Momentum-Resolved Sublattice Phase Measurements

• Momentum-resolved sublattice phase measurements are advanced techniques that reveal phase relationships in multi-sublattice quantum systems.
• They utilize methods like STM, k-space photoluminescence, and TOF imaging to extract form factor-specific signatures and quantum geometric information.
• These techniques enable direct observation of symmetry breaking, topological order, and emergent band structures, offering critical insights for quantum materials research.

Momentum-resolved sublattice phase measurements constitute a class of experimental techniques and analytical protocols designed to probe the internal phase structure of multi-sublattice quantum materials, cold atom systems, and engineered photonic lattices with momentum (k-) resolution. The goal is to directly access the phase relationships between components associated with different sublattices (or pseudo-spinors) as a function of crystal momentum, thereby elucidating the nature of symmetry breaking, topological order, or emergent band structure effects. Representative implementations span scanning tunneling microscopy (STM) in correlated oxides, k-space photoluminescence in photonic lattices, and momentum- or Bragg-resolved probes in ultracold atomic gases.

1. Conceptual Framework and Motivation

Momentum-resolved sublattice phase measurements seek to determine not merely the spectral or occupation properties, but the microscopic phase correlations between internal degrees of freedom (such as sublattice, pseudo-spin, or valley index) in a momentum-resolved fashion. For materials and systems described by two (or more) coupled sublattices per unit cell, Bloch wavefunctions naturally assume a multi-component spinor structure. The relative phase between these components often carries direct physical meaning, encoding the order parameter symmetry (e.g., s, extended-s′, d) of density waves or the local quantum geometry of Bloch bands. Standard density or spectroscopic measurements alone lack phase sensitivity and cannot distinguish between competing order parameters or geometric/topological effects.

Direct detection of these phase relationships is therefore critical for distinguishing, for example, d-form factor density waves in cuprates (Fujita et al., 2014), sublattice pseudospin winding in photonic graphene (Guillot et al., 22 Jul 2025), or checkerboard/stripe order in cold atomic mixtures (Maska et al., 2011).

2. Experimental Protocols

STM-based Sublattice Phase Measurement in Correlated Oxides

One key implementation involves site-specific STM imaging on complex oxides such as underdoped cuprates (Fujita et al., 2014). Spatially resolved measurements acquire, with sub-ångström precision, the topograph $T(\mathbf{r})$, differential conductance $g(\mathbf{r},E) = dI/dV$, and DC tunnel current $I(\mathbf{r},E)$. Artifacts arising from spatially varying tunneling matrix elements are suppressed by considering ratio images $$Z(\mathbf{r},|E|)=g(\mathbf{r},+E)/g(\mathbf{r},-E)$$ or $$R(\mathbf{r},|E|)=I(\mathbf{r},+E)/I(\mathbf{r},-E)$$ that isolate the particle–hole symmetric component.

Images are drift-corrected using the Lawler–Fujita displacement field $u(\mathbf{r})$, ensuring atomic registry at a precision better than $0.01 a_0$. The field of view is then segmented into masks corresponding to each sublattice: Cu, O$_x$, O$_y$. For each, an image is produced containing only the signal from its respective lattice sites. The discrete Fourier transforms $\tilde{Cu}(\mathbf{q})$, $\tilde{O}_x(\mathbf{q})$, $\tilde{O}_y(\mathbf{q})$ are then computed for each channel, establishing the basis for momentum-resolved phase analysis.

Sublattice Stokes Polarimetry in Bipartite Photonic Lattices

An alternative protocol employs k-space photoluminescence of bipartite photonic lattices, such as GaAs–AlGaAs honeycomb micropillar arrays, to reconstruct the sublattice pseudospin structure (Guillot et al., 22 Jul 2025). Here, a spatial light modulator (SLM) placed in the far field, configured with small “window” masks centered on each micropillar, imparts independently tunable phase shifts ($\phi^\sigma$) and amplitude attenuations ($\alpha^\sigma$) to each sublattice (A/B).

