Probing many-body dynamics in a two-dimensional dipolar spin ensemble

Nat. Phys. 19, 836-844 (2023) · DOI: 10.1038/s41567-023-01944-5

License: CC BY 4.0.

Abstract

The most direct approach for characterizing the quantum dynamics of a strongly interacting system is to measure the time evolution of its full many-body state. Despite the conceptual simplicity of this approach, it quickly becomes intractable as the system size grows. An alternate approach is to think of the many-body dynamics as generating noise, which can be measured by the decoherence of a probe qubit. Here we investigate what the decoherence dynamics of such a probe tells us about the many-body system. In particular, we utilize optically addressable probe spins to experimentally characterize both static and dynamical properties of strongly interacting magnetic dipoles. Our experimental platform consists of two types of spin defects in nitrogen delta-doped diamond: nitrogen-vacancy colour centres, which we use as probe spins, and a many-body ensemble of substitutional nitrogen impurities. We demonstrate that the many-body system’s dimensionality, dynamics and disorder are naturally encoded in the probe spins’ decoherence profile. Furthermore, we obtain direct control over the spectral properties of the many-body system, with potential applications in quantum sensing and simulation.

Figures

31 panels with data across 10 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

• panel (d): Secondary ion mass spectrometry depth profile of the $^{14}$N density $\rho$ (a.u.) against depth (nm) for sample S1. The printed 8 nm width marker is not drawn. data.csv

Fig. 2

• panel (a): Normalized coherence of DEER measurements, plotted as $-\log C(t)$ against free evolution time $t$ ($\mu$s) on log-log axes, for sample S1 (blue) and sample S2 (orange), mean $\pm$ s.e.m. Dashed lines mark stretch powers 2/3 and 1/3 (S1) and 1 (S2); the vertical dashed line marks the crossover. The insets are hosted as a-inset-1 and a-inset-2. data.csv
• panel (a-inset-1): Lower-right inset 1 of panel (a): the sample S1 DEER coherence $C(t)$ against $t$ ($\mu$s) on linear axes, mean $\pm$ s.e.m., with the dashed curve drawn through it. data.csv
• panel (a-inset-2): Lower-right inset 2 of panel (a): the sample S2 DEER coherence $C(t)$ against $t$ ($\mu$s) on linear axes, mean $\pm$ s.e.m., with the dashed curve drawn through it. data.csv
• panel (b): Spin echo coherence plotted as $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for samples S3 (teal) and S4 (red), mean $\pm$ s.e.m., with dashed lines of stretch power 3/2. The inset is hosted as b-inset. data.csv
• panel (b-inset): Lower-right inset of panel (b): spin echo coherence $C(t)$ against $t$ ($\mu$s) on linear axes for samples S3 (teal) and S4 (red), mean $\pm$ s.e.m., with a dashed curve through each. data.csv

Fig. 3

• panel (a-1): $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for sample S1 (two-dimensional) under an incoherent drive of $\Omega/2\pi$ = 0.61 MHz: DEER and spin echo, mean $\pm$ s.e.m., each with its dashed fit curve, and solid lines of slope 2/3, 1/3, 1; the vertical dashed line marks where the two signals overlap. data.csv
• panel (a-2): $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for sample S1 (two-dimensional) under an incoherent drive of $\Omega/2\pi$ = 1.22 MHz: DEER and spin echo, mean $\pm$ s.e.m., each with its dashed fit curve; the vertical dashed line marks where the two signals overlap. data.csv
• panel (a-3): $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for sample S1 (two-dimensional) under an incoherent drive of $\Omega/2\pi$ = 2.44 MHz: DEER and spin echo, mean $\pm$ s.e.m., each with its dashed fit curve; the vertical dashed line marks where the two signals overlap. data.csv
• panel (b-1): $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for sample S2 (three-dimensional) under an incoherent drive of $\Omega/2\pi$ = 0.26 MHz: DEER and spin echo, mean $\pm$ s.e.m., each with its dashed fit curve, and solid lines of slope 1, 3/2. data.csv
• panel (b-2): $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for sample S2 (three-dimensional) under an incoherent drive of $\Omega/2\pi$ = 1.18 MHz: DEER and spin echo, mean $\pm$ s.e.m., each with its dashed fit curve; the vertical dashed line marks where the two signals overlap. data.csv
• panel (b-3): $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for sample S2 (three-dimensional) under an incoherent drive of $\Omega/2\pi$ = 2.35 MHz: DEER and spin echo, mean $\pm$ s.e.m., each with its dashed fit curve, and solid lines of slope 1/2; the vertical dashed line marks where the two signals overlap. data.csv
• panel (d): Correlation time $\tau_c$ ($\mu$s) against drive Rabi frequency $\Omega/2\pi$ (MHz) for sample S1: best-fit values $\pm$ fitting error (purple), with the dashed grey theoretical model curve. data.csv
• panel (e): Correlation time $\tau_c$ ($\mu$s) against drive Rabi frequency $\Omega/2\pi$ (MHz) for sample S2: best-fit values $\pm$ fitting error (green), with the dashed grey theoretical model curve. data.csv

