Single-Phonon Interference & Berry Phase

• Single-phonon interference is a quantum phenomenon in which individual phonons exhibit interference by accumulating both dynamical and Berry phases that reflect the underlying band topology.
• Experimental methods like SdH oscillations and quantum Hall fan diagrams allow precise extraction of Berry phases, linking interference patterns to pseudospin textures.
• The tunable Berry phase serves as a bulk fingerprint, confirming topological order and providing actionable insights for probing quantum coherence in condensed matter systems.

Single-phonon interference refers to the quantum-coherent behavior of phonons—quanta of lattice vibrations—in settings where their wave nature is manifested through phase evolution, path superposition, and corresponding geometric or topological Berry phases. While the canonical context for single-particle interference historically centers on photons and electrons, the extension of quantum interference principles to phononic excitations (acoustic or optical phonons) is underlying contemporary research at the intersection of condensed matter physics, quantum transport, and topological phases. This entry synthesizes the theoretical formalism and experimental protocols for diagnosing and utilizing single-phonon interference effects, focusing explicitly on the role of Berry phases and related quantum geometry in two-band systems, particularly those displaying nontrivial band topology such as quantum wells with inverted bands (Couëdo et al., 2019).

1. Quantum Description of Phonons and Interference

In the quantum regime, phonons—elementary vibrational excitations of a periodic solid—are associated with quantized modes of the lattice displacement field. A single-phonon state may be represented as a quantum superposition of creation operators acting on the vacuum, $$\lvert 1_{\mathbf{q}}\rangle = \hat{b}^\dagger_{\mathbf{q}} \lvert 0 \rangle$$ for wavevector $\mathbf{q}$. Interference emerges when a phonon follows coherent pathways that later recombine, such that the resulting state reflects the phase difference accumulated along distinct trajectories.

The phase arising from the energy-time evolution (dynamical phase) is complemented by an additional geometric phase—the Berry phase—when the system’s parameters (momentum, lattice coordinates, external fields) are varied cyclically (Sprinkart et al., 2024). For phonons coupled to electronic or pseudospin degrees of freedom in a multiband system, the geometric phase is not merely a simple function of path length and energy, but encodes the underlying quantum geometry of the system’s band structure.

2. Two-Band Model: Pseudospin Berry Phase and Phonon Dynamics

The prototypical two-band Hamiltonian formalism, $$H(\mathbf{k}) = \mathbf{d}(\mathbf{k})\cdot \boldsymbol{\sigma}$$, with $\mathbf{d}(\mathbf{k})$ a vector field in parameter space and $\boldsymbol{\sigma}$ the vector of Pauli matrices, captures the essential physics of solids with coupled electron and hole subbands, including phonon-coupled systems in semiconductors and topological insulators (Couëdo et al., 2019). The eigenstates are spinors on a Bloch sphere parameterized by polar and azimuthal angles $(\theta, \phi)$ set by the components of $\mathbf{d}(\mathbf{k})$.

The Berry connection for such an eigenstate is $$A(\mathbf{k}) = i \langle u(\mathbf{k}) | \nabla_\mathbf{k} u(\mathbf{k}) \rangle$$, and the Berry phase along a closed path $C$ in momentum space is $\Gamma = \oint_C A(\mathbf{k}) \cdot d\mathbf{k}$ (Sprinkart et al., 2024). For cyclotron orbits or constant-energy contours relevant to single-phonon dynamics, this phase reflects the winding of the pseudospin texture induced by electron-phonon (or phonon-phonon) coupling and, in certain regimes, encodes topological information such as Chern numbers (Couëdo et al., 2019, Fuchs et al., 2010).

3. Tunable Berry Phase: Interplay of Band Topology and Fermi Level

In the context of quantum wells exhibiting inverted band structure (BHZ model), the Berry phase associated with quasiparticles—or in general, any single-exciton (phonon/electron) following a closed $k$-space trajectory—is analytically given by

$$\Gamma_\pm(k) = \pi[ 1 \pm \frac{M - Bk^2}{\sqrt{A^2k^2 + (M - Bk^2)^2}} ]$$

for upper ($+$) and lower ($-$) bands (Couëdo et al., 2019). Here, $A$, $B$, and $M$ are material-dependent parameters, and $k$ is the modulus of the in-plane momentum. As the probe energy (or Fermi level) is tuned across the hybridization gap, the Berry phase $\Gamma$ evolves smoothly from $0$ to $2\pi$, passing through $\pi$ at the gap minimum. This variation is a bulk signature of the topological band inversion: the locus of $\Gamma = \pi$ coincides with the momentum at which the pseudospin vector sweeps across the Bloch sphere equator.

