In physics, a squeezed coherent state is any state of the quantum mechanical Hilbert space such that the uncertainty principle is saturated. That is, the product of the corresponding two operators takes on its minimum value: This article needs additional references or sources for verification. ... Fig. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...

The simplest such state is the ground state of the quantum harmonic oscillator. The next simple class of states that satisfies this identity are the family of coherent states . The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... In 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. ...

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 quadrature diagram (see below) has been "squeezed" to an ellipse of the same area. Look up quadrature in Wiktionary, the free dictionary. ... For other uses, see Ellipse (disambiguation). ...

Mathematical definition

The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with ) A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ...

where C,x₀,w₀,p₀ are constants (a normalization constant, the center of the wavepacket, its width, and its average momentum). The new feature relative to a coherent state is the free value of the width w₀, which is the reason why the state is called "squeezed". In quantum mechanics, for particles in a region of space where no forces act, the Schrödinger equation admits solutions known as wave packets. ... In classical mechanics, momentum (pl. ... In 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 squeezed state above is an eigenstate of a linear operator In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

and the corresponding eigenvalue equals . In this sense, it is a generalization of the ground state as well as the coherent state. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

Examples of squeezed coherent states

Depending on at which phase the state's quantum noise 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 Heisenbergs 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 π / 2. Shot noise consists of random fluctuations of the electric current in an electrical conductor, which are caused by the fact that the current is carried by discrete charges (electrons). ... Werner Heisenberg Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ... The word complement (with an e in the second syllable, not to be confused with a different word, compliment with an i) has a number of uses. ...

Figure 1: Measured quantum noise of the electric field of different squeezed states in dependence of the phase of the light field. For the first two states a 3π-interval is shown, for the last three states, belonging to a different set of measurements it is a 4π-interval. (source: link 1 and ref. 3)

Figure 2: Oscillating wave packets of the five states.

Figure 3: Wigner functions of the five states. The ripples are due to experimental inaccuracies.

From the top: Image File history File links Download high resolution version (423x1053, 72 KB) Please see the file description page for further information. ... Image File history File links Download high resolution version (423x1053, 72 KB) Please see the file description page for further information. ... Image File history File links Download high resolution version (1085x3420, 298 KB) Please see the file description page for further information. ... Image File history File links Download high resolution version (1085x3420, 298 KB) Please see the file description page for further information. ... Image File history File links Download high resolution version (932x1325, 371 KB) Please see the file description page for further information. ... Image File history File links Download high resolution version (932x1325, 371 KB) Please see the file description page for further information. ... The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. ...

• 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 state 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 packet 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 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. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye, or in a more general sense, any electromagnetic radiation in the range from infrared to ultraviolet. ... The wave packet is one of the most widely misunderstood and misused concepts in physics. ...

In phase space quantum mechanical uncertainties can be depicted by Wigner distributions. 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. The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. ...

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 photon number distribution of the light wave and its phase distribution as well.

Figure 4: Measured photon number distributions for an amplitude-squeezed state, a coherent state, and a phase squeezed state. Bars refer to theory, dots to experimental values. (source: link 1 and ref. 2)

Figure 5: Pegg-Barnett phase distribution of the three states.

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-poissonian 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. Image File history File links Download high resolution version (619x946, 60 KB) Please see the file description page for further information. ... Image File history File links Download high resolution version (619x946, 60 KB) Please see the file description page for further information. ... Image File history File links Please see the file description page for further information. ... Image File history File links Please see the file description page for further information. ...

Figure 4: Measured photon number distributions for a squeezed-vacuum state. (source: link 1 and ref. 3)

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 generation and annihilation processes. Photons in a squeezed vacuum state are more likely to appear in pairs. Image File history File links Please see the file description page for further information. ... Image File history File links Please see the file description page for further information. ... The squeeze operator for a single mode is where the operators inside the exponential are the ladder operators. ...

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 lasers and non-linear optics (see optical parametric oscillator). 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 ion in a trap, phonon states in crystal lattices or atom ensembles. Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states. Experiment with a laser (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: Light Amplification by Stimulated Emission of Radiation. ... Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. ... An optical parametric oscillator (OPO) converts a input laser wave (called pump) into two output waves of lower frequency () by means of nonlinear optical interaction. ... This article is about the electrically charged particle. ... A trap is a device or tactic intended to harm, capture, detect, or inconvenience an intruder. ... Normal modes of vibration progression through a crystal. ... In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle still characterizing a chemical element. ...

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 measurements (see for example gravitational waves). Amplitude-squeezed light can improve the read out of very weak spectroscopic signals. It has been suggested that Optical interferometry be merged into this article or section. ... For the concept in fluid dynamics and meteorology, see Gravity wave. ... Extremely high resolution spectrogram of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter. ...

Various squeezed coherent states, generalized to the case of many degrees of freedom, are used in various calculations in quantum field theory, for example Unruh effect and Hawking radiation (generally: particle production in curved backgrounds and Bogoliubov transformation). Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... Quantum field theory (QFT) is the quantum theory of fields. ... The Unruh effect, discovered 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, that is, the accelerating observer will find themselves in a warm background. ... In physics, Hawking radiation (also known as Bekenstein-Hawking radiation) is a thermal radiation thought to be emitted by black holes due to quantum effects. ... In 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 isomorphism of the CCR/CAR algebra. ...

See also

• Quantum optics
• Nonclassical light

Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ... Nonclassical light refers to states of light that can not be described using classical electromagnetism; its characteristics are described by the quantised electromagnetic field and quantum mechanics. ...

External links

• An introduction to quantum optics of the light field

References

1. Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), [ISBN 0-19-850177-3]
2. D.F. Walls and G.J. Milburn, Quantum Optics, Springer Berlin 1994
3. G. Breitenbach, S. Schiller, and J. Mlynek, "Measurement of the quantum states of squeezed light", Nature, 387, 471 (1997)