Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2602.04793

Correspondence between classical and quantum resonances

F. J. Arranz, F. Borondo

nlin.CDquant-phSubmitted Feb 4, 2026• Announced Feb 5, 2026
v1 · Feb 4, 2026v2 · Feb 5, 2026
arXivHTMLPDF

Abstract

Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, this leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li$\leftrightarrows$Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series to $\hbar=0$ unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limit $\hbar\to0$. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.

Journal reference
Physical Review E 103, 062207 (2021)
DOI
10.1103/PhysRevE.103.062207

Related papers

Semiclassical behavior at a quantum avoided crossing

Marc Joyeux

For a polynomial potential with resonant fundamental frequencies (1:2 and 1:3 resonances), quantum avoided crossings can occur when quantum eigenvalues are plotted versus a parameter in the Hamiltonian. In the present paper, primitive (EBK) semiclassical behavior at the quantum avoided crossing is reinvestigated, using the exact analytical calculation of the action integrals, which was devised recently [Chem. Phys. 185, 263 (1994)] for an approximate resonance Hamiltonian that can be deduced from the exact polynomial Hamiltonian by low order perturbation theory. The previously reported behavior, that is semiclassical levels passing through the intersection instead of avoiding each other, is shown to happen if there exist two superimposed branches in the plot of the second action integral ℐ2 as a function of the energy. These results are interpreted in terms of semiclassical diabatic basis and of quantum dynamical tunneling. In contrast, if the semiclassical system enters the (anti)crossing region with semiclassical quantum numbers ℐ2 which do not lie on superimposed branches of the plot, it is shown that at least one, and possibly two, level(s) must cross the separatrix, that is pass from the inside to the outside of the resonance region (or conversely) in order to adapt to the quantum avoided crossing. This causes (i) corresponding semiclassical quantum number ℐ2 to change (ii) the close correspondence between quantum and semiclassical mechanics to break down.

The Journal of Chemical Physics•Feb 15, 1995•7 citations

Regular and Irregular Correspondences: Adiabatic Invariants in Classical and Quantum Mechanics

William P. Reinhardt

We outline a rather extraordinary series of similarities between classical and quantal behavior in the limit of adiabatic time changes. These include the power laws for the goodness of the respective invariants for isolated eigenstates and invariant tori for integrable systems, the nature of the breakdown of the invariances—level crossing in quantum systems and the role of ever present non-linear resonances is examined in the case of generically non-integrable classical dynamics—and the perhaps surprising relationship for fully chaotic systems where sufficiently slow switching in either classical or quantal systems precisely preserves the number of energy levels up to a given energy. For suitably small values of Planck's constant these similarities yield clear examples of the Bohr correspondence principle linking classical and quantum mechanics; for larger values the details in the classical picture are quenched in the quantum.

Progress of Theoretical Physics Supplement•May 16, 2013•0 citations

Semiclassical spectrum of small Bose-Hubbard chains: A normal-form approach

A. P. Itin, Peter Schmelcher

We analyze the spectrum of the three-site Bose-Hubbard model with periodic boundary conditions using a semiclassical method. The Bohr-Sommerfeld quantization is applied to an effective classical Hamiltonian which we derive using resonance normal form theory. The derivation takes into account the 1:1 resonance between frequencies of a linearized classical system and brings nonlinear terms into a corresponding normal form. The obtained expressions reproduce the exact low-energy spectrum of the system remarkably well even for a small number of particles $N$ corresponding to fillings of just two particles per site. Such small fillings are often used in current experiments, and it is inspiring to get insight into this quantum regime using essentially classical calculations.

arXiv:1109.1304•Physical Review A•Dec 7, 2011•11 citations

The influence of classical resonances on quantum energy levels

B. Ramachandran, Kenneth G. Kay

In this paper, we examine the behavior of the quantum energy levels of a coupled oscillator system as the zero-order frequencies are varied to carry the corresponding classical system through a resonance. We find that the levels exhibit a pattern that is characteristic of the resonance. This pattern consists of clusters of levels, each containing a number of curves that run roughly parallel to one another and a number of curves that undergo pairwise narrowly avoided crossings. Adiabatic switching calculations show that the ‘‘parallel’’ curves are associated with states within the classical resonance region, while the narrowly avoiding curves are associated with states that are outside this region. It is further shown that the curves describing resonance states are formed from zero-order nonresonant curves by the overlap of many avoided crossings. This reorganization of multiply intersecting lines into parallel curves reflects the classical reorganization of phase space at a resonance.

