scholarly journals Toolbox for non-classical state calculations

2021 ◽  
Author(s):  
Filippus Stefanus Roux
Keyword(s):  
Generating Functions ◽  
Photon Statistics ◽  
Wigner Functions ◽  
Ladder Operators ◽  
Fock States ◽  
Classical State ◽  
Experimental Implementation ◽  
Photon Subtraction ◽  
Definition Of ◽  
Thermal States

Abstract Computational challenges associated with the use of Wigner functions to identify non-classical properties of states are addressed with the aid of generating functions. It allows the computation of the Wigner functions of photon-subtracted states for an arbitrary number of subtracted photons. Both the formal definition of photon-subtracted states in terms of ladder operators and the experimental implementation with heralded photon detections are analyzed. These techniques are demonstrated by considering photon subtraction from squeezed thermal states as well as squeezed Fock states. Generating functions are also used for the photon statistics of these states. These techniques reveal various aspects of the parameter dependences of these states.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

scholarly journals Gaussian and Non-Gaussian operations on non-Gaussian state: engineering non-Gaussianity

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Stefano Olivares ◽  
Alessia Allevi ◽  
Maria Bondani
Keyword(s):  
Coherent State ◽  
Direct Detection ◽  
The State ◽  
Gaussian State ◽  
Classical State ◽  
Phase Sensitive ◽  
Photon Subtraction ◽  
Non Gaussian

AbstractMultiple photon subtraction applied to a displaced phase-averaged coherent state, which is a non-Gaussian classical state, produces conditional states with a non trivial (positive) Glauber-Sudarshan Prepresentation. We theoretically and experimentally demonstrate that, despite its simplicity, this class of conditional states cannot be fully characterized by direct detection of photon numbers. In particular, the non-Gaussianity of the state is a characteristics that must be assessed by phase-sensitive measurements. We also show that the non-Gaussianity of conditional states can be manipulated by choosing suitable conditioning values and composition of phase-averaged states.

scholarly journals On Multiple Interpolation Functions of the Nörlund-Typeq-Euler Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Mehmet Acikgoz ◽  
Yilmaz Simsek
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Multiple Interpolation ◽  
Definition Of ◽  
Interpolation Functions

In (2006) and (2009), Kim defined new generating functions of the Genocchi, Nörlund-typeq-Euler polynomials and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz typeq-zeta function. This function interpolates Nörlund-typeq-Euler polynomials at negative integers. We also give some identities related to these polynomials and functions. Furthermore, we give some remarks about approximations of Bernoulli and Euler polynomials.

scholarly journals CANONICAL TRANSFORMATIONS IN THREE-DIMENSIONAL PHASE-SPACE

2009 ◽  
Vol 24 (25n26) ◽  
pp. 4769-4788 ◽  
Author(s):  
TEKİN DERELİ ◽  
ADNAN TEĞMEN ◽  
TUĞRUL HAKİOĞLU
Keyword(s):  
Phase Space ◽  
Canonical Transformation ◽  
Generating Functions ◽  
General Framework ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Canonical Transformations ◽  
Dimensional Phase Space ◽  
Nambu Bracket ◽  
Definition Of

Canonical transformation in a three-dimensional phase-space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed based on canonoid transformations. It is shown that generating functions, transformed Hamilton functions and the transformation itself for given generating functions can be determined by solving Pfaffian differential equations corresponding to that quantities. Types of the generating functions are introduced and all of them are listed. Infinitesimal canonical transformations are also discussed. Finally, we show that the decomposition of canonical transformations is also possible in three-dimensional phase space as in the usual two-dimensional one.

scholarly journals Some results on generalised voigt functions

2002 ◽  
Vol 44 (2) ◽  
pp. 299-303 ◽  
Author(s):  
K. C. Gupta ◽  
S. P. Goyal ◽  
Rohit Mukherjee
Keyword(s):  
Generating Functions ◽  
Special Cases ◽  
Neutron Reactions ◽  
Definition Of

Recently, Srivastava, Pathan and Kamarujjama established several results for generalised Voigt functions which play an important role in several diverse fields of physics—such as astrophysical spectroscopy and the theory of neutron reactions. In the present paper we aim to generalise some partly bilateral and partly unilateral representations and generating functions of Srivastava et al. by considering a specialised version of the Srivastava-Chen definition of the unified Voigt functions. Several special cases of our main results are mentioned briefly.

