Abstract

We present a field-theoretical description of the squeezed states of the electromagnetic field. A definition of the squeezed state is introduced that is a natural generalization of the definition for one or two modes. We show that the squeezing produced by the medium with space- and time-dependent material coefficients and μ is directly related to the photon-pair production. In Appendix A we describe the time evolution of the Gaussian squeezed states in terms of the field variables.

© 1987 Optical Society of America

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References

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  1. I. Bialynicki-Birula, “Beyond the coherent states: Gaussons of the electromagnetic field,” in Coherence, Cooperation and Fluctuations, F. Haake, L. M. Narducci, and D. Walls, eds. (Cambridge U. Press, Cambridge, 1986), pp. 159–170.
  2. B. L. Schumaker, “Quantum mechanical pure states with Gaussian wave functions,” Phys. Rep. 135, 317–408 (1986).
    [CrossRef]
  3. D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983).
    [CrossRef]
  4. The fact that the D field and not the E field is the canonically conjugate momentum was previously recognized by M. Born and L. Infeld in their study of nonlinear electrodynamics [M. Born and L. Infeld, “On the quantization of the new field equations I,” Proc. R. Soc. London Ser. A 147, 522–546 (1934); “On the quantization of the new field equations II,” Proc. R. Soc. London Ser. A 150, 141–166 (1935)]. Cf. also I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum Electrodynamics (Pergamon, Oxford, 1975).
    [CrossRef]
  5. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), pp. 158–161.
  6. H. A. Kramers, Quantum Mechanics (North-Holland, Amsterdam, 1957), pp. 418–441.
  7. There is a systematic error in Ref. 1 where these equations were derived; viz., the imaginary unit i should be replaced by −i everywhere.

1986 (1)

B. L. Schumaker, “Quantum mechanical pure states with Gaussian wave functions,” Phys. Rep. 135, 317–408 (1986).
[CrossRef]

1983 (1)

D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983).
[CrossRef]

1934 (1)

The fact that the D field and not the E field is the canonically conjugate momentum was previously recognized by M. Born and L. Infeld in their study of nonlinear electrodynamics [M. Born and L. Infeld, “On the quantization of the new field equations I,” Proc. R. Soc. London Ser. A 147, 522–546 (1934); “On the quantization of the new field equations II,” Proc. R. Soc. London Ser. A 150, 141–166 (1935)]. Cf. also I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum Electrodynamics (Pergamon, Oxford, 1975).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula, “Beyond the coherent states: Gaussons of the electromagnetic field,” in Coherence, Cooperation and Fluctuations, F. Haake, L. M. Narducci, and D. Walls, eds. (Cambridge U. Press, Cambridge, 1986), pp. 159–170.

Born, M.

The fact that the D field and not the E field is the canonically conjugate momentum was previously recognized by M. Born and L. Infeld in their study of nonlinear electrodynamics [M. Born and L. Infeld, “On the quantization of the new field equations I,” Proc. R. Soc. London Ser. A 147, 522–546 (1934); “On the quantization of the new field equations II,” Proc. R. Soc. London Ser. A 150, 141–166 (1935)]. Cf. also I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum Electrodynamics (Pergamon, Oxford, 1975).
[CrossRef]

Infeld, L.

The fact that the D field and not the E field is the canonically conjugate momentum was previously recognized by M. Born and L. Infeld in their study of nonlinear electrodynamics [M. Born and L. Infeld, “On the quantization of the new field equations I,” Proc. R. Soc. London Ser. A 147, 522–546 (1934); “On the quantization of the new field equations II,” Proc. R. Soc. London Ser. A 150, 141–166 (1935)]. Cf. also I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum Electrodynamics (Pergamon, Oxford, 1975).
[CrossRef]

Kramers, H. A.

H. A. Kramers, Quantum Mechanics (North-Holland, Amsterdam, 1957), pp. 418–441.

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), pp. 158–161.

Schumaker, B. L.

B. L. Schumaker, “Quantum mechanical pure states with Gaussian wave functions,” Phys. Rep. 135, 317–408 (1986).
[CrossRef]

Walls, D. F.

