Abstract

The vacuum field’s role in the radiative properties of two-level atoms located externally to a phase-conjugate mirror (PCM) is investigated. The basic result of Gaeta and Boyd [ Phys. Rev. Lett. 60, 2618 ( 1988)], that the PCM emits an average of R (where R is reflectivity) noise photons per mode of the radiation field, is modified. Stimulated emission of the combined PCM–atom system takes place when the atom is in its ground state; this emission is caused by a scattering–conjugation process of virtual photons. The expression for the power emitted by a system of N (N > 1) two-level atoms also contains terms that are the expectation values of products of two atomic raising or lowering operators in addition to the usual superradiant terms. The case N = 2 is solved exactly in the limit of a vanishing separation-to-wavelength ratio. The expectation values of individual and pair-product operators are derived from the Heisenberg equations of motion, and the solutions are classified according to the value of an integral of motion. When R < 1, the time evolution is characterized by four real-time constants, of which two are a complex-conjugate pair for R > 1, indicating relaxation by damped oscillations.

© 1992 Optical Society of America

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1991 (1)

H. F. Arnoldus and T. F. George, Phys. Rev. A 43, 3675 (1991).
[Crossref] [PubMed]

1989 (3)

P. W. Milonni, E. J. Bochove, and R. J. Cook, J. Opt. Soc. Am. B 6, 1932 (1989).
[Crossref]

B. H. W. Hendriks and G. Nienhuis, Phys. Rev. A 40, 1892 (1989).
[Crossref] [PubMed]

P. W. Milonni, E. J. Bochove, and R. J. Cook, Phys. Rev. A 40, 4100 (1989).
[Crossref] [PubMed]

1988 (2)

R. J. Cook and P. W. Milonni, IEEE J. Quantum Electron. QE-24, 1383 (1988).
[Crossref]

A. L. Gaeta and R. W. Boyd, Phys. Rev. Lett. 60, 2618 (1988).
[Crossref] [PubMed]

1987 (3)

1980 (2)

1979 (2)

1978 (3)

R. J. Cook and P. W. Milonni, Phys. Rev. A 35, 5081 (1978).
[Crossref]

D. M. Pepper and R. W. Abrams, Opt. Lett. 3, 212 (1978).
[Crossref] [PubMed]

J. H. Marburger, Appl. Phys. Lett. 32, 372 (1978).
[Crossref]

1977 (1)

1976 (1)

1974 (1)

J. R. Ackerhalt and J. H. Eberly, Phys. Rev. D 10, 3350 (1974); H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
[Crossref]

1973 (1)

P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
[Crossref]

1970 (1)

H. Kuhn, J. Chem. Phys. 53, 101 (1970).
[Crossref]

1969 (1)

H. Morawitz, Phys. Rev. 187, 1792 (1969).
[Crossref]

1963 (1)

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[Crossref]

1954 (1)

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[Crossref]

1946 (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

1932 (1)

E. Fermi, Rev. Mod. Phys. 4, 87 (1932).
[Crossref]

1891 (1)

G. Lippmann, C. R. Acad. Sci. Paris 112, 274 (1891).

1890 (1)

O. Wiener, Ann. Phys. (Leipzig) 40, 203 (1890).

Abrams, R. W.

Ackerhalt, J. R.

J. R. Ackerhalt and J. H. Eberly, Phys. Rev. D 10, 3350 (1974); H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
[Crossref]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

Arnoldus, H. F.

H. F. Arnoldus and T. F. George, Phys. Rev. A 43, 3675 (1991).
[Crossref] [PubMed]

Bochove, E. J.

P. W. Milonni, E. J. Bochove, and R. J. Cook, J. Opt. Soc. Am. B 6, 1932 (1989).
[Crossref]

P. W. Milonni, E. J. Bochove, and R. J. Cook, Phys. Rev. A 40, 4100 (1989).
[Crossref] [PubMed]

E. J. Bochove, Phys. Rev. Lett. 59, 2547 (1987).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 7.4.

Boyd, R. W.

Caves, C. M.

C. M. Caves, Phys. Rev. D 26, 1817 (1979).
[Crossref]

Cook, R. J.

