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I'm specifically looking for Schrödinger's Cat states involving superpositions of two, or if it's been done more, coherent states, i.e. monomodal states of the form $$|\psi\rangle=a|\alpha\rangle+b|\beta\rangle.$$ What states of this form have been produced in experiment? How even can the weights be? What regions of the $\hat{a}$-eigenvalue $\alpha$ and $\beta$ are accessible? If more than two coherent states can be superposed, how many? What phase-space geometries are possible so far?

I'm also interested in what techniques are currently used to generate these states.

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As far as I can tell, despite the recent achievements the experimental toolbox in this field is quite limited. The two main techniques are photon-subtraction from squeezed vaquum and generation from Fock states by conditional homodyne detection.

First technique is based on the fact, that an odd cat state may be expressed as: $$\left(\left|\alpha\right>-\left|-\alpha\right>\right)\propto \alpha\left|1\right>+\frac{\alpha^3}{\sqrt{6}}\left|3\right>+\ldots,$$ which for small $\alpha$ resembles squeezed vacuum state with one photon removed. Experimentally photon subtraction is realized with a low reflectivity beam splitter and single photon detection in the reflected port. Detecting a single photon in this port heralds the preparation of the desired state. This is only applicable for states with small $\alpha$ - so called "Schrödinger kittens". These kittens may be later "breeded" on a beamsplitter to increase $\alpha$.

Second technique uses homodyne detection of $p$ quadrature of a Fock state $\left|n\right>$ splitted on a 50/50 beamsplitter to conditionally prepare cats. A detection of $p\sim0$ heralds the preparation of a cat state with $\alpha=\sqrt{n}$.

The closest to arbitrary superposition preparation is described here. This is not exactly a cat state, but a superposition of a squeezed vacuum and squeezed single-photon states of the form: $$\left|\psi\right>=\cos\theta \hat{S}(r)\left|0\right>+e^{i\varphi}\sin\theta \hat{S}(r)\left|1\right>,$$ with $\hat{S}(r)$ being the squeezing operator. It is probably the most general continious-variable superposition experimentally generated so far.

I have to say that I do not specifically keep track of experiments in this field, that is just those which I heard of. As a good reference on experiemtal techniques I can recommend this review by Lvovsky and Raymer. This paper also contains a lot of references.

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