Mechanisms and Möbius strips: understanding dynamic processes in annulenes

Exploratory computational studies on annulenes with planar, Möbius, and two‐twist topologies have resulted in new mechanisms to explain facile thermal configuration change (cis‐trans isomerization) for medium‐sized annulenes ([12]‐ to [16]annulene). Möbius π‐bond shifting through both aromatic and antiaromatic transition states, two‐twist π‐bond shifting, and planar nondegenerate π‐bond shifting can all be invoked to explain experimental results. Moreover, a simple bond‐shift rule, which is based on the change in number of trans C?C double bonds (Δtrans), was developed that predicts the topology of the transition state(s) necessary to effect the desired cis‐trans isomerization. The bond‐shift rule was also applied to configuration change in dehydro[12]annulene. Finally, extensive investigation of the [14]annulene hypersurface revealed that numerous Möbius minima exist within 10 kcal/mol of the global minimum. Copyright © 2012 John Wiley & Sons, Ltd.     

Magnetic and charge orders in zigzag nanographene ribbons

We theoretically study the electronic states in graphene ribbons which are strongly affected by the edge states, the peculiar non-bonding molecular orbitals localized along the zigzag edges of the ribbons. New kinds of edge localized electronic states with spin and charge polarizations are found in the mean field solutions of the extended Hubbard model with onsite and nearest-neighbor Coulomb repulsions. These novel states appear due to the interplay between the edge states and the Fermi instabilities. We also examine the competition between the charge polarized state and the spin polarized state to draw a phase diagram depending on Coulomb parameters. The results obtained by the mean field calculations with the extended Hubbard model modified to include Coulomb integrals provide useful insights to understand and functionalize the nanoscale materials.     

Geometric phases and coherent states

A cyclic evolution of a pure quantum state is characterized by a closed curve γ in the projective Hilbert space , equipped with the Fubini-Study geometry. It is known that the geometric phase for this evolution is given by the integral of the symplectic form of the Fubini-Study geometry over an arbitrary surface spanning γ. This result extends to an infinite-dimensional Hilbert space for a bosonic quantum field. We prove that is bounded above by the infimum area over all surfaces spanning γ, and that the bound is attained if γ can be spanned by a holomorphic curve. Using an earlier result concerning the intrinsic Euclidean geometry of the coherent state submanifold , we derive an expression for the geometric phase for a cyclic evolution amongst coherent states. We indicate how the intensity of a classical configuration can be inferred from the winding number of the exponential geometric phase about the origin in the complex plane. In the case of photon states we present group theoretic and 2-component spinor representations of . We derive an expression for in the case of a sequence of measurements such that the resulting states are coherent at each step, in terms of a sequence of projection operators. The situation in relation to some earlier experiments of Pancharatnam and Tomita–Chiao is explained.     

Geometric curvature and phase of the Rabi model

We study the geometric curvature and phase of the Rabi model. Under the rotating-wave approximation (RWA), we apply the gauge independent Berry curvature over a surface integral to calculate the Berry phase of the eigenstates for both single and two-qubit systems, which is found to be identical with the system of spin-1/2 particle in a magnetic field. We extend the idea to define a vacuum-induced geometric curvature when the system starts from an initial state with pure vacuum bosonic field. The induced geometric phase is related to the average photon number in a period which is possible to measure in the qubit–cavity system. We also calculate the geometric phase beyond the RWA and find an anomalous sudden change, which implies the breakdown of the adiabatic theorem and the Berry phases in an adiabatic cyclic evolution are ill-defined near the anti-crossing point in the spectrum.     

Geometric Phase in a Generalized Time-Dependent Jaynes-Cummings Model

By using the Lewis–Riesenfeld invariant theory, we have studied the dynamical and the geometric phases in a generalized time-dependent Jaynes-Cummings model. It is found that the geometric phases in a cycle case have nothing to do with the frequency of the electromagnetic wave, the energy difference between two levels of the atom, and the coupling strength between the atom and the light field.     

Geometric phasea la Pancharatnam

In mid-1950s, Pancharatnam [1] encountered the geometric phase associated with the evolution along a geodesic triangle on the Poincaré sphere. We generalize his 3-vertex phase and employ it as the fundamental building block, to geometrically construct a general ray-space expression for geometric phase. In terms of a reference ray used to specify geometric phase, we delineate clear geometric meanings for gauge transformations and gauge freedom, which are generally regarded as mere mathematical abstractions.     

Geometric phase-shifting for low-coherence interference microscopy

A low-coherence Linnik interference microscope using high numerical aperture optics has been constructed. The system uses a tungsten halogen lamp and Köhler illumination, with separate control over field and aperture stops, so that experiments can be conducted with a range of different operating conditions. The novel feature of the system is the use of an achromatic phase-shifter operating on the principle of the geometric phase, achieved by using a polarising beam splitter, a quarter wave plate and a rotating polariser. Image information is extracted from the visibility of the fringes, the position of the visibility peak along the scanning axis yielding the height of the test surface at the corresponding point.     

Geometric Phase in Fluctuating Magnetic Field

Geometric phase in a two-level atom with a fluctuating magnetic field is calculated by a nonunit vector ray in a complex projective Hilbert space, where the nonunit vector is a map connecting with density matrices of a quantum open system. We find that the Pancharatnam phase oscillates with evolving time. The Berry phase depends on the fluctuating parameter but it is proportional to the area spanned in the Bloch parameter space.     

