Quantum Metrology with Strongly Interacting Spin Systems

TL;DR

This paper introduces a robust quantum control method for dense spin ensembles in diamond, significantly enhancing coherence time and surpassing interaction-imposed limits in magnetic field sensing, advancing solid-state quantum metrology.

Contribution

It presents a novel robust quantum control technique that mitigates spin-spin interactions, disorder, and control imperfections in dense spin ensembles for improved quantum sensing.

Findings

Achieved a five-fold increase in coherence time.

Demonstrated breaking the sensitivity limit imposed by interactions.

Enabled development of highly sensitive solid-state magnetometers.

Abstract

Quantum metrology makes use of coherent superpositions to detect weak signals. While in principle the sensitivity can be improved by increasing the density of sensing particles, in practice this improvement is severely hindered by interactions between them. Using a dense ensemble of interacting electronic spins in diamond, we demonstrate a novel approach to quantum metrology. It is based on a new method of robust quantum control, which allows us to simultaneously eliminate the undesired effects associated with spin-spin interactions, disorder and control imperfections, enabling a five-fold enhancement in coherence time compared to conventional control sequences. Combined with optimal initialization and readout protocols, this allows us to break the limit for AC magnetic field sensing imposed by interactions, opening a promising avenue for the development of solid-state ensemble magnetometers with unprecedented sensitivity.

Equations

$$H=H_s+H_{\Omega }(t)+H_{\text{AC}}(t),$$

$$H_{\text{avg,AC}}$$

$$={\gamma }_{\text{NV}}{\vec{B}}_{\text{eff}}\cdot \sum _i{\vec{S}}_i,$$

Full text

Electronic spins associated with color centers in diamond have recently emerged as a promising platform for nanoscale precision sensing and imaging, with superior sensitivity and spatial resolution Schirhagl et al. (2014); Rondin et al. (2014). A common approach to improving sensitivity of such quantum systems is to utilize a dense ensemble of individual sensors and take advantage of parallel averaging. However, beyond a certain density, undesired interactions between sensors result in a rapid decay of the ensemble coherence Mitchell (2019), limiting the overall sensitivity of the quantum system (Fig. 1A). Moreover, in practice, disorder and control errors further deteriorate coherence and metrological sensitivity of such interacting many-body quantum systems. Over the past few decades, pulsed control techniques realizing dynamical decoupling and motional averaging have been developed and deployed to manipulate interacting ensembles of quantum systems. While extremely successful in the context of nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI) Haeberlen and Waugh (1968); Waugh et al. (1968); Mansfield (1971); Burum and Rhim (1979); Chmelka and Pines (1989); Cory et al. (1990) and atomic gas magnetometers Allred et al. (2002), the efficacy of these techniques is severely limited in the presence of strong disorder and other imperfections. Therefore, these techniques are not directly applicable to quantum sensors based on electronic spin ensembles SM , where such effects are prominent.

In this Letter, we demonstrate a new approach for quantum sensing with disordered interacting spin ensembles. Our method uses periodic pulsed manipulation (Floquet engineering) of a spin ensemble Haeberlen and Waugh (1968) to detect an external signal of interest with high sensitivity, while simultaneously decoupling the effects of interactions and disorder, and being fault-tolerant against the leading-order imperfections arising from finite pulse durations and experimental control errors. Specifically, we introduce a set of simple rules imposed on the pulse sequence for disordered, interacting systems and a new, generalized picture of AC field sensing that is essential in the interacting regime. These go beyond existing dynamical decoupling techniques for non-interacting spin systems Suter and Álvarez (2016); Degen et al. (2017); Lang et al. (2017); Genov et al. (2017) and low-disorder NMR systems Haeberlen and Waugh (1968); Waugh et al. (1968); Mansfield (1971); Burum and Rhim (1979); Chmelka and Pines (1989); Cory et al. (1990), allowing us to design and implement a sequence that breaks the sensitivity limit on AC sensing imposed by spin-spin interactions for the first time.

