Operator representation
edit
The general form of a displaced squeezed state for a quantum harmonic oscillator is given by
|
ζ
,
α
⟩
=
D
(
α
)
|
ζ
⟩
=
D
(
α
)
S
(
ζ
)
|
0
⟩
{\displaystyle |\zeta ,\alpha \rangle =D(\alpha )|\zeta \rangle =D(\alpha )S(\zeta )|0\rangle }
where
|
0
⟩
{\displaystyle |0\rangle }
vacuum state ,
D
(
α
)
{\displaystyle D(\alpha )}
displacement operator and
S
(
ζ
)
{\displaystyle S(\zeta )}
squeeze operator , given by
D
(
α
)
=
exp
(
α
a
^
†
−
α
∗
a
^
)
and
S
(
ζ
)
=
exp
[
1
2
(
ζ
∗
a
^
2
−
ζ
a
^
†
2
)
]
{\displaystyle D(\alpha )=\exp(\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}})\qquad {\text{and}}\qquad S(\zeta )=\exp {\bigg [}{\frac {1}{2}}(\zeta ^{*}{\hat {a}}^{2}-\zeta {\hat {a}}^{\dagger 2}){\bigg ]}}
where
α
=
|
α
|
e
i
ϕ
{\displaystyle \alpha =|\alpha |e^{i\phi }}
ζ
=
r
e
i
θ
{\displaystyle \zeta =re^{i\theta }}
a
^
{\displaystyle {\hat {a}}}
a
^
†
{\displaystyle {\hat {a}}^{\dagger }}
quantum harmonic oscillator of angular frequency
ω
{\displaystyle \omega }
a
^
†
=
m
ω
2
ℏ
(
x
−
i
p
m
ω
)
and
a
^
=
m
ω
2
ℏ
(
x
+
i
p
m
ω
)
{\displaystyle {\hat {a}}^{\dagger }={\sqrt {\frac {m\omega }{2\hbar }}}\left(x-{\frac {ip}{m\omega }}\right)\qquad {\text{and}}\qquad {\hat {a}}={\sqrt {\frac {m\omega }{2\hbar }}}\left(x+{\frac {ip}{m\omega }}\right)}
For a real
ζ
{\displaystyle \zeta }
r is squeezing parameter),[clarification needed the uncertainty in
x
{\displaystyle x}
p
{\displaystyle p}
(
Δ
x
)
2
=
ℏ
2
m
ω
e
−
2
ζ
and
(
Δ
p
)
2
=
m
ℏ
ω
2
e
2
ζ
{\displaystyle (\Delta x)^{2}={\frac {\hbar }{2m\omega }}\mathrm {e} ^{-2\zeta }\qquad {\text{and}}\qquad (\Delta p)^{2}={\frac {m\hbar \omega }{2}}\mathrm {e} ^{2\zeta }}
Therefore, a squeezed coherent state saturates the Heisenberg uncertainty principle
Δ
x
Δ
p
=
ℏ
2
{\displaystyle \Delta x\Delta p={\frac {\hbar }{2}}}
Some expectation values for displaced squeezed state are
⟨
ζ
,
α
|
a
^
|
ζ
,
α
⟩
=
α
{\displaystyle \langle \zeta ,\alpha |{\hat {a}}|\zeta ,\alpha \rangle =\alpha }
⟨
ζ
,
α
|
a
^
2
|
ζ
,
α
⟩
=
α
2
−
e
i
θ
c
o
s
h
(
r
)
s
i
n
h
(
r
)
{\displaystyle \langle \zeta ,\alpha |{\hat {a}}^{2}|\zeta ,\alpha \rangle =\alpha ^{2}-e^{i\theta }cosh(r)sinh(r)}
⟨
ζ
,
α
|
a
^
†
a
^
|
ζ
,
α
⟩
=
|
α
|
2
+
s
i
n
h
2
(
r
)
{\displaystyle \langle \zeta ,\alpha |{\hat {a}}^{\dagger }{\hat {a}}|\zeta ,\alpha \rangle =|\alpha |^{2}+sinh^{2}(r)}
squeezed coherent state for a quantum harmonic oscillator is given by
|
α
,
ζ
⟩
=
S
^
(
ζ
)
|
α
⟩
=
S
^
(
ζ
