Abstract

A dissipative beam-splitter (BS) has been analyzed by modeling the losses in the BS due to the excitation of optical phonons. The losses are obtained in terms of the BS medium properties. The model simplifies the picture by treating the loss mechanism as a perturbation on the photon modes in a linear, non-lossy medium in the limit of small losses, instead of using the full field quantization in lossy, dispersive media. The model uses second order perturbation in the Markoff approximation and yields the Beer’s law for absorption in the first approximation, thus providing a microscopic description of the absorption coefficient. It is shown that the fluctuations in the modes get increased because of the losses. We show the existence of quantum interferences due to phase correlations between the input beams and it is shown that these correlations can result in loss quenching. Hence in spite of having such a dissipative medium, it is possible to design a lossless 50–50 BS at normal incidence which may have potential applications in laser optics and dielectric-coated mirrors.

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References

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  1. For a review see the tutorial by M. C. Teich and B. A. E. Saleh, "Squeezed states of light", Quantum. Opt. 1, 151(1989).
    [CrossRef]
  2. Also see the special issue of J. Mod. Opt. 34 (1987).
  3. Also see the special issue of J. Opt. Soc. Am. B 4(1987).
  4. S. Prasad, M. O. Scully and W. Martienssen, "A quantum description of the beam-splitter", Opt. Commun., 62, 139 (1987).
    [CrossRef]
  5. B. Yurke, S. L. McCall and J. R. Klauder, "SU(2) and SU(1,1) interferometers", Phys. Rev. A 33, 4033 (1986).
    [CrossRef] [PubMed]
  6. R. A. Campos, B. E. A. Saleh and M. C. Teich, "Quantumum mechanical lossless beam splitter: SU(2) symmetry and photon statistics", Phys. Rev. A 40, 1371 (1989).
    [CrossRef] [PubMed]
  7. B. Huttner and Y. Ben-Aryeh, "In uence of a beam splitter on photon statistics", Phys. Rev. A 38, 204 (1988).
    [CrossRef] [PubMed]
  8. J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen and M. O. Scully, "A beam splitting experiment with correlated photons", Europhys. Lett., 5, 223 (1988).
    [CrossRef]
  9. C. K. Hong, Z. Y. Ou and L. Mandel, "Measurement of sub-picosecond time intervals between two photons by interference", Phys. Rev. Lett., 59, 2044 (1987).
    [CrossRef] [PubMed]
  10. M. Dakna, T. Anhut, T. Opatrny, L. Knoll and D.-G. Welsh, "Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter", Phys. Rev. A 55, 3184 (1997).
    [CrossRef]
  11. J. R. Jeers, N. Imoto and R. Loudon, "Quantumum optics of traveling wave attenuators and ampliers", Phys. Rev. A 47, 3346 (1993).
    [CrossRef] [PubMed]
  12. U. Leonhardt, " Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space", Phys. Rev. A 48, 3265 (1993).
    [CrossRef] [PubMed]
  13. S.-T. Ho and P. Kumar, "Quantumum optics in a dielectric: macroscopic electromagnetic field and medium operators for a linear dispersive lossy medium - a microsopic derivation of the operators and their commutation relations", J. Opt. Soc. Am. B 10, 1620 (1993).
    [CrossRef]
  14. B. Huttner and S. M. Barnett, "Dispersion and loss in a Hopeld dielectric", Europhys. Lett. 18, 487 (1992).
    [CrossRef]
  15. B. Huttner and S. M. Barnett, "Quantumization of the electromagnetic field in dielectrics", Phys. Rev. A 46, 4306 (1992).
    [CrossRef] [PubMed]
  16. C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems : quantum stochastic dierential equation and the master equation", Phys. Rev. A 31, 3761 (1985).
    [CrossRef] [PubMed]
  17. U. Leonhardt, "In uence of a dispersive and dissipative medium on spectral squeezing", J. Mod. Opt. 42, 1165 (1995).
  18. R. Matloob and R. Loudon, "Electromagnetic field quantization in absorbing dielectrics", Phys. Rev. A 52, 4823 (1995).
    [CrossRef] [PubMed]
  19. T. Gruner and D.-G. Welsch, "Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics", Phys. Rev. A 54, 1661 (1996).
    [CrossRef] [PubMed]
  20. Y. Aharanov, D. Falko, E. Lerner and H. Pendleton, "A quantum characterization of classical radiation", Ann. Phys. 39, 498 (1966).
    [CrossRef]
  21. N. W. Ashcroft and N. D. Mermin, Solid State Physics, International ed., (Saunders College, Philadelphia, 1976), Appendix-L .
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  23. W. H. Louisell, Quantumum Statistical Properties of Radiation, (John Wiley and Sons, NY, 1973).