Four canonical measurement configurations (A+B, A+iB, A, B) correspond to distinct projections of the sublattice pseudospin onto the Stokes sphere, analogous to polarimetric measurement of light. The resultant k-resolved intensity maps $$I_{A+B}(k, E), I_{A+iB}(k, E), I_A(k, E), I_B(k, E)$$, collected via Fourier-imaging optics and spectrometer, serve as the raw data for extracting amplitude and relative phase across the Brillouin zone.

Momentum- and Bragg-Resolved Probes in Cold Atom Lattices

In mixtures of ultracold fermions on optical lattices, such as those modeled by the Falicov–Kimball Hamiltonian, momentum-resolved time-of-flight (TOF) imaging yields $n(\mathbf{k})$, the single-particle momentum distribution (Maska et al., 2011). While $n(\mathbf{k})$ is only indirectly sensitive to sublattice phase ordering (due to spatial inhomogeneity and strong trap effects), Bragg scattering probes directly the static structure factor $S(\mathbf{k})$, whose finite-$\mathbf{k}$ peaks reveal sublattice (density-wave) order. The specific momentum at which these peaks emerge identifies checkerboard, stripe, or more complex orders.

3. Extraction and Interpretation of Sublattice Phase Information

Fourier-Space Analysis and Form Factor Decomposition

For systems with multiple sublattices per unit cell, Fourier transforms of sublattice-segregated signals provide the central data. Defining: $$\tilde{Cu}(\mathbf{q}) = \sum_{r\in \mathrm{Cu}} Cu(r) e^{-i\mathbf{q}\cdot r}, \quad \tilde{O}_x(\mathbf{q}) = \sum_{r\in O_x} O_x(r) e^{-i\mathbf{q}\cdot r}, \quad \tilde{O}_y(\mathbf{q}) = \sum_{r\in O_y} O_y(r) e^{-i\mathbf{q}\cdot r}$$ one can form distinct intra-unit-cell “form factor channels”:

• s-form factor: $S(\mathbf{q}) \propto \tilde{Cu}(\mathbf{q})$
• extended-s′ form factor: $$S_{s'}(\mathbf{q}) \propto [\tilde{O}_x(\mathbf{q}) + \tilde{O}_y(\mathbf{q})]/2$$
• d-form factor: $$S_d(\mathbf{q}) \propto [\tilde{O}_x(\mathbf{q}) - \tilde{O}_y(\mathbf{q})]/2$$

The presence of DW peaks exclusively in $S_d(\mathbf{q})$, with vanishing spectral weight in $S$ and $S_{s'}$ at the DW wavevector, establishes the d-form factor character. The momentum-resolved relative phase,

$$\phi(\mathbf{q}) = \arg[\tilde{O}_x(\mathbf{q})] - \arg[\tilde{O}_y(\mathbf{q})]$$

provides a direct measure of the internal phase structure; for d-form factor order, $\phi(\mathbf{q}=\mathbf{Q}) \approx \pi$.

Stokes Vector and Poincaré Sphere in Photonic Lattices

For bipartite lattices, the sublattice pseudospinor Bloch state $(u^A_{n\mathbf{k}}, u^B_{n\mathbf{k}})^T$ is mapped onto Stokes parameters $(S_1, S_2, S_3)$, extracted via combinations of the four measurement masks. From these,

$$S_1 = 2\,\mathrm{Re}(u^{A*} u^B), \quad S_2 = -2\,\mathrm{Im}(u^{A*} u^B), \quad S_3 = |u^A|^2 - |u^B|^2$$

the spherical angles $\theta(\mathbf{k}), \phi(\mathbf{k})$ fully define the sublattice phase: $$\theta(\mathbf{k}) = \arccos[S_3(\mathbf{k})], \quad \phi(\mathbf{k}) = \mathrm{atan2}[S_2(\mathbf{k}), S_1(\mathbf{k})]$$ and the Bloch spinor is reconstructed as $(\cos[\theta/2], \sin[\theta/2] e^{i\phi})^T$.

