Exact Cahill-Glauber Hierarchy for the Double-Morse Ground State
Abstract
Phase-space and quasiprobability methods now play operational roles in quantum technologies, characterizing localization, non-Gaussianity, nonclassical resources, and coarse-graining. We develop an exact, representation-consistent analysis of the lowest quasi-exact ground state of the symmetric double-Morse oscillator. In the double-Morse potential, the dimensionless parameter A controls the separation of the minima and the central barrier, thereby changing the physical ground state. At fixed A, the Cahill--Glauber parameter s labels the quasiprobability WA(s)(q,p): s=0, −1, and $1$ give the Wigner, Husimi Q, and Glauber--Sudarshan P representations, respectively. Although the potential is double-welled for 0<A<1, the exact ground-state amplitude is single-peaked at the origin and lies above the barrier; as A approaches unity, the merged well remains locally quartic rather than harmonic. Closed analytical expressions are obtained for the Wigner function and Weyl characteristic function. The Wigner function displays the A-dependent exchange between position and momentum localization and retains negative regions, certifying nonclassicality and, for this pure state, non-Gaussianity. The Weyl function is its Fourier dual, generates symmetrically ordered moments and cumulants, and yields the full s-ordered hierarchy. For s<0, isotropic Gaussian smoothing suppresses fine sign-changing structure while preserving the large-scale localization envelope. The Husimi endpoint is nonnegative without implying classicality, whereas the P representation remains distributional. Thus, A controls the physical phase-space geometry, while s controls how the same non-Gaussian and nonclassical state is resolved across complementary representations
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BibTeX
@misc{chogle2026exact,
title = {{Exact Cahill-Glauber Hierarchy for the Double-Morse Ground State}},
author = {Firoz Chogle and Berihu Teklu and Mauro F. Pereira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.23316},
eprint = {2608.23316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23316},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Cahill-Glauber Hierarchy for the Double-Morse Ground State
AU - Firoz Chogle
AU - Berihu Teklu
AU - Mauro F. Pereira
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.23316
UR - https://arxiv.org/abs/2608.23316
AB - Phase-space and quasiprobability methods now play operational roles in quantum technologies, characterizing localization, non-Gaussianity, nonclassical resources, and coarse-graining. We develop an exact, representation-consistent analysis of the lowest quasi-exact ground state of the symmetric double-Morse oscillator. In the double-Morse potential, the dimensionless parameter A controls the separation of the minima and the central barrier, thereby changing the physical ground state. At fixed A, the Cahill--Glauber parameter s labels the quasiprobability W_A^(s)(q,p): s=0, −1, and $1$ give the Wigner, Husimi Q, and Glauber--Sudarshan P representations, respectively. Although the potential is double-welled for 0<A<1, the exact ground-state amplitude is single-peaked at the origin and lies above the barrier; as A approaches unity, the merged well remains locally quartic rather than harmonic. Closed analytical expressions are obtained for the Wigner function and Weyl characteristic function. The Wigner function displays the A-dependent exchange between position and momentum localization and retains negative regions, certifying nonclassicality and, for this pure state, non-Gaussianity. The Weyl function is its Fourier dual, generates symmetrically ordered moments and cumulants, and yields the full s-ordered hierarchy. For s<0, isotropic Gaussian smoothing suppresses fine sign-changing structure while preserving the large-scale localization envelope. The Husimi endpoint is nonnegative without implying classicality, whereas the P representation remains distributional. Thus, A controls the physical phase-space geometry, while s controls how the same non-Gaussian and nonclassical state is resolved across complementary representations
ER -