## Relative intensity squeezing by four-wave mixing with loss: an analytic model and experimental diagnostic |

Optics Express, Vol. 19, Issue 4, pp. 3765-3774 (2011)

http://dx.doi.org/10.1364/OE.19.003765

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### 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 ^{85}Rb vapor and demonstrate the analytic model’s utility as an experimental diagnostic.

© 2011 Optical Society of America

## 1. Introduction

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]

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]

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]

## 2. Relative intensity squeezing

*â*,

*b̂*and

*ĉ*respectively and the interaction strength by

*ξ*, the interaction picture Hamiltonian is In the “undepleted pump” approximation, the intense pump beam remains in its initial coherent state |

*ψ*〉 and the substitution

_{c}*ĉ*→

*ψ*can be made: The time-evolution of this Hamiltonian over the interaction time-scale

_{c}*τ*is This is the two-mode squeezing operator for modes

*â*and

*b̂*, where

*s*is the “squeezing parameter” [9]. The four-wave mixing system therefore produces a two-mode squeezed state, reducing amplitude difference noise at the expense of increasing phase difference noise [5

5. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science **321**, 544–547 (2008). [CrossRef] [PubMed]

*s*results in a rotation of the (arbitrary) measurement quadratures, so

*s*may be taken as real and positive. The probe and conjugate modes

*â*and

*b̂*are then transformed as Defining the number operator of the incident probe beam as

*N̂*

_{0}〉 ≫ 1, the number operators after squeezing become where

*G*≡ cosh

^{2}

*s*is the increase in probe intensity, termed the “mixing gain”.

## 3. Optical losses after squeezing

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]

*x̂*and

*ŷ*respectively, the standard beam-splitter input-output relations [12] give where

*η*and

_{a}*η*are the fractions of the probe and conjugate intensities transmitted. The relative intensity noise can then be expressed in terms of the individual beam variances and covariance to give Computing the variances using Eq. (2), the noise figure corresponding to four-wave mixing followed by optical losses is This expression highlights the importance of balanced beam detection, as unbalanced losses (

_{b}*η*≠

_{a}*η*) result in detection of amplified noise instead of squeezing.

_{b}## 4. Optical losses during squeezing: Interleaved gain/loss model

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]

*N*discrete interleaved stages of gain and loss (Fig. 2). Distributed models of this type were first proposed by Loudon [13

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]

*â*and

_{N}*â*

_{0}and

*x̂*and

_{i}*ẑ*consisting of the probe annihilation and conjugate creation operators

_{i}*ẑ*

_{1}=

*â*

_{0}and

*ẑ*

_{2i+1}=

*x̂*and

_{i}*i*≤

*N*. Expanding Eq. (7) in terms of this operator and a set of coefficients

*α*and

_{i}*β*gives

_{i}*N̂*

_{0}≫ 1), this variance simplifies to The shot-noise limit is

*N̂*

_{0}) = 〈

*N̂*

_{0}〉 is It remains to express the

*α*,

_{i}*β*coefficients in terms of the model parameters

_{i}*T*,

_{a}*T*and

_{b}*S*, and hence obtain an ab-initio expression for the degree of squeezing.

*N*vacuum modes

*x̂*contribute a term to the variance in Eq. (10):

_{i}*i*= 1, 2 contributions to the variance in Eq. (10), and summing over the vacuum contributions gives The continuum behaviour is recovered in the limit

*N*→ ∞. To obtain a closed form expression for the sum, the infinitesimal parameters are expanded as a power series in 1/

*N*. Expanding the elements of

**A**in Eq. (8) gives Similarly taking

*N*using Eq. (13) gives Taking the limit

*N*→ ∞, the neglected

*O*(1/

*N*

^{2}) terms vanish and

**A**

*→ exp(*

^{N}**A**

_{0}). Hence the sum

**X**converges and obeys This is a system of four linear equations for the elements of

**X**in terms of the model parameters

*T*,

_{a}*T*and

_{b}*S*, and can be solved algebraically.