P. W. Milonni, E. J. Bochove, and R. J. Cook, J. Opt. Soc. Am. B 6, 1932 (1989).
[Crossref]

P. W. Milonni, E. J. Bochove, and R. J. Cook, Phys. Rev. A 40, 4100 (1989).
[Crossref] [PubMed]

R. J. Cook and P. W. Milonni, IEEE J. Quantum Electron. QE-24, 1383 (1988).
[Crossref]

R. J. Cook and P. W. Milonni, Phys. Rev. A 35, 5081 (1978).
[Crossref]

Dicke, R. H.

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[Crossref]

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Oxford U. Press, New York, 1958).

Eberly, J. H.

J. R. Ackerhalt and J. H. Eberly, Phys. Rev. D 10, 3350 (1974); H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
[Crossref]

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

Faizullov, F. S.

O. Yu Nosach, V. I. Popovichev, V. V. Ragul’skiy, and F. S. Faizullov, Zh. Eksp. Teor. Fiz. Pis’ma Red.16, 617 [Sov. Phys. JETP 16, 435 (1972)].

Fermi, E.

E. Fermi, Rev. Mod. Phys. 4, 87 (1932).
[Crossref]

Gaeta, A. L.

A. L. Gaeta and R. W. Boyd, Phys. Rev. Lett. 60, 2618 (1988).
[Crossref] [PubMed]

George, T. F.

H. F. Arnoldus and T. F. George, Phys. Rev. A 43, 3675 (1991).
[Crossref] [PubMed]

Glauber, R. J.

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[Crossref]

R. J. Glauber, in Quantum Optics and Electronics, C. de Witt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965).

Habashy, T. M.

Hellwarth, R. W.

Hendriks, B. H. W.

B. H. W. Hendriks and G. Nienhuis, Phys. Rev. A 40, 1892 (1989).
[Crossref] [PubMed]

Hopf, F. A.

Jacobs, A. A.

Knight, P. L.

P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
[Crossref]

Kuhn, H.

H. Kuhn, J. Chem. Phys. 53, 101 (1970).
[Crossref]

Lippmann, G.

G. Lippmann, C. R. Acad. Sci. Paris 112, 274 (1891).

Loudon, R.

R. Loudon, The Quantum Theory of Light, 2nd ed. (Oxford U. Press, Oxford, 1983).

Mandel, L.

Marburger, J. H.

J. H. Marburger, Appl. Phys. Lett. 32, 372 (1978).
[Crossref]

Milonni, P. W.

P. W. Milonni, E. J. Bochove, and R. J. Cook, Phys. Rev. A 40, 4100 (1989).
[Crossref] [PubMed]

P. W. Milonni, E. J. Bochove, and R. J. Cook, J. Opt. Soc. Am. B 6, 1932 (1989).
[Crossref]

R. J. Cook and P. W. Milonni, IEEE J. Quantum Electron. QE-24, 1383 (1988).
[Crossref]

R. J. Cook and P. W. Milonni, Phys. Rev. A 35, 5081 (1978).
[Crossref]

P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
[Crossref]

Morawitz, H.

H. Morawitz, Phys. Rev. 187, 1792 (1969).
[Crossref]

Nienhuis, G.

B. H. W. Hendriks and G. Nienhuis, Phys. Rev. A 40, 1892 (1989).
[Crossref] [PubMed]

Nieto-Vesperinas, M.

Pepper, D. M.

Popovichev, V. I.

O. Yu Nosach, V. I. Popovichev, V. V. Ragul’skiy, and F. S. Faizullov, Zh. Eksp. Teor. Fiz. Pis’ma Red.16, 617 [Sov. Phys. JETP 16, 435 (1972)].

Purcell, E. M.

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

Ragul’skiy, V. V.

O. Yu Nosach, V. I. Popovichev, V. V. Ragul’skiy, and F. S. Faizullov, Zh. Eksp. Teor. Fiz. Pis’ma Red.16, 617 [Sov. Phys. JETP 16, 435 (1972)].

Serber, R.

R. Serber and C. H. Townes, in Quantum Electronics: A Symposium, C. H. Townes, ed. (Columbia, New York, 1960), pp. 233–255.

Shapiro, J. H.

Tompkin, W. R.

Townes, C. H.

R. Serber and C. H. Townes, in Quantum Electronics: A Symposium, C. H. Townes, ed. (Columbia, New York, 1960), pp. 233–255.

Wiener, O.

