Dynamical-Decoupling-Based Quantum Sensing: Floquet Spectroscopy

Sensing the internal dynamics of individual nuclear spins or clusters of nuclear spins has recently become possible by observing the coherence decay of a nearby electronic spin: the weak magnetic noise is amplified by a periodic, multipulse decoupling sequence. However, it remains challenging to robustly infer underlying atomic-scale structure from decoherence traces in all but the simplest cases. We introduce Floquet spectroscopy as a versatile paradigm for analysis of these experiments and argue that it offers a number of general advantages. In particular, this technique generalizes to more complex situations, offering physical insight in regimes of many-body dynamics, strong coupling, and pulses of finite duration. As there is no requirement for resonant driving, the proposed spectroscopic approach permits physical interpretation of striking, but overlooked, coherence decay features in terms of the form of the avoided crossings of the underlying quasienergy eigenspectrum. This is exemplified by a set of “diamond-shaped” features arising for transverse-field scans in the case of single-spin sensing by nitrogen-vacancy centers in diamond. We also investigate applications for donors in silicon, showing that the resulting tunable interaction strengths offer highly promising future sensors.

Popular Summary

The development of magnetic resonance imaging revolutionized medical and biological science. Now, recent technical developments offer the comparable prospect of imaging single molecular and biological structures. As in magnetic resonance imaging, single-spin sensing also exploits measurements of the temporal decay of the coherence of quantum spins to infer detailed information. The field of single-spin sensing is at a comparatively early stage, but one day it could have a transformative effect on biological and chemical physics. Here, we show how one can exploit the well-known fact that a quantum system exposed to a periodic potential acquires very different spectral characteristics. We analyze time-periodic sensing protocols using Floquet theory and show it is potentially useful for a wider range of scenarios than standard methods, including many-body interactions and strong quantum entanglement.

In dynamical-decoupling sensing, noise from environmental spins is amplified by a sequence of microwave pulses and detected by a sensor (such as a nitrogen-vacancy color center in diamond). The coherence behavior is often analyzed using signal-processing ideas that rely on the frequency of the pulses becoming resonant with a characteristic frequency in the environment, which leads to a single sharp “dip” in the coherence. We show here that such a restriction is unnecessary. By looking at the nonresonant regime, we identify new information-rich structures that can be interpreted using Floquet spectroscopy but might be overlooked in current experiments since they lie in regimes that do not yield single sharp dips. In particular, we show that transverse magnetic field scans of coherence of nitrogen-vacancy centers in diamond reveal a set of striking diamond-shaped coherence envelopes with sharp boundaries that depend on the interesting frequencies. The power of Floquet theory is that it is applicable to both resonant and nonresonant driving, and it works for a range of coupling strengths.

We expect that our findings will motivate future studies of sensor spins, other than nitrogen-vacancy centers, such as spin-1/2 systems that have long coherence times and allow further control over the Floquet states.

Figure 1

(a) Current experiments have employed the $S=1$ electronic spins of NV centers to successfully detect (i) single nuclear spins [15, 21, 22], (ii) the internal dynamics of nuclear spin pairs [16], as well to characterize on the atomic scale, by estimating parameters such as electronic-nuclear dipolar couplings $A$ and internuclear dipolar couplings $C_{12}$. (b) Generic sensor detecting a pair of nuclear spins. The electronic spin state is in a superposition of “up” $|u⟩$ and “down” $|d⟩$ states. The nuclear dynamics and its characteristic frequencies ${\omega }_{u,d}$ depend on the electronic state. In turn, the electronic coherence is sensitive to the resulting weak ac noise from the nuclei. This may be amplified by dynamical decoupling control such as CPMG, leading to observed “dips” in coherence. These are at well-defined frequencies in typical weak-coupling regimes where the nuclear dynamics is not too different in the $u$, $d$ subspaces. However, strong-coupling regimes do not necessarily yield sharp dips. (c) Additional complexities occur for pulses of finite durations. (d) It is also challenging to differentiate between (i) independent pairs of spins and (ii) many-body effects from an equivalent interacting cluster. Floquet theory is not restricted to single spins or spin pairs and can be applied also to analysis of larger, correlated spin clusters, strong coupling, and off-resonant driving.

