Hyperfine interaction in a quantum dot: Non-Markovian electron spin dynamics

Abstract

We have performed a systematic calculation for the non-Markovian dynamics of a localized electron spin interacting with an environment of nuclear spins via the Fermi contact hyperfine interaction. This work applies to an electron in the s-type orbital ground state of a quantum dot or bound to a donor impurity, and is valid for arbitrary polarization p of the nuclear spin system, and arbitrary nuclear spin I in high magnetic fields. In the limit of p = 1 and I = 1 2 , the Born approximation of our perturbative theory recovers the exact electron spin dynamics. We have found the form of the generalized master equation (GME) for the longitudinal and transverse components of the electron spin to all orders in the electron spin-nuclear spin flip-flop terms. Our perturbative expansion is regular, unlike standard time-dependent perturbation theory, and can be carried-out to higher orders. We show this explicitly with a fourth-order calculation of the longitudinal spin dynamics. In zero magnetic field, the fraction of the electron spin that decays is bounded by the smallness parameter δ = 1/p 2 N , where N is the number of nuclear spins within the extent of the electron wave function. However, the form of the decay can only be determined in a high magnetic field, much larger than the maximum Overhauser field. In general the electron spin shows rich dynamics, described by a sum of contributions with non-exponential decay, exponential decay, and undamped oscillations. There is an abrupt crossover in the electron spin asymptotics at a critical dimensionality and shape of the electron envelope wave function. We propose a scheme that could be used to measure the non-Markovian dynamics using a standard spin-echo technique, even when the fraction that undergoes non-Markovian dynamics is small.

Single molecule charge transport: from a quantum mechanical to a classical description

“…72,73,79 There are also numerous non-Markovian generalizations of the master equation. [80][81][82][83] In the above derivation it was assumed that the system is weakly coupled to the environment. Indeed, for practical purposes this is the most interesting case: if the coupling of the system to the environment is strong, the system behaves classically and can be described with classical equations of motion, while if the coupling is absent altogether, eqn (1) is valid.…”

The Decoherence of the Electron Spin and Meta-Stability of 13C Nuclear Spins in Diamond

“…Consequently, it has been reported that (inhomogeneous) electron spin-dephasing times of T * 2 = 0.5 − 6 μs can be achieved in Diamond, at 300 K-without the need for cryogenic cooling [1,4,18], which makes Diamond an attractive candidate for Quantum Information Processing. Following [12], however, we argue that whilst it is clear that a successful qubit can be constructed from the spin polarization the 3 A 2 state (as has been successfully demonstrated in [1,4,9,16,17]) it is not at all clear why this state should be so highly stable from a calculational standpoint, with such a relatively long decoherence time, given the current understanding of the mechanism of spectral diffusion [19,20].…”

“…However, it is our new argument that this qubit does not simply decohere to become entangled with the bath, but with a bath which is also decohering to become entangled with itself and, therefore, that this slowly changes the definition of the SO(3) rotational symmetry of the electron spin to make it evolve into an oblate spheroid (relative to its initial spin projection). Whilst previous calculational schemes have treated the nuclear spin bath dynamics [19,20], these schemes have been restricted to short-time regimes and localised electron states, which we are now able to go beyond in our new approach. The limitation we have resolved is that there is generally a two-scale process involved in bath dynamics: The range of the nuclear dipole interaction strength and the cluster size of the frozen cores of nuclear spins that form through the flip-flop processes [19], and it was not previously understood to be possible to define a expansion program that is simultaneously valid at two very different scales.…”

“…However, it is possible to treat the intersystem mixing model in (3) via the General Master Equation prescription in [20] by making the simple, yet rigorous, step of changing time from being a linear to a nonlinear variable t → t ν .…”

“…The basic differential conductance transport calculation that is important for building a Quantum Computing device using these NV centers Diamond (and manipulating these centers via electrical currents) is the decoherence of the single electron spin associated with the NV center [20]. As shown above, at least in principle, this NV center can then be manipulated as a qubit for Quantum Information processing [1][2][3][4].…”