Abstract

Four-wave mixing near resonance in an atomic vapor can produce relative intensity squeezed light suitable for precision measurements beyond the shot-noise limit. We develop an analytic distributed gain/loss model to describe the competition of mixing and absorption through the non-linear medium. Using a novel matrix calculus, we present closed-form expressions for the degree of relative intensity squeezing produced by this system. We use these theoretical results to analyze experimentally measured squeezing from a 85Rb vapor and demonstrate the analytic model’s utility as an experimental diagnostic.

© 2011 Optical Society of America

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  1. C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, "Strong relative intensity squeezing by four-wave mixing in rubidium vapor," Opt. Lett. 32, 178-180 (2007).
    [CrossRef]
  2. Q. Glorieux, L. Guidoni, S. Guibal, J.-P. Likforman, and T. Coudreau, "Strong quantum correlations in four wave mixing in 85Rb vapor," (SPIE, 2010), vol. 7727 of Proc. SPIE, p. 772703.
  3. C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, "Strong low-frequency quantum correlations from a four-wave-mixing amplifier," Phys. Rev. A 78, 043816 (2008).
    [CrossRef]
  4. V. Boyer, A. M. Marino, and P. D. Lett, "Generation of spatially broadband twin beams for quantum imaging," Phys. Rev. Lett. 100, 143601 (2008).
    [CrossRef] [PubMed]
  5. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from four-wave mixing," Science 321, 544 (2008).
    [CrossRef] [PubMed]
  6. V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett, "Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor," Phys. Rev. Lett. 99, 143601 (2007).
    [CrossRef] [PubMed]
  7. R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, "Low-noise amplification of a continuous-variable quantum state," Phys. Rev. Lett. 103, 010501 (2009).
    [CrossRef] [PubMed]
  8. Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
    [CrossRef]
  9. C. C. Gerry, and P. L. Knight, Introductory quantum optics, (Cambridge University Press, 2005).
  10. H. A. Bachor, and T. C. Ralph, A guide to experiments in quantum optics, (Wiley-VCH Verlag, 2004).
    [CrossRef]
  11. C. M. Caves, "Quantum-mechanical radiation-pressure fluctuations in an interferometer," Phys. Rev. Lett. 45, 75-79 (1980).
    [CrossRef]
  12. R. Loudon, The quantum theory of light (Oxford University Press, 1983), 2nd ed.
  13. R. Loudon, "Theory of noise accumulation in linear optical-amplifier chains," IEEE J. Quantum Electron. 21, 766-773 (1985).
    [CrossRef]
  14. C. M. Caves, and D. D. Crouch, "Quantum wideband traveling-wave analysis of a degenerate parametric amplifier," J. Opt. Soc. Am. B 4, 1535-1545 (1987).
    [CrossRef]

2010 (1)

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

2009 (1)

R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, "Low-noise amplification of a continuous-variable quantum state," Phys. Rev. Lett. 103, 010501 (2009).
[CrossRef] [PubMed]

2008 (3)

C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, "Strong low-frequency quantum correlations from a four-wave-mixing amplifier," Phys. Rev. A 78, 043816 (2008).
[CrossRef]

V. Boyer, A. M. Marino, and P. D. Lett, "Generation of spatially broadband twin beams for quantum imaging," Phys. Rev. Lett. 100, 143601 (2008).
[CrossRef] [PubMed]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from four-wave mixing," Science 321, 544 (2008).
[CrossRef] [PubMed]

2007 (2)

V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett, "Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor," Phys. Rev. Lett. 99, 143601 (2007).
[CrossRef] [PubMed]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, "Strong relative intensity squeezing by four-wave mixing in rubidium vapor," Opt. Lett. 32, 178-180 (2007).
[CrossRef]

1987 (1)

1985 (1)

R. Loudon, "Theory of noise accumulation in linear optical-amplifier chains," IEEE J. Quantum Electron. 21, 766-773 (1985).
[CrossRef]

1980 (1)

C. M. Caves, "Quantum-mechanical radiation-pressure fluctuations in an interferometer," Phys. Rev. Lett. 45, 75-79 (1980).
[CrossRef]

Arimondo, E.

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, "Strong relative intensity squeezing by four-wave mixing in rubidium vapor," Opt. Lett. 32, 178-180 (2007).
[CrossRef]

V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett, "Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor," Phys. Rev. Lett. 99, 143601 (2007).
[CrossRef] [PubMed]

Boyer, V.