O. Wiener, Ann. Phys. (Leipzig) 40, 203 (1890).

Wolf, E.

Yariv, A.

Yu Nosach, O.

O. Yu Nosach, V. I. Popovichev, V. V. Ragul’skiy, and F. S. Faizullov, Zh. Eksp. Teor. Fiz. Pis’ma Red.16, 617 [Sov. Phys. JETP 16, 435 (1972)].

Yuen, H. P.

Ann. Phys. (Leipzig) (1)

O. Wiener, Ann. Phys. (Leipzig) 40, 203 (1890).

Appl. Phys. Lett. (1)

J. H. Marburger, Appl. Phys. Lett. 32, 372 (1978).
[Crossref]

C. R. Acad. Sci. Paris (1)

G. Lippmann, C. R. Acad. Sci. Paris 112, 274 (1891).

IEEE J. Quantum Electron. (1)

R. J. Cook and P. W. Milonni, IEEE J. Quantum Electron. QE-24, 1383 (1988).
[Crossref]

J. Chem. Phys. (1)

H. Kuhn, J. Chem. Phys. 53, 101 (1970).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
[Crossref]

Opt. Lett. (3)

Phys. Rev. (4)

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[Crossref]

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[Crossref]

H. Morawitz, Phys. Rev. 187, 1792 (1969).
[Crossref]

Phys. Rev. A (4)

R. J. Cook and P. W. Milonni, Phys. Rev. A 35, 5081 (1978).
[Crossref]

P. W. Milonni, E. J. Bochove, and R. J. Cook, Phys. Rev. A 40, 4100 (1989).
[Crossref] [PubMed]

B. H. W. Hendriks and G. Nienhuis, Phys. Rev. A 40, 1892 (1989).
[Crossref] [PubMed]

H. F. Arnoldus and T. F. George, Phys. Rev. A 43, 3675 (1991).
[Crossref] [PubMed]

Phys. Rev. D (2)

J. R. Ackerhalt and J. H. Eberly, Phys. Rev. D 10, 3350 (1974); H. J. Kimble and L. Mandel, Phys. Rev. A 13, 2123 (1976).
[Crossref]

C. M. Caves, Phys. Rev. D 26, 1817 (1979).
[Crossref]

Phys. Rev. Lett. (2)

A. L. Gaeta and R. W. Boyd, Phys. Rev. Lett. 60, 2618 (1988).
[Crossref] [PubMed]

E. J. Bochove, Phys. Rev. Lett. 59, 2547 (1987).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

E. Fermi, Rev. Mod. Phys. 4, 87 (1932).
[Crossref]

Other (9)

O. Yu Nosach, V. I. Popovichev, V. V. Ragul’skiy, and F. S. Faizullov, Zh. Eksp. Teor. Fiz. Pis’ma Red.16, 617 [Sov. Phys. JETP 16, 435 (1972)].

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

R. Serber and C. H. Townes, in Quantum Electronics: A Symposium, C. H. Townes, ed. (Columbia, New York, 1960), pp. 233–255.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

R. Loudon, The Quantum Theory of Light, 2nd ed. (Oxford U. Press, Oxford, 1983).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 7.4.

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Oxford U. Press, New York, 1958).

R. J. Glauber, in Quantum Optics and Electronics, C. de Witt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965).

Ref. 23, p. 9.

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Figures (6)

Fig. 1
Fig. 1

Illustration of the directly emitted radiation field (solid circles and arrows) and the phase-conjugated field (dashed circles and arrows) of a source atom. At any point to the right of the atom the fields are phase conjugates of each other, whereas to the left they are identical except for a constant amplitude factor and phase difference.

Fig. 2
Fig. 2

Probability of steady-state excitation of N Dicke atoms as function of the quantum number m for various values of the PCM reflectivity R. The constants are N = 10 and j = 5, where N is the number of atoms and j is the cooperation number.

Fig. 3
Fig. 3

Steady-state values of the functions K = S i S 2 , M = r*〈S1S2〉, N = 〈σz1σz2〉, and q = 〈σz〉 for a pair of atoms in any combination of the triplet states are plotted as functions the reflectivity R.

Fig. 4
Fig. 4

Fluorescence intensity of a pair of atoms as function of R for the correlated state corresponding to Figs. 3, 5, and 6, denoted by intensity I, and the uncorrelated state, denoted by Ia. Both intensities are divided by Io, the intensity radiated by an atom in free space when in the upper energy level.

