Pair-Density Modulation (PDM)

• Pair-Density Modulation (PDM) is a phenomenon where the local density or amplitude of paired entities, such as electrons, Cooper pairs, or pulses, is periodically modulated, preserving lattice translations while breaking intra-unit-cell symmetries.
• Theoretical frameworks use tight-binding models, BdG calculations, and Landau-Ginzburg analyses to elucidate PDM's emergence via mechanisms like glide-mirror symmetry breaking and nematic order.
• Experimental realizations in superconductors, high-energy plasmas, and neuromorphic devices validate PDM through STM imaging, particle-in-cell simulations, and pulse-density modulation, with implications for quantum materials and signal processing.

Pair-Density Modulation (PDM) represents a family of phenomena across physics in which the spatial or temporal density of a fundamental entity—most commonly electron or Cooper pairs, but also particle pairs in quantum field or plasma contexts, or pulses in signal processing—is modulated, forming periodic or otherwise structured patterns. The specific physical realization and microscopic mechanism underlying PDM depends strongly on context, with major classes now established in condensed matter physics (modulation of the superconducting pairing amplitude, with or without translational symmetry breaking), high-energy-density plasmas (periodic structuring of electron-positron pairs), quantum kinetic pair production (density modulation via amplitude-modulated fields), and information processing hardware (modulation of pulse density for encoding and conversion).

1. Microscopic Definitions and Physical Manifestations

PDM arises whenever the local density or amplitude of a pair-related quantity, such as the Cooper-pair amplitude in a superconductor or the density of electron-positron pairs in a plasma, develops a static or dynamical modulation:

• In superconductors, PDM refers to a modulation of the superconducting order parameter on microscopic (usually unit-cell) length scales. Unlike conventional pair-density-wave (PDW) states, where the order parameter varies at a nonzero wavevector Q and breaks lattice translation, PDM states exhibit modulation at reciprocal Bragg vectors, preserving lattice translations but breaking certain intra-unit-cell symmetries (Kong et al., 15 Apr 2024, Papaj et al., 24 Jun 2025, Chan et al., 5 Dec 2025).
• In relativistic plasma physics, PDM describes the periodic concentration (“bunching”) of electron-positron pairs in space and time, driven by trapping in laser-induced standing wave (SW) electromagnetic fields, with the modulation wavelength typically at half the driving laser’s wavelength (Liu et al., 2017).
• In quantum kinetic pair production, PDM emerges when an oscillating electric field is amplitude-modulated, opening extra (“side-band”) channels for pair creation, leading to enhanced and modulated particle densities (Sitiwaldi et al., 2017).
• In digital hardware and neuromorphic engineering, PDM denotes pulse-density modulation: the density of output pulses encodes the amplitude of an analog or digital signal, which can then be mapped onto spike-based computation or power-processing architectures (Jimenez-Fernandez et al., 2019, Li et al., 1 Sep 2025).

2. Theoretical Frameworks and Order Parameters

PDM in superconductors exhibits several distinct theoretical signatures, differing fundamentally from both PDW and charge/spin-density-wave orders.

• Order Parameter Structure: For PDW, $\Delta(r) = \sum_Q \Delta_Q e^{i Q\cdot r}$ with $Q\neq 0$ breaks translational symmetry. For PDM, the modulation is at Bragg vectors (e.g., $Q=(\pi,~\pi)$ in the 2-Fe iron-based unit cell), preserving lattice translation: $$\Delta(r) = \Delta_0 + \sum_{Q_{Bragg}} |\Delta_Q| \cos(Q\cdot r + \phi_Q)$$ (Kong et al., 15 Apr 2024).
• Hamiltonian Construction: PDM can result from a tight-binding model where sublattice inequivalence (e.g., via glide-mirror symmetry breaking and nematicity) leads to distinct local superconducting gaps on symmetry-equivalent but physically inequivalent sites, e.g., $\Delta_A \neq \Delta_B$ for sites $A,B$ in Fe(Te,Se) (Papaj et al., 24 Jun 2025).
• Landau Theory: The Landau free energy for PDM in Fe-based systems must include two order parameters $\eta_1,\eta_2$ of opposite glide/screw parity, hybridized only in the presence of local nematic order, with the PDM phase appearing as a hybridized state when $\beta_{12}^{\rm eff} < \sqrt{\beta_1 \beta_2}$ is satisfied and the coupling $\psi\,\eta_1\eta_2$ becomes symmetry-allowed due to glide symmetry breaking at the surface (Chan et al., 5 Dec 2025).

