Planck’s Constant in the Light of Quantum Logic

The goal of quantum logic is the “bottom-top” reconstruction of quantum mechanics. Starting from a weak quantum ontology, a long sequence of arguments leads to quantum logic, to an orthomodular lattice, and to the classical Hilbert spaces. However, this abstract theory does not yet contain Planck’s constant ℏ. We argue, that ℏ can be obtained, if the empty theory is applied to real entities and extended by concepts that are usually considered as classical notions. Introducing the concepts of localizability and homogeneity we define objects by symmetry groups and systems of imprimitivity. For elementary systems, the irreducible representations of the Galileo group are projective and determined only up to a parameter z, which is given by z=m/ℏ, where m is the mass of the particle and ℏ Planck’s constant. We show that ℏ has a meaning within quantum mechanics, irrespective of use the of classical concepts in our derivation.     

Entanglement dynamics for two interacting spins

We study the dynamical generation of entanglement for a very simple system: a pair of interacting spins, s₁ and s₂, in a constant magnetic field. Two different situations are considered: (a) s₁ → ∞, s₂ = 1/2 and (b) s₁ = s₂ → ∞, corresponding, respectively, to a quantum degree of freedom coupled to a semiclassical one (a qubit in contact with an environment) and a fully semiclassical system, which in this case displays chaotic behavior. Relations between quantum entanglement and classical dynamics are investigated.     

Massive spin-momentum entanglement measured in an atomic beam spin echo experiment

In this paper we present an experiment performed with an atomic beam spin echo interferometer, in which massive intraparticle entanglement is demonstrated. In the longitudinal Stern-Gerlach arrangement the nuclear spin and linear momentum of ³He particles are inextricably linked, such that the overall system state cannot be written as the tensor product of the corresponding Hilbert spaces. The measured data show maximal entanglement between ℋ _I and ℋ _p . This hybrid system of one quantum and one classical degree of freedom is a textbook example of entanglement between discrete and continuous observables.     

Transverse Ising model: Markovian evolution of classical and quantum correlations under decoherence

The transverse Ising Model (TIM) in one dimension is the simplest model which exhibits a quantum phase transition (QPT). Quantities related to quantum information theoretic measures like entanglement, quantum discord (QD) and fidelity are known to provide signatures of QPTs. The issue is less well explored when the quantum system is subjected to decoherence due to its interaction, represented by a quantum channel, with an environment. In this paper we study the dynamics of the mutual information I(ρ _AB ), the classical correlations C(ρ _AB ) and the quantum correlations Q(ρ _AB ), as measured by the QD, in a two-qubit state the density matrix of which is the reduced density matrix obtained from the ground state of the TIM in 1d. The time evolution brought about by system-environment interactions is assumed to be Markovian in nature and the quantum channels considered are amplitude damping, bit-flip, phase-flip and bit-phase-flip. Each quantum channel is shown to be distinguished by a specific type of dynamics. In the case of the phase-flip channel, there is a finite time interval in which the quantum correlations are larger in magnitude than the classical correlations. For this channel as well as the bit-phase-flip channel, appropriate quantities associated with the dynamics of the correlations can be derived which signal the occurrence of a QPT.     

Dynamics of vibrational chaos and entanglement in triatomic molecules： Lie algebraic model

In this paper, the dynamics of chaos and the entanglement in triatomic molecular vibrations are investigated. On the classical aspect, we study the chaotic trajectories in the phase space. We employ the linear entropy to examine the dynamical entanglement of the two bonds on the quantum aspect. The correspondence between the classical chaos and the quantum dynamical entanglement is also investigated. As an example, we apply our algebraic model to molecule H2O.     

Information-Geometric Indicators of Chaos in Gaussian Models on Statistical Manifolds of Negative Ricci Curvature

A new information-geometric approach to chaotic dynamics on curved statistical manifolds based on Entropic Dynamics (ED) is proposed. It is shown that the hyperbolicity of a non-maximally symmetric 6N-dimensional statistical manifold ℳ _s underlying an ED Gaussian model describing an arbitrary system of 3N degrees of freedom leads to linear information-geometric entropy growth and to exponential divergence of the Jacobi vector field intensity, quantum and classical features of chaos respectively.     

