title: Non-linear coherent states
text: # Non-linear coherent states
  
  Coherent states are quasi-classical states that may be defined in different ways, for instance as eigenstates of the annihilation operator
  
      $a|\alpha \rangle =\alpha |\alpha \rangle$,
  
  or as a displacement from the vacuum
  
      $|\alpha \rangle =D(\alpha )|0\rangle$,
  
  where $D(\alpha )=\exp(\alpha a^{\dagger }-\alpha ^{*}a)$ is the Sudarshan-Glauber displacement operator.
  
  One may think of a non-linear coherent state by generalizing the annihilation operator:
  
      $A=af(a^{\dagger }a)$,
  
  and then using any of the above definitions by exchanging $a$ by $A$ . The above definition is also known as an $f$-deformed annihilation operator.
  
  