Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems

Joonhee Choi; Hengyun Zhou; Helena S. Knowles; Renate Landig; Soonwon Choi; Mikhail D. Lukin

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Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems

Joonhee Choi, Hengyun Zhou, Helena S. Knowles, Renate Landig, Soonwon Choi, Mikhail D. Lukin

arXiv:1907.03771v3Jul 8, 2019quant-phcond-mat.dis-nncond-mat.mes-hall

TL;DR

Robust control of strongly interacting many-body spin dynamics lacks a systematic framework that handles disorder, unwanted interactions, and pulse imperfections across applications. The paper introduces a toggling-frame matrix formalism that yields algebraic design conditions for robust global pulse sequences, and demonstrates broad theoretical, numerical, and experimental applicability.

Problem
Periodic driving is vulnerable to spin inhomogeneities, finite pulse durations, and imperfect manipulation, while a general framework for designing sequences robust to these errors remains lacking.
Method
The framework represents pulse sequences through toggling-frame transformations of Pauli operators and derives algebraic conditions for designing robust periodic global-control sequences.
Results
The approach supports robust Hamiltonian engineering for coherence protection, quantum sensing, and quantum simulation, with efficient sequence-search strategies and experimental confirmation in a disordered, dipolar-interacting NV-center ensemble.
Takeaways & Limitations
The framework provides a systematic way to adapt pulse sequences to disorder, interactions, and control inhomogeneities across experimental platforms and application regimes.

Abstract

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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.

I. INTRODUCTION AND MOTIVATION

The paper introduces a systematic framework for designing globally controlled pulse sequences that engineer robust dynamics across systems with differing disorder, interactions, and control imperfections. It applies this framework to coherence protection, quantum sensing, quantum simulation, and experimental validation.

Motivation: Periodic control pulses can protect coherence, engineer qubit interactions, enable quantum sensing, and create nonequilibrium many-body phenomena.These applications include dynamical phase transitions, quantum chaos, and discrete time crystals.
Motivation: Conventional sequences optimized for dipolar-dominated nuclear-spin systems often do not apply when disorder dominates or interactions have more general forms.The paper identifies this mismatch across electronic spin ensembles and coupled-qubit arrays.
Motivation: A systematic framework is needed to handle spin inhomogeneity, finite pulse durations, and imperfect state manipulation while customizing sequences for different platforms.Existing robust sequences address some imperfections, but the paper describes a general treatment as lacking.
Approach: The proposed matrix representation uses interaction-picture transformations of S_z to derive intuitive algebraic conditions for robust, self-correcting global pulse sequences.The representation is designed to connect pulse construction with application-specific system properties.
Applications: Applications include coherence protection, robust quantum sensing, quantum simulation, and experimental demonstration in a disordered, dipolar-interacting NV-center ensemble.The sensing design combines effective-field optimization with disorder and interaction suppression, while simulation targets desired Hamiltonians.
Advances: The framework covers robustness, generic interaction types, energy-hierarchy-aware design, shortest sequence construction, and constrained numerical searches.It addresses on-site disorder, Ising, spin-exchange, and complex three-body interactions, while optimizing sequence length and sensing sensitivity.

II. GENERAL FORMALISM AND FRAME REPRESENTATION

The formalism represents pulse sequences through the interaction-picture rotation of the S_z operator rather than directly through applied pulses. This representation links sequence design to average-Hamiltonian dynamics and supports concise criteria for pulse imperfections.

General formalism: The framework specifies how the S_z operator rotates in the toggling frame, providing a one-to-one correspondence with the system’s average Hamiltonian.It extends related control-matrix and vector-modulation approaches to settings with general interactions and robust interacting-regime decoupling.
General formalism: The same matrix representation describes ideal decoupling performance and formulates concise criteria for incorporating pulse imperfections.The representation is based on interaction-picture transformations during the sequence’s free-evolution periods.

II.1. Frame Representation

The frame representation encodes pulse sequences as time-domain transformations of S_z, with a matrix and frame-duration vector determining the toggling-frame evolution. Weighted matrix sums then specify the engineered leading-order average Hamiltonian.