Our experimental system consists of a dense electronic spin ensemble of NV centers in diamond Kucsko et al. (2018), as shown in Fig. 1B. NV centers exhibit long-lived spin coherence even at room temperature and are excellent sensors of magnetic fields, electric fields, pressure, and temperature Schirhagl et al. (2014); Rondin et al. (2014); Taylor et al. (2008); Degen (2008). Our sample has a high density of NV centers ($\sim$15 ppm, see SM ), with long-range magnetic dipolar interactions between the spins as well as strong on-site disorder originating from other paramagnetic impurities, inhomogeneous strain in the diamond lattice, and local electric fields Kucsko et al. (2018). The bulk diamond is etched into a nanobeam to improve control homogeneity and confine the probing volume to $V=8.1(9)\times 10^{-3}~{}\mu$m3 SM . The NV center ground state is an electronic $S=1$ spin, and we apply a static magnetic field to isolate an ensemble of effective two-level systems formed of NVs with the same crystallographic orientation. We initialize and detect the spin states optically and use resonant microwave excitation to drive coherent spin dynamics. See SM for further details of the measurement sequences.

Spin echo measurements Slichter (2013) reveal that the coherence decay time $T_{2}$ in our dense NV ensemble is limited to only $1.0~{}\mu s$ (Fig. 2C, gray crosses). The conventional method to extend $T_{2}$ beyond the spin echo is the XY-8 dynamical decoupling sequence Suter and Álvarez (2016), consisting of equally-spaced $\pi$ pulses along the $\hat{x}$ and $\hat{y}$ axes (Fig. 2A, top row). In our system, however, the XY-8 sequence only provides a small improvement ($T_{2}=1.6~{}\mu$s, Fig. 2C, blue circles), since the XY-8 $T_{2}$ is limited by strong spin-spin interactions, which are not affected by $\pi$ rotations (see SM for density scaling measurements which further confirm that the XY-8 $T_{2}$ is interaction-limited).

In order to significantly extend $T_{2}$ in the presence of interactions and control imperfections, we use a novel approach to design pulse sequences Choi et al. (2019). We model our spin system, including the control fields and the external AC magnetic field target signal, by the Hamiltonian SM ; Kucsko et al. (2018)

[TABLE]

where the internal system Hamiltonian is $H_{s}=\sum_{i}h_{i}S_{i}^{z}+\sum_{ij}\frac{J_{ij}}{r_{ij}^{3}}(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y}-S_{i}^{z}S_{j}^{z})$, global spin-control pulses are given by $H_{\Omega}(t)=\sum_{i}(\Omega_{i}^{x}(t)S_{i}^{x}+\Omega_{i}^{y}(t)S_{i}^{y})$ and the external target signal is $H_{\text{AC}}(t)=\gamma_{\text{NV}}B_{\text{AC}}\cos(2\pi f_{\text{AC}}t-\phi)\sum_{i}S_{i}^{z}$. Here, $S_{i}^{\mu}$ ($\mu=x,y,z$) are spin-1/2 operators, $h_{i}$ is a random on-site disorder potential with zero mean that follows a Gaussian distribution with standard deviation $W=(2\pi)~{}4.0$ MHz, $J_{ij}/r_{ij}^{3}$ is the anisotropic dipolar interaction strength between two spins of the same crystallographic orientation at a distance $r_{ij}$, with strength $J=(2\pi)~{}35$ kHz at a typical separation of 11 nm, $\Omega_{i}^{x,y}(t)$ are the global control amplitudes exhibiting weak position dependence due to spatial field inhomogeneities, $\gamma_{\text{NV}}$ is the gyromagnetic ratio of the NV center, and $B_{\text{AC}},f_{\text{AC}}$ and $\phi$ are the amplitude, frequency and phase of the target AC signal, respectively.

Our approach employs average Hamiltonian theory to engineer the system evolution through pulsed periodic manipulation of the spins Haeberlen and Waugh (1968); Slichter (2013). A sequence composed of $n$ equidistant control pulses {$P_{k};k=1,2,..,n$} with spacing $\tau$ defines a unitary time-evolution operator $\mathcal{U}(T)=P_{k}e^{-iH_{s}\tau}\cdots P_{1}e^{-iH_{s}\tau}$ over the Floquet period $T$. If the pulse spacing $\tau$ is much shorter than the timescales of the system Hamiltonian ($\tau\ll\frac{1}{W},\frac{1}{J}$), the unitary operator $\mathcal{U}(T)$ can effectively be approximated by a time-independent average Hamiltonian as $\mathcal{U}(T)\approx e^{-iH_{\text{avg}}T}$, with $H_{\text{avg}}=\frac{1}{T}\sum_{k=1}^{n}\tilde{H}_{k}$, and $\tilde{H}_{k}=(P_{k-1}\cdots P_{1})^{\dagger}H_{s}(P_{k-1}\cdots P_{1})$, $\tilde{H}_{1}=H_{s}$. Motivated by this picture, we develop a pulse sequence which generates a desirable form of $H_{\text{avg}}$ from the $H_{s}$ intrinsic to the system.

