Adiabatic Geometric Phase Research Articles

Adiabatic geometric phases in hydrogenlike atoms

We examine the effect of spin-orbit coupling on geometric phases in hydrogenlike atoms exposed to a slowly varying magnetic field. The marginal geometric phases associated with the orbital angular momentum and the intrinsic spin fulfill a sum rule that explicitly relates them to the corresponding geometric phase of the whole system. The marginal geometric phases in the Zeeman and Paschen-Back limit are analyzed. We point out the existence of nodal points in the marginal phases that may be detected by topological means.

Effect of intersubsystem coupling on the geometric phase in a bipartite system

The influence of intersubsystem coupling on the cyclic adiabatic geometric phase in bipartite systems is investigated. We examine the geometric phase effects for two uniaxially coupled spin$-{1/2}$ particles, both driven by a slowly rotating magnetic field. It is demonstrated that the relation between the geometric phase and the solid angle enclosed by the magnetic field is broken by the spin-spin coupling, in particular leading to a quenching effect on the geometric phase in the strong coupling limit.

Geometric phase gate without dynamical phases

A general scheme for an adiabatic geometric phase gate is proposed which is maximally robust against parameter fluctuations. While in systems with SU(2) symmetry geometric phases are usually accompanied by dynamical phases and are thus not robust, we show that in the more general case of a SU(2) x SU(2) symmetry it is possible to obtain a non-vanishing geometric phase without dynamical contributions. The scheme is illustrated for a phase gate using two systems with dipole-dipole interactions in external laser fields which form an effective four-level system.

Decoherence of geometric phase gates

We consider the effects of certain forms of decoherence applied to both adiabatic and non-adiabatic geometric phase quantum gates. For a single qubit we illustrate path-dependent sensitivity to anisotropic noise and for two qubits we quantify the loss of entanglement as a function of decoherence.

NMR C-NOT gate through the Aharanov-Anandan phase shift

Recently, it has been proposed to perform quantum computation through the Berry phase (adiabatic cyclic geometric phase) shift with NMR. This geometric quantum gate will hopefully be fault tolerant to certain types of error because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through the Aharonov-Anandan phase (non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. Moreover, in the scheme we have chosen the zero dynamic phase time evolution loop in obtaining the geometric phase shift; we need not perform any extra operation to cancel the dynamic phase.

Nonadiabatic conditional geometric phase shift with NMR.

A conditional geometric phase shift gate, which is fault tolerant to certain types of errors due to its geometric nature, was realized recently via nuclear magnetic resonance (NMR) under adiabatic conditions. However, in quantum computation, everything must be completed within the decoherence time. The adiabatic condition makes any fast conditional Berry phase (cyclic adiabatic geometric phase) shift gate impossible. Here we show that by using a newly designed sequence of simple operations with an additional vertical magnetic field, the conditional geometric phase shift gate can be run nonadiabatically. Therefore geometric quantum computation can be done at the same rate as usual quantum computation.

Geometric quantum computation with Josephson qubits

The quest for large scale integrability and flexibility has very recently stimulated an increasing interest in designing quantum computing devices. In this contribution we discuss a proposal based on small-capacitance Josephson junctions in the charge regime in which quantum gates are implemented by means of adiabatic geometric phases.

A new class of adiabatic cyclic states and geometric phases for non-Hermitian Hamiltonians

For a T-periodic non-Hermitian Hamiltonian H( t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field.

Perturbative calculation of the adiabatic geometric phase and particle in a well with moving walls

We use the Rayleigh-Schrodinger perturbation theory to calculate the corrections to the adiabatic geometric phase due to a perturbation of the Hamiltonian. We show that these corrections are at least of second order in the perturbation parameter. As an application of our general results we address the problem of the adiabatic geometric phase for a one-dimensional particle which is confined to an infinite square well with moving walls.

Noncyclic geometric phase and its non-Abelian generalization

We use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the Abelian and non-Abelian geometric phases. This derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the Abelian noncyclic geometric phase. We also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non-Abelian noncyclic geometric phase for a spin-1 magnetic (or electric) quadrupole interacting with a precessing magnetic (electric) field.

Relativistic adiabatic approximation and geometric phase

A relativistic analogue of the quantum adiabatic approximation is developed for Klein-Gordon fields minimally coupled to electromagnetism, gravity and an arbitrary scalar potential. The corresponding adiabatic dynamical and geometrical phases are calculated. The method introduced in this paper avoids the use of an inner product on the space of solutions of the Klein-Gordon equation. Its practical advantages are demonstrated in the analysis of the relativistic Landau level problem and the rotating cosmic string.

Extending the quantal adiabatic theorem: Geometry of noncyclic motion

We show that a noncyclic phase of geometric origin has to be included in the approximate adiabatic wave function. The adiabatic noncyclic geometric phase for systems exhibiting a conical intersection as well as for an Aharonov–Bohm situation is worked out in detail. A spin-12 experiment to measure the adiabatic noncyclic geometric phase is discussed. We also analyze some misconceptions in the literature and textbooks concerning noncyclic geometric phases.

Adiabatic Geometric Phases and Response Functions

Treating a many-body Fermi system in terms of a single particle in a deforming mean field, we relate adiabatic geometric phase to susceptibility for the noncyclic case, and to its derivative for the cyclic case. Employing the semiclassical expression of the susceptibility, the expression for geometric phase for chaotic quantum system immediately follows. Exploiting the well-known association of the absorptive part of susceptibility with dissipation, our relations may provide a quantum mechanical origin of the damping of collective excitations in Fermi systems.

BERRY'S PHASE IN PILOT-WAVE THEORY

The projective-geometric derivation of Berry's adiabatic geometric phase for a single trajectory in the de Broglie-Bohm pilot-wave theory is given. The relation between this phase and the first ord ...

Integrability of the quantum adiabatic evolution and geometric phases

We show that the cyclic adiabatic evolution of a quantum system is completely integrable as a classical Hamiltonian system. In this context the Berry phases arise naturally as cohomology of the invariant tori.

CONDITIONS FOR THE EXISTENCE OF NON-ZERO GEOMETRIC PHASE AND ITS FORMULA

First we put forward an essential condition, for the existence of the non-zero adiabatic geometric phase of a quantum-mechanical Hamiltonian system, that the Hamiltonian operators at different times do not commute. Then it is shown that constraint =0 determines completely the phase relation of normalized eigenstate vectors │n' (t)> at different times. According to this property, we advance the sufficient and necessary condition for the existence of the non-zero adiabatic geometric phase, which is |n′(T)>≠|n′(0)> when =0. And also we derive a general time-integral formula for the adiabatic geometric phase along the line. Finally, as an application of it, we calculate the geometric phase of a spin 1/2 system once discussed by Solem and Biedenharn. It is pointed out that the problem encountered by them lies in the multi-valuedness of the eigenstate vector in parameter space.

Deviation from Berry's adiabatic geometric phase in a 131Xe nuclear gyroscope.

The concept of geometric phase is demonstrated in a nuclear gyroscope using $^{131}\mathrm{Xe}$ nuclear spins (I=3/2) as sensors for quantum-phase accumulation. By spatial rotation sub-Hertz splittings due to geometric phases are resolved in nuclear-quadrupole spectra. Deviations from Berry's adiabatic geometric phase appear in the regime of nonadiabatic rotation. The observed frequency splittings are no longer linear in the rotational frequency, as expected from adiabatic rotations, and all possible transitions, namely, six in this partially degenerate spin-3/3 system, are observed experimentally.