Position- and Momentum-Space Quantum Information Measures of the Double-Morse Oscillator

AI summary

Overview

Research area: Quantum information theory applied to exactly solvable quantum models — specifically the quasi-exactly solvable double-Morse oscillator, analyzed through position- and momentum-space information measures. Technical level: Intermediate (requires comfort with wavefunctions, Fourier transforms, and information-theoretic quantities, but the presentation is largely graphical and parameter-driven). Scope: A systematic, parameter-resolved study of Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher–Shannon products for the ground and first excited states of the double-Morse potential as it crosses over from a double well to a single well.

What This Paper Is About

The double-Morse potential can be tuned continuously, via a single dimensionless shape parameter (A), from two well-separated wells to one merged well. The authors ask how this geometric change redistributes uncertainty, localization, and structural complexity between position and momentum space. Their goal is to provide an information-theoretic diagnostic of the double-well-to-single-well crossover for the two states that the model's quasi-exact solvability makes analytically accessible.

Key Contributions

1. Closed-form information measures for two quasi-exact states. Using the (n=1) quasi-exact sector, the authors derive explicit Bessel-function expressions for the position-space Shannon entropies, Onicescu energies, and Fisher informations of the ground and first excited states, together with integral representations for the corresponding momentum-space quantities.
2. A parameter sweep of the double-well-to-single-well crossover. All five measure families — Shannon entropy, Onicescu energy, Fisher information, Catalan–Garay–López-Ruiz statistical complexity, and Fisher–Shannon products — are evaluated as functions of (A \in (0,1)), with representative plots at (A = 0.1, 0.4, 0.7) and (\mu = 2).
3. Benchmark-based reading of non-Gaussianity. The results are organized against the Gaussian reference values (1+\ln\pi) (entropic bound), (1/(2\pi)) (Onicescu product), (4) (Fisher product), and (\mathcal{P}=1) (Fisher–Shannon), with the ground-state Fisher–Shannon excess (\Delta_{\rm FS} = \mathcal{P}-1) used as a quantitative non-Gaussianity indicator.
4. A full-state complement to marginal diagnostics. The authors compute relative-entropy non-Gaussianity (\delta_{\rm NG}) from the covariance matrix and symplectic eigenvalue, and contrast what it requires experimentally (state tomography) with what the marginal-density measures require (a single calibrated profile).

Main Findings

• Complementary position–momentum trends. As (A \to 0^{+}) the position distribution becomes delocalized over two wells and position entropy rises, while the momentum distribution is sharply peaked near (k=0) with lower momentum entropy. As (A \to 1^{-}) the trends reverse: the particle localizes near the origin and the momentum distribution spreads.
• Entropic uncertainty is respected, with the ground state approaching the limit. The total entropy satisfies (\mathcal{S}{\rho} + \mathcal{S}{\gamma} \geq 1 + \ln\pi) and remains nearly constant as (A) increases. The ground-state total entropy approaches, but does not attain, (1+\ln\pi), supporting near-Gaussian behavior in the merged-well regime.
• Onicescu product stays below the Gaussian benchmark in the studied interval. Position-space Onicescu energy increases with (A) (density concentrates near the origin), while momentum-space Onicescu energy decreases. The ground-state product approaches (1/(2\pi)) as (A) increases; the authors explicitly caution that this is a model-specific numerical observation, not a proof of a universal bound.
• Fisher information traces local gradients. Momentum-space Fisher information is large at small (A) and decreases as the momentum distribution spreads. Position-space Fisher information grows as a sharper central structure develops near (y=0) when the wells merge. The Fisher product satisfies (\mathcal{I}{\rho}\mathcal{I}{\gamma} \geq 4).
• Statistical complexity distinguishes ground and excited states. For small (A) the position-space complexities of the two states start close together; as (A) increases the ground-state complexity shows a steep decrease followed by an increase as its two lobes merge, whereas the excited state varies more weakly because it retains its two-lobe structure. Momentum-space complexities exceed their position-space counterparts owing to the oscillatory momentum wavefunctions.
• Ground state becomes Gaussian-like; the excited state does not. The ground-state Fisher–Shannon excess decreases toward zero as (A \to 1^{-}), while the excited state retains stronger non-Gaussian structure. The relative-entropy non-Gaussianity decreases with (A) and is larger for the excited state, consistent with its persistent nodal structure.
• Energy-level behavior. The excited-state energy (\epsilon_1) decreases monotonically as (A) decreases, while (\epsilon_0) reaches a minimum at (A = 0.5); the two levels coincide as (A \to 0^{+}), where the particle is symmetrically localized in two separated wells.
• Wavefunction morphology. Near (A \to 1^{-}) the ground and excited states have one and two main peaks respectively; as (A \to 0^{+}) the ground-state peak splits into two lobes and the excited-state lobes move farther apart.

