Question

I'm having some trouble with the physical implementation of a phase vs rotation in phase-space for a coherent state.Say I have a laser pulse which yields a coherent state |α⟩, I then put that light into a Mach-Zender interferometer with a 50/50 beamsplitter to the point where one of the arms adds an additional phase of ϕ, does that mean the state is

|eiϕα2√⟩
or is it
eiϕ|α2√⟩
I thought it was the first instance, but then thinking about it, I know MZIs can create destructive interference so when ϕ=π, the output state has zero amplitude, which would suggest we get |α⟩−|α⟩
and not |α⟩+|−α⟩Which leads me to my next question, if I had state |α⟩
and wanted to transform it to |−α⟩, how would I go about physically implementing that? Can it be done with a MZI?

Answer 1

Adding a phase shift to a coherent state |α⟩ in a Mach-Zehnder interferometer affects the entire state, resulting in e^iφ|α/√2⟩. For creating destructive interference, one could obtain |α⟩ - |α⟩. To transform |α⟩ into |-α⟩, a phase shifter introducing a π phase shift would be used, similar to the concept of angular momentum quantization.

Step-by-step explanation:

When a coherent laser state |α⟩ is put through a Mach-Zehnder interferometer (MZI) with a 50/50 beam splitter, and one arm induces an additional phase φ, the state alteration depends on where the phase is applied. In the MZI context, adding a phase shift affects the entire state, leading to eiφ|α/√2⟩. However, for creating destructive interference, such as when φ=π, the output can be thought of as |α⟩ - |α⟩ effectively canceling out the state because the phases oppose each other. To transform |α⟩ into |-α⟩, you typically would need something like a phase shifter to introduce a π phase shift, effectively inverting the state in phase space.

The underlying quantum mechanical concepts involve the quantization of angular momentum, as with the Zeeman effect, where the z-component of angular momentum can have discrete values and projections, Lz = miℏ/2π. In a similar fashion, phase space manipulations in quantum optics consider discrete alterations of phase which result in quantized states of the light field, and such manipulations play a crucial role in quantum communications and computing.