- Squeezed coherent state
In

physics , a**squeezed coherent state**is any state of thequantum mechanical Hilbert space such that theuncertainty principle is saturated. That is, the product of the corresponding twooperator s takes on its minimum value::$Delta\; x\; Delta\; p\; =\; frac\{hbar\}2$

The simplest such state is the ground state $|0\; angle$ of the

quantum harmonic oscillator . The next simple class of states that satisfies this identity are the family ofcoherent state s $|alpha\; angle$.Often, the term "squeezed state" is used for any such state with $Delta\; x\; eq\; Delta\; p$. The idea behind this is that the circle denoting a coherent state in a

quadrature diagram (see below) has been "squeezed" to anellipse 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 $hbar=1$):$psi(x)\; =\; C,expleft(-frac\{(x-x\_0)^2\}\{2\; w\_0^2\}\; +\; i\; p\_0\; x\; ight)$

where $C,x\_0,w\_0,p\_0$ are constants (a normalization constant, the center of the

wavepacket , its width, and its averagemomentum ). The new feature relative to acoherent state is the free value of the width $w\_0$, which is the reason why the state is called "squeezed".The squeezed state above is an

eigenstate of a linear operator:$hat\; x\; +\; ihat\; p\; w\_0^2$

and the corresponding

eigenvalue equals $x\_0+ip\_0\; w\_0^2$. 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 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 andHeisenberg 'suncertainty relation : Diminishing the quantum noise at a specific quadrature (phase) of the wave has as a direct consequence an enhancement of the noise of thecomplementary quadrature, that is, the field at the phase shifted by $pi/2$.From the top:

*Vacuum state

*Squeezed vacuum state

*Phase-squeezed state

*arbitrary squeezed state

*Amplitude-squeezed stateAs can be seen at once, in contrast to the

coherent state the quantum noise is not independent of the phase of thelight wave anymore. A characteristic broadening and narrowing of the noise during one oscillation period can be observed. Thewave 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 phase space, quantum mechanical uncertainties can be depicted by Wigner distribution

Wigner quasi-probability distribution . 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 photon number distribution of the light wave and its phase 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-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.

thumb|400px|center|Figure 4: Measured photon number distributions for a squeezed-vacuum state."> (source: link 1)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.**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

laser s andnon-linear optics (seeoptical 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 anion in atrap ,phonon states incrystal lattice s oratom 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. [*H. Vahlbruch et al., Observation of squeezed light with 10 dB quantum noise reduction, Phys. Rev. Lett., 25. Januar 2008*]**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 wave s). Amplitude-squeezed light can improve the readout of very weak spectroscopic signals.Various squeezed coherent states, generalized to the case of many degrees of freedom, are used in various calculations in

quantum field theory , for exampleUnruh effect andHawking radiation , and generally, particle production in curved backgrounds andBogoliubov transformation ).**See also***

Quantum optics

*Nonclassical light **External links*** [

*http://gerdbreitenbach.de/gallery An introduction to quantum optics of the light field*]

* [*http://www.squeezed-light.de/ www.squeezed-light.de*]**References**# Loudon, Rodney, "The Quantum Theory of Light" (Oxford University Press, 2000), [ISBN 0-19-850177-3]

# D. F. Walls and G.J. Milburn, "Quantum Optics", Springer Berlin 1994

#C W Gardiner andPeter Zoller , "Quantum Noise", 3rd ed, Springer Berlin 2004

# D. Walls, "Squeezed states of light", Nature 306, 141 (1983)

# 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)

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