Quantum Geometry and Band Structure

The full k-resolved measurement of sublattice amplitude and phase enables reconstruction of the Bloch Hamiltonian $H(\mathbf{k})$ and the quantum geometric tensor $Q^n_{\mu\nu}(\mathbf{k})$. The Berry curvature and quantum metric, calculated from gauge-invariant finite differences of the Bloch vectors, permit the assignment of valley Chern numbers or mapping of quantum distance in reciprocal space.

4. Signature Observables and Ordering Phenomena

The direct outcomes of momentum-resolved sublattice phase measurements are the phase differences at selected $\mathbf{q}$-vectors, form factor-specific spectral weight, and characteristic features in the structure factor or Stokes vector maps. Key engineered and natural signatures include:

• For underdoped cuprates, fundamental density wave peaks visible only in the d-form factor channel and a measured oxygen-oxygen phase difference $\phi(\mathbf{q})\approx \pi$ unequivocally signal d-symmetry order, excluding simple s- or extended s′-wave scenarios (Fujita et al., 2014).
• In photonic graphene, mapping $(S_1, S_2, S_3)$ over the Brillouin zone reconstructs pseudospin winding and enables full Hamiltonian tomography, including quantification of phenomena such as trigonal warping or topologically nontrivial band structures (Guillot et al., 22 Jul 2025).
• In cold atom settings, sharp finite-$\mathbf{k}$ Bragg peaks in $S(\mathbf{k})$ identify checkerboard or stripe order; phase coexistence or inhomogeneity is evident as superpositions or broadened features. The temperature dependence of Bragg peak height provides a sensitive low-$T$ thermometric method (Maska et al., 2011).

5. Practical Applications and Extensions

Momentum-resolved sublattice phase measurements have established themselves as central to:

• Identifying and classifying unconventional density-wave and symmetry broken phases in strongly correlated electron systems, as demonstrated in high-$T_c$ cuprates (Fujita et al., 2014).
• Reconstructing and engineering topological invariants and quantum geometric properties in artificial lattices, with implications for Chern insulator engineering and novel optical device design (Guillot et al., 22 Jul 2025).
• Distinguishing genuine ordering from trap-generated artifacts or phase separation in cold atom experiments, with potential application as a robust thermometer for quantum simulators (Maska et al., 2011).

These methods are extensible to systems with higher internal symmetry (e.g., multiple sublattices, valleys, spin-orbit coupled bands), where analogous phase-resolved tomography can access richer forms of quantum geometry, such as Euler-class topology in multigap bands and non-Abelian Berry curvature.

6. Limitations and Special Considerations

Several classes of systematic effects bear consideration in these protocols:

• In STM-based measurements, accuracy hinges on proper drift correction and unambiguous sublattice identification. Potential tip-matrix-element and surface termination effects are ruled out by verifying universality across compounds (Fujita et al., 2014).
• In photonic implementations, sublattice selectivity depends on the spatial resolution and phase-purity of the SLM masks; eigenenergies are fit with sub-linewidth precision but require deconvolution in regions of degeneracy via density matrix analysis (Guillot et al., 22 Jul 2025).
• In cold atom lattices, spatial inhomogeneity smears momentum features, and the single-band approximation plus neglect of interactions during expansion limits the interpretation of $n(\mathbf{k})$, making $S(\mathbf{k})$ the unambiguous marker for sublattice order (Maska et al., 2011).

Optimal results require aligning experimental protocols to the intrinsic spatial and spectral scales of the system under paper and careful discrimination of ordering signatures from extrinsic effects.

7. Outlook

Momentum-resolved sublattice phase measurement protocols are poised for further development as tools for exploring correlated, topological, and engineered quantum matter. Their extension to multi-component and strongly interacting systems enables direct access to geometric and topological invariants beyond conventional observables. A plausible implication is the routine extraction of quantum geometric tensors and higher-order Berry phase phenomena from experimental data, facilitating precision Hamiltonian engineering and symmetry classification in quantum materials and photonic platforms.