In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, a
squeezed coherent state is any state of the quantum mechanical
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
such that the
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known...
is saturated. That is, the product of the corresponding two
operatorIn mathematics, an operator is a type of function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the...
s takes on its minimum value:
The simplest such state is the ground state of the
quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point...
. The next simple class of states that satisfies this identity are the family of
coherent stateIn quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system...
s .
Often, the term
squeezed state is used for any such state with .
In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, a
squeezed coherent state is any state of the quantum mechanical
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
such that the
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known...
is saturated. That is, the product of the corresponding two
operatorIn mathematics, an operator is a type of function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the...
s takes on its minimum value:
The simplest such state is the ground state of the
quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point...
. The next simple class of states that satisfies this identity are the family of
coherent stateIn quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system...
s .
Often, the term
squeezed state is used for any such state with . The idea behind this is that the circle denoting a coherent state in a
quadratureQuadrature, derived from Latin quadrare, may refer to:In signal processing:*Quadrature amplitude modulation , a modulation method of using both a carrier wave and a 'quadrature' carrier wave that is 90° out of phase with the main carrier...
diagram (see below) has been "squeezed" to an
ellipseIn mathematics, an ellipse is the bounded case of a conic section, the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane...
of the same area.
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, its width, and its average
momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
). The new feature relative to a
coherent stateIn quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system...
is the free value of the width , which is the reason why the state is called "squeezed".
The squeezed state above is an eigenstate of a linear operator
and the corresponding eigenvalue equals . In this sense, it is a generalization of the ground state as well as the coherent state.
Examples of squeezed coherent states
Depending on at which phase the state's
quantum noiseQuantum noise is uncertainty of some physical quantity due to its quantum origin.In the case of number of particles , the quantum noise is also called shot noise. Most optical communications use amplitude modulation...
is reduced, one can distinguish amplitude-squeezed and phase-squeezed states or general quadrature squeezed states. If no coherent excitation exists the state is called a squeezed vacuum. The figures below give a nice visual demonstration of the close connection between squeezed states and Heisenberg's uncertainty relation: Diminishing the quantum noise at a specific quadrature (phase) of the wave has as a direct consequence an enhancement of the noise of the complementary quadrature, that is, the field at the phase shifted by .
From the top:
- Vacuum state
- Squeezed vacuum state
- Phase-squeezed state
- arbitrary squeezed state
- Amplitude-squeezed state
As can be seen at once, in contrast to the
coherent stateIn quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system...
the quantum noise is not independent of the phase of the light wave anymore. A characteristic broadening and narrowing of the noise during one oscillation period can be observed. The
wave packetIn physics, a wave packet is a short "burst" or "envelope" of 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 different wavenumbers, with phases and amplitudes such that they interfere...
of a squeezed state is defined by the square of the wave function introduced in the last paragraph. They correspond to the probability distribution of the electric field strength of the light wave. The moving wave packets display an oscillatory motion combined with the widening and narrowing of their distribution: the "breathing" of the wave packet. For an amplitude-squeezed state, the most narrow distribution of the wave packet is reached at the field maximum, resulting in an amplitude that is defined more precisely than the one of a coherent state. For a phase-squeezed state, the most narrow distribution is reached at field zero, resulting in an average phase value that is better defined than the one of a coherent state.
In phase space, quantum mechanical uncertainties can be depicted by Wigner distribution
Wigner quasi-probability distributionThe Wigner quasi-probability distribution is a special type of quasi-probability distribution. It was introduced by Eugene Wigner in 1932 to study quantum...
. The intensity of the light wave, its coherent excitation, is given by the displacement of the Wigner distribution from the origin. A change in the phase of the squeezed quadrature results in a rotation of the distribution.
Photon number distributions and phase distributions of squeezed states
The squeezing angle, that is the phase with minimum quantum noise, has a large influence on the
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
number distribution of the light wave and its
phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
distribution as well.
For amplitude squeezed light the photon number distribution is usually narrower than the one of a coherent state of the same amplitude resulting in
sub-PoissonianIn probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event...
light, whereas its phase distribution is wider. The opposite is true for the phase-squeezed light, which displays a large intensity (photon number) noise but a narrow phase distribution.
For the squeezed vacuum state the photon number distribution displays odd-even-oscillations. This can be explained by the mathematical form of the squeezing operator, that resembles the operator for
two-photon generationSpontaneous parametric down-conversion is an important process in quantum optics. A nonlinear crystal splits incoming photons into pairs of photons of lower energy whose combined energy and momentum are equal to the energy and momentum of the original photon...
and annihilation processes. Photons in a squeezed vacuum state are more likely to appear in pairs.
Experimental realizations of squeezed coherent states
There has been a whole variety of successful demonstrations of squeezed states. The most prominent ones were experiments with light fields using
laserA laser is a device that emits light through a process called stimulated emission. Laser light is usually spatially coherent, which means that the light either is emitted in a narrow, low-divergence beam, or can be converted into one with the help of optical components such as lenses...
s and non-linear optics (see
optical parametric oscillatorAn optical parametric oscillator is a parametric oscillator which oscillates at optical frequencies. It converts an input laser wave into two output waves of lower frequency by means of nonlinear optical interaction. The sum of the output waves frequencies is equal to the input wave frequency:...
). This is achieved by a simple process of four-wave mixing with a Chi-3 crystal. Squeezed states have also been realized via motional states of an
ionAn ion is an atom or molecule where the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge...
in a
trapA trap is a device or tactic intended to catch an intruder or an enemy.Trap may also refer to:*Bat and trap, an old game related to cricket*Flood basalt or trap basalt, a type of rock formation*Tartrate resistant acid phosphatase , in chemics...
,
phononIn physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's...
states in crystal lattices or
atomThe atom is a basic unit of matter consisting of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
ensembles. Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states. Current state of the art in noise suppression, for laser radiation using squeezed light, amounts to 10 dB.
Applications
Squeezed states of the light field can be used to enhance precision measurements. For example phase-squeezed light can improve the phase read out of
interferometric measurementsInterferometry is the technique of diagnosing the properties of two or more waves by studying the pattern of interference created by their superposition. The instrument used to interfere the waves together is called an interferometer...
(see for example
gravitational waveIn physics, a gravitational wave is a fluctuation in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted by Einstein's theory of general relativity, the waves transport energy known as gravitational radiation...
s). Amplitude-squeezed light can improve the readout of very weak
spectroscopic signalsSpectroscopy was originally the study of the interaction between radiation and matter as a function of wavelength . In fact, historically, spectroscopy referred to the use of visible light dispersed according to its wavelength, e.g. by a prism. Later the concept was expanded greatly to comprise...
.
Various squeezed coherent states, generalized to the case of many
degrees of freedomDegrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters...
, are used in various calculations in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, for example
Unruh effectThe Unruh effect, described in 1976 by Bill Unruh of the University of British Columbia, is the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none...
and
Hawking radiationHawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects...
, and generally, particle production in curved backgrounds and
Bogoliubov transformationIn theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an...
).
External links