Abstract

The relationships between certain important nonclassical states of the quantized field and the coherent states associated with the SU(2) and SU(1,1) Lie groups and associated Lie algebras is briefly reviewed. As an example of the utility of group theoretical methods in quantum optics, a method for generating maximally entangled photonic states is discussed. These states may be of great importance for achieving Heisenberg-limited interferometry and in beating the diffraction limit in lithography.

© 2001 Optical Society of America

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  1. B. G. Wybourne, “The ‘Gruppen Pest’ yesterday, today, and tomorrow,” Intl. J. Quant. Chem. Symp. 7, 35–43 (1973).
    [CrossRef]
  2. K. Wódkiewicz and J. H. Eberly, “Coherent states, squeezed fluctuations, and the SU(2) and SU(1,1) groups in quantum optics,” J. Opt. Soc. Am. B 2, 458–466 (1985).
    [CrossRef]
  3. A. M. Perelomov, “Coherent states for an arbitrary Lie group,” Commun. Math. Phys. 26, 222–236 (1972)
    [CrossRef]
  4. A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
    [CrossRef]
  5. C. C. Gerry and E. E. Hach, “Generation of even and odd coherent states in a competitive two-photon process,” Phys. Lett. A 117, 185–189 (1993).
    [CrossRef]
  6. V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics XXXIV, E. Wolf, ed. (Elesevier, Amsterdam, 1995).
  7. G. S. Agarwal, “Nonclassical statistics of fields in pair coherent states,” J. Opt. Soc. Am. B 5, 1940–1947 (1988).
    [CrossRef]
  8. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A. 40, 1371–1348 (1989).
    [CrossRef] [PubMed]
  9. J. M. Radcliffe, “Some properties of spin coherent states,” J. Phys. A 4, 313–323 (1971).
    [CrossRef]
  10. B. Yurke and D. Stoler, “Quantum behavior of a four-way mixer operated in nonlinear regime,” Phys. Rev. A 35, 4846–4849 (1987).
    [CrossRef] [PubMed]
  11. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811-1–043811-7 (2000).
    [CrossRef]
  12. K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
    [CrossRef]
  13. A. Boto et al., “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
    [CrossRef] [PubMed]

2000 (2)

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811-1–043811-7 (2000).
[CrossRef]

A. Boto et al., “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[CrossRef] [PubMed]

1999 (1)

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
[CrossRef]

1993 (1)

C. C. Gerry and E. E. Hach, “Generation of even and odd coherent states in a competitive two-photon process,” Phys. Lett. A 117, 185–189 (1993).
[CrossRef]

1989 (1)

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A. 40, 1371–1348 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

B. Yurke and D. Stoler, “Quantum behavior of a four-way mixer operated in nonlinear regime,” Phys. Rev. A 35, 4846–4849 (1987).
[CrossRef] [PubMed]

1985 (1)

1973 (1)

B. G. Wybourne, “The ‘Gruppen Pest’ yesterday, today, and tomorrow,” Intl. J. Quant. Chem. Symp. 7, 35–43 (1973).
[CrossRef]

1972 (1)

A. M. Perelomov, “Coherent states for an arbitrary Lie group,” Commun. Math. Phys. 26, 222–236 (1972)
[CrossRef]

1971 (2)

A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[CrossRef]

J. M. Radcliffe, “Some properties of spin coherent states,” J. Phys. A 4, 313–323 (1971).
[CrossRef]

Agarwal, G. S.

Barut, A. O.

A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[CrossRef]

Boto, A.

A. Boto et al., “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[CrossRef] [PubMed]

Buzek, V.

V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics XXXIV, E. Wolf, ed. (Elesevier, Amsterdam, 1995).

Campos, R. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A. 40, 1371–1348 (1989).
[CrossRef] [PubMed]

Eberly, J. H.

Gerry, C. C.

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811-1–043811-7 (2000).
[CrossRef]

C. C. Gerry and E. E. Hach, “Generation of even and odd coherent states in a competitive two-photon process,” Phys. Lett. A 117, 185–189 (1993).
[CrossRef]

Girardello, L.

A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[CrossRef]

Hach, E. E.

C. C. Gerry and E. E. Hach, “Generation of even and odd coherent states in a competitive two-photon process,” Phys. Lett. A 117, 185–189 (1993).
[CrossRef]

Knight, P. L.

V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics XXXIV, E. Wolf, ed. (Elesevier, Amsterdam, 1995).

Mølmer, K.

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
[CrossRef]

Perelomov, A. M.

A. M. Perelomov, “Coherent states for an arbitrary Lie group,” Commun. Math. Phys. 26, 222–236 (1972)
[CrossRef]

Radcliffe, J. M.

J. M. Radcliffe, “Some properties of spin coherent states,” J. Phys. A 4, 313–323 (1971).
[CrossRef]

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A. 40, 1371–1348 (1989).
[CrossRef] [PubMed]

Sørensen, A.

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
[CrossRef]

Stoler, D.

B. Yurke and D. Stoler, “Quantum behavior of a four-way mixer operated in nonlinear regime,” Phys. Rev. A 35, 4846–4849 (1987).
[CrossRef] [PubMed]

Teich, M. C.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A. 40, 1371–1348 (1989).
[CrossRef] [PubMed]

Wódkiewicz, K.

Wybourne, B. G.

B. G. Wybourne, “The ‘Gruppen Pest’ yesterday, today, and tomorrow,” Intl. J. Quant. Chem. Symp. 7, 35–43 (1973).
[CrossRef]

Yurke, B.

B. Yurke and D. Stoler, “Quantum behavior of a four-way mixer operated in nonlinear regime,” Phys. Rev. A 35, 4846–4849 (1987).
[CrossRef] [PubMed]

Commun. Math. Phys. (2)

A. M. Perelomov, “Coherent states for an arbitrary Lie group,” Commun. Math. Phys. 26, 222–236 (1972)
[CrossRef]

A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[CrossRef]

Intl. J. Quant. Chem. Symp. (1)

B. G. Wybourne, “The ‘Gruppen Pest’ yesterday, today, and tomorrow,” Intl. J. Quant. Chem. Symp. 7, 35–43 (1973).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. A (1)

J. M. Radcliffe, “Some properties of spin coherent states,” J. Phys. A 4, 313–323 (1971).
[CrossRef]

Phys. Lett. A (1)

C. C. Gerry and E. E. Hach, “Generation of even and odd coherent states in a competitive two-photon process,” Phys. Lett. A 117, 185–189 (1993).
[CrossRef]

Phys. Rev. A (2)

B. Yurke and D. Stoler, “Quantum behavior of a four-way mixer operated in nonlinear regime,” Phys. Rev. A 35, 4846–4849 (1987).
[CrossRef] [PubMed]

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811-1–043811-7 (2000).
[CrossRef]

Phys. Rev. A. (1)

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A. 40, 1371–1348 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
[CrossRef]

A. Boto et al., “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[CrossRef] [PubMed]

Other (1)

V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics XXXIV, E. Wolf, ed. (Elesevier, Amsterdam, 1995).

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Equations (45)

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z 1 2 + z 2 2 ,
z 1 2 z 2 2 .
[ J + , J ] = 2 J 3 , [ J 3 , J ± ] = ± J ± ,
C 2 = J 3 2 + 1 2 ( J + J + J J + ) ,
[ K + , K ] = 2 K 0 , [ K 0 , K ± ] = ± K ± .
C 11 = K 0 2 1 2 ( K + K + K K + ) .
C 2 j , m = j ( j + 1 ) j , m , J 3 j , m = m j , m
J ± j , m = ( j m ) ( j ± m + 1 ) j , m ± 1 ,
j = 1 2 , 1 , 3 2 , 2 , , m = j , j + 1 , , j .
C 11 k , m = k ( k 1 ) k , m , K 0 k , m = ( k + m ) k , m ,
K + k , m = ( m + 1 ) ( m + 2 k ) k , m + 1 ,
K k , m = m ( m + 2 k 1 ) k , m 1 ,
k = 1 2 , 1 , 3 2 , 2 , ; m = 0 , 1 , 2 , .
K 0 = 1 2 ( a + a + 1 2 ) , K + = 1 2 a + 2 , K = 1 2 a 2 ,
n k , m for n = 2 ( m + k ) 1 2 .
ξ , k = S ( z ) k , 0
ξ , k = ( 1 ξ 2 ) k m = 0 [ Γ ( 2 k + m ) m ! Γ ( 2 k ) ] 1 2 ξ m k , m
ξ sv = 1 cosh ( θ 2 ) m = 0 ( 1 ) m ( 2 m ) ! 2 m m ! e im ϕ ( tanh ( θ 2 ) ) m 2 m ,
H I = i ( λ a + 2 λ * a 2 ) = 2 i ( λ K + λ * K )
U I ( t ) = exp [ iH I t ] = exp [ 2 t ( λ K + λ * K ) ] ,
K η , k = η η , k ,
η , k = N k m = 0 η m m ! Γ ( 2 k + m ) k , m
N k = [ Γ ( 2 k ) η 2 k + 1 I 2 k 1 ( 2 η ) ] 1 2 ,
η , 1 4 = N + ( α + α ) ,
η , 3 4 = N ( α α ) ,
N ± = [ 2 ± 2 exp ( 2 α 2 ) ] 1 2 .
ρ t = i [ H I , ρ ] κ 2 ( a + 2 a 2 ρ 2 a 2 ρ a + 2 + ρ a + 2 a 2 )
= 2 [ λ K + λ * K , ρ ] 2 κ ( K + K ρ 2 K ρ K + + ρ K + K ) ,
K 0 = 1 2 ( a + a + b + b + 1 ) , K + = a + b + , K = ab
C 11 = 1 4 ( Δ 2 1 ) , Δ = a + a b + b .
n + q , n k , m , k = 1 2 ( 1 + q ) , m = n .
ξ , 1 2 ( 1 + q ) = ( 1 ξ 2 ) ( 1 + q ) 2 n = 0 [ ( n + q ) ! n ! q ! ] 1 2 ξ n n + q , n .
ξ tmsv = 1 cosh θ n = 0 ( 1 ) n e in ϕ [ tanh ( θ 2 ) ] n n , n .
H I = i ( λ a + b + λ ab ) = i ( λ K + λ * K ) .
η , 1 2 ( 1 + q ) = N q n = 0 η n [ n ! ( n + q ) ! ] n + q , n ,
N q = [ q ! η q I q ( 2 η ) ] 1 2 .
J 3 = 1 2 ( a + a b + b ) , J + = a + b , J = ab + , J 0 = 1 2 ( a + a + b + b )
j , m n a n b , j = 1 2 ( n a + n b ) , m = 1 2 ( n a n b ) ,
U 1 ( 2 ) ( θ ) = exp ( i θ J 1 ( 2 ) )
H I = i ( λ a + b λ * ab + ) = i ( λ J + λ * J ) .
ζ , j = exp ( β J + β * J ) j , j
= ( 1 + ζ 2 ) j m = j j ( 2 j j + m ) 1 2 ζ j + m j , m
= ( 1 + ζ 2 ) N n = 0 N ( N n ) 1 2 ζ n n , N n
H I = Ω 4 ( a + b + ab + ) 2 = Ω J 1 2 .
1 2 ( n 0 + e i Φ n 0 n ) , Φ n = ( n + 1 ) π 2 ,

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