Nonclassical State Generation and Quantum Metrology in the Double-Morse Potential

Abstract: In this paper, we investigate the nonlinear properties of the double-Morse potential as a possible resource for single-mode quantum states because of its double-well structure and anharmonicity. We derive the ground state wave function and the associated energy spectrum analytically, using the asymmetry (width parameter) $α$ as the primary control parameter. These results show a systematic and evident influence on $α$. We assess the non-Gaussianity and non-classicality measures, quantifying their nonlinearity and quantum behavior. In particular, we discover that both metrics rise monotonically with $α$. Furthermore, we examine the metrological performance for estimating $α$. By calculating the pertinent Fisher information and building workable estimators, we show that optimal strategies can saturate the Cramér-Rao bound, with straightforward position measurements on shallow wells already producing high precision. These results collectively demonstrate that the double-Morse potential is a genuine, controllable source of non-Gaussianity, whose non-classicality and metrological applications increase with $α$. We highlight the potential applications of this model in real-world quantum technologies and discuss the implications for continuous-variable quantum information processing and computation.

• The paper presents analytic ground-state solutions for the double-Morse potential, revealing tunable non-Gaussian and nonclassical behavior with varying α.
• It quantifies state properties using measures like Bures nonlinearity, Wigner negativity, and entanglement potential, demonstrating monotonic scaling with potential parameters.
• The study establishes optimal quantum metrology protocols via quantum Fisher information, showcasing enhanced parameter estimation and practical implications for quantum sensing.

Nonclassical State Generation and Quantum Metrology in the Double-Morse Potential

Introduction and Motivation

This paper rigorously analyzes the quantum properties of the double-Morse (DM) potential, focusing on its utility for generating nonclassical states and enhancing quantum metrology. The DM potential, constructed by superposing two Morse wells, offers tunable anharmonicity and bistability via the control parameter $\alpha$ (width/asymmetry). Exploiting analytic tractability, the authors derive ground-state solutions, compute measures quantifying nonlinearity, non-Gaussianity, nonclassicality, entanglement generation, and parameter estimation performance, systematically mapping how these properties respond to variations in $\alpha$. The investigation demonstrates that the DM potential’s ground state provides an experimentally relevant single-mode non-Gaussian resource, with implications for continuous-variable quantum information and quantum-enhanced sensing.

Figure 1: Double Morse potential profiles for various values of $\alpha$.

Analytical Ground State and Potential Landscape

The DM potential is expressed as $$V_{\mathrm{DM}}(x) = D\left(A\cosh{(\alpha x)} -1\right)^2$$ with $A = 2e^{-\alpha x_0}$ defining well separation. For $A < 1$, the system exhibits a symmetric double-well; increasing $\alpha$ deepens the barrier and widens the minima separation. The analytically derived ground-state wavefunction, $\psi_0(x)$ in terms of modified Bessel functions, allows evaluation of all relevant expectation values and statistical descriptors. The ground-state probability density $|\psi_0(x)|^2$ sharpens with increasing $\alpha$, indicating enhanced localization at the minima and pronounced suppression in the inter-well region.

Figure 2: Ground-state probability density $|\psi_0(x)|^2$ becomes narrower and more localized at the well minima as $\alpha$ increases.

Quantification of Non-Gaussianity and Nonclassicality

The paper employs both the Bures-based ground-state nonlinearity measure and quantum relative entropy-based non-Gaussianity, capitalizing on covariance-matrix analytics. It is observed that $\eta_{\mathrm{NG}}$ grows monotonically with $\alpha$ (Figure 3), with larger well separations $x_0$ yielding higher non-Gaussianity. For sufficiently large $\alpha$, $\eta_{\mathrm{NG}}$ exhibits insensitivity to $x_0$, signifying universal behavior in the strongly anharmonic regime.

Figure 3: Non-Gaussianity measure $\eta_\mathrm{NG}$ increases monotonically with $\alpha$ for all tested $x_0$, converging for large width.