On entanglement distillation and quantum error correction for unknown states and channels

We consider the problem of invariance of distillable entanglement D and quantum capacities Q under erasure of information about single copy of quantum state or channel respectively. We argue that any 2 ⊗N two-way distillable state is still two-way distillable after erasure of single copy information. For some known distillation protocols the obtained two-way distillation rate is the same as if Alice and Bob knew the state from the very beginning. The isomorphism between quantum states and quantum channels is also investigated. In particular it is pointed out that any transmission rate down the channel is equal to distillation rate with formal LOCC-like superoperator that uses in general nonphysical Alice actions. This allows to we prove that if given channel Λ has nonzero capacity (Q _→ or Q _⟺) then the corresponding quantum state ϱ(Λ) has nonzero distillable entanglement (D _→ or D _⟺). Follwoing the latter arguments are provided that any channel mapping single qubit into N level system allows for reliable two-way transmission after erasure of information about single copy. Some open problems are discussed.     

Quantum entanglement and chaos in kicked two-component Bose-Einstein condensates

We study two-component Bose-Einstein condensates that behave collectively as a spin system obeying the dynamics of a quantum kicked top. Depending on the nonlinear interaction between atoms in the classical limit, the kicked top exhibits both regular and chaotic dynamical behavior. The quantum entanglement is physically meaningful if the system is viewed as a bipartite system, where the subsystem is any one of the two modes. The dynamics of the entanglement between the two modes in this classical chaotic system has been investigated. The chaos leads to rapid rise and saturation of the quantum entanglement. Furthermore, the saturated values of the entanglement fall short of its maximum. The mean entanglement has been used to clearly display the close relation between quantum entanglement and underlying chaos.     

Oscillator rule of the trap activation energies in NaCl crystals

The energy spectra of traps in NaCl crystals have been studied in detail by the method of thermoluminescence. Crystals of NaCl were undoped but treated thermally in different ways. The activation energies of traps form a single oscillator series, E _n =ℏω _TL (n+1/2), ℏω _TL =903 cm^-1. Contrary to other previously studied crystals with complex lattices, the corresponding line ℏω _Ram =ℏω _TL was not found in Raman spectra of NaCl. It is assumed that the oscillator rule is governed by the polaron nature of traps. The trap activation energy is determined by the vibration level from which the transition of the charge carrier to the excited luminescence centre is made possible and depends on the distance between these centres.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

Wavevector and frequency dependent dielectric function dynamic structure factor and the instability of plasma waves in two component hot rare quantum and classical plasmas

The full wavevector and frequency dependent complex dielectric function for two component classical and quantum rare hot plasmas have been derived. The real part of dielectric function is obtained in the form of a series. Difference between quantum and classical real and imaginary parts of dielectric function have been brought out by making explicit calculations. The quantum nature of the plasma brings about significant changes in both parts depending upon the magnitude of quantum parameter,R (= 8.93(λ_th)/λ). Expressions for the dynamic structure factors for both two component classical and quantum plasma have been evaluated for different values of the mass of the positive componentm ₊, temperature T₊ and wavevector k. It is found that the plasma exhibits well defined collective modes for certain values of |k| accompanied by varying disorder which depends upon the values of m₊ as well as on |k| and T₊. For the quantum case the collective modes are less well defined as compared to the corresponding classical case, thus proving that quantum nature introduces inherent disorder in the system. But for both the cases, increase in temperature destroys collective modes. Another feature is the appearance of a hump near Ω = 0 which becomes smaller and vanishes as the quantum parameter is decreased. Instability of plasma modes in the presence of constant electric field has also been worked out for the quantum case.     

Semiclassical Behaviour of Expectation Values in Time Evolved Lagrangian States for Large Times

We study the behaviour of time evolved quantum mechanical expectation values in Lagrangian states in the limit 0 and t. We show that it depends strongly on the dynamical properties of the corresponding classical system. If the classical system is strongly chaotic, i.e. Anosov, then the expectation values tend to a universal limit. This can be viewed as an analogue of mixing in the classical system. If the classical system is integrable, then the expectation values need not converge, and if they converge their limit depends on the initial state. An additional difference occurs in the timescales for which we can prove this behaviour; in the chaotic case we get up to Ehrenfest time, t ln (1/), whereas for integrable system we have a much larger time range.     

Exciton magnetotransport in two-dimensional systems: Weak-localization effects

The paper considers the effect of a magnetic field B on the transport of neutral composite particles, namely excitons, in weakly disordered two-dimensional (2D) systems. In the case of classical transport (when the interference of different paths is neglected), the magnetic field suppresses exciton transport, and the static diffusion constant D(B) monotonically drops with B. When quantum-mechanical corrections due to weak localization are taken into account, D(B) becomes a nonmonotonic function of B. In weak magnetic fields, where the magnetic length is much larger than the exciton Bohr radius, ℓ_B=(ℏc/eB)^1/2≫a _B =ε ℏ²/μe ²,a positive magnetodiffusion effect is predicted, i.e., the exciton mobility should increase with B. Zh. éksp. Teor. Fiz. 114, 359–378 (July 1998)     

