Abstract

It has been previously shown [Phys. Rev. A 50, 89 (1994) [CrossRef]  ] that cavity-based electromagnetic confinement leads to an “anomalous” fields operators commutation relation that is undetectable by probing the cavity with a beam splitter. However, using this commutator in the case of parametric fluorescence (spontaneous parametric down conversion) when it occurs inside an open cavity implies a strong intensification of this process. This prediction can validate, or not, this commutation relation. The ab initio approach used is based entirely on vacuum field fluctuations and does not resort to the concept of density of states. Finally, through a generalization of creation and annihilation operators in the presence of noise, this approach raises fundamental questions about quantum modes. We expect this work to stimulate new theoretical developments and related experiments, which might lead to new applications in quantum nonlinear optics.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2005 (6)

J. C. Martinez, “Dynamics of frequency conversion of an optical pulse in a microcavity,” Phys. Rev. A 71, 015801 (2005).
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E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett. 95, 067401 (2005).
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F. Intravaia and A. Lambrecht, “Surface plasmon modes and the Casimir energy,” Phys. Rev. Lett. 94, 110404 (2005).
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2004 (3)

D. Angelakis, P. Knight, and E. Paspalakis, “Photonic crystals and inhibition of spontaneous emission: an introduction,” Contemp. Phys. 45, 303–318 (2004).
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J. P. Reithmaier, G. Sęk, A. Lffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004).
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2003 (10)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
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C. Viviescas and G. Hackenbroich, “Field quantization for open optical cavities,” Phys. Rev. A 67, 013805 (2003).
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2002 (3)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
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2001 (3)

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565–582 (2001).
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2000 (3)

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1998 (3)

E. P. Petrov, V. N. Bogomolov, I. I. Kalosha, and S. V. Gaponenko, “Spontaneous emission of organic molecules embedded in a photonic crystal,” Phys. Rev. Lett. 81, 77–80 (1998).
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A. Aiello, D. Fargion, and E. Cianci, “Parametric fluorescence and second-harmonic generation in a planar Fabry-Perot microcavity,” Phys. Rev. A 58, 2446–2459 (1998).
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1996 (1)

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1995 (1)

S. W. Koch, F. Jahnke, and W. W. Chow, “Physics of semiconductor microcavity lasers,” Semicond. Sci. Technol. 10, 739–751 (1995).
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1994 (1)

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1993 (2)

G. Rempe, “Atoms in an optical cavity: quantum electrodynamics in confined space,” Contemp. Phys. 34, 119–129 (1993).
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1992 (5)

Y. Yamamoto, S. Machida, and G. Björk, “Micro-cavity semiconductor lasers with controlled spontaneous emission,” Opt. Quantum Electron. 24, S215–S243 (1992).
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1991 (2)

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1990 (1)

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1989 (3)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
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1988 (2)

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1985 (2)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
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1984 (1)

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
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1981 (1)

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1979 (1)

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1975 (1)

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1974 (1)

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1973 (1)

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1968 (2)

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1967 (1)

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1964 (1)

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1961 (1)

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1946 (1)

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J.-M. Gérard, E. Moreau, I. Robert, I. Abram, and B. Gayral, “Les boîtes quantiques semi-conductrices : des atomes artificiels pour l’optique quantique,” C. R. Physique 3, 29–40 (2002).
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Aiello, A.

A. Aiello, G. Nienhuis, and J. P. Woerdman, “Subthreshold optical parametric oscillator with nonorthogonal polarization eigenmodes,” Phys. Rev. A 67, 043803 (2003).
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A. Aiello, “Input-output relations in optical cavities: a simple point of view,” Phys. Rev. A 62, 063813 (2000).
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A. Aiello, D. Fargion, and E. Cianci, “Parametric fluorescence and second-harmonic generation in a planar Fabry-Perot microcavity,” Phys. Rev. A 58, 2446–2459 (1998).
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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
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Figures (2)

Fig. 1.
Fig. 1. Physical model accounting for the universe as a closed cavity, which includes the open cavity of interest. The perfectly conducting walls (intensity reflectance, r , of 1 and amplitude reflectance, ρ , of 1 ), at z = d and z = L , define the universe. The space between z = d and z = 0 comprises the cavity, while the rest of the universe is between z = 0 and z = L . The semi-reflecting mirror at z = 0 is assumed lossless and characterized by ρ = r and τ = i 1 r , with r 1 for a high-finesse cavity. The subscripts “in” and “ex” refer to the open cavity and the rest of the universe, respectively. The a ^ correspond to annihilation operators for each electric field component, E .
Fig. 2.
Fig. 2. Shape of the “modulation function” Λ I versus angular frequency, ω , for various reflectance values, r . The intensifications of vacuum field fluctuations are located at the vicinity of eigenmodes, which are located at integer values of π c / d .

