The standard process for the production of strongly squeezed states of light is degenerate optical parametric amplification (OPA) below threshold in nonlinear dielectric media such as LiNbO3 or periodically poled potassium titanyl phosphate (KTP). Here, we present a graphical description of squeezed-light generation via OPA, visualizing the interaction between the nonlinear dielectric polarization of the medium and the electromagnetic quantum field. We explicitly focus on the transfer from the field's ground state to a squeezed vacuum state and from a coherent state to a bright squeezed state by the medium's second-order nonlinearity, respectively. Our pictures illustrate the phase-dependent amplification and deamplification of quantum uncertainties and give the phase relations between all propagating electromagnetic fields as well as the internally induced dielectric polarizations. The graphical description can also be used to describe the generation of nonclassical states of light via higher-order effects of the nonlinear dielectric polarization such as four-wave mixing and the optical Kerr effect.
REFERENCES
1.
H. P.
Yuen
and J. H.
Shapiro
, “Optical communication with two-photon coherent states—Part I: Quantum-state propagation and quantum-noise reduction
,” IEEE Trans. Inf. Theory
24
(6
), 657
–668
(1978
).2.
Y.
Yamamoto
and H. A.
Haus
, “Preparation, measurement and information capacity of optical quantum states
,” Rev. Mod. Phys.
58
, 1001
–1020
(1986
).3.
B. E. A.
Saleh
and M. C.
Teich
, “Can the channel capacity of a light-wave communication system be increased by the use of photon-number–squeezed light?
,” Phys. Rev. Lett.
58
, 2656
–2659
(1987
).4.
S. L.
Braunstein
and P.
v. Loock
, “Quantum information with continuous variables
,” Rev. Mod. Phys.
77
, 513
–577
(2005
).5.
G.
Smith
, J. A.
Smolin
, and J.
Yard
, “Quantum communication with Gaussian channels of zero quantum capacity
,” Nature Photon.
5
, 624
–627
(2011
).6.
A.
Furusawa
, J. L.
Sørensen
, S. L.
Braunstein
, C. A.
Fuchs
, H. J.
Kimble
, and E. S.
Polzik
, “Unconditional quantum teleportation
,” Science
282
, 706
–709
(1998
).7.
W. P.
Bowen
, N.
Treps
, B. C.
Buchler
, R.
Schnabel
, T. C.
Ralph
, H.-A.
Bachor
, T.
Symul
, and P. K.
Lam
, “Experimental investigation of continuous-variable quantum teleportation
,” Phys. Rev. A
67
, 032302
(2003
).8.
J.
DiGuglielmo
, B.
Hage
, A.
Franzen
, J.
Fiurášek
, and R.
Schnabel
, “Experimental characterization of Gaussian quantum-communication channels
,” Phys. Rev. A
76
, 012323
(2007
).9.
A.
Einstein
, B.
Podolsky
, and N.
Rosen
, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?
,” Phys. Rev.
47
, 777
–780
(1935
).10.
Z. Y.
Ou
, S. F.
Pereira
, H. J.
Kimble
, and K. C.
Peng
, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables
,” Phys. Rev. Lett.
68
(25
), 3663
–3666
(1992
).11.
W. P.
Bowen
, R.
Schnabel
, and P. K.
Lam
, “Experimental Investigation of Criteria for Continuous Variable Entanglement
,” Phys. Rev. Lett.
90
(4
), 043601
(2003
).12.
S.
Steinlechner
, J.
Bauchrowitz
, T.
Eberle
, and R.
Schnabel
, “Strong Einstein-Podolsky-Rosen steering with unconditional entangled states
,” Phys. Rev. A
87
, 022104
(2013
).13.
R.
Schnabel
, N.
Mavalvala
, D. E.
McClelland
, and P. K.
Lam
, “Quantum metrology for gravitational wave astronomy
,” Nat. Commun.
1
(121
), 1
–10
(2010
).14.
The LIGO Scientific Collaboration
, “A gravitational wave observatory operating beyond the quantum shot-noise limit
,” Nat. Phys.
7
(12
), 962
–965
(2011
).15.
H. P.
Yuen
, “Two-photon coherent states of the radiation field
,” Phys. Rev. A
13
, 2226
–2243
(1976
).16.
D. F.
Walls
, “Squeezed states of light
,” Nature
306
, 141
–146
(1983
).17.
G.
Breitenbach
, S.
Schiller
, and J.
Mlynek
, “Measurement of the quantum states of squeezed light
,” Nature
387
, 471
–475
(1997
).18.
V. V.
Dodonov
, “ ‘Nonclassical’ states in quantum optics: A ‘squeezed’ review of the first 75 years
,” J. Opt. B Quantum Semiclassical Opt.
4
, R1
–R33
(2002
).19.
R. E.
Slusher
, L. W.
Hollberg
, B.
Yurke
, J. C.
Mertz
, and J. F.
Valley
, “Observation of squeezed states generated by four-wave mixing in an optical cavity
,” Phys. Rev. Lett.
55
(22
), 2409
–2412
(1985
).20.
S. F.
Pereira
, M.
Xiao
, H. J.
Kimble
, and J. L.
