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Husimi Q Representation
The Husimi Q representation, introduced by Kôdi Husimi in 1940, is a quasiprobability distribution commonly used in quantum mechanics to represent the phase space distribution of a quantum state such as light in the phase space formulation. It is used in the field of quantum optics and particularly for tomographic purposes. It is also applied in the study of quantum effects in superconductors. Definition and properties The Husimi Q distribution (called Q-function in the context of quantum optics) is one of the simplest distributions of quasiprobability in phase space. It is constructed in such a way that observables written in ''anti''-normal order follow the optical equivalence theorem. This means that it is essentially the density matrix put into normal order. This makes it relatively easy to calculate compared to other quasiprobability distributions through the formula : Q(\alpha)=\frac\langle\alpha, \hat, \alpha\rangle, which is effectively a trace of the densit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kôdi Husimi
Kōji Husimi (June 29, 1909 – May 8, 2008, ja, 伏見康治) was a Japanese theoretical physicist who served as the president of the Science Council of Japan.. Husimi trees in graph theory, the Husimi Q representation in quantum mechanics, and Husimi's theorem in the mathematics of paper folding are named after him. Education and career Husimi studied at the University of Tokyo, graduating in 1933. He spent a year there as an assistant, and then moved to Osaka University in 1934, where he soon began working with Seishi Kikuchi. At Osaka, he became Dean of the Faculty of Science. He moved to Nagoya University in 1961, and directed the plasma institute there. He retired in 1973, and became a professor emeritus of both Nagoya and Osaka. Contributions Physics A 1940 paper by Husimi introduced the Husimi Q representation in quantum mechanics. Husimi also gave the name to the kagome lattice, frequently used in statistical mechanics. Graph theory In the mathematical area of graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent States
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems.J.R. Klauder and B. Skagerstam, ''Coherent States'', World Scientific, Singapore, 1985. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g. Schiff's textbook). The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be relate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonclassical Light
Nonclassical light is light that cannot be described using classical electromagnetism; its characteristics are described by the quantized electromagnetic field and quantum mechanics. The most common described forms of nonclassical light are the following: *Photon statistics of Nonclassical Light is Sub-PoissonianM. Fox, ''Quantum Optics: An Introduction'', Oxford University Press, New York, 2006 in the sense that the average number of photons in a photodetection of this kind of light shows a standard deviation that is less than the mean number of the photons. *Squeezed light exhibits reduced noise in one quadrature component. The most familiar kinds of squeezed light have either reduced amplitude noise or reduced phase noise, with increased noise of the other component. * Fock states (also called photon number states) have a well-defined number of photons (stored e.g. in a cavity), while the phase is totally undefined. Glauber–Sudarshan P representation The density matrix for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiprobability Distribution
A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, ''the ability to yield expectation values with respect to the weights of the distribution''. However, they can violate the ''σ''-additivity axiom: integrating over them does not necessarily yield probabilities of mutually exclusive states. Indeed, quasiprobability distributions also have regions of negative probability density, counterintuitively, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time-frequency analysis, and elsewhere. Introduction In the most general form, the dynamics of a quantum-mechanical system are determined by a master equation in Hilbert space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Axioms
The Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem. Axioms The assumptions as to setting up the axioms can be summarised as follows: Let (\Omega, F, P) be a measure space with P(E) being the probability of some event E'','' and P(\Omega) = 1. Then (\Omega, F, P) is a probability space, with sample space \Omega, event space F and probability measure P. First axiom The probability of an event is a non-negative real number: :P(E)\in\mathbb, P(E)\geq 0 \qquad \forall E \in F where F is the event space. It follows that P(E) is always finite, in contrast with more general measure theory. Theories which assign negative probability relax the first axiom. Second axiom This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Probability Distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s). In the formal mathematical setup of measure theory, the joint distribution is given by the pushforward measure, by the map obtained by pairing together the given random variables, of the sample space's probability measure. In the case of real-valued random variables, the joint distribution, as a particular multivariate distribution, may be expressed by a multivariate cumulativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segal–Bargmann Space
In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions ''F'' in ''n'' complex variables satisfying the square-integrability condition: :\, F\, ^2 := \pi^ \int_ , F(z), ^2 \exp(-, z, ^2)\,dz < \infty, where here ''dz'' denotes the 2''n''-dimensional Lebesgue measure on It is a with respect to the associated inner product: : The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see and . Basic information about the material in this section may be found in and . Segal worked from the beginning in the infinite-dimensional setting; see and Sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution Theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms. Functions of a continuous variable Consider two functions g(x) and h(x) with Fourier transforms G and H: \begin G(f) &\triangleq \mathcal\(f) = \int_^g(x) e^ \, dx, \quad f \in \mathbb\\ H(f) &\triangleq \mathcal\(f) = \int_^h(x) e^ \, dx, \quad f \in \mathbb \end where \mathcal denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically 2\pi or \sqrt) will appear in the convolution theorem below. The convolution of g and h is defined by: r(x) = \(x) \triangleq \int_^ g(\t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Filter
In electronics and signal processing mainly in digital signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would have infinite impulse response). Gaussian filters have the properties of having no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay. A Gaussian filter will have the best combination of suppression of high frequencies while also minimizing spatial spread, being the critical point of the uncertainty principle. These properties are important in areas such as oscilloscopes and digital telecommunication systems. Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform. Definition The one-dimensional Gaussian filter has an impulse res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner Quasiprobability Distribution
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction . Thus, it maps on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927, in a context related to representation theory in mathematics (see Weyl quantization). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Transform
In mathematics, the Weierstrass transform of a function , named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of , weighted with a Gaussian centered at ''x''. Specifically, it is the function defined by :F(x)=\frac\int_^\infty f(y) \; e^ \; dy = \frac\int_^\infty f(x-y) \; e^ \; dy~, the convolution of with the Gaussian function :\frac e^~. The factor 1/√(4 π) is chosen so that the Gaussian will have a total integral of 1, with the consequence that constant functions are not changed by the Weierstrass transform. Instead of one also writes . Note that need not exist for every real number , when the defining integral fails to converge. The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
