Nonadiabatic Geometric Phase Research Articles

Nonadiabatic noncyclic geometric phase and ensemble average spectrum of conductance in disordered mesoscopic rings with spin-orbit coupling

We generalize Yang's theory from the U(1) gauge field to the non-Abelian U(1)xSU(2)(spin) gauge field. Based on this generalization and taking into account the geometric Pancharatnam phase as well as an effective Aharonov-Bohm (AB) phase in nonadiabatic noncyclic transport, we calculate the ensemble average Fourier spectrum of the conductance in disordered mesoscopic rings connected to two leads. Our theory can explain the experimental results reported by Morpurgo et al. [Phys. Rev. Lett. 80, 1050 (1998)] more satisfactorily. We indicate that the observed splitting stems from the nonadiabatic noncyclic Pancharatnam phase and the effective AB phase, both being dependent on spin-orbit coupling.

Nonadiabatic noncyclic geometric phase of a spin-12particle subject to an arbitrary magnetic field

We derive a formula of the nonadiabatic noncyclic Pancharatnam phase for a quantum spin-$\frac{1}{2}$ particle subject to an arbitrary magnetic field. The formula is applied to three specific kinds of magneic fields. (i) For an orientated magnetic field, the Pancharatnam phase is derived exactly. (ii) For a rotating magnetic field, the evolution equation is solved analytically. The Aharonov-Anandan phase is obtained exactly and the Pancharatnam phase is computed numerically. (iii) We propose a kind of topological transition in one-dimensional mesoscopic ring subject to an in-plane magnetic field, and then address the nonadiabatic noncyclic effect on this phenomenon.

Exact solutions of the nonstationary Schrödinger equation

On the basis of exactly solvable stationary models for the Schrodinger equation, we develop a procedure for solving the nonstationary Schodinger equation in an explicit analytic form. We investigate the formation of the nonadiabatic geometric phase during cyclic evolution of a quantum system.

Nonadiabatic noncyclic geometric phase and persistent current in one-dimensional rings

The total geometric phase is composed of the nonadiabatic noncyclic Pancharatnam phase, the usual Aharonov-Bohm (AB) phase, and the effective AB phase. It is found that the persistent current in one-dimensional rings is determined from this phase. As applications, we address first the geometric phase and the persistent current in a ring subject to a cylindrically symmetric electromagnetic field. We show that the Pancharatnam phase recovers the Aharanov-Anandan phase in the case of cyclic evolution, as well as the Berry phase in the adiabatic evolution. Moreover, we discuss the persistent current induced by the spin-orbit-induced geometric phase in the presence of a local magnetic field. Generalization to many-body cases is also addressed.

Geometric phase in a mesoscopic Josephson junction with classical driving source

We examine a mesoscopic Josephson junction driven by the time periodic field and solve its time-dependent Schr\"odinger equation with the help of the time dependent invariant of the Hamiltonian operator. We show that the state of the system can evolve a generalized squeezed state if the junction is prepared initially in its $n\mathrm{th}$ excited state. It also turns out to be possible to produce coherent and ordinary squeezed states for the mesoscopic junction system. The explicit expression of the nonadiabatic geometric phase of the system is also obtained.

Non-Abelian geometric phase, Floquet theory and periodic dynamical invariants

For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field.

Nonadiabatic geometric phase for the cyclic evolution of a time-dependent Hamiltonian system

The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states are expressed as a superposition of an infinte number of squeezed states. Their geometric phases are found to be independent of $\hbar$, and equal to $-(n+1/2)$ times the classical nonadiabatic Hannay angle.

Nonadiabatic geometric phase in periodic linear systems

Nonadiabatic geometric phase in periodic linear systems

Use of four mirrors to rotate linear polarization but preserve input-output collinearity. II.

We report on the design, construction, and testing of a four-mirror reflective polarization rotator, proposed by Smith and Koch [J. Opt. Soc. Am. A 13, 2102 (1996)], that rotates by an angle phi the input linear polarization while preserving the input-output beam collinearity. We correct errors in the previous work that led to an incorrect design for a phi = pi/2 rotator. This type of pure rotator is simple and inexpensive, and it is a direct application of the concept of the nonadiabatic geometric phase to polarization rotation. We also present measurements of the polarization rotation for the case of three metallic mirrors with antiparallel input and output beams, a test of geometric phase in polarization optics not done before.

Duality and scaling in quantum mechanics

The nonadiabatic geometric phase in a time dependent quantum évolution is shown to provide an intrinsic concept of time having dual properties relative to the external time. A nontrivial extension of the ordinary quantum mechanics is thus obtained with interesting scaling laws. A fractal-like structure in time is thus revealed.

