Berry phases in the three-level atoms driven by quantized light fields

A theoretical analysis of Berry’s phases is given for the three-level atoms interacting with external one-mode and two-mode quantized light fields. Three main results are obtained: (i) There is a Berry phase which vanishes in the classical limit or this Berry phase is completely induced by the field quantization; (ii) Berry’s phases for the one-mode field and the two-mode field can be equal so long as the photon numbers of the two-mode field are properly chosen; (iii) In the two-mode case, Berry phases of the atom interacting with one mode is affected by the other mode even if the photon number of the other mode is zero.      

Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models

The semiclassical quantization of cyclotron orbits for two-dimensional Bloch electrons in a coupled two band model with a particle-hole symmetric spectrum is considered. As concrete examples, we study graphene (both mono and bilayer) and boron nitride. The main focus is on wave effects – such as Berry phase and Maslov index – occurring at order (^h/_2p)\hbar in the semiclassical quantization and producing non-trivial shifts in the resulting Landau levels. Specifically, we show that the index shift appearing in the Landau levels is related to a topological part of the Berry phase – which is basically a winding number of the direction of the pseudo-spin 1/2 associated to the coupled bands – acquired by an electron during a cyclotron orbit and not to the complete Berry phase, as commonly stated. As a consequence, the Landau levels of a coupled band insulator are shifted as compared to a usual band insulator. We also study in detail the Berry curvature in the whole Brillouin zone on a specific example (boron nitride) and show that its computation requires care in defining the “k-dependent Hamiltonian” H(k), where k is the Bloch wavevector.     

Berry phase and Hannay’s angle in the Born–Oppenheimer hybrid systems

In this paper, we investigate the Berry phase and Hannay’s angle in the Born–Oppenheimer (BO) hybrid systems and obtain their algebraic expressions in terms of one form connection. The semiclassical relation of Berry phase and Hannay’s angle is discussed. We find that, besides the usual connection term, the Berry phase of quantum BO composite system also contains a novel term brought forth by the coupling induced effective gauge potential. This quantum modification can be viewed as an effective Aharonov–Bohm effect. Moreover, the similar phenomenon is founded in Hannay’s angle of classical BO composite system, which indicates that the Berry phase and Hannay’s angle possess the same relation as the usual one. An example is used to illustrate our theory. This scheme can be used to generate artificial gauge potentials for neutral atoms. Besides, the quantum–classical hybrid BO system is also studied to compare with the results in full quantum and full classical composite systems.     

Berry phase in a nonisolated system

We investigate the effect of the environment on a Berry phase measurement involving a spin-half. We model the spin + environment using a biased spin-boson Hamiltonian with a time-dependent magnetic field. We find that, contrary to naive expectations, the Berry phase acquired by the spin can be observed, but only on time scales which are neither too short nor very long. However this Berry phase is not the same as for the isolated spin-half. It does not have a simple geometric interpretation in terms of the adiabatic evolution of either bare spin states or the dressed spin resonances. This result is crucial for proposed Berry phase measurements in superconducting nanocircuits.     

Geometric nature of the environment-induced Berry phase and geometric dephasing

We investigate the geometric phase or Berry phase acquired by a spin half which is both subject to a slowly varying magnetic field and weakly coupled to a dissipative environment (either quantum or classical). We study how this phase is modified by the environment and find that the modification is of a geometric nature. While the original Berry phase (for an isolated system) is the flux of a monopole field through the loop traversed by the magnetic field, the environment-induced modification of the phase is the flux of a quadrupolelike field. We find that the environment-induced phase is complex, and its imaginary part is a geometric contribution to dephasing. Its sign depends on the direction of the loop. Unlike the Berry phase, this geometric dephasing is gauge invariant for open paths of the magnetic field.     

Testing Born’s Rule in Quantum Mechanics for Three Mutually Exclusive Events

We present a new experimental approach using a three-path interferometer and find a tighter empirical upper bound on possible violations of Born’s Rule. A deviation from Born’s rule would result in multi-order interference. Among the potential systematic errors that could lead to an apparent violation we specifically study the nonlinear response of our detectors and present ways to calibrate this error in order to obtain an even better bound.     

Born-Oppenheimer近似下谐振子场驱动电磁模系统的Berry相和Hannay角

研究了Born-Oppenheimer近似下谐振子场驱动电磁模系统的Berry相和Hannay角, 通过理论计算得到了其表达式, 并讨论了这二者之间的半经典关系.结果表明, 这一量子Born-Oppenheimer复合系统的Berry相包含两部分: 第一部分与通常几何相的定义相同, 另一项则是由耦合造成的有效规范式引入的.这一量子修正可以被看作一个等效的Aharonov-Bohm效应.不仅如此, 其对应经典系统的Hannay角的定义中也存在类似的现象. 由此可见, 这一复合系统的Berry相与Hannay角之间也存在半经典关系, 并与文献[16] 中通常情况下的半经典关系相同.此外, 上述理论也可以运用于解决产生中性原子的人造规范势等物理问题. 关键词： Berry相 Hannay角 量子经典对应 Born-Oppenheimer近似     

Nonlinear Topological Effects in Optical Coupled Hexagonal Lattice

Topological physics in optical lattices have attracted much attention in recent years. The nonlinear effects on such optical systems remain well-explored and a large amount of progress has been achieved. In this paper, under the mean-field approximation for a nonlinearly optical coupled boson–hexagonal lattice system, we calculate the nonlinear Dirac cone and discuss its dependence on the parameters of the system. Due to the special structure of the cone, the Berry phase (two-dimensional Zak phase) acquired around these Dirac cones is quantized, and the critical value can be modulated by interactions between different lattices sites. We numerically calculate the overall Aharonov-Bohm (AB) phase and find that it is also quantized, which provides a possible topological number by which we can characterize the quantum phases. Furthermore, we find that topological phase transition occurs when the band gap closes at the nonlinear Dirac points. This is different from linear systems, in which the transition happens when the band gap closes and reopens at the Dirac points.     

