Characterizing spin-bath parameters using conventional and time-asymmetric Hahn-echo sequences

Spin-bath noise characterization, which is typically performed by multipulse control sequences, is essential for understanding most spin dynamics in the solid state. Here, we theoretically propose a method for extracting the characteristic parameters of a noise source with a known spectrum, using modified Hahn-echo pulses. By varying the application time of the pulse, measuring the coherence curves of an addressable spin, and fitting these curves to a theoretical function derived by us, we extract parameters characterizing the physical nature of the noise. Assuming a Lorentzian noise spectrum, we illustrate this method for extracting the correlation time of a bath of nitrogen paramagnetic impurities in diamond, and its coupling strength to the addressable spin of a nitrogen-vacancy center. First, we demonstrate that fitting conventional Hahn-echo measurements to the explicit coherence function is essential for extracting the correct parameters in the general physical regime, for which common methods relying on the assumption of a slow bath are inaccurate. Second, considering a realistic experimental scenario with a $5\%$ noise floor, we simulate the extraction of these parameters utilizing the asymmetric Hahn-echo scheme. The scheme is effective for samples having a natural homogeneous coherence time $\left(T_2\right)$ up to two orders of magnitude greater than the inhomogeneous coherence time $\left(T_2^{*}\right)$. In the presence of realistic technical drifts for which averaging capabilities are limited, we simulate more than a factor of 3 improvement of the extracted parameter uncertainties over conventional Hahn-echo measurements. Beyond its potential for reducing experiment times by an order of magnitude, such single-pulse noise characterization could minimize the effects of long timescale drifts and accumulating pulse imperfections and numerical errors.

Figure 1

Conventional free induction decay and Hahn-echo sequences, and the time-asymmetric Hahn-echo sequence.

Figure 2

Coherence curves as a function of time generated from Eq. (9) considering a spin-bath with ${\tau }_c=100\mu \text{s},b=5\mathrm{kHz}$, and no typical experimental readout errors (blue, solid), and their corresponding fits to stretched exponential decay functions (red, dashed) for conventional (a) Hahn-echo $\left(\alpha =\frac{1}{2}\right)$ and (b) FID $\left(\alpha =0\right)$ experiments. Despite fitting experimental data, the conventional stretched exponential analysis under the assumption of a slow bath may lead to the incorrect evaluation of spin-bath parameters when this slow noise condition is not satisfied.

Figure 3

Fitting to a conventional Hahn-echo $\left(\alpha =\frac{1}{2}\right)$ and coherence curve as a function of time. The simulated data (dots) were generated considering a spin-bath with ${\tau }_c=100\mu \text{s},b=5\mathrm{kHz}$, and a typical experimental readout error of $5\%$. Different decay curves (lines) fitted to the data using Eq. (9) and different spin-bath parameters fit the data within an uncertainty of $\mathrm{\Delta }{\chi }_{\nu }^2=0.14$.

Figure 4

Parameter pairs fitting the Hahn-echo $\left(\alpha =\frac{1}{2}\right)$ explicit coherence curve [Eq. (9)] for a spin-bath with parameters ${\tau }_c=100\mu \text{s},b=5\mathrm{kHz}$ within the ${\chi }_{\nu }^2$ uncertainty of 0.14 (yellow regions) as a function of the number of averages. Initial shot-noise simulated with a signal-to-noise ratio of 1. (a) 25, (b) 100, (c) 400, (d) 1e4, (e) 4e4, and (f) 2.5e5 averages. The convergence to a single point indicates the unlimited precision achievable by averaging.

Figure 5

Parameter pairs fitting the Hahn-echo $\left(\alpha =\frac{1}{2}\right)$ explicit coherence curve [Eq. (9)] for a spin-bath with parameters ${\tau }_c=100\mu \text{s},b=5\mathrm{kHz}$ within the ${\chi }_{\nu }^2$ uncertainty of 0.14 (yellow regions) and a $5\%$ noise floor as a function of the number of averages. Initial shot-noise simulated with a signal-to-noise ratio of 1. (a) 25, (b) 100, (c) 400 (corresponding to a signal-to-noise ratio of $5\%)$, (d) 1e4, (e) 4e4, and (f) 2.5e5 averages. Further averaging no longer reduces the extracted parameter uncertainties when the noise floor dominates.

Figure 6

Parameter pairs fitting the explicit coherence curves [Eq. (9)] of asymmetric Hahn-echo experiments with $\alpha =$ (a) 0, (b) 0.1, (c) 0.2, (d) 0.3, (e) 0.4, and (f) 0.5, for a spin-bath with parameters ${\tau }_c=100\mu \text{s},b=5\mathrm{kHz}$ within the ${\chi }_{\nu }^2$ uncertainty of 0.14 (yellow regions), a $5\%$ noise floor, and an initial shot-noise with a signal-to-noise ratio of 1 and 2.5e5 averages. (g) The shared parameter pairs obtained from the intersection of the regions in (a)–(f), and resulting in the extracted parameters ${\tau }_c=95\pm 15\mu \text{s},b=5.00\pm 0.17\mathrm{kHz}$.