Witnessing non-stationary and non-Markovian environments with a quantum sensor

TL;DR

This work develops an operational framework to classify nanoscale environmental noise as stationary/non-stationary and Markovian/non-Markovian using quantum sensors, without requiring full bath reconstruction. It introduces a Gaussian Langevin-type noise model with inertia $m$ that captures memory effects and initial-bath quenches, yielding analytical Ramsey decay predictions $S(t)=e^{-\chi(t)}$ with $\chi(t)$ determined by noise correlations. The authors validate the theory experimentally using a single NV center with tunable injected noise, demonstrating distinct short- and long-time signatures for each dynamical regime and showing how complementary stationary and quenched measurements enable unambiguous parameter extraction ($\omega_0$, $t_c$, $A/m^2$). This sensor-based approach provides a practical pathway to characterize complex environmental dynamics at the nanoscale and informs strategies to mitigate decoherence while exploiting bath memory for enhanced sensing.

Abstract

Quantum sensors offer exceptional sensitivity to nanoscale magnetic fluctuations, where non-stationary effects -- such as spin diffusion -- and non-Markovian dynamics arising from coupling to few environmental degrees of freedom play critical roles. Because fully reconstructing the microscopic structure of realistic spin baths is often infeasible, a practical challenge is to identify the dynamical features that are actually encoded in the sensor's decoherence signal. Here, we demonstrate how quantum sensors can operationally characterize the statistical nature of environmental noise, distinguishing between stationary and non-stationary behaviors, as well as Markovian and non-Markovian dynamics. Using nitrogen-vacancy (NV) centers in diamond as a platform, we develop a physical noise model that captures the essential dynamical features of realistic environments relevant to sensor observables -- independently of the microscopic bath details -- and provides analytical predictions for Ramsey decay across different regimes. These predictions are experimentally validated through controlled noise injection with tunable correlation properties. Our results showcase the capability of quantum sensors to isolate and identify key dynamical properties of complex environments, without requiring full microscopic bath reconstruction. This work clarifies the operational signatures of non-stationarity and non-Markovian behavior at the nanoscale and lays the foundation for strategies that mitigates decoherence while exploiting environmental dynamics for enhanced quantum sensing.

Figures (12)

• Figure 1: Experimental setup to probe environmental noise using a single NV center as a quantum sensor via Ramsey decay measurements. (a) The setup consists of a diamond membrane containing nanopillars with single NV centers placed on a printed circuit board. A 520 nm laser is focused through a hole in the board to initialize and read out the NV spin state. The NV is subjected to a stationary magnetic field $B_0=298~\text{G}$, while environmental field fluctuations $n(t) = {\gamma}_\mathrm{nv}\Delta B_z\left(t\right)\ $ are generated by injecting noise through a spiral antenna beneath the diamond via a variable voltage. Radiofrequency (RF) pulses for NV spin manipulation are applied via a copper wire above the diamond membrane. (b,c) Representative samples (10 sample trajectories shown as colored curves for each) of the four noise processes $n(t)$ considered in this study: stationary Markovian, quenched Markovian, stationary non-Markovian, and quenched non-Markovian (from top to bottom). We distinguish between Markovian (panel b, $m = 0$) and non-Markovian (panel c, $m \neq 0$). (d) Schematic representation of how magnetic field fluctuations $n(t)$, sensed by the quantum sensor, are modeled as the position of a stochastically driven harmonic oscillator by a Langevin-type equation: $mn"+\Gamma n'+\kappa n=\eta(t)$. The parameters $m$, $\Gamma$, $\kappa$ describe the effective dynamics of the spins that directly couple to the qubit, capturing both oscillatory (non-Markovian) and purely relaxational (Markovian) regimes. The stochastic force $\eta(t)$, illustrated by the red arrow, represents the influence of more distant environmental spins that do not couple directly to the sensor but modulate the collective spin-bath dynamics. This model unifies non-Markovian and Markovian spin-bath behaviors within a single physical framework.
• Figure 2: Ramsey decay signals, ${S(t)}$, for equilibrium and quenched Markovian noise, with correlation times $t_c = 1.25, 2.5, 5,$ and $10\,\upmu\mathrm{s}$, and noise standard deviation $\Delta = 0.10\,\mathrm{V}$. Symbols represent experimental data; solid curves are analytical fits using $t_c$ as a free parameter, consistent with the injected values. For equilibrium noise, all curves collapse onto a single decay due to identical short-time behavior, making them visually indistinguishable. In contrast, quenched noise curves remain clearly distinct, demonstrating the Ramsey signal’s sensitivity to the correlation time in non-stationary regimes.
• Figure 3: Measured Ramsey signals, ${S(t)}$, used to probe the onset of quenches in Markovian noise environments. Symbols represent experimental data and solid lines are fits to the analytical expressions. (a) Ramsey decays for noise with a standard deviation $\Delta$ = 0.10 V and correlation time $t_c = 10\,\upmu\mathrm{s}$ measured at equilibrium ($t_d = \infty$) and with quenches occurring at delays $t_d$ = 0, 250 ns, 625 ns, 1250 ns, and 2500 ns prior to the start of the Ramsey sequence. (b) Ramsey decays for noise with fixed standard deviation $\Delta$ = 0.10 V and a correlation time that switches from $t_a$ = 15 ns to $t_b$ = 150 ns at times $t_s$ = 248 ns, 500 ns, 748 ns, and 1000 ns.
• Figure 4: Measured Ramsey decay signals ${S(t)}$ under equilibrium and quenched underdamped non-Markovian noise ($m \neq 0$), used to probe non-Markovian features and their interplay with non-stationary effects. The effective driving strength is fixed at $A/m^2$ = 0.054 V$^2$/$\upmu$s$^3$. (a) Ramsey signals for a fixed damping coefficient $2t_c^{-1}$ = 0.1 MHz, and restoring frequencies $\omega_0$ = 5, 6, and 7 MHz. (b) Ramsey signals for a fixed restoring frequency $\omega_0$ = 6 MHz and damping coefficients $2t_c^{-1}$ = 0.025, 0.05, 0.1, 0.15, and 0.2 MHz. (c) Comparison of equilibrium and quenched non-Markovian noise for $\omega_0$ = 6 MHz and $2t_c^{-1}$ = 0.05, 0.1, and 0.2 MHz. In all cases, solid curves are fits to analytical expressions for Ramsey decay under non-Markovian noise (Appendix \ref{['sec:underdamped_Ramsey_exponents']}), using the experimental values of $\omega_0$ and $2t_c^{-1}$, with the effective driving strength of the magnetic field as the only free parameter to account for the antenna's impedance.
• Figure 5: Analytical calculation of the Ramsey signal for equilibrium (light red, light blue) and quenched (red, blue) underdamped non-Markovian ($m\neq 0$) injected noise with a driving strength $A/m^2$ of 400 MHz$^5$, restoring frequency 5 MHz, and damping coefficient 0.2 MHz, both without (red) and with (blue) an independent $T_2^*$ of 1.67 $\upmu$s from non-injected white noise.