In
quantum physics
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, qua ...
, the squeeze operator for a single mode of the electromagnetic field is
:
where the
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
inside the
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
are the
ladder operators
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
. It is a unitary operator and therefore obeys
, where
is the identity operator.
Its action on the annihilation and creation operators produces
:
The squeeze operator is ubiquitous 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 ...
and can operate on any state. For example, when acting upon the vacuum, the squeezing operator produces the squeezed vacuum state.
The squeezing operator can also act on
coherent states
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 ...
and produce
squeezed coherent states. The squeezing operator does not commute with the
displacement operator In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics,
:\hat(\alpha)=\exp \left ( \alpha \hat^\dagger - \alpha^\ast \hat \right ) ,
where \alpha is the amount of dis ...
:
:
nor does it commute with the ladder operators, so one must pay close attention to how the operators are used. There is, however, a simple braiding relation,
[ M. M. Nieto and D. Truax (1995), Eqn (15). Note that in this reference, the definition of the squeeze operator (eqn. 12) differs by a minus sign inside the exponential, therefore the expression of is modified accordingly (). ]
Application of both operators above on the vacuum produces
squeezed coherent states
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position x and momentum p of a particle, and the (dimension-less) electri ...
:
:
.
Derivation of action on creation operator
As mentioned above, the action of the squeeze operator
on the annihilation operator
can be written as
To derive this equality, let us define the (skew-Hermitian) operator
, so that
.
The left hand side of the equality is thus
. We can now make use of the
general equality which holds true for any pair of operators
and
. To compute
thus reduces to the problem of computing the repeated commutators between
and
.
As can be readily verified, we have
Using these equalities, we obtain
so that finally we get
See also
*
Squeezed coherent state
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position x and momentum p of a particle, and the (dimension-less) electri ...
References
Quantum optics
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