Questions tagged [wigner-transform]

The Wigner transform is the bridge between Hilbert space operators to phase-space quantities (c-numbers). Use for issues relating to the Weyl correspondence (the inverse of the Wigner transform), the Wigner function (the Wigner-transform of the density matrix) and, in general, Quantum Mechanics in phase space issues, such as the *-product, the Wigner transform of the operator multiplication operation. May also use for distributions such as the Husimi.

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Wigner-Weyl transform for a function of coordinates only

I am reading this paper by Tatarskii, which serves as an introduction to the Wigner representation of quantum mechanics. There is a step in the paper involving the Weyl transform that does not seem ...
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Change of variables for momenta [closed]

http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stud_seminar/Wigner_function.pdf From the Appendix in the above PDF (page 945), below equation (A3) the following expressions are given: $$ u = ...
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Is the Wigner function a signed measure?

I have read in Wikipedia that quasiprobability distributions in phase space quantum mechanics may fail to be $\sigma$-additive, but I don't know in which sense this is true. If I have a Wigner ...
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How does the Wigner function differ from quantum distribution functions?

I understand that Wigner functions are quantum-mechanical phase-space distribution functions (quasi-distribution to be more specific). For spin-1/2 particles the Wigner function is a $4\times 4$ ...
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Prove that $f_\psi(x,p)$ is the Wigner Function of a pure state iff $H\star f_\psi= E f_\psi$

Given a pure state $|\psi\rangle$ with position wavefunction $x\mapsto\psi(x)$, define its Wigner function as $$f_\psi(x,p) = \frac{1}{2\pi} \int dy e^{-iyp} \psi(x+y/2)\psi^*(x-y/2) \equiv \frac{1}{2\...
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Wigner Function and Spin in the Classical Limit?

This is something I got curious about. Let's say I have the Wigner function for an $n$ particle system: $$W \equiv W(x_1,\dots,x_n,;p_1,\dots,p_n) $$ Now, let's say this system obeys has spin. As far ...
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79 views

Generalization of Wigner overlap formula

I want to generalize the Wigner overlap formula, $Tr( F G ) = 2 \pi \int_{-\infty}^{\infty} dq \int_{-\infty}^{\infty} dq W_F(q,p) W_G(q,p)$, where $W_F(q,p)$ and $W_G(q,p)$ are the Wigner functions ...
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How does the Weyl transform take into account which quasiprobability distribution was used?

I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation ...
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Why does the star product satisfy the “Bopp Shift relations”: $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?

In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts": $$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
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What are the Fock-state probabilities of general Gaussian states?

A general (pure) Gaussian state has the form $\newcommand{\on}[1]{\operatorname{#1}}\newcommand{\ket}[1]{\lvert #1\rangle}\ket{\alpha,\xi}\equiv D(\alpha)S(\xi)\ket{\on{vac}}$, with $\ket{\on{vac}}$ ...
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Is there a list of computed Wigner distributions of notable states?

I'm looking for a list of explicit expressions of Wigner distributions of notable functions (e.g., Fock states, Gaussian states, thermal states, etc). Is there a paper, book, or other online source ...
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Bounding derivatives of the Wigner function using phase-space tails

Suppose I have a Wigner function that falls off faster than any polynomial for all directions in phase space. That is, for all $a,b>0$, $$\lim_{|x|\to\infty} |x^a p^b W(x,p)| =0=\lim_{|p|\to\infty} ...
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107 views

Time Evolution of Wigner Function

The Wigner Function is defined as: $$W(x,p,t)=\frac{1}{2\pi\hbar}\int dy \rho(x+y/2, x-y/2, t)e^{-ipy/\hbar}\tag{1}$$ Where $\rho(x, y, t)=\langle x|\hat{\rho}|y\rangle$. I am supposed to find the ...
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What is wrong with Weyl-Wigner representation?

