Universal scaling of hyperfine-induced electron spin echo decay

The decoherence of a localized electron spin in a lattice of nuclear spins is an important problem for potential solid-state implementations of a quantum computer. We demonstrate that even at high fields, virtual electron spin-flip processes due solely to the hyperfine interaction can lead to complex nuclear spin dynamics. These dynamics, in turn, can lead to single electron spin phase fluctuation and decoherence. We show here that remarkably, a spin echo pulse sequence can almost completely reverse these nuclear dynamics except for a small visibility loss, thereby suppressing contribution of the hyperfine interaction to $T_2$ processes. For small systems, we present numerical evidence which demonstrates a universal scaling of the magnitude of visibility loss that depends only on the inhomogeneous linewidth of the system and the magnetic field.

Figure 1

The expectation value of $P_{{\mathbf{z}}_0}={\mathcal{I}}_S\otimes \mid {\mathbf{z}}_0⟩⟨{\mathbf{z}}_0\mid$ as a function of $t$ at $B=0.030,0.121,0.482,1.93,\text{and}7.72\mathrm{T}$ for the initial state $\mid {\psi }_0⟩$ [see Eq. (30)]. In this example $N=9$ and ${\Delta }_c=0.482\mathrm{T}$. As the external field increases, nuclear dynamics are slowly “frozen out.” However, nuclear dynamics clearly persist well above the critical field.

Figure 2

The expectation value of the magnitude of the in-plane magnetization $\mid ⟨S_{+}\left(t\right)⟩\mid$ as a function of $t$ at $B=0.030,0.121,0.482,1.93,7.72\mathrm{T}$ for the same system as in Fig. 1, i.e., the initial state is again the pure state $\mid {\psi }_0⟩$. As the nuclear dynamics are slowly “frozen out,” there is a corresponding decrease in the magnitude of in-plane magnetization decay.

Figure 3

The expectation value of the magnitude of the in-plane magnetization $\mid ⟨S_{+}\left(t\right)⟩\mid$ as a function of $t$ at $B=0.030$, 0.121, 0.482, 1.93, and $7.72\mathrm{T}$ for an initial density matrix $\rho \left(0\right)$ corresponding to a completely mixed nuclear state [see Eq. (31)]. Inhomogeneous broadening causes the in-plane magnetization to decay on approximately the same time scale $\left(T_2^{*}\right)$ independent of the external field.

Figure 4

The expectation value of the magnitude of the spin echo envelope $\mid ⟨{\tilde{S}}_{+}(\tau ;\tau )⟩\mid$ as a function of $\tau$ at $B=0.030$, 0.121, 0.482, 1.93, and $7.72\mathrm{T}$. The initial nuclear state is again the mixed state specified by $\rho \left(0\right)$ [see Eq. (31)]. Above the critical field ${\Delta }_c$, the spin echo experiment removes nearly all decay of the in-plane magnetization, even for systems displaying substantial nuclear dynamics (i.e., $B=1.93\mathrm{T}$). Note that this effect persists even at long times.

Figure 5

The matrix representation of the operator ${\tilde{S}}_{+}({\tau }^{\prime };\tau )$ at $B={\Delta }_c=0.482\mathrm{T}$ over the course of a spin echo experiment where a $\pi$ pulse is applied at $\tau =100000\mathrm{ns}$. The operators are shown in the $z$-basis representation of the nuclear states; the intensity of each point on the plot represents the amplitude of the corresponding matrix element between nuclear states. For clarity, we consider only the primary contributing block of the operator (i.e., the block connecting electron spin down to electron spin up states) as the other blocks are much smaller in magnitude. At short times, nuclear dynamics are negligible. At longer times, nuclear dynamics are substantial, leading to potential decay of $S_{+}$. However, the final panel shows that the spin echo experiment nearly reverses all nuclear dynamics, leading to a recovery of in-plane magnetization.

Figure 6

The universal scaling of visibility loss $v$ [see Eq. (33)] versus external field $B∕{\Delta }_c$ evaluated for systems of $N=5$, $N=9$, $N=11$, $N=13$ with randomly generated sets of hyperfine coupling contstants. Five systems were simulated for the system sizes $N=5$, $N=9$, and $N=11$. Only one system was simulated for $N=13$ because the large size of the system required substantial computation time (approximately five days on a multiple-node workstation). Above the critical field $B>{\Delta }_c$, the visibility loss scales as ${({\Delta }_c∕B)}^2$, independent of the selection of the hyperfine constant values and the size of the system $N$.