Dynamical Structure Factors in Quantum Systems

• Dynamical Structure Factors (DSFs) are two-point correlation functions that capture the quantum and thermal dynamics of density and local operators in many-body systems.
• They are measured using advanced techniques such as weak measurement protocols and Fourier analysis to reconstruct excitation spectra without destroying quantum coherence.
• DSFs provide practical insights into collective excitations, phase transitions, and serve as benchmarks for both theoretical models and quantum simulation experiments.

The dynamical structure factor (DSF) is a fundamental two-point correlation function that encodes the quantum and thermal dynamics of density (or other local operators) in many-body systems. It is central in diverse contexts ranging from inelastic neutron and X-ray scattering to cold-atom Bragg spectroscopy and quantum simulation. The DSF accesses the spectral content and momentum structure of operator fluctuations and collective excitations, providing direct links between microscopic theory and experimental observables.

1. Formal Definition and Physical Significance

In a translationally invariant quantum lattice system, the DSF for the density operator is defined as

$$S(q, \omega) = \int_{-\infty}^{\infty} dt \, e^{i\omega t} \langle \rho_q(t) \, \rho_{-q}(0) \rangle,$$

where $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$ is the spatial Fourier transform of the number operator, and the average is taken at thermal equilibrium (or on the ground state at $T=0$). This form captures the propagation of collective excitations at wavevector $q$ and frequency $\omega$ (Altuntas et al., 2024).

The DSF more generally extends to any local operator $O_j$, with

$$S_O(q, \omega) = \int_{-\infty}^{\infty} dt \, e^{i\omega t} \langle O_q^\dagger(t) \, O_q(0) \rangle$$

and admits the Lehmann spectral representation

$$S_O(q, \omega) = \sum_f |\langle\psi_f | O_q | \psi_0\rangle|^2 \delta(\omega - (E_f - E_0)),$$

where $|\psi_0\rangle$ is the ground state and $|\psi_f\rangle$ are excited states (Baez et al., 2019). DSFs provide direct access to the excitation spectra, collective mode structure, and many-body response of the system.

In experimental terms, $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$0 is the central quantity accessed by inelastic scattering, e.g., the differential cross section of neutron or X-ray scattering is proportional to the DSF. In cold-atom systems, DSFs quantify the excitability of the system by weak density probes, encoding signals such as phonons, gaps, and spinon continua.

2. Measurement Theory: From Projective to Weak Measurements

Conventional measurement of unequal-time, two-point correlators underlying the DSF is fundamentally limited by projective (strong) measurement: a strong measurement of $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$1 collapses the wavefunction and destroys the quantum coherence required to access $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$2 for $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$3. Therefore, standard protocols relying on projective detection are insufficient for reconstructing the DSF (Altuntas et al., 2024).

To circumvent this, weak measurement protocols have been developed, notably in cold-atom experiments. The approach is structured as follows:

• Weak measurement protocol: Two brief, spatially resolved weak (homodyne) measurements are performed, separated in time by $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$4. Each measurement yields a noisy outcome,

$\rho_q(t) = \sum_j e^{-iqj} n_j(t)$5

with $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$6 the measurement strength and $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$7 Gaussian white noise.

• Ensemble cross-correlation: Repeatedly sampling the noisy outputs, the noise-averaged cross-correlation

$\rho_q(t) = \sum_j e^{-iqj} n_j(t)$8

reconstructs $\rho_q(t) = \sum_j e^{-iqj} n_j(t)$9 to leading order in $T=0$0.

• Spatial and temporal Fourier analysis: By Fourier transforming both in space and time,

$T=0$1

with $T=0$2 the spatially averaged cross-correlation, one recovers the DSF (Altuntas et al., 2024).

This protocol enables direct, minimally invasive measurement of $T=0$3 in quantum gases without the need for explicit external driving at $T=0$4.

3. Computational and Experimental Protocols

Several operational routes to DSF estimation in quantum simulators and experiments have emerged:

