Use of the poincare sphere in polarization optics and classical and quantum mechanics. Review

The method of the Poincaré sphere, which was proposed by Henri Poincaré in 1891–1892, is a convenient approach to represent polarized light. This method is graphical: each point on the sphere corresponds to a certain polarization state. Apart from the obvious representation of polarized light, the method of the Poincaré sphere permits efficient solution of problems that result from the use of a set of phase plates or a combination of phase plates and ideally homogeneous polarizers. Recently, to calculate the geometric phase (which is often called the Berry phase) in polarization optics and quantum and classical mechanics, the method of the Poincaré sphere has drawn much attention, since it allows us to carry out these calculations very efficiently and intuitively using the solid angle resting, on a closed curve on the Poincaré sphere that corresponds to the change in the state of light polarization or in the state of spin of an elementary particle or its orientation in space from the viewpoint of systems in classical mechanics. The review considers papers on the above problems. Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 3, pp. 265–307, March. 1997.     

The parity operator in quantum optical metrology

Photon number states are assigned a parity of +1 if their photon number is even and a parity of ?1 if odd. The parity operator, which is minus one to the power of the photon number operator, is a Hermitian operator and thus a quantum mechanical observable although it has no classical analogue, the concept being meaningless in the context of classical light waves. In this paper we review work on the application of the parity operator to the problem of quantum metrology for the detection of small phase shifts with quantum optical interferometry using highly entangled field states such as the so-called N00N states, and states obtained by injecting twin Fock states into a beam splitter. With such states and with the performance of parity measurements on one of the output beams of the interferometer, one can breach the standard quantum limit, or shot-noise limit, of sensitivity down to the Heisenberg limit, the greatest degree of phase sensitivity allowed by quantum mechanics for linear phase shifts. Heisenberg limit sensitivities are expected to eventually play an important role in attempts to detect gravitational waves in interferometric detection systems such as LIGO and VIRGO.     

偏振光学系统中相位延迟机理及其应用

为了实现偏振编码的自由空间量子密钥分发实验,研制了偏振保持的光机系统,并对该系统所采用的相位延迟传输机理及应用进行了研究,建立了偏振误码率在允许范围内的量子链路.首先,采用矩阵光学理论对偏振光的方位角、相位延迟与消光比的关系进行了介绍.接着,通过矩阵光学理论及实验验证了偏振光学系统的相位延迟线性叠加原理.然后,在相位延迟线性叠加原理的基础上,设计了一套偏振保持光学系统,并通过理论分析及实验验证了此系统具备良好的偏振保持效果.最后,将偏振保持光学系统的设计机理应用于量子通信光机系统的设计之中,并取得了良好的设计效果.实验结果表明:相位之间的相互抵消可以有效地进行偏振保持设计,最终设计的量子通信光机系统的偏振消光比优于500∶1.满足了自由空间量子通信实验中对偏振误码率的要求.     

Quantum mechanical version of the classical Liouville theorem

In terms of the coherent state evolution in phase space,we present a quantum mechanical version of the classical Liouville theorem.The evolution of the coherent state from |z>to|sz-rz*> corresponds to the motion from a point z(q,p) to another point sz-rz* with |s|2-|r|2=1.The evolution is governed by the so-called Fresnel operator U(s,r) that was recently proposed in quantum optics theory,which classically corresponds to the matrix optics law and the optical Fresnel transformation,and obeys group product rules.In other words,we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space,which seems to be a combination of quantum statistics and quantum optics.     

Modeling and simulation of polarization errors in Sagnac fiber optic current sensor

Sagnac fiber optic current sensor (S-FOCS) is a kind of optical interferometer based on Sagnac structure, optical polarization states of sensing light wave in Sagnac fiber optic current sensor are limited. However, several factors induce optical polarization error, and non-ideal polarized light waves cause the interference signal crosstalk in sensor, including polarizer, quarter-wave retarder, splice angular, birefringence and so on. With these errors, linearly polarized light wave in PM fiber and circularly polarized light wave in sensing fiber become elliptically polarized light wave, then, nonreciprocal phase shift induced by magnetic field of the current is interrupted by wrong polarization state. To clarify characteristics of optical polarization error in fiber optic current sensor, we analyze the evolution process of random optical polarization state, linear optical polarization state and circular optical polarization state in Sagnac fiber optic current sensor by using Poincare sphere, then, build optical polarization error models by using Jones matrix. Based on models of polarization state in Sagnac fiber optic current sensor, we investigate the influence of several main error factors on optical polarization error characteristics theoretically, including extinction ratio in polarizer, phase delay in quarter-wave retarder, splice angular between quarter-wave retarder and polarization maintaining fiber. Finally, we simulate and quantify nonreciprocal phase shift to be detected in fiber optic current sensor related with optical polarization errors. In the end, we demonstrate S-FOCS in test. The results show that transfer matrix errors are induced by inaccurate polarization properties during polarization state conversion, then, the stability and accuracy of the S-FOCS are affected, and it is important to control the polarization properties at each step of the polarization state conversion precisely.     

Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes

Quantum systems such as, for example, photons, atoms, or Bose-Einstein condensates, prepared in complex states where entanglement between distinct degrees of freedom is present, may display several intriguing features. In this Letter we introduce the concept of such complex quantum states for intense beams of light by exploiting the properties of cylindrically polarized modes. We show that already in a classical picture the spatial and polarization field variables of these modes cannot be factorized. Theoretically it is proven that by quadrature squeezing cylindrically polarized modes one generates entanglement between these two different degrees of freedom. Experimentally we demonstrate amplitude squeezing of an azimuthally polarized mode by exploiting the nonlinear Kerr effect in a specially tailored photonic crystal fiber. These results display that such novel continuous-variable entangled systems can, in principle, be realized.     

Poincaré's theorem and unitary transformations for classical and quantum systems

Poincaré's celebrated theorem on the nonexistence of analytical invariants of motion is extended to the case of a continuous spectrum to deal with large classical and quantum systems. It is shown that Poincaré's theorem applies to situations where there exist continuous sets of resonances. This condition is equivalent to the nonvanishing of the asymptotic collision operator as defined in modern kinetic theory. Typical examples are systems presenting relaxation processes or exhibiting unstable quantum levels. As the result of Poincaré's theorem, the unitary transformation, leading to a cyclic Hamiltonian in classical mechanics or to the diagonalization of the Hamiltonian operator in quantum mechanics, diverges. We obtain therefore a dynamical classification of large classical or quantum systems. This is of special interest for quantum systems as, historically, quantum mechanics has been formulated following closely the patterns of classical integrable systems. The well known results of Friedrichs concerning the coupling of discrete states with a continuum are recovered. However, the role of the collision operator suggests new ways of eliminating the divergence in the unitary transformation theory.     

Quantum dynamics, measurement and entropy

The aim of this paper is to present a line of ideas, centred around entropy production andquantum dynamics, emerging from von Neumann's work on foundations of quantum mechanics and leading to current research. The concepts of measurement, dynamical evolution and entropy were central in J. von Neumann's work. Further developments led to the introduction of generalized measurements in terms of positive operator-valued measures, closely connected to the theory of open systems. Fundamental properties of quantum entropy were derived and Kolmogorov and Sinai related the chaotic properties of classical dynamical systems with asymptotic entropy production. Finally, entropy production in quantum dynamical systems was linked with repeated measurement processes and a whole research area on nonequilibrium phenomena in quantum dynamical systems seems to emerge.     

Constraints of Cluster Separability and Covariance on Current Operators

Realistic models of hadronic systems should be defined by a dynamical unitary representation of the Poincaré group that is also consistent with cluster properties and a spectral condition. All three of these requirements constrain the structure of the interactions. These conditions can be satisfied in light-front quantum mechanics, maintaining the advantage of having a kinematic subgroup of boosts and translations tangent to a light front. The most straightforward construction of dynamical unitary representations of the Poincaré group due to Bakamjian and Thomas fails to satisfy the cluster condition for more than two particles. Cluster properties can be restored, at significant computational expense, using a recursive method due to Sokolov. In this work we report on an investigation of the size of the corrections needed to restore cluster properties in Bakamjian–Thomas models with a light-front kinematic symmetry. Our results suggest that for models based on nucleon and meson degrees of freedom these corrections are too small to be experimentally observed.     

Random matrices as models for the statistics of quantum mechanics

Random matrices from the Gaussian unitary ensemble generate in a natural way unitary groups of evolution in finite-dimensional spaces. The statistical properties of this time evolution can be investigated by studying the time autocorrelation functions of dynamical variables. We prove general results on the decay properties of such autocorrelation functions in the limit of infinite-dimensional matrices. We discuss the relevance of random matrices as models for the dynamics of quantum systems that are chaotic in the classical limit.     

基于压缩光的量子精密测量

对任何物理量的测量都有一定的噪声, 经典测量所能达到的最小噪声一般称为散粒噪声, 对应着测量的标准量子极限. 利用压缩光可以突破标准量子极限, 从而提高测量精度. 本文介绍了压缩态光场用于突破标准量子极限的基本原理, 以及压缩态光场在相位测量、光学横向小位移及倾斜测量、磁场测量以及时钟同步等精密测量领域的应用和最新进展.     

