Phase Rigidity Research Articles

Phase rigidity breaking in open Aharonov-Bohm ring coupled to a cantilever

The conductance and the transmittance phase shifts of a two-terminal Aharonov-Bohm ring are analyzed in the presence of mechanical displacements due to coupling to an external cantilever. We show that phase rigidity is broken, even in the linear response regime, by means of inelastic scattering due to phonons. Our device provides a way of observing continuous variation of the transmission phase through a two-terminal nanoelectromechanical system.

Projective Hilbert space structures at exceptional points

A non-Hermitian complex symmetric 2 × 2-matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase-jump behaviour are analysed, and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of -symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.

Wave functions, nodal domains, flow, and vortices in open microwave systems

The signatures of wave functions in open cylindrical microwave billiards are investigated. The wave functions are obtained by means of a transmission measurement from an attached lead to a probe antenna inside the billiard. One can deduce complex wave functions and current densities of the system from the measurement. We investigate distributions and correlations of wave functions, currents and other quantities and compare this results with predictions from the random plane wave approach. Vortices and saddles are created and annihilated as a function of frequency and show a rich dynamic. Additionally we investigate nodal domains of the real and imaginary part of the wave function, their relation to the phase rigidity, and its parametric dependance on global phase shifts.

Nodal domains in open microwave systems

Nodal domains are studied both for real psiR and imaginary part psiI of the wave functions of an open microwave cavity and found to show the same behavior as wave functions in closed billiards. In addition we investigate the variation of the number of nodal domains and the signed area correlation by changing the global phase phig according to psiR+ipsiI=eiphig(psiR'+ipsiI'). This variation can be qualitatively, and the correlation quantitatively explained in terms of the phase rigidity characterizing the openness of the billiard.

Tunable Fano‐Kondo effect in a quantum dot with an Aharonov‐Bohmring

We report experimental results and a theoretical model of the Fano-Kondo effect in a quantum dot embedded in an Aharonov-Bohm (AB) ring. A clear Fano-Kondo lineshape is observed in the coherent conductance as a function of the dot gate voltage. And the phase shift locking to π /2, which is a touchstone of the SU(2) Kondo theory is clearly observed in the middle of Kondo valleys. The variation of the lineshape versus the magnetic flux piercing the ring is similar to that expected for a simple double-slit system rather than that under “phase rigidity” for two-terminal quantum interference devices. We have extended the theoretical model of single-channel AB ring to that of a multi-channel case and succeeded to explain the above puzzling results. This demonstrates that useful information can be extracted even from transport through two terminal devices. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase rigidity and avoided level crossings in the complex energy plane

We consider the effective Hamiltonian of an open quantum system, its biorthogonal eigenfunctions phi(lambda), and define the value r(lambda)=(phi(lambda)|phi(lambda))/<phi(lambda)|phi(lambda)> that characterizes the phase rigidity of the eigenfunctions phi(lambda). In the scenario with avoided level crossings, r(lambda) varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of r(lambda) is an internal property of an open quantum system. In the literature, the phase rigidity rho of the scattering wave function Psi(C)(E) is considered. Since Psi(C)(E) can be represented in the interior of the system by the phi(lambda), the phase rigidity rho of the Psi(C)(E) is related to the r(lambda) and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity rho to values smaller than 1 should be considered, at least partly, as an internal property of an open quantum system in the overlapping regime. The relation to measurable values such as the transmission through a quantum dot, follows from the fact that the transmission is, in any case, resonant at energies that are determined by the real part of the eigenvalues of the effective Hamiltonian. We illustrate the relation between phase rigidity rho and transmission numerically for small open cavities.

Charge detection in a closed-loop Aharonov-Bohm interferometer

We report on a study of complementarity in a two-terminal closed-loop Aharonov-Bohm interferometer. In this interferometer, the simple picture of two-path interference cannot be applied. We introduce a nearby quantum point contact to detect the electron in a quantum dot inserted in the interferometer. We found that charge detection reduces but does not completely suppress the interference even in the limit of perfect detection. We attribute this phenomenon to the unique nature of the closed-loop interferometer. That is, the closed-loop interferometer cannot be simply regarded as a two-path interferometer because of multiple reflections of electrons. As a result, there exist indistinguishable paths of the electron in the interferometer and the interference survives even in the limit of perfect charge detection. This implies that charge detection is not equivalent to path detection in a closed-loop interferometer. We also discuss the phase rigidity of the transmission probability for a two-terminal conductor in the presence of a detector.

Spectroscopic properties of large open quantum-chaotic cavities with and without separated time scales

The spectroscopic properties of an open large Bunimovich cavity are studied numerically in the framework of the effective Hamiltonian formalism. The cavity is opened by attaching two leads to it in four different ways. In some cases, the transmission takes place via standing waves with an intensity that closely follows the profile of the resonances. In other cases, short-lived and long-lived resonance states coexist. The short-lived states cause traveling waves in the transmission while the long-lived ones generate superposed fluctuations. The traveling waves oscillate as a function of energy. They are not localized in the interior of the large chaotic cavity. In all considered cases, the phase rigidity fluctuates with energy. It is mostly near to its maximum value and agrees well with the theoretical value for the two-channel case.

