Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems

We introduce a new approach for the robust control of quantum dynamics of strongly interacting many-body systems. Our approach involves the design of periodic global control pulse sequences to engineer desired target Hamiltonians that are robust against disorder, unwanted interactions, and pulse imperfections. It utilizes a matrix representation of the Hamiltonian engineering protocol based on time-domain transformations of the Pauli spin operator along the quantization axis. This representation allows us to derive a concise set of algebraic conditions on the sequence matrix to engineer robust target Hamiltonians, enabling the simple yet systematic design of pulse sequences. We show that this approach provides an efficient framework to (i) treat any secular many-body Hamiltonian and engineer it into a desired form, (ii) target dominant disorder and interaction characteristics of a given system, (iii) achieve robustness against imperfections, (iv) provide optimal sequence length within given constraints, and (v) substantially accelerate numerical searches of pulse sequences. Using this systematic approach, we develop novel sets of pulse sequences for the protection of quantum coherence, optimal quantum sensing, and quantum simulation. Finally, we experimentally demonstrate the robust operation of these sequences in a system with the most general interaction form.

Popular Summary

Controlling and manipulating quantum systems opens the door to new applications that harness quantum-mechanical effects, such as quantum information processing, metrology, and quantum simulation. However, efficient and robust manipulation of large-scale interacting many-body systems remains an outstanding challenge. We introduce and develop a new, systematic framework for robust engineering of quantum many-body systems via periodic pulsed driving. We illustrate its use with example applications such as protecting quantum information and sensing external signals with high sensitivity.

We introduce a new set of tools for the analysis and engineering of effective Hamiltonians in periodically driven quantum systems. Our framework is based on a simple matrix-based representation, where the creation of desired Hamiltonians and their robustness to imperfections translate to concise algebraic rules imposed on the matrix. This approach is widely applicable to a broad range of different systems and can be flexibly tailored to particular parameters of physical systems and target applications. We apply our framework to design improved pulse sequences for dynamical decoupling, quantum sensing, and quantum simulation and verify its performance on an experimental platform of interacting nitrogen-vacancy centers in diamond with the most general form of interactions.

Our work enables new approaches for robust quantum system engineering and opens new avenues for applications of quantum many-body quantum systems. Specifically, our framework enables realizations of entanglement-assisted quantum metrology and studies of out-of-equilibrium phenomena in a new regime of strongly interacting quantum systems.

Figure 1

Optimal pulse sequence design for robust dynamic Hamiltonian engineering. (a) Illustration of the interplay between disorder, interactions, and control errors in different quantum systems, with examples of a disorder-dominated system (system A) and an interaction-dominated system (system B). (b) Applications of driven quantum many-body systems. (c) Our Hamiltonian engineering approach is based on “Sudoku-puzzle-like” design rules, imposed on the matrix $F$ that represents a periodic pulse sequence. (d) Resulting robust periodic pulse sequence, optimized for a target application with system-targeted design. The sequence, characterized by $n$ finite-duration pulses $\{P_1,\dots ,P_n\}$ with free-evolution intervals between the pulses $\{{\tau }_1,\dots ,{\tau }_n\}$, is periodically applied to a system to dynamically engineer the Hamiltonian.

Figure 2

Efficient representation of pulse sequences. (a)–(c) Conventional illustrations of pulse sequences performing $\pm \pi$ and $\pm \pi /2$ rotations around the $\widehat{x}$ axis (red) and $\widehat{y}$ axis (blue). The spheres below the sequences describe time-domain transformations of spin frames (spin operators instead of states) in the interaction picture, periodically rotated by pulses from each sequence. In our framework, we focus only on the rotation of the $S^z$ operator in the time domain, whose orientation is highlighted as yellow and green arrows for positive and negative axis directions, respectively. The rotations employ the standard convention of a right-handed coordinate system. (a) CPMG sequence designed to decouple spins from on-site disorder. (b) WAHUHA sequence designed to suppress spin-spin interactions and (c) echo+WAHUHA sequence designed to cancel both disorder and interaction effects. (d)–(f) Efficient matrix-based representation of each periodic pulse sequence. The 3-by-$n$ matrix $\mathbf{F}=[F_{\mu ,k}]$ is employed to describe such time-domain $S^z$ operator transformations in a simple form; for example, $F_{\mu ,k}$, a nonzero matrix element at $(\mu ,k)$, means that $S^z$ transforms to $S^{\mu }$ in the $k$th frame. The bottom insets illustrate different decoupling characteristics of each sequence, where checks and crosses indicate success and failure in suppressing disorder (left circle) and interaction (right circle) effects, respectively.

