Editing Squeezed coherent state (section)

====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>.