Homo and Heteronuclear Spin Pair Dynamics

Homo and Heteronuclear Spin Pair Dynamics

The following section is referenced in the main text in Section Spin Hamiltonian. In order to be coherent with the published paper (Yang et al., 2014), the same notation as in the paper is used.

Electron Spin Hamiltonian

The electron Spin Hamiltonian can be written as

where the $\gamma_{e}=-1.76\times10^{11}$ rad/(s·T) is the gyromagnetic ratio of the electron, and the known $D=35$ MHz. Note that the zfs ($=2D$) is large enough to prevent spin flipping due to weak hyperfine coupling strength. Typical hyperfine strength are smaller than 100 kHz.

Nuclear Spin Bath Hamiltonian

The nuclear spin bath is described by the following Hamiltonian:

where $i$ is the $i$-th nuclear spin ($i=1,2,...,N_{\xi}$) and $\xi$ the type of spin, meaning $\xi \in \left\{ \textrm{C},\textrm{Si} \right\}$, and $N_{\xi}$ is the number of $\xi$-type nuclear spin. The corresponding gyromagnetic ratio of Si and C are $\gamma_{\textrm{C}}=6.73\times10^{7}$ rad/(s·T) and $\gamma_{\textrm{Si}}=-5.32\times10^{7}$ rad/(s·T), respectively. As the nuclear spins are coupled via magnetic dipole-dipole interaction, the Hamiltonian can be written as:

where the different nuclear spins located at $\mathbf{r}_{i,\xi}$ and $\mathbf{r}_{i,\xi'}$ interact via the magnetic dipolar coupling tensor $\mathbf{D}$, which reads as:

with $\mu_0$ being the vacuum permeability, and $\mathbf{r}_{ij}=\mathbf{r}_{i,\xi'}-\mathbf{r}_{i,\xi}$ defines the relative displacement between the nuclei.

The hyperfine interaction between the electron spin and the nuclear spin couples the spin species, that interaction is defined as:

where $\mathbb{A}_{i,\xi}$ represents the hyperfine coupling tensor, which is generally composed of an anisotropic dipole part and an isotropic part, known as Fermi contact interaction. The Fermi contact interaction arises for electrons close to the nuclear spins. Such interactions cause fast ($>$MHz) oscillations in the spin echo. However, the main source of the electron spin decoherence is caused by a large number of distant nuclei. For such nuclei the Fermi interaction can be neglected, and this is the reason why the hyperfine coupling tensor is assumed to be of dipolar form:

The regular hyperfine coupling strengths for natural abundance isotopes is $\lessapprox$ 100 kHz, which corresponds to a distance of $r_{i,\xi}\gtrapprox$ 0.5 nm. Giving the fact that the hyperfine coupling strength is greatly smaller than the zfs, electron spin flip processes, eg. between $S_x$ and $S_y$ can be neglected. That is of course only valid when the system is far away from the level-crossing point. Hence $S_z$ can be seen as a good quantum number with allowed half-integer values ranging from $S_z=-3/2,...,3/2$. Based on this approximation, the hyperfine interaction can be expanded to the eigenstates $|m\rangle$ of $\hat{H}_{\textrm{int}}$ in the following way:

The eigenvalue of the eigenstate $|m\rangle$ is $\omega_m=m^2 \Delta -m \gamma_e B$, with $\Delta$ being the nuclear Zeeman splitting (see also Figure pseudospin). The components of $b_m$ are defined by

where $\hat{\mathbf{z}}$ defines the unit vector in $z$ direction.

Electron Spin Coherence

In order to study the electron spin coherence, a starting point needs to be defined. The electron spin system is initially prepared into:

(B.1)

As explained in Section Open Quantum Systems assuming separability, the initial state (at $t=0$) of nuclear-electron spin system can be expressed as a product state:

The density matrix $\rho_{\textrm{bath}}$ of the nuclear spin bath is composed of the total bath spin number $N = N_\textrm{C}+N_\textrm{Si}$. For temperatures higher than the Zeeman energy ($\leqslant \mu$K) the density matrix can be approximated as: $\rho_{\textrm{bath}} = \mathbb{1}^{N}/2^N$, where $\mathbb{1}$ is the identity matrix with a dimension of $2 \times 2$. The composite system will then evolve as a function of time to the density matrix $\rho(t)$.

The spin coherence at a time-point $t$ is defined by the mean value of the transversal electron spin component, and can be written as:

where $S_+$ is the raising operator, simply given by $S_+=S_x+iS_y$. The electron spin may have an effect on the bath as well. Known as back-action, that effect depends on the superimposed state defined in Equation B.1. Hence, various states can have distinct decay rates. The effect of the back-action has been shown for the NV in diamond in theory (Zhao et al., 2011) and in experiment (Huang et al., 2011). However, the back-action has no drastic effect on the overall decoherence time, and can be neglected as a reasonable approximation.

In order to match the theory to the experimental observation shown in Figure spin_bath and in Section Experiment the magnetic quantum numbers are chosen accordingly to $m=3/2$ and $n=1/2$.

Homo and Heteronuclear Spin Pair Dynamics

Decoherence of electron spin is mainly caused by strong nuclear spin pair flip-flop processes, which have been shown for various quantum systems, such as quantum dots (Liu et al., 2007; Yang et al., 2014). The effect has been shown to be even stronger at high magnetic fields.

With assuming an electron spin in the state $|m\rangle$, consider a nuclear spin $\mathbf{I}$ of type $\xi$ at a position $i$ in an external magnetic field $B$ which is aligned along $\hat{\mathbf{z}}$. The effective magnetic field $\mathbf{b}_{(i,\xi)}^{(m)}$ experienced by the nucleus $(i,\xi)$ is the sum of the external magnetic field $\mathbf{B}$ and the hyperfine field $\mathbf{A}$:

where the hyperfine field is defined as:

The nuclear spin pair Hamiltonian is composed of the individual interactions given by the effective fields and their dipole-dipole interaction:

In contrast to the homo-nuclear case eg. in diamond, where $\xi = \xi'$ the spin pair dynamics are different for the hetero-nuclear case eg. in SiC, where $\xi \neq \xi'$.

This previous section is referenced in the main text in Section Spin Hamiltonian.