non-Abelian Geometric Phases Research Articles

Entanglement gauge and the non-Abelian geometric phase with two photonic qubits

We introduce the entanglement gauge describing the combined effects of local operations and nonlocal unitary transformations on bipartite quantum systems. The entanglement gauge exploits the invariance of nonlocal properties for bipartite systems under local (gauge) transformations. This new formalism yields observable effects arising from the gauge geometry of the bipartite system. In particular, we propose a non-Abelian gauge theory realized via two separated spatial modes of the quantized electromagnetic field manipulated by linear optics. In this linear optical realization, a bipartite state of two separated spatial modes can acquire a non-Abelian geometric phase.

Torus quantization for spinning particles.

We derive semiclassical quantization conditions for particles with spin. These generalize the Einstein-Brillouin-Keller quantization in such a way that, in addition to the Maslov correction, there appears another term which is a remnant of a non-Abelian geometric or Berry phase. This correction is interpreted in terms of a rotation angle for a classical spin vector.

Holonomic Quantum Computation with Josephson Networks

It is well known that it is possible to build quantum gates based entirely on the action of geometric phases. This approach is called holonomic quantum computation. Here we show that it is possible to generate and detect non-Abelian geometric phases in the charge dynamics of a network of mesoscopic Josephson junctions. The scheme can be used to realize a universal set of quantum gates.

Geometric phases of mesoscopic spin in Bose-Einstein condensates

We propose a possible scheme for generating spin-J geometric phases using a coupled two-mode Bose-Einstein condensate (BEC). First we show how to observe the standard Berry phase using Raman coupling between two hyperfine states of the BEC. We find that the presence of intrinsic interatomic collisions creates degeneracy in energy that allows implementation of the non-Abelian geometric phases as well. The evolutions produced can be used to produce interference between different atomic species with high numbers of atoms or to fine control the difference in atoms between the two species. Finally, we show that errors in the standard Berry phase due to elastic collisions may be corrected by controlling inelastic collisions between atoms.

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.

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.

Non-Abelian geometric phases and conductance of spin-32holes

Angular momentum $J=3/2$ holes in semiconductor heterostructures are showed to accumulate nonabelian geometric phases as a consequence of their motion. We provide a general framework for analyzing such a system and compute conductance oscillations for a simple ring geometry. We also analyze a figure-8 geometry which captures intrinsically nonabelian interference effects.

Non-Abelian geometric phase for general three-dimensional quantum systems

Adiabatic U(2) geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians and their linear combinations for spin j = 1 systems. In particular it is shown that the nuclear quadrupole Hamiltonian does actually lead to non-Abelian geometric phases for j = 1. This system, being bosonic, is time-reversal invariant. Therefore, it cannot support Abelian adiabatic geometrical phases.

Thomas rotation and polarized light: A non-abelian geometric phase in optics

We describe anon-abelian Berry phase in polarization optics, suggested by an analogy due to Nityananda between boosts in special relativity and the effect of elliptic dichroism on polarized light. The analogy permits a simple optical realization of the non-abelian gauge field describing Thomas rotation. We also show how Thomas rotation can be understood geometrically on the Poincare sphere in terms of the Pancharatnam phase.

Modified symmetry generators and the geometric phase

Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately Sd (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.

Symplectic structure for the non-Abelian geometric phase

We split the non-Abelian geometric factor in such a way that part of it may be derived from the symplectic structure in an appropriate space of states.

Algebraic scattering theory and the geometric phase.

A nonstandard realization of the su(1,1) algebra is used to extract a two-parameter class of scattering potentials as well as to calculate the reflection coefficient of the associated one-dimensional scattering problem in the spirit of the algebraic scattering theory. The nontrivial geometric content of such realizations is discussed, and an interesting connection with geometric phases is pointed out. It is argued that using larger noncompact groups, realizations related to non-Abelian geometric phases may be useful for obtaining analytical expressions for interaction terms corresponding to higher-dimensional scattering problems.

Non-Abelian geometric phase and generalized invariant method.

We quantitatively formulate the nonadiabatic non-Abelian geometric phase by generalizing the Lewis invariant method to a degenerate case, which is reduced to Wilczek and Zee's [Phys. Rev. Lett. 52, 2111 (1984)] non-Abelian geometric phase in the adiabatic limit. We also derive the effective propagator accommodating the nonadiabatic non-Abelian geometric phase and the semiclassical quantization rule in the adiabatic limit.

Energy spectrum of the total system for a NQR experiment

We obtain the effective propagator and the semiclassical quantization rule for the external system in two adiabatically interacting systems, which accommodate the non-Abelian geometric phase of Wilczek and Zee. By using this formalism we calculate the energy spectrum of the total system which consists of the internal spin quadropole and the external agent rotating the internal system.

Non-Abelian geometric phase and long-range atomic forces.

Using fully quantal methods, we investigate various manifestations of the geometric phase in bound and free diatom systems. Adiabaticity and degeneracy are not invoked in the discussion.Received 3 May 1989DOI:https://doi.org/10.1103/PhysRevLett.64.256©1990 American Physical Society

Non-abelian geometric phase from incomplete quantum measurements

We study the evolution of a quantum system due to a sequence of incomplete measurements, each of which projects the state into an n-dimensional subspace V n of the Hilbert space of dimension n+ m. It is shown that the effect of a measurement that projects V n to V' n is to parallel transport an orthonormal basis of V n along the shortest geodesic joining V n and V' n in the Grassmann manifold G n,m and transform it by a Hermitian matrix whose eigenvalues are in (0,1]. The effect of a dense sequence of measurements leading to a cyclic evolution of V n is the holonomy transformation associated with the corresponding curve in G n,m , with respect to the natural U( n) connection over G n,m .

Non-adiabatic non-abelian geometric phase

The non-integrable geometric phase factor is generalized to a non-abelian phase factor which in the adiabatic limit is the Wilczek-Zee generalization of the Berry phase. It is then extended to the evolution of mixed states. A theorem of Narasimhan and Ramanan is used to relate the gauge field connection to the generalized geometric connection.