Coherence of single spins coupled to a nuclear spin bath of varying density

The dynamics of single electron and nuclear spins in a diamond lattice with different ${}^{13}\text{C}$ nuclear spin concentration is investigated. It is shown that coherent control of up to three individual nuclei in a dense nuclear spin cluster is feasible. The free-induction decays of nuclear spin Bell states and single nuclear coherences among ${}^{13}\text{C}$ nuclear spins are compared and analyzed. Reduction in a free-induction-decay time $T_2^{\ast }$ and a coherence time $T_2$ upon increase in nuclear spin concentration has been found. For pure diamond, $T_2^{\ast }$ as long as $30\mu \text{s}$ and $T_2$ of up to 0.65 ms for the electron spin has been observed. The ${}^{13}\text{C}$ concentration dependence of $T_2^{\ast }$ is explained by Fermi contact and dipolar interactions with nuclei in the lattice. It has been found that $T_2$ decreases approximately as $1/n$, where $n$ is ${}^{13}\text{C}$ concentration, which corresponds to the reported theoretical line of $T_2$ for an electron spin interacting with a nuclear spin bath.

Figure 1

(Color online) Electron-spin resonance (ESR) spectra of single NVs with (a) zero, (b) one, (c) two, and (d) three ${}^{13}\text{C}$ in the first shell. Lower solid lines (blue) in (b-d) are simulation spectra. ESR spectra with (e) one and (f) two ${}^{13}\text{C}$ in the third shell, and (g) one ${}^{13}\text{C}$ in the first shell and one ${}^{13}\text{C}$ in the third shell. Blue dashed lines are fitting lines with Gaussian. (h) Atomic structure of NV center. The numbers 1, 2, and 3 mark C in the first shell.

Figure 2

(Color online) (a) Energy levels for nuclear spins in $M_S=-1$. Rabi oscillation of single ${}^{13}\text{C}$ in the first shell of neighbors around the NV with (b) two ${}^{13}\text{C}$ in the first shell and (c) three ${}^{13}\text{C}$ in the first shell. (d) Rabi oscillation of single ${}^{13}\text{C}$ in the third shell with one ${}^{13}\text{C}$ in the first shell and one ${}^{13}\text{C}$ in the third shell. The pulse sequence is $\pi \left(\text{MW}\right)\text{-Rabi}\left(\text{RF}\right)\text{−}\pi \left(\text{MW}\right)$ (Ref. 1). The ESR transitions of the MW $\pi$ pulse are those at lowest frequency in Figs. 1c, 1d, 1g and are indicated by dark gray (blue) arrows in (a). The NMR transitions of the RF pulse are indicated by light gray (orange) arrows in (a). The recording the data in (b)–(d) required about 20 min of averaging. (e) Electron nuclear double resonance (ENDOR) spectrum of ${}^{13}\text{C}$ at first shell in (b) with fitted Gaussian line (blue dashed line) by the pulse sequence of $\pi \left(\text{MW}\right)\text{−}\pi \left(\text{RF}\right)\text{−}\pi \left(\text{MW}\right)$. Free-induction decay (FID) of (f) ${\Psi }^{-}$, (g) ${\Phi }^{-}$, and (h) nuclear coherence between $|00⟩\leftrightarrow |10⟩$ and (j) $|00⟩\leftrightarrow |01⟩$. (k) Pulse sequence for ${\Phi }^{-}$ generation and detection between two nuclear spins. $E$ and $N_{1,2}$ marks the electron and the two nuclear spins, respectively. Spin selective pulses are represented by squares, operating on a target qubit. Vertical lines represent logical connections. The control qubit state $|1⟩$ and the state $|0⟩$ are displayed as filled (●) and open (○) circles. For example, (○) indicates that the pulse is applied to the target qubit if the quantum state of the controlling qubit is $|0⟩$.

Figure 3

(Color online) ESR spectra of single NV in (a) 20.7% and (b) 8.4% ${}^{13}\text{C}$ diamond with fitted Gaussian lines (blue dashed lines). (c) Fourier-transformed spectrum of FID of 1.1% ${}^{13}\text{C}$ diamond shown in (d) on expanded frequency axis. The splitting is due to HFC of distant ${}^{13}\text{C}$ nuclear spins. The hyperfine splitting to $N$ is not visible in this spectrum due to polarization of the N nuclear spin (Ref. 28). (e) Dependence of Inhomogeneous linewidth on ${}^{13}\text{C}$ concentration. The error bars indicate the distributions measured.

Figure 4

(Color online) Echo decays of electron spin in (a) 20.7%, (b) 8.4%, and (c) 1.1% ${}^{13}\text{C}$ diamond. The MW pulse sequence is $\pi /2\text{−}\tau \text{−}\pi \text{−}\tau \text{−}\pi /2$ where $\tau$ is delay (Ref. 8). Solid (red) lines are curves fitted with $\text{exp}\left[-{\left(t/T_2\right)}^3\right]$. (d) Plot of $T_2$ over ${}^{13}\text{C}$ concentration $n$. The solid (red) line is fitted with a $1/n$ dependence.