This continuous tunability of the Berry phase in a single-particle context (phonon or electronic quasiparticle) signifies the nontrivial quantum geometry underpinning single-phonon interference. The phase acquired from traversing different $k$-space loops directly manifests as observable quantum interference—modulating transport or scattering phenomena such as Shubnikov–de Haas (SdH) and quantum Hall (QH) oscillations (Couëdo et al., 2019).

4. Experimental Probes: Magneto-Oscillation and Quantum Hall Phase Offsets

Experimentally, the observation of single-phonon interference via its Berry phase proceeds by extracting phase offsets from quantum oscillations in transport:

• SdH oscillations: Measuring longitudinal resistance $R_{xx}(B)$ at fixed gate voltage, the Fourier transform of $R_{xx}(1/B)$ yields dominant frequencies mapping the Fermi surface area sampled by the phonon/electron. The position of resistance minima reflects the phase offset $\gamma$ (where $\gamma = \Gamma / 2\pi$), directly linking the measured interference to the accrued geometric phase (Couëdo et al., 2019, Fuchs et al., 2010).
• Quantum Hall fan diagrams: The pattern of minima and plateaus in $R_{xx}(V_g,B)$, indexed via filling fractions, provides another route to quantifying the Berry phase encoded in the interference of single quasiparticle trajectories.

The formula used to extract the Berry phase from minima positions is

$$\Gamma^{(j)} = 2\pi [N^{(j)} - F^{(j)}/B_{\text{min}}]$$

with $N^{(j)}$ the Landau level index and $F^{(j)}$ the oscillation frequency. By tuning the control parameter (typically the gate voltage), the transition of $\Gamma$ through $\pi$ as the Fermi level crosses the hybridization gap is observed, offering a direct map of single-phonon phase interference as a bulk probe of band topology (Couëdo et al., 2019).

5. Topological and Physical Implications: Bulk Fingerprint and Protection

The winding nature of the pseudospin polarization in two-band models ensures that, in the topological (inverted) regime, the Berry phase associated with single-phonon interference unavoidably passes through $\pi$. In trivial (non-inverted) systems, the phase remains near $0$ or $2\pi$ and never crosses $\pi$. This behavior encodes a $\mathbb{Z}_2$ classification at the level of single-excitation wavefunctions and generalizes the role of Berry phase as a foundation for protected edge states and robust transport in quantum (spin) Hall systems.

The bulk measurement of Berry phase via single-phonon (or single-electron/hole) interference thus provides a fingerprint of the same band topology that, in the fully gapped regime, enforces protected edge modes (Couëdo et al., 2019, Fuchs et al., 2010). This establishes the utility of single-particle phase interference as a sensitive diagnostic of topological order and quantum-geometric phases.

6. Broader Context: Berry Phases, Interference, and Quantum Geometry

Single-phonon interference embodies the fundamental interplay between wave coherence, quantum geometry, and topological invariants. The Berry phase accumulated by such interference processes is not only a hallmark of quantum coherence at the single-excitation level but also a gateway to experimentally probing and utilizing topological phases in condensed matter settings.

These conceptual frameworks parallel those established for electrons, photons, and other quasiparticles, demonstrating that the principles of geometric phase and quantum interference generalize naturally to phononic degrees of freedom—so long as quantum coherence and relevant energy/momentum resolution are maintained (Sprinkart et al., 2024, Couëdo et al., 2019).

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References

• "Pseudospin Berry phase as a signature of nontrivial band topology in a two-dimensional system" (Couëdo et al., 2019)
• "Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models" (Fuchs et al., 2010)
• "Tutorial: From Topology to Hall Effects -- Implications of Berry Phase Physics" (Sprinkart et al., 2024)