The displacement operator for one mode in quantum optics is the shift operator
D
^
(
α
)
=
exp
(
α
a
^
†
−
α
∗
a
^
)
,
where
α
is the amount of displacement in optical phase space,
α
∗
is the complex conjugate of that displacement, and
a
^
and
a
^
†
are the lowering and raising operators, respectively. The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude
α
. It may also act on the vacuum state by displacing it into a coherent state. Specifically,
D
^
(
α
)
|
0
⟩
=
|
α
⟩
where
|
α
⟩
is a coherent state, which is the eigenstates of the annihilation (lowering) operator.
The displacement operator is a unitary operator, and therefore obeys
D
^
(
α
)
D
^
†
(
α
)
=
D
^
†
(
α
)
D
^
(
α
)
=
1
^
, where
1
^
is the identity operator. Since
D
^
†
(
α
)
=
D
^
(
−
α
)
, the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude (
−
α
). The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.
D
^
†
(
α
)
a
^
D
^
(
α
)
=
a
^
+
α
D
^
(
α
)
a
^
D
^
†
(
α
)
=
a
^
−
α
The product of two displacement operators is another displacement operator, apart from a phase factor, has the total displacement as the sum of the two individual displacements. This can be seen by utilizing the Baker-Campbell-Hausdorff formula.
e
α
a
^
†
−
α
∗
a
^
e
β
a
^
†
−
β
∗
a
^
=
e
(
α
+
β
)
a
^
†
−
(
β
∗
+
α
∗
)
a
^
e
(
α
β
∗
−
α
∗
β
)
/
2
.
which shows us that:
D
^
(
α
)
D
^
(
β
)
=
e
(
α
β
∗
−
α
∗
β
)
/
2
D
^
(
α
+
β
)
When acting on an eigenket, the phase factor
e
(
α
β
∗
−
α
∗
β
)
/
2
appears in each term of the resulting state, which makes it physically irrelevant.
Two alternative ways to express the displacement operator are:
D
^
(
α
)
=
e
−
1
2
|
α
|
2
e
+
α
a
^
†
e
−
α
∗
a
^
D
^
(
α
)
=
e
+
1
2
|
α
|
2
e
−
α
∗
a
^
e
+
α
a
^
†
The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as
A
^
ψ
†
=
∫
d
k
ψ
(
k
)
a
^
†
(
k
)
,
where
k
is the wave vector and its magnitude is related to the frequency
ω
k
according to
|
k
|
=
ω
k
/
c
. Using this definition, we can write the multimode displacement operator as
D
^
ψ
(
α
)
=
exp
(
α
A
^
ψ
†
−
α
∗
A
^
ψ
)
,
and define the multimode coherent state as
|
α
ψ
⟩
≡
D
^
ψ
(
α
)
|
0
⟩
.
