Classical Systems, Standard Quantum Systems, and Mixed Quantum Systems in Hilbert Space

Traditionally, there has been a clear distinction between classical systems and quantum systems, particularly in the mathematical theories used to describe them. In our recent work on macroscopic quantum systems, this distinction has become blurred, making a unified mathematical formulation desirable, so as to show up both the similarities and the fundamental differences between quantum and classical systems. This paper serves this purpose, with explicit formulations and a number of examples in the form of superconducting circuit systems. We introduce three classes of physical systems with finite degrees of freedom: classical, standard quantum, and mixed quantum, and present a unified Hilbert space treatment of all three types of system. We consider the classical/quantum divide and the relationship between standard quantum and mixed quantum systems, illustrating the latter with a derivation of a superselection rule in superconducting systems.     

某些力学系统与电路系统的相似性

1力学与电路理论中定理的相似性 牛顿第二定律与电容的伏安关系是极其相似的 牛顿第二定律 F外=mdv质/dt 电容的伏安关系 Is=CdUc/dt 其中F外与电源Is对应,m与C对应,v质与Uc对应.     

Heisenberg-Integrable Spin Systems

We investigate certain classes of integrable classical or quantum spin systems. The first class is characterized by the recursively defined property P saying that the spin system consists of a single spin or can be decomposed into two uniformly coupled or disjoint subsystems with property P. For these systems the time evolution can be explicitly calculated. The second class consists of spin systems where all non-zero coupling constants have the same strength (spin graphs) possessing N − 1 independent, commuting constants of motion of Heisenberg type. These systems are shown to have the above property P and can be characterized as spin graphs not containing chains of length four as vertex-induced sub-graphs. We completely enumerate and characterize all spin graphs up to N = 5 spins. Applications to the construction of symplectic numerical integrators for non-integrable spin systems are briefly discussed.      

Zero Entropy Systems

This paper introduces the notion of entropy dimension to measure the complexity of zero entropy dynamical systems, including the probabilistic and the topological versions. These notions are isomorphism invariants for measure-preserving transformation and continuity. We discuss basic propositions for entropy dimension and construct some examples to show that the topological entropy dimension attains any value between 0 and 1. This paper also gives a symbolic subspace to achieve zero topological entropy, but with full entropy dimension.     

Large Few-Body Systems

Signatures of large few-body systems were recently observed in experiments with ultracold gases. This paper presents a brief introductory review of related theoretical concepts and recent experimental advances.     

Alice-Bob Peakon Systems

We study the Alice-Bob peakon system generated from an integrable peakon system using the strategy of the socalled Alice-Bob non-local KdV approach [Scientific Reports 7(2017) 869]. Nonlocal integrable peakon equations are obtained and shown to have peakon solutions.     

Dispersive Quantum Systems

A dispersive quantum system is a quantum system which is both isolated and non-time reversal invariant. This article presents precise definitions for those concepts and also a characterization of dispersive quantum systems within the class of completely positive Markovian quantum systems in finite dimension (through a homogeneous linear equation for the non-Hamiltonian part of the system’s Liouvillian). To set the framework, the basic features of quantum mechanics are reviewed focusing on time evolution and also on the theory of completely positive Markovian quantum systems, including Kossakowski–Lindblad’s standard form for Liouvillians. After those general considerations, a simple two-dimensional example is presented and then applied to describe the neutrino oscillation, with the introduction of a new “dispersive parameter.”     

Reconfigurable MEMS OADM Systems

This paper proposes a serial reconfigurable OADM consisting of optical circulator and 2 × 2 MEMS optical switch and tunable FBG. Based on MEMS technology, the OADM is demonstrated to increase the flexibility, decrease the cost andimprove the reliability.     

Singularities of Bi-Hamiltonian Systems

We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degeneracy of singular points of bi-Hamiltonian systems and describe their types.     

