Geometric Phase Gate Based on Both Displacement Operator and Squeezed Operators with a Superconducting Circuit Quantum Electrdynamics

We give the brief review on the related definition of the geometric phase independent of specific physical system based on the displacement opreator and
the sqeezed operator, then show how the displacement operator and the squeezed operator can induce the general geometric phase. By means of the displacement operator and the squeezed operator concerning the circuit cavity mode state along a closed path in the phase space, we discuss specifically how to implement a two-qubit geometric phase gate in circuit quantum electrodynamics with both single photon interaction and two-photon interaction between the superconducting qubits and the circuit cavity modes. The experimental feasibility is discussed in detail.     

Transition of Bery Phase and Pancharatnam Phase and Phase Change

Berry Phase and time-dependent Pancharatnam phase are investigated for nuclear spin polarization in a liquid by a rotation magnetic field, where two-state mixture effect is exactly included in the geometric phases. We find that when the system of nuclear spin polarization is in the unpolarized state, the transitive phenomena of both Berry phase and Pancharatnam phase are taken place. For the polarized system, in contrast, such a transition is not taken place. It is obvious that the transitions of geometric phase correspond to the phase change of physical system.     

Off-diagonal Geometric Phase of Two-Qubit System

This paper focuses on the off-diagonal geometric phase of the thermal state of a two-qubit system under the external magnetic field. The properties of geometric phases of the state in critical and non-critical regions are discussed respectively. The sudden change of structure of degeneracy at the critical point do not affect the geometric phase of the model. Increasing temperature tends to suppress the off-diagonal geometric phase. The relationship between the geometric phase and external magnetic field is also discussed.     

Geometric Phase in the Breit-Rabi System

By using of the invariant theory, we have studied the geometric phase in a time-dependent system with Higgs algebra structure, the dynamical and geometric phases are given, respectively. The Aharonov-Anandan phase is also obtained under the cyclical evolution. The disappearing condition of the geometric phase is given.     

Generalization of Susskind-Glogower Phase Operators and Inverse Field Operators to q-Deformed Case

We analyze some properties of Susskind-Glogower (SG) phase operators exp(iΦ) and exp(-iΦ) by making use of inverse field operators. The generalization of the analysis to q-deformed case is given.     

Vertex Operators, Grassmannians, and Hilbert Schemes

We approximate the infinite Grassmannian by finite-dimensional cutoffs, and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups, with respect to a one-dimensional torus action. We prove that in the localization basis, these are the well-known fermionic vertex operators on the infinite wedge representation. Furthermore, the boson-fermion correspondence, locality, and intertwining properties with the Virasoro algebra are the limits of relations on the finite-dimensional cutoff spaces, which are true for geometric reasons. We then show that these operators are also, almost by definition, the vertex operators defined by Okounkov and the author in Carlsson and Okounkov ( [math.AG], 2009), on the equivariant cohomology groups of the Hilbert scheme of points on \mathbb C²{\mathbb C^2} , with respect to a special torus action.     

Basic Properties of Symplectic Dirac Operators

Symplectic Dirac operators, acting on symplectic spinor fields introduced by B.~Kostant in geometric quantization, are canonically defined in a similar way as the Dirac operator on Riemannian manifolds. These operators depend on a choice of a metaplectic structure as well as on a choice of a symplectic covariant derivative on the tangent bundle of the underlying manifold. This paper performs a complete study of these relations and shows further basic properties of the symplectic Dirac operators. Various examples are given for illustration. Received: 1 July 1996 / Accepted: 24 September 1996     

Phase Dynamics in Nonlinear Coupled Oscillators

We discuss the phase dynamics in the system of a set of nonlinear coupled oscillators. We find a direct and simple geometric method to observe the geometric phase for this system, and it provides a possibility for the experimental measurements.     

The Geometry of Recursion Operators

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser’s theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures that have not been considered before. One of these is the symplectic formulation of hyper-Kähler geometry, which turns out to be a strict generalization of the usual definition in terms of differential or Kähler geometry.     

Berezin-Toeplitz Operators, a Semi-Classical Approach

This article is devoted to the Toeplitz Operators [4] in the context of the geometric quantization [11, 15]. We propose an ansatz for their Schwartz kernel. From this, we deduce the main known properties of the principal symbol of these operators and obtain new results : we define their covariant and contravariant symbols, which are full symbols, and compute the product of these symbols in terms of the Kähler metric. This gives canonical star products on the Kählerian manifolds. This ansatz is also useful to introduce the notion of microsupport.     