R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, "Low-noise amplification of a continuous-variable quantum state," Phys. Rev. Lett. 103, 010501 (2009).
[CrossRef] [PubMed]

C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, "Strong low-frequency quantum correlations from a four-wave-mixing amplifier," Phys. Rev. A 78, 043816 (2008).
[CrossRef]

V. Boyer, A. M. Marino, and P. D. Lett, "Generation of spatially broadband twin beams for quantum imaging," Phys. Rev. Lett. 100, 143601 (2008).
[CrossRef] [PubMed]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from four-wave mixing," Science 321, 544 (2008).
[CrossRef] [PubMed]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, "Strong relative intensity squeezing by four-wave mixing in rubidium vapor," Opt. Lett. 32, 178-180 (2007).
[CrossRef]

V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett, "Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor," Phys. Rev. Lett. 99, 143601 (2007).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, and D. D. Crouch, "Quantum wideband traveling-wave analysis of a degenerate parametric amplifier," J. Opt. Soc. Am. B 4, 1535-1545 (1987).
[CrossRef]

C. M. Caves, "Quantum-mechanical radiation-pressure fluctuations in an interferometer," Phys. Rev. Lett. 45, 75-79 (1980).
[CrossRef]

Coudreau, T.

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

Crouch, D. D.

Dubessy, R.

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

Glorieux, Q.

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

Guibal, S.

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

Guidoni, L.

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

Jones, K. M.

R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, "Low-noise amplification of a continuous-variable quantum state," Phys. Rev. Lett. 103, 010501 (2009).
[CrossRef] [PubMed]

Lett, P. D.

R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, "Low-noise amplification of a continuous-variable quantum state," Phys. Rev. Lett. 103, 010501 (2009).
[CrossRef] [PubMed]

C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, "Strong low-frequency quantum correlations from a four-wave-mixing amplifier," Phys. Rev. A 78, 043816 (2008).
[CrossRef]

V. Boyer, A. M. Marino, and P. D. Lett, "Generation of spatially broadband twin beams for quantum imaging," Phys. Rev. Lett. 100, 143601 (2008).
[CrossRef] [PubMed]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from four-wave mixing," Science 321, 544 (2008).
[CrossRef] [PubMed]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, "Strong relative intensity squeezing by four-wave mixing in rubidium vapor," Opt. Lett. 32, 178-180 (2007).
[CrossRef]

V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett, "Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor," Phys. Rev. Lett. 99, 143601 (2007).
[CrossRef] [PubMed]

Likforman, J.-P.

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

Loudon, R.

R. Loudon, "Theory of noise accumulation in linear optical-amplifier chains," IEEE J. Quantum Electron. 21, 766-773 (1985).
[CrossRef]

Marino, A. M.

R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, "Low-noise amplification of a continuous-variable quantum state," Phys. Rev. Lett. 103, 010501 (2009).
[CrossRef] [PubMed]

C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, "Strong low-frequency quantum correlations from a four-wave-mixing amplifier," Phys. Rev. A 78, 043816 (2008).
[CrossRef]

V. Boyer, A. M. Marino, and P. D. Lett, "Generation of spatially broadband twin beams for quantum imaging," Phys. Rev. Lett. 100, 143601 (2008).
[CrossRef] [PubMed]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from four-wave mixing," Science 321, 544 (2008).
[CrossRef] [PubMed]

McCormick, C. F.

C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, "Strong low-frequency quantum correlations from a four-wave-mixing amplifier," Phys. Rev. A 78, 043816 (2008).
[CrossRef]

V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett, "Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor," Phys. Rev. Lett. 99, 143601 (2007).
[CrossRef] [PubMed]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, "Strong relative intensity squeezing by four-wave mixing in rubidium vapor," Opt. Lett. 32, 178-180 (2007).
[CrossRef]

Pooser, R. C.