Figure 2

(a) Usual geometric approach. Under CPMG-N control (b) the detected spins represent two-state systems that precess about effective magnetic field, depending on the “up” $|u⟩$ and “down” $|d⟩$ states of the probe spin. The coherence dips are understood by following the precessions and relative angles between these spins, with increasing $N$. (c) Spectroscopic picture. The dips in coherence occur at avoided crossings of the Floquet eigenstates. Both the position and contrast of the decoherence dip are related to the curvature of the crossing. This is characterized by the splitting between the states $2\delta$ and the deviation from the early $\tau \to 0$ linear evolution. The early time evolution (the ${\epsilon }_0$ quasienergy) gives the dip position ${\overline{\tau }}_{\text{dip}}$ for average Hamiltonian theory; the coherence minimum is given by $\mathcal{E}({\tau }_{\text{dip}}/2)=\pi /2$. (d) The dip contrast depends on the degree of curvature of the crossing, characterized by the level-repulsion strength, $\delta =2[\mathcal{E}({\tau }_{\text{dip}})/2-\mathcal{E}({\tau }_{\text{dip}}/2)]$. (e) NV-center decoherence “diamonds.” While typical experimental studies scan along parallel field (${\omega }_z$) component (thus remaining in weak-coupling, single-dip regime), scanning the transverse magnetic field (${\omega }_x$) would produce diamond pattern of high-decoherence regions, as avoided crossings widen (and even overlap) then narrow (here, ${\omega }_z=0$ and $A_{\parallel }=50\mathrm{kHz}$). The upper panel shows full oscillating coherence function, for $N_p=10$ pulse pairs; the lower panel shows coherence envelopes, filled as $N_p\to \infty$. Here, ${\omega }_z=0$. Boundaries of the diamonds trace out (green) $\tau =\pi /2({\omega }_d+{\omega }_u)$ and (cyan) $\tau =\pi /2({\omega }_d-{\omega }_u)$ (see text). (f) Expanded version of low-field region showing shape of avoided crossings versus coherence traces corresponding to two cuts (i),(ii), indicated in the upper panel.

Figure 3

Coherence decay behavior for an electron spin detecting a flip-flopping pair of nuclear spins, for a donor in silicon system (see Ref. [25]) with tunable interactions. $\mathcal{L}(B_0,t)$ exhibits a rich structure in the two-dimensional $\tau$, $B_0$ plane, which is not evident in the normal traces at constant $B_0$. Decoherence map is shown for different $R=\mathrm{\Delta }A/C_{12}$, $(\mathrm{\Delta }A=A_1-A_2)$. Large $R$ corresponds to weaker dipolar coupling $C_{12}$, and the maps trace the locus of a set of isolated sharp dips in coherence. For smaller $R$, there are no longer single dips; nevertheless, the envelopes [given by $F(\tau )$] are well defined and track the behavior of the underlying Floquet avoided crossings. The background oscillatory structure depends on $N_p$, the envelopes do not. Time $t\equiv 4N_p\tau$ (color scale linear, where black is 1, yellow is $<0.5$). Similar behavior is obtained for several transitions of Si:Bi and other donors, but specific parameters are for $12\to 9$ ESR transition of Si:Bi and $2N_p=40$.

Figure 4

Comparisons between the dip positions obtained with Eq. (4) (blue line) and average Hamiltonian theory Eq. (7) (red line) for the full coherence function (left) as well as its envelope (right). Within the field sweep there are global weak-coupling points (e.g., $B_0\approx 0.19\mathrm{T}$) where there is weak coupling regardless of the cluster properties and where the decoherence envelope collapses into a single sharp dip, a useful feature if high resolution is required: here, there is always good agreement with average Hamiltonian theory. These points correspond to so-called optimal working points [31, 32] of silicon donors. Hence, the advantage of such systems as future spin sensors, in addition to their very long $\sim 1\mathrm{s}$ coherence times, is that a magnetic field sweep could tune the dynamics from the weak- to strong-coupling regimes. (a),(d) $R=100$; (b),(e) $R=20$; (c),(f) $R=10$.

Figure 5

Fingerprinting multiple environmental spin cluster pairs via their decoherence “bar codes” illustrates the effect of 3-body correlations. The figure shows the coherence as a function of magnetic field $B_0$ and pulse interval $\tau$, calculated with a full numerical propagation under the total Hamiltonian for $N_p=100$. Panel (a) denotes three independent pairs while panel (b) shows three interacting spins, with otherwise equivalent dipolar couplings and intrabath interactions as illustrated in Fig. 1. One evident difference (and signature of a cluster of three spins) is the doublets due to the two separate subspaces of the three interacting spins. The splittings are directly related to the interactions. For the 3-cluster, in fact, there is a secular contribution from interactions between spins, greatly amplifying their contribution. The two right-hand panels (i) and (ii) show single traces corresponding to the cuts in (b) as well as the six corresponding eigenphases: in case (i), in a weak-coupling regime the dips are narrow and the eigenphases behave like three independent pairs; the eigenvalues correspond to conjugate pairs (with blue, red, and green lines denoting the three pairs). In case (ii) there is stronger coupling, the avoided crossings of the corresponding eigenphases are broader, giving rise to the features shown in the coherence maps.

Figure 6

Decoherence for an interacting cluster of three spins (3-cluster). The colored lines show comparisons with Eq. (10), showing excellent agreement with numerics obtained by diagonalization of the full joint sensor-cluster Hamiltonian.