Topology induced anomalous plasmon modes in metallic Möbius nanorings

We report on the theoretical investigation of plasmonic resonances in metallic Möbius nanorings. Half‐integer numbers of resonant modes are observed due to the presence of an extra phase π provided by the topology of the Möbius nanostrip. Anomalous plasmon modes located at the non‐orientable surface of the Möbius nanoring break the symmetry that exist in conventional ring cavities, thus enable far‐field excitation and emission as bright modes. The far‐field resonant wavelength as well as the feature of half‐integer mode numbers is constant to the change of charge distribution on the Möbius nanoring due to the topology of Möbius ring. Owing to the ultra‐small mode volume induced by the remaining dark feature, an extremely high sensitivity as well as a remarkable figure of merit is obtained in our numerical calculations for sensing performance. The topological metallic nanostructure provides a novel platform for the investigation of localized surface plasmon modes exhibiting unique phenomena for potential plasmonic applications.

     

Dynamical and Geometric Phases of Exciton Emission in a Semiconductor Microcavity

By using of the invariant theory, we have studied the phase of exciton emission in a semiconductor microcavity, the dynamical and geometric phases are presented respectively. The Aharonov-Anandan phase is also obtained in the case of cyclical evolution.     

Quantum renormalization group approach to geometric phases in spin chains

A relation between geometric phases and criticality of spin chains are studied using the quantum renormalization-group approach. I have shown how the geometric phase evolve as the size of the system becomes large, i.e., the finite size scaling is obtained. The renormalization scheme demonstrates how the first derivative of the geometric phase with respect to the field strength diverges at the critical point and maximum value of the first derivative, and its position, scales with the exponent of the system size.     

Quantum phases for a generalized harmonic oscillator

An effective Hamiltonian for the generalized harmonic oscillator is determined by using squeezed state wavefunctions. The equations of motion over an extended phase space are determined and then solved perturbatively for a specific choice of the oscillator parameters. These results are used to calculate the dynamic and geometric phases for the generalized oscillator with this choice of parameters.      

Geometric phase in Bohmian mechanics

Using the quantum kinematic approach of Mukunda and Simon, we propose a geometric phase in Bohmian mechanics. A reparametrization and gauge invariant geometric phase is derived along an arbitrary path in configuration space. The single valuedness of the wave function implies that the geometric phase along a path must be equal to an integer multiple of 2π. The nonzero geometric phase indicates that we go through the branch cut of the action function from one Riemann sheet to another when we locally travel along the path. For stationary states, quantum vortices exhibiting the quantized circulation integral can be regarded as a manifestation of the geometric phase. The bound-state Aharonov-Bohm effect demonstrates that the geometric phase along a closed path contains not only the circulation integral term but also an additional term associated with the magnetic flux. In addition, it is shown that the geometric phase proposed previously from the ensemble theory is not gauge invariant.     

Geometric phase and Pancharatnam phase induced by light wave polarization

We use the quantum kinematic approach to revisit geometric phases associated with polarizing processes of a monochromatic light wave. We give the expressions of geometric phases for any, unitary or non-unitary, cyclic or non-cyclic transformations of the light wave state. Contrarily to the usually considered case of absorbing polarizers, we found that a light wave passing through a polarizer may acquire in general a nonzero geometric phase. This geometric phase exists despite the fact that initial and final polarization states are in phase according to the Pancharatnam criterion and cannot be measured using interferometric superposition. Consequently, there is a difference between the Pancharatnam phase and the complete geometric phase acquired by a light wave passing through a polarizer. We illustrate our work with the particular example of total reflection based polarizers.     

Geometric Phase in a Two Energy Level Jaynes-Cummings Model with Imaginary Photon Process

By using the Lewis–Riesenfeld invariant theory, we have studied the dynamical and the geometric phases in a two energy level Jaynes-Cummings model with imaginary photon process. We find that the geometric phases in a cycle case have nothing to do with the frequency of the photon field, the coupling coefficient between photons and atoms, and the atom transition frequency. If we use the more accuracy device, the geometric phases in the imaginary photon process may be observed, and the geometric phases in this process have the observable physical effect.     

Geometric Phase in a Generalized Time-Dependent Karassiov-Klimov Model

By using of the invariant theory, we have studied the generalized time-dependent Karassiov-Klimov model, the dynamical and geometric phases are given, respectively. The Aharonov-Anandan phase is also obtained under the cyclical evolution.     

Electron reorganization in allowed and forbidden pericyclic reactions: multicenter bond indices as a measure of aromaticity and/or anti‐aromaticity in transition states of pericyclic electrocyclizations

The multicenter bond indices (MCI), recently proposed as quantitative measures of cyclic delocalization in aromatic systems, have been applied to characterize the differences in the nature of the electron reorganization in a series of allowed and forbidden electrocyclic reactions of linear neutral polyenes of general formula C_nH_n+2 and related charged systems of formula C_nH_n+2(+) and C_nH_n+2(?) for n ranging from 4 to 7. The proposed methodology, which is based on the monitoring of the variation of the extent of cyclic delocalization along the concerted reaction paths, is shown to be completely consistent with the empirical Evans/Dewar classification anticipating aromatic transition states for allowed and anti‐aromatic transition states for forbidden electrocyclic reactions. Although the study reports the results of the analysis of electron reorganization just for the above‐mentioned particular class of electrocyclic reactions, the proposed approach is completely general and its conclusions remain valid for any pericyclic reaction. Copyright © 2009 John Wiley & Sons, Ltd.     

Geometric phase as a simple closed-form integral

For a general evolution of a quantal system, the geometric phase measured with reference to a given initial state is derived as an integral of a function of the pure state density operator by invoking the Pancharatnam connection continuously.