Equations (57)

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E p ( z , t ) = E ˜ p u p ( z ) cos ( ω p t ) := E p ( z ) cos ( ω p t ) .
ε ( z , t ) = { ε ex + δ ε d z 0 ε ex 0 z L ,
δ ε = ε o χ ( 2 ) E p ( z , t ) = ε o χ ( 2 ) E p ( z ) cos ( ω p t ) = ε o χ ( 2 ) E ˜ p cos ( ω p t ) u p ( z ) .
Λ E ( ω ) | E in ( ω ) | | E ex | .
Λ I ( ω ) I in I ex = | E in ( ω ) | 2 | E ex | 2 = Λ E 2 ( ω ) .
Λ I ( ω ) = 1 r 1 + r 2 r cos ( 2 d ω / c ) .
P fi = 2 π 1 | f | H ^ in | i | 2 ρ ( U f ) = 2 π 1 | f | Λ E H ^ ex | i | 2 ρ ( U f ) = 2 π 1 | f | H ^ ex | i | 2 Λ I ρ ( U f ) ,
F ( ω ) P fi P fi ex = Λ I ( ω ) .
Λ I max = 1 + r 1 r .
F = π r 4 1 r ,
F = 1 2 λ d Q ,
Λ I max = 1 2 π 1 + r r 4 λ d Q .
F p 1 π λ d Q .
0 N Δ ω Λ I ( ω ) d ω = N Δ ω .
[ a ^ ex + ( ω ) , a ^ ex + ( ω ) ] = [ a ^ ex ( ω ) , a ^ ex ( ω ) ] = δ ( ω ω ) ,
[ a ^ in + ( ω ) , a ^ in + ( ω ) ] = [ a ^ in ( ω ) , a ^ in ( ω ) ] = Λ ( ω ) δ ( ω ω ) .
A ^ ( z , t ) = { A ^ in ( z , t ) d z 0 A ^ ex ( z , t ) 0 z L .
A ^ in ( z , t ) = l 1 4 ε o 1 ω l { [ a ^ in l + e i ( k l z ω l t ) + h.c. ] + [ a ^ in l e i ( k l z + ω l t ) + h.c. ] }
A ^ ex ( z , t ) = l 1 4 ε o 1 ω l { [ a ^ ex l + e i ( k l z ω l t ) + h.c. ] + [ a ^ ex l e i ( k l z + ω l t ) + h.c. ] } .
tan ( k l d ) tan ( k l L ) = 1 r 1 + r .
p ^ in l ( t ) = 2 ω l d [ a ^ in l e i ( k l d ω l t ) + h.c. ] ,
q ^ in l ( t ) = i 2 d ω l [ a ^ in l e i ( k l d ω l t ) h.c. ] ,
p ^ ex l ( t ) = 2 ω l L [ a ^ ex l e i ( k l L + ω l t ) + h.c. ] ,
q ^ ex l ( t ) = i 2 L ω l [ a ^ ex l e i ( k l L + ω l t ) h.c. ] .
H ^ tot ( t ) = 1 2 l [ p ^ in l 2 ( t ) + ω l 2 q ^ in l 2 ( t ) ] + 1 2 l [ p ^ ex l 2 ( t ) + ω l 2 q ^ ex l 2 ( t ) ] { 1 2 l sinc ( 2 k l d ) [ p ^ in l 2 ( t ) ω l 2 q ^ in l 2 ( t ) ] + 1 2 l sinc ( 2 k l L ) [ p ^ ex l 2 ( t ) ω l 2 q ^ ex l 2 ( t ) ] } Correcting terms .
H ^ tot ( t ) = 1 2 l [ p ^ in l 2 ( t ) + ω l 2 q ^ in l 2 ( t ) ] + 1 2 l [ p ^ ex l 2 ( t ) + ω l 2 q ^ ex l 2 ( t ) ] .
a ^ in l = i 1 r 1 r e i 2 k l d a ^ ex l
a ^ in l + = i 1 r e i 2 k l d r a ^ ex l + .
Λ E in l i 1 r 1 r e i 2 k l d
Λ E in l + i 1 r e i 2 k l d r ,
