Editing Squeezed coherent state (section)

{{Short description|Type of quantum state}}
{{Use American English|date = April 2019}}
{{Use mdy dates|date = April 2019}}

In [[physics]], a '''squeezed coherent state''' is a quantum state that is usually described by two [[Commutator|non-commuting]] [[Observable|observables]] having continuous spectra of [[Eigenvalues and eigenvectors|eigenvalues]]. Examples are position <math>x</math> and momentum <math>p</math> of a particle, and the (dimension-less) electric field in the amplitude <math>X</math> (phase 0) and in the mode <math>Y</math> (phase 90°) of a light wave (the wave's [[Optical phase space|quadratures]]).  The product of the standard deviations of two such [[Operator (physics)|operator]]s obeys the [[uncertainty principle]]:

:<math>\Delta x \Delta p \geq \frac{\hbar}2\;</math>     and    <math>\;\Delta X \Delta Y \geq \frac{1}4</math> ,  respectively.

[[File:Wignerfunction squeezed 0.50.png|thumb|[[Wigner quasiprobability distribution|Wigner]] phase space distribution of a squeezed state of light with ζ=0.5.]]

Trivial examples, which are in fact not squeezed, are the ground state <math>|0\rangle</math> of the [[quantum harmonic oscillator]] and the family of [[coherent state]]s  <math>|\alpha\rangle</math>. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with <math>\Delta x_g = \Delta p_g</math> in "natural oscillator units" and <math>\Delta X_g = \Delta Y_g = 1/2</math>.{{NoteTag|note=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 <math>\Delta^2 X_g + \Delta^2 Y_g = 1/2</math>}}

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]] of the same area.<ref>
Loudon, Rodney, ''The Quantum Theory of Light'' (Oxford University Press, 2000), {{ISBN|0-19-850177-3}}
</ref><ref>
[[C W Gardiner]] and [[Peter Zoller]], "Quantum Noise", 3rd ed, Springer Berlin 2004
</ref><ref>{{Cite journal|last=Walls|first=D. F.|s2cid=4325386|date=November 1983|title=Squeezed states of light|journal=Nature|language=En|volume=306|issue=5939|pages=141–146|doi=10.1038/306141a0|issn=1476-4687|bibcode=1983Natur.306..141W}}</ref> Note that a squeezed state does not need to saturate the uncertainty principle.

[[Squeezed states of light]] were first produced in the mid 1980s.<ref name="slusher">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)
</ref><ref>{{Cite journal|last=Wu|first=Ling-An|date=1986|title=Generation of Squeezed States by Parametric Down Conversion|journal=Physical Review Letters|volume=57|issue=20|pages=2520–2523|doi=10.1103/physrevlett.57.2520|pmid=10033788|bibcode=1986PhRvL..57.2520W|url=https://authors.library.caltech.edu/6028/1/WULprl86.pdf|type=Submitted manuscript|doi-access=free}}</ref> At that time, quantum noise squeezing by up to a factor of about 2 (3&nbsp;dB) in variance was achieved, i.e. <math>\Delta^2 X \approx \Delta^2 X_g/2</math>. As of 2017, squeeze factors larger than 10 (10&nbsp;dB) have been directly observed.<ref>{{Cite journal|last1=Vahlbruch|first1=Henning|last2=Mehmet|first2=Moritz|last3=Chelkowski|first3=Simon|last4=Hage|first4=Boris|last5=Franzen|first5=Alexander|last6=Lastzka|first6=Nico|last7=Goßler|first7=Stefan|last8=Danzmann|first8=Karsten|last9=Schnabel|first9=Roman|s2cid=3938634|date=2008-01-23|title=Observation of Squeezed Light with 10-dB Quantum-Noise Reduction|journal=Physical Review Letters|volume=100|issue=3|pages=033602|doi=10.1103/PhysRevLett.100.033602|pmid=18232978|arxiv=0706.1431|bibcode=2008PhRvL.100c3602V|hdl=11858/00-001M-0000-0013-623A-0}}</ref><ref name=":0" /><ref>{{Cite journal|last=Schnabel|first=Roman|s2cid=119098759|title=Squeezed states of light and their applications in laser interferometers|journal=Physics Reports|volume=684|pages=1–51|doi=10.1016/j.physrep.2017.04.001|arxiv=1611.03986|bibcode=2017PhR...684....1S|year=2017}}</ref>