Identifying and decoupling many-body interactions in spin ensembles in diamond

We simulate the dynamics of varying density quasi-two-dimensional spin ensembles in solid-state systems, focusing on the nitrogen-vacancy centers in diamond. We consider the effects of various control sequences on the averaged dynamics of large ensembles of spins, under a realistic “spin-bath” environment. We reveal that spin locking is efficient for decoupling spins initialized along the driving axis, both from coherent dipolar interactions and from the external spin-bath environment, when the driving is two orders of magnitude stronger than the relevant coupling energies. Since the application of standard pulsed dynamical decoupling sequences leads to strong decoupling from the environment, while other specialized pulse sequences can decouple coherent dipolar interactions, such sequences can be used to identify the dominant interaction type. Moreover, a proper combination of pulsed decoupling sequences could lead to the suppression of both interaction types, allowing additional spin manipulations. Finally, we consider the effect of finite-width pulses on these control protocols and identify improved decoupling efficiency with increased pulse duration, resulting from the interplay of dephasing and coherent dynamics.

Figure 1

Cluster-based simulations of the dynamics of a spin ensemble, dominated by internal dipolar interactions. (a) Spin concentration of ${10}^{10}{\mathrm{cm}}^{-2}$ within a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface, representing 464 spins with a typical interaction strength of $\sim 60$ Hz. Different curves represent different numbers of interacting spins taken into account in a cluster. (b) Varying typical dipolar strengths, using clusters of six spins: 60 Hz and 10 kHz typical interactions within a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface (464 and 9980 spins), and 1 MHz within a $\approx 0.46\mu {\mathrm{m}}^2$ measurement surface (9980 spins).

Figure 2

Cluster-based simulations of the spin dynamics of an ensemble consisting of 9980 spins, under spin-lock driving at different intensities, for dipolar interactions of (a) 10 kHz (a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface) and (b) 1 MHz (a $\approx 0.46\mu {\mathrm{m}}^2$ measurement surface) within the ensemble. For a dipolar interaction of 60 Hz, the evolution was unity even for the weakest examined spin-lock intensity. (c) Effect of spin locking on an ensemble with dipolar interactions of 60 Hz (464 spins), in a spin-bath environment with ${\tau }_c=5\mu \mathrm{s}$ correlation time and $b=20$ kHz coupling strength to the ensemble, averaging only five dipolar realizations.

Figure 3

Cluster-based simulations of the spin dynamics of a spin ensemble, under DD and the WAHUHA sequence consisting of 1000 pulses. The dominant interactions are (a) A spin-bath environment, correlation time ${\tau }_c=5\mu \mathrm{s}$ and coupling strength $b=20$ kHz, and (b) dipolar interactions among the spins in the ensemble, with a typical interaction strength of 60 Hz (464 spins within a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface). (c) Combined interaction of spin-bath and dipolar couplings with the same parameter as in (a) and (b), utilizing a total number of $\sim 21$ 000 pulses: total 21 000 CPMG, WAHUHA, and a combination of 5 WAHUHA within 1000 CPMG pulses, averaging only five dipolar realizations.

Figure 4

Cluster-based simulations of the spin dynamics of an ensemble consisting of 464 spins, under a realistic spin-bath environment and dipolar coupling (same parameters as before), utilizing 4000 realistic CPMG pulses with various finite durations. Since weaker driving results in shorter free-evolution times for dephasing between pulses, the spin state is better initialized along the driving axis, which leads to a more efficient decoupling of the MW driving from dipolar interactions and longer decay times.

Figure 5

Average Hamiltonian calculation of a spin-1/2 ensemble dipolar Hamiltonian, and the addition term in the case of two-level subspace of a spin-one system, under conventional and modified WAHUHA sequences.

Figure 6

Oscillation frequencies and their populations for different numbers of spins in the limit of couplings with equal strength ${\omega }_0$, calculated from Eq. (B4). Larger numbers of spins introduce additional higher frequencies and a significant drop in the populations of all terms.

Figure 7

Probability distribution generated for the dominant interactions in a quasi-2D NV ensemble with a spin concentration of ${10}^{10}{\mathrm{cm}}^{-2}$ within a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface, obtained by averaging over many realizations of randomly generated 464 spins. The resulting average interaction strength is ${\omega }_0\approx 60$ Hz.

Figure 8

Fourier transform of the dynamics of a spin ensemble consisting of 464 spins with a typical interaction strength ${\omega }_0\approx 60$ Hz, under the dipolar Hamiltonian (1). The interactions within clusters consisting of different numbers of spins are generated from the probability distribution in Fig. 7.

Figure 9

Cluster-approximation simulations of the dynamics of a spin ensemble consisting of 464 spins with a typical interaction strength ${\omega }_0\approx 60$ Hz, under a spin-lock driving with an intensity of $\mathrm{\Omega }\approx 12{\omega }_0$. The interactions within clusters consisting of different numbers of spins are generated from the probability distribution in Fig. 7.

Figure 10

(a) Interaction strength distributions and (b) cluster-based simulations of the spin dynamics of spin ensembles with different properties: Spin concentration of ${10}^{10}{\mathrm{cm}}^{-2}$ within a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface, representing 464 spins with a typical interaction strength of ${\omega }_0\approx 60$ Hz, ${\omega }_0\approx 10$ kHz typical interaction within a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface (9980 spins), and ${\omega }_0\approx 1$ MHz within a $\approx 0.46\mu {\mathrm{m}}^2$ measurement surface (9980 spins). The horizontal axes are normalized to ${\omega }_0$ to emphasize similarities between the results, with small differences due to the slightly different strength distributions.

Figure 11

Cluster-based simulations of the spin dynamics of an ensemble consisting of 464 spins, under spin-lock driving with the strength of 0.1 MHz, for dipolar NV-NV interactions with a typical interaction strength of 60 Hz.

Figure 12

Cluster-based simulations of the spin dynamics under the application of the WAHUHA sequence with 100 repetitions on a spin ensemble dominated by NV-NV dipolar interactions at various concentrations: Spin concentration of ${10}^{10}{\mathrm{cm}}^{-2}$ within a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface, representing 464 spins with a typical interaction strength of ${\omega }_0\approx 60$ Hz, ${\omega }_0\approx 10$ kHz typical interaction within a $\approx 4.5\mu {\mathrm{m}}^2$ measurement surface (9980 spins), and ${\omega }_0\approx 1$ MHz within a $\approx 0.46\mu {\mathrm{m}}^2$ measurement surface (9980 spins).

Figure 13

Cluster-based simulations of the spin dynamics of an ensemble consisting of 464 spins, under a realistic spin-bath environment and dipolar coupling (same parameters in Fig. 4), utilizing 500 repetitions of the XY8 sequence (total of 4000 pulses) with various finite durations. Since weaker driving results in shorter free-evolution times for dephasing between pulses, the spin state is better initialized along the driving axis, which leads to a more efficient decoupling of the MW driving from dipolar interactions and longer decay times.

Figure 14

Different realizations for the (a) WAHUHA, (b) CPMG, and (c) combined WAHUHA and CPMG expressed in Fig. 3.