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I need to apply a hermitian phase operator $\dfrac{1}{\sqrt{1+\hat{a}^\dagger\hat{a}}}\hat{a}$ to the nth harmonic oscillator state, but I have no idea how to interpret the square root of a combination of ladder operators in a denominator. How is this operator to be interpreted and applied to a state? I would also appreciate any book references or links to information on hermitian phase operators.

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