In plasma contexts, the relevant order parameter is the pair density $n_\pm(x, t)$, which develops a modulation of the form $$n_\pm(x, t) \simeq n_0 + \Delta n \cos(2k_L x)\cos(\omega_L t + \phi)$$ under standing-wave excitation (Liu et al., 2017).

3. Experimental Realizations and Key Observations

PDM has been experimentally accessed and characterized in several distinct systems:

• Iron-based superconductor FeTe$_{0.55}$Se$_{0.45}$: STM imaging reveals a superconducting gap amplitude that oscillates by more than 30% at a wavelength equal to the Fe-Fe atomic spacing, with the gap maximum on one sublattice and minimum on the other (Kong et al., 15 Apr 2024). This modulation persists up to strong magnetic fields and high impurity density but vanishes in thick and bulk samples, indicating its surface-sensitive and symmetry-broken character.
• Relativistic laser-driven foils: In high-energy-density targets (thin foils with $n_e=200$–$280\,n_c$, $L\approx 1\,\mu$m) irradiated by ultraintense laser pulses, pair-density modulation manifests as spatial “stripes” and periodic bunching of electron-positron plasmas with $\lambda/2$ spatial periodicity and temporal oscillation at $T_0/2$ ($T_0$ the laser period). The modulation amplitude can reach $\Delta n / n_0 \sim 0.5$–$1$, and is optimal under specific conditions of foil density and laser intensity (Liu et al., 2017).
• Quantum kinetic processes: In modulated fields, enhancement of pair production is observed when the modulation frequency assists in bridging the multiphoton pair-creation threshold. The resulting momentum distributions and number densities show modulated structure, with enhancement factors $R > 10^3$ and power-law density scaling $n(N) \sim N^{1.6}$ for $N$-pulse subcycle trains (Sitiwaldi et al., 2017).
• MEMS microphones and neuromorphic sensors: PDM is exploited in low-power microphones to directly convert analog signals into 1-bit pulse streams, with density proportional to instantaneous amplitude, and further mapped to pulse-frequency modulated (PFM) spike trains with predictable inter-spike intervals (Jimenez-Fernandez et al., 2019).
• Wireless power transfer systems: Targeted-subharmonic-eliminating PDM is used to suppress abnormal oscillations in resonant current, achieved by shaping the noise-transfer function to notch out problematic subharmonic frequency components (Li et al., 1 Sep 2025).

4. Microscopic Mechanisms and Symmetry Breaking

The emergence and stability of PDM depend on system-specific symmetry breaking and order parameter interactions:

• Condensed Matter (Fe-based PDM):
  • Glide-mirror symmetry breaking: Induced on cleaved or exfoliated surfaces where chalcogen heights differ above and below the Fe plane, allowing the nematic order parameter to hybridize otherwise orthogonal pairing channels, yielding a PDM. Restoration of glide symmetry in the bulk prevents this mixing, explaining the surface confinement of PDM (Kong et al., 15 Apr 2024, Chan et al., 5 Dec 2025).
  • Nematic superconductivity: The admixture of $s_\pm$ and $d$-wave order parameters, aligned with the sublattice texture generated by glide-mirror breaking, leads to the observed two-sublattice modulation (Papaj et al., 24 Jun 2025). The magnitude of PDM is governed by the amplitude ratio $f = (\Delta_A - \Delta_B)/(\Delta_A + \Delta_B)$, experimentally reaching up to 40%.
  • Site-local pairing: The symmetry analysis favors a local (e.g., Hund's coupling-driven) pairing model over bond-based pairing, as only site-local order parameters can have the required opposite glide parity, naturally accounting for the absence of PDM in the bulk (Chan et al., 5 Dec 2025).
• Plasma and Quantum Kinetics:
  • Standing-wave field trapping: The overlap of counter-propagating laser pulses in a transparent foil forms a SW, trapping freshly Breit–Wheeler pairs at the SW nodes, which modulates $n_\pm(x, t)$ (Liu et al., 2017).
  • Amplitude modulation: In strong-field QED, amplitude modulation of the driving field creates sidebands in the frequency spectrum of the field, opening new absorption channels for pair creation and resulting in strongly modulated momentum and real-space densities (Sitiwaldi et al., 2017).

