Abstract

When a light beam propagates through an ensemble of two-level quantum systems, the resonant interaction between these systems and the radiation field may be described by a complex susceptibility. Even if the medium is of a type ordinarily considered isotropic, the susceptibility will be a tensor. In this paper, explicit expressions for the elements of this tensor are derived from quantum-statistical mechanics. A physical explanation of the induced anisotropy is given, and the experimental conditions under which it can be neglected are described.

© 1971 Optical Society of America

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References

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  1. C. P. Slichter, Principles of Magnetic Resonance (Harper and Row, New York, 1963), pp. 127–134, 156–159.
  2. R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
    [CrossRef]
  3. E. U. Condon, G. H. Shortley, Theory of Atomic Spectra (Maemillan, New York, 1935), pp. 90–93, 100–103.
  4. R. P. Dicke, Phys. Rev. 93, 99 (1954).
    [CrossRef]
  5. W. Gröbner, N. Hofreiter, Integraltafel (Springer-Verlag, Vienna, 1961), Part 2, p. 99.

1957 (1)

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

1954 (1)

R. P. Dicke, Phys. Rev. 93, 99 (1954).
[CrossRef]

Condon, E. U.

E. U. Condon, G. H. Shortley, Theory of Atomic Spectra (Maemillan, New York, 1935), pp. 90–93, 100–103.

Dicke, R. P.

R. P. Dicke, Phys. Rev. 93, 99 (1954).
[CrossRef]

Feynman, R. P.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Gröbner, W.

W. Gröbner, N. Hofreiter, Integraltafel (Springer-Verlag, Vienna, 1961), Part 2, p. 99.

Hellwarth, R. W.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Hofreiter, N.

W. Gröbner, N. Hofreiter, Integraltafel (Springer-Verlag, Vienna, 1961), Part 2, p. 99.

Shortley, G. H.

E. U. Condon, G. H. Shortley, Theory of Atomic Spectra (Maemillan, New York, 1935), pp. 90–93, 100–103.

Slichter, C. P.

C. P. Slichter, Principles of Magnetic Resonance (Harper and Row, New York, 1963), pp. 127–134, 156–159.

Vernon, F. L.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

J. Appl. Phys. (1)

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Phys. Rev. (1)

R. P. Dicke, Phys. Rev. 93, 99 (1954).
[CrossRef]

Other (3)

W. Gröbner, N. Hofreiter, Integraltafel (Springer-Verlag, Vienna, 1961), Part 2, p. 99.

E. U. Condon, G. H. Shortley, Theory of Atomic Spectra (Maemillan, New York, 1935), pp. 90–93, 100–103.

C. P. Slichter, Principles of Magnetic Resonance (Harper and Row, New York, 1963), pp. 127–134, 156–159.

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Figures (1)

Fig. 1
Fig. 1

The energy density associated with the component of the polarization wave oscillating in the yz plane divided by the energy density associated with the component oscillating in the xz plane, ρp (relative to the same energy distribution coefficient for the components of the electric field wave, ρE) as a function of the difference between the frequency of the waves and the frequency at which the absorptance is a maximum, δ. The half-width of the absorption band at half-maximum is 1/T2. The parameters chosen for the calculation are appropriate for a Q-switched laser beam, elliptically polarized so that ρ E = 1 2, irradiating a sample consisting of organic dye molecules in dilute solution.

