Depolarization Dynamics in a Strongly Interacting Solid-State Spin Ensemble

We study the depolarization dynamics of a dense ensemble of dipolar interacting spins, associated with nitrogen-vacancy centers in diamond. We observe anomalously fast, density-dependent, and nonexponential spin relaxation. To explain these observations, we propose a microscopic model where an interplay of long-range interactions, disorder, and dissipation leads to predictions that are in quantitative agreement with both current and prior experimental results. Our results pave the way for controlled many-body experiments with long-lived and strongly interacting ensembles of solid-state spins.

Figure 1

Experimental system. (a) Level diagram for NV centers. Red arrows indicate possible spin decay channels, ${\gamma }_1$ and ${\gamma }_2$. (b) Schematic electron spin resonance spectrum of four subensembles of NV centers $A$, $B$, $C$, $D$ with spectral separation $\delta$ between $B$ and $C$ (upper curve). The effective density of resonant spins can be tuned by changing the orientation of the external magnetic field (lower curve). (c) Experimental sequence used to measure NV dynamics. I, charge equilibration ($\sim 100\mu \mathrm{s}$ duration); II, spin polarization ($10\mu \mathrm{W}$ laser power, $200\mu \mathrm{s}$ duration) and subsequent microwave manipulation to modify the initial state; III, time evolution; IV, spin state readout. The red $\pi$ pulse is used to measure the population in an orthogonal state. (d) Comparison of depolarization time scales between a single NV (red data, exponential fit) and a dense ensemble of NVs (45 ppm, blue data, stretched-exponential fit with $T_1\sim 67\mu \mathrm{s}$). The dashed line is a simple exponential curve with a time constant of $100\mu \mathrm{s}$ for comparison.

Figure 2

Depolarization dynamics. (a) NV depolarization time scale probed for different initial states. Solid lines represent stretched-exponential fits with corresponding $T_1$ of $56\pm 2\mu \mathrm{s}$ ($|m_s=0⟩$, blue data) and $80\pm 2\mu \mathrm{s}$ ($|m_s=-1⟩$, red data). (b) Population difference between $|m_s=0⟩$ and $|m_s=1⟩$ as a function of time for initialization into $|m_s=-1⟩$. The solid line corresponds to a rate equation model of magnetic-noise-induced spin depolarization [26]. (c) Measured depolarization rates $1/T_1$ as a function of the spectral distance $\delta$ between two subensembles $B$ and $C$. A Lorentzian fit (dashed orange line) is used to extract the full width at half maximum of $(2\pi )$ $25\pm 6\mathrm{MHz}$. (d) Spin-lock lifetime $T_1^{\rho }$ as a function of driving strength $\mathrm{\Omega }$. In (c) and (d), red lines indicate the predictions from the spin-fluctuator model at an optimized value of $n_f=16\mathrm{ppm}$ and ${\gamma }_f=(2\pi )3.3\mathrm{MHz}$.

Figure 3

Spin-fluctuator model. (a) Level diagram of a single spin and fluctuator in two different spin states (green disk). Red arrows indicate fast depolarization channels of a fluctuator. Solid gray arrows depict spin exchange via dipolar interactions between the spin and fluctuator. (b) Schematic representation of several spins I, II, and III (dark blue) in the ensemble with different depolarization rates owing to random positions of the surrounding fluctuators (red crosses). Ensemble averaging of such depolarization rates gives rise to a stretched exponential (red solid line).

Figure 4

Charge-state dynamics. (a) Level diagram, showing optical ground state $|g⟩$ and excited state $|e⟩$, for ${\mathrm{NV}}^0$ and ${\mathrm{NV}}^{-}$ under illumination. A yellow laser ($\lambda =594\mathrm{nm}$) can off-resonantly excite ${\mathrm{NV}}^{-}$, leading to a strong fluorescence signal. ${\mathrm{NV}}^0$, however, remains in its ground state due to a higher transition frequency, allowing the optical detection of NV charge states. (b) Pulse sequence used to measure charge distribution. A green laser is used to create an out-of-equilibrium initial state. The resulting charge distribution can be measured via short yellow laser pulses. (c) Relative charge distribution measured via a yellow scanning laser ($27\mu \mathrm{W}$) after a strong green laser illumination at the center ($100\mu \mathrm{W}$). (d) Relaxation of the charge distribution at the center over time (open circles) and a theoretical fit based on a classical diffusion model (solid line).