R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, "Low-noise amplification of a continuous-variable quantum state," Phys. Rev. Lett. 103, 010501 (2009).
[CrossRef] [PubMed]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from four-wave mixing," Science 321, 544 (2008).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

R. Loudon, "Theory of noise accumulation in linear optical-amplifier chains," IEEE J. Quantum Electron. 21, 766-773 (1985).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. A (2)

C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, "Strong low-frequency quantum correlations from a four-wave-mixing amplifier," Phys. Rev. A 78, 043816 (2008).
[CrossRef]

Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, "Double-λ microscopic model for entangled light generation by four-wave mixing," Phys. Rev. A 82, 033819 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett, "Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor," Phys. Rev. Lett. 99, 143601 (2007).
[CrossRef] [PubMed]

R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, "Low-noise amplification of a continuous-variable quantum state," Phys. Rev. Lett. 103, 010501 (2009).
[CrossRef] [PubMed]

V. Boyer, A. M. Marino, and P. D. Lett, "Generation of spatially broadband twin beams for quantum imaging," Phys. Rev. Lett. 100, 143601 (2008).
[CrossRef] [PubMed]

C. M. Caves, "Quantum-mechanical radiation-pressure fluctuations in an interferometer," Phys. Rev. Lett. 45, 75-79 (1980).
[CrossRef]

Science (1)

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from four-wave mixing," Science 321, 544 (2008).
[CrossRef] [PubMed]

Other (4)

Q. Glorieux, L. Guidoni, S. Guibal, J.-P. Likforman, and T. Coudreau, "Strong quantum correlations in four wave mixing in 85Rb vapor," (SPIE, 2010), vol. 7727 of Proc. SPIE, p. 772703.

C. C. Gerry, and P. L. Knight, Introductory quantum optics, (Cambridge University Press, 2005).

H. A. Bachor, and T. C. Ralph, A guide to experiments in quantum optics, (Wiley-VCH Verlag, 2004).
[CrossRef]

R. Loudon, The quantum theory of light (Oxford University Press, 1983), 2nd ed.

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Figures (5)

Fig. 1
Fig. 1

(A) Four-wave mixing energy-level transitions and (B) and experimental schematic.

Fig. 2
Fig. 2

Competing gain and loss processes modelled by interleaved stages of squeezing (SQZ) and loss.

Fig. 3
Fig. 3

Predicted squeezing for four-wave mixing with negligible conjugate absorption and net detection efficiency η = 85%.

Fig. 4
Fig. 4

(A) Four-wave mixing in the “reverse” configuration. (B) Predicted squeezing for four-wave mixing in the normal/reverse configurations with intrinsic gain G = 3.

Fig. 5
Fig. 5

(A) Measured probe and conjugate gain across four-wave mixing resonance and (B) inferred model parameters. (C) Experimentally measured squeezing and (D) associated noise powers compared to model predictions for these parameters. Detunings are for pump beam and are measured above the centre of the 85Rb 5S1/2(F = 2) → 5P1/2 transition.

Equations (30)