Fig. 5
Fig. 5

Four decay rates of the functions K, M, N, q in units of β, plotted in the range 0 ≤ R ≤ 1.

Fig. 6
Fig. 6

Same decay rates as in Fig. 6, plotted for 1 ≤ R ≤ 10. Two of the rate constants are real, and the other two are a pair of complex conjugates, indicating relaxation with oscillations.

Equations (132)

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b - k = r a k + τ a - k ,
b ¯ - k = r a ¯ k * ;
n ¯ b = R ( n ¯ 3 + 1 ) ;
δ n b 2 - n ¯ b = R 2 ( δ n a 2 + n ¯ a ) .
A = ( 1 + R ) A .
ω probe + ω con = 2 ω pump .
E = 0 ,             H n = 0.
E k σ ( z , t ) = E k σ sin k z sin ( ω t + ϕ ) e ^ k σ ,             σ = 1 , 2 ,
I k σ ( z ) = E k σ ( z , t ) 2 t = ½ E k σ 2 sin 2 k z ,
E k σ ( z , t ) = i ( 2 π ω / V ) 1 / 2 [ a k σ ( t ) - a k σ ( t ) ] sin k z ,
[ a k σ ( t ) , a k σ ( t ) ] = 0 ,             [ a k σ ( t ) , a k σ ( t ) ] = δ kk δ σ σ .
E ( r , t ) = E ( + ) ( r , t ) + E ( - ) ( r , t ) ,
E ( + ) ( r , t ) = i k , σ ( 2 π ω k / V ) 1 / 2 U k ( x , y ) a k σ ( t ) [ e ^ k σ 1 exp ( i k z z ) + e ^ k σ 2 exp ( - i k z z ) ]
I ( r , t ) = E ( - ) ( r , t ) E ( + ) r , t ) ,
I ( r , t ) = k , σ k σ ( 2 π / V ) ( ω k ω k ) 1 / 2 U k ( x , y ) U k * ( x , y ) × a k σ ( t ) a k σ ( t ) { e ^ k σ 1 · e ^ k σ 1 exp [ i ( k z - k z ) z ] + e ^ k σ 2 · e ^ k σ 2 exp [ i ( k z - k z ) ] + e ^ k σ 2 · e ^ k σ 1 exp [ - i ( k z + k z ) z ] + e ^ k σ 1 · e ^ k σ 2 exp [ i ( k z + k z ) z ] } .
I ( r , t ) = 2 k , σ ( 2 π ω k / V ) n k σ [ 1 + e ^ k σ 1 · e ^ k σ 2 cos ( 2 k z z ) ] .
a k σ α k σ = α k σ α k σ ,
I ( r , t ) = k , σ k σ ( 2 π / V ) ( ω k ω k ) 1 / 2 U k ( x , y ) U k * ( x , y ) × α k σ ( t ) α k σ ( t ) { e ^ k σ 1 · e ^ k σ 1 exp [ i ( k z - k z ) z ] + e ^ k σ 2 · e ^ k σ 2 × exp [ i ( k z - k z ) ] + e ^ k σ 2 · e ^ k σ 1 exp [ - i ( k z + k z ) z ] + e ^ k σ 1 · e ^ k σ 2 exp [ i ( k z + k z ) z ] } = ( + ) ( r , t ) 2 ,
( + ) ( r , t ) = i k , σ ( 2 π ω k / V ) 1 / 2 U k ( x , y ) α k σ ( t ) [ e ^ k σ 1 exp ( i k z z ) + e ^ k σ 2 exp ( - i k z z ) ] ,
E con ( + ) ( r ) = r E in ( - ) ( r ) = r E in ( + ) ( r ) * ,
E con ( + ) ( r , t ) = r E in ( - ) ( r , t - 2 τ o ) exp [ - 2 i ω o ( t - τ o ) ] ,
E con ( d , t ) r E in ( d , t - T ) * ,
E ( + ) ( r , t ) = E exp [ i ( k · r - ω t ) ] + r E exp [ i ( - k · r - ω t ) ] ,
I ( r , t ) = [ 1 + r 2 + 2 r e 2 cos ( 2 k · r - v t + 2 ϕ E - ϕ PCM ) ] E 2 ,
