Geometric Phase Effects for Wave-Packet Revivals

Jarzynski, Christopher

doi:10.1103/physrevlett.74.1264

Cited by 16 publications

(16 citation statements)

References 15 publications

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“…Not only is the theory firmly based, and simulations convincing, but even an application, based on this phenomenon and aimed at separation of isotopes, has been proposed [199]. Elsewhere, it was shown that the effect of slow cycling on the evolving wave-packet is to leave the revival period unchanged, but to cause a shift in the position of the revived wave packet [200].…”

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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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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“…Most such applications apply to time-independent Hamiltonians, and the observation of cyclic wave packet has been of interest [2,3]. Wave-packet revival was also studied in the context of adiabatic time-dependent Hamiltonians, and the link between Berry's phase and Hannay's angle was confirmed [4]. The purpose of this Letter is to explore GWP dynamics for Hamiltonians with nonadiabatic time dependence, find conditions for the occurrence of cyclic GWP, and demonstrate an explicit connection between the nonadiabatic geometrical phase effects in quantum and classical mechanics.…”

mentioning

confidence: 98%

Nonadiabatic Geometrical Phase during Cyclic Evolution of a Gaussian Wave Packet

Ge¹,

Child²

1997

Phys. Rev. Lett.

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The Gaussian wave packet solution to the Schrödinger equation is studied for time-dependent Hamiltonians. The geometrical phase is obtained for a cyclic wave packet solution of the generalized harmonic oscillator with a nonadiabatic time-periodic Hamiltonian. It is found that the geometrical phase is independent ofh, and is equal to one-half of the classical nonadiabatic Hannay angle. The Hannay angle is shown to be independent of the classical action and does not involve averaging. [S0031-9007(97)02766-X] PACS numbers: 03.65.Bz, 03.65.GeSemiclassical Gaussian wave packets (GWP) have been applied to many problems in atomic and molecular physics including Franck-Condon spectra, photodissociation, spectral quantization, and Rydberg dynamics [1,2]. Most such applications apply to time-independent Hamiltonians, and the observation of cyclic wave packet has been of interest [2,3]. Wave-packet revival was also studied in the context of adiabatic time-dependent Hamiltonians, and the link between Berry's phase and Hannay's angle was confirmed [4]. The purpose of this Letter is to explore GWP dynamics for Hamiltonians with nonadiabatic time dependence, find conditions for the occurrence of cyclic GWP, and demonstrate an explicit connection between the nonadiabatic geometrical phase effects in quantum and classical mechanics. An ideal system for the study of timedependent GWP dynamics is the generalized harmonic oscillator, the adiabatic Hannay angle of which was studied in [5]. Since the time-independent counterpart of this system admits exact GWP solutions, it is natural to consider a time-dependent generalization. The relevance of this system is evident from its well-known special cases including Mathieu's and Hill's equations, which govern motions in the neighborhood of classical periodic orbits. The generalized harmonic oscillator Hamiltonian is H͑x, p x , t͒where ͑x, p x ͒ are the quantum operators corresponding to ͑q, p͒, the conjugate variables of the phase space, and the real parameters ͕a͑t͒, b͑t͒, c͑t͖͒ are periodic functions of time with a common period t. The exact GWP solution is extended to the time-dependent Hamiltonian (1), and a simple cyclic GWP is found. The original approach of Aharonov and Anandan [6] is generalized in order to facilitate our calculation of the geometrical phase. It is found that the geometrical phase of the cyclic GWP is equal to one-half of the nonadiabatic Hannay angle. It is also shown that the classical Hannay angle is independent of the action, and may be reduced to the area of a pair of ears, thus the ensemble averaging over action contour involved in the definition of Berry and Hannay [7] may be removed. These results indicate a unified formulation of the classical geometrical angle and quantum geometrical phase, which will be presented elsewhere [8].

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“…It can be seen that Hannay's angles (13) are defined by the similar one-form connection with the Berry phase in Eq. (7).…”

Section: Hannay's Angle In Bo Composite Systemmentioning

confidence: 97%

“…For the same reason, the contribution of the fast quantum subsystem to φ k is written as the mean value of the one in Eq. (13). For A hyb 2 (m, n; X ), different with the full quantum phase one-form or classical angle one-form, the part…”

Section: Berry Phase and Hannay Angle In Bo Hybrid Systemmentioning

confidence: 98%