D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983).
[CrossRef]

Nature (1)

D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983).
[CrossRef]

Phys. Rep. (1)

B. L. Schumaker, “Quantum mechanical pure states with Gaussian wave functions,” Phys. Rep. 135, 317–408 (1986).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

The fact that the D field and not the E field is the canonically conjugate momentum was previously recognized by M. Born and L. Infeld in their study of nonlinear electrodynamics [M. Born and L. Infeld, “On the quantization of the new field equations I,” Proc. R. Soc. London Ser. A 147, 522–546 (1934); “On the quantization of the new field equations II,” Proc. R. Soc. London Ser. A 150, 141–166 (1935)]. Cf. also I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum Electrodynamics (Pergamon, Oxford, 1975).
[CrossRef]

Other (4)

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), pp. 158–161.

H. A. Kramers, Quantum Mechanics (North-Holland, Amsterdam, 1957), pp. 418–441.

There is a systematic error in Ref. 1 where these equations were derived; viz., the imaginary unit i should be replaced by −i everywhere.

I. Bialynicki-Birula, “Beyond the coherent states: Gaussons of the electromagnetic field,” in Coherence, Cooperation and Fluctuations, F. Haake, L. M. Narducci, and D. Walls, eds. (Cambridge U. Press, Cambridge, 1986), pp. 159–170.

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Equations (63)

