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arXiv:2511.07591 · EPJ Quantum Technology

Nonlinearity and Quantum Metrology in the Double-Morse Potential

Firoz Chogle, Berihu Teklu, Jorge Zubelli, Ernesto Damiani

quant-phSubmitted Nov 10, 2025• Announced Nov 12, 2025• 2 citations• Revised Jul 28, 2026 (v3)
v1 · Nov 10, 2025v2 · May 15, 2026v3 · Jul 28, 2026
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Abstract

We address the nonlinear properties of the double-Morse potential as a resource for single-mode quantum states due to its double-well structure and anharmonicity. We obtain analytical expressions for the ground-state wavefunction and the corresponding ground-state energy, using the asymmetry (width) parameter $α$ as the primary control parameter. We then assess non-Gaussianity and nonclassicality as quantitative signatures of nonlinearity and quantumness, and we find that both increase monotonically with $α$. Furthermore, we analyze the metrological performance of the model for estimating the structural parameter $α$. By evaluating the corresponding Fisher information, we show that position measurements are optimal and can saturate the Cramér-Rao bound. In particular, the estimation of $α$ is most precise in the shallow-well regime, where the quantum Fisher information is largest. For deep wells, enhanced sensitivity is instead obtained for the reparameterized control variable $A=2e^{-αx_0}$, provided that $x_0$ is independently calibrated. These results establish the double-Morse potential as a controllable source of non-Gaussianity and nonclassicality, with a metrological behavior that depends on the chosen estimation parameter. We highlight possible applications of this model in quantum sensing, continuous-variable quantum information, and quantum simulation.

Comments
16 pages, 12 figures

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We address the nonlinear properties of the double-Morse potential as a resource for single-mode quantum states due to its double-well structure and anharmonicity. We obtain analytical expressions for the ground-state wavefunction and the corresponding ground-state energy, using the asymmetry (width) parameter ααα as the primary control parameter. We then assess non-Gaussianity and nonclassicality as quantitative signatures of nonlinearity and quantumness, and we find that both increase monotonically with ααα. Furthermore, we analyze the metrological performance of the model for estimating the structural parameter ααα. By evaluating the corresponding Fisher information, we show that position measurements are optimal and can saturate the Cramér-Rao bound. In particular, the estimation of ααα is most precise in the shallow-well regime, where the quantum Fisher information is largest. For deep wells, enhanced sensitivity is instead obtained for the reparameterized control variable A=2e−αx0A=2e^{-αx_0}A=2e−αx0​, provided that x0x_0x0​ is independently calibrated. These results establish the double-Morse potential as a controllable source of non-Gaussianity and nonclassicality, with a metrological behavior that depends on the chosen estimation parameter. We highlight possible applications of this model in quantum sensing, continuous-variable quantum information, and quantum simulation.
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 AAA, 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.
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 AAA controls the separation of the minima and the central barrier, thereby changing the physical ground state. At fixed AAA, the Cahill--Glauber parameter sss labels the quasiprobability WA(s)(q,p)W_A^{(s)}(q,p)WA(s)​(q,p): s=0s=0s=0, −1-1−1, and 111 give the Wigner, Husimi QQQ, and Glauber--Sudarshan PPP representations, respectively. Although the potential is double-welled for 0<A<10<A<10<A<1, the exact ground-state amplitude is single-peaked at the origin and lies above the barrier; as AAA 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 AAA-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 sss-ordered hierarchy. For s<0s<0s<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 PPP representation remains distributional. Thus, AAA controls the physical phase-space geometry, while sss controls how the same non-Gaussian and nonclassical state is resolved across complementary representations
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
We address the precise characterization of coupling strength of nonlinear media in continuous-variable (CV) quantum systems using coherent and squeezed vacuum states as Gaussian probes, together with their photon-added and photon-subtracted counterparts as non-Gaussian probes. We consider three main classes of nonlinear Hamiltonians, namely quadrature nonlinearities, generalized squeezing and Kerr-type interactions. By analytically evaluating the quantum Fisher information (QFI), we compare the performance of Gaussian and non-Gaussian probes and assess the optimal probe based on the probe parameters, energy resource and non-Gaussianity. Our results are twofold as follows: first, for coherent-state family, the improvement provided by photon addition at fixed coherent amplitude originates mainly from the extra energy carried by the probe and does not provide a genuine metrological resource, since the same precision can be achieved by a Gaussian coherent-state signal of a larger energy, which can be more easily produced. Second, in contrast, photon addition and subtraction become effective resources when applied to already nonclassical states such as squeezed vacuum states. In this case, they lead to a significant enhancement of the QFI, particularly for higher-order interactions. Although Gaussian squeezed states remain optimal at equal energy constraint, photon-added and photon-subtracted squeezed states achieve comparable sensitives with significantly lower squeezing requirements. Since large squeezing level remains experimentally challenging, these non-Gaussian probes offer a practical route towards enhanced estimation of the nonlinear coupling strength within currently accessible squeezing regimes.
The well-known metrological linear squeezing parameters (such as quadrature or spin squeezing) efficiently quantify the sensitivity of Gaussian states. Yet, these parameters are insufficient to characterize the much wider class of highly sensitive non-Gaussian states. Here, we introduce a class of metrological nonlinear squeezing parameters obtained by analytical optimization of measurement observables among a given set of accessible (possibly nonlinear) operators. This allows for the metrological characterization of non-Gaussian quantum states of discrete and continuous variables. Our results lead to optimized and experimentally feasible recipes for a high-precision moment-based estimation of a phase parameter and can be used to systematically construct multipartite entanglement and nonclassicality witnesses for complex quantum states.
As one of the main pillars of quantum technologies, quantum metrology aims to improve measurement precision using techniques from quantum information. The two main strategies to achieve this are the preparation of nonclassical states and the design of optimized measurement observables. We discuss precision limits and optimal strategies in quantum metrology and sensing with a single mode of quantum continuous variables. We focus on the practically most relevant cases of estimating displacements and rotations and provide the sensitivities of the most important classes of states that includes Gaussian states and superpositions of Fock states or coherent states. Fundamental precision limits that are obtained from the quantum Fisher information are compared to the precision of a simple moment-based estimation strategy based on the data obtained from possibly sub-optimal measurement observables, including homodyne, photon number, parity and higher moments. Finally, we summarize some of the main experimental achievements and present emerging platforms for continuous-variable sensing. These results are of particular interest for experiments with quantum light, trapped ions, mechanical oscillators, and microwave resonators.