Nonadiabatic Geometrical Phase during Cyclic Evolution of a Gaussian Wave Packet

The Gaussian wave packet solution to the Schr{umlt o}dinger 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 of {h_bar}, 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. {copyright} {ital 1997} {ital The American Physical Society}

Nonadiabatic geometric phase in a textured mesoscopic ring subject to a crown-like magnetic field

With the aid of the exact solution for a quantum spin-1/2 in a rotating magnetic field and by imposing a commensurability condition, we propose a novel experiment to test a kind of Aharonov-Anandan (AA) geometric phase in a textured mesoscopic ring subject to a crown-like magnetic field.

Nonadiabatic geometric phase in quaternionic Hilbert space

We develop the theory of the nonadiabatic geometric phase, in both the Abelian and non-Abelian cases, in quaternionic Hilbert space.

Relation between the nonadiabatic geometric phase and the nonclassical photon statistics for two-mode quantum light traveling through a dispersive fiber

A general relation between the nonadiabatic geometric phase and the nonclassical photon statistics of an initial two-mode field propagating through a nonlinear dispersive fiber has been explicity established. Also are derived the expressions of the nonadiabatic geometric phase when the system is prepared initially in a SU(1,1) coherent state.

Geometric phase, bundle classification, and group representation

The line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel–Weil–Bott theorem of the representation theory. The remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for establishing the relevance of the Borel–Weil–Bott theorem to Berry’s adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. These charges signify the topological content of the phase. They can be explicitly computed. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. It is shown that, in general, the parameter space is either a flag manifold or one of its submanifolds. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry’s connection. The results about the fiber bundles and group theory are used to introduce a procedure to reduce the problem of the nonadiabatic (geometric) phase to Berry’s adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-Abelian monopoles is pointed out.

Geometric phase in NMR interferometry experiment

Experimental detection of geometric phase effects is based on interferometry techniques. The observation of a nonadiabatic geometric phase in nuclear magnetic resonance has been reported by D. Suter, K. T. Muelter, and A. Pines (1988, Phys. Rev. Lett., 60, 1218). A rigorous interpretation of their results is presented here. The experiment exploits the concept of fictitious spin-l/2 subsystems embedded in a spin system of higher order. A relevant expression for the geometric phase is derived here using the Euler rotation representation for the SU(2) group. In order to correlate the experimental data to the geometric phase, a decomposition of the density matrix in a basis of pure states is used and the isomorphism between the cyclic evolution of pure states and mixed states is constructed. Rigorous treatment of the dynamic phase, along with technical aspects of the experiment, shows that the observed echo shift is related directly to the geometric phase, in agreement with Suter et al. The observed geometric phase is classified as a frame-related phase.

Nonadiabatic geometric phases of multiphoton transitions in dissipative systems and spin-Jsystems

Abstract We present new developments in nonadiabatic geometric phases along two lines for systems undergoing changes of quantum state in intense fields. We first present a geometric representation of the non-Hermitian Schrodinger equation and introduce the notion of a complex multiphoton Aharonov-Anandan (AA) phase associated with dissipative two-level systems driven by periodic fields. The concept is further extended to include field modulation effects. We then develop the AA phase for spin-j systems in periodic fields and find conditions for cyclic evolution for general multi-level systems. In both cases, generalizations of the Floquet formalism lead to general analytical expressions for geometric phases that can be tested by experiments.

Geometric phase with photon statistics and squeezed light for the dispersive fiber.

The nonadiabatic geometric phase for an arbitrary cyclic evolution of the state vector for a system described by the quantum theory of the propagation of a single-mode off-resonant field through the dispersive fiber is discussed here. The sensitivity of the geometric phase on the initial field statistics has been explicitly shown in the calculations. We also derive the expressions of the geometric phase when the system is prepared initially in a squeezed coherent state.

Spin-orbit interaction and Aharonov-Anandan phase in mesoscopic rings.

We show the existence of a nonadiabatic geometric phase, i.e., an Aharonov-Anandan (AA) phase, in the Aharonov-Casher (AC) topological interference effect in one-dimensional mesoscopic rings. We find the AC phase is the phase accumulated by the spin wave function during a cyclic evolution, and show it is the sum of a geometric AA phase and a dynamical phase. In the adiabatic limit, the AA phase becomes the spin-orbit Berry phase introduced by Aronov and Lyanda-Geller. By solving exactly the model of a quasi-one-dimensional ring formed by the 2DEG on a semiconductor heterostructure, we discuss the observability of the AA phase in the AC effect.

Interpretation of geometric phase via geometric distance and length during cyclic evolution.

We cast the nonadiabatic geometric phase in terms of the geometric distance function and the geometric length of the curve for arbitrary cyclic evolution of the quantum states. An interpretation is given to the geometric phase as the value of the integral of the contracted length of the curve along which the system traverses. It is found that for arbitrary cyclic evolution of the quantum states the geometric phase β(C) acquired by the system cannot be greater than the total length of the curve l(C). We have argued that the geometric phase arises because of the fundamental inequality between the length of the curve and the distance function