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X 1 [ f ] = d 3 r D ( r , 0 ) · f ( r ) ,
X 2 [ g ] = d 3 r B ( r , 0 ) · g ( r ) ,
[ B i ( r , t ) , D j ( r , t ) ] = i i j k k δ ( r - r ) ,
[ B i ( r , t ) , B j ( r , t ) ] = 0 = [ D i ( r , t ) , D j ( r , t ) ] .
[ X 1 [ f ] , X 2 [ g ] ] = i d 3 r f ( r ) · [ × g ( r ) ] .
Δ X 1 [ f ] Δ X 2 [ g ] ½ | d 3 r f ( r ) · [ × g ( r ) ] | .
D ( r ) = i [ 2 ( 2 π ) 3 ] 1 / 2 d 3 k ω 1 / 2 { [ e ( k ) a R ( k ) - e * ( k ) a L ( k ) ] e i k · r - h . c . } ,
B ( r ) = 1 [ 2 ( 2 π ) 3 ] 1 / 2 d 3 k ω 1 / 2 { [ e ( k ) a R ( k ) + e * ( k ) a L ( k ) ] e i k · r + h . c . } ,
e * ( k ) = e ( - k ) ,
k × e ( k ) = - i ω e ( k ) ,
e ( k ) · e ( k ) = 0 ,
e i ( k ) e j * ( k ) = ½ ( δ i j - ω - 2 k i k j - i ω - 1 i j k k k ) ,
[ a λ ( k ) , a λ ( k ) ] = δ λ λ δ ( k - k ) .
Δ X 1 [ f ] = [ ½ d 3 k ω f ˜ ( k ) 2 ] 1 / 2 ,
Δ X 2 [ g ] = [ ½ d 3 k ω g ˜ ( k ) 2 ] 1 / 2 .
d 3 k 2 ω f ˜ ( k ) 2 = 1 = d 3 k 2 ω g ˜ ( k ) 2 .
ω g ˜ ( k ) = ± i k × f ˜ ( k ) .
Δ X 1 [ f ] Δ X 2 [ g ] ¼ .
F ( r , t ) = ½ [ D ( r , t ) + i B ( r , t ) ] ,
t F + i × F = - i × ( κ + F + κ - F ) ,
· F = 0 ,
κ + = ½ ( - 1 - 1 + μ - 1 - 1 ) ,
κ - = ½ ( - 1 - μ - 1 ) .
F ( r , t ) = F in ( r , t ) - i d 3 r d t G R ( r , t ; r , t ) · × [ κ + ( r , t ) · F ( r , t ) + κ - ( r , t ) · F ( r , t ) ] 1 .
G R i j ( r , t ; r , t ) = ( δ i j t - i i j k k ) D R ( r - r , t - t ) ,
F out ( r , t ) = F in ( r , t ) - i × ( t - i × ) d 3 r d t D R × ( r - r , t - t ) [ κ + ( r , t ) · F in ( r , t ) + κ - ( r , t ) · F in ( r , t ) ] .
F in out ( r , t ) = i [ 2 ( 2 π ) 3 ] - 1 / 2 d 3 k ω 1 / 2 e ( k ) [ a R in out ( k ) × exp ( - i ω t + i k · r ) + a L in out ( k ) exp ( i ω t - i k · r ) ] .
a λ out ( k 1 ) = a λ in ( k 1 ) + i ( 2 π ) - 1 d 3 k 2 ( ω 1 ω 2 ) 1 / 2 × [ M λ λ ( k 1 , k 2 ) a λ in ( k 2 ) + N λ λ ( k 1 , k 2 ) a λ int ( k 2 ) ] ,
M R R ( k 1 , k 2 ) = - e * ( k 1 ) · κ ˜ + ( k 1 - k 2 ) · e ( k 2 ) ,
M R L ( k 1 , k 2 ) = e * ( k 1 ) · κ ˜ - ( k 1 - k 2 ) · e * ( k 2 ) = M L R * ( k 2 , k 1 ) ,
M L L ( k 1 , k 2 ) = - e ( k 1 ) · κ ˜ + ( k 1 - k 2 ) · e * ( k 2 ) ,
N R R ( k 1 , k 2 ) = e * ( k 1 ) · κ ˜ - ( k 1 + k 2 ) · e * ( k 2 ) ,
N R L ( k 1 , k 2 ) = e * ( k 1 ) · κ ˜ + ( k 1 + k 2 ) · e ( k 2 ) ,
N L L ( k 1 , k 2 ) = e ( k 1 ) · κ ˜ - ( k 1 + k 2 ) · e ( k 2 ) .
κ ˜ ± ( k 1 ± k 2 ) = κ ˜ ± ( k 1 ± k 2 , ω 1 ± ω 2 ) .
( Δ X 1 out [ f ] ) 2 - ( Δ X 1 in [ f ] ) 2 = i ( 4 π ) - 1 d 3 k 1 ω 1 d 3 k 2 ω 2 × [ - 2 f ˜ + ( k 1 ) · κ ˜ + ( k 1 + k 2 ) · f ˜ - ( k 2 ) - f ˜ + ( k 1 ) · κ ˜ - ( k 1 + k 2 ) · f ˜ + ( k 2 ) - f ˜ - ( k 1 ) · κ ˜ - ( k 1 + k 2 ) · f ˜ - ( k 2 ) - c . c ] = i ( 8 π ) - 1 d 3 k 1 d 3 k 2 { ω 1 ω 2 × f ˜ ( k 1 ) · ˜ - 1 ( k 1 + k 2 ) · f ˜ ( k 2 ) + [ k 1 × f ˜ ( k 1 ) ] · μ ˜ - 1 ( k 1 + k 2 ) · [ k 2 × f ˜ ( k 2 ) ] - c . c . } ,
