scholarly journals On the diagonalization of quadratic Hamiltonians

2021 ◽  
Author(s):  
Ville Härkönen ◽  
Ivan Gonoskov
Keyword(s):  
Unitary Transformation ◽  
Frame Of Reference ◽  
Quadratic Hamiltonians ◽  
Special Cases ◽  
Quadratic Hamiltonian ◽  
General Method

Abstract A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can establish the diagonalization of a quadratic Hamiltonian by changing the frame of reference by a unitary transformation. We give a general method to diagonalize an arbitrary quadratic Hamiltonian and derive a few of the simplest special cases in detail.

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.

scholarly journals Quantum key distribution with correlated sources

2020 ◽  
Vol 6 (37) ◽  
pp. eaaz4487 ◽  
Author(s):  
Margarida Pereira ◽  
Go Kato ◽  
Akihiro Mizutani ◽  
Marcos Curty ◽  
Kiyoshi Tamaki
Keyword(s):  
Quantum Key Distribution ◽  
Key Distribution ◽  
The Other ◽  
Correlated Sources ◽  
Information Theoretic ◽  
Security Proofs ◽  
Information Theoretic Security ◽  
Special Cases ◽  
New Framework ◽  
General Method

In theory, quantum key distribution (QKD) offers information-theoretic security. In practice, however, it does not due to the discrepancies between the assumptions used in the security proofs and the behavior of the real apparatuses. Recent years have witnessed a tremendous effort to fill the gap, but the treatment of correlations among pulses has remained a major elusive problem. Here, we close this gap by introducing a simple yet general method to prove the security of QKD with arbitrarily long-range pulse correlations. Our method is compatible with those security proofs that accommodate all the other typical device imperfections, thus paving the way toward achieving implementation security in QKD with arbitrary flawed devices. Moreover, we introduce a new framework for security proofs, which we call the reference technique. This framework includes existing security proofs as special cases, and it can be widely applied to a number of QKD protocols.

Another Look at Nonholonomic Systems

1973 ◽  
Vol 40 (1) ◽  
pp. 101-104 ◽  
Author(s):  
C. E. Passerello ◽  
R. L. Huston
Keyword(s):  
Analytical Methods ◽  
Classical Problem ◽  
Nonholonomic Systems ◽  
Constraint Forces ◽  
Advantages And Disadvantages ◽  
Arbitrary Choice ◽  
Special Cases ◽  
Dependent Variables ◽  
Automatic Elimination ◽  
General Method

The relative advantages and disadvantages of various analytical methods for nonholonomic systems are briefly presented and discussed. The techniques of Kane’s method are then used to develop a derivation of a general method which consolidates and employs the advantages of the various classical methods. These advantages include the automatic elimination of nonworking constraint forces while avoiding the computation of vector components of acceleration. The method also provides for the arbitrary choice of dependent variables so that it may be adapted to a variety of nonholonomic systems. Two special cases are considered and the method is then illustrated in the classical problem of the rolling coin.

scholarly journals A Soft Interval Based Decision Making Method and Its Computer Application

2021 ◽  
Vol 46 (3) ◽  
pp. 273-296
Author(s):  
Gözde Yaylalı ◽  
Nazan Çakmak Polat ◽  
Bekir Tanay
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Soft Set ◽  
Computer Application ◽  
Complex Data ◽  
Soft Sets ◽  
Mathematical Algorithm ◽  
Huge Data ◽  
Special Cases ◽  
General Method

Abstract In today’s society, decision making is becoming more important and complicated with increasing and complex data. Decision making by using soft set theory, herein, we firstly report the comparison of soft intervals (SI) as the generalization of interval soft sets (ISS). The results showed that SIs are more effective and more general than the ISSs, for solving decision making problems due to allowing the ranking of parameters. Tabular form of SIs were used to construct a mathematical algorithm to make a decision for problems that involves uncertainties. Since these kinds of problems have huge data, constructing new and effective methods solving these problems and transforming them into the machine learning methods is very important. An important advance of our presented method is being a more general method than the Decision-Making methods based on special situations of soft set theory. The presented method in this study can be used for all of them, while the others can only work in special cases. The structures obtained from the results of soft intervals were subjected to test with examples. The designed algorithm was written in recently used functional programing language C# and applied to the problems that have been published in earlier studies. This is a pioneering study, where this type of mathematical algorithm was converted into a code and applied successfully.

