Geometric phasea la Pancharatnam

In mid-1950s, Pancharatnam [1] encountered the geometric phase associated with the evolution along a geodesic triangle on the Poincaré sphere. We generalize his 3-vertex phase and employ it as the fundamental building block, to geometrically construct a general ray-space expression for geometric phase. In terms of a reference ray used to specify geometric phase, we delineate clear geometric meanings for gauge transformations and gauge freedom, which are generally regarded as mere mathematical abstractions.     

Geometric phase and Pancharatnam phase induced by light wave polarization

We use the quantum kinematic approach to revisit geometric phases associated with polarizing processes of a monochromatic light wave. We give the expressions of geometric phases for any, unitary or non-unitary, cyclic or non-cyclic transformations of the light wave state. Contrarily to the usually considered case of absorbing polarizers, we found that a light wave passing through a polarizer may acquire in general a nonzero geometric phase. This geometric phase exists despite the fact that initial and final polarization states are in phase according to the Pancharatnam criterion and cannot be measured using interferometric superposition. Consequently, there is a difference between the Pancharatnam phase and the complete geometric phase acquired by a light wave passing through a polarizer. We illustrate our work with the particular example of total reflection based polarizers.     

光声互作用模型中的Pancharatnam相位

通过解析求解简单极化激元模型的Pancharatnam相位, 研究了温度、耦合强度、声子与光子频率差、平均光子数等对其演化的影响. 结果表明,Pancharatnam相位随时间振荡,且振荡频率和振荡波形随时间变化,这种变化随着温度升高和耦合强度、声子与光子频率差增加而加大. 系统Pancharatnam相位随时间的演化在平均光子数较小时表现比较有规律,但随着平均光子数的增加,它趋于混沌化. 关键词： 极化激元 Pancharatnam 相位 Mach-Zehnder干涉仪 相干态     

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 Poincaré sphere in terms of the Pancharatnam phase.     

Geometric phase as a simple closed-form integral

For a general evolution of a quantal system, the geometric phase measured with reference to a given initial state is derived as an integral of a function of the pure state density operator by invoking the Pancharatnam connection continuously.     

The quantum geometric phase as a transformation invariant

The kinematic approach to the theory of the geometric phase is outlined. This phase is shown to be the simplest invariant under natural groups of transformations on curves in Hilbert space. The connection to the Bargmann invariant is brought out, and the case of group representations described.     

Engineering geometric phase in semiconductor microcavities

We present rigorous investigations of the geometric phase in semiconductor microcavities. The effects of excitonic spontaneous emission, initial state setting and cavity dissipation have been discussed. It is shown that the geometric phase decays exponentially due to the presence of excitonic spontaneous emission. More importantly, the inclusion of the phase shift leads to an enhanced sensitivity for the control of the geometric phase evolution and system dynamics.     

利用量子几何相位制作光纤偏振旋转器

利用光纤中的量子几何相位(又称Berry相位),制作了单模光纤的线偏振光偏振面旋转器,并对器件进行了测量.实验数据表明,这种光纤偏振旋转器对光偏振面的旋转基本符合理论预计.对偏差进行处理后得到修正的定标公式,可更精确地反映出此光纤偏振旋转器的特性.     

Use of non-adiabatic geometric phase for quantum computing by NMR

Geometric phases have stimulated researchers for its potential applications in many areas of science. One of them is fault-tolerant quantum computation. A preliminary requisite of quantum computation is the implementation of controlled dynamics of qubits. In controlled dynamics, one qubit undergoes coherent evolution and acquires appropriate phase, depending on the state of other qubits. If the evolution is geometric, then the phase acquired depend only on the geometry of the path executed, and is robust against certain types of error. This phenomenon leads to an inherently fault-tolerant quantum computation. Here we suggest a technique of using non-adiabatic geometric phase for quantum computation, using selective excitation. In a two-qubit system, we selectively evolve a suitable subsystem where the control qubit is in state |1, through a closed circuit. By this evolution, the target qubit gains a phase controlled by the state of the control qubit. Using the non-adiabatic geometric phase we demonstrate implementation of Deutsch-Jozsa algorithm and Grover's search algorithm in a two-qubit system.     

Dynamics of the geometric phase in the adiabatic limit of a quench induced quantum phase transition

The geometric phase associated with a many body ground state exhibits a signature of quantum phase transition. In this context, we have studied the behavior of the geometric phase during a linear quench caused by a gradual turning off of the magnetic field interacting with a spin chain.     