Other (23)

For a review see the tutorial by M. C. Teich and B. A. E. Saleh, "Squeezed states of light", Quantum. Opt. 1, 151(1989).
[CrossRef]

Also see the special issue of J. Mod. Opt. 34 (1987).

Also see the special issue of J. Opt. Soc. Am. B 4(1987).

S. Prasad, M. O. Scully and W. Martienssen, "A quantum description of the beam-splitter", Opt. Commun., 62, 139 (1987).
[CrossRef]

B. Yurke, S. L. McCall and J. R. Klauder, "SU(2) and SU(1,1) interferometers", Phys. Rev. A 33, 4033 (1986).
[CrossRef] [PubMed]

R. A. Campos, B. E. A. Saleh and M. C. Teich, "Quantumum mechanical lossless beam splitter: SU(2) symmetry and photon statistics", Phys. Rev. A 40, 1371 (1989).
[CrossRef] [PubMed]

B. Huttner and Y. Ben-Aryeh, "In uence of a beam splitter on photon statistics", Phys. Rev. A 38, 204 (1988).
[CrossRef] [PubMed]

J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen and M. O. Scully, "A beam splitting experiment with correlated photons", Europhys. Lett., 5, 223 (1988).
[CrossRef]

C. K. Hong, Z. Y. Ou and L. Mandel, "Measurement of sub-picosecond time intervals between two photons by interference", Phys. Rev. Lett., 59, 2044 (1987).
[CrossRef] [PubMed]

M. Dakna, T. Anhut, T. Opatrny, L. Knoll and D.-G. Welsh, "Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter", Phys. Rev. A 55, 3184 (1997).
[CrossRef]

J. R. Jeers, N. Imoto and R. Loudon, "Quantumum optics of traveling wave attenuators and ampliers", Phys. Rev. A 47, 3346 (1993).
[CrossRef] [PubMed]

U. Leonhardt, " Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space", Phys. Rev. A 48, 3265 (1993).
[CrossRef] [PubMed]

S.-T. Ho and P. Kumar, "Quantumum optics in a dielectric: macroscopic electromagnetic field and medium operators for a linear dispersive lossy medium - a microsopic derivation of the operators and their commutation relations", J. Opt. Soc. Am. B 10, 1620 (1993).
[CrossRef]

B. Huttner and S. M. Barnett, "Dispersion and loss in a Hopeld dielectric", Europhys. Lett. 18, 487 (1992).
[CrossRef]

B. Huttner and S. M. Barnett, "Quantumization of the electromagnetic field in dielectrics", Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems : quantum stochastic dierential equation and the master equation", Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

U. Leonhardt, "In uence of a dispersive and dissipative medium on spectral squeezing", J. Mod. Opt. 42, 1165 (1995).

R. Matloob and R. Loudon, "Electromagnetic field quantization in absorbing dielectrics", Phys. Rev. A 52, 4823 (1995).
[CrossRef] [PubMed]

T. Gruner and D.-G. Welsch, "Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics", Phys. Rev. A 54, 1661 (1996).
[CrossRef] [PubMed]

Y. Aharanov, D. Falko, E. Lerner and H. Pendleton, "A quantum characterization of classical radiation", Ann. Phys. 39, 498 (1966).
[CrossRef]

N. W. Ashcroft and N. D. Mermin, Solid State Physics, International ed., (Saunders College, Philadelphia, 1976), Appendix-L .

P. Bruesch, Phonons : Theory and Experiments, Vol-I and II, (Springer-Verlag, Heidelberg, 1983).

W. H. Louisell, Quantumum Statistical Properties of Radiation, (John Wiley and Sons, NY, 1973).

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Figures (2)

Figure 1.
Figure 1.

A beam-splitter with input and output light beams

Figure 2.
Figure 2.

Lattice displacements due to optical phonons : (a) transverse, (b) longitudinal.