5. Methodologies for Theoretical and Experimental Characterization

Techniques for analyzing and detecting PDM include:

• Scanning Tunneling Microscopy (STM): Resolves sublattice-resolved SC gap modulations and identifies domain structures in thin Fe-based superconductors (Kong et al., 15 Apr 2024).
• Tight-binding and BdG Calculations: Model Hamiltonians incorporating symmetry-breaking terms and nematic distortion reproduce self-consistent sublattice gap differences and predict the spatial distribution of the order parameter (Kong et al., 15 Apr 2024, Papaj et al., 24 Jun 2025).
• Landau-Ginzburg Expansion: Landau-theoretic analysis establishes phase diagrams, coexistence criteria for different pairing components, and surfaces' critical thickness for PDM stabilization (Chan et al., 5 Dec 2025).
• Particle-in-cell (PIC) Simulations: Reveal dynamic bunching, trapping, and modulation of pair plasma in SW nodes under relativistic laser irradiation (Liu et al., 2017).
• Quantum Kinetic/Vlasov Simulations: Batched solution of non-Markovian kinetic equations for pair production in modulated electromagnetic fields allows extraction of production rates, resonance conditions ($N\omega_c + k\omega_m = 2m_*$), scaling exponents, and pulse-train effects (Sitiwaldi et al., 2017).
• Distribution Matching in Communication Theory: Product distribution matching (PDM) for Amplitude-Shift Keying (ASK) constellations optimizes input symbol probabilities by decomposing the target distribution into independent binary marginals, reducing complexity and approaching channel capacity with bit loading and power allocation algorithms (Steiner et al., 2018).

6. Applications and Broader Implications

PDM has concrete and emerging applications as well as profound theoretical ramifications:

• Superconductivity: The identification and modeling of PDM open new avenues for understanding and classifying symmetry breaking in unconventional superconductors, particularly in systems with strong electronic correlations, sublattice degeneracy, and nematic order. The possible tie to Hund's physics points to new mechanisms for robust pairing (Kong et al., 15 Apr 2024, Chan et al., 5 Dec 2025).
• Plasma Physics: Structured pair plasmas produced via PDM enable exploration of collective phenomena (Weibel instability, two-stream instability) and laboratory simulation of astrophysical conditions such as pulsar wind nebulae (Liu et al., 2017).
• High-field QED: PDM in quantum kinetic pair production suggests experimental strategies for enhancing pair yields below ordinary thresholds, efficient pulse-train structuring, and reveals novel nonlinear scaling in the produced density distributions (Sitiwaldi et al., 2017).
• Neuromorphic Engineering and Signal Processing: Pulse-density modulation, as a hardware-level coding and conversion mechanism, enables low-power, low-latency interfacing between analog, digital, and spike-computation domains, with precise control over time-domain resolution and implementation complexity (Jimenez-Fernandez et al., 2019, Li et al., 1 Sep 2025).
• Wireless Power Transfer: Notch design in PDM enables suppression of detrimental subharmonic resonances, significantly improving stability and ensuring soft-switching in resonant WPT systems (Li et al., 1 Sep 2025).
• Communications: Product-density modulation facilitates practical high-order modulation with optimal probabilistic shaping, achieving waterfilling capacity in coded communication systems with reduced implementation cost (Steiner et al., 2018).

7. Distinctions from and Connections to Related Orders

PDM must be distinguished sharply from conventional PDWs and charge/spin density waves:

• Translational symmetry: PDW/charge-density wave order modulates at non-Bragg wavevectors, breaking lattice translation; PDM modulates at Bragg vectors or intra-cell and so preserves translation (Kong et al., 15 Apr 2024).
• Order parameter structure: The PDM amplitude is related to the difference in gap magnitude or pair density between symmetry-inequivalent but crystallographically equivalent sites, as opposed to PDW where center-of-mass momentum of pairs is finite.
• Microscopic diagnostic: STM and other local probes are sensitive to PDM via intra-unit-cell gap imaging, while ARPES and single-particle measurements may miss both PDW and PDM phenomenology except in the presence of field- or impurity-induced domain structures (Kong et al., 15 Apr 2024).
• Symmetry constraints: Only particular combinations of symmetry breaking (e.g., glide-mirror and nematic) enable or suppress PDM, highlighting its sensitivity to both local environment and global crystal structure (Chan et al., 5 Dec 2025).

These concepts collectively position PDM as a unifying framework for periodic structuring of paired entities across physical contexts, with both applied and fundamental implications for quantum materials, strong-field plasmas, and information-engineering systems.