Electron spin noise under the conditions of nuclei-induced frequency focusing

We study theoretically the electron spin noise in quantum dots under nonequilibrium conditions caused by the pumping by a train of circularly polarized optical pulses. In such a situation, the nuclear spins are known to adjust in such a way that the electron spin precession frequencies become multiples of the pump pulse repetition frequency. This so-called phase synchronization effect was uncovered in A. Greilich et al. [Science 317, 1896 (2007)] and termed nuclei-induced frequency focusing of electron spin coherence. Using the classical approach to the central spin model, we evaluate the nuclear spin distribution function and the electron spin noise spectrum. We show that the electron spin noise spectrum consists of sharp peaks corresponding to the phase synchronization conditions and directly reveal the distribution of the nuclear spins. We discuss the effects of nuclear spin relaxation after the pumping is over and analyze the corresponding evolution of nuclear spin distributions and electron spin noise spectra.

Figure 1

Distributions of Cartesian components of the Overhauser field for the steady state of a periodically pulsed system. The initial distributions after the sequence of pump pulses has finished are shown in red. The Overhauser field distributions after approximately $10000T^{*}$ are shown in blue. The Gaussian envelopes are depicted in black.

Figure 2

Central spin dynamics starting from the steady state of a pulsed system. The red line gives the envelope of the electron spin revival decay. The inset shows spin dynamics on the shorter timescale.

Figure 3

(a) Spin noise spectrum of the full numerical simulation. (b) The initial Overhauser field distribution taken from Fig. 1 at $t^{\prime }=1000T_{\mathrm{R}}\approx 10000T^{*}$.

Figure 4

The distribution of the total nuclear spin in $x$ direction $M_x$ corresponding to the Overhauser field shown in Fig. 1 is conserved.

Figure 5

Left side: normalized Overhauser field distribution. Right side: normalized spin noise spectra for (a) $t^{\prime }/T_{1,\mathrm{N}}=0.001$, (b) 0.01, and (c) 0.1. The spin noise spectra are shifted by the frequency of the external magnetic field for better comparability.