Micro-electromechanical Systems for Nano-science

Micro-electromechanical systems are ideal tools for nano-science because they bridge the gap between the nano- and the macro-world. Moreover, several of these instruments can be operated in parallel to either increase the throughput or to provide redundancy. The majority of the components of such a system have dimensions above the nanometer scale. Still, some require placement and pattern accuracy well below this limit. This will be highlighted in a short review of a few examples: Scanning optical near field microscope probe fabrication, where an aperture of 50 nm in diameter was incorporated at the probing tip; parallel scanning force microscope for measuring dust particles on Mars where redundancy is essential; and a miniaturized electron column for parallel electron beam lithography, where the throughput needs to be increased.     

Orthocomplementation and Compound Systems

In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice _sep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms” of the lab, and before any interaction takes place. In that case, the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore _sep cannot satisfy all Piron’s axioms. In previous works, assuming that _sep is orthocomplemented, it was argued that _sep is not orthomodular and fails to have the covering property. Here we prove that _sep cannot admit an orthocomplementation. Moreover, we propose a natural model for _sep which has the covering property. PACS: 03.65.Ta, 03.65.Ca     

Rotating Minimal Thermodynamic Systems

Minimal rotating thermodynamic systems are addressed. Particle m placed into the rotating symmetrical double-well potential (bowl), providing binary logical system is considered. The condition providing the transfer of the particle from one frictionless half-well to another, and, in this way, the possibility to record 1 bit of information is derived. The procedure of recording turns out to be irreversible; it is impossible to return the particle to its initial state under rotation about the same axis. The same rotating double-well system exerted to the thermal noise is considered. A minimal rotating thermal engine built of the rotating chamber, movable partition, and the particle confined within the chamber is treated. Rotation of the system displaces the partition, thus enabling erasing of one bit information. Erasing of 1 bit of information is due to the inertia (centrifugal force) acting on the partition. Isothermal expansion of the “minimal gas” expectedly gives rise to the Landauer bound. Compression of the “gas” with the rotation around the same axis is impossible and demands the additional axis of rotation. The interrelation between the possibility of recording/erasing information and the symmetry of the system is considered.     

Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems

Integrable non-linear Hamiltonian systems perturbed by additive noise develop a Lyapunov instability, and are hence chaotic, for any amplitude of the perturbation. This phenomenon is related, but distinct, from Taylor’s diffusion in hydrodynamics. We develop expressions for the Lyapunov exponents for the cases of white and colored noise. The situation described here being ‘multi-resonance’—by nature well beyond the Kolmogorov–Arnold–Moser regime, it offers an analytic glimpse on the regime in which many near-integrable systems, such as some planetary systems, find themselves in practice. We show with the aid of a simple example, how one may model in some cases weakly chaotic deterministic systems by a stochastically perturbed one, with good qualitative results.     

On Generalized Non-holonomic Systems

Generalized non-holonomic mechanical systems are analyzed from a geometric point of view. The existence and uniqueness of solutions, D’Alembert principle, Gauss principle of minimal constraint, the non-holonomic momentum and Gibbs–Appell equations for such systems are studied in an invariant Lagrangian framework.      

Reichenbachian Common Cause Systems

A partition C_{i i∈ I} of a Boolean algebra $\mathcal{S}$ in a probability measure space $(\mathcal{S},p)$ is called a Reichenbachian common cause system for the correlated pair A,B of events in $\mathcal{S}$ if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set I is called the size of the common cause system. It is shown that given any correlation in $(\mathcal{S},p)$ , and given any finite size n>2, the probability space $(\mathcal{S},p)$ can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of $\mathcal{S}$ contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated.     

Thermodynamics of Self-Gravitating Systems

This work assembles some basic theoretical elements on thermal equilibrium, stability conditions, and fluctuation theory in self-gravitating systems illustrated with a few examples. Thermodynamics deals with states that have settled down after sufficient time has gone by. Time dependent phenomena are beyond the scope of this paper. While thermodynamics is firmly rooted in statistical physics, equilibrium configurations, stability criteria and the destabilizing effect of fluctuations are all expressed in terms of thermodynamic functions. The work is not a review paper but a pedagogical introduction which may interest theoreticians in astronomy and astrophysicists. It contains sufficient mathematical details for the reader to redo all calculations. References are only to seminal works or readable reviews. Delicate mathematical problems are mentioned but are not discussed in detail.