Toggling-frame construction: Average Hamiltonian theory describes the periodically driven dynamics as a weighted average of toggling-frame Hamiltonians over the free-evolution intervals.Each interval is governed by the system Hamiltonian transformed by the preceding global pulses.
Scope: The framework targets secular systems whose interactions conserve total magnetization along the quantization axis and includes Ising, exchange, dipolar, and three-body interactions.The underlying Hamiltonian contains on-site disorder and arbitrary interaction strengths for the supported interaction forms.
Toggling-frame construction: The method assumes periodic short pulses built from π/2 rotations about the x and y axes and tracks the interaction-picture transformation of S_z.Under ideal instantaneous pulses, S_z becomes a signed S_x, S_y, or S_z operator in each toggling frame.
Matrix representation: The 3×n matrix F records the signed axis reached by S_z in each frame, while τ=[τ_1,τ_2,…,τ_n] records the corresponding frame durations.A nonzero F_μ,k indicates that S_z transforms into S_μ during interval τ_k.
Canonical sequences: CPMG alternates the S_z orientation to suppress on-site disorder, whereas WAHUHA cycles through all three axes to cancel dipole-dipole interactions.Echo+WAHUHA combines these mechanisms to suppress disorder while symmetrizing interactions.
Pulse decomposition: Decomposing larger rotations into π/2 building blocks fixes rotation-axis ambiguities and naturally represents finite-duration and composite pulses.Intermediate frames can have zero duration and distinguish the axes used within a π pulse.
Hamiltonian engineering: Weighted row sums and weighted absolute row sums of F determine the leading-order average Hamiltonian for general two-body interactions.The resulting expressions cover disorder, Ising, symmetric exchange, and antisymmetric exchange terms.

II.2. Decoupling Conditions for Ideal Pulses

For ideal pulses, the frame matrix yields separate conditions for cancelling disorder, symmetrizing interactions, and decoupling higher-body terms. These conditions explain why different pulse sequences realize different effective Hamiltonians.

Disorder decoupling: Linear frame-matrix terms vanish when each axis has equal positive and negative entries, cancelling on-site disorder.For equidistant sequences, this is equivalent to each row summing to zero and produces a spin-echo-type structure.
Disorder decoupling: CPMG and echo+WAHUHA cancel on-site disorder, whereas WAHUHA leaves a residual chemical-shift term.The distinction follows from whether the sequence provides balanced frame signs for each axis.
Interaction symmetrization: Quadratic frame-matrix terms cannot generally be eliminated because the isotropic Heisenberg interaction is invariant under global rotations.They can instead be symmetrized into a Heisenberg Hamiltonian.
Interaction symmetrization: Interaction symmetrization requires equal time-weighted coverage of the x, y, and z axes.For equidistant pulses, this reduces to equal sums of absolute frame entries across axes.
Higher-body interactions: WAHUHA fully cancels spin-1/2 dipolar interactions to leading order, while the framework also extends the decoupling conditions to general three-body interactions for polarized initial states.The three-body extension uses results from unitary t-designs.

III. ROBUST PULSE SEQUENCE DESIGN

Robust pulse-sequence design represents a periodic sequence with a frame-duration vector and frame matrix, then expresses dynamical-decoupling and fault-tolerance requirements as conditions on that matrix. The goal is to cancel leading-order errors from finite-duration pulses and rotation imperfections.

Error sources: Finite pulse duration and imperfect spin manipulation add an error Hamiltonian δHavg to the driven dynamics.These effects include dynamics during pulses and over- or under-rotations.
Matrix representation: A periodic sequence is encoded by a frame-duration vector τ and a 3-by-n frame matrix F describing the toggling-frame Sz operator.The matrix representation turns robustness requirements into algebraic and pictorial conditions.
Robust design objective: Robust Hamiltonian engineering designs leading-order fault-tolerant, self-correcting sequences that suppress δHavg to zero.The cancellation target is δHavg = 0.

Average Hamiltonian Analysis for Finite Pulse Duration

Finite pulse duration causes additional average-Hamiltonian terms because the toggling frame rotates continuously between discrete frames. The matrix representation computes these terms and exposes cancellation rules for disorder, interactions, cross-terms, and pulse duration.

Finite-pulse analysis: The frame matrix directly provides an efficient route to the leading-order finite-pulse error Hamiltonian δHavg.The calculation accounts for the continuous frame evolution during each π/2 pulse.
Frame interpolation: During a finite pulse, the toggling-frame Sz operator smoothly interpolates between neighboring frames as a function of the rotation angle.The pulse unitary rotates spins globally along the pulse axis, with no rotation-angle errors assumed in this step.
Error-Hamiltonian structure: The pulse contribution is not only the weighted average of neighboring toggling-frame Hamiltonians; it also contains an interaction cross-term determined by neighboring frame columns.Distinct weighting factors apply to different interaction types.
Cross-term cancellation: Cross-terms cancel when the parity of neighboring frames sums to zero for every axis pair.Parity records whether neighboring frame signs remain the same or change when switching axes.
Effective dynamics: The effective Hamiltonian combines the ideal average Hamiltonian with finite-pulse corrections over the full sequence duration.The sequence representation provides the toggling-frame disorder and interaction terms entering this combination.

