Nimrod Moiseyev

Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling

Abstract Complex scaling enables one to associate the resonance phenomenon, as it appears in atomic, molecular, nuclear physics and in chemical reactions, with a single square integrable eigenfunction of the complex-scaled Hamiltonian, rather than with a collection of continuum eigenstates of the unscaled hermitian Hamiltonian. In this report, we illustrate the complex-scaling method by giving examples of simple analytically soluble models. We describe the computational algorithms which enable the use of complex scaling for the calculations of the energy positions lifetimes and partial widths of atomic and molecular autoionization resonance states, of small polyatomic molecules and van der Waals molecules in predissociation resonance states, of atoms and molecules which are temporarily trapped on a solid surface and of atoms and molecules which ionized/dissociate when they are exposed to high intensity laser field. We focus on the properties of the complex scaled Hamiltonian and on the extension of theorems and principles, which were originally proved in quantum mechanics for hermitian operators to non-hermitian operators and also on the development of the complex coordinate scattering theory.

Visualization of Branch Points in PT-Symmetric Waveguides

The visualization of an exceptional point in a PT-symmetric directional coupler (DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schrödinger perturbation theory we prove that the spectrum of a PT-symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a PT-symmetric directional coupler to measure the radius of convergence for non-PT-symmetric structures. For such systems the physical meaning of the rather mathematical term radius of convergence is exemplified.

The solution of the time‐dependent Schrödinger equation by the (t,t’) method: Theory, computational algorithm and applications

A new powerful computational method is introduced for the solution of the time dependent Schrodinger equation with time‐dependent Hamiltonians (not necessarily time‐periodic). The method is based on the use of the Floquet‐type operator in an extended Hilbert space, which was introduced by H. Sambe [Phys. Rev. A 7, 2203 (1973)] for time periodic Hamiltonians, and was extended by J. Howland [Math Ann. 207, 315 (1974)] for general time dependent Hamiltonians. The new proposed computational algorithm avoids the need to introduce the time ordering operator when the time‐dependent Schrodinger equation is integrated. Therefore it enables one to obtain the solution of the time‐dependent Schrodinger equation by using computational techniques that were originally developed for cases where the Hamiltonian is time independent. A time‐independent expression for state‐to‐state transition probabilities is derived by using the analytical time dependence of the time evolution operator in the generalized Hilbert space. Ill...

Cusps, θ trajectories, and the complex virial theorem

The conditions are discussed under which cusps in ϑ trajectories of complex resonance energies correspond to eigenvalues which satisfy the complex virial theorem. (AIP)

Derivations of universal exact complex absorption potentials by the generalized complex coordinate method

On the basis of the Moiseyev-Hirschfelder generalization of the complex coordinate method, a universal energy-independent complex absorbing potential (CAP), is derived. It is proven that the universal CAP consists of flux and diffusion-type operators. When a smooth exterior scaling is used, the CAP gets non-zero values in the region where the interaction potential vanishes. An illustrative numerical example is given where narrow and broad, isolated and overlapping resonances were all calculated with more than nine digits of accuracy.

The solution of the time dependent Schrödinger equation by the (t,t’) method: The use of global polynomial propagators for time dependent Hamiltonians

Using the (t,t’) method as introduced in Ref. [J. Chem. Phys. 99, 4590 (1993)] computational techniques which originally were developed for time independent Hamiltonians can be used for propagating an initial state for explicitly time dependent Hamiltonians. The present paper presents a time dependent integrator of the Schrodinger equation based on a Chebychev expansion, of the operator U(x,t’,t0→t), and the Fourier pseudospectral method for calculating spatial derivatives [(∂2/∂x2),(∂/∂t’)]. Illustrative numerical examples for harmonic and Morse oscillators interacting with CW and short pulsed laser fields are given.

On the observability and asymmetry of adiabatic state flips generated by exceptional points

In open quantum systems where the effective Hamiltonian is not Hermitian, it is known that the adiabatic (or instantaneous) basis can be multivalued: by adiabatically transporting an eigenstate along a closed loop in the parameter space of the Hamiltonian, it is possible to end up in an eigenstate different from the initial eigenstate. This ‘adiabatic flip’ effect is an outcome of the appearance of a degeneracy known as an ‘exceptional point’ inside the loop. We show that contrary to what is expected of the transport properties of the eigenstate basis, the interplay between gain/loss and non-adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect.

Representation of several complex coordinate methods by similarity transformation operators

The complex coordinate method (CCM) can be presented by carrying out a similarity transformation of the Hamiltonian S+1HS−1 in order to correct the asymptotic behavior of the resonance eigenfunction, such that Sψres→0, whereas ψres→∞, as r→∞. Therefore, after the similarity transformation, the number of the particles is conserved in the coordinate space for any given time. Here, several different possibilities for S are presented, emphasizing the advantage of the representation of CCM by similarity transformation operators. A new extension of the complex coordinate method is that S∼exp[−θf 1/2(r)(∂/∂r) f 1/2(r)], where f(r) can be any function for which f(r)/r→1 as r→∞. Whereas, in the conventional CCM, f(r)=r. This new method enables one to select the ‘‘optimal’’ path in the complex coordinate plane, which gets past the intrinsic nonanalyticities of the potential and provides the most stable resonance solution which can be obtained by a given number of basis functions.

Dynamically encircling an exceptional point for asymmetric mode switching.

Physical systems with loss or gain have resonant modes that decay or grow exponentially with time. Whenever two such modes coalesce both in their resonant frequency and their rate of decay or growth, an ‘exceptional point’ occurs, giving rise to fascinating phenomena that defy our physical intuition. Particularly intriguing behaviour is predicted to appear when an exceptional point is encircled sufficiently slowly, such as a state-flip or the accumulation of a geometric phase. The topological structure of exceptional points has been experimentally explored, but a full dynamical encircling of such a point and the associated breakdown of adiabaticity have remained out of reach of measurement. Here we demonstrate that a dynamical encircling of an exceptional point is analogous to the scattering through a two-mode waveguide with suitably designed boundaries and losses. We present experimental results from a corresponding waveguide structure that steers incoming waves around an exceptional point during the transmission process. In this way, mode transitions are induced that transform this device into a robust and asymmetric switch between different waveguide modes. This work will enable the exploration of exceptional point physics in system control and state transfer schemes at the crossroads between fundamental research and practical applications.

On the interatomic Coulombic decay in the Ne dimer

The interatomic Coulombic decay (ICD) in the Ne dimer is discussed in view of the recent experimental results. The ICD electron spectrum and the kinetic energy release of the Ne+ fragments resulting after Coulomb explosion of Ne2 (2+) are computed and compared to the measured ones. A very good agreement is found, confirming the dynamics predicted for this decay mechanism. The effect of the temperature on the electron spectrum is briefly investigated.