Dynamical decoupling design for identifying weakly coupled nuclear spins in a bath

Identifying weakly coupled nuclear spins around single electron spins is a key step toward implementing quantum information processing using coupled electron-nuclei spin systems or sensing like single-spin nuclear magnetic resonance detection using diamond defect spins. Dynamical decoupling control of the center electron spin with periodic pulse sequences [e.g., the Carre-Purcell-Meiboom-Gill (CPMG) sequence] has been successfully used to identify single nuclear spins and to resolve structure of nuclear spin clusters. Here, we design a type of pulse sequence by replacing the repetition unit (a single $\pi$ pulse) of the CPMG sequence with a group of nonuniformly spaced $\pi$ pulses. Using the nitrogen-vacancy center system in diamond, we theoretically demonstrate that the designed pulse sequence improves the resolution of nuclear spin noise spectroscopy, and more information about the surrounding nuclear spins is extracted. The principle of dynamical decoupling design proposed in this paper is useful in many systems (e.g., defect spin qubit in solids, trapped ion, and superconducting qubit) for high-resolution noise spectroscopy.

Figure 1

(a) The spin coherence under Hahn echo control in a $B=10$ Gauss magnetic field. The inset shows the coherence collapse and revival in a longer time scale, and the main panel zooms in the first collapse. (b) The spin coherence under 50-pulse CPMG control. Only the first collapse is shown. Oscillations in green, red, and blue (shades of gray denoted by A, B, and C in turn) are caused by three individual ${}^{13}\mathrm{C}$ spins with hyperfine coupling strengths $\sim 100 \mathrm{kHz}$. Their contributions to the coherence are singled out in panels (c)–(e) with corresponding colors or shades of gray.

Figure 2

(a) The noise spectrum of a nuclear spin bath coupled to an NV center spin. Circle symbols represent the noise frequency and the dimensionless spectrum weight pairs $(f_j,w_j)$ with $w_j\equiv {\left|b_j\right|}^2/f_j^2$ [right axis; see Eq. (8)]. The solid curve is obtained by replacing the $\delta$ function in Eq. (8) by a Lorentzian lineshape with a phenomenological broadening of $\sim 0.1 \mathrm{kHz}$ (corresponding to the electron spin relaxation $T_1\gtrsim 1$ ms). A large number of ${}^{13}\mathrm{C}$ spins contribute to the quasicontinuous band (the gray shaded region) around the ${}^{13}\mathrm{C}$ Larmor frequecy $f_L=10.7$ kHz at $B=10$ G. Several ${}^{13}\mathrm{C}$ spins close to the NV center contribute to the discrete peaks. Three peaks, denoted by A, B, and C in turn, with frequencies higher than $f_L$ correspond to the oscillations shown in Figs. 1–1(e). Discrete peaks D and E close to the noise band are not resolved by the standard CPMG sequence or by the designed DD sequence (see Fig. 3). (b) The filter function of $F_{50}\left(\omega t\right)$ 50-pulse CPMG control sequence. The inset shows a global view in logarithm scale.

Figure 3

(a) The filter function $F_{50}\left(\omega t\right)$ of 50-pulse CPMG control. (b) The filter function $F_{150}\left(\omega t\right)$ of 150-pulse CPMG control. The 150-pulse sequence can be regarded as 50 repetitions of 3-pulse unit shown in the inset. (c) The filter function of a modified $(50\times 3$)-pulse sequence. The repetition unit is shown in the inset with parameters $r=0.31$. (d) The same as (c) but for a modified $(50\times 5)$-pulse sequence. The parameters are $p=0.209$ and $q=0.384$. (e) The noise spectrum in reciprocal axis. (f)–(h) The electron spin coherence under the DD control with sequences shown in panels (b)–(d) in turn.