Geometric Phase and Bose-Einstein Condensate

We show that a geometric phase may appear in the Bose-Einstein condensate (BEC) in which an adiabatic procedure happens, then a perturbation expression of geometric phase is obtained for the case of time-averaged orbiting potential trap. The phase caused by the adiabatic bias magnetic field in one BEC may interfere with another, which is similar to the phase interference of Aharonov-Susskind effect, and can be observed by experiments.     

Non-Adiabatic Geometric Phase in a Dispersive Interaction System

We investigate the geometric phase and dynamic phase of a two-level fermionic system with dispersive interaction, driven by a quantized bosonic field which is simultaneously subjected to parametric amplification. It is found that the geometric phase is induced by a counterpart of the Stark shift. This effect is due to distinct shifts in the field frequency induced by interaction between different states （｜e〉 and ｜g〉 ） and cavity field, and a simple geometric interpretation of this phenomenon is given, which is helpful to understand the natural origin of the geometric phase.     

Geometric Phase and Sidebands

Geometric phase of mixed state is expanded for a two-level atom interacting with a quantized field mode, where master equation is a modified Bloch equation with constant terms that are used to explain emergence of sidebands in the spectrum of the fluorescent light. The results show that geometric phase transition is observed for medium initial angle when the Rabi frequency associated with the driving field becomes comparable to the spectral width of the atom. We find that the geometric phase transition depends on population inversion and Bloch radius, which is helpful to understand mechanism of sideband.     

Geometric Phase in a Time-Dependent Coupled Atom-Heteronuclear-Molecule Condensate

The geometric phase in a time-dependent coupled atom-heteronuclear-molecule condensate is investigated, and the disappearing condition of the geometric phase is given.     

相位解码的时-空重建算法

基于相位映射的三维传感技术对几何形状和拓扑结构复杂或表面梯度很大的物体进行绝对相位测量及相位重建仍然是一个困难的问题。近年来国际上提出了一种时间维度相位重建算法可以对此提供一种解决方案。然而,该算法对结构光照明系统提出了很高的要求,当系统无法满足算法要求时,重建结果存在严重的噪声。针对这一问题,提出了一种利用分段函数构造的相位解码的时空重建算法。该算法在相位重建过程中同时考虑时间维度和空间维度相位的相对关系,使得空间频率非严格按指数增长的条纹序列可以得到正确的重建,消除了跳变边界的相位模糊问题,从而可以更加有效地解决深度表面不连续和存在相互孤立物表面拓扑结构的景物相位重建问题。实验结果证明了此算法的可行性和有效性。     

Geometric Phase in a Time-Dependent Bose-Fermi System

By using the Lewis-Riesenfeld invariant theory, the geometric phase in a time-dependent Bose-Fermi system has been studied. It is found that the geometric phase has nothing to do with the field frequency and the coupling coefficient between the Boson and Fermion.     

Influence of Bipartite Qubit Coupling on Geometric Phase

The geometric phase of the bipartite Heisenberg spin-1/2 system with one spin driven by rotating magnetic field is investigated. It is found that in the one-site drive case, the intersubsystem coupling can be equivalent to a static quasi-magnetic field in the parameter space. This perspective has satisfactorily explained the irregular asymptote effect of geometric phase. We discuss the property of the two-site magnetic drive spin system and discover that a stationary state with no geometric phase shift is generated.     

Influence of Bipartite Qubit Coupling on Geometric Phase

The geometric phase of the bipartite Heisenberg spin-1/2 system with one spin driven by rotating magnetic field is investigated. It is found that in the one-site drive case, the intersubsystem coupling can be equivalent to a static quasi-magnetic field in the parameter space. This perspective has satisfactorily explained the irregular asymptote effect of geometric phase. We discuss the property of the two-site magnetic drive spin system and discover that a stationary state with no geometric phase shift is generated.     

Spatial-Variant Geometric Phase of Hybrid-Polarized Vector Optical Fields

We investigate a novel spatial geometric phase of hybrid-polarized vector fields consisting of linear, elliptical and circular polarizations by Young's two-slit interferometer instead of the widely used Mach-Zehnder interferometer.This spatial geometric phase can be manipulated by engineering the spatial configuration of hybrid polarizations,and is directly related to the topological charge, the local states of polarization and the rotational symmetry of hybrid-polarized vector optical fields. The unique feature of geometric phase has implications in quantum information science as well as other physical systems such as electron vortex beams.     

Geometric Phase in a Time-Dependent System with Higgs Algebra Structure

By using of the invariant theory, we have studied the geometric phase in a time-dependent system with Higgs algebra structure, the dynamical and geometric phases are given, respectively. The Aharonov-Anandan phase is also obtained under the cyclical evolution. The disappearing condition of the geometric phase is given.