potentials. For small angular momenta, the contribution due to coupling is of the same order of magnitude as the Coulomb contribution. but rapidly decreases with higher angular momentum.

Abstract. In this note we extend the WKB inverse problem which involved a short-ranged potential to include a combined long and short-ranged potential.The inversion problem as presented by Vollmer [1] and Buck [-2] is an extremely interesting and practical problem. If one has knowledge of the phase shifts produced, one can use the inversion problem to find the potential involved. However, as we shall see later, because the WKB method is used, one is restricted to molecular scattering (i.e. a short ranged potential only). From the nature of the problem associated with the WKB method, one cannot expect the method to be applicable to the Coulomb Potential. Here, we will extend the above approach to include the longranged Coulomb Potential by use of the generalized WKB method [-3, 4]. The reason for the limitation which occurs in the ordinary WKB method has been pointed out in both Ref. 1 and 2. The ordinary WKB phase shift iswhere ,~2 =/(/+ 1), k2=2t~E/h 2. ~l(E'b)-(sZ-b2)~lnInNow, in order to make use the inversion formulas, which are, s and from this known relationship one sees that V(r) = E(1 -s2/r2).Eq. (7)