Central spin problem in a quantum dot:

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Introduction:

There is a world-wide effort underway to control individual quantum mechanical degrees of freedom experimentally. One of the best-studied model systems in this quest is a single electron spin, confined to a quantum dot. Such single spins have been manipulated and measured in various conditions. One of the most important properties to understand about such a quantum dot spin is its decoherence: how does the spin relax its orientation by coupling to other degrees of freedom. It is known that at low temperatures in GaAs quantum dots, the dominant mechanism of spin relaxation is via hyperfine coupling of the electron spin to nuclear moments,

To a good approximation, the electronic spin is coupled to a large number of nuclei (e.g. >10⁴) with a non-uniform profile of hyperfine interactions: largest for those nuclei in the center of the dot, and smallest for those further away. Since the nuclear magnetic moment is very small, each individual coupling is weak, but since there are many, the net effect is not. However, dipolar coupling between two nuclei is extremely small and at a first cut negligible. Thus the problem is that of a "central" electron spin coupled to many nuclear spins, which are not coupled to one another. The nuclei are unpolarized unless specifically prepared otherwise, due to their small Zeeman energy.

To a first approximation, because the electron spin is coupled to many nuclei with random orientations, it feels a hyperfine exchange field which is "large" and of order N^1/2, while the nuclear feel a small order 1 field due only to the single electron. Thus on short times the electron precesses while the nuclei remain static. However, if one observes the system for times N^1/2 times long than the electron precession time, the nuclei do evolve and so does the electron precession axis. On this longer timescale, the electron spin does relax due to the wandering of the precession axis. It has been noticed numerically by various authors that the relaxation of the electronic spin is non-exponential . In fact, if the electron spin is initially polarized in the +z direction, after a long time t, the probability that the electron remains in the +z state decays like 1/(ln t) ^a, where a is an exponent observed numerically to be about 1.

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Our work:

We developed a theory of this long-time evolution. First, we showed that it is indeed correct to treat the problem semi-classically, as implicit in the above discussion. This can be proved by field-theory methods. Second, we showed that the long-time logarithmic relaxation has a simple physical origin. It reflects the slow transfer of spin angular momentum from the strongly coupled nuclei at the center of the dot to the weakly coupled ones at its fringe, which occurs indirectly via the electron spin. We showed that the exponent "a" above depends upon the geometry of the dot, through the distribution of the hyperfine couplings. Some explicit formulas were derived to obtain results for specific and general situations.

We believe these results resolve some questions that have been unanswered in the literature for several years, and provide some simple means of addressing more complex and experimentally interesting situations that may be useful in the future.