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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 \...
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Wigner function for composite systems?

How is it possible to find the Wigner function for composite systems? In particulr, for a composite system which in the product state we know that $\psi_{AB}(x_A,x_B)=\psi_{A}(x_A)\,\psi_{B}(x_B)$. ...
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Measurements in the phase space picture of quantum mechanics

Suppose we are dealing with non relativistic quantum mechanics of point particles, that is we are in the realm of 'classic' quantum mechanics (no quantum fields ect.). In the Heisenberg picture, ...
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Proof of “non-existence” of marginals of the Husimi $Q$-function

There are many ways to consider the Husimi ($Q$) quasi-probability distribution function, e.g. as the expectation of the density operator in a coherent state or as the Weirstrass transform of the ...
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Is the Moyal-Liouville equation $\frac{\partial \rho}{\partial t}= \frac{1}{i\hbar} [H\stackrel{\star}{,}\rho]$ used in applications?

This answer by Qmechanic shows that the classical Liouville equation can be extended to quantum mechanics by the use of Moyal star products, where it takes the form $$ \frac{\partial \rho}{\partial t}~...
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Wigner function of position eigenket?

I am trying to derive the Wigner function of position eigenket $\rho = |x'\rangle \langle x'|$. One method is to use the formal expression for the Wigner function and then solve: $$ W(q,p) = \frac{1}...
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Deriving the Husimi Function of Harmonic Oscillator Eigenstates by Convolution

In phase space formulation of quantum mechanics, the so-called Husimi function can be defined as the convolution of the Wigner function by an appropriate Gaussian. There are apparently alternative ...
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What is the Wigner function of $|n\rangle\langle m|$?

I have been searching in the literature for the Wigner function of $|n \rangle \langle m|$. For $n=m$ it can be found in page 120 of Barnett and Radmore's Methods in Theoretical Quantum Optics and it ...
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Basic confusion with quantum mechanical operators

Given a classical observable, $a(x,p),$ Weyl quantization gives the correspondent QM observable as: $$\langle x | \hat{A} | \phi \rangle=\hbar^{-3}\int \int a \left(\frac{x+y}{2},p\right)\phi(y) e^{2\...
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Can the Wigner function be described using coherent states?

The Wigner function for a wave function $\Psi(\vec{r})$ is $$ W(\vec{r},\vec{k}) = \frac{1}{2\pi} \int dy e^{-i \vec{k} \cdot \vec{y}} \Psi^{*}(\vec{r}-\vec{y}/2) \Psi(\vec{r}+\vec{y}/2) . \tag{1} $$ ...
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Can states with negative Wigner function exhibit no quantumness?

I always thought that the negativity of the Wigner function is a direct manifestation of nonclassicality, but the accepted answer to the question "Are negativity of the Wigner function and quantum ...
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Groenewold's derivation of the star product (On the principles of elementary Quantum Mechanics) [closed]

$\newcommand{\dd}{{\rm d}}$ In the paper "On the principles of elementary Quantum Mechanics" am trying to get from equation EQN 4.25 to EQN 4.27. I need help on exponential identities and integration ...
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Wigner transformation of operator $X_1P_2$

Suppose I have a gaussian wigner function $W(X1, X2,p1, p1)$. For example it could be the wigner function of two modes squeezed vaccume state. I need to find the expectation value of operator $...
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Definition of symmetrically ordered operator for multi-mode case?

As I know, Wigner function is useful for evaluating the expectation value of an operator. But first you have to write it in a symmetrically ordered form. For example: $$a^\dagger a = \frac{a^\dagger ...
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Efficient computation of Wigner function from density matrix

I would like to visualize $W(X,P)=W(\alpha,\alpha^*)$ from a given density matrix $\hat{\rho}$ that has been obtained before e.g. from the master equation. I am especially interested in density ...
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Wigner function for a Lorentzian function

I am calculating the Wigner function for a Lorentzian function, which is the Fourier transform of the exp(-|x|) damping function. But I am having a problem normalizing this function, as I am getting ...
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Classical limit in deformation quantization

In deformation quantization, when dealing with the Moyal product, the classical limit is \begin{align} \lim_{\hbar\to0}\frac{1}{i\hbar}[f,g]_{\star_M}=\{f,g\}\, \end{align} which is just the Poisson ...
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Fourier transform in the complex plane

I see the following formula when reading a textbook on quantum optics: $$g(u)=\int f(\alpha)\, e^{\alpha^*u-\alpha u^*} \, \mathrm d^2\alpha,$$ $$f(\alpha)=\frac{1}{\pi^2}\int g(u)\, e^{\alpha u^*-\...
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Wigner's function in geometric quantisation

Let $\overleftarrow{a}$ and $\overrightarrow{a}$ represent the action of the operator $a$ in arguments to the left and to the right of it, respectively. Define, then, $$\star := \exp \left \{ \frac{i ...
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quantum state homodyne tomography

I'm confused as to whether quantum state homodyne tomography can be successfully performed for detector efficiency less than 1/2, in the limit of infinite number of trials. My understanding is that ...
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How would I get a Boltzmann equation in quantum field theory?

Statistical phase space distributions are related with Wigner functions defined by: $f(x,k,t) = \int \frac{d^3x'}{(2 \pi)^3}e^{ikx'}\psi^*(x+x'/2,t)\psi(x-x'/2,t)$. This definition holds only for ...
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Relation between Diffusion in Momentum and Decoherence

I have noticed in several sources (for example, eq. (3.151) of Decoherence and the Appearance of a Classical World in Quantum Theory and eq. (5.40) of Decoherence, einselection, and the quantum ...
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Is there a Schrödinger equation for phase space?

The Schrödinger equation is generally formulated in position space $$ i \hbar \frac{\partial}{\partial t}\psi(x,t) = \hat H_x \psi(x,t) = \left [ \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + ...
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Can I swap quantum mechanical ground state for some classical trajectory distribution and have it sit still after the swap?

Suppose that I have a single massive quantum mechanical particle in $d$ dimensions ($1\leq d\leq3$), under the action of a well-behaved potential $V(\mathbf r)$, and that I let it settle on the ground ...
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Energy Spectrum of Star Products

We have that a property of the transition operator defining c-equivalence (or star equivalence from equation 1 in Bertelson) is \begin{align*} T(f\star_Mg)=T(f)\star'T(g)\,, \end{align*} where $\...