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The Journal of Chemical Physics

Time development of geometric phases in the Longuet-Higgins model

Vı́ctor Romero-Rochı́n, Jeffrey A. Cina

Published Nov 15, 1989• 29 citations
Publisher

Abstract

Using a time-dependent variational method, we study the evolution of nonstationary states in Longuet-Higgin’s model of a Jahn–Teller molecule. Conditions are found for the nuclear motion to be adiabatic. The effects of wave-packet spreading are neglected upon specializing to the case of nearly harmonic motion. It is shown explicitly how the effective vector potential introduced by Mead and Truhlar gives rise to an electronic Berry phase. In a semiclassical approximation sufficient to produce the electronic adiabatic phase anticipated from the result for a given sequence of nuclear configurations, it is demonstrated that the effective vector potential has a negligible effect on the nuclear motion; the effective vector potential, the source of an effective field proportional to ℏ, is seen to affect the nuclear trajectory only in higher order. For the special case of periodic nuclear motion the electronic adiabatic phase is seen as a contribution to the geometric phase attending an arbitrary cyclic evolution. It is demonstrated that a molecular state prepared with identically pseudorotating nuclear wave packets in both electronic levels corresponds, in a weak coupling limit, to a spin 1/2 in a conically varying external field. Geometrical phase differences are shown to make discernible contributions to the frequencies of oscillation of the electronic charge and current densities, which may serve as classical sources for superradiant emission. Our results are shown to be gauge invariant.

