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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67 views

Vanishing of terms in Wigner transform of operators

I have been given the following proof for the fact that the Moyal bracket is the Wigner-Weyl transform of the quantum commutator. However, I am unsure about the vanishing of some boundary terms in an ...
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43 views

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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14 views

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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59 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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44 views

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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28 views

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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28 views

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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33 views

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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111 views

Wigner function of 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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121 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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148 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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60 views

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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50 views

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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50 views

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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83 views

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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113 views

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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104 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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159 views

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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36 views

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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109 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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100 views

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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180 views

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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200 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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117 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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96 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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100 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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107 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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77 views

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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459 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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94 views

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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193 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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79 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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117 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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57 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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224 views

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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99 views

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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174 views

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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153 views

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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1answer
206 views

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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228 views

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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819 views

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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106 views

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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277 views

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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176 views

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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1answer
142 views

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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208 views

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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508 views

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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2answers
471 views

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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297 views

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 ...