Hamiltonian engineering can be understood as the result of a sequence of frame transformations (also known as toggling frame transformations) which rotate the spin operators in the interaction picture (Fig. 2B): for example, a $\pi$-pulse flips $S_{i}^{z}\rightarrow-S_{i}^{z}$, while a $\pi/2$-pulse rotates $S_{i}^{z}\rightarrow\pm S_{i}^{x,y}$ depending on the rotation axis. Importantly, the average Hamiltonian is uniquely specified by such toggling frame transformations of the $S^{z}$ operator SM , resulting in simple decoupling conditions that facilitate the procedure to find desired pulse sequences.

For example, any pulse sequence in which the transformed $S^{z}$ operator spends equal time along the positive and negative direction for each axis—effectively producing a spin echo along all three axes—suppresses the on-site disorder Hamiltonian. Similarly, in order to symmetrize the dipolar interaction into a Heisenberg interaction Hamiltonian (where polarized states are eigenstates and coherence is preserved Choi et al. (2017a)), we require that the transformed $S^{z}$ operator spend an equal amount of time in each direction $\hat{x}$, $\hat{y}$, $\hat{z}$ SM ; Waugh et al. (1968). Furthermore, we can prioritize one condition over another to find a pulse sequence that better suits a given system; since disorder is dominant in our spin ensemble ($W\gg J$), we perform the echo operation more frequently than interaction symmetrization.

In realistic situations, the above strategy will be affected by various imperfections, such as disorder and interactions acting during the finite pulse durations and errors of each control pulse, resulting in imperfections $\delta H_{\text{avg}}$ to the target effective Hamiltonian, $H_{\text{eff}}=H_{\text{avg}}+\delta H_{\text{avg}}$. Indeed, if we simply design a pulse sequence that decouples disorder and interactions only in the ideal pulse limit (Seq. A in Fig. 2A), we see only a marginal increase in coherence time compared to XY-8, yielding $T_{2}=2.8~{}\mu$s (Fig. 2C, green diamonds). A careful examination of Seq. A reveals that pulse-related imperfections play a dominant role in the dynamics, illustrating the importance of robust sequence design  SM .

To address this key challenge, a number of strategies have been proposed that aim to cancel or control the undesired Hamiltonian terms acting during the finite pulse duration Mansfield (1971); Burum and Rhim (1979); Chmelka and Pines (1989); Cory et al. (1990); Lang et al. (2017); Genov et al. (2017); however, a simple and systematic approach to treating the imperfections in a general setting, particularly for interacting systems, is still lacking. Remarkably, we find that the transformations of the $S^{z}$ operator during the free evolution intervals are in fact sufficient to predict and suppress the errors during pulse rotations. This is a consequence of the fact that the unwanted residual Hamiltonian acting within the finite pulse duration can be uniquely determined by the two toggling-frame Hamiltonians on either side of the control pulse SM . This insight motivates us to describe the $S^{z}$ operator transformations using the matrix, ${\bf F}=[F_{\mu,k}]=2\textrm{Tr}[S^{\mu}\tilde{S}^{z}_{k}]$, $\mu=\hat{x},\hat{y},\hat{z}$, where $\tilde{S}^{z}_{k}$ is the transformed spin operator within the $k$-th free evolution period. Crucially, this allows us to construct a simple set of algebraic conditions imposed on the matrix ${\bf F}$ to formalize the above decoupling rules, enabling not only the suppression of disorder and interaction effects during free evolution periods, but also the cancellation of dominant imperfections arising from finite pulses as well as rotation angle errors (see Eqs. (S8-S11) in SM for the exact expressions of decoupling conditions). This also enables system-customized design for optimal AC-field sensing, taking into account the energetic hierarchy between on-site disorder, spin-spin interactions and control imperfections in a given system, significantly extending beyond existing techniques.

By satisfying these rules, we can thus systematically generate robust pulse sequences that to first order yield a pure Heisenberg Hamiltonian with $\delta H_{\text{avg}}=0$. Seq. B, as shown in Fig. 2A, is an example of a robust pulse sequence designed with our formalism SM . Due to its robustness against all leading-order effects, it shows a significant extension of coherence time compared to the sequences described above, reaching $T_{2}=7.9$ $\mu$s (Fig. 2C, squares). Moreover, the coherence time is independent of the initial state.