Figure 3

Understanding control imperfections due to finite pulse duration and spin-manipulation error. (a) $\pi$-pulse treatment. A $\pi$ pulse should be interpreted as two consecutive $\pi /2$ pulses with zero time separation. The intermediate frame after the first $\pi /2$ pulse is represented as a thin line at the interface of the sequence matrix. (b) Finite pulse duration effect. Under a $\pi /2$ pulse, the spin frame smoothly rotates over the finite pulse duration $t_p$, which introduces errors in dynamic Hamiltonian engineering. (c)-(f) Leading-order error terms added to target effective Hamiltonian due to finite pulse duration and rotation angle errors; (c) On-site disorder Hamiltonians during the finite-duration pulse. To cancel them, the sequence matrix should contain an equal number of positive (yellow) and negative (green) intermediate frames for each axis. (d) Two-body Ising interactions during the finite-duration pulse. To suppress their effects, the sequence matrix should contain an equal number of intermediate frames for all three axes. (e) Two-body interaction cross-terms. To cancel them, the sequence matrix should contain an equal number of even and odd parities for every pair of neighboring frames between two axes; if the spin frame maintains (changes) its sign when switching to a different axis, the parity is defined as even (odd). (f) Spin-rotation angle error effects. To cancel them, the matrix should contain an equal number of positive and negative chiralities for every axis; the rotation axis and sign of the spin frame rotation in the toggling frame define positive/negative chirality along that axis.

Figure 4

Sequence symmetrization for higher-order suppression and numerical validation of the fault-tolerant decoupling rules. (a) Symmetrized $\mathrm{echo}+\mathrm{WAHUHA}$ sequence [Fig. 2] and (b) its matrix-based sequence representation. The $\pi /2$ and $\pi$ pulses are depicted in the same way as in Fig. 2. In (b), the intermediate frames are explicitly shown to incorporate finite pulse effects. The middle yellow bar at the interface introduces a small, residual disorder and interaction in the zeroth-order average Hamiltonian, which can be compensated by slightly adjusting free-evolution periods of neighboring toggling frames. Because of the reflection symmetry with respect to the center of the sequence, all odd-order expansion terms vanish. (c) Histograms of $1/e$ coherence decay times for classes of pulse sequences exhibiting different degrees of robustness. The sequences are generated by respecting the decoupling rules in Table 1, which help to efficiently reduce the size of the sequence search space. For sequence evaluation, we performed exact diagonalization studies for a disordered, interacting eight-spin system (see text for simulation details). The different sequences are classified into four distinct categories: class I (blue), class II (red), class III (green), and class IV (yellow), according to their robustness to control imperfections (see text for class definitions).

Figure 5

System-targeted dynamical decoupling. (a),(b) Sequence details and their matrix representations. The inset illustrates the pulse legend used to denote $\pi /2$ and $\pi$ pulses. Decoupling sequences can be characterized by their different timescales of disorder and interaction decoupling, denoted as $T_W$ and $T_J$, respectively. In (a), Seq. A (Cory-48) is designed for interaction-dominated systems, where axis permutation is performed on the fastest timescale to prioritize interaction symmetrization over disorder cancellation ($T_W\gg T_J$). In (b), Seq. B (DROID-60) is designed for disorder-dominated systems where echolike operations are performed on the fastest timescale to prioritize disorder suppression over interaction symmetrization ($T_W\ll T_J$). (c) Numerical simulation results of the dynamical decoupling performance of Seq. A and Seq. B for a wide range of disorder $W$ and interaction $J$ strengths. The figure of merit for the comparison is the coherence time $T_2$, extracted as the $1/e$ decay time of initially polarized spin states. More specifically, we perform an exact diagonalization simulation of a disordered, interacting ensemble of 12 spins. In the simulations, we initialize all spins along the $\widehat{x}$ axis, periodically drive them with the pulse sequences, and measure the spin decoherence profile at stroboscopic times $t=NT$. Each parameter set is averaged over 100 disorder realizations and the averaged profile is used to identify $T_2$. The $\pi$-pulse duration and spacing are chosen to be $t_p=25\mathrm{ns}$ and $\tau =20\mathrm{ns}$, respectively. We have verified that choosing different $t_p$ values does not qualitatively change the results.

Figure 6

Optimal ac signal sensing with robust pulse sequences. (a),(b) Sequence details for (a) the conventional XY-8 sequence optimized for noninteracting systems and (b) the robust sensing sequence, Seq. C, optimized for interacting systems. The matrix representations, $\mathbf{F}=[{\overrightarrow{F}}_x;{\overrightarrow{F}}_y;{\overrightarrow{F}}_z]$, for each sequence are shown with resonant ac signals (blue curves) at the frequency $f_0$ to be detected. The sequence length $T$ includes both free-evolution periods and finite pulse durations. The $\pi /2$ and $\pi$ pulses are depicted in the same way as in the pulse legend of Fig. 5. In (b), composite pulses consisting of two $\pi /2$ pulses are extensively used to preserve sequence robustness while performing ac-field sensing (see Sec. 6b for a detailed discussion).