The Weyl-Wigner representation is a useful tool to study QM from a semiclassical, phase-space point of view. My question is simple: if this method is so close to classical mechanics, why don't we use ...
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Free evolution of a density matrix in position space

I have a density matrix $\rho$ in momentum representation at time $t=0$: \begin{equation} \langle p' |\rho(0) |p\rangle = \sum_{n=1}^{1000} p_n \Psi_n^*(p',0) \Psi_n(p,0) \end{equation} ...
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Intuitive derivation of the Husimi Q function from the Wigner function

The Wigner function is given by $$W(\alpha)=\frac{1}{\pi^2}\int \text{e}^{\alpha \beta^*-\alpha^*\beta}\text{Tr}\left(\hat \rho \hat D(\beta) \right) \text{d}^2\beta,$$ where $\hat D(\beta)=\text e^...
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Post measurement state after heterodyne measurement

I want to understand the phase space formulation of quantum mechanics better. Specifically, I am considering the following situation: A quantum state $\rho$ on two modes can be described by its ...
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Integral of Huygens principle for a quantum wave function to find the Wigner function

I'm trying to figure out a new, deterministic interpretation for quantum mechanics using the Wigner function. As a test case, I'm using a plane wave for photons hitting an infinite thin absorbing slab ...
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64 views

Wigner-Weyl transformation for particle in magnetic field

I consider particle in external magnetic field, ${\bf A}=(-yB,0,0)$ and find wave functions (may be up to normalization factors), $$\psi(x,y,z)=\sum_n\sum_s\int\frac{dp_xdp_z}{(2\pi)^2}f_s\left(eBy+...
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Wigner distribution harmonic oscillator

I am interested in the Wigner distribution of a quantum harmonic oscillator \begin{equation*} W_n(q,p) = \int_{\mathbb R} \mathrm d x e^{ipx}\;\psi_n(q-x/2)\psi_n(q+x/2) \end{equation*} with ...
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Trace in correlations to compute Wigner transform

In the derivation of Wigner-transformed quantum time correlation functions, the following identity is used (in the case of a one-dimensional particle, for simplicity): \begin{align} C(t) &\equiv \...
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Continuous variables: Probability of outcome in a quadrature measurement

In the phase-space formulation of QM over continuous variables, how can I determine the probability of obtaining a particular measurement outcome $m$ in the following setting. Given a quantum state $\...
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Density matrix and wigner function from first and second moments

Let's say I know the first and second moments of position and momentum for all times. $\langle x\rangle $, $\langle p\rangle $,$\langle x^2\rangle $, $\langle p^2\rangle $, $\langle xp\rangle $, $\...
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What is the Wigner function of a thermal state?

I am wondering how you would compute the Wigner Function of a Thermal State with average phonon number $\bar{n}_{\mathrm{th}}$. I know the result should be a Gaussian with variance in position $\...
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164 views

Obstruction in quantization. Weyl Ordering

What is an obstruction in quantization? I've found that obstructions object of the study of a mathematical theory, previously concerned with homotopy. The problem is that to explain what an ...
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186 views

Wigner-Weyl ordering in exponential

If the particle number is $\hat{a}^\dagger\hat{a}\leftrightarrow|\alpha_w|^2-1/2 $, it can be mapped on the Wigner fields by assuming symmetric ordering:$|\alpha_w|^2\leftrightarrow\hat{a}^\dagger\hat{...
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Computation of Wigner Functions

The Wigner function can be computed as the Fourier transform of the Weyl-ordered characteristic function: $$ W(\alpha) = \frac{1}{\pi^2} \int e^{\lambda^* \alpha - \lambda \alpha^*} C_W(\lambda) d^2\...
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Wigner phase space operator correspondence: how to order?

According to Gardiner-Zoller (Quantum Noise), operators acting on the density matrix can be mapped via e.g. (I'm taking Wigner space as an example, but the same holds for P and Q) $$a\rho\...
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Normalising squeezed position eigenket?

I want to find the effect of squeezing operator $S(r) = \exp \big[r(\hat{a}^2 - \hat{a}^{{\dagger}^2})\big]$ on $|q\rangle$ i.e. $S(r)|q\rangle$. I proceed as follows: $$S(r)\hat{q}|q\rangle = S(r)...
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Wigner map of the product of two operators

Does anyone know how to prove that for the product of two operators $\hat{A}\hat{B}$ the Weyl-Wigner correspondence reads $$ (AB)(x,p) = A\left (x-\frac{\hbar}{2i}\frac{\partial}{\partial p}, p+\frac{\...
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Non-commutative Fourier transform of an operator

Wigner-Weyl transform relates an operator to its distribution function in phase space through an operator Fourier transform which is said to be non-commutative. $$ \hat{\rho} \xrightarrow[non-comm]{...
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107 views

Fourier transform of cross-spectral density space matrix elements

In order to derive phase space like equation of motion (e.g. the equation of motion for the Wigner function of a single particle in one-dimension), it is an advantage to work with the Fourier ...
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How can one get the density operator from the characteristic function?