Breakdown of phase rigidity and variations of the Fano effect in closed Aharonov-Bohm interferometers

Although the conductance of a closed Aharonov-Bohm interferometer, with a quantum dot on one branch, obeys the Onsager symmetry under magnetic field reversal, it needs not be a periodic function of this field: the conductance maxima move with both the field and the gate voltage on the dot, in an apparent breakdown of `phase rigidity'. These experimental findings are explained theoretically as resulting from multiple electronic paths around the interferometer ring. Data containing several Coulomb blockade peaks, whose shapes change with the magnetic flux, are fitted to a simple model, in which each resonant level on the dot couples to a different path around the ring.

Magnetic Field Symmetry and Phase Rigidity of the Nonlinear Conductance in a Ring

We have performed nonlinear transport measurements as a function of a perpendicular magnetic field in a semiconductor Aharonov-Bohm ring connected to two leads. While the voltage-symmetric part of the conductance is symmetric in the magnetic field, the voltage-antisymmetric part of the conductance is not symmetric. These symmetry relations are compatible with the scattering theory for nonlinear mesoscopic transport. The observed asymmetry can be tuned continuously by changing the gate voltages near the arms of the ring, showing that the phase of the nonlinear conductance in a two-terminal interferometer is not rigid, in contrast with the case for the linear conductance.

Measurement of Long-Range Wave-Function Correlations in an Open Microwave Billiard

We investigate the statistical properties of wave functions in an open chaotic cavity. When the number of channels in the openings of the billiard is increased by varying the frequency, wave functions cross over from real to complex. The distribution of the phase rigidity, which characterizes the degree to which a wave function is complex, and long-range correlations of intensity and current density are studied as a function of the number of channels in the openings. All measured quantities are in perfect agreement with theoretical predictions.

Efficient Divergent Synthesis of Dendronized Polymers with Extremely High Molecular Weight: Structural Characterization by SEC-MALLS and SFM and Novel Organic Gelation Behavior

High molecular weight poly(p-hydroxystyrene) (PHS) was dendronized in high yield by the divergent grafting of aliphatic polyester dendrons using acetal and ketal protected anhydride derivatives of bis(hydroxymethyl)propionic acid. Relatively low polydispersities (PDI = 1.1-1.3) were maintained throughout the dendronization process. Detailed characterization of a series of these polymers by size exclusion chromatography (SEC) and SEC with multi-angle laser light scattering (SEC-MALLS) clearly shows the effect of dendron steric bulk on the persistence length (ξ) of the polymer main chain in solution. As a result of the fixed degree of polymerization (DP), the main chain rigidification that accompanies lateral dendron growth causes an unusual relationship between R g,z and molecular weight across a series of generations. The rodlike structure and contour length of the alkylated dendronized polymers was also characterized by scanning force microscopy (SFM) on mica, highly oriented pyrolytic graphite (HOPG), and molybdenum disulfide (MoS 2 ) surfaces. One of the G3 dendronized polymers with an alkylated periphery showed significant physical gelation behavior at concentrations below 1 wt % in several organic solvents, which constitutes additional evidence that the solution phase rigidity of a polymer chain can be enhanced by dendronization.

Wave function statistics in open chaotic billiards.

We study the statistical properties of wave functions in a chaotic billiard that is opened up to the outside world. Upon increasing the openings, the billiard wave functions cross over from real to complex. Each wave function is characterized by a phase rigidity, which is itself a fluctuating quantity. We calculate the probability distribution of the phase rigidity and discuss how phase rigidity fluctuations cause long-range correlations of intensity and current density. We also find that phase rigidities for wave functions with different incoming wave boundary conditions are statistically correlated.

Sublimated decoupling of the vortex lattice in extreme type-II layered superconductors

The question of whether layer decoupling and vortex-lattice melting occur separately or not in the mixed phase of pristine layered superconductors in the extreme type-II limit is studied through a partial duality analysis of the layered XY model with uniform frustration. We find that both transitions occur simultaneously if the normal/superconducting transition of the vortex lattice in an isolated layer is first order and if a sufficient degree of layer anisotropy exists. We also find that a crossover to a highly entangled vortex lattice phase with relatively low phase rigidity across layers does not occur in practice under any circumstances at temperatures below the two-dimensional vortex-lattice melting transition.

High field phase diagram of cuprates derived from the Nernst effect.

Measurements of the Nernst signal in the vortex-liquid state of the cuprates to high fields (33 T) reveal that vorticity extends to very high fields even close to the zero-field critical temperature T(c0). In overdoped La2-xSrxCuO4, we show that the upper critical field H(c2)(T) curve does not end at T(c0), but at a much higher temperature. These results imply that T(c0) corresponds to a loss in phase rigidity rather than a vanishing of the pairing amplitude. An intermediate field H*(T)<<H(c2)(T) is shown to be the field scale for the flux-flow resistivity.