Equations (45)

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E 1 ( + ) = α 11 E 1 ( + ) + α 12 E 2 ( + )
E 2 ( + ) = α 21 E 1 ( + ) α 22 E 2 ( + )
a 1 a 2 = α 11 α 12 α 21 α 22 a 1 a 2
α 11 2 + α 21 2 = 1
α 12 2 + α 22 2 = 1
α 11 α 12 * + α 21 α 22 * = 0
a 1 a 2 = a 1 a 2 = α 11 α 12 α 21 α 22 a 1 a 2
= exp [ i ( ξ a 1 a 2 + η a 2 a 1 ) ]
α 11 α 12 α 21 α 22 = cos ηξ i ξ η sin ηξ i η ξ sin ηξ cos ηξ
1 , 0 = α 11 1,0 + α 21 0,1
n 1 , n 2 = 1 n 1 ! n 2 ! ( α 11 a 1 + α 21 a 2 ) n 1 ( α 12 a 1 + α 22 a 2 ) n 2 0,0
n 1 , n 2 ( Δ n 1 ) 2 n 1 , n 2 = α 11 2 α 12 2 ( n 1 + n 2 + 2 n 1 n 2 )
α , β ( Δ n 1 ) 2 α , β = ( α 11 2 + α 12 2 ) α 11 α + α 12 β 2
b k = 1 N k e i k · R [ M ω k 2 ħ u ( R ) + i 1 2 ħM ω k P ( R ) ] e ̂ k
b k = 1 N k e i k · R [ M ω k 2 ħ u ( R ) i 1 2 ħM ω k P ( R ) ] e ̂ k
R = l ħ ω l ( b l b l + 1 2 )
V i = q M A ( R ) · p
ψ = | j j f j ( r )
A ( R ) = k , λ ħ 2 ω V int [ e i k · R ̂ a + e i k . R ̂ a ] e ̂ λ
p ( R ) = l i ħ ω l M 2 N ( b l e i ω l t b l e i ω l t ) e ̂ l
V I = i = 1,2 l γ [ a i b l e i ( ω + ω l ) t a i b l e i ( ω ω l ) t a i b l e i ( ω ω l ) t + a i b l e i ( ω + ω l ) t ] e ̂ λ i · e ̂ l
T = o + R + V I
b l R = b l R = 0
b l b k R = b l b k R = 0
b l b k R = n ¯ ph ( ω l ) δ lk
b l b k R = [ n ¯ ph ( ω l ) + 1 ] δ lk
s = s ( 1 / 2 ) i , j ( a i a j s a i s a j a j s a i + s a j a i ) S ij
+ ( a i a j s a i s a j a j s a i + s a j a i ) S ij *
+ ( a i a j s a j s a i ) L ij ( a i s a j s a j a i ) L ij *
L ij = γ ( ω ) 2 cos θ il cos θ jl g ( ω )
S ij = n ¯ ph ( ω ) L ij
n 1 , n 2 a 1 a 1 n 1 , n 2 = n 1 ( 1 L 11 ) + S 11
n 1 , n 2 a 2 a 2 n 1 , n 2 = n 2 ( 1 L 22 ) + S 22
a i L = j ( δ ij L ij 2 ) a j NL
n 1 , n 2 Δ n 1 2 n 1 , n 2 = L 11 ( 4 L 11 ) n 2 + [ 2 S 11 + L 11 ( 2 S 11 1 ) ] n + S 11 ( 1 S 11 )
Δ I = I 0 [ 1 exp ( αl ) ] I 0 [ 1 ( 1 αl ) ] = I 0 αl
δ n 1 = L 11 n 1 = [ π 2 12 q 2 ρg ( ω ) ∊M V x ] A n 1 l
n 1 , n 2 a 1 a 1 n 1 , n 2 = ( α 11 2 n 1 + α 12 2 n 2 ) ( 1 L 11 )
Re [ L 12 ( α 11 | * α 21 n 1 + α 12 * α 22 n 2 ) ] + S 11
= L 2 [ ( n 1 + n 2 ) sin ξ sin θ sin 2 δ ξ ( n 1 n 2 ) ]
α , β a 1 a 1 α , β = α 11 α + α 12 β 2 ( 1 L 11 )
Re [ L 12 ( α 11 * α * + α 12 * β * ) ( α 21 α + α 22 β ) ]
= L 2 [ α 2 + β 2 + 2 αβ sin θ [ cos ( δ β δ α ) ( cos 2 ξ sin 2 ξ cos 2 δ ξ )
+ sin ( δ β δ α ) sin 2 ξ sin 2 δ ξ ]
= L 2 [ α 11 α + α 12 β 2 + α 21 α + α 22 β 2 + 2 sin θ Re { ( α 11 * α * + α 12 * β ) ( α 21 α + α 22 β ) } ]

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