Λ E in l Λ E in l = Λ E in l + Λ E in l + = Λ I ( ω l ) .
H ^ tot = H ^ in + H ^ ex .
H ^ tot ( t ) = 1 2 [ p ^ in 2 ( ω , t ) + ω 2 q ^ in 2 ( ω , t ) ] d ω + 1 2 [ p ^ ex 2 ( ω , t ) + ω 2 q ^ ex 2 ( ω , t ) ] d ω .
P ^ 2 ( ω , t ) p ^ in 2 ( ω , t ) + p ^ ex 2 ( ω , t )
Q ^ 2 ( ω , t ) q ^ in 2 ( ω , t ) + q ^ ex 2 ( ω , t ) .
P ^ ( ω , t ) 1 + d L Λ ( ω ) p ^ ex ( ω , t ) ,
Q ^ ( ω , t ) 1 + d L Λ ( ω ) q ^ ex ( ω , t ) .
H ^ tot ( t ) = 1 2 P ^ 2 ( ω , t ) + ω 2 Q ^ 2 ( ω , t ) d ω = 1 2 [ 1 + d L Λ ( ω ) ] [ p ^ ex 2 ( ω , t ) + ω 2 q ^ ex 2 ( ω , t ) ] d ω .
[ Q ^ ( ω , t ) , P ^ ( ω , t ) ] = i δ ( ω ω ) ,
[ Q ^ ( ω , t ) , Q ^ ( ω , t ) ] = [ P ^ ( ω , t ) , P ^ ( ω , t ) ] = 0 .
[ a ^ ex + ( ω ) , a ^ ex + ( ω ) ] = [ a ^ ex ( ω ) , a ^ ex ( ω ) ] = 1 L + Λ ( ω ) d δ ( ω ω ) ,
[ a ^ in + ( ω ) , a ^ in + ( ω ) ] = [ a ^ in ( ω ) , a ^ in ( ω ) ] = Λ ( ω ) [ a ^ ex ( ω ) , a ^ ex ( ω ) ] .
a ^ in ± | n = n + Λ | n + Λ
a ^ in ± | n = n | n Λ .
N ^ in | n = a ^ in ± a ^ in ± | n = n a ^ in ± | n Λ = n n Λ + Λ | n Λ + Λ = n | n .
n | E ^ in l + 2 | n = 1 2 ε o ω l ( n + Λ 2 ) .
P Λ + = n | E ^ in l + E ^ in l + | n n | E ^ ex l + E ^ ex l + | n .
P Λ + = n + Λ / 2 n + 1 / 2 .
H ^ int ( t ) = 1 2 κ [ a ^ s ( t ) a ^ i ( t ) e i ω p t + a ^ s ( t ) a ^ i ( t ) e i ω p t ] .
κ = 1 2 ε o ε ex χ ( 2 ) ω s ω i d 0 E p ( z ) u s ( z ) u i ( z ) d z .
N ¯ s ( t ) = N s ( 0 ) cosh 2 ( κ t ) + ( N i ( 0 ) + 1 ) sinh 2 ( κ t ) .
H ^ tot ( t ) = 1 2 d 0 ε ex | E ^ in ( z , t ) | 2 + 1 μ in | B ^ in ( z , t ) | 2 d z + 1 2 0 L ε ex | E ^ ex ( z , t ) | 2 + 1 μ ex | B ^ ex ( z , t ) | 2 d z + 1 2 ε o χ ( 2 ) E ˜ p cos ( ω p t ) d 0 u p ( z ) | E ^ in ( z , t ) | 2 d z ,
H ^ int ( t ) = cos ( ω p t ) κ [ a ^ in ( ω , t ) e i ω d c + a ^ in ( ω , t ) e i ω d c ] [ a ^ in ( ω , t ) e i ω d c + a ^ in ( ω , t ) e i ω d c ] d ω d ω ,
κ = κ ( ω , ω ) = 1 2 π ε o c χ ( 2 ) ε ex E ˜ p L ω ω d 0 u p ( z ) sin ( ω z + d c ) sin ( ω z + d c ) d z .
N ¯ s ( t ) = N s ( 0 ) cosh 2 ( Λ κ t ) + [ N i ( 0 ) + Λ ] sinh 2 ( Λ κ t ) .
H ^ in l ( t ) = ω l ( a ^ in l a ^ in l + Λ l 2 ) = ω l ( n in l + Λ l 2 ) ,
Λ l = 1 r 1 + r 2 r cos ( 2 k l / c ) .