In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a squeezed coherent state is a quantum state that is usually described by two
non-commuting observables
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physi ...
having continuous spectra of
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
. Examples are position
and momentum
of a particle, and the (dimension-less) electric field in the amplitude
(phase 0) and in the mode
(phase 90°) of a light wave (the wave's
quadratures). The product of the standard deviations of two such
operators obeys the
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
:
:
and
, respectively.

Trivial examples, which are in fact not squeezed, are the ground state
of the
and the family of
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 ...
s
. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with
in "natural oscillator units" and
. (In literature different normalizations for the quadrature amplitudes are used. Here we use the normalization for which the sum of the ground state variances of the quadrature amplitudes directly provide the zero point quantum number
).
The term squeezed state is actually used for states with a standard deviation below that of the ground state for one of the operators or for a linear combination of the two. The idea behind this is that the circle denoting the uncertainty of a coherent state in the
quadrature phase space (see right) has been "squeezed" to an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
of the same area. Note that a squeezed state does not need to saturate the uncertainty principle.
Squeezed states of light
In quantum physics, light is in a '' squeezed state'' if its electric field strength ''Ԑ'' for some phases \vartheta has a quantum uncertainty smaller than that of a coherent state. The term ''squeezing'' thus refers to a reduced quantum un ...
were first produced in the mid 1980s.
[R. E. Slusher et al., ''Observation of squeezed states generated by four wave mixing in an optical cavity'', Phys. Rev. Lett. 55 (22), 2409 (1985)
] At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i.e.
. As of 2017, squeeze factors larger than 10 (10 dB) have been directly observed.
Mathematical definition

The most general
wave function that satisfies the identity above is the squeezed coherent state (we work in units with
)
:
where
are constants (a normalization constant, the center of the
wavepacket
In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diffe ...
, its width, and the expectation value of its
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
). The new feature relative to 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 ...
is the free value of the width
, which is the reason why the state is called "squeezed".
The squeezed state above 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 a linear operator
:
and the corresponding
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
equals
. In this sense, it is a generalization of the ground state as well as the coherent state.
Operator representation
The general form of a squeezed coherent state for a quantum harmonic oscillator is given by
:
where
is the
vacuum state,
is the
displacement operator and
is the
squeeze operator In quantum physics, the squeeze operator for a single mode of the electromagnetic field is
:\hat(z) = \exp \left ( (z^* \hat^2 - z \hat^) \right ) , \qquad z = r \, e^
where the operators inside the exponential are the ladder operators. It is a ...
, given by
: