Phase-modulus relations in cyclic wave functions

Englman, R.; Yahalom, Asher; Baer, Michael

doi:10.1016/s0375-9601(98)00897-4

Cited by 16 publications

(34 citation statements)

References 14 publications

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“…It is also important to mention that when we are far from the adiabatic limit the fit is less satisfactory. However, we also found that for the choices of k which make the function χ 1 (t) periodic, namely when ( k ω ) = integer, the agreement resurfaces [12].…”

Section: The Degenerate Casementioning

confidence: 65%

“…In a representation, adopted from [17], this doublet is described in terms of the electronic functions cos θ and sinθ and therefore ψ can be expressed as: ([6], [12])…”

Section: The Basic Equationsmentioning

confidence: 99%

“…Versions of these relations in other contexts were given earlier ( [10]- [12]). The existence of these relations has the remarkable consequence that the associated open path phase, defined by them, is a "physical observable" (and inter alia gauge invariant) as a function of the path, a quality heretofore associated with the closed path (Berry) phase.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

Englman

Yahalom

Baer³

2000

Eur. Phys. J. D

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We calculate the open path phase in a two state model with a slowly (nearly adiabatically) varying time-periodic Hamiltonian and trace its continuous development during a period. We show that the topological (Berry) phase attains π or 2π depending on whether there is or is not a degeneracy in the part of the parameter space enclosed by the trajectory. Oscillations are found in the phase. As adiabaticity is approached, these become both more frequent and less pronounced and the phase jump becomes increasingly more steep. Integral relations between the phase and the amplitude modulus (having the form of Kramers-Kronig relations, but in the time domain) are used as an alternative way to calculate open path phases. These relations attest to the observable nature of the open path phase.

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Section: The Degenerate Casementioning

confidence: 65%

“…In a representation, adopted from [17], this doublet is described in terms of the electronic functions cos θ and sinθ and therefore ψ can be expressed as: ([6], [12])…”

Section: The Basic Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

Englman

Yahalom

Baer³

2000

Eur. Phys. J. D

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“…This, unlike the Berry-phase proper, is not gauge-invariant, but is, nevertheless (partially) accessible by experiments ( [30] - [32]). The Berry phase for non-stationary states was given in [13], the interchange between dynamic and geometric phases is treated in [118].…”

Section: Aspects Of Phase In Moleculesmentioning

confidence: 99%

“…The treatment by Garrison and Wright of complex topological phases for non-Hermitean Hamiltonians [170] was extended in [171] - [173]. Further advances on Berry-phases are corrections due to non-adiabatic effects (resulting, mainly, in a decrease from the value of the phase in the adiabatic, infinitely slow limit) [30,174,175]. In [176] the complementarity between local and nonlocal effects is studied by means of some examples.…”

Section: Aspects Of Phase In Moleculesmentioning

confidence: 99%

Complex States of Simple Molecular Systems

Englman¹,

Yahalom²

2002

Advances in Chemical Physics

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A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities ("the phase problem"), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow.When the stationary Born-Oppenheimer description for nearly degenerate state is regarded as "embedded" in a broader dynamic formulation, namely through solution of the time dependent Schrödinger equation, the wave function becomes necessarily complex. The analytic behavior of this function in the complex time-plane can be exploited to gain information about the connection between phases and moduli of the componentamplitudes. One form of this connection are a pair of reciprocal (Kramers-Kronig type) relations in the complex time-domain. These show that certain phase changes in a component amplitude (such as, e.g., lead to a non-zero Berry-phase) require changes in the amplitude-moduli, too, and imply degeneracies of the electronic state.The subject of conical degeneracy, or intersection, of adjacent potential surface is extended to nonlinear nuclear-electronic coupling, which can then result in multiple conical degeneracies on a nuclear coordinate plane. We treat cases of double and fourfold intersections, the latter under trigonal symmetry, and employ an analytic-graphical phase tracing method to obtain the resulting Berry phases as the system circles around some or all of the degeneracy points. These phases can take up values that are all integral multiples of π, with integers that vary with the physical situation (or with the model postulated for it).A further type of invariance, with respect to gauge transformation, is also tied to the complex form of the wave function. For a many-component state this leads to a consideration of tensorial (Yang-Mills type) fields F for molecular systems. It is shown that it is the truncation of the Born-Oppenheimer electron-nuclear superposition that generates the molecular Yang-Mills fields.The equations of motion for the nuclear degrees of freedom are derived from a Lagr...

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Experimental and calculational consequences of phases in molecules with multiple conical intersections

Englman

Yahalom

Baer³

et al. 2003

Int J of Quantum Chemistry

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ABSTRACT:We set up a theoretical model for treating several degeneracies between two adjacent potential energy surfaces in molecular systems, e.g., cases of four (or more) conically intersecting degeneracies, located in a plane formed by two-fold molecular displacement coordinates, in trigonal (or cubic) symmetry and twin conical intersections (CIs) for molecules with two-fold symmetries. When the system circles (in a time-variant manner) entirely inside or entirely outside these CIs, it picks up phases (the geometric phases) that are zero or 2N. Here, N is (to a good approximation) an integer whose value depends on the Hamiltonian for the system, on the distance of circling from the CI, and on the speed of circling. For the adiabatic (slow circling) limit, we develop a simple algebraic-graphical method that yields the value of N in each case. Even in this limit, the value of N is model dependent and is not uniquely given by the number of CIs that are circled. Further, we suggest an experiment (based on a pumpprobe method) for tracing the time development of a system that is subject to a periodically varying field carrying it around the CI. The experiment leads to relations between the spectroscopic transition strengths and state amplitudes, such that the (open-path) phases are fully given (including the additive 2N term and excluding the dynamic phase). Finally, we compare geometric phases that differ by integral multiples of 2 and, therefore, cannot be distinguished by phase interference. We show that the different cases have, nevertheless, observational correlates in the number of times that zero or unity occurs in a wave function component amplitude during circling. The rules of correspondence are given.

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Phase-modulus relations in cyclic wave functions

Cited by 16 publications

References 14 publications

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

Complex States of Simple Molecular Systems

Experimental and calculational consequences of phases in molecules with multiple conical intersections

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