• Weak measurement in cold atoms (homodyne/PCI): Time-separated weak phase-contrast imaging of site populations, as detailed above, allows direct DSF estimation in systems modeled by, e.g., the Bose–Hubbard Hamiltonian (Altuntas et al., 2024).
• Quantum simulation protocols: For generic lattice models, a Ramsey-style sequence can map unequal-time correlators to single-time observables. For example, in spin systems, applying a local $T=0$5 pulse and evolving under the Hamiltonian, measurement of single-spin observables yields retarded correlators whose spatial and temporal Fourier transform reconstructs the DSF (Baez et al., 2019).
• Simulation and scaling considerations:
  • Statistical errors in weak measurement scale as $T=0$6; systematic back-action errors scale $T=0$7. An experimentally practical compromise has been demonstrated at $T=0$8 for sample sizes on the order of tens.
  • The frequency and momentum resolution depend on the sampling interval ($T=0$9) and total observation time ($q$0), as well as numerical aperture (NA) and depth-of-field for imaging systems.
  • In practical cold-atom experiments, phase-contrast imaging with sufficient NA captures long-wavelength modes with percent-level accuracy, with artifacts confined to high-$q$1 components (Altuntas et al., 2024).
• Matrix product state (MPS) and classical simulations: Numerical verification employs MPS-based sampling of measurement protocols, where ground-state preparation and time evolution are performed on, e.g., the 1D Bose–Hubbard model. The reconstruction matches exact DSF results both in the gapless superfluid and gapped Mott-insulating regimes (Altuntas et al., 2024).

4. Theoretical Structure and Exact Results

The formal theoretical structure of DSFs reveals rich connections to many-body physics:

• Sum rules: The DSF satisfies the $q$2-sum rule,

$q$3

ensuring consistency with total spectral weight from fluctuation–dissipation relations (Baez et al., 2019).

• Spectral properties: The DSF encodes the spectrum of elementary excitations, including phonon branches, Mott gaps, multi-magnon continua, and signatures of fractionalization or topology, depending on the model. In the 1D Bose–Hubbard model, the DSF directly displays the transition from a gapless linear phonon branch in the superfluid to a finite Mott gap in the insulating regime (Altuntas et al., 2024).
• Computational complexity: For general local Hamiltonians, estimation of the DSF is BQP-hard—classically intractable—since accurate evaluation of unequal-time correlators would allow simulation of arbitrary quantum circuits. This computational hardness is inherited by DSFs under any experimental or numerically practical approximation within polynomial precision (Baez et al., 2019).
• Practical quantum advantage: Quantum simulators operating with moderate system sizes ($q$4–70), commonly accessed in noisy intermediate-scale quantum (NISQ) hardware, already explore DSFs beyond the reach of classical diagonalization or Krylov subspace approaches (Baez et al., 2019).

5. Specific Applications, Error Analysis, and Experimental Implementation

The DSF has been computed and measured in a variety of contexts:

• Cold-atom systems: The DSF measured from PCI/weak-measurement protocols exhibits quantitative agreement with exact MPS correlators for the 1D Bose–Hubbard model, including the identification of phononic and Mott-insulating signatures (Altuntas et al., 2024).
• Error sources and trade-offs:
  • Small $q$5 improves linearity but requires more trajectories for statistical convergence; large $q$6 increases backaction errors. An empirical optimum is near $q$7 with $q$8 trajectories.
  • Limited imaging NA primarily affects high-$q$9 response; phonon features and low-$\omega$0 response remain robust, even with finite depth-of-field and aberrations (Altuntas et al., 2024).
• Temporal and spatial resolution: Adequate frequency resolution ($\omega$1) requires long observation windows, while momentum resolution is determined by the Fourier-space coverage of the imaging system.
• Broader platforms: Analogous protocols apply to trapped ions (e.g., Raman addressing and fluorescence imaging), Rydberg atom arrays (site-addressed pulses and van-der-Waals evolution), and superconducting qubits (microwave pulse sequences and dispersive readout), all yielding DSFs via similar Fourier analysis of measured correlators (Baez et al., 2019).

6. Physical Interpretation and Impact

Measurement of $\omega$2 provides direct access to the fundamental dynamical properties:

• Dynamical response and excitation spectrum: Peaks and gaps in $\omega$3 map onto elementary collective modes (e.g., phonons, rotons, magnons) and many-body spectral features (e.g., Mott gaps, spinon continua).
• Probing of phase transitions: The DSF provides clear spectroscopic fingerprints for phase transitions, such as closing of the phonon gap at the superfluid–Mott-insulator transition or the appearance of critical signatures and mode softening.
• Benchmarking and quantum simulation: The DSF is a stringent benchmark for both theoretical models and experimental platforms, enabling verification of quantum advantage and the study of real-time many-body dynamics in regimes inaccessible to classical computation.

The development of weak measurement protocols, along with scalable quantum-simulation-based measurement, now offers a minimally invasive, all–measurement-based route to the full dynamical response function, positioning the DSF as a central observable in quantum many-body physics (Altuntas et al., 2024, Baez et al., 2019).

---

References:

• "Dynamical Structure Factor from Weak Measurements" (Altuntas et al., 2024)
• "Dynamical structure factors of dynamical quantum simulators" (Baez et al., 2019)