Adiabatic anholonomy and canonical transformations

Biswas and Soni [4] have surmised a semiclassical formula for Berry’s phase in terms of a generating function. We derive this formula apart from showing that it is not true in general and investigate its domain of validity. We also derive transformation formulae for Berry’s phase (Hannay’s angle) under general canonical transformations. A simpler proof for total angle invariance than hitherto available, is given.     

Berry phase for spin-1/2 particles moving in a space-time with torsion

Berry phase for a spin-1/2 particle moving in a flat space-time with torsion is investigated in the context of the Einstein–Cartan–Dirac model. It is shown that if the torsion is due to a dense polarized background, then there is a Berry phase only if the fermion is massless and its momentum is perpendicular to the direction of the background polarization. The order of magnitude of this Berry phase is discussed in other theoretical frameworks. Received: 12 February 2001 / Revised version: 2 May 2001 / Published online: 29 June 2001     

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order ħ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.     

The grounds for time dependent market potentials from dealers’ dynamics

We apply the potential force estimation method to artificial time series of market price produced by a deterministic dealer model. We find that dealers’ feedback of linear prediction of market price based on the latest mean price changes plays the central role in the market’s potential force. When markets are dominated by dealers with positive feedback the resulting potential force is repulsive, while the effect of negative feedback enhances the attractive potential force.     

Adiabatic Berry phase in an atom-molecule conversion system

We investigate the Berry phase of adiabatic quantum evolution in the atom-molecule conversion system that is governed by a nonlinear Schrödinger equation. We find that the Berry phase consists of two parts: the usual Berry connection term and a novel term from the nonlinearity brought forth by the atom-molecule coupling. The total geometric phase can be still viewed as the flux of the magnetic field of a monopole through the surface enclosed by a closed path in parameter space. The charge of the monopole, however, is found to be one third of the elementary charge of the usual quantized monopole. We also derive the classical Hannay angle of a geometric nature associated with the adiabatic evolution. It exactly equals minus Berry phase, indicating a novel connection between Berry phase and Hannay angle in contrast to the usual derivative form.     

Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model

We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.     

Tunneling time distribution by means of Nelson’s quantum mechanics and wave-particle duality

We construct a tunneling time distribution by means of Nelson’s quantum mechanics and investigate statistical properties of the tunneling time distribution. As a result, we find that the relationship between the average and the variance of the tunneling time shows ‘wave-particle duality’.     

Massive particle’s tunneling radiation from higher-dimensional Schwarzchild de Sitte and Anti-de Sitter black holes

We investigate the massive particle’s tunneling radiation from Schwarzchild black holes in higher-dimensional de Sitter and Anti-de Sitter space-times. Difference from the mass-less particle, the geodesics of the massive particle is not light-like, but decided by the phase velocity. We focus on s-waves, extend Parikh and Wilczek’s semi-classical tunneling method, and calculate the massive particle’s emission rate. It is shown that the emission rate is relevant to the change of the black hole’s entropy respectively, and the result takes the same functional form as that of the mass-less particle.     

Berry curvature in graphene: a new approach

In the present paper we have directly computed the Berry curvature terms relevant for graphene in the presence of an inhomogeneous lattice distortion. We have employed the generalized Foldy–Wouthuysen framework, developed by some of us. We show that a non-constant lattice distortion leads to a valley–orbit coupling which is responsible for a valley–Hall effect. This is similar to the valley–Hall effect induced by an electric field proposed in the literature and is the analogue of the spin–Hall effect in semiconductors. Our general expressions for Berry curvature, for the special case of homogeneous distortion, reduce to the previously obtained results. We also discuss the Berry phase in the quantization of cyclotron motion.     

Topological nature of the phonon Hall effect

We provide a topological understanding of the phonon Hall effect in dielectrics with Raman spin-phonon coupling. A general expression for phonon Hall conductivity is obtained in terms of the Berry curvature of band structures. We find a nonmonotonic behavior of phonon Hall conductivity as a function of the magnetic field. Moreover, we observe a phase transition in the phonon Hall effect, which corresponds to the sudden change of band topology, characterized by the altering of integer Chern numbers. This can be explained by touching and splitting of phonon bands.     

Cartan–Weyl Dirac and Laplacian Operators,Brownian Motions: The Quantum Potential and Scalar Curvature,Maxwell’s and Dirac-Hestenes Equations,and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q     

Quantization,coherent states and geometric phases of a generalized nonstationary mesoscopic RLC circuit

We present an alternative quantum treatment for a generalized mesoscopic RLC circuit with time-dependent resistance, inductance and capacitance. Taking advantage of the Lewis and Riesenfeld quantum invariant method and using quadratic invariants we obtain exact nonstationary Schrödinger states for this electromagnetic oscillation system. Afterwards, we construct coherent and squeezed states for the quantized RLC circuit and employ them to investigate some of the system’s quantum properties, such as quantum fluctuations of the charge and the magnetic flux and the corresponding uncertainty product. In addition, we derive the geometric, dynamical and Berry phases for this nonstationary mesoscopic circuit. Finally we evaluate the dynamical and Berry phases for three special circuits. Surprisingly, we find identical expressions for the dynamical phase and the same formulae for the Berry’s phase.