Emergent Decoherence Dynamics in Doubly Disordered Spin Networks

TL;DR

Emergent decoherence dynamics in a doubly disordered spin network reveal a universal two-channel law $M(t)=e^{-\,\sqrt{R_p t}}e^{-R_d t}$ linking reversible microscopic quantum dynamics to irreversible macroscopic relaxation. A combination of Floquet engineering of the spin Hamiltonian and all-optical bath control tunes two independent channels, $R_p$ and $R_d$, while subdiffusive nuclear transport ($\alpha\approx0.85$) and disorder-generated polarization traps extend lifetimes. The authors support this picture with a minimal Markov-Chain Monte Carlo model, eigenmode analysis, and a detailed relaxation landscape, demonstrating that disorder can enhance coherence by forming electron-free polarization reservoirs. Together, these results establish a programmable framework to manipulate decoherence pathways, offering a design principle for long-lived quantum memories and sensors in disordered solid-state spin systems.

Abstract

Elucidating the emergence of irreversible macroscopic laws from reversible quantum many-body dynamics is a question of broad importance across all quantum science. Many-body decoherence plays a key role in this transition, yet connecting microscopic dynamics to emergent macroscopic behavior remains challenging. Here, in a doubly disordered electron-nuclear spin network, we uncover an emergent decoherence law for nuclear polarization, $e^{-\sqrt{R_{p}t}}e^{-R_{d}t}$, that is robust across broad parameter regimes. We trace its microscopic origins to two interdependent decoherence channels: long-range interactions mediated by the electron network and spin transport within the nuclear network exhibiting anomalous, sub-diffusive dynamics. We demonstrate the capacity to control--and even eliminate--either channel individually through a combination of Floquet engineering and (optical) environment modulation. We find that disorder, typically viewed as detrimental, here proves protective, generating isolated electron-free clusters that localize polarization and prolong coherence lifetimes. These findings establish a microscopic framework for manipulating decoherence pathways and suggests engineered disorder as a new design principle for realizing long-lived quantum memories and sensors.

Figures (20)

