scholarly journals Local scaling asymptotics in phase space and time in Berezin–Toeplitz quantization

2014 ◽  
Vol 25 (06) ◽  
pp. 1450060 ◽  
Author(s):  
Roberto Paoletti
Keyword(s):  
Phase Space ◽  
Evolution Operator ◽  
Quantum Evolution ◽  
Classical Dynamics ◽  
Semiclassical Asymptotics ◽  
Local Scaling ◽  
Space And Time

This paper deals with the local semiclassical asymptotics of a quantum evolution operator in the Berezin–Toeplitz scheme, when both time and phase space variables are subject to appropriate scalings in the neighborhood of the graph of the underlying classical dynamics. Global consequences are then drawn regarding the scaling asymptotics of the trace of the quantum evolution as a function of time.

Exploring scalar quantum walks on Cayley graphs

2008 ◽  
Vol 8 (1&2) ◽  
pp. 68-81
Author(s):  
O.L. Acevedo ◽  
J. Roland ◽  
N.J. Cerf
Keyword(s):  
Evolution Operator ◽  
Cayley Graphs ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Constructive Method ◽  
Quantum Evolution ◽  
Necessary Condition ◽  
Internal Space ◽  
Special Cases ◽  
Scalar Quantum

A quantum walk, \emph{i.e.}, the quantum evolution of a particle on a graph, is termed \emph{scalar} if the internal space of the moving particle (often called the coin) has dimension one. Here, we study the existence of scalar quantum walks on Cayley graphs, which are built from the generators of a group. After deriving a necessary condition on these generators for the existence of a scalar quantum walk, we present a general method to express the evolution operator of the walk, assuming homogeneity of the evolution. We use this necessary condition and the subsequent constructive method to investigate the existence of scalar quantum walks on Cayley graphs of groups presented with two or three generators. In this restricted framework, we classify all groups -- in terms of relations between their generators -- that admit scalar quantum walks, and we also derive the form of the most general evolution operator. Finally, we point out some interesting special cases, and extend our study to a few examples of Cayley graphs built with more than three generators.

scholarly journals Strong Phase-Space Semiclassical Asymptotics

2011 ◽  
Vol 43 (5) ◽  
pp. 2116-2149 ◽  
Author(s):  
Agissilaos Athanassoulis ◽  
Thierry Paul
Keyword(s):  
Phase Space ◽  
Semiclassical Asymptotics ◽  
Strong Phase

THE QUADRATIC TIME-DEPENDENT HAMILTONIAN: EVOLUTION OPERATOR, SQUEEZING REGIONS IN PHASE SPACE AND TRAJECTORIES

1994 ◽  
Vol 08 (11n12) ◽  
pp. 1563-1576 ◽  
Author(s):  
S.S. MIZRAHI ◽  
M.H.Y. MOUSSA ◽  
B. BASEIA
Keyword(s):  
Phase Space ◽  
Unitary Transformation ◽  
Uniform Magnetic Field ◽  
Evolution Operator ◽  
Single Mode ◽  
Time Dependent ◽  
Special Cases ◽  
Complete Set ◽  
Hamiltonian Evolution ◽  
General Time

We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a+, which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.

QUANTIZATION OF NON-LAGRANGIAN SYSTEMS

2009 ◽  
Vol 24 (28n29) ◽  
pp. 5319-5340 ◽  
Author(s):  
DENIS KOCHAN
Keyword(s):  
Variational Principle ◽  
Open Systems ◽  
Transition Probability ◽  
Functional Integral ◽  
Quantum Evolution ◽  
Lagrangian Systems ◽  
Classical Dynamics ◽  
Standard Case ◽  
Calculus Of Variation ◽  
Novel Method