Analysis of Rotation Angle Error

Rotation-angle errors are described in the toggling frame through the chirality of each pulse transition. Opposite chiralities along an axis provide the cancellation mechanism for these errors.

Chirality picture: A rotation-angle error around one toggling-frame direction can be compensated by a rotation around the opposite direction.This gives an intuitive cancellation rule for imperfect global spin manipulation.
Chirality picture: The rotation axis in each toggling frame is obtained from the cross product of the frame vectors before and after the pulse.This cross product characterizes the chirality of the frame transition.
Error cancellation: The frame matrix supplies a cancellation condition for the rotation-angle error contribution δHrot.The condition is identified as condition 4 in Table 1.

Decoupling Conditions for Finite Duration Pulses

The framework incorporates finite pulse-duration effects into the effective Hamiltonian and expresses robust decoupling conditions through simple matrix-based rules. It can also analyze waveform and pulse-shape imperfections.

Finite pulse durations add corrections to effective free-evolution intervals and parity- and chirality-dependent terms from toggling-frame changes.
For equidistant pulses, disorder decoupling requires balanced positive and negative entries within each matrix row.
Interaction decoupling requires different matrix rows to contain equal numbers of positive and negative entries.
The same matrix-based approach can analyze waveform transients, pulse-shape imperfections, and rotation-axis errors.

IV.1. Suppression of Higher Order Effects

The framework extends beyond zeroth-order average Hamiltonians by evaluating Magnus-expansion terms and incorporating symmetry-based strategies to suppress higher-order effects. Reflection symmetry, concatenation, and second averaging provide complementary design tools.

The frame-matrix representation numerically evaluates higher-order Magnus terms, which involve commutators between Hamiltonians at different toggling-frame times.
The first-order contribution is computable from time-weighted toggling-frame Hamiltonians and yields algebraic conditions involving second-order polynomials in matrix entries.
For echo+WAHUHA with uniform intervals, the first-order term can be derived analytically under the stated short-pulse and interaction assumptions.
Reflection symmetry: Reflection-symmetric toggling-frame sequences cancel all odd-order Magnus terms and can be constructed by mirroring a sequence, with care at the central interface.
Concatenated sequence symmetrization: Concatenated symmetrization builds long sequences from systematically arranged short blocks to suppress higher-order effects.
Second averaging: Second averaging suppresses dominant noncommuting error terms by alternating control-pulse axes between Floquet cycles or using off-resonant driving.

IV.2. Enhanced Numerical Search of Pulse Sequences

The decoupling rules sharply reduce the pulse-sequence search space before full numerical simulation, making systematic searches more tractable. Additional application-specific constraints can narrow it further and organize sequences by robustness.

The decoupling rules provide a rapid way to restrict numerical searches to sequences likely to perform well before full-dynamics simulations.
612 ≈ 10^9 possible 12-interval sequences make exhaustive simulation prohibitively large when each toggling frame has six configurations.
Application-specific algebraic constraints, including phase-locked fast echoes for AC sensing, further reduce the candidate space and impose a target-frequency bandwidth filter.
14,080 constrained sequences were identified and sorted into four robustness classes, with Class I containing 448 sequences satisfying all decoupling rules.

IV.3. Extensions to Multi-Body Interactions

The formalism extends to secular three-body interactions and connects decoupling conditions with unitary t-designs. Under ideal pulses, the resulting symmetrized dynamics can preserve polarized states, while finite pulse durations require additional conditions.

Extensions to Multi-Body Interactions: The framework develops robust protocols for three-body interactions, including finite-pulse corrections and new decoupling conditions beyond the one- and two-body case.
Ideal pulses: For ideal pulses, the existing decoupling conditions suffice to fully suppress dynamics from secular three-body interactions for a polarized initial state.
Unitary t-design connection: The unitary t-design connection explains why averaging over the Clifford group symmetrizes interactions involving three particles or fewer into symmetric-group terms.
Any globally polarized initial state is an eigenstate with eigenvalue 0 under the symmetrized Hamiltonian in the stated example.
Finite pulse durations: Finite-pulse analysis expands the Hamiltonian as a polynomial in sequence-matrix entries to derive generalized three-body decoupling conditions.
System-targeted decoupling: System-targeted sequences distinguish disorder and interaction timescales, prioritizing faster disorder cancellation or interaction symmetrization according to the regime.

V. APPLICATION: DYNAMICAL DECOUPLING

The framework designs pulse sequences tailored to disorder, interactions, sensing, and control imperfections, while providing algebraic routes to efficient and robust Hamiltonian engineering. Applications include dynamical decoupling, optimal AC sensing, tunable quantum simulation, and experimental robustness demonstrations.