I ( r ) = ( 1 + R ) I o ( r ) ,
a k a k ,             k z > 0 ,
a k r a - k + τ a k ,             k z < 0.
τ 2 = 1 + R .
r r exp [ 2 i ( ω o - ω k ) d ] .
E F ( + ) ( r , t ) = i k σ ( 2 π ω k / V ) 1 / 2 e ^ k σ a k σ exp [ i ( k · r - ω k t ) ] ,
E B ( + ) ( r , t ) = i k σ ( 2 π ω k / V ) 1 / 2 e ^ k σ ( r a - k σ + τ a k σ ) × exp [ i ( k · r - ω k t ) ] = r E o F ( - ) ( r , t ) + τ E o B ( + ) ( r , t ) ,
I = E F ( - ) E F ( + ) + R E F ( + ) E F ( - ) + τ 2 E B ( - ) E B ( + ) + [ τ E F ( - ) E B ( + ) + r E F ( - ) E F ( - ) + r τ E F ( - ) E B ( - ) + c . c . ] ,
R E F ( + ) E F ( - ) = R E F ( - ) E F ( + ) + R [ E F ( + ) , E F ( - ) ] = R I F + k σ R ( 2 π ω k / V ) .
I o = ( 1 + R ) ( I o F + I o B ) + I N ,
I N = 8 π 2 f o 3 w R o / c 3 ,
w = R o - 1 0 ( ω / ω o ) 3 r ( ω ) 2 d ω / 2 π R o - 1 0 r ( ω ) 2 d ω / 2 π ,
I = o F ( + ) + r o F ( - ) + τ o B ( + ) 2 + I N ,
E F ( + ) ( r , t ) = E o F ( + ) ( r , t ) + E s F ( + ) ( r , t ) ,
E B ( + ) ( r , t ) = E s B ( + ) ( r , t ) + r E o F ( - ) ( r , t ) + r E s F ( - ) ( r , t ) + τ E o B ( + ) ( r , t ) .
D ( r ) = - ω o 2 r ^ x r ^ x μ / ( c 2 r ) ,
E s , con ( + ) ( r , t ) = r D ( r ) σ ( t - r / c ) exp ( 2 i ω o r / c ) ,
E s F ( + ) ( r , t ) = D ( r ) σ ( t + r / c )             ( z < 0 ) ,
E ( + ) = E o F ( + ) + E s ( + ) + τ E o B ( + ) + r E o F ( - ) + r E s F ( - ) ,
I = E s ( - ) E s ( + ) + R [ E o F ( + ) E o F ( - ) + E s F ( + ) E s F ( - ) + E s F ( - ) E o F ( + ) ] + [ τ E F ( - ) E F ( + ) + E F ( - ) E s B ( + ) + r E F ( - ) E F ( - ) + τ E o B ( - ) E s B ( + ) + τ r E o B ( - ) E F ( - ) + r E s B ( - ) E F ( - ) + c . c . ] + τ 2 E o B ( - ) E o B ( + ) .
E ( + ) vac = vac E ( - ) = 0 ;
I = I s + R I s ( a ) + I s s + I s n + I N ,
I s s = 2 Re [ r E s ( - ) E s ( - ) ] ,
I s n = 2 Re [ r E s ( - ) E o F ( - ) + R E s F ( + ) E o F ( - ) ] .
I s s = r i j D ( r - r i ) D ( r - r j ) σ i ( t - τ i ) σ j ( t - τ j ) ,
I s n 1 = i = 1 N D ( r - r i ) r σ ( t - τ i ) E o F ( - ) ( r , t ) ret ,
S i ( t ) = σ i ( t ) exp ( i ω o t ) .
S ˙ i ( t ) = - β S i ( t ) - β r S i + ( t ) - σ z i ( t ) Ω ( + ) ( r i , t ) ,
Ω ( + ) ( r i , t ) = i ( μ / ) · [ E o F ( + ) ( r i , t ) + r E o F ( - ) ) ( r i , t ) ret + τ E o B ( + ) ( r i , t ) + j = 1 N D ( r i - r j ) σ j ( t ) + r j = 1 N D ( r i - r j ) σ j ( t ) ] exp ( i ω o t ) .