TIME EVOLUTION OF THE NONCLASSICALITY FOR THE FOCK STATES AND THEIR SUPERPOSITION STATES BY THE WIGNER FUNCTION

2011 ◽  
Vol 25 (16) ◽  
pp. 1401-1415 ◽  
Author(s):  
KUN SI ◽  
NING-LI ZHU ◽  
HUAN-YU JIA
Keyword(s):  
Time Evolution ◽  
Wigner Function ◽  
Gain Coefficient ◽  
Wigner Functions ◽  
Fock States ◽  
Negative Region ◽  
Cavity Radiation ◽  
Superposition States

The Wigner functions of the Fock states and their superposition states have negative value clearly. We focus on discussing the time evolution of the Fock states and their superposition states by the Wigner function under the coherent drive (pumping laser) and dissipation channels (cavity radiation). The results show that the negative region of their Wigner function gradually diminishes as the time kt or the gain coefficient g increasing. In addition, the disappearing time is related to the value of g - k. The loss of non-classical features becomes slower when g increases with the k > g; there is the "frozen zone" when k = g; then the loss of non-classical features becomes faster when g increases with k < g.

scholarly journals Induction Models on $\mathbb{N}$

10.29007/kvp3 ◽  
2020 ◽  
Author(s):  
A. Dileep ◽  
Kuldeep S. Meel ◽  
Ammar F. Sabili
Keyword(s):  
Computer Science ◽  
Generating Function ◽  
Generating Functions ◽  
Mathematical Induction ◽  
Base Case ◽  
And Mathematics ◽  
Definition Of ◽  
Induction Model

Mathematical induction is a fundamental tool in computer science and mathematics. Henkin [12] initiated the study of formalization of mathematical induction restricted to the setting when the base case B is set to singleton set containing 0 and a unary generating function S. The usage of mathematical induction often involves wider set of base cases and k−ary generating functions with different structural restrictions. While subsequent studies have shown several Induction Models to be equivalent, there does not exist precise logical characterization of reduction and equivalence among different Induction Models. In this paper, we generalize the definition of Induction Model and demonstrate existence and construction of S for given B and vice versa. We then provide a formal characterization of the reduction among different Induction Models that can allow proofs in one Induction Models to be expressed as proofs in another Induction Models. The notion of reduction allows us to capture equivalence among Induction Models.

Durfee squares in compositions

2018 ◽  
Vol 28 (6) ◽  
pp. 359-367 ◽  
Author(s):  
Margaret Archibald ◽  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Asymptotic Analysis ◽  
Generating Functions ◽  
Integer Partitions ◽  
Definition Of ◽  
Durfee Square ◽  
Durfee Squares

Abstract We study compositions (ordered partitions) of n. More particularly, our focus is on the bargraph representation of compositions which include or avoid squares of size s × s. We also extend the definition of a Durfee square (studied in integer partitions) to be the largest square which lies on the base of the bargraph representation of a composition (i.e., is ‘grounded’). Via generating functions and asymptotic analysis, we consider compositions of n whose Durfee squares are of size less than s × s. This is followed by a section on the total and average number of grounded s × s squares. We then count the number of Durfee squares in compositions of n.

Patterns in Repeated Trials

1962 ◽  
Vol 88 (3) ◽  
pp. 360-366 ◽  
Author(s):  
M. T. L. Bizley
Keyword(s):  
General Theory ◽  
Generating Functions ◽  
Recurrent Events ◽  
General Pattern ◽  
Recurrent Event ◽  
Expected Number ◽  
Definition Of ◽  
Regular Patterns

In a series of trials with constant probability p of ‘success’ (S) and q ( = I–p) of ‘failure’ (F), the problems arise of determining the expected number of trials required to obtain a specified pattern of results (e.g. SSFSFFSSSFF), and of calculating the probability that such a pattern will appear in a given number of trials. We are here concerned essentially with ‘general’ patterns which may exhibit no clear regularity; many methods are available, and well known, for dealing with regular patterns (e.g. SSSSSS or SFSFSF), which do not apply to irregular ones. Feller, has shown how to solve the problems for general patterns, using an ingenious definition of the event corresponding to the appearance of the pattern whereby it becomes a recurrent event; his powerful general theory of recurrent events then applies and yields the required information by means of generating functions. A different method is given on page 171 of Bizley for finding the expected number of trials required to obtain a general pattern; this uses only the simplest mathematical tools but involves rather a lot of work for long patterns. Under each of these methods, however, every given pattern has to be treated individually and a separate calculation performed.

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