Quantum description of nuclear spin cooling in a quantum dot

We study theoretically the cooling of an ensemble of nuclear spins coupled to the spin of a localized electron in a quantum dot. We obtain a master equation for the state of the nuclear spins interacting with a sequence of polarized electrons that allows us to study quantitatively the cooling process including the effect of nuclear spin coherences, which can lead to “dark states” of the nuclear system in which further cooling is inhibited. We show that the inhomogeneous Knight field mitigates this effect strongly and that the remaining dark-state limitations can be overcome by very few shifts of the electron wave function, allowing for cooling far beyond the dark-state limit. Numerical integration of the master equation indicates that polarizations larger than 90% can be achieved within a millisecond time scale.

Figure 1

(Color online) Exact polarization dynamics. Left: Homogeneous case, $g_j=1∕\sqrt{N}$. Right: In the inhomogeneous case, $g_j\propto \exp (-{(j-N∕2-1∕4)}^2∕w^2)$. The term $1∕4$ is added to account for asymmetry between electron wave function and the lattice and avoid symmetry effects for this small scale system.

Figure 2

(Color online) Comparison of different approximation schemes for the homogeneous situation with $N=100$ (left) and the case of Gaussian couplings (as in Fig. 1) and $N=10$ nuclear spins (right).

Figure 3

(Color online) The polarization dynamics for $N=1000$ spins coupled with a 2D Gaussian wave function, which is shifted from the origin by $1∕3$ in the $x$ and $y$ directions.

Figure 4

(Color online) Polarization dynamics in the enhanced cooling protocol for $N=196$ (upper plots) and $N=1000$ (lower plot). In the upper plots approximation schemes (ii) (left) and (v) (right) have been invoked; the lower plot is based on the bosonic model and the partial rotating-wave approximation (see text). In all plots the different lines are representing cooling procedures with different numbers of mode changes. In the upper plots the randomly chosen Gaussian modes with width $w=N∕4$ are defined by the centers $\{(1∕3,1∕3),(1.35,-0.81),(0.32,-0.04),(1.17,0.79),(-0.13,-1.44),(0.96,-0.17),(0.35,0.88),(1.27,0.71)\}$. In the lower plot only two modes with centers $\{(1∕3,1∕3),(-3.15,-1.5)\}$ have been iterated.