To solve analytically the master equation of two qubits interacting with a cavity mode through their environment we use the charactristic function, $$\chi (\beta)=\operatorname{tr}[\rho D(\beta)],$$ ...
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Quasiprobability for position operators at different times

Quasiprobability distribution (e.g. the Winger distribution) are usually defined in phase-space, i.e., they give the joint distribution of the momentum and the position operators $P$ and $Q$. I am ...
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154 views

Efficient numerical evaluation of Wigner function

Suppose we want to calculate the Wigner function of some state $|\Psi\rangle = \sum_{n=0}^{N_{max}} c_n|n\rangle$ ($|n\rangle$ are the eigenstates of the Harmonic oscillator) numerically. Starting ...
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How is the complex integration done for the Wigner function in coherent state representation?

$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{Tr}\left[ \hat{\rho}e^{\lambda\hat{a}^\dagger} e^{-\lambda^* \hat{a}} \right] e^{-\frac{|\lambda|^2}{2}} \, d^2\lambda....
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Negative probabilities with Wigner quasi-probability distributions

I was toying with Wigner corrections to thermodynamic equilibrium. The semiclassical correction for the position probability density to second order in $\hbar$ is: $$P(x)= \text{e}^{-\beta V(x)}\left(...
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269 views

Klein Gordon equation in the nonrelativistic and semiclassical limit in a Wigner approach

I would like to analyse the semiclassical and nonrelativistic limit of the Klein-Gordon equation, \begin{equation} \frac{1}{c^2} \partial_t^2 \phi - \Delta \phi + \frac{M^2 c^2}{\hbar^2} \phi =0. ...
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135 views

Phase space representations and coherent state representation of $\rho$, W, P and Q functions in quantum optics and related questions

What is the difference between phase space representations (with real x & p as variables) and coherent state representation (with complex $\alpha$ as a variable) of $\rho$, W, P and Q functions in ...
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104 views

Wigner Function for an Entangled Composite System

How is it possible to compute the Wigner function for a composite system that is prepared in an entangled state? In particular, consider the state $|ψ_{AB}\rangle=\frac{1}{\sqrt{2}}(|0_A\rangle|1_B\...
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108 views

Wigner flow for open quantum systems

In the paper by Friedman and Blencowe the Wigner flow for an open quantum system is derived. On page 3 the Wigner flow for the harmonic oscillator is derived. Substituting the potential $V = \frac{1}{...
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119 views

Wigner image of the product of two operators

If we know the Wigner image of $\hat{A}$ and $\hat{B}$, how do we calculate the Wigner transform of $\hat{A}\hat{B}$?
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What would be in the Kernel of a Dequantization Map?

Consider forming a symplectic map between all the Hamiltonians on Hilbert Space and all the Hamiltonians on Phase Space. (I understand that taking the Converse of the Groenewold Van-Hove Theorem this ...
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574 views

How to plot Wigner functions for a quantum state [closed]

Given some quantum state $\rho$, I would like to plot its Wigner function. Taking some examples, the Wigner function for number states, coherent states are well known, but it is not clear how one can ...
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Wigner function in quantum teleportation

I'm starting to manipulate Wigner function and I'm studying the Wigner function approach in Quantum Teleportation. My reference for this is page 29-30 of Braunstein and Van Loock "Quantum ...
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198 views

Diagram versus gradient expansion

Suppose one starts from the Dyson equation $\int dy\left[G^{-1}\left(x_{1},y\right)\cdot G\left(y,x_{2}\right)\right]=\delta\left(x_{1}-x_{2}\right)$ with $G$ some Green's function. One may usually ...
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84 views

Photon number distribution

Is there any relationship to write the "photon number distribution" in terms of "Wigner's function"? I have a wavefunction for a specific quantum system and I have calculated corresponding ground ...
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123 views

Star Product and Poisson Brackets

I have the following definition of star product, \begin{equation} \star=\exp\left[\frac{i\hbar}{2}\left(\frac{\overleftarrow{\partial}}{\partial Q^{I}}\frac{\overrightarrow{\partial}}{\partial P_{I}}-\...
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69 views

Distribution of zero-mean, independent, complex-valued, white noise terms

In this paper (open access here), in equations (13) and (14) they state that $W(\mathbf{x})$ is a zero-mean, independent, complex-valued, white noise term such that $$\overline{W(\mathbf{x})W(\...
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Solving the *-genvalue equation of a free particle

The background I want to solve the $\star$-genvalue equation $$ H(x,p) \star \psi(x,p) = E~\psi(x,p),$$ where $\star$ denotes the Moyal star product given by $$ \star \equiv \exp \left\lbrace \...