• Figure 1: Emergent decoherence law in a doubly disordered spin network. (a) Protocol and setup. Floquet engineering pulse sequence consists of one $\vartheta_{y}$ pulse followed by $\sim$8M $\vartheta_{x}$ pulses. $^{13}$C magnetization is monitored quasi-continuously after each $\vartheta_{x}$ pulse via inductive cavity readout. (b) Doubly disordered spin network illustrating disordered $^{13}$C network embedded within disordered electron network (NV and P1 centers; concentrations not to scale). Pink lines indicate inter-spin couplings; transparent circles show spinless $^{12}$C; blue spheres mark electron frozen cores where inward diffusion is suppressed (see Methods). (c) Experimentally measured decoherence traces vs. pulse detuning, $\Delta\omega$ (colorbar). Experimental data (smoothed with a 100-point moving average) plotted on log vs. $\sqrt{t}$ scale sampled every $\approx$80 $\mu$s over $>$500 s. Top $x$-axis shows $t$. Horizontal dashed line marks $1/e$ crossing with surrogate $T_{2}'$ values labeled. Black dashed lines are fits to emergent law, $e^{-\sqrt{R_{p}t}}e^{-R_{d}t}$, showing excellent agreement across entire dataset (for residuals see SI Sec. \ref{['section_fitting']}). At $\Delta\omega \simeq 2.25$ kHz (top pink trace), decay reduces to $e^{-\sqrt{R_{p}t}}$. (d)-(e) Microscopic decoherence mechanisms. (d) Paramagnetic relaxation $R_{p}$: direct relaxation pathway via dipolar coupling to disordered electron environment (blue waves) (e) Diffusive relaxation $R_{d}$: indirect pathway via polarization transport through nuclear spin network toward electron "sinkholes" (red arrows). Nuclei near electrons relax rapidly, generating polarization gradients that drive spin transport.
• Figure 2: Controlling emergent dynamics via simultaneous system and environment engineering. (a) System control: engineering microscopic Hamiltonians enables selective elimination of either decoherence channel. In Regime I ($\Delta\omega=0$ kHz, $\vartheta=90^{\circ}$), both paramagnetic (blue waves) and diffusive (red arrows) pathways are active. In Regime II ($\Delta\omega=2.25$ kHz, $\vartheta=90^{\circ}$), $H_{nn}^{(1)} \rightarrow 0$ effectively "turns off" diffusive pathway, leaving only paramagnetic relaxation. In Regime III ($\Delta\omega=5$ kHz, $\vartheta=5^{\circ}$), energy mismatch suppresses paramagnetic channel, isolating diffusive pathway (see Methods). (b) Environmental modulation via simultaneous laser illumination. Decay curves (log scale) for each regime are shown with laser turned on 50 s into Floquet sequence at various powers (colorbar). Increasing power causes curves to "fan out." Laser-driven electron fluctuations reshape spectral distribution of magnetic noise. (c)-(e) Probing relaxation rates $R_{p}$ and $R_{d}$ via fits to Eq. (\ref{['eqn:emerge']}). Laser remains on throughout; error bars denote standard error from three random trials. (c) Regime I: $R_{p}$ increases then decreases ($>$4 W), while $R_{d}$ increases linearly. (d) Regime II: diffusive channel "turned off" (flat line) yielding decay as $e^{-\sqrt{R_{p}t}}$. $R_{p}$ increases linearly with power. (e) Regime III: paramagnetic channel suppressed (flat line) yielding decay as $e^{-R_{d}t}$. $R_{d}$ increases linearly with power, with overall smaller rates. Shaded regions indicate simulated relaxation rates from Markov chain Monte Carlo analysis, with thickness representing standard error (see SI Sec. \ref{['section_montecarlo']}).
• Figure 3: Concentration-dependent transport and relaxation. (a) Polarization transport. Simulations track mean squared displacement and fit to $6Dt^{\alpha}$ (see Methods, SI Sec. \ref{['subsection_transport']}). Main plot (purple) shows diffusion exponent $\alpha$ increases with $^{13}$C concentration; solid line is guide to eye. Each point is mean of 5 independent runs of 100 trajectories; error bars denote standard error. Inset (yellow) shows corresponding diffusion coefficient $D$. At 1.1% $^{13}$C (this work), $D=3.86$Å/s$^{0.85}$, and $\alpha=0.85$, indicating subdiffusive behavior that deviates from hydrodynamic expectations. (b) Relative contributions, $\rm{log_{10}}(R_p/R_d)$, of the two relaxation channels shown as heat map (colorbar) across varying $^{13}$C ($x$-axis) and electron ($y$-axis) concentrations; top $x$-axis indicates corresponding diffusion coefficient $D$. Blue regions denote "diffusion-limited" regimes (${\rightarrow} e^{-\sqrt{R_{p}t}}$), while red indicates "diffusion-dominated" regimes (${\rightarrow} e^{-R_{d}t}$) (see SI Sec. \ref{['section_relaxationlandscape']}). Dashed lines mark $R_p/R_d = 3$ or $1/3$. Star (this work) marks a newly accessed regime, (see comparison with literature in SI Sec. \ref{['section_context']}).
• Figure 4: Microscopic origins of emergent dynamics. (a)-(b) Eigenvalue and rate scaling with electron concentration. (a) Mean slowest eigenvalue $\langle \lambda_0 \rangle$ alongside $2R_d$; solid line is linear guide to eye. Scaling indicates $\langle \lambda_0 \rangle$ is origin of monoexponential component in Eq. (\ref{['eqn:emerge']}). (b) $R_p$ from full Monte Carlo model with $R_p^{\text{dep}}$ (excluding diffusion); solid line is quadratic guide to eye. Results show $R_p$ is electron-mediated on-site decoherence channel independent of diffusion. (c) Random walks in random environments conceptual schematic, drawing analogy of polarization diffusion to particle walks in media with randomly distributed static traps (electrons). (d) Polarization decay averaged over 100 $^{13}$C configurations for fixed electron positions, decay follows Eq. (\ref{['eqn:emerge']}). (e)-(f) Polarization heatmaps showing polarization (colorbar) projected onto $xy$-plane at $t=3$ s and $t=200$ s (dashed lines in d). Black contours mark equal polarization levels. At late times polarization is confined to electron-free domains. (g) Profile of slowest eigenmode displayed similarly, shows close match to the confinement pattern in (f). (h-j) Slowest eigenmode profile versus $^{13}$C concentration. 2D projections as in (g) for single $^{13}$C configurations at 0.2%, 1.1%, and 10% concentrations, respectively. Panels (h-i) show trapped regions (orange) similar to (f-g). At higher concentrations (j), increased network connectivity progressively eliminates trap-free domains, marking transition to diffusion dominated regime and hydrodynamic behavior (Fig. \ref{['fig:fig3']}a). For eigenvalue spectra, see SI Sec. \ref{['subsection_eigenvalues']}.
• Figure 5: A detailed breakdown of the frozen core, into regions that fall within the excitation bandwidth and detection window, relative to their distance from the electron. Frozen-core spins within the red inner-most region are neither excited nor detected. A small shell of frozen-core spins (orange, middle region) may in principle be observed, but are not excited. The outer layer of frozen-core spins (light orange) are excited and observed, but decay rapidly due to their proximity to the electron.