Figure 7

Hamiltonian engineering for quantum simulations. (a) Floquet-engineered many-body Hamiltonians with tunable disorder strengths and interaction types. The various interaction types, including Ising ($c=0$), Heisenberg ($c=1/3$), XY ($c=1/2$), and dipolar interactions ($c=1$), can be realized by varying the single parameter $c$, which is defined as the relative fraction of the total time evolution along the $\widehat{z}$ axis in the toggling frame. In this example, the system is chosen to be a disordered dipolar interacting spin ensemble with $J_{ij}^S=-J_{ij}^I$ and $J_{ij}^A=0$, naturally realized with NV centers in diamond [80]. The dashed line indicates the maximum effective disorder strength $W_{\mathrm{eff}}$ achievable for a given interaction type, $W_{\mathrm{eff}}/W=\sqrt{c^2+(1-c{)}^2/2}$, where $W$ is the native on-site disorder strength. The gray area denotes the Hamiltonian regimes that are not accessible. (b) Examples of robust periodic pulse sequences that generate (I) XY interactions with strong disorder and (II) Ising interactions with weak disorder. These sequences correspond to the two green markers in (a).

Figure 8

Experimental demonstration. (a) Sample used in experiments, containing strongly disordered, interacting NV centers in black diamond. We isolate two levels from the spin-1 ground state, $\{|0⟩,|\pm 1⟩\}$, by a static Zeeman shift (red arrow), to define an effective spin-$1/2$ (qubit) system, $\{|0⟩,|-1⟩\}$. Resonant pulsed driving at frequency ${\omega }_0$ is applied to the spin to manipulate its quantum states. In the sample, NV centers at a distance interact with one another via magnetic dipolar interactions (circled diagram). (b) Comparison of coherence decay of driven spins under Seq. A (Cory-48, red), Seq. B (blue), and XY-8 (green). For sequence evaluation, we use two degenerate NV groups, which exhibit position-dependent random couplings with various interaction types including Ising, symmetric, and antisymmetric spin-exchange interactions [80], thus representing the most general class of one- and two-body interaction Hamiltonians. The pulse spacing and duration are fixed to $\tau =15\mathrm{ns}$ and $t_p=6\mathrm{ns}$, respectively, and resulting decay profiles are fitted to a stretched exponential function. (c) Sequence robustness against spin-manipulation error. Here, we tested sequences on a single, isolated NV group. In the presence of systematic spin-rotation-angle deviation $0.925(\pi /2)$ [gray points in (d)], the nonrobust Seq. D (right) shows modulations in the coherence profile for all three initial states polarized along the $\widehat{x}$ (blue), $\widehat{y}$ (red), and $\widehat{z}$ (yellow) axes; the robust Seq. B is insensitive to such errors, corroborated by the absence of the modulation. (d) Modulation frequency for Seq. B and Seq. D as a function of systematic rotation angle error relative to the perfect $\pi /2$ rotation, ${\mathrm{\Omega }}_0{t}_p=\pi /2$. The small lateral offset observed in Seq. D at zero modulation frequency is due to a slight Rabi frequency calibration error.

Figure 9

Complete representation of periodic pulse sequences. All sequences are shown in the conventional pulse representation and our toggling-frame transformation-based representation. In the toggling-frame representation, squares indicate free-evolution times of length $\tau$ and narrow lines indicate short intermediate frames; yellow is positive and green is negative. Seq. A, the Cory-48 sequence, which decouples interactions on a faster timescale and disorder on a slower timescale, and is robust against leading-order imperfections; Seq. B, pulse sequence designed to decouple disorder on a faster timescale and interactions on a slower timescale, and is robust against leading-order imperfections; XY-8, standard pulse sequence for dynamical decoupling with noninteracting spins; Seq. C, pulse sequence to illustrate robust dynamical decoupling of interactions and disorder, and well-aligned sensing resonances; Seq. D, pulse sequence that has identical free-evolution frames as Seq. B, but with intermediate frames in the second half permuted to remove the robustness against rotation angle errors. Seq. E, simple sequence to fully symmetrize interactions; however, the transverse spin operators do not return to themselves at the end of this sequence; Seq. F, modified version of Seq. E, in which the composite pulses ensure that the transverse spin operators return at the end of the sequence; Seq. G, minimal pulse sequence that meets all dynamical decoupling and robustness requirements (Table 1 in the main text) without composite pulses; Seq. H, minimal pulse sequence that meets all dynamical decoupling and robustness requirements without composite pulses, and has a fast spin echo structure; Seq. I, minimal pulse sequence that achieves optimal vector sensitivity under interaction-decoupling constraints, and illustration of phase accumulation along each axis.