S i ( t ) = S i o ( t ) - j = 1 N 0 t g i j ( t - t ) σ z j ( t ) Ω ( + ) ( r j , t ) d t - j = 1 N 0 t h i j ( t - t ) Ω ( - ) ( r j , t ) σ z j ( t ) d t ,
A = r σ i ( t - τ i ) E o F ( - ) ( r , t ) ret = i exp [ - i ω o ( t + τ i ) ] ( μ / ) R × [ j = 1 N 0 t - τ i g i j ( t - τ - t ) E o F ( + ) ( r j , t ) σ z j ( t ) × E o F ( - ) ( r , t ) ret exp ( i ω o t ) d t + j = 1 N t - τ i h i j ( t - τ j - t ) × σ z j ( t ) E o F ( - ) ( r j , t ) E o F ( - ) ( r , t ) ret exp ( - i ω o t ) d t ] .
[ σ z j ( t ) , E o F ( - ) ( r , t ) ] = 0
t - t + r - r j / c > 0.
B = i ( μ / ) R J = 1 n 0 t - τ j g i j ( t - t - τ j ) E o F ( + ) ( r j , t ) × E o F ( - ) ( r , t - 2 τ j ) σ z ( t ) exp [ i ω o ( t - t + τ j ) ] d t .
B j = i ( μ / ) R k , σ ( 2 π ω k / V ) ( e ^ k σ · μ ^ j ) ( e ^ k σ · n ^ j ) × 0 t - τ i g i j ( t - t - τ ) σ z ( t ) × exp [ - i k · ( r - r j ) ] exp [ i ω k ( t - t - 2 τ j ) ] × exp [ i ω o ( t - t + τ j ) ] d t .
B j = i ( μ ) r 2 ( 2 π / V ) V ( 2 π c ) 3 d Ω k d ω k ω k 3 P × 0 t - τ j d t g i j ( t - t - τ j ) σ z j ( t ) × exp [ - i ω k ( t - t + 2 τ j - Q τ j ) + i ω o ( t - t + τ j ) ] ,
P = σ ( e ^ k σ · μ ^ j ) ( e ^ k σ · n ^ j ) , ω k τ j Q = - k · ( r - r j ) .
B j = i μ R ( 2 π ) - 1 ( ω o / c ) 3 d Ω k P 0 t - τ j d t g i j ( t - t - τ j ) × σ z j ( t ) δ [ t - t + ( Q - 2 ) τ j ] exp [ i ω o ( t - t + τ j ) ] .
B i = 1 2 i μ R ( 2 π ) - 1 ( ω o c ) 3 σ z ( t - τ i ) × d ϕ exp [ - i 1 2 ω o τ i sin 2 θ ϕ 2 ] × d θ sin 2 θ exp [ - i 1 2 ω o τ i ( θ - θ ) 2 ] = 1 2 μ R ( ω o c ) 2 sin θ σ z i ( t - τ i ) r - r i .
I s n 1 ( r ) = R i = 1 N D ( r - r i ) 2 σ z i ( t - τ i ) .
I s n 1 ( r ) = R i = 1 N D ( r - r i ) 2 σ z i ( t - τ i ) cos ( 2 ω o τ i ) .
I s n 2 ( r ) = - R i = 1 N D ( r - r i ) 2 σ z i ( t - τ i )
S n ( r ) = i = 1 N j = 1 N D ( r - r i ) · D ( r - r j ) { σ i ( t - τ i ) σ j ( t - τ j ) ± R σ i ( t - τ i ) σ j ( t - τ j ) ± 2 Re [ r σ i ( t - τ i ) σ j ( t - τ j ) ] } + I s n + I N = I s + i = 1 N D ( r - r i ) 2 × { P 2 i ( t - τ i ) + R P 1 i ( t - τ i ) + R ( P 2 i - P 1 i ) [ 0 1 ] } + i = 1 N j = 1 j i N D ( r - r i ) · D ( r - r j ) × { σ i ( t - τ i ) σ j ( t - τ j ) ± R σ i ( t - τ i ) σ j ( t - τ j ) ± 2 Re [ r σ i ( t - τ i ) σ j ( t - τ j ) ] } + I N ,
S n ( r ) = I o ( r ) [ P 2 ( t - r / c ) + R P 1 ( t - r / c ) ]