A novel method for quantization of non-Lagrangian (open) systems is proposed. It is argued that the essential object, which provides both classical and quantum evolution, is a certain canonical two-form defined in extended velocity space. In this setting classical dynamics is recovered from the stringy-type variational principle, which employs umbilical surfaces instead of histories of the system. Quantization is then accomplished in accordance with the introduced variational principle. The path integral for the transition probability amplitude (propagator) is rearranged to a surface functional integral. In the standard case of closed (Lagrangian) systems the presented method reduces to the standard Feynman's approach. The inverse problem of the calculus of variation, the problem of quantization ambiguity and the quantum mechanics in the presence of friction are analyzed in detail.

scholarly journals Time Recurrence Analysis of a Near Singular Billiard

2019 ◽  
Vol 24 (2) ◽  
pp. 50 ◽  
Author(s):  
Rodrigo Simile Baroni ◽  
Ricardo Egydio de Carvalho ◽  
Bruno Castaldi ◽  
Bruno Furlanetto
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Singular Limit ◽  
Largest Lyapunov Exponent ◽  
Classical Dynamics ◽  
Sharp Drop ◽  
Nonlinear Part ◽  
Return Times

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.

scholarly journals A PATH INTEGRAL FOR CLASSICAL DYNAMICS, ENTANGLEMENT, AND JAYNES-CUMMINGS MODEL AT THE QUANTUM-CLASSICAL DIVIDE

2011 ◽  
Vol 09 (supp01) ◽  
pp. 203-224 ◽  
Author(s):  
HANS-THOMAS ELZE ◽  
GIOVANNI GAMBAROTTA ◽  
FABIO VALLONE
Keyword(s):  
Time Evolution ◽  
Path Integral ◽  
Statistical Physics ◽  
Hamiltonian Dynamics ◽  
Path Integrals ◽  
Classical Mechanics ◽  
Quantum Evolution ◽  
Classical Dynamics ◽  
Von Neumann ◽  
Feynman Path Integrals

The Liouville equation differs from the von Neumann equation "only" by a characteristic superoperator. We demonstrate this for Hamiltonian dynamics, in general, and for the Jaynes-Cummings model, in particular. Employing superspace (instead of Hilbert space), we describe time evolution of density matrices in terms of path integrals, which are formally identical for quantum and classical mechanics. They only differ by the interaction contributing to the action. This allows us to import tools developed for Feynman path integrals, in order to deal with superoperators instead of quantum mechanical commutators in real time evolution. Perturbation theory is derived. Besides applications in classical statistical physics, the "classical path integral" and the parallel study of classical and quantum evolution indicate new aspects of (dynamically assisted) entanglement (generation). Our findings suggest to distinguish intra- from inter-space entanglement.

scholarly journals Complex Systems in Phase Space

Entropy ◽  
2020 ◽  
Vol 22 (10) ◽  
pp. 1103
Author(s):  
David K. Ferry ◽  
Mihail Nedjalkov ◽  
Josef Weinbub ◽  
Mauro Ballicchia ◽  
Ian Welland ◽  
...  
Keyword(s):  
Phase Space ◽  
Complex Systems ◽  
Transport Equation ◽  
Semiconductor Device ◽  
Quantum Effects ◽  
Classical Dynamics ◽  
Quantum Processes ◽  
Single Digit ◽  
Efficient Simulation ◽  
Quantum Phenomena

The continued reduction of semiconductor device feature sizes towards the single-digit nanometer regime involves a variety of quantum effects. Modeling quantum effects in phase space in terms of the Wigner transport equation has evolved to be a very effective approach to describe such scaled down complex systems, accounting from full quantum processes to dissipation dominated transport regimes including transients. Here, we discuss the challanges, myths, and opportunities that arise in the study of these complex systems, and particularly the advantages of using phase space notions. The development of particle-based techniques for solving the transport equation and obtaining the Wigner function has led to efficient simulation approaches that couple well to the corresponding classical dynamics. One particular advantage is the ability to clearly illuminate the entanglement that can arise in the quantum system, thus allowing the direct observation of many quantum phenomena.

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