Equations (82)

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W = - μ · Re ( F ) .
F ( t ) = F exp ( - i ω t ) .
( H 1 ) o p = - μ o p · Re ( F ) .
μ o p = γ I o p .
( H 1 ) o p = - γ [ ( I x ) o p Re ( F x ) + ( I y ) o p Re ( F y ) ] .
( I x ) o p = 1 2 ( I o p + + I o p - )
( I y ) o p = 1 2 ( I o p + - I o p - ) / i .
( 1 ( H 1 ) o p 2 ) = - m [ Re ( F x ) - i Re ( F y ) ] .
m γ / 2.
Ω ± = m ( F x ± i F y ) .
[ 1 ( H 1 ) o p 2 ] = - [ Ω ˜ + exp ( i ω t ) + Ω - exp ( - i ω t ) ] / 2.
i ( ρ o p / t ) = [ ( H 0 ) o p , ρ o p ] + [ ( H 1 ) o p , ρ o p ] .
( ρ o p ) ρ j k .
ρ 12 = ρ 12 * exp ( i ω 0 t )
ρ 21 = ρ 21 * exp ( - i ω 0 t ) ,
ω 0 = [ ( H 0 ) 22 - ( H 0 ) 11 ] / .
ω 0 = γ F z .
- i ( ρ 11 / t ) = 1 2 [ Ω ˜ + exp ( - i δ + t ) + Ω - exp ( - i δ - t ) ] ρ 21 - 1 2 [ Ω ˜ - exp ( i δ - t ) + Ω + exp ( i δ + t ) ] ρ 12 * ,
( ρ 22 / t ) = ( ρ 11 / t ) ,
- i ( ρ 12 * / t ) = 1 2 ( ρ 11 - ρ 22 ) × [ Ω ˜ + exp ( - i δ + t ) + Ω - exp ( - i δ - t ) ] ,
ρ 21 * = p ˜ 12 * .
δ ± = ω 0 ω .
( R 1 / t ) = - R 3 Im [ Ω + exp ( i δ + t ) ] ,
( R 2 / t ) = R 3 Re [ Ω + exp ( i δ + t ) ] ,
( R 3 / t ) = R 1 Im [ Ω + exp ( i δ + t ) ] - R 2 Re [ Ω + exp ( i δ + t ) ] ,
( Σ / t ) = 0.
R 1 = ρ 12 * + ρ 21 * ,
R 2 = i ( ρ 12 * - ρ 21 * ) ,
R 3 = ρ 11 - ρ 22 ,
Σ = ρ 11 + ρ 22 .
Ω ± = Ω ± exp ( i φ ± ) ,
( R 1 R 2 R 3 ) = [ - sin ( φ + + δ t ) - cos ( φ + + δ t ) 0 cos ( φ + + δ t ) - sin ( φ + + δ t ) 0 0 0 1 ]             ( R 1 + R 2 + R 3 + ) .
( R 1 + / t ) = Ω + R 3 + + δ + R 2 + - R 1 + / T 2 ,
( R 2 + / t ) = - + R 1 + - R 2 + / T 2 + ,
( R 3 + / t ) = - Ω + R 1 + - ( R 3 + - R 0 + ) / T 1 + ,
R 1 ± = R 0 ± Ω ± T 2 ± / S ± ,
R 2 ± = - δ ± T 2 ± R 1 ± ,
R 3 ± = R 0 ± [ 1 + δ ± 2 ( T 2 ± ) 2 ] / S ± ,
S ± = 1 + Ω ± 2 T 1 ± T 2 ± + δ ± 2 ( T 2 ± ) 2 .
M = N T r ( ρ o p μ o p ) .
M x ± = N ± m ± 2 R 0 ± T 2 ± [ δ ± T 2 ± Re ( F x ) Im ( F x ) - Re ( F y ) δ ± T 2 ± Im ( F y ) ] / h S ±
M y ± = N ± m ± 2 R 0 ± T 2 ± [ Re ( F x ) ± δ ± T 2 ± Im ( F x ) + δ ± T 2 ± Re ( F y ) Im ( F y ) ] / S ± .