Methodology in Plain English

The authors solve the time-independent Schrödinger equation for a particle of mass (m) in the double-Morse potential (V_{\rm DM} = D(A\cosh(\alpha x) - 1)^2), where (A = 2e^{-\alpha x_0}). Dimensionless variables ((\alpha x = 2y), plus rescaled (\mu^2) and (\epsilon)) turn this into a cleaner equation. Because the model is quasi-exactly solvable at (\mu = n+1), choosing (n=1) yields exactly two analytic bound states, written in terms of modified Bessel functions of the second kind. Fourier-transforming these wavefunctions gives the momentum-space counterparts, which are expressed as finite integrals rather than closed forms. The information measures are then computed — analytically where Bessel-function identities allow, and by adaptive numerical quadrature otherwise.

For the numerics, the super-exponentially decaying integrands let the semi-infinite domain be truncated at a cutoff (y_{\max}) chosen so that (|f(y)| \le \epsilon) beyond it, with (\epsilon = 10^{-6}). Every measure is plotted against (A) and compared with the corresponding Gaussian reference line, so that deviations from Gaussianity can be read directly off the figures.

Why This Matters

The paper supplies a transferable effective-potential language in which information measures quantify how localization, momentum spreading, and non-Gaussian structure respond to changes in double-well shape. It complements earlier work on the same model that treated (\alpha) and (A) as metrological parameters and that derived the phase-space quasiprobability hierarchy of the lowest quasi-exact state, filling a gap in the systematic information-theoretic characterization of the model.

Real-world applications:

• Ultracold atoms in optical double wells, where double wells have been used for condensate interferometry and bosonic Josephson dynamics, and where painted optical potentials and optical superlattices provide tunable one-dimensional landscapes. Position densities come from in-situ or reconstructed profiles; momentum densities from time-of-flight expansion.
• Semiconductor and laser-dressed quantum wells, where double-Morse-type confinement has been analyzed for optical absorption and nonlinear optical response under changes of structure and laser parameters.
• Segmented trapped-ion devices, where electrode voltages reshape axial confinement, with demonstrated controlled ion separation and measurable anharmonic trap effects.
• Hydrogen-bonded and proton-transfer systems, where two-Morse or double-minimum potentials arise naturally and NMR and related spectroscopy have been used to infer microscopic Morse-like proton potentials in KDP-type materials.

Industry relevance: The measures give experimentally calibratable diagnostics — barrier height (D(1-A)^2), well separation (\pm\frac{1}{\alpha}\operatorname{arcosh}(1/A)), and local oscillation frequency (\omega_{\rm loc} = \alpha\sqrt{2D(1-A^2)/m}) — that connect to quantum sensing and metrology, cold-atom and trapped-ion hardware calibration, and the design of semiconductor heterostructures. The distinction between marginal-density diagnostics (accessible from one calibrated profile) and full-state non-Gaussianity (requiring tomography) is directly relevant to what a realistic experiment can certify.

Future Directions

• Extend beyond the (n=1) sector. The authors state their conclusions apply only to the ground and first excited states and should not be extrapolated to higher bound states, whose additional nodes can substantially alter Fisher information, momentum oscillations, and complexity. Treating those requires numerical diagonalization or a different quasi-exact sector.
• Move from effective potential to device-specific protocol. The fixed-(\mu) curves represent either co-tuning (D/\alpha^2) to remain constant or a local benchmark; a full device-specific estimation protocol remains open.
• Propagate experimental noise into the measures. Finite resolution, limited field of view, and counting noise should be handled by detector-response deconvolution or resampling, with particular attention to Fisher information because it involves derivatives.
• Test the diagnostics against tomography. Comparing marginal-density Fisher–Shannon curves with relative-entropy non-Gaussianity reconstructed from measured states would establish how much state-level information the cheaper marginal diagnostics capture.

Target Audience

Quantum information and mathematical physics researchers interested in exactly solvable potentials and information-theoretic descriptors of quantum states; graduate students seeking a worked example of how Shannon, Onicescu, Fisher, and complexity measures behave across a tunable double-well crossover; and experimentalists in cold-atom, trapped-ion, semiconductor quantum-well, and proton-transfer spectroscopy settings who want calibration-linked diagnostics of non-Gaussianity and localization.

Authors’ abstract

We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter $A$, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.