Coherent states in minimal-length Quantum Mechanics: inequivalent characterizations and emergent squeezing

Technical Summary: Coherent States in Minimal-Length Quantum Mechanics

Problem Statement
The paper addresses a fundamental gap in the phenomenology of quantum gravity: the consistent characterization of coherent states (CSs) within theories featuring a minimal measurable length. While the Generalized Uncertainty Principle (GUP), derived from deformed Heisenberg algebras (specifically the Kempf-Mangano-Mann or KMM model), is a standard framework for exploring Planck-scale effects, the equivalence of the standard definitions of CSs in ordinary Quantum Mechanics (QM) has not been rigorously tested in this context. In standard QM, CSs are uniquely defined by three equivalent properties: (a) being minimum-uncertainty wave packets, (b) being eigenstates of the annihilation operator, and (c) being displaced vacuum states. The authors investigate whether these characterizations remain equivalent when the canonical commutator is deformed by a minimal length scale.

Methodology
The authors employ a systematic analysis of the one-dimensional harmonic oscillator within the GUP framework. Their approach relies on the following methodological steps:

1. Canonical Transformation: Following recent results [30], they distinguish between the physical position operator $x$ (which satisfies the deformed commutator $[x, p] = i\hbar f(p)$) and an auxiliary canonical coordinate $q$ (satisfying $[q, p] = i\hbar$). The Hamiltonian is formulated in terms of the canonical pair $(q, p)$ to preserve Galilean invariance, while physical observables are mapped back to $x$.
2. Definition of Generalized CSs: They define generalized coherent states $|\alpha\rangle_f$ as eigenstates of the annihilation operator constructed from the auxiliary variables $(q, p)$. This preserves the formal algebraic structure of standard CSs (properties b and c).
3. Variational Approach: To identify states that saturate the generalized uncertainty relation (property a), they employ a variational method [59] to solve the differential equation for minimum-uncertainty states in the presence of the deformation function $f(p) = 1 + \beta p^2$.
4. Comparative Dynamics: The authors compare the time evolution, uncertainty profiles, and energy spectra of the algebraically defined CSs ($|\alpha\rangle_f$) against the states that saturate the GUP ($|\phi_{sat}\rangle$). They analyze these states in both position and momentum representations, utilizing the momentum-space wave functions derived from the deformed commutator.

Key Contributions and Results
The central finding of the work is the generic inequivalence of the standard characterizations of coherent states in minimal-length quantum mechanics.

• Breakdown of Equivalence: The authors demonstrate that states defined as eigenstates of the generalized annihilation operator (derived from the auxiliary canonical sector) do not saturate the physical GUP uncertainty relation involving the physical position $x$. Conversely, the states that do saturate the GUP are not eigenstates of the standard annihilation operator and cannot be generated by a simple displacement operator acting on the vacuum.
• Emergent Squeezing: The GUP-saturating states exhibit an intrinsic "squeezing" mechanism absent in ordinary QM. In the momentum representation, these states become increasingly narrow and sharply peaked as the deformation parameter $\beta$ increases, while their position-space representation broadens. This contrasts with the standard vacuum, which is a minimum-uncertainty Gaussian state.
• Dynamical Deformations:
  • Phase Space Trajectories: The physical position expectation values of the algebraic CSs follow trajectories that are deformed relative to the standard harmonic motion due to the non-linear relationship between $x$ and $q$.
  • Uncertainty Evolution: Unlike standard CSs where uncertainties remain constant, the physical position uncertainty of GUP-saturating states in a harmonic potential undergoes periodic "breathing" (oscillation) with a period of $\pi/\omega$. This oscillation involves higher harmonics (e.g., $4\omega, 6\omega$) arising from the non-linear momentum dependence of the physical position operator.
• Energetic Properties: The GUP-saturating states are not stationary states of the harmonic oscillator Hamiltonian. Consequently, they possess non-vanishing energy uncertainties and mean energies that increase monotonically with the deformation parameter $\beta$. The authors note that for maximally localized states (where $\Delta p = 1/\sqrt{\beta}$), higher-order momentum moments may diverge, limiting their interpretation as semiclassical energy states.

Significance and Claims
The paper establishes a unified framework for coherence in GUP-based quantum theories, highlighting that the concepts of "coherence" (algebraic definition) and "localization" (uncertainty minimization) are disentangled in the presence of a minimal length.

• Semiclassical Signatures: The authors claim that the observed phenomena—specifically the intrinsic squeezing, the periodic breathing of uncertainty, and the emergence of higher harmonics in the uncertainty dynamics—constitute distinctive semiclassical signatures of minimal-length physics.
• Probing Quantum Gravity: These results are presented as a theoretical basis for probing quantum-gravitational effects. The authors suggest that high-precision semiclassical systems, such as optomechanical setups or analogue-gravity platforms, could potentially detect these deviations through frequency measurements and noise spectroscopy.
• Future Directions: The work opens avenues for investigating the role of these generalized states in gravity-induced decoherence and the quantum-to-classical transition. It also suggests potential applications in condensed matter analog systems (e.g., graphene), where effective GUP-like structures may arise from lattice geometry, potentially influencing wave-packet propagation and coherence.

The authors maintain a modest tone regarding experimental verification, framing their results as establishing a theoretical framework and identifying specific signatures that could be tested in future high-precision or analogue experiments, rather than claiming immediate experimental confirmation.