DOI
10.1063/1.457429

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Using a time-dependent variational method, we study the evolution of nonstationary states in Longuet-Higgin’s model of a Jahn–Teller molecule. Conditions are found for the nuclear motion to be adiabatic. The effects of wave-packet spreading are neglected upon specializing to the case of nearly harmonic motion. It is shown explicitly how the effective vector potential introduced by Mead and Truhlar gives rise to an electronic Berry phase. In a semiclassical approximation sufficient to produce the electronic adiabatic phase anticipated from the result for a given sequence of nuclear configurations, it is demonstrated that the effective vector potential has a negligible effect on the nuclear motion; the effective vector potential, the source of an effective field proportional to ℏ, is seen to affect the nuclear trajectory only in higher order. For the special case of periodic nuclear motion the electronic adiabatic phase is seen as a contribution to the geometric phase attending an arbitrary cyclic evolution. It is demonstrated that a molecular state prepared with identically pseudorotating nuclear wave packets in both electronic levels corresponds, in a weak coupling limit, to a spin 1/2 in a conically varying external field. Geometrical phase differences are shown to make discernible contributions to the frequencies of oscillation of the electronic charge and current densities, which may serve as classical sources for superradiant emission. Our results are shown to be gauge invariant.
Kinematical and dynamical aspects of the noncyclic geometric phase for coherent states are analyzed. It is shown for Glauber and SU(2) coherent states that the geometric phase obeys a geodesic closure rule associated with the coset space of the respective Lie group. In the former case the geodesic is a straight line in phase space, while in the latter the geodesic is a great circle on the Bloch sphere (S2).({\mathcal{S}}^{2}).(S2). An alternative expression for the geometric phase is derived from the time-dependent variational principle by making a specific choice of gauge. The noncyclic geometric phase is numerically calculated in a time-dependent variational treatment of the $E\ensuremath{\bigotimes}\ensuremath{\varepsilon}$ Jahn-Teller model first introduced by Longuet-Higgins and co-workers. The electronic and nuclear degrees of freedom are parametrized by SU(2) and Glauber coherent states respectively. In particular, the adiabatic limit is studied and shown to yield the anticipated Berry geometric phase.
The Born-Oppenheimer electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$ picks up a topological phase factor $\ifmmode\pm\else\textpm\fi{}1$, a special case of Berry phase, when it is transported around a conical intersection of two adiabatic potential energy surfaces in RRR space. We show that this topological quantity reverts to a geometric quantity ${e}^{i\ensuremath{\gamma}}$ if the geometric phase $\ensuremath{\gamma}=\ensuremath{\oint}\mathrm{Im}\ensuremath{\langle}{\mathrm{\ensuremath{\Phi}}}_{R}|{\mathbf{\ensuremath{\nabla}}}_{\ensuremath{\mu}}{\mathrm{\ensuremath{\Phi}}}_{R}\ensuremath{\rangle}\ifmmode\cdot\else\textperiodcentered\fi{}d{\mathbf{R}}_{\ensuremath{\mu}}$ is evaluated with the conditional electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}(r)$ from the exact electron-nuclear factorization ${\mathrm{\ensuremath{\Phi}}}_{R}(r)\ensuremath{\chi}(R)$ instead of the adiabatic function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$. A model of a pseudorotating triatomic molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides examples of nontrivial induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current that cannot be removed by a gauge transformation. The exact potential energy surface is calculated and found to contain a term depending on the Fubini-Study metric for the conditional electronic wave function.
The fate of the molecular geometric phase in an exact dynamical framework is investigated with the help of the exact factorization of the wave function and a recently proposed quantum hydrodynamical description of its dynamics. An instantaneous, gauge-invariant phase is introduced for arbitrary paths in nuclear configuration space in terms of hydrodynamical variables, and shown to reduce to the adiabatic geometric phase when the state is adiabatic and the path is closed. The evolution of the closed-path phase over time is shown to adhere to a Maxwell-Faraday induction law, with nonconservative forces arising from the electron dynamics that play the role of electromotive forces. We identify the pivotal forces that are able to change the value of the phase, thereby challenging any topological argument. Nonetheless, negligible changes in the phase occur when the local dynamics along the probe loop is approximately adiabatic. That is, the geometric phase effects that arise in an adiabatic limiting situation remain suitable to effectively describe certain dynamic observables.
This thesis concerns the following topics: geometric phase in the context of Galilean invariance and quantum measurements, Rydberg states of hydrogen atoms, vibronic coupling in the E A e Jahn-Teller system and realism in quantum computations. In the analyses the de Broglie-Bohm pilot-wave formulation of quantummechanics is mainly used.It is shown that geometric phase is not Galilean invariant. Experimental implications are discussed and it is found that the experiments performed to date are frame independent. An experiment which is in principle able to detect the noninvariance is sketched. By adopting the measurement theory of the pilot-wave formulation it is shown how the measurement induced geometric phase continuously emerges. The Samuel-Bhandari geometric phase is identified as the nonrandom part of the total geometric phase induced in the measurement.Ensembles of particles for a circular Rydberg wave packet are studied. The trajectories of pilot-wave particles are shown to accurately imitate the behaviour of the wave packet in the high quantum number limit. The nonclassical features of the wave packet are intuitively explained by the nonvanishing quantum potential.Vibronic coupling in the Longuet-Higgins model of the E A e Jahn-Teller system is investigated by means of quantal trajectories. The pilot-wave picture provides an intuitive tool for discussing time-scales. An argument based on ergodicity leads to an understanding of the averaging procedure over the electronic motion which provides the approximate nuclear motion.The existence of efficient quantum algorithms triggers questions on Natures ability of storing and processing information during quantum computations. The role of elements of reality in quantum computations is addressed using quantal trajectories. It is found that there is a many-to-one relationship between quantal trajectories and performed computations when quantum parallelism is utilized.
Jahn-Teller systems and the Jahn-Teller effect are discussed in terms of cavity QED models. By expressing the field modes in a quadrature representation, it is shown that certain setups of a two-level system interacting with a bimodal cavity are described by the Jahn-Teller $E\ifmmode\times\else\texttimes\fi{}\ensuremath{\epsilon}$ Hamiltonian. We identify the corresponding adiabatic potential surfaces and the conical intersection. The effects of a nonzero geometrical Berry phase, governed by encircling the conical intersection, are studied in detail both theoretically and numerically. The numerical analysis is carried out by applying a wave packet propagation method, more commonly used in molecular or chemical physics, and analytic expressions for the characteristic time scales are presented. It is found that the collapse-revival structure is greatly influenced by the geometrical phase and as a consequence, the field intensities contain direct information about this phase. We also mention the link between the Jahn-Teller effect and the Dicke phase transition in cavity QED.
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrödinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.