Spin Environments & the Suppression of Quantum Coherence

Abstract

We give details of the solvable limits for a “central spin” model, which we have previously used to analyze the behaviour of environmental spins on MQC in magnetic grains (see Prokof’ev & Stamp, J. Phys. CM 5, L667 (1993)). As discussed previously, both nuclear spins in the grain and nuclear and electronic spins outside the grain will almost invariably suppress MQC, by a variety of decoherence mechanisms. The central spin model is also useful in describing systems like Si:P, or quantum spin glasses.

We show how one may find analytic results for the behaviour of the central spin (which describes, e.g., the magnetic grain) in 3 limits. Each limit displays a different kind of decoherence effect. “Topological decoherence” arises solely from the transfer of the topological “Haldane” phase from the environment to the central spin. “Orthogonality blocking” arises from the incomplete overlap of initial and final environmental spin states. Finally “degeneracy blocking” suppresses coherence by removing the degeneracy between initial and final states for the combined system.

Based on these results, the prognosis for MQC in magnetic grains is rather gloomy — but we offer some methods for avoiding the decoherence in MQC experiments.

The problem we wish to address in this article can be described most simply as the problem of a “central spin” $\vec S$, interacting with a set $\{ {\vec \sigma _k}\}$ of surrounding spins. There is no particular restriction on the quantum number S of the central spin, although often we will assume that S ≫ 1. In the present article we will assume for simplicity that the $\{ {\vec \sigma _k}\}$ axe spin-1/2 objects; they may represent nuclear spins or electronic spins. There is no difficulty in generalizing the discussion to higher spin quantum numbers for the $\{ {\vec \sigma _k}\}$, except one of formal complexity. The $\{ {\vec \sigma _k}\}$ do not mutually interact.

The reasons for studying such a model are quite varied, and it is worth enumerating some of them, as follows

• (i) There are several important physical realizations of such a model. Quite apart from the problem of MQC (Macroscopic Quantum Coherence) for magnetic grains [1], of particular interest at this conference, similar models have been adopted to discuss Si:P near the metal-insulator transition [2], and for Kagomé spin systems [3]. The model can also be applied to the new magnetic cantilever microscope, and with some modification to discuss multiple quantum coherence in NMR. In all of these cases the dynamics of a central spin $\vec S$ is modified in a peculiar way by the presence of the environment $\{ {\vec \sigma _k}\}$, as discussed for magnetic grains in Ref.[4].

• (ii) As discussed elsewhere [5, 6], the effect of the $\{ {\vec \sigma _k}\}$ on the dynamics of $\vec S$ can be thought of as due to the coupling of $\vec S$ to an “unconventional environment”. The physics of these unconventional environments is interesting theoretically, because they involve peculiar “infrared” effects, often worse than those in the conventional “IR catastrophe” which arises in Bremsstrahlung, or in quantum impurity problems, or in Ohmic quantum dissipation. As a consequence one sees peculiar long-time dynamics [4] for the system coupled to them. We shall see this very clearly in the case of the central spin, where a combination of exact solutions, perturbation expressions, and computer simulations shows a quite remarkable zoology of long-time behaviour for $\vec S(t)$, often quite counterintuitive. The resulting correlation functions are very unconventional.

• (iii) The central spin model is actually integrable — one can write down Bethe ansatz equations for it [7]. To date we see no means of using this integrability in any useful way. However the techniques we have devised give a virtually complete description of the low-energy dynamics of $\vec S$. This makes our results interesting from a purely theoretical standpoint, as a means of understanding an important integrable system.

• (iv) As previously emphasized, our results in general show a complete suppression of coherent behaviour on the part of $\vec S$. This result has important consequences for the possibility of observing MQC in magnetic systems, or indeed elsewhere, as already emphasized [4, 5, 6]. However our techniques also suggest [4] possible ways of escaping this quandary — we concentrate on this point, and on possible ways one might reveal MQC in grains, at the end of the present article.

In this article we intend to try and explain, without too much in the way of formal calculations, the basic physics of the central spin model; and to illustrate some of the zoology of behaviour which $\vec S$ can display, depending on the nature of the spin environment and its coupling to $\vec S$, as well as external conditions (such as an applied field). Readers (particularly theorists) who wish to go to the gory calculations should go to the original papers [4, 5, 6] (particularly the most recent long paper [8]); the space we have here only allows an overview.

The plan of the article is as follows. In section I we describe the model, and how it applies to some of the systems mentioned above. The model involves various parameters, including the coupling energies between $\vec S$ and the $\{ {\vec \sigma _k}\}$, the spread in these energies, as well as the energy scales involved in the “local” Hamiltonian $H_k^o({\vec \sigma _k};{\vec H_o})$, in the absence of $\vec S$ but in the possible presence of a magnetic field ${\vec H_o}$. Then there are the energy scales involved in the central spin dynamics, these being Δ _o (S) (the tunneling splitting, in the case where we are interested in coherent motion of $\vec S$) and Ω _o (the “bounce frequency” for tunneling of $\vec S$, comparable to the frequency of small oscillations of $\vec S$). A qualitative description is given in section I of the effect of the $\{ {\vec \sigma _k}\}$ on the quantum dynamics of $\vec S$, and how this depends on these parameters.

In section II we describe the various cases of the model that can be treated exactly. These involve choosing particular values or limits of the above parameters. They are very illuminating because they bring out the different physical effects arising from the coupling of the $\{ {\vec \sigma _k}\}$ to $\{ {\vec \sigma _k}\}$. The 3 most important of these effects are the “topological decoherence” which arises through randomisation of the phase in the action, the “degeneracy blocking”, whereby coupling to the $\{ {\vec \sigma _k}\}$ lifts the degeneracy between the initial and final states of $\vec S$, and the “orthogonality blocking”, whereby incomplete overlap between initial and final environmental states suppresses coherence. These have been discussed previously [4], but here we shall see their interplay, as well as some other effects not dealt with before. We do not have space here to discuss the “generic case” away from these limits (see Ref.[8]).

Finally in section III we comment on possible strategies for evading the decoherence effects discussed here, as well as the implications of our results for experiments (particularly the recent MQC experiments on magnetic proteins [9]). Reader not interested in the details of section II can read I and III only.