Mathematical definition Squeezed coherent state

animated position-wavefunction of 2db amplitude-squeezed coherent state of α=3.

the general wave function satisfies identity above squeezed coherent state (we work in units



ℏ
=
1


{\displaystyle \hbar =1}

)

ψ
(
x
)
=
c

exp
⁡

(
−



(
x
−

x

0



)

2




2

w

0


2





+
i

p

0


x
)



{\displaystyle \psi (x)=c\,\exp \left(-{\frac {(x-x_{0})^{2}}{2w_{0}^{2}}}+ip_{0}x\right)}

where



c
,

x

0


,

w

0


,

p

0




{\displaystyle c,x_{0},w_{0},p_{0}}

constants (a normalization constant, center of wavepacket, width, , expectation value of momentum). new feature relative coherent state free value of width




w

0




{\displaystyle w_{0}}

, reason why state called squeezed .

the squeezed state above eigenstate of linear operator

x
^



+
i



p
^




w

0


2




{\displaystyle {\hat {x}}+i{\hat {p}}w_{0}^{2}}

and corresponding eigenvalue equals




x

0


+
i

p

0



w

0


2




{\displaystyle x_{0}+ip_{0}w_{0}^{2}}

. in sense, generalization of ground state coherent state.