Nonclassicality is assessed via the negativity of the ground-state Wigner function, $W_0(x,p)$, for which a closed-form analytic expression in terms of imaginary-order Bessel functions is provided. The Wigner distribution reveals strong negative regions—quantum interference fringes—near the central barrier and along the $p$ direction, directly indicating nonclassical behavior per Hudson's theorem.

Figure 4: Ground-state Wigner distribution $W_0(x,p)$ for $\alpha=5$ and $x_0=1$, exhibiting regions of negativity.

The integrated nonclassicality measure, $\eta_{\mathrm{NC}}$, likewise increases monotonically with $\alpha$. All curves for different well separations originate from a minimal baseline (Gaussian-like at small $\alpha$) and grow significantly with stronger double-well character.

Figure 5: Nonclassicality measure $\eta_{\mathrm{NC}}$ as a function of $\alpha$, monotonically increasing for all $x_0$.

A parametric sweep of $\eta_{\mathrm{NC}}$ versus $\eta_{\mathrm{NG}}$ shows a universal monotonic and superlinear co-variation, regardless of $x_0$. This operational dependence enables robust quantification of nonclassical resources from ground-state statistics alone.

Figure 6: Correlation between non-Gaussianity $\eta_{\mathrm{NG}}$ and nonclassicality $\eta_{\mathrm{NC}}$ for various $x_0$, fit with a composite power law.

Entanglement Generation via Linear Optics

Entanglement potential (EP) is used to operationally benchmark single-mode nonclassicality: interference of the DM ground state with vacuum at a $50{:}50$ beam splitter produces bipartite output entanglement, quantified by von Neumann entropy. As the control parameter $\alpha$ increases, $EP(\rho)$ rises monotonically and saturates, with higher $x_0$ consistently increasing the entanglement for all $\alpha$.

Figure 7: Entanglement potential $EP(\rho)$ generated by beam splitter versus $\alpha$; higher $x_0$ boosts entanglement output.

Quantum Metrological Performance

Estimation protocols targeting the DM width/asymmetry $\alpha$ are cast in the framework of quantum estimation theory (QET). The authors derive a compact analytical formula for the quantum Fisher information (QFI) of the DM ground state using the closed-form solution for $\psi_0(x)$. The QFI is shown to decrease monotonically with $\alpha$ for all tested values of $x_0$, reflecting reduced sensitivity in the deep double-well regime. Direct position measurements on the ground state are proven to be optimal, achieving the quantum Cramér–Rao bound for estimator precision in $\alpha$.

Figure 8: Fisher information of $\psi_0(x)$ as a function of $\alpha$ for multiple $x_0$; information content reduces as wells deepen.

Implications and Prospects

This study establishes the double-Morse potential as a structurally tunable, analytically tractable platform for continuous-variable nonclassical state engineering. The ability to transition between Gaussian and strongly non-Gaussian regimes by manipulating $\alpha$ enables fine control over quantum features required in quantum information processing, quantum simulation, and precision metrology. The systematic monotonic enhancements in non-Gaussianity, nonclassicality, and output entanglement position the DM oscillator as a versatile resource for quantum technologies.

Operational implementation would benefit from experimental studies of decoherence effects and structural imperfections, investigating the resilience of Wigner negativity and output entanglement under realistic conditions. Extensions to mixed-state scenarios and verification of entanglement generation by linear optics will broaden applicability. Quantum estimation strategies leveraging shallow-well sensitivity could yield improved protocols for parameter sensing in chemical and condensed matter contexts.

Conclusion

The paper demonstrates that the double-Morse potential is a controllable generator of non-Gaussian, nonclassical single-mode quantum states, with monotonic scaling of operational resource measures with the width parameter $\alpha$. Analytic tractability enables closed-form assessments of state properties and metrological bounds. The DM oscillator’s capacity for entanglement generation and parameter estimation advocates its adoption in experimental quantum technology platforms and supports further exploration of complex, tunable nonlinear potentials for quantum information science.