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Borrowing some description for the setup from a question I posted earlier here;

Suppose we have $N$ bosonic modes (or quantum harmonic ocsillators) with the usual commutation relations. Now define the position and momentum operators for the $k^{th}$ mode as, $$\hat{q}_k=\frac{1}{\sqrt{2}}(\hat{a}^\dagger + \hat{a}), \, \hat{p}_k=\frac{-i}{\sqrt{2}}(\hat{a}^\dagger - \hat{a}).$$ Let $\hat{\boldsymbol{X}}=(\hat{q}_1,\hat{p}_1,...,\hat{q}_N,\hat{p}_N)^T,$ then the first statistical moment is the vector of means $\langle \hat{\boldsymbol{X}} \rangle=:\boldsymbol{X}$. The second statistical moment is the covariance matrix $\sigma$ with entries $\sigma_{jk}=\frac{1}{2}\langle \hat{X}_j\hat{X}_k +\hat{X}_k\hat{X}_j\rangle -\langle \hat{X}_j\rangle \langle\hat{X}_k \rangle.$

A Gaussian state is fully characterized by its vector of means and its covariance matrix. And this in fact gives us it its Wigner function as 2N-dim Gaussian distribution.

$$ W_{\boldsymbol{X},\sigma}(\boldsymbol{Y}) =\frac{1}{(2\pi \hbar)^N}\exp\left( \frac{1}{2}(\boldsymbol{Y}-\boldsymbol{X})^T\sigma^{-1}(\boldsymbol{Y}-\boldsymbol{X})\right)\, \text{ Wigner Function for a Gaussian state defined by } \boldsymbol{X},\sigma.$$

A good starting point is the definition of the wigner function below. $$ W(\vec{q},\vec{p})=\frac{1}{(2\pi \hbar)^N}\int d^N\vec{x} \,e^{\frac{-i}{\hbar} \vec{p}\cdot\vec{x} }\langle \vec{q} +\frac{1}{2} \vec{x}|\hat{\rho}|\vec{q}-\frac{1}{2}\vec{x}\rangle$$

But I'm not quite sure this will lead to what I'm seeking.

My question is, Given the wigner function $W_{\boldsymbol{X},\sigma}(\boldsymbol{Y})$ mentioned earlier, how can I get the density matrix in the Fock state representation? Explicitly let's assume $N=1$, I want to go from

$$ W_{\boldsymbol{X},\sigma}(\boldsymbol{Y}) \longrightarrow \hat{\rho}=\sum_{m,n=0}^\infty \rho_{mn}|m\rangle\langle n|$$

So how can one obtain $\rho_{mn}$ from the Wigner function.

Any help is appreciated, thanks!

Lost In Euclids 5th Postulate
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    You neglected to reassure the reader you are completely, unhesitatingly in control of the N=1 case... Label your oscillators by k. Do you appreciate the Weyl map? – Cosmas Zachos Mar 16 '22 at 19:35
  • Do not even think of going past N=1, until then. Only then, consult this. – Cosmas Zachos Mar 16 '22 at 19:57
  • Well yes I forgot to extend the fock states to k oscillators. But I think the process for N=1 would straightforwardly translate to general N (perhaps my naive optimisim). – Lost In Euclids 5th Postulate Mar 17 '22 at 13:28
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    That's my very point: do not conflate general N with your problem. If, indeed, you can do the N=1 case in your sleep, ask about extension to general N; it is straightforward, as you suspect. But it is not good mental hygiene to throw all pots on the fire without losing track of your problem. Show what you know and have tried for N=1. This is a standard chapter/section in any and all decent textbooks and reviews on the subject. You start with the least usable formulation of the Wigner function, and then roll your eyes that it does not tell you anything about your problem, in your mind. – Cosmas Zachos Mar 17 '22 at 13:45
  • Do the right thing: understand the Weyl map, the inverse of the Wigner map, and translate it to creation/annihilation operators and coherent states. – Cosmas Zachos Mar 17 '22 at 13:47
  • @CosmasZachos I think a related problem is the proliferation of textbooks that start with the least usable form of the Wigner function... at least you can use it to show it has the correct marginals – Quantum Mechanic Mar 17 '22 at 14:58
  • @CosmasZachos I think I have a good understanding on the Wigner-Weyl correspondence. It's not thing more than a fancy way of doing Fourier transforms on a quasiprobability distribution. Knowing this and even the review article/tutorial you mentioned, does not allow one to construct the density matrix from the Wigner function. If you can provide a helpful comment or even a hint it would be more useful than this strange handwavy sass. – Lost In Euclids 5th Postulate Mar 17 '22 at 15:21
  • @Quantum Mechanic Well, our book uses it, and everything follows essentially trivially, of course; but the OP refuses to reassure us that his problem is not the translation from phase space to optical phase space, or the simple Gaussian cigar, or if he wants the Weyl map strictly within the creation/annihilation formalism, or coherent states and squeezers... So his question multiplexes to about 5-6... If he reassures us that he appreciates the 1940s basics, one could send them to sec 4.3 of Olivares, cited .... – Cosmas Zachos Mar 17 '22 at 15:23
  • @CosmasZachos My question is precisely: how does one obtain the number state/fock state representation of a density matrix from the phase space (position, momentum not the coherent representation) Wigner function?

