Cat State - in Quantum Optics

In Quantum Optics

In quantum optics, a cat state is defined as the coherent superposition of two coherent states with opposite phase:

|\mathrm{cat}_e\rangle \propto|\alpha\rangle+|{-}\alpha\rangle ,

where

|\alpha\rangle =e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{\alpha^n\over\sqrt{n!}}|n\rangle ,

and

|{-}\alpha\rangle =e^{-{|{-}\alpha|^2\over2}}\sum_{n=0}^{\infty}{({-}\alpha)^n\over\sqrt{n!}}|n\rangle ,

are coherent states defined in the number (Fock) basis. Notice that if we add the two states together, the resulting cat state only contains even Fock state terms

|\mathrm{cat}_e\rangle \propto 2e^{-{|\alpha|^2\over2}}\left({\alpha^0\over\sqrt{0!}}|0\rangle+{\alpha^2\over\sqrt{2!}}|2\rangle+{\alpha^4\over\sqrt{4!}}|4\rangle+\dots\right) .

As a result of this property, the above cat state is often refereed to as an even cat state. Alternatively, we can define an odd cat state as

|\mathrm{cat}_o\rangle \propto|\alpha\rangle-|{-}\alpha\rangle ,

which only contains odd Fock states

|\mathrm{cat}_o\rangle \propto 2e^{-{|\alpha|^2\over2}}\left({\alpha^1\over\sqrt{1!}}|1\rangle+{\alpha^3\over\sqrt{3!}}|3\rangle+{\alpha^5\over\sqrt{5!}}|5\rangle+\dots\right) .

Read more about this topic:  Cat State

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