All Questions
Tagged with phase-space quantum-optics
27
questions
1
vote
0
answers
29
views
Can we get quasiprobability distributions other than $P,Q,W$ from generalised characteristic functions?
It's a standard result that the three well-known quasiprobability distributions can all be expressed in terms of the "$s$-ordered characteristic functions" as
$$
W(\alpha) = \int\frac{d^2\...
0
votes
0
answers
31
views
Unitary evolution of composite system in phase space
Given a quantum state $\rho$ in a Hilbert space $\mathcal H_S$, we can always write it in terms of the displacement operator $D_\alpha$ using the characteristic function $\chi_\rho(\alpha)=\text{Tr}[\...
0
votes
0
answers
62
views
The Lorentz-non-covariance of the Wigner Function
What does the fact that the Wigner function is not Lorentz-covariant imply?
My analysis so far led me to the (probably naive) understanding that there really is nothing special about it, just that it ...
2
votes
0
answers
89
views
Fourier Transform of $s$-ordered Characteristic Function
In the book, "Quantum Continuous Variables (A Primer of Theoretical Methods)" by Alessio Serafini, on page 70, he defines an $s$-ordered characteristic function to be:
$$
\chi_s(\alpha)=\...
1
vote
1
answer
81
views
What is the Weyl transform of narrow Gaussians and/or the Dirac delta?
Consider the family of Gaussians in $q$, $p$ with decreasing widths $σ$
$$Φ_σ(q,p) = \frac{2}{π σ^2} e^{-\frac{2}{σ^2}(q^2+p^2)}$$
or in complex plane coordinates
$$\tilde Φ_σ(α) = \frac{1}{π σ^2} e^{-...
1
vote
1
answer
217
views
What is the most general wave function of a minimum uncertainty (Gaussian) state in quantum mechanics?
For some state $|\psi\rangle$ it is possible to recover the uncertainty principle using the fact that $$\left|(\hat{\sigma_{Q}}-i\lambda\hat{\sigma_{P}})|\psi\rangle\right|^{2}\geq0,$$where$$\hat{\...
0
votes
0
answers
56
views
Husimi $Q$-function of Infinite Square well
Eigen-Wavefunction of infinite square well is
$$\psi(x)=\sqrt{2/l}\sin(n\pi x/l).$$ I want to write Husimi $Q$ function for infinite square well. General expression of Q function is $$Q=(1/2)\pi \...
5
votes
1
answer
449
views
How can the Wigner function of squeezed states be non-negative?
It is always said that when the Wigner function of quantum states takes a negative value, then it is a clear signature of non-classicality of this particular state. It is also well-known that the ...
3
votes
2
answers
561
views
What does it mean for $P$ functions to be "more singular than a delta"?
Consider the Glauber-Sudarshan $P$ representation of a state $\rho$, which is the function $\mathbb C\ni\alpha\mapsto P_\rho(\alpha)\in\mathbb R$ such that
$$\rho = \int d^2\alpha \, P_\rho(\alpha) |\...
2
votes
1
answer
444
views
Understanding derivation of Wigner function for the Harmonic oscillator
In the document https://www.hep.anl.gov/czachos/aaa.pdf, they derive the Wigner functions, $f_n$ for the harmonic oscillator. However, I have some problems understanding some of the steps. On page 37 ...
0
votes
1
answer
351
views
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
$$f_\psi(x,p) = \frac{1}{2\pi} \int dy e^{-iyp} \psi(x+y/2)\psi^*(x-y/2)
\equiv \frac{1}{2\...
0
votes
1
answer
273
views
How does the Weyl transform take into account which quasiprobability distribution was used?
I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation ...
1
vote
1
answer
318
views
Why does the star product satisfy the "Bopp Shift relations": $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?
In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts":
$$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
1
vote
0
answers
509
views
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 $\ket{\on{vac}}$ ...
1
vote
0
answers
522
views
$P$ representation of a general Gaussian state
Let $\rho$ be the density operator of a Gaussian quantum state on $M$ modes. This implies that its Wigner function can be written as
$$ W_{\text{Gaussian}}\left(\boldsymbol{q},\boldsymbol{p}\right)=\...
0
votes
1
answer
79
views
Phase space formulation: "Representation" vs "function" vs "quasi-probability distribution"
In the phase space formulation, the terms "representation", "function, and "quasi-probability distribution" (as in Glauber–Sudarshan P representation, $P$-function) seem to be used interchangeably.
I ...
5
votes
2
answers
2k
views
What is the Wigner function of a thermal state?
I am wondering how you would compute the Wigner Function of a Thermal State with
average phonon number $\bar{n}_{\mathrm{th}}$.
I know the result should be a Gaussian with variance in position $\...
1
vote
1
answer
124
views
Wigner phase space operator correspondence: how to order?
According to Gardiner-Zoller (Quantum Noise), operators acting on the density matrix can be mapped via e.g. (I'm taking Wigner space as an example, but the same holds for P and Q)
$$a\rho\...
1
vote
1
answer
347
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
$$
(AB)(x,p) = A\left (x-\frac{\hbar}{2i}\frac{\partial}{\partial p}, p+\frac{\...
1
vote
1
answer
163
views
Fourier transform of cross-spectral density space matrix elements
In order to derive phase space like equation of motion (e.g. the equation of motion for the Wigner function of a single particle in one-dimension), it is an advantage to work with the Fourier ...
0
votes
1
answer
476
views
Plotting quadrature uncertainties in phase space
In most books like in the picture given below, the uncertainties regarding quantum states like coherent and squeezed states are represented in phase space plot by some area enclosed within a circle or ...
4
votes
1
answer
916
views
Why exactly is the Husimi-Q distribution not a real probability distribution?
From this question I understood that the uncertainty principle is causing a problem because two points $x,p$ and $x',p'$ in phase space can be confused. Why exactly is this a problem? I don't grasp ...
1
vote
1
answer
281
views
Pegg-Barnett phase implementation does not seem to work
I attempt to monitor the phase of a wavevector $|\psi\rangle$.
As I found (e.g. here ), a matrix representation for the Pegg-Barnett phase operator in Fock base can be obtained as
$$\Phi=\sum_{m,n,...
5
votes
1
answer
596
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 ...
15
votes
1
answer
3k
views
Understanding the relationship between Phase Space Distributions (Wigner vs Glauber-Sudarshan P vs Husimi Q)
I am moving into a new field and after thorough literature research need help appreciating what is out there.
In the continuos variable formulation of optical state space.
(Quantum mechanical/Optical) ...
2
votes
1
answer
574
views
Wigner functions, symmetry
I'm trying to get more insight into quasiprobability distributions, as for example the Wigner function.
There are some Wigner functions, which are symmetric.
Symmetric:
Fock state
Thermal states
...
4
votes
1
answer
497
views
Are the Wigner and Husimi transforms injective?
I am wondering if the Wigner function is injective. By injective I mean, that, for every density matrix $\rho$, there is a different Wigner distribution. The same question applies to the Husimi ...