All Questions
Tagged with phase-space quantum-optics
27
questions
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) ...
- 630
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 ...
- 135
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 $\...
- 141
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 ...
- 266
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 ...
user164828
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 ...
- 2,090
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
$$\rho = \int d^2\alpha \, P_\rho(\alpha) |\...
- 14.8k
2
votes
1
answer
444
views
Understanding derivation of Wigner function for the Harmonic oscillator
In the document https://www.hep.anl.gov/czachos/aaa.pdf, they derive the Wigner functions, $f_n$ for the harmonic oscillator. However, I have some problems understanding some of the steps. On page 37 ...
- 261
2
votes
1
answer
574
views
Wigner functions, symmetry
I'm trying to get more insight into quasiprobability distributions, as for example the Wigner function.
There are some Wigner functions, which are symmetric.
Symmetric:
Fock state
Thermal states
...
- 560
2
votes
0
answers
89
views
Fourier Transform of $s$-ordered Characteristic Function
In the book, "Quantum Continuous Variables (A Primer of Theoretical Methods)" by Alessio Serafini, on page 70, he defines an $s$-ordered characteristic function to be:
$$
\chi_s(\alpha)=\...
1
vote
1
answer
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$$\hat{\...
- 941
1
vote
1
answer
347
views
Wigner map of the product of two operators
Does anyone know how to prove that for the product of two operators $\hat{A}\hat{B}$ the Weyl-Wigner correspondence reads
$$
(AB)(x,p) = A\left (x-\frac{\hbar}{2i}\frac{\partial}{\partial p}, p+\frac{\...
- 793
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
$$\tilde Φ_σ(α) = \frac{1}{π σ^2} e^{-...
- 1,347
1
vote
1
answer
318
views
Why does the star product satisfy the "Bopp Shift relations": $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?
In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts":
$$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
- 14.8k
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\...
- 1,620