Nuclear spin pumping and electron spin susceptibilities

In this work we present a new formalism to evaluate the nuclear spin dynamics driven by hyperfine interaction with nonequilibrium electron spins. To describe the dynamics up to second order in the hyperfine coupling it suffices to evaluate the susceptibility and fluctuations of the electron spin. Our approach does not rely on a separation of electronic energy scales or the specific choice of electronic basis states, thereby overcoming practical problems which may arise in certain limits when using a more traditional formalism based on rate equations.

Figure 1

Energy diagram for a trivial single-quantum dot setup in which nuclear spin pumping is to be expected. The dot is coupled to two leads with different chemical potentials. One single-electron level lies within the bias window, which is split into two sublevels by the Zeeman energy (we assumed a negative $g$ factor). The tunnel rate from the left lead into the dot, ${\Gamma }_{\text{in}}$ is large and equal for both spin directions. The outward tunnel rates ${\Gamma }_{\uparrow ,\downarrow }$ are different for spin up and down.

Figure 2

Energy diagram of a more complicated setup. Two quantum dots are coupled to each other and to two leads with different chemical potentials. The two dots have different effective $g$ factors, so that the application of an external magnetic field yields different Zeeman energies in the dots. The double dot is tuned to the $(0,0)\to (1,0)\to (0,1)\to (0,0)$ transport regime. The outward tunneling rates $\Gamma$ are equal and much smaller than the inward rate ${\Gamma }_{\text{in}}$.