The Journal of Chemical Physics•Sep 1, 1993•23 citations

Bifurcation Structure of Eigenstates and Periodic Trajectories in TDHF Phase Space: Weak Nonlinearity Case in SU(3) Model

Hidehiko Tsukuma, Yukio Hashimoto, Fumihiko Sakata +1 more

To find out how quantum bifurcation phenomena manifest themselves in the space of states, various periodic trajectories accompanied with several sets of bifurcated trajectories are studied for a simple quantum model Hamiltonian under a time-dependent mean-field approximation. Making use of the quantum-classical correspondence, a method of classifying the eigenstates and of visualizing the global structures of the whole space of eigenstates is proposed. By means of the profile functions of the eigenstates, appropriate sets of eigenstates representing the quantum correspondents of the bifurcations are found. The idea of quantum bifurcations is supported by similar behavior of the energy vs period time ( E-T ) plots of the periodic trajectories and the eigenstates.

Progress of Theoretical Physics•Dec 1, 1998•0 citations

Manifestation of classical bifurcation in the spectrum of the integrable quantum dimer

S. Aubry, Sergej Flach A, K. Kladko +1 more

We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model.

arXiv:cond-mat/9808332•Aug 30, 1998•0 citations
Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, this leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li⇆\leftrightarrows⇆Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series to ℏ=0\hbar=0ℏ=0 unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limit ℏ→0\hbar\to0ℏ→0. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.
For a polynomial potential with resonant fundamental frequencies (1:2 and 1:3 resonances), quantum avoided crossings can occur when quantum eigenvalues are plotted versus a parameter in the Hamiltonian. In the present paper, primitive (EBK) semiclassical behavior at the quantum avoided crossing is reinvestigated, using the exact analytical calculation of the action integrals, which was devised recently [Chem. Phys. 185, 263 (1994)] for an approximate resonance Hamiltonian that can be deduced from the exact polynomial Hamiltonian by low order perturbation theory. The previously reported behavior, that is semiclassical levels passing through the intersection instead of avoiding each other, is shown to happen if there exist two superimposed branches in the plot of the second action integral ℐ2 as a function of the energy. These results are interpreted in terms of semiclassical diabatic basis and of quantum dynamical tunneling. In contrast, if the semiclassical system enters the (anti)crossing region with semiclassical quantum numbers ℐ2 which do not lie on superimposed branches of the plot, it is shown that at least one, and possibly two, level(s) must cross the separatrix, that is pass from the inside to the outside of the resonance region (or conversely) in order to adapt to the quantum avoided crossing. This causes (i) corresponding semiclassical quantum number ℐ2 to change (ii) the close correspondence between quantum and semiclassical mechanics to break down.
We outline a rather extraordinary series of similarities between classical and quantal behavior in the limit of adiabatic time changes. These include the power laws for the goodness of the respective invariants for isolated eigenstates and invariant tori for integrable systems, the nature of the breakdown of the invariances—level crossing in quantum systems and the role of ever present non-linear resonances is examined in the case of generically non-integrable classical dynamics—and the perhaps surprising relationship for fully chaotic systems where sufficiently slow switching in either classical or quantal systems precisely preserves the number of energy levels up to a given energy. For suitably small values of Planck's constant these similarities yield clear examples of the Bohr correspondence principle linking classical and quantum mechanics; for larger values the details in the classical picture are quenched in the quantum.
We analyze the spectrum of the three-site Bose-Hubbard model with periodic boundary conditions using a semiclassical method. The Bohr-Sommerfeld quantization is applied to an effective classical Hamiltonian which we derive using resonance normal form theory. The derivation takes into account the 1:1 resonance between frequencies of a linearized classical system and brings nonlinear terms into a corresponding normal form. The obtained expressions reproduce the exact low-energy spectrum of the system remarkably well even for a small number of particles NNN corresponding to fillings of just two particles per site. Such small fillings are often used in current experiments, and it is inspiring to get insight into this quantum regime using essentially classical calculations.
In this paper, we examine the behavior of the quantum energy levels of a coupled oscillator system as the zero-order frequencies are varied to carry the corresponding classical system through a resonance. We find that the levels exhibit a pattern that is characteristic of the resonance. This pattern consists of clusters of levels, each containing a number of curves that run roughly parallel to one another and a number of curves that undergo pairwise narrowly avoided crossings. Adiabatic switching calculations show that the ‘‘parallel’’ curves are associated with states within the classical resonance region, while the narrowly avoiding curves are associated with states that are outside this region. It is further shown that the curves describing resonance states are formed from zero-order nonresonant curves by the overlap of many avoided crossings. This reorganization of multiply intersecting lines into parallel curves reflects the classical reorganization of phase space at a resonance.
To find out how quantum bifurcation phenomena manifest themselves in the space of states, various periodic trajectories accompanied with several sets of bifurcated trajectories are studied for a simple quantum model Hamiltonian under a time-dependent mean-field approximation. Making use of the quantum-classical correspondence, a method of classifying the eigenstates and of visualizing the global structures of the whole space of eigenstates is proposed. By means of the profile functions of the eigenstates, appropriate sets of eigenstates representing the quantum correspondents of the bifurcations are found. The idea of quantum bifurcations is supported by similar behavior of the energy vs period time ( E-T ) plots of the periodic trajectories and the eigenstates.
We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model.