)
D
^
(
α
)
|
0
⟩
{\displaystyle |\alpha ,\zeta \rangle ={\hat {S}}(\zeta )|\alpha \rangle ={\hat {S}}(\zeta ){\hat {D}}(\alpha )|0\rangle }
Some expectation values for squeezed coherent states are
⟨
α
,
ζ
|
a
^
|
α
,
ζ
⟩
=
α
c
o
s
h
(
r
)
−
α
∗
e
i
θ
s
i
n
h
(
r
)
{\displaystyle \langle \alpha ,\zeta |{\hat {a}}|\alpha ,\zeta \rangle =\alpha cosh(r)-\alpha ^{*}e^{i\theta }sinh(r)}
⟨
α
,
ζ
|
a
^
2
|
α
,
ζ
⟩
=
α
2
c
o
s
h
2
(
r
)
+
α
∗
2
e
2
i
θ
s
i
n
h
2
(
r
)
−
(
1
+
2
|
α
|
2
)
e
i
θ
c
o
s
h
(
r
)
s
i
n
h
(
r
)
{\displaystyle \langle \alpha ,\zeta |{\hat {a}}^{2}|\alpha ,\zeta \rangle =\alpha ^{2}cosh^{2}(r)+{\alpha ^{*}}^{2}e^{2i\theta }sinh^{2}(r)-(1+2{|\alpha |}^{2})e^{i\theta }cosh(r)sinh(r)}
⟨
α
,
ζ
|
a
^
†
a
^
|
α
,
ζ
⟩
=
|
α
|
2
c
o
s
h
2
(
r
)
+
(
1
+
|
α
|
2
)
s
i
n
h
2
(
r
)
−
(
α
2
e
−
i
θ
+
α
∗
2
e
i
θ
)
c
o
s
h
(
r
)
s
i
n
h
(
r
)
{\displaystyle \langle \alpha ,\zeta |{\hat {a}}^{\dagger }{\hat {a}}|\alpha ,\zeta \rangle =|\alpha |^{2}cosh^{2}(r)+(1+{|\alpha |}^{2})sinh^{2}(r)-({\alpha }^{2}e^{-i\theta }+{\alpha ^{*}}^{2}e^{i\theta })cosh(r)sinh(r)}
Since
S
^
(
ζ
)
{\displaystyle {\hat {S}}(\zeta )}
D
^
(
α
)
{\displaystyle {\hat {D}}(\alpha )}
S
^
(
ζ
)
D
^
(
α
)
≠
D
^
(
α
)
S
^
(
ζ
)
{\displaystyle {\hat {S}}(\zeta ){\hat {D}}(\alpha )\neq {\hat {D}}(\alpha ){\hat {S}}(\zeta )}
|
α
,
ζ
⟩
≠
|
ζ
,
α
⟩
{\displaystyle |\alpha ,\zeta \rangle \neq |\zeta ,\alpha \rangle }
where
D
^
(
α
)
S
^
(
ζ
)
=
S
^
(
ζ
)
S
^
†
(
ζ
)
D
^
(
α
)
S
^
(
ζ
)
=
S
^
(
ζ
)
D
^
(
γ
)
{\displaystyle {\hat {D}}(\alpha ){\hat {S}}(\zeta )={\hat {S}}(\zeta ){\hat {S}}^{\dagger }(\zeta ){\hat {D}}(\alpha ){\hat {S}}(\zeta )={\hat {S}}(\zeta ){\hat {D}}(\gamma )}
γ
=
α
cosh
r
+
α
∗
e
i
θ
sinh
r
{\displaystyle \gamma =\alpha \cosh r+\alpha ^{*}e^{i\theta }\sinh r}
[1]
^ M. M. Nieto and D. Truax (1995), Nieto, Michael Martin; Truax, D. Rodney (1997). "Holstein‐Primakoff/Bogoliubov Transformations and the Multiboson System". Fortschritte der Physik/Progress of Physics . 45 (2): 145–156. arXiv :quant-ph/9506025 doi :10.1002/prop.2190450204 . S2CID 14213781 .
γ
{\displaystyle \gamma }
θ
→
θ
+
π
{\displaystyle \theta \rightarrow \theta +\pi }
![{\displaystyle D(\alpha )=\exp(\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}})\qquad {\text{and}}\qquad S(\zeta )=\exp {\bigg [}{\frac {1}{2}}(\zeta ^{*}{\hat {a}}^{2}-\zeta {\hat {a}}^{\dagger 2}){\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38db3b55c5540db90927a7bc911cc38b9ef2a243)