Degradation of phase coherence by defects in a two-dimensional vortex lattice.

The thermodynamic nature of two-dimensional vortex matter is studied through a duality analysis of the XY model over the square lattice with low uniform frustration. A phase-coherent vortex lattice state is found at low temperature if rigid translations are prohibited. It shows a nonzero phase rigidity that is degraded exclusively by the creation of dislocation pairs. The unbinding of such pairs causes the vortex lattice to simultaneously lose phase coherence and to melt at a continuous (Kosterlitz-Thouless) phase transition. General phase autocorrelation functions are also computed, and these results are used to argue for the existence of a continuous melting transition of vortex matter in layered superconductors.

Dynamics of conversion of supercurrents into normal currents and vice versa

The generation and destruction of the supercurrent in a superconductor (S) between two resistive normal (N) current leads connected to a current source is computed from the source equation for the supercurrent density. This equation relates the gradient of the pair potential's phase to electron and hole wavepackets that create and destroy Cooper pairs in the N/S interfaces. Total Andreev reflection and supercurrent transmission of electrons and holes are coupled together by the phase rigidity of the non-bosonic Cooper-pair condensate. The calculations are illustrated by snapshots from a computer film.

Dynamical and wave chaos in the Bose–Einstein condensate

Within the past five years Albert Einstein's concept of a dilute atomic Bose Condensate has been realized in many experimental laboratories. Temperatures in the nano-Kelvin regime have been achieved using magnetic and optical trapping of laser and evaporatively cooled atoms. At such temperatures the relative de Broglie wavelengths of the gaseous trapped atoms can become long compared to their mean spacing, and through a process of bosonic amplification a “quantum phase transition” takes place involving 103 to 1011 atoms, most of which end up in identical single particle quantum states, whose length scales are determined by the external trap. Rb, Na, Li and atomic H have been trapped at variable densities of the order of 1013/cm3, and in traps of varying geometries. Of these all but Li have an effective repulsive atomic pair interaction, but utilization of molecular Feshbach resonances allows the interactions of other species to be tuned over wide ranges of strengths, including control of the sign of the effective atomic pair interactions. Fully quantum and macroscopic systems at such low densities are a theorist's dream: simple Hartree type mean field theory provides a startlingly accurate description of density profiles, low energy excitation frequencies, and such a description, commonly called the Non-linear Schrödinger Equation (NLSE) or the Gross-Pitaevskii (GP) Equation, will be explored here. The NLSE appropriate for attractive atomic interactions is known to lead to chaotically unstable dynamics, and eventual implosion of the condensate should the local number density exceed a critical value. In this work we illustrate this type of chaotic collapse for attractive condensates, and then explore the types of chaotic dynamics of solitons and vortices, which are the signatures of dynamical non-linearity in the repulsive case. Finally the implications of the symmetry-breaking associated with phase rigidity are explored in model simulations of repulsive condensates: condensates with repulsive atomic interactions break into phase domains when subject to weak shocks and, perhaps surprisingly, break into chaotic “laser-speckle” type patterns as the shock level increases. The fully quantum mechanical NLSE thus displays a full range of chaotic types of motion: from particle-like chaotic collisions of solitons and vortices to fully developed time dependent wave chaos.

Breaking of phase rigidity by a time-varying field for a two-terminal modified Aharonov-Bohm ring

Breaking of phase rigidity by a time-varying field for a two-terminal modified Aharonov-Bohm ring

Effect of initial take-up speed on properties and structure of as-spun and drawn/heat-set poly(ethylene terephthalate) filaments

The effect of initial take-up speed on the properties and structure of both as-spun and drawn/heat-set poly(ethylene terephthalate) filaments was characterized through measurements of birefringence, percent crystallinity, tensile properties, high temperature shrinkage, loss tangent temperature dependence, DSC melting behavior, and wide-angle (WAXS) and small-angle X-ray scattering (SAXS). While a steady trend toward improved as-spun filament orientation and tensile properties occurred with increasing initial take-up speed, the reduced drawability of these more structured precursor filaments resulted in corresponding drawn/heat-set filaments that were of relatively lower overall orientation and tensile strength. The observed trends in tenacity, initial modulus, and high temperature shrinkage of the drawn/heat-set filaments appeared to be well correlated with the extent and distribution of amorphous phase rigidity as perceived through inferences made from the loss tangent temperature dependence. The WAXS patterns of the drawn/heat-set samples indicated that these filaments all possess a well-developed and highly oriented crystalline structure. Application of a simple two phase model allowed the determination of an amorphous orientation factor, which for the drawn/heat-set filaments was generally found to decrease as the draw ratio imposed in order to achieve comparable levels of elongation to break decreased. The SAXS patterns of the drawn/heat-set filaments indicated that comparable long period spacings exist in all cases and that a transition from a four-point pattern to a two-point bar-shaped pattern occurred when the precursor filament possessed some significant amount of as-spun crystallinity. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 69: 2115–2131, 1998