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^ i = i ( ξ b ^ c ^ a ^ c ^ ξ * c ^ a ^ c ^ b ^ ) .
^ i = i ( ξ ψ c 2 b ^ a ^ ξ * ( ψ c * ) 2 a ^ b ^ ) .
S ^ exp ( i H ^ i τ / ) = exp ( s b ^ a ^ s * a ^ b ^ ) , where s = ξ ψ c 2 τ .
a ^ S ^ a ^ S ^ = cosh ( s ) a ^ + sinh ( s ) b ^ and b ^ S ^ b ^ S ^ = sinh ( s ) a ^ + cosh ( s ) b ^ .
N ^ a a ^ a ^ G N ^ 0 and N ^ b b ^ b ^ ( G 1 ) N ^ 0 ,
NF Var ( N ^ a N ^ b ) N ^ a + N ^ b = N ^ 0 G N ^ 0 + ( G 1 ) N ^ 0 = 1 2 G 1 .
a ^ η a a ^ + 1 η a x ^ and b ^ η b b ^ + 1 η b y ^ ,
Var ( N ^ a N ^ b ) = η a 2 Var ( N ^ a ) + η a ( 1 η a ) N ^ a + η b 2 Var ( N ^ b ) + η a ( 1 η b ) N ^ b 2 η a η b CoVar ( N ^ a , N ^ b ) .
NF = 1 + 2 ( G 1 ) ( G ( η a η b ) 2 η b 2 ) G η a + ( G 1 ) η b .
a ^ n + 1 = t a ( cosh s a ^ n + sinh s b ^ n ) + 1 t a 2 x ^ n + 1 , b ^ n + 1 = t b ( sinh s a ^ n + cosh s b ^ n ) + 1 t b 2 y ^ n + 1 .
( a ^ N b ^ N ) = A ( a ^ N 1 b ^ N 1 ) + ( 1 t a 2 x ^ N 1 t b 2 y ^ N ) = A N ( a ^ 0 b ^ 0 ) + i = 1 N A N i ( 1 t a 2 x ^ i 1 t b 2 y ^ i ) ,
A = ( t a cosh s t a sinh s t b sinh s t b cosh s ) .
a ^ N = α 1 a ^ 0 + α 2 b ^ 0 + α 3 x ^ 1 + α 4 y ^ 1 + + α 2 N + 1 x ^ N + α 2 N + 2 y ^ N = i α i z ^ i , b ^ N = β 1 a ^ 0 + β 2 b ^ 0 + β 3 x ^ 1 + β 4 y ^ 1 + + β 2 N + 1 x ^ N + β 2 N + 2 y ^ N = i β i z ^ i .
N ^ a N ^ b = a ^ N a ^ N b ^ N b ^ N = i , j ( α i α j β i β j ) z ^ i z ^ j + 1 and Var ( N ^ a N ^ b ) = i , j , k , l ( α i α j β i β j ) ( α k α l β k β l ) CoVar ( z ^ i z ^ j , z ^ k z ^ l ) .
Var ( N ^ a N ^ b ) = ( α 1 2 β 1 2 ) 2 Var ( N ^ 0 ) + i > 1 ( α 1 α i β 1 β i ) 2 N ^ 0 .
NF Var ( N ^ a N ^ b ) N ^ a + N ^ b = i = 1 N ( α 1 α i β 1 β i ) 2 α 1 2 + β 1 2 .
( α 1 α 2 β 1 β 2 ) = A N and ( α 2 i + 1 α 2 i + 2 β 2 i + 1 β 2 i + 2 ) = A N i ( 1 t a 2 0 0 1 t b 2 ) .
( α 1 α 2 i + 1 β 1 β 2 i + 1 ) 2 = ( α 1 , β 1 ) ( α 2 i + 1 β 2 i + 1 ) ( α 2 i + 1 , β 2 i + 1 ) ( α β 1 ) = ( α 1 , β 1 ) A N i ( 1 t a 2 0 ) ( 1 t a 2 , 0 ) ( A T ) N i ( α 1 β 1 ) = ( α 1 , β 1 ) A N i ( 1 t a 2 0 0 0 ) A N i ( α 1 β 1 ) .
i > 2 ( α 1 α i β 1 β i ) 2 = ( α 1 , β 1 ) { i = 1 N A N i ( 1 t a 2 0 0 1 t b 2 ) A N i } ( α 1 β 1 ) .
A = 1 + 1 N A 0 + O ( 1 N 2 ) where A 0 = ( 1 2 log T a S S 1 2 log T b ) .
X = i = 1 N A N i ( 1 N T ) A N i = 1 N i = 0 N 1 A i T A i with T = ( log T a 0 0 log T b ) .
AXA X = 1 N ( A N T A N T ) .
AXA X = 1 N { A 0 X + X A 0 } A 0 X + X A 0 = A N T A N T .
A 0 X + X A 0 = exp ( A 0 ) T exp ( A 0 ) T .
( α 1 2 β 1 2 ) 2 = { ( α 1 , β 1 ) ( α 1 β 1 ) } 2 = ( α 1 , β 1 ) e A 0 ( 1 0 0 0 ) e A 0 ( α 1 β 1 ) ,
Var ( N ^ a N ^ b ) = ( α 1 , β 1 ) { e 2 A 0 + X } ( α 1 β 1 ) with ( α 1 β 1 ) = ( 1 0 0 1 ) e A 0 ( 1 0 ) .
Var ( N ^ a N ^ b ) = ( α 1 , β 1 ) { P ( e 2 A 0 + X ) P + ( 1 P ) P } ( α β 1 ) N ^ 0 where P = ( η a 0 0 η b ) .
G a N ^ a N ^ 0 = η a α 1 2 and G b N ^ b N ^ 0 = η b α 2 2 with ( α 1 α 2 ) = e A 0 ( 1 0 ) ,
NF = 1 η 2 S sinh 2 ξ ξ cosh ( 2 ξ + χ ) + η T a S log 2 T a sinh 4 ξ 2 ξ 3 cosh ( 2 ξ + χ ) ,
NF = 1 η 2 S cosh 2 ( ξ + χ ) ξ cosh ( 2 ξ + χ ) + η T b S ( 4 S log T b sinh ( 2 ξ + χ ) ) 2 8 ξ 3 cosh ( 2 ξ + χ ) .

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