In the
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
study of
optical phase space
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 displacement operator for one mode is the
shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift o ...
in
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have b ...
,
:
,
where
is the amount of displacement in
optical phase space
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 ...
,
is the complex conjugate of that displacement, and
and
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
. It may also act on the vacuum state by displacing it into a
coherent state
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
. Specifically,
where
is a
coherent state
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
, which is an
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
of the annihilation (lowering) operator.
Properties
The displacement operator is a
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
, and therefore obeys
,
where
is the identity operator. Since
, the
hermitian conjugate
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
wher ...
of the displacement operator can also be interpreted as a displacement of opposite magnitude (
). The effect of applying this operator in a
similarity transformation of the ladder operators results in their displacement.
:
:
The product of two displacement operators is another displacement operator whose total displacement, up to a phase factor, is the sum of the two individual displacements. This can be seen by utilizing the
Baker–Campbell–Hausdorff formula.
:
which shows us that:
:
When acting on an eigenket, the phase factor
appears in each term of the resulting state, which makes it physically irrelevant.
[Christopher Gerry and Peter Knight: ''Introductory Quantum Optics''. Cambridge (England): Cambridge UP, 2005.]
It further leads to the braiding relation
:
Alternative expressions
The Kermack-McCrae identity gives two alternative ways to express the displacement operator:
:
:
Multimode displacement
The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as
:
,
where
is the wave vector and its magnitude is related to the frequency
according to
. Using this definition, we can write the multimode displacement operator as
:
,
and define the multimode coherent state as
:
.
See also
*
Optical phase space
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 ...
References
{{Physics operators
Quantum optics