S n ( r ) = I o ( r ) [ ( 1 + R ) P 2 - 2 R P 1 ] ,
P ˙ 2 = - ½ β [ ( 2 + R ) P 2 - R P 1 ] ,
I s n 2 ( r ) = 0
S n ( r ) = I o ( r ) ( 1 + R ) P 2 ( t - r / c )
S n ( r ) = I o ( r ) ( P 2 - R P 1 ) .
Ψ = i = 1 N θ , ϕ i ,
θ , ϕ i = cos ( θ / 2 ) exp ( i ϕ / 2 ) 2 i + sin ( θ / 2 ) exp ( - i ϕ / 2 ) 1 i .
I = I 1 [ N cos 2 θ 2 + N R sin 2 θ 2 + N 2 4 ( 1 + R ) P sin 2 θ + N 2 2 Re r Q sin 2 θ exp ( 2 i ϕ ) ] ,
β = ( 1 + R ) β ,
J k = j = 1 N σ j k ,             k = 1 , 2 , 3 ,
m j N / 2.
P ˙ ( j , m ) = β ( j + m ) ( j - m + 1 ) [ ½ R P ( j , m - 1 ) - ½ ( 1 + R ) P ( j , m ) ] + β ( 1 + ½ R ) × ( j + m + 1 ) ( j - m ) P ( j , m + 1 ) - ½ β R ( j + m + 1 ) ( j - m ) P ( j , m ) .
P ( j , - j ) = P ( j , - j + 1 ) = = P ( j , j ) = 1 ( 2 j + 1 )             for             R .
S ˙ i = - σ z i Ω i ( + ) ,
σ ˙ z i = 2 [ S Ω ( + ) + Ω ( - ) S ] ,
E i ( + ) = E R R i ( + ) + E o F i ( + ) + r E o F i ( - ) + τ E o B i ( + ) ,
Ω R R i ( + ) = - β j = 1 N S j + β r j = 1 N S j ,
K ˙ i j = S i S ˙ j + S ˙ i S j = - S i σ z j Ω j ( + ) - Ω i ( - ) σ z i S j .
S i σ z j Ω j ( + ) = - β l i j [ S i σ z j S l + r S i σ z j S l ] - β S i σ z j S i - β S i σ z j S j + β r S i σ z j S i + β r S i σ z j S j + β r S i σ z j S i + r S i σ z j Ω o F j ( - ) + S i σ z j Ω o F j ( + ) + τ S i σ z j Ω o B j ( + ) ,
S i σ z j Ω j ( + ) = - β S i σ z j S i - β S i σ z j S j + β r S i σ z j S i + β r S i σ z j S j - r S i σ z j Ω o F j ( - ) .
r S i σ z j Ω o F j ( - ) = r Ω o F j ( - ) S i σ z j + r [ S i σ z j , Ω o F j ( - ) ] = r S i [ σ z j , Ω o F j ( - ) ] + r [ S i , Ω o F j ( - ) ] σ z j .
σ z i ( t ) = σ z i ( 0 ) + 2 t [ S i ( t ) Ω i ( + ) ( t ) + Ω i ( - ) ( t ) S i ( t ) ] d t .
[ σ z j , Ω o F j ( - ) ] = 2 0 t { [ S j ( t ) Ω j ( + ) ( t ) , Ω o F j ( - ) ( t ) ] + [ Ω j ( - ) ( t ) S j ( t ) , Ω o F j ( - ) ( t ) ] } d t .
r [ σ z j , Ω o F j ( - ) ] = 2 r 0 t { S j ( t ) [ Ω o F j ( + ) ( t ) , Ω o F j ( - ) ( t ) ] + r * [ Ω o F j ( + ) ( t ) , Ω o F j ( - ) ( t ) ] S j ( t ) } d t .
[ Ω o F i ( + ) ( t ) , Ω o F j ( - ) ( t ) ] β δ ( t - t ) ,
r [ σ z j , Ω o F j ( - ) ] = β r ( 2 S j - r * S j ) .
r [ S i , Ω o F j ( - ) ] = - r 0 t [ Ω i ( - ) ( t ) σ z i ( t ) , Ω o F j ( - ) ( t ) ] d t R 0 t [ Ω o F i ( + ) ( t ) , Ω o F j ( - ) ( t ) ] σ z i ( t ) d t ½ R β σ z i .