Fig. 4

• panel (a): Coherence $C(t)$ against time $t$ ($\mu$s) for spin echo (blue), XY-8 (orange) and DROID (red) sequences with interpulse spacing $\tau_p$ = 100 ns, mean $\pm$ s.e.m., each with its dashed fit curve; the grey horizontal line marks $1/e$. data.csv
• panel (b): Spin-locked NV depolarization: $C(t)$ against time $t$ ($\mu$s) with only the NV centres driven (orange) or both NV and P1 centres driven (green), mean $\pm$ s.e.m., each with its dashed fit curve; the grey horizontal line marks $1/e$. The top inset is hosted as b-inset. data.csv
• panel (b-inset): Top inset of panel (b): the same data as $-\log C(t)$ against $t$ ($\mu$s) on log-log axes, only NV driven (orange) and NV and P1 driven (green), mean $\pm$ s.e.m., each with its dashed fit curve. data.csv
• panel (c): Hartmann-Hahn polarization exchange resonance: contrast $C$ against the P1 Rabi frequency $\Omega_{\mathrm{P1}}/2\pi$ (MHz) at fixed NV Rabi frequency $\Omega_{\mathrm{NV}}/2\pi$ = 5 MHz, mean $\pm$ s.e.m., with a Lorentzian fit. data.csv

Ext. Data Fig. 1

• panel (a): XY-8 sequence: $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes, measured XY-8 data for sample S1 (orange points, with error bars), computed early-time dynamics for NV areal densities $n^{\mathrm{NV}}_{\mathrm{3D}}w$ = 17 to 21 ppm nm (dashed) and the interpolation (red). The grey late-time region of the print is marked by its left edge. data.csv
• panel (b): DEER sequence: $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes, DEER data for sample S1 after removing the NV contribution (points, with error bars), and computed early-time dynamics for P1 areal densities $n^{\mathrm{P1}}_{\mathrm{3D}}w$ = 75 to 95 ppm nm (dashed). The grey late-time region of the print is marked by its left edge. data.csv
• panel (c): Effect of a finite-thickness layer for the XY-8 data: $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes, XY-8 data for sample S1 (orange points, with error bars), computed dynamics at fixed areal density 19 ppm nm for layer widths $w$ = 2.0, 4.0, 6.0 and 8.0 nm (dashed) and the interpolation (red). data.csv
• panel (d): Effect of a finite-thickness layer for the DEER data: $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes, DEER data for sample S1 (points, with error bars) and computed dynamics at P1 density 85 ppm nm for layer widths $w$ = 2.0, 4.0, 6.0 and 8.0 nm (dashed). The grey late-time region of the print is marked by its left edge. data.csv

Ext. Data Fig. 2

• panel (1): P1 spectrum: contrast (a.u.) against microwave frequency (GHz) for three of the five P1 groups, measured points with error bars and the dashed fit to a sum of three Lorentzians. The printed dip labels 1/3, 1/4 and 1/12 are not drawn. data.csv

Ext. Data Fig. 3

• panel (1): Coherence $C(t)$ against time $t$ ($\mu$s) for DROID-60 and its variants Seq H ($\tau_p$ = 100, 200, 300 ns), Seq G, Cory and XY8 ($\tau_p$ = 100 ns), points with error bars; DROID-60 points are joined by a dashed line, the others carry dashed fit curves, and the grey line marks $1/e$. data.csv

Ext. Data Fig. 4

• panel (a): Sample S1 with a clean surface: $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for DEER (blue) and spin echo (red), with error bars, and dashed lines of slope 2/3 (two) and 1/3. data.csv
• panel (b): Sample S1 after worsening the surface quality: $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for DEER (purple) and spin echo (green), with error bars, and dashed lines of slope 2/3 and 1. data.csv
• panel (c): Sample S2: $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for DEER (orange) and spin echo (teal), with error bars, and dashed lines of slope 1 (two). data.csv

Ext. Data Fig. 5

Illustrative figure, no extractable data. Shown in the paper PDF.

Ext. Data Fig. 6

• panel (a): Reduced $\chi^2_{\mathrm{fit}}$ of fits to the DEER measurements on sample S1 against drive Rabi frequency $\Omega/2\pi$ (MHz) for four fit models: $D = 3$ and $D = 2$, each fitted to $-\log C(t)$ and to $C(t)$, points joined by lines; the dashed line marks 1. Values above the printed axis range are hosted. data.csv
• panel (b): Reduced $\chi^2_{\mathrm{fit}}$ of fits to the DEER measurements on sample S2 against drive Rabi frequency $\Omega/2\pi$ (MHz) for four fit models: $D = 3$ and $D = 2$, each fitted to $-\log C(t)$ and to $C(t)$, points joined by lines; the dashed line marks 1. Values above the printed axis range are hosted. data.csv

Ext. Data Fig. 7

• panel (a): Sample S1 under a fast incoherent drive ($\Omega/2\pi$ = 3.45 MHz): $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for spin echo (red) and DEER (blue), with error bars and a dashed curve through each, and black dashed lines of slope 2/3, 1 and 1/3. data.csv
• panel (b): Sample S2 under a fast incoherent drive ($\Omega/2\pi$ = 4.05 MHz): $-\log C(t)$ against time $t$ ($\mu$s) on log-log axes for spin echo (teal) and DEER (orange), with error bars, and a black dashed line of slope 1. data.csv

Cite

E. J. Davis et al. (16 authors). Probing many-body dynamics in a two-dimensional dipolar spin ensemble. Nat. Phys. 19, 836-844 (2023). https://doi.org/10.1038/s41567-023-01944-5

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