Converting a real quantum spin bath to an effective classical noise acting on a central spin

We present a cluster expansion method for approximating quantum spin-bath dynamics in terms of a classical Gaussian stochastic process. The cluster expansion produces the two-point correlation function of the approximate classical bath, permitting rapid evaluation of noise-mitigating quantum control strategies without resorting to computationally intensive dynamical decoupling models. Our approximation is valid for the wide class of models possessing negligible back-action and nearly-Gaussian noise. We study several instances of the central spin decoherence problem in which the central spin and randomly-located bath spins are alike and dipolarly coupled. For various pulse sequences, we compare the coherence echo decay computed explicitly quantum mechanically versus those computed using our approximate classical model, and obtain agreement in most, but not all, cases. We demonstrate the utility of these classical noise models by efficiently searching for the 4-pulse sequences that maximally mitigate decoherence in each of these cases, a computationally expensive task in the explicit quantum model.

Figure 1

Relative correlation function calculation results, $C_{\mathrm{Q}}\left(t\right)-C_{\mathrm{Q}}\left(0\right)$ in ${(\text{rad}/\text{s})}^2$, for the five cases of bath spatial configurations labeled A–F studied in Ref. [8] with linear scales (left) and logarithmic scales (right). The $C_{\mathrm{Q}}\left(0\right)$ values are $55594$, $14.3$, $59.8$, $19.1$, $5.93$, and 287 ${(\text{rad}/\text{s})}^2$, respectively. Black dashed curves are O-U type of the form $Aexp(-Bt)+C$, loosely fitting the calculations over some respective time ranges. Thick (and colored) solid, dashed, and dotted curves on the right plot are 2-cluster, 3-cluster, and 4-cluster results, respectively.

Figure 2

Comparison of echo decays (plotted as 1-echo versus total pulse sequence time on a log-log scale) calculated directly using the CCE versus via the correlation functions from Fig. 1 for cases A–F. The solid black curves are direct calculation results. The yellow dashed curves are derived from correlation functions. Both include up to 4-cluster contributions. Each case shows results for 1- to 4-pulse UDD sequences (UDD1 is a Hahn echo). Initial coherence behavior improves with an increased number of pulses; that is, the 1–4 pulse curves are seen left to right. Additionally for cases B–F we show echo decay for optimized 4-pulse sequences that perform slightly better than UDD4; the red dashed curves are the correlation function derived results for these pulse sequences, in excellent agreement with direct calculations in solid black. Dotted black curves show direct 2-cluster results; we find that corrections from larger clusters are important, even at short times, for some of the multipulse sequences.

Figure 3

Scatter plot of the extreme values for the relative discrepancy of echo infidelity (1-echo) estimates versus the maximum coupling constant to the central spin. Hahn echo (UDD1) and UDD4 cases are represented from about 900 and 750 random spatial configurations, respectively. The relative discrepancy is computed as (estimated - expected)/min(estimated, expected) where “estimated” uses the correlation function and “expected” uses the direct CCE approach. Positive values correspond with pessimistic estimates. Relative discrepancies shown are the two extremes (per spatial configuration) over 100 logarithmically spaced time points between 1 ms and $0.3$ s, skipping infidelities below ${10}^{-7}$ where the numerical precision is suspect. These are plotted on a symmetric positive and negative logarithmic scale with a linear region between $\pm 0.1$. The top figure shows cumulative counts of instances versus maximum coupling constants. The right figure shows cumulative probability distributions of relative discrepancies.

Figure 4

Optimal, symmetric, 4-pulse refocusing sequences for cases B–F found efficiently using the correlation functions of Fig. 1. The curves indicate the fractional times of $\pi$ pulses as a function of the total pulse sequence time. For comparision, we show the 4-pulse UDD and Carr-Purcell-Meiboom-Gill sequence as dashed and dotted lines respectively.