Resonance microwave photoresistance of a two-subband electron system at large filling factors

The microwave photoresistance of a double GaAs quantum well with two occupied size-quantization sub-bands E ₁ and E ₂ has been studied at the temperatures T = 1.6–4.2 K in the magnetic fields B < 0.5 T. The microwave photoresistance of such a system has been found to have a maximum amplitude when the maximum of the magneto-intersubband oscillations with the number k = (E ₂ − E ₁)ℏω_c coincides with the maximum or minimum of the ω/ω_c oscillations, where ω is the microwave frequency and ω_c is the cyclotron frequency. It has been shown that the resonance photoresistance that appears in the kth maximum of the magneto-intersubband oscillations is determined by the condition ℏω/(E ₂ − E ₁) = (j ± 0.2)/k, where k and j are positive integers.     

Persistent entanglement in two coupled squid rings in the quantum to classical transition: a quantum jumps approach

We explore the quantum–classical crossover of two coupled, identical, superconducting quantum interference device (SQUID) rings. The motivation for this work is based on a series of recent papers. In [1] we showed that the entanglement characteristics of chaotic and periodic (entrained) solutions of the Duffing oscillator differed significantly and that in the classical limit entanglement was preserved only in the chaotic-like solutions. However, Duffing oscillators are a highly idealized toy system. Motivated by a wish to explore more experimentally realizable systems, we extended our work in [2, 3] to an analysis of SQUID rings. In [3] we showed that the two systems share a common feature. That is, when the SQUID ring’s trajectories appear to follow (semi)classical orbits, entanglement persists. Our analysis in [3] was restricted to the quantum-state diffusion unraveling of the master equation – representing unit efficiency heterodyne detection (or ambi-quadrature homodyne detection). Here we show that very similar behavior occurs using the quantum jumps unraveling of the master equation. Quantum jumps represents a discontinuous photon counting measurement process. Hence, the results presented here imply that such persistent entanglement is independent of measurement process and that our results may well be quite general in nature.     

2N qubit “mirror states” for optimal quantum communication

We introduce a new genuinely 2N qubit state, known as the “mirror state” with interesting entanglement properties. The well known Bell and the cluster states form a special case of these “mirror states”, for N = 1 and N = 2 respectively. It can be experimentally realized using SWAP and multiply controlled phase shift operations. After establishing the general conditions for a state to be useful for various communicational protocols involving quantum and classical information, it is shown that the present state can optimally implement algorithms for the quantum teleportation of an arbitrary N qubit state and achieve quantum information splitting in all possible ways. With regard to superdense coding, one can send 2N classical bits by sending only N qubits and consuming N ebits of entanglement. Explicit comparison of the mirror state with the rearranged N Bell pairs and the linear cluster states is considered for these quantum protocols. We also show that mirror states are more robust than the rearranged Bell pairs with respect to a certain class of collisional decoherence.     

Chaotic dynamics in the electroconvection system of a nematic liquid crystal

The transition from a steady domain structure to turbulence in the electroconvection system of a nematic under the action of a constant electric field is studied using the methods of optical and acoustic responses. The chaotic dynamics is investigated both by conventional methods (Fourier signal spectrum) and by methods of nonlinear dynamics. From the quantitative estimates of basic characteristics of the chaotic behavior (namely, the correlation dimension, leading Lyapunov exponent, K-entropy, and embedding dimension), one can conclude that temporal chaos arises in the system, giving rise to a strange attractor, as the control parameter increases at ɛ ≥ ɛ _c ≈ 0.5. The fact that the distribution of laminar domains in the liquid-crystal layer depends on their length under the conditions of developed turbulence indicates that the dynamics of the nematic demonstrates the intermittent behavior.     

Quantum geometry and quantum mechanics of integrable systems

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ^∞) and are invariant under quantum dynamics in the long-time range O(ħ ^−∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.     

Quantum correlations in the dimerized spin chain at zero and finite temperatures

By using the method of exact diagonalization, we investigate the quantum correlation measured by quantum discord of the dimerized spin chain at both zero and finite temperatures. The results disclose that the quantum discord is robust at any finite parameter α and temperature T, in contrast to entanglement which shows a sudden death when the parameter α or the temperature T reaches a critical point. At finite temperature, it is interesting to find that the quantum discord QD _{2i−1, 2i} can increase with temperature T no matter if the entanglement EoF _{2i−1, 2i} exists or not. The research on the relation between the quantum discord and the quantum phase transition in the dimerized spin chain indicates that the transition can be characterized by the first derivation of the quantum discord at zero and low temperatures.