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I recently read about Schrödinger cat states, which are basically a superposition of two coherent states $|\alpha\rangle$ with opposite phases, that is, $$ |\mathrm{cat}\rangle = |\alpha\rangle \pm |{-\alpha}\rangle $$ which gives us an even Shcrödinger cat states with only even parts and an odd cat state with only odd parts, \begin{align} \left| \mathrm{cat}_\mathrm{even} \right\rangle & \propto 2 e^{-\frac{|\alpha|^2}{2}} \left( \frac{\alpha^0}{\sqrt{0!}}| 0 \rangle + \frac{\alpha^2}{\sqrt{2!}}| 2 \rangle + \frac{\alpha^4}{\sqrt{4!}}| 4 \rangle + \dots \right), \quad\text{and} \\ \left| \mathrm{cat}_\mathrm{odd} \right\rangle & \propto 2 e^{- \frac{|\alpha|^2}{2}} \left( \frac{\alpha^1}{\sqrt{1!}} | 1 \rangle + \frac{\alpha^3}{\sqrt{3!}} | 3 \rangle + \frac{\alpha^5}{\sqrt{5!}} | 5 \rangle + \dots\right). \end{align} My question is, how can I visualize something like this? I've seen a few pictures of density matrices and Wigner functions, but I'm not exactly knowledgeable about those. Is there an easier way to understand this/plot it myself? I'm thinking of something similar to the probabilities for a coherent state, which is basically a Poisson distribution. Or do I explicitly need Wigner functions and such?

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The visualization method you choose is directly and completely determined by the information you need to see regarding your state. For the states of a single bosonic mode, there are multiple different visualization methods, and they all have their pros and cons. In particular, there is a direct trade-off between the amount of information you can display on a plot and the work you will need to put in to understand the plots. Simply put, plots which are easy to understand won't depict the states in sufficient detail.

Thus, you might want to plot

  • the photon-number distribution, which is completely insensitive to the phases and therefore cannot distinguish between coherent superpositions of those number states and incoherent ones;
  • the position- and momentum-space probability distributions, which are insensitive to the phase of the wavefunction in those spaces and therefore do not convey where the state is moving towards, or whether the state is pure or mixed;
  • the position- and momentum-space wavefunctions, which are only applicable to pure states and for which momentum and position encodings can be hard to unentangle;
  • the phase-space based functions (i.e. the Wigner function as well as the Sudarshan $P$ and Husimi $Q$ representations), which graphically and explicitly depict the entire phase-space dependence of the state, and which take all of about ten minutes to understand.

If you really want to properly understand a state like a coherent-state cat superposition, there really is no way to get around a phase-space-based approach, because you are specifically selecting the states you superpose based on their phase-space characteristics.

This leaves you with the Wigner function, which isn't that hard. In particular, if you integrate over vertical (horizontal) strips, the resulting marginal distributions are exactly the position-(momentum-)space probability distributions. This is fairly easy and intuitive to display graphically:

Wigner function of a cat state, with marginal distributions for position and momentum

Mathematica source code for the image in the revision history.

The variables on the plane are position (horizontal) and momentum (vertical). Each cat-state coherent-state component is a blob which rotates once around the origin over the oscillator period. In between any two such cat "eyes", you get a "smile": a fringe pattern along their separation. These fringes are precisely where the information about the superposition is encoded, and they are crucial to get the correct interference between the two components.

In particular:

  • When the two blobs are spatially separate, you integrate across the fringes, and the result is essentially zero, so there is no probability between the blobs.
  • When the two blobs overlap spatially, then they must interfere because they have different phase profiles. Here the Wigner function fringes are integrated over longitudinally, so the positive regions give you constructive interference and the negative regions account for the destructive interference.

It should be easy to see how this can account perfectly naturally for mixed states, i.e. states where the coherence of the superposition is less than perfect. The two blobs will remain - they represent the population at each $\alpha$ - but the contrast in the interference fringes will go down, which means that the amplitude of the "smile" will also decrease. Easy!

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  • $\begingroup$ I gotta say, in my book you are becoming notorious as the Physics SX user with the most profound answers. Hasn't been the first time you answered a question of mine in such detail! $\endgroup$ – John W. Jun 25 '15 at 13:26

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