Optical properties of polarization-dependent geometric phase elements with partially polarized light

The behavior of geometric phase elements illuminated with partially polarized monochromatic beams is investigated both theoretically and experimentally. The element discussed in this paper is composed of wave plates with π-retardation and a space-variant orientation angle. We found that a beam emerging from such an element comprises two polarization orders; right-and left-handed circularly polarized states with conjugate geometric phase modification. This phase equals twice the orientation angle of the space-variant wave plate comprising the element. Apart from the two polarization orders, the emerging beam coherence polarization matrix includes a “vectorial interference matrix” which contains information concerning the correlation between the two orthogonal, circularly polarized portions of the incident beam. In this paper we measure this correlation by a simple interference experiment. In addition, we found that the equivalent mutual intensity of the emerging beam is modulated according to the geometric phase induced by the element. Other interesting phenomena concerning propagation will be discussed theoretically and demonstrated experimentally. The experiment made use of a spherical geometric phase element that was realized by use of a space-variant subwavelength grating illuminated with CO₂ laser radiation of 10.6 μm wavelength.     

Double transmission-mediums based geometric phase analysis for determining the two surface profiles of transparent object

The accurate measurement for the surface profiles of transparent object is of significance for quality control in optical devices and precision instruments. Here, a double transmission-mediums based geometric phase analysis method has been developed to evaluate both the upper and lower surface profiles of transparent object. To do this, the tested transparent object is placed above a preprinted lattice pattern. When viewed from above with a CCD camera, any slope variations of the surfaces will lead to distortions of the transmission-lattice patterns. And when changing one side of object’s contact medium, the lattice virtual image with modulated phase is distorted once again. Combined with the derived relationship between phase variations of transmission-lattice patterns and out-of-plane heights of two surfaces, the double-sided surface profiles of transparent object can be reconstructed successfully. With this, the technique, which is verified experimentally, is demonstrated to be a feasible and reliable method. The advantage of this method is that it simplifies the setup and allows a fast estimation of the geometry of a transparent specimen. The double-sided profiles can be decoupled easily according to the big difference of refractive indexes between contact mediums. And the calculation accuracy can be guaranteed by the weighted average from four directions.     

The geometric phase of a two-level atom in a narrow-bandwidth squeezed vacuum

In this paper, we derive the time dependent solution of the effective master equation for the reduced density matrix operator of a two level atom driven by a coherent laser field and damped by a finite bandwidth squeezed vacuum. The results show that the initial state setting, detuning parameter and Rabi frequency play important roles in the evolution of the system dynamics and geometric phase. We present a useful way for controlling the geometric phase variation for the system under consideration.     

部分偏振光经界面反射时的偏振度分析

定量计算了轴外点光源发出的部分偏振光束经界面反射后反射光偏振度与空间坐标及点光源坐标间关系，定量计算了特定参数的入射部分偏振光束经界面反射后的偏振度分布，给出了偏振度变化的数量级。计算表明一定偏振度的入射部分偏振光束经界面反射后，反射光束偏振度由于界面反射在反射光束截面上将产生一定分布。偏振度的变化不仅与光线入射角有关，还与点光源位置有关     

Observing build up of geometric phase in an adiabatic RF flipper with the amplitude of its rotating field

To demonstrate that adiabatic RF flippers impose an inherent geometric phase on the neutron polarization vector, we built a NSE setup consisting of two pairs of such flippers in a pulsed neutron beam. As is well known, the combined gradient and RF fields in each flipper—in the rotating frame—behave as a rotating field. The amplitude of this field in the first three flippers was kept maximum. For various amplitudes of the rotating field in the remaining flipper we measured the NSE pattern. Besides the shift of the NSE-point due to the variation of the dynamic phase, the NSE patterns show the development of the geometric phase.     

Three-dimensional correlated-fermion phase separation from analysis of the geometric mean of the individual susceptibilities

A quasi-Gaussian approximation scheme is formulated to study the strongly correlated imbalanced Fermions thermodynamics, where the mean-field theory is not applicable. The non-Gaussian correlation effects are understood to be captured by the statistical geometric mean of the individual susceptibilities. In the three-dimensional unitary fermions ground state, a universal nonlinear scaling transformation relates the physical chemical potentials with the individual Fermi kinetic energies. For the partial polarization phase separation to full polarization, the calculated critical polarization ratio is δ _C = [1−(1−ξ)^6/5]/[1+(1−ξ)^6/5] ≐ 0.34. ξ = 4/9 gives the ratio of the symmetric ground state energy density to that of the ideal fermion gas. Supported by the National Natural Science Foundation of China (Grant Nos. 10875050 and 10675052), the Fund of Central China Normal University, and the Foundation of Ministry of Education of China (Grant No. IRT0624)