M ± = Re ( χ ± F ) .
( M x ± M y ± ) = Re [ ( χ x x ± χ x y ± χ y x ± χ y y ± )             ( F x F y ) ] .
M x ± = Re ( χ x x ± ) Re ( F x ) - Im ( χ x x ± ) Im ( F x ) + Re ( χ x y ± ) Re ( F y ) - Im ( χ x y ± ) Im ( F y ) .
χ ± = N ± m ± 2 R 0 ± T 2 ± S ± [ ( δ ± T 2 ± - 1 1 δ ± T 2 ± ) + i ( 1 δ ± T 2 ± - δ ± T 2 ± 1 ) ] ,
N + = N - = N / 2 ,
m + = - m - = m ,
T 2 + = T 2 - = T 2 ,
δ + = δ - = δ ,
R 0 + = - R 0 - = R 0 .
R 0 = [ 1 - exp ( - ω 0 / k T ) ] / [ 1 + exp ( - ω 0 / k T ) ] .
S ± = Δ ( 1 λ ) ,
Δ = 1 + δ 2 T 2 2 + T 1 T 2 m 2 ( F x 2 + F y 2 ) / h 2
λ = 2 T 1 T 2 m 2 F x F y sin ( φ y - φ x ) / Δ 2 .
F x = F x exp ( i φ x )
F y = F y exp ( i φ y ) .
T 1 + = T 1 - = T 1 .
χ χ + + χ - .
χ = N m 2 R 0 T 2 Δ ( 1 - λ 2 ) [ ( δ T 2 - λ λ δ T 2 ) + i ( 1 δ T 2 λ - δ T 2 λ 1 ) ] .
n = N m e 2 R 0 T 2 0 Δ e ( 1 - λ e 2 ) [ ( δ T 2 - λ c λ e δ T 2 ) + i ( 1 δ T 2 λ e - δ T 2 λ e 1 ) ] .
Ψ j ( q , t ) = ϕ j ( q ) exp ( - i E j t / ) .
Ψ = c j ( t ) Ψ j + c j ( t ) Ψ j ,
μ = 2 Re [ c ˜ j ( t ) c j ( t ) exp ( - i ω 0 t ) ϕ ˜ j ( q ) μ o p ϕ j ( q ) d q ] ,
ω 0 ( E j - E j ) / .
μ = m e [ x ^ cos ( ω 0 t ) + y ^ sin ( ω 0 t ) ] .
E = { [ E x x ^ + E y y ^ exp [ i ( φ y - φ x ) ] } × exp [ i ( φ x - ω t ) ] .
P = 0 nE .
ρ E = E y 2 / E x 2
ρ P = P y 2 / P x 2 .
ρ P ρ E .
ρ P = λ e 2 E x 2 + E y 2 - 2 λ e E x E y sin ( φ y - φ x ) E x 2 + λ e 2 E y 2 - 2 λ e E x E y sin ( φ y - φ x ) .
θ T 1 T 2 ( E x 2 + E y 2 ) m e 2 / 2 ,
D 1 + δ 2 T 2 2 ,
ξ 2 sin ( φ y - φ x ) .
λ e = ξ ρ E 1 2 θ / ( 1 + ρ E ) ( θ + D ) .
ρ P = ρ E ξ 2 θ 2 + ( 1 + ρ E ) 2 ( θ + D ) 2 - ξ 2 θ ( 1 + ρ E ) ( θ + D ) ξ 2 θ 2 ρ E 2 + ( 1 + ρ E ) 2 ( θ + D ) 2 - ξ 2 ρ E ( 1 + ρ E ) ( θ + D ) .
ρ P = ρ E [ 2 θ - ( 1 + ρ E ) ( θ + D ) 2 θ ρ E - ( 1 + ρ E ) ( θ + D ) ] 2 .
( φ y - φ x ) / z 2 π / t .
F x z , F y z ( φ y - φ x ) z .
1 2 π 0 2 π d x 1 + b cos x = 1 ( 1 - b 2 ) 1 2 .
χ = N m 2 R 0 T 2 ( δ T 2 + i ) / Δ [ 1 - 4 ρ θ 2 / ( 1 + ρ ) 2 ( θ + D ) 2 ] 1 2 .

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