    Sec 4.3 of your cited article does not have anything to do with what I'm asking. I'm aware of ways to construct the density matrix using the covariance matrix and the vector of means due to the correspondence between symplectic transformations and gaussian unitaries. But I'm wonder if one can get this density matrix using only the Wigner function.

     – Lost In Euclids 5th Postulate Mar 17 '22 at 15:31
  • The density matrix is the Weyl transform of the Wigner function! If you completely forfeit creation/annihilation operators, you can do it essentially trivially. But I cannot start explaining something you know and would have no use for... The Weyl transform was introduced before its inverse, the Wigner transform... – Cosmas Zachos Mar 17 '22 at 15:38
  • The way I've encountered it, the Weyl transform gives the characteristic function which is $$ \chi(\eta) = Tr(\hat{D}(\eta)\hat{\rho}) $$ In this instance how can I obtain the density matrix when its being traced over? Unless you mean something else by the Weyl map. I feel that the language I have been taught is not the same you are using, so I would appreciate a bit more explicitness. – Lost In Euclids 5th Postulate Mar 17 '22 at 15:43
  • Give me an explicit Gaussian cigar $W(q,p;a,b; s)$ in terms of the means a and b of q and p, and the widths of the respective Gaussians. Do you want the operator density matrix that this cigar is the Wigner function of, in terms of operators $\hat q, \hat p$ ? If you feel confident in translating it to Fock space, it's a cinch. – Cosmas Zachos Mar 17 '22 at 15:47
  • Ah! this is the source of the miscommunication! what you wrote down is a Weyl symbol, not an operator! The Weyl transform is the operator this symbol basically maps to... Eqn (134) of our linked booklet. – Cosmas Zachos Mar 17 '22 at 15:58
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    Okay, looking at the quoted "position representation" of the Weyl map on the Wiki page and your referenced equation 134 in your book, I now understand what you mean. Yes in this fashion I can use this, and expand the identity operator twice in the fock state representation on the right and left of the density matrix , and integrate the coefficients resulting from the overlap of the position basis states and fock states. That was helpful, many thanks! Though I will have some work to do on Mathematica to do this efficiently :) – Lost In Euclids 5th Postulate Mar 17 '22 at 16:06
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    @LostInEuclids5thPostulate a quick word of caution in this arena: sometimes Mathematica runs into challenges with phase space integrals because it doesn't recognize integrals of $\exp(i kx)$ as delta functions, so I always recommend a heavy dose of pen+paper when doing these things alongside Mathematica – Quantum Mechanic Mar 17 '22 at 16:36

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OK, I assume you addressed your question and you are on your way... Just a reminder of language to bridge the distracting communications gap.

For a "Gaussian cigar" (W Schleich's book's nomenclature, not ours!) Wigner function, $$ W(x,p;a,b;s) =\frac{\pi}{\hbar} \exp \left ( -(x-a)^2/2s^2 -2s^2 (p-b)^2/\hbar^2 \right ), $$ with $\langle x\rangle =a; \langle p\rangle =b$ and variance/squeezing of the phase-space ellipse, σ= diag($s^2,\hbar^2/s^2$), the operator density matrix which produces this Wigner function is given by the Weyl transform of it, eqn (3) of our booklet, $$ \hat \rho =\int\!\! dydxdp ~~|x+y/2\rangle ~ W(x,p;a,b;s) e^{ipy/\hbar}~\langle x-y/2| ~, $$ in the position representation.

You should plug this into your "starting point definition" to confirm that it, in fact, is but its inverse, $$ {1\over 2\pi \hbar} \int\!\! dz e^{-iuz/\hbar}\langle q+z/2|\hat \rho|q-z/2\rangle \\ = {1\over 2\pi \hbar} \int\!\! dzdydxdp ~ e^{i(py-uz)/\hbar} W(x,p)\langle q+z/2| x+y/2\rangle \langle x-y/2| q-z/2\rangle\\ ={1\over 2\pi \hbar} \int\!\! dzdydxdp ~ e^{i(py-uz)/\hbar} W(x,p) \delta(q-x)\delta(z-y) \\ ={1\over 2\pi } \int\!\! dw dp ~ e^{iw(p-u) } W(q,p) =W(q,u) ~. $$

You are probably familiar with the standard change of basis from position to energy/number eigenstates, $$\left\langle x \mid n \right\rangle = {1 \over \sqrt{2^n n!}}~ \pi^{-1/4} \exp(-x^2 / 2)~ H_n(x),$$ as here. You may gingerly plug in number basis completeness relations on either side of the above density matrix.

Potentially relevant questions here include this or that.

Cosmas Zachos
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  • Thanks for further posting the details, that's indeed exactly what I did following our exchanges in the comments. I'm now actually working on obtaining the coherent representation (in terms of a complex number alpha as the displacement argument) rather than the typical q,p representation. If you have any pointers on that without spoiling that would be good (I only got started an hour ago, I would like to work on it for a day or two first). – Lost In Euclids 5th Postulate Mar 18 '22 at 18:29
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    I'd be shocked if Ulf Leonhardt's book, or Wolfgang Schleich's don't go through this in gory detail... – Cosmas Zachos Mar 18 '22 at 18:31