Hall
, “Generation of squeezed light by intracavity frequency doubling
,” Phys. Rev. A
38
(9
), 4931
–4934
(1988
).21.
H.
Tsuchida
, “Generation of amplitude-squeezed light at 431 nm from a singly resonant frequency doubler
,” Opt. Lett.
20
(21
), 2240
–2242
(1995
).22.
K.
Bergman
and H. A.
Haus
, “Squeezing in fibers with optical pulses
,” Opt. Lett.
16
(9
), 663
–665
(1991
).23.
K. S.
Zhang
, T.
Coudreau
, M.
Martinelli
, A.
Maître
, and C.
Fabre
, “Generation of bright squeezed light at 1.06 μm using cascaded nonlinearities in a triply resonant cw periodically-poled lithium niobate optical parametric oscillator
,” Phys. Rev. A
64
(3
), 033815
(2001
).24.
R.
Dong
, J.
Heersink
, J. F.
Corney
, P. D.
Drummond
, U. L.
Andersen
, and G.
Leuchs
, “Experimental evidence for Raman-induced limits to efficient squeezing in optical fibers
,” Opt. Lett.
33
(2
), 116
–118
(2008
).25.
H.
Vahlbruch
, M.
Mehmet
, S.
Chelkowski
, B.
Hage
, A.
Franzen
, N.
Lastzka
, S.
Goßler
, K.
Danzmann
, and R.
Schnabel
, “Observation of Squeezed Light with 10-dB Quantum-Noise Reduction
,” Phys. Rev. Lett.
100
, 033602
(2008
).26.
T.
Eberle
, S.
Steinlechner
, J.
Bauchrowitz
, V.
Händchen
, H.
Vahlbruch
, M.
Mehment
, H.
Müller-Ebhardt
, and R.
Schnabel
, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection
,” Phys. Rev. Lett.
104
(25
), 251102
(2010
)27.
L.-A.
Wu
, H. J.
Kimble
, J. L.
Hall
, and H.
Wu
, “Generation of squeezed states by parametric down conversion
,” Phys. Rev. Lett.
57
(20
), 2520
–2523
(1986
).28.
P. K.
Lam
, T. C.
Ralph
, B. C.
Buchler
, D. E.
McClelland
, H.-A.
Bachor
, and J.
Gao
, “Optimization and transfer of vacuum squeezing from an optical parametric oscillator
,” J. Opt. B
1
, 469
–474
(1999
).29.
J.
Laurat
, T.
Coudreau
, N.
Treps
, A.
Maître
, and C.
Fabre
, “Conditional preparation of a quantum state in the continuous variable regime: Generation of a sub-Poissonian state from twin beams
,” Phys. Rev. Lett.
91
(21
), 213601
(2003
).30.
S.
Feng
and O.
Pfister
, “Quantum interference of ultrastable twin optical beams
,” Phys. Rev. Lett.
92
(20
), 203601
(2004
).31.
A. S.
Villar
, L. S.
Cruz
, K. N.
Cassemiro
, M.
Martinelli
, and P.
Nussenzveig
, “Generation of bright two-color continuous variable entanglement
,” Phys. Rev. Lett.
95
, 243603
(2005
).32.
A.
Ourjoumtsev
, R.
Tualle-Brouri
, J.
Laurat
, and P.
Grangier
, “Generating optical Schrödinger kittens for quantum information processing
,” Science
312
, 83
–86
(2006
).33.
J. S.
Neergaard-Nielsen
, B. M.
Nielsen
, C.
Hettich
, K.
Mølmer
, and E. S.
Polzik
, “Generation of a superposition of Odd photon number states for quantum information networks
,” Phys. Rev. Lett.
97
, 083604
(2006
).34.
Y.
Takeno
, M.
Yukawa
, H.
Yonezawa
, and A.
Furusawa
, “Observation of –9 dB quadrature squeezing with improvement of phase stability in homodyne measurement
,” Opt. Express
15
(7
), 4321
–4327
(2007
).35.
C. K.
Hong
, Z. Y.
Ou
, and L.
Mandel
, “Measurement of subpicosecond time intervals between two photons by interference
,” Phys. Rev. Lett.
59
(18
), 2044
–2046
(1987
).36.
D.
Bouwmeester
, J.-W.
Pan
, K.
Mattle
, M.
Eibl
, H.
Weinfurter
, and A.
Zeilinger
, “Experimental quantum teleportation
,” Nature
390
, 575
–579
(1997
).37.
38.
H. A.
Haus
, Electromagnetic Noise and Quantum Optical Measurements
(Springer
, Berlin
, 2000
), pp. 387
–392
.39.
M. J.
Collett
and C. W.
Gardiner
, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification
,” Phys. Rev. A
30
(3
), 1386
–1391
(1984
).40.
41.
Bahaa E. A.
Saleh
and Malvin C.
Teich
, Fundamentals of Photonics
, 1st ed. (Wiley
, New York
, 1991
), pp. 739
–746
.42.
F. A. M.
Oliveira
and P. L.
Knight
, “Bright squeezing
,” Phys. Rev. Lett.
61
(7
), 830
–833
(1988
).© 2013 American Association of Physics Teachers.
2013
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
