page_title: Optical phase space
section_title: Phase-shifting operator
hierarchical_section_title: Optical phase space / Operators on phase space / Phase-shifting operator
caption_attribution_description: English: Rotation operator on a coherent state in phase space
context_page_description: In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an optical system. For any such system, a plot of the quadratures against each other, possibly as functions of time, is called a phase diagram. If the quadratures are functions of time then the optical phase diagram can show the evolution of a quantum optical system with time.
  An optical phase diagram can give insight into the properties and behaviors of the system that might otherwise not be obvious. This can allude to qualities of the system that can be of interest to an individual studying an optical system that would be very hard to deduce otherwise. Another use for an optical phase diagram is that it shows the evolution of the state of an optical system. This can be used to determine the state of the optical system at any point in time.
context_section_description: The phase-shifting operator rotates the coherent state by an angle  in the optical phase space. This operator is given by:
   
  The important relationship
  is derived as follows:
  and solving this differential equation yields the desired result.
  Thus, using the above it becomes clear that
  ,
  or a rotation by an angle theta on the coherent state in phase space. The following illustrates this more clearly:
  (which is obtained using the fact that the phase-shifting operator is unitary
  Thus,
  is the eigenpair of
  .
  From this it is possible to see that
  which is another way of expressing the eigenpair which more clearly illustrates the effects of the phase-shifting operator on coherent states.
caption: Phase shifting operator acting on a coherent state rotating it by an angle 
    
      
        
          θ
        
      
      {\displaystyle \theta }
    
   in phase space.
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