Quantum Systems Connected by a Time-Dependent Canonical Transformation

We study both classical and quantum relation between two Hamiltoniansystems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other istime-dependent Hamiltonian system. The quantum unitary operatorrelevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.     

Dynamics of geometric phase for composite quantum system under nonlocal unitary transformation: a case study of dimer system

In this paper, we explore the dynamical properties of geometric phase for a composite quantum system under the nonlocal unitary evolution. As an illustrative example, the analytical expressions of geometric phase are derived for the dimer system. We find that geometric phase presents some interesting properties with coupling strengths (corresponding to nonlocal unitary evolution), such as dynamical oscillation behavior with time evolution, monotonicity, symmetry, etc. We show that the geometric phase and entanglement have the same period for some conditions. Moreover, we discuss geometric phase of the whole system and its subsystems. Our investigations show that geometric phase can reflect some inherent properties of the system: it signals a transition from self-trapping to delocalization.     

Quantum mechanics in noninertial reference frames: Violations of the nonrelativistic equivalence principle

In previous work we have developed a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group that includes transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as is the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. A special feature of these previously constructed representations is that they all respect the non-relativistic equivalence principle, wherein the fictitious forces associated with linear acceleration can equivalently be described by gravitational forces. In this paper we exhibit a large class of cocycle representations of the Galilean line group that violate the equivalence principle. Nevertheless the classical mechanics analogue of these cocycle representations all respect the equivalence principle.     

Classical Mechanics Is not the ħ, → 0 Limit of Quantum Mechanics

Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied. It is shown that the sets have a common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vice versa, there exist tomograms of quantum states which are not admissible in classical mechanics. The role of different transformations of reference frames in the phase space of classical and quantum systems (scaling and rotation) determining the admissibility of tomograms as well as the role of quantum uncertainty relations are elucidated. The union of all admissible tomograms of both quantum and classical states is discussed in the context of interaction of quantum and classical systems. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively. The role of the semigroup of scaling transforms of the Planck's constant is discussed.     

Multipartite State Representations in Multi-mode Fock Space and Their Squeezing Transformations

We present the continuous state vector of the total coordinate of multi-partlcle and the state vector of their total momentum, respectively, which possess completeness relation in multi-mode Fock space by virtue of the integration within an order product （IWOP） technique. We also calculate the transition from classical transformation of variables in the states to quantum unitary operator, deduce a new multi-mode squeezing operator, and discuss its squeezing effect. In progress, it indicates that the IWOP technique provides a convenient way to construct new representation in quantum mechanics.     

Constitution of objects in classical mechanics and in quantum mechanics

The constitution of objects is discussed in classical mechanics and in quantum mechanics. The requirement of objectivity and the Galilei invariance of classical and quantum mechanics leads to the postulate of covariance which must be fulfilled by observable quantities. Objects are then considered as carriers of these covariant observables and turn out to be representations of the Galilei group. Individual systems can be defined in classical mechanics by their trajectories in phase space. However, in quantum mechanics the characterization of individuals can only be achieved approximately by means of unsharp observables.     

Zwitters: Particles between quantum and classical

We describe both quantum particles and classical particles in terms of a classical statistical ensemble, with a probability distribution in phase space. By use of a wave function in phase space both can be treated in the same quantum formalism. Quantum particles are characterized by a specific choice of observables and time evolution of the probability density. Then interference and tunneling are found within classical statistics. Zwitters are (effective) one-particle states for which the time evolution interpolates between quantum and classical particles. Experimental bounds on a small parameter can test quantum mechanics.     

非阿贝尔腔量子电动力学模型下偏振光场的影响

通过使用场正交算符,而不是传统的玻色算符,研究了非阿贝尔腔量子电动力学(QED)模型中原子和偏振光场的相互作用。讨论了初始双模偏振光场对于原子布居数反转以及偏振光场的压缩特性的影响。结果表明,原子布居数反转的演化不仅与偏振椭圆的相位角有关,也与偏振椭圆的椭率角有关;只有当偏振椭圆是右旋圆偏振光时,原子布居数反转随时间的演化基本不变,趋近于初始值0,而当偏振椭圆是左旋圆偏振光时,原子布居数反转随时间的演化呈现周期性的崩塌复苏变化。另外,当初始光场是左旋圆偏振光时,光场可以出现周期性的压缩;而当初始光场是右旋圆偏振光时,光场的压缩不会持续出现。