Editing Squeezed coherent state (section)

==Classification==

===Based on the number of modes===
Squeezed states of light are broadly classified into single-mode squeezed states and two-mode squeezed states,<ref>{{cite arXiv |first=A. I. |last=Lvovsky |title=Squeezed light |year=2014 |class=quant-ph |eprint=1401.4118 }}</ref> depending on the number of [[Normal mode|modes]] of the [[electromagnetic field]] involved in the process. Recent studies have looked into multimode squeezed states showing quantum correlations among more than two modes as well.

====Single-mode squeezed states====
Single-mode squeezed states, as the name suggests, consists of a single mode of the electromagnetic field whose one quadrature has fluctuations below the shot noise level {{clarify|reason=What is the "shot noise level"? A Poisson distribution?|date=September 2016}} and the orthogonal quadrature has excess noise. Specifically, a single-mode squeezed ''vacuum'' (SMSV) state can be mathematically represented as,
:<math> |\text{SMSV}\rangle =  S(\zeta)|0\rangle  </math>
where the squeezing operator S is the same as introduced in the section on operator representations [[#Operator representation|above]]. In the photon number basis, writing <math>\zeta = r e^{i\phi}</math> this can be expanded as,
:<math> |\text{SMSV}\rangle = \frac{1}{\sqrt{\cosh r}} \sum_{n=0}^\infty (- e^{i\phi} \tanh r)^n \frac{\sqrt{(2n)!}}{2^n n!} |2n\rangle</math>
which explicitly shows that the pure SMSV consists entirely of even-photon [[Fock state]] superpositions.
Single mode squeezed states are typically generated by degenerate parametric oscillation in an optical parametric oscillator,<ref>{{cite journal |first1=L.-A. |last1=Wu |first2=M. |last2=Xiao |first3=H. J. |last3=Kimble |title=Squeezed states of light from an optical parametric oscillator |journal=J. Opt. Soc. Am. B |volume=4 |pages=1465 |year=1987 |issue=10 |doi=10.1364/JOSAB.4.001465 |bibcode=1987JOSAB...4.1465W }}</ref> or using four-wave mixing.<ref name="slusher"/>

====Two-mode squeezed states====
Two-mode squeezing involves two modes of the electromagnetic field which exhibit quantum noise reduction below the [[Shot noise|shot noise level]]{{clarify|reason=What is the "shot noise level"? A Poisson distribution?|date=September 2016}} in a linear combination of the quadratures of the two fields. For example, the field produced by a nondegenerate parametric oscillator above threshold shows squeezing in the amplitude difference quadrature. The first experimental demonstration of two-mode squeezing in optics was by Heidmann ''et al.''.<ref>{{cite journal | last1 = Heidmann | first1 = A. | last2 = Horowicz | first2 = R. | last3 = Reynaud | first3 = S. | last4 = Giacobino | first4 = E. | last5 = Fabre | first5 = C. | last6 = Camy | first6 = G. | year = 1987 | title = Observation of Quantum Noise Reduction on Twin Laser Beams | journal = Physical Review Letters | volume = 59 | issue = 22| pages = 2555–2557 | doi=10.1103/physrevlett.59.2555| pmid = 10035582 | bibcode = 1987PhRvL..59.2555H }}</ref> More recently, two-mode squeezing was generated on-chip using a four-wave mixing OPO above threshold.<ref>{{cite journal | last1 = Dutt | first1 = A. | last2 = Luke | first2 = K. | last3 = Manipatruni | first3 = S.|author4-link=Alexander Gaeta | last4 = Gaeta | first4 = A. L. | last5 = Nussenzveig | first5 = P. | last6 = Lipson | first6 = M. | year = 2015 | title = On-Chip Optical Squeezing | journal = Physical Review Applied | volume = 3 | issue = 4| page = 044005 | doi = 10.1103/physrevapplied.3.044005 | arxiv = 1309.6371 | bibcode = 2015PhRvP...3d4005D | doi-access = free }}</ref> Two-mode squeezing is often seen as a precursor to continuous-variable entanglement, and hence a demonstration of the [[Einstein-Podolsky-Rosen paradox]] in its original formulation in terms of continuous position and momentum observables.<ref>{{cite journal | last1 = Ou | first1 = Z. Y. | last2 = Pereira | first2 = S. F. | last3 = Kimble | first3 = H. J. | last4 = Peng | first4 = K. C. | year = 1992 | title = Realization of the Einstein-Podolsky-Rosen paradox for continuous variables | url = https://authors.library.caltech.edu/6493/1/OUZprl92.pdf| journal = Phys. Rev. Lett. | volume = 68 | issue = 25| pages = 3663–3666 | doi=10.1103/physrevlett.68.3663 | pmid=10045765| bibcode = 1992PhRvL..68.3663O | type = Submitted manuscript }}</ref><ref>{{cite journal | last1 = Villar | first1 = A. S. | last2 = Cruz | first2 = L. S. | last3 = Cassemiro | first3 = K. N. | last4 = Martinelli | first4 = M. | last5 = Nussenzveig | first5 = P. | s2cid = 13815567 | year = 2005 | title = Generation of Bright Two-Color Continuous Variable Entanglement | journal = Phys. Rev. Lett. | volume = 95 | issue = 24| page = 243603 | doi=10.1103/physrevlett.95.243603 | pmid=16384378| arxiv = quant-ph/0506139 | bibcode = 2005PhRvL..95x3603V }}</ref> A two-mode squeezed vacuum (TMSV) state can be mathematically represented as,
:<math> |\text{TMSV}\rangle =  S_2(\zeta)|0,0\rangle = \exp(\zeta^* \hat a \hat b - \zeta \hat a^\dagger \hat b^\dagger) |0,0\rangle </math>,

