The influence of classical resonances on quantum energy levels (original) (raw)

Classical resonance overlapping and quantum avoided crossings

Physical Review Letters, 1987

It is pointed out that for pulsed rotators the classical resonance-overlapping criterion for the onset of chaotic motion implies the existence of avoided crossings among an arbitrarily large number of quantum levels at the classical limit 6 small, nb =2 close to a classical resonance.

Correspondence between classical and quantum resonances

Physical Review E, 2021

Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, 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 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 toh = 0 unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limith → 0. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.

Crossover phenomena and resonances in quantum systems

Physical Review A, 2001

The finite-size scaling method is used to calculate the critical parameters of a two-parameter model Hamiltonian that exhibit resonances. We show the existence of a crossover phenomenon for the energy spectrum; the transition from a bound state to a continuum is a ''second-order phase transition'' in one region and a ''firstorder phase transition'' in another region. As the parameters varied, the numerical value of the critical exponent of the energy levels changed from two to one. In the zone where the critical exponent equals one, the system has narrow resonances and they disappear when the exponent is two. The method has potential applicability in predicting stable and metastable atomic and molecular states. We also show that finite-size scaling methods are capable of detecting multicritical points.

Classical resonances in quantum mechanics

Physical Review A, 1992

%e study the role of classical resonances in quantum systems subjected to a periodic external force. It is demonstrated that classical invariant vortex tubes determine the structure of Floquet states "captured" by resonances in the classical phase space. In addition, resonance quantum numbers are introduced. The analysis of simple model calculations leads to a qualitative description of nonperturbative phenomena relevant for the interaction of atoms or molecules with strong, short laser pulses.

Avoided crossing resonances: Structural and dynamical aspects

Physical Review A, 2009

We examine structural and dynamical properties of quantum resonances associated with an avoided crossing and identify the parameter shifts where these properties attain extreme values. Thus the concept of avoided crossing resonance can be defined in different ways, which do not coincide in the general case. These definitions are described first at a general level, and then for a two-level system coupled to a harmonic oscillator, of the type commonly found in quantum optics. Finally the results obtained are exemplified and applied to optimize the fidelity and speed of quantum gates in trapped ions.

Phase-space picture of resonance creation and avoided crossings

Physical Review A, 2001

Complex coordinate scaling (CCS) is used to calculate resonance eigenvalues and eigenstates for a system consisting of an inverted Gaussian potential and a monochromatic driving field. Floquet eigenvalues and Husimi distributions of resonance eigenfunctions are calculated using two different versions of CCS. The number of resonance states in this system increases as the strength of the driving field is increased, indicating that this system might have increased stability against ionization when the field strength is very high. We find that the newly created resonance states are scarred on unstable periodic orbits of the classical motion. The behavior of these periodic orbits as the field strength is increased may explain why there are more resonance states at high field strengths than at low field strengths. Close examination of an avoided crossing between resonance states shows that this type of avoided crossing does not delocalize the resonance states, although it may lead to interesting effects at certain field strengths.

Widths of resonances above an energy-level crossing

Journal of Functional Analysis, 2021

We study the existence and location of the resonances of a 2 × 2 semiclassical system of coupled Schrödinger operators, in the case where the two electronic levels cross at some point, and one of them is bonding, while the other one is anti-bonding. Considering energy levels just above that of the crossing, we find the asymptotics of both the real parts and the imaginary parts of the resonances close to such energies. This is a continuation of our previous works where we considered energy levels around that of the crossing.

Phase rigidity and avoided level crossings in the complex energy plane

Physical Review E, 2006

We consider the effective Hamiltonian of an open quantum system, its biorthogonal eigenfunctions φ λ and define the value r λ = (φ λ |φ λ)/ φ λ |φ λ that characterizes the phase rigidity of the eigenfunctions φ λ. In the scenario with avoided level crossings, r λ varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of r λ may be considered as an internal property of an open quantum system. In the literature, the phase rigidity ρ of the scattering wave function Ψ E C is considered. Since Ψ E C can be represented in the interior of the system by the φ λ , the phase rigidity ρ of the Ψ E C is related to the r λ and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity ρ to values smaller than 1 should be considered, at least partly, as an internal property of an open quantum system in the overlapping regime. The relation to measurable values such as the transmission through a quantum dot, follows from the fact that the transmission is, in any case, resonant with respect to the effective Hamiltonian. We illustrate the relation between phase rigidity ρ and transmission numerically for small open cavities.

Semiclassical complex energy theory of orbiting resonances in curve crossing systems

Journal of Physics B: Atomic and Molecular Physics, 1986

We discuss orbiting resonances in two-state curve crossing systems. A semiclassical quantisation condition determining complex-energy resonance poles is derived. Applications to the electron capture collision process N3+ + H-+ N2+ + H+ show good agreement with recent exact quantum computations.

Adiabatic Theorems for Quantum Resonances

Communications in Mathematical Physics, 2007

We study the adiabatic time evolution of quantum resonances over time scales which are small compared to the lifetime of the resonances. We consider three typical examples of resonances: The first one is that of shape resonances corresponding, for example, to the state of a quantum-mechanical particle in a potential well whose shape changes over time scales small compared to the escape time of the particle from the well. Our approach to studying the adiabatic evolution of shape resonances is based on a precise form of the time-energy uncertainty relation and the usual adiabatic theorem in quantum mechanics. The second example concerns resonances that appear as isolated complex eigenvalues of spectrally deformed Hamiltonians, such as those encountered in the N-body Stark effect. Our approach to study such resonances is based on the Balslev-Combes theory of dilatation-analytic Hamiltonians and an adiabatic theorem for nonnormal generators of time evolution. Our third example concerns resonances arising from eigenvalues embedded in the continuous spectrum when a perturbation is turned on, such as those encountered when a small system is coupled to an infinitely extended, dispersive medium. Our approach to this class of examples is based on an extension of adiabatic theorems without a spectral gap condition. We finally comment on resonance crossings, which can be studied using the last approach. ysis can be extended to the case of degenerate resonances; (see for a discussion of the latter in the time-independent case).