K ˙ i j = - 2 β K i j - β ( K i j + K j i * ) + R β σ z i σ z j + 2 β P 2 i σ z j + β M i j + M j i * .
L i j = K j i .
M ˙ i j = r * [ S ˙ i S j + S i S ˙ j ] = - r * [ S i σ z j Ω j ( + ) - σ z i Ω i ( + ) S j ] = r * [ β i j l ( S i σ z j S l + σ z i S i S j - r S i σ z j S l - r σ z i S l S j ) + β S i σ z j S j + β S i σ z j S i - r β S i σ z j S j - r β S i σ z j S i + β σ z i S i S j + β σ z i S j S j - r β σ z i S i S j - r β σ z i S j S j - S i σ z j Ω o F j ( + ) - σ z i Ω o F i ( + ) S j + r S i σ z j Ω o F j ( - ) + r σ z i Ω o F i ( - ) S j - τ S i σ z j Ω o B j ( + ) - τ σ z i Ω o B i ( + ) S j ] .
M ˙ i j = - 2 β ( 1 + R ) M i j - R β ( σ z i P 2 j + P 1 i σ z j + σ z i σ z j - K i j - L i j ) .
K ˙ = β ( - 2 τ 2 K + τ 2 N + q + 2 M ) ,
L = K ,
M ˙ = β [ - 2 τ 2 M - R ( N + q - 2 K ) ] ,
N ˙ = 2 [ σ z S + Ω ( + ) + σ z Ω ( - ) S + S Ω ( + ) σ z + Ω ( - ) S σ z ] = - 4 β [ 2 σ z S + S + σ z S + S + σ S + S - r σ z S + S - r * σ z S S ] + 2 [ - r σ z S + Ω o F ( + ) + σ z Ω o ( - ) S + c . c . ] .
N ˙ = - 4 β ( τ 2 N + q - 2 τ 2 K + 2 M ) .
u = N + 4 K = const .
U = σ z σ z + 2 ( σ σ + σ σ ) ,
q ˙ = - 2 β ( τ 2 q + 1 + 2 K - 2 M ) .
- 2 K + N + q = 0 ,
M = 0 ,
- N - q + 2 K - 2 M = 0 ,
- q - 1 - 2 K + 2 M = 0.
q = - 1 - 2 K ,
N = 1 + 4 K .
K s = 0 ,             N s = - q s = 1 ,
K a = - 1 / 2 ,             N a = - 1 ,             q a = 0.
- 2 τ 2 K + τ 2 N + q + 2 M = 0 ,
2 τ 2 M + R [ q + N - 2 K ] = 0 ,
τ 2 N + q 2 - 2 τ K + 2 M = 0 ,
τ 2 q + 1 + 2 K - 2 M = 0.
K s = τ 4 - τ 2 6 τ 4 - 6 R + ( 2 - 6 τ 2 ) / ( τ 2 + 1 ) .
S n / S n o = 1 + q + R ( 1 - q ) + 2 ( 1 + R ) K + 4 M .
d p / d t = - i [ S grad Ω ( + ) - grad Ω ( - ) S ] ,
d a k σ / d t = ( 2 π ω k / V ) 1 / 2 μ · e ^ k σ S exp ( - i k · r ) ;
d p / d t = - i r [ S grad Ω s F ( - ) - S grad Ω o F ( - ) ] + c . c . ,
F = d p ( t ) / d t = i r S ( t ) grad Ω o F ( - ) ( t ) + c . c .
F = i R 0 t d t Ω o F ( + ) ( t ) σ z ( t ) grad Ω o F ( - ) ( t ) + c . c . = i R 0 t d t Ω o F ( + ) ( t ) grad Ω o F ( - ) ( t ) σ z ( t ) + c . c .
[ Ω o F ( + ) ( t ) , grad Ω o F ( - ) ( t ) ] 9 16 i β ω 0 c δ ( t - t ) n ^ ,
F = 9 32 β R ( P 1 - P 2 ) ω o c n ^ ,
F se = - R P 2 w k k z = 9 / 32 P 2 β R ω o ,
F a = R P 1 w k k z = ( 9 / 32 ) P 1 β R ω o ,
σ z ss = 2 P 2 ss - 1 = - 1 / ( 1 + R ) ,
F ss = 9 32 β ω o c - 1 R 1 + R .

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