and, writing down <math>\zeta = r e^{i\phi}</math>, in the photon number basis as,<ref>{{Cite journal|last1=Schumaker|first1=Bonny L.|last2=Caves|first2=Carlton M.|date=1985-05-01|title=New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation|journal=Physical Review A|volume=31|issue=5|pages=3093–3111|doi=10.1103/PhysRevA.31.3093|pmid=9895863|bibcode=1985PhRvA..31.3093S}}</ref>

:<math> |\text{TMSV}\rangle = \frac{1}{\cosh r} \sum_{n=0}^\infty (-e^{i \phi}\tanh r)^n  |nn\rangle</math>
If the individual modes of a TMSV are considered separately (i.e., <math>|nn\rangle=|n\rangle_1 |n\rangle_2</math>), then tracing over or absorbing one of the modes leaves the remaining mode in a [[thermal state]]

:<math>\begin{align}\rho_1 &= \mathrm{Tr}_2 [| \mathrm{TMSV} \rangle \langle \mathrm{TMSV} | ]\\ &= \frac{1}{\cosh^2(r)} \sum_{n=0}^\infty \tanh^{2n}(r) 
|n \rangle \langle n|, \end{align} </math>

with an effective average number of photons <math>\widetilde{n} = \sinh^2(r)</math>.

===Based on the presence of a mean field===
Squeezed states of light can be divided into squeezed vacuum and bright squeezed light, depending on the absence or presence of a non-zero mean field (also called a carrier), respectively. An [[optical parametric oscillator]] operated below threshold produces squeezed vacuum, whereas the same OPO operated above threshold produces bright squeezed light. Bright squeezed light can be advantageous for certain quantum information processing applications as it obviates the need of sending [[local oscillator]] to provide a phase reference, whereas squeezed vacuum is considered more suitable for quantum enhanced sensing applications. The [[LIGO#Advanced LIGO|AdLIGO]] and [[GEO600]] gravitational wave detectors use squeezed vacuum to achieve enhanced sensitivity beyond the standard quantum limit.<ref>{{cite journal | last1 = Grote | first1 = H. | last2 = Danzmann | first2 = K. | last3 = Dooley | first3 = K. L. | last4 = Schnabel | first4 = R. | last5 = Slutsky | first5 = J. | last6 = Vahlbruch | first6 = H. | s2cid = 3566080 | year = 2013 | title = First Long-Term Application of Squeezed States of Light in a Gravitational-Wave Observatory | journal = Phys. Rev. Lett. | volume = 110 | issue = 18| page = 181101 | doi=10.1103/physrevlett.110.181101| pmid = 23683187 | arxiv = 1302.2188 | bibcode = 2013PhRvL.110r1101G }}</ref><ref>{{cite journal | author = The LIGO Scientific Collaboration | year = 2011 | title = A gravitational wave observatory operating beyond the quantum shot-noise limit | journal = Nature Physics | volume = 7 | issue = 12| page = 962 | arxiv = 1109.2295 | bibcode = 2011NatPh...7..962L | doi = 10.1038/nphys2083 | s2cid = 209832912 }}</ref>