System-Targeted Dynamical Decoupling: System-targeted sequences can outperform one another as disorder and interaction strengths vary, with Seq. B remaining more stable across the studied regime.Seq. A performs better at small disorder, whereas Seq. B overtakes it as disorder increases; the crossover occurs within 0.009 < W(τ + tp) < 0.09 and 0.009 < J(τ + tp) < 0.023.
Design Efficiency: The algebraic conditions simplify sequence design, certify shortest lengths for target requirements, and constrain numerical searches to promising pulse sequences.They also support combinatorial optimization of sensing sequences and maximum sensitivity under specified constraints.
Quantum Sensing: Robust sensing sequences decouple disorder and interactions while recoupling resonant AC signals through an effective toggling-frame sensing field.The effective field’s orientation and strength are determined by the sequence’s frequency-domain modulation characteristics.
Quantum Simulation: The framework continuously interpolates among Ising, Heisenberg, XY, and dipolar-like interactions by tuning c, while independently controlling on-site disorder.Representative robust sequences access XY interactions with strong disorder and Ising interactions with weak disorder.
Experimental Demonstration: Experiments in disordered, interacting NV ensembles show that robust sequences suppress rotation-error-induced coherence modulations that appear for a non-robust sequence.The comparison uses the most general one- and two-body interaction form represented in the experiment.
Extensions: The formalism also extends to three-body interactions, enabling robust decoupling under the secular approximation for polarized initial states despite finite pulse durations.The construction relies on symmetrization over toggling frames covering the six ±x, ±y, and ±z directions.

D. Analysis of Rotation Axis Errors

The framework models rotation-axis errors and converts their cancellation into algebraic conditions on pulse-frame parities. It also identifies cases requiring an expanded history-dependent formalism and applies the broader engineering rules to many-body Hamiltonians.

Rotation-axis error model: Rotation-axis errors are modeled as small deviations of nominal x and y rotations toward the orthogonal transverse axis, with a shared-direction assumption for AWG-synthesized pulses.The assumed error modifies exp[−iπSx/2] and exp[−iπSy/2] by cross-axis terms proportional to ζ.
Rotation-axis error model: The error contribution is expressed through cross products between the disorder axis and the ideal rotation axis, then transformed into the toggling frame.This representation permits calculation of the rotation-axis error Hamiltonian during each pulse.
Cancellation condition: Cancellation requires equal sums of frame parities before and after π/2 pulses for each x, y, and z axis direction.Including π pulses makes the corresponding sum vanish, consistent with global phase rotations being removable by changing the axis.
Limitations: Different axis deviations between x and y rotations cannot be read directly from adjacent free-evolution frames and require tracking full frame configurations or rotation history.The paper expects this error to be small when both rotations are directly synthesized with an AWG.
Hamiltonian engineering: The framework independently engineers disorder and interaction terms, supports tunable Ising, Heisenberg, XY, and dipolar-like interactions, and can generate higher-body terms through higher-order Magnus commutators.The Heisenberg component remains invariant, some terms cannot be independently controlled, magnitudes may rescale, and higher-order contributions can matter.

H. Proof of Optimal Sensitivity Under Interaction Decoupling Constraints

Under interaction-decoupling constraints, the framework optimizes sensing by redistributing phase accumulation across toggling-frame axes rather than requiring equal phase accumulation. The resulting sequence reaches an effective-field limit above fast-echo sensing and can be further improved without an immediate echo structure.

Interaction-decoupling constraint: Interaction decoupling requires equal evolution times along the x, y, and z toggling-frame axes, while sensing optimization depends on unequal phase accumulation.The target signal’s sinusoidal modulation determines how phase should be distributed across the axes.
Optimal sensitivity: The optimized sequence is optimal for disorder-dominated interacting spin systems despite a relative sensing-field reduction compared with XY-8 without interaction decoupling.A fast spin-echo structure is needed both to detect AC signals and to suppress disorder effects in this regime.
Beyond immediate echoes: Sequences without an immediate echo structure can improve the result further and are provably optimal in the infinitely short-pulse limit.The improvement follows from allowing axis-dependent phase accumulation while preserving equal axis evolution times.
Optimization strategy: The effective field is optimized by minimizing By while keeping Bz minimal, thereby maximizing the phase-accumulation imbalance across axes.The optimization favors concentrating phase near sinusoidal antinodes along x and nodes along z.
Optimal sensitivity: |Beff| ≈0.634BXY-8 is the theoretical upper limit under interaction decoupling, about 10% above fast-echo-based sensing sequences.This bound is stated for the zero-pulse-duration limit.