A unified approach to limit theorems for urn models

1979 ◽  
Vol 16 (01) ◽  
pp. 154-162 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Characteristic Function ◽  
Negative Binomial Distribution ◽  
Limit Theorems ◽  
Negative Binomial ◽  
Random Variable ◽  
Unified Approach ◽  
Special Cases ◽  
Occupancy Problems ◽  
General Method

An urn contains A balls of each of N colours. At random n balls are drawn in succession without replacement, with replacement or with replacement together with S new balls of the same colour. Let Xk be the number of drawn balls having colour k, k = 1, …, N. For a given function f the characteristic function of the random variable ZM = f(X 1)+ … + f(XM ), M ≦ N, is derived. A limit theorem for ZM when M, N, n → ∞is proved by a general method. The theorem covers many special cases discussed separately in the literature. As applications of the theorem limit distributions are obtained for some occupancy problems and for dispersion statistics for the binomial, Poisson and negative-binomial distribution.

A General Method for a Piezothermoelastic Plate of Crystal Class 6mm Subjected to Axisymmetric Loading

1999 ◽  
Author(s):  
Xiangqing Wang ◽  
Om Prakash Agrawal
Keyword(s):  
Cadmium Selenide ◽  
Solution Method ◽  
Piezoelectric Materials ◽  
Electrical Potential ◽  
Special Cases ◽  
Crystal Class ◽  
Piezoelectric Solids ◽  
And Control ◽  
Electrical Equilibrium ◽  
General Method

Abstract Applications of piezoelectric materials for the development of “intelligent” structural and mechanical systems through sensing, actuation, and control have received considerable recent interest. In this document, we present a general solution method for piezothermoelasticity for hexagonal piezoelectric solids of class 6 mm. In the formulation presented, potential functions are introduced to represent the coupled thermal, elastic, and mechanical fields, which satisfy the thermal, mechanical, and electrical equilibrium and prescribed boundary conditions. The formulation is similar to those presented by Ashida, Tauchert, and Noda (1993, 1994), however, it is simpler and direct, and it eliminates the need to discuss special cases. To demonstrate applications of the technique, a piezothermoelasticity problem subjected to axisymmetric thermal, electrical and mechanical loads on a plate is considered. Numerical calculations for the stresses and the electrical potential are carried out for a cadmium selenide body exhibiting class 6mm symmetry. Results of these calculations are presented graphically.

Sequences of Linear Birth, Death, and Immigration Processes with Applications to Cascades of Optical Amplifiers

1997 ◽  
Vol 11 (3) ◽  
pp. 387-394 ◽  
Author(s):  
Peter J. Smith
Keyword(s):  
Population Size ◽  
Fiber Optics ◽  
Optical Amplifiers ◽  
Fiber Amplifiers ◽  
Immigration Process ◽  
Special Cases ◽  
Erbium Doped ◽  
The Absolute ◽  
General Method ◽  
Birth Death

Consider a population having size X(t) at time t that undergoes a sequence of periods of linear birth, death, and immigration. This paper provides a general method for calculating the absolute probabilities pi, (t) = P(X(t) = i) at any stage in this sequence. In general, the different periods of birth, death, and immigration will have different parameters, although some special cases are also considered where certain parameters are common to all periods. Applications of this model can be found in fiber optics when considering a general cascade of erbium-doped fiber amplifiers. In this application, population size represents photon numbers, transmission through fibers causes attenuation (death), and amplification can be described by a linear birth, death, and immigration process.

Use of a General Dose–Response Model for Rockfish Fecundity–Length Relationships

1990 ◽  
Vol 47 (6) ◽  
pp. 1148-1156 ◽  
Author(s):  
Laura J. Richards ◽  
Jon T. Schnute
Keyword(s):  
Dose Response ◽  
Regression Models ◽  
Data Sets ◽  
Response Model ◽  
Exact Inference ◽  
Special Cases ◽  
Wide Range ◽  
Length Data ◽  
The Relationship ◽  
General Method

In this paper we describe a general method for determining the relationship between fecundity and another fish attribute, such as size or age. Our methods include linear and logarithmic regression models as special cases and are applicable to a wide range of situations. The model we propose is based on the univariate form of the Schnute–Jensen dose–response model. However, we extend the Schnute–Jensen analysis by describing exact inference regions obtained from likelihood contours, to which we assign nominal probability levels. We also provide a method for obtaining an inference band for the predicted curve. We examine the issue of model adequacy as it relates to fecundity–length data from two rockfish (Sebastes) species. We show that the extra complexity of our model is justified, as none of the traditional regression models are appropriate for all three of our data sets. Further, we use inference bands to distinguish fecundity–length relationships for quillback rockfish (S. maliger) from two areas, but we are unable to distinguish one of these relationships from a similar relationship for copper rockfish (S. caurinus).

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