( Δ X 2 out [ g ] ) 2 - ( Δ X 2 in [ g ] ) 2 = i ( 4 π ) - 1 d 3 k 1 ω 1 d 3 k 2 ω 2 × { - 2 g ˜ + ( k 1 ) · κ ˜ + ( k 1 + k 2 ) · g ˜ - ( k 2 ) + g ˜ + ( k 1 ) · κ ˜ - ( k 1 + k 2 ) · g ˜ + ( k 2 ) + g ˜ - ( k 1 ) · κ ˜ - ( k 1 + k 2 ) · g ˜ - ( k 2 ) - c . c . } = - i ( 8 π ) - 1 d 3 k 1 d 3 k 2 × { ( k 1 × g ˜ ( k 1 ) ) · ˜ - 1 ( k 1 + k 2 ) · [ ( k 2 × g ˜ ( k 2 ) ] + ω 1 ω 2 g ˜ ( k 1 ) · μ ˜ - 1 ( k 1 + k 2 ) · g ˜ ( k 2 ) - c . c . } ,
f ˜ + ( k ) = ( f ˜ ( k ) · e * ( k ) ) e ( k ) = ½ [ f ˜ ( k ) + i ω - 1 k × f ˜ ( k ) ] ,
f ˜ - ( k ) = [ f ˜ ( k ) · e ( k ) ] e * ( k ) = ½ [ f ˜ ( k ) - i ω - 1 k × f ˜ ( k ) ] .
Δ X 1 in [ f ] = ½ = Δ X 2 in [ g ] .
a λ out ( k ) = exp ( i Φ ) a λ in ( k ) exp ( - i Φ ) .
Φ = - ( 4 π ) - 1 d 3 k 1 ω 1 1 / 2 d 3 k 2 ω 2 1 / 2 [ a R ( k 1 ) M R R ( k 1 , k 2 ) a R ( k 2 ) + 2 a R ( k 1 ) M R L ( k 1 , k 2 ) a L ( k 2 ) + a L ( k 1 ) M L L ( k 1 , k 2 ) a L ( k 2 ) + a R ( k 1 ) N R R ( k 1 , k 2 ) a R ( k 2 ) + 2 a R ( k 1 ) N R L × ( k 1 , k 2 ) a L ( k 2 ) + a L ( k 1 ) N L L ( k 1 , k 2 ) a L ( k 2 ) + h . c . ] ,
Φ = d t d 3 r ½ { D in ( r , t ) · [ - 1 ( r , t ) - 1 ] · D in ( r , t ) + B in ( r , t ) · [ μ - 1 ( r , t ) - 1 ] · B in ( r , t ) } .
F out ( r , t ) = S F in ( r , t ) S .
S = T exp [ - i d t H I ( t ) ] ,
a 1 out [ f ] = i d 3 k ( 2 ω ) 1 / 2 { [ f ˜ * ( k ) · e ( k ) ] a L out ( k ) - [ f ˜ * ( k ) · e * ( k ) ] a R out ( k ) } ,
a 2 out [ g ] = d 3 k ( 2 ω ) 1 / 2 { [ g ˜ * ( k ) · e ( k ) ] a L out ( k ) + [ g ˜ * ( k ) · e * ( k ) ] a R out ( k ) } ,
A 11 [ f , f ] = ( 2 ) - 1 / 2 0 out ( a 1 out [ f ] ) 2 0 in .
A 11 [ f , f ] = - i ( 2 ) - 1 / 2 0 in ( a 1 in [ f ] ) 2 Φ 0 in ,
( Δ X 1 out [ f ] ) 2 - ( Δ X 1 in [ f ] ) 2 = ( 2 ) - 1 / 2 Re A 11 [ f , f ] .
A 22 [ g , g ] = - i ( 2 ) - 1 / 2 0 in ( a 2 in [ g ] ) 2 Φ 0 in ,
( Δ X 2 out [ g ] ) 2 - ( Δ X 2 in [ g ] ) 2 = ( 2 ) - 1 / 2 Re A 22 [ g , g ] .
i t Ψ ( t ) = H Ψ ( t ) .
Ψ ( t ) = Ψ [ A t ] ,
B ( r ) Ψ [ A ] = × A ( r ) Ψ [ A ] ,
D ( r ) Ψ [ A ] = - i δ / δ A ( r ) Ψ [ A ] .
H = ½ d 3 r { - [ δ / δ A ( r ) ] 2 + [ × A ( r ) ] 2 } .
Ψ 0 [ A ] = N exp { - ( 2 π ) - 2 d 3 r d 3 r × [ A ( r ) ] r - r - 2 [ × A ( r ) ] } .
Ψ [ A t ] = N ( t ) exp { - ½ d 3 r d 3 r [ × A ( r ) - B ( r , t ) ] i × W i j ( r , r ; t ) [ × A ( r ) - B ( r , t ) ] j + i d 3 r E ( r , t ) · A ( r ) } .
t B = - × E ,             · B = 0 ,
t E = × B ,             · E = 0 ,
t K i j ( r , r ; t ) = - i d 3 r 1 K i k ( r , r 1 ; t ) K k j ( r 1 ; r ; t ) + i ( i j - δ i j Δ ) δ ( r - r ) ,
K i j ( r , r ; t ) = - i k l k W l m ( r , r ; t ) n m n j .

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