Simulating decoherence of two coupled spins using the generalized cluster correlation expansion

Abstract

We simulate the coherence of two coupled electron spins interacting with a bath of nuclei using the generalized cluster correlation expansion (gCCE) method. An exchange interaction between the electrons facilitates a family of entangling gates that can be spoiled by nuclear-induced dephasing. Consequently, we study the dephasing of the coherent two-electron system by characterizing the $T_2$ and $T_2^*$ of the two-electron reduced density matrix for various system parameters in the range mimicking magnetic molecules, including magnetic field strength and orientation, exchange interaction strength, distance between the two spins, minimum distance between electron and nuclei and between nuclei, and nuclei density. We find the optimal regime for each parameter in which the coherence time is maximized and provide a physical understanding of it.

Figures (18)

• Figure 1: A sketch of the model. The electron spins and the nuclear bath spins are represented by the red and black arrows, respectively.
• Figure 2: A sketch of the effective fields $\overrightarrow{\chi}_{\alpha k}$ on the pseudospins. Their cartesian components are also labeled on axis. The black arrow represents the Bloch vector of $\left|\downarrow_{k}\right>$.
• Figure 3: (a) $f_k$ and (b) $g_k$ as functions of $C_{\alpha k}$ and $E_{-1,k}$. For these plots, $D_{\alpha k}=5\textrm{kHz}$, $\tau=20\,\mu\textrm{s}$.
• Figure 4: (a) Time evolution of coherences $L^0_{\alpha\beta}$, in the scenario of FID ($N=0$); (b) Time evolution of coherences $L^1_{\alpha\beta}$, in the scenario of Hahn-echo ($N=1$). In both cases $N=0$ and $N=1$, $L^N_{-1,0}=L^N_{1,0}=L^N_{-1,S}=L^N_{1,S}\neq L^N_{-1,1}$, and these coherences exhibit much faster decay in contrast to $L^N_{S,0}$, which exhibits essentially no change on the decay time scale of other coherences. In the example plotted here, the model parameters are $J=10\,\textrm{GHz}$, $d=5\,\textrm{\AA}$, $R_S=5\,\textrm{\AA}$, $R_B=2\,\textrm{\AA}$, $n_B=0.01/\,\textrm{\AA}^{3}$, $B=1\,\textrm{T}$, and two electrons are aligned along $z$, i.e. parallel to the field.
• Figure 5: Dependence of (a) $T_2^*$ and (b) $T_2$ on the field strength $B$ and relative orientation between the $\vec{B}$ field and the position vector $\vec{d}$ joining the two electron spins. The dashed blue and purple lines in (a) are $0.5T^*_{2;-1,S}$ for $\vec{B}\parallel\vec{d}$ and $\vec{B}\perp\vec{d}$, respectively. In (c), the time evolution of $L^1_{-1,S}$ and $L^1_{-1,1}$ at $B=0.3\,\textrm{T}$ and $B=0.05\,\textrm{T}$ for $\vec{B}\parallel\vec{d}$ are shown with ESEEM developed significantly at the smaller field between the two. Other parameters of the model used for the results in this figure are $J=10\,\textrm{GHz}$, $d=5\,\textrm{\AA}$, $R_S=5\,\textrm{\AA}$, $R_B=2\,\textrm{\AA}$, $n_B=0.01/\,\textrm{\AA}^{3}$.