Optical Phase Space - Quadratures

Quadratures

Operators given by

and

are called the quadratures and they represent the real and imaginary parts of the complex amplitude represented by . The commutation relation between the two quadratures can easily be calculated:

This looks very similar to the commutation relation of the position and momentum operator. Thus, it can be useful to think of and treat the quadratures as the position and momentum of the oscillator although in fact they are the "in-phase and out-of-phase components of the electric field amplitude of the spatial-temporal mode", or u, and have nothing really to do with the position or momentum of the electromagnetic oscillator (as it is hard to define what is meant by position and momentum for an electromagnetic oscillator).