We now apply this method to quantum sensing, where our goal is to robustly engineer the dynamics of the spin ensemble to be sensitive to the target sensing signal. AC magnetic field sensing typically uses periodic inversions of the spin operator between $S^{z}$ and $-S^{z}$ in the interaction picture, driven by a train of equidistant $\pi$ pulses at a separation of $\frac{1}{2f_{\mathrm{AC}}}$. This modulation causes cumulative precession of the sensor spin when the AC field sign change coincides with the frame inversion, resulting in high sensitivity to a signal field at $f_{\mathrm{AC}}$. Our interaction-decoupling sequences explore all three frame directions $S^{x}$, $S^{y}$, $S^{z}$ and AC selectivity requires synchronized periodic frame inversions in each of the three axes, while preserving the desired $H_{\text{avg}}$ and suppressing $\delta H_{\text{avg}}$ to maintain long coherence times.

In Fig. 3A we illustrate how this is achieved in Seq. B. The pulses lead to periodic changes in the sign and orientation of the interaction-picture $S^{z}$ operator, depicted by the time-domain modulation functions for each axis direction, $F_{x},F_{y}$ and $F_{z}$. The detailed resonance properties of the pulse sequence can be characterized by the Fourier transforms $\tilde{F}_{\mu}(f)=|\tilde{F}_{\mu}(f)|e^{-i\tilde{\phi}_{\mu}(f)}$ of $F_{\mu}$ for $\mu=x,y,z$, where $\tilde{\phi}_{\mu}(f)$ is the spectral phase for a given axis $\mu$. Fig. 3B shows the calculated spectral intensities along different axes, $|\tilde{F}_{x}(f)|^{2}$,$|\tilde{F}_{y}(f)|^{2}$, and $|\tilde{F}_{z}(f)|^{2}$, as well as the total intensity $|\tilde{F}_{t}(f)|^{2}$ SM . At the dominant resonance of the total intensity (red arrow in Fig. 3B), all three axes exhibit a phase-locked periodic sign modulation (Fig. 3A), leading to constructive phase accumulation and high sensitivity.

In order to intuitively understand our sensing protocol and quantify its sensitivity, we generalize the average Hamiltonian analysis to incorporate AC signal fields, finding SM

[TABLE]

where $\vec{B}_{\text{eff}}$ is an effective magnetic field vector in the interaction picture which appears static to the driven spins. This allows a simple interpretation of our scheme: the spins undergo a precession around $\vec{B}_{\text{eff}}$, with the field orientation and magnitude determined by the frequency-domain modulation functions $\tilde{F}_{\mu}$ and $|\tilde{F}_{t}|$, respectively. For conventional sequences with $\vec{B}_{\text{eff}}\parallel\hat{z}$, the requirements for interaction symmetrization dictate that only $1/3$ of the total sensing time can be spent along any given axis, resulting in a significant loss of sensitivity. In contrast, for our optimal Seq. B, the signal at the principal resonance $f_{\text{AC}}$ gives rise to $|\tilde{F}_{x}|=|\tilde{F}_{y}|=|\tilde{F}_{z}|$ with $\tilde{\phi}_{x}=\tilde{\phi}_{y}=\tilde{\phi}_{z}$, leading to $\vec{B}_{\text{eff}}\propto[\frac{1}{3},\frac{1}{3},\frac{1}{3}]$ with the total strength $|\vec{B}_{\text{eff}}|$ reduced by a factor of only $1/\sqrt{3}$. This reduction is fundamental and, in fact, close to optimal Choi et al. (2019), given the requirements to suppress the effects of spin-spin interactions via symmetrization. However, despite the reduction, the sensitivity is still improved because the coherence time is extended by interaction suppression. Moreover, the sensitivity to the signal $B_{\text{AC}}$ is maximized when the spins are initialized perpendicular to the $\vec{B}_{\text{eff}}$ direction—to allow the largest precession orbit (Fig. 3C)—and the corresponding optimal readout requires an unconventional rotation axis $[-1,1,0]$ and angle $\arccos(\sqrt{2/3})$ to bring the precession plane parallel to the $\hat{z}$ axis. Indeed, as shown in Fig. 3D, we observe a larger contrast when spins are initialized in an optimal direction along $[1,1,-2]$, compared to an initialization in the conventional $\hat{x}$ direction.

We now proceed to characterize the AC magnetic field sensitivity, defined as the minimum detectable signal amplitude per unit time, $\eta=\frac{\sigma_{S}}{|dS/dB_{\text{AC}}|}$. Here, $S$ is the spin contrast, $\sigma_{S}$ is the uncertainty of $S$ for one second of averaging, and $|dS/dB_{\text{AC}}|$ is the gradient of $S$ with respect to the field amplitude $B_{\text{AC}}$.

Figure 4A shows the measured contrast $S$ as a function of $B_{\text{AC}}$ under optimal conditions for Seq. B and the conventional sequence XY-8, where we choose acquisition parameters for each of the two sequences that optimize their respective absolute sensitivities SM . We find that the spin contrast shows faster oscillations and a significantly steeper maximum slope under Seq. B, indicating that it is more sensitive to the external signal than XY-8. This is due to a combination of optimal state preparation and readout schemes as well as significantly improved coherence times, despite a reduced effective signal strength $|\vec{B}_{\text{eff}}|$. In Fig. 4B, we show the sensitivity scaling with phase accumulation time, for a fixed signal frequency and integration time, finding good agreement with the theoretical prediction SM . We find that the volume-normalized sensitivity $\eta_{V}=\eta\sqrt{V}$ of Seq. B, designed with our optimal sensing approach, reaches more than 40% improvement in sensitivity over the conventional XY-8 sequence. Moreover, we find that the improvement persists even for sizable (10%) systematic rotation angle errors, confirming the sequence robustness SM . With these enhancements, we demonstrate $\eta_{V}=8.3(9)$ $\textrm{nT}\cdot\mu\textrm{m}^{3/2}/\sqrt{\textrm{Hz}}$ for Seq. B, among the best volume-normalized AC sensitivities for solid-state magnetometers measured thus far Wolf et al. (2015) (see Ref. SM for a direct comparison).

Our work establishes a novel approach to quantum sensing by utilizing robust interaction decoupling, and provides the first demonstration of a solid-state ensemble quantum sensor surpassing the interaction limit. Additionally, our design formalism and generalized effective field picture are also directly applicable to robust DC field sensing. While our approach already yields a significant improvement in sensitivity, the $T_{2}$ reached here is still shorter than the depolarization time $T_{1}\sim 100~{}\mu$s in our dense spin ensemble. This $T_{2}$ is likely limited by waveform distortions in control pulses and higher-order terms in the average Hamiltonian analysis. The former can be mitigated by waveform engineering Khaneja et al. (2005), and the latter by sequence symmetrization Mansfield (1971) or disorder-reduction via spin-bath engineering de Lange et al. (2012); Bauch et al. (2018). Together with new diamond growth techniques Edmonds et al. (2012), double-quantum magnetometry Fang et al. (2013); Bauch et al. (2018) and improved photon collection Wolf et al. (2015), these improvements may push the volume-normalized sensitivity to single-digit picotesla level in a $\mu$m3 volume SM , opening the door to many applications, such as high-sensitivity nanoscale NMR Aslam et al. (2017); Glenn et al. (2018) and investigations of strongly correlated condensed matter systems Casola et al. (2018). Beyond applications to diamond magnetometry, the robust sequence design presented in this work can be extended to engineer a broad class of many-body Hamiltonians Choi et al. (2017a, 2019) in a wide variety of quantum hardware platforms, providing a useful tool for quantum information processing Stajic (2013), simulation Zhang et al. (2017); Choi et al. (2017b), and metrology Kitagawa and Ueda (1993); Cappellaro and Lukin (2009); Choi et al. (2018).

The authors thank W. W. Ho, A. Omran, C. Ramananthan, T. Sumarac and E. Urbach for helpful discussions, A. Sushko for experimental assistance, and F. Machado, L. Martin for critical reading of the manuscript. This work was supported in part by CUA, NSSEFF, ARO MURI, DARPA DRINQS, Moore Foundation GBMF-4306, Samsung Fellowship, Miller Institute for Basic Research in Science, NSF PHY-1506284, Japan Society for the Promotion of Science KAKENHI (No. 26246001), EU (FP7, Horizons 2020, ERC), DFG, SNSF, and BMBF. H.Z., J.C., S.C., M.D.L. conceived the idea. H.Z., J.C., S.C. performed the theoretical analysis and pulse sequence design. H.Z., J.C., R.L., A.M.D., H.S.K. performed the experiment and data analysis. J.I., F.J., S.O., H.S. provided the sample. H.Z., J.C., H.S.K. wrote the manuscript, with input from all authors. M.D.L. and H.P. supervised the research. The authors declare no competing financial interests. All data needed to evaluate the conclusions are present in the paper and/or the supplementary materials. Additional data related to this paper may be requested from the authors.