**X**contains all terms in the variance of Eq. (10) except

*i*= 1,2 which correspond to the probe and conjugate coefficients. The probe contribution is while the conjugate contribution (

*α*

_{1}

*α*

_{2}–

*β*

_{1}

*β*

_{2})

^{2}has diagonal ⌈0 1⌋. Computing the full variance sum in Eq. (10) yields We introduce one final stage of loss to model optical losses after mixing (as in §4), scaling each coefficient by the relevant transmission factor (

*T*,

_{a}*T*and

_{b}*S*, but runs to a dozen typeset lines. However, special cases are readily derived and provide physical insight not readily accessible from numerical models.

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]

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]

*η*=

_{a}*η*≡

_{b}*η*) and the far-detuned conjugate experiences negligible absorption (

*T*= 1). The corresponding degree of squeezing is with parameters

_{b}*χ*= (log

*T*– log

_{a}*T*)/4

_{b}*ξ*. The three terms describe the shot-noise limit, correlations from four-wave mixing, and injected vacuum noise.

*T*and intrinsic mixing gain

_{a}*G*= cosh

^{2}

*S*(in excellent agreement with the numerical model of Ref. [3

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]

*T*< 1). This is because the shot noise carried by the incident probe beam is also amplified by the mixing process, and a small “optimal” level of probe loss decreases this contribution to the measured noise power before injected vacuum noise dominates. This optimal level is easily obtained by minimizing Eq. (16).

_{a}*T*= 1) compared to the conjugate (

_{a}*T*< 1), and the predicted squeezing is The vacuum noise term in Eq. (17) is considerably larger than in Eq. (16), resulting in several decibels difference for moderate levels of absorption (Fig. 4B). Unlike the probe losses discussed above, losses on the conjugate only destroy correlations and introduce noise, so squeezing by four-wave mixing in the reverse configuration is always less effective for the same level of intrinsic mixing gain.

_{b}*T*,

_{a}*T*→ 1, the post-mixing optical-loss result of Eq. (5) is obtained.

_{b}## 5. Experimental diagnostic

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]

*μ*W probe beam with 1/

*e*

^{2}beam waists of 630

*μ*m and 375

*μ*m respectively at an angle of 0.3° within a pure

^{85}Rb vapor cell of internal length 7mm heated to 130°C (Fig. 1B). The probe was generated by an AOM with fixed detuning 3040MHz below the pump, which was scanned across the Doppler broadened

*D*

_{1}resonance at 795nm. The relative intensity between probe and conjugate was measured with a balanced photodetector (Thorlabs PDB150A), refitted with high-efficiency photodiodes (Hamamatsu S3883, net efficiency 95%). The overall detection efficiency of the system was

*η*= 85 ± 1%. The relative intensity noise was measured with a Rhode & Schwarz FSP7 spectrum analyzer at an analysis frequency of 1MHz with 30kHz resolution bandwidth.

*G*and

_{a}*G*) were measured as a function of pump beam detuning (Fig. 5A) and used to simultaneously solve Eq. (15) for the intrinsic gain

_{b}*G*and probe transmission

*T*(Fig. 5B) via the coefficients

_{a}*α*

_{1}and

*α*

_{2}. Note that the gain resonance extends well into the Doppler-broadened absorption resonance for detunings below 600MHz from the line-centre, demonstrating strong competition between the processes.

## 6. Conclusions

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]

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**, 544–547 (2008). [CrossRef] [PubMed]

## References and links

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. |

2. | Q. Glorieux, L. Guidoni, S. Guibal, J.-P. Likforman, and T. Coudreau, “Strong quantum correlations in four wave mixing in |

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 |

4. | V. Boyer, A. M. Marino, and P. D. Lett, “Generation of spatially broadband twin beams for quantum imaging,” Phys. Rev. Lett. |

5. | V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science |

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. |

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. |

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 |

9. | C. C. Gerry and P. L. Knight, |

10. | H. A. Bachor and T. C. Ralph, |

11. | C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. |

12. | R. Loudon, |

13. | R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. |

14. | C. M. Caves and D. D. Crouch, “Quantum wideband traveling-wave analysis of a degenerate parametric amplifier,” J. Opt. Soc. Am. B |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 14, 2010

Revised Manuscript: January 24, 2011

Manuscript Accepted: January 24, 2011

Published: February 11, 2011

**Citation**

M. Jasperse, L. D. Turner, and R. E. Scholten, "Relative intensity squeezing by four-wave mixing with loss: an analytic model and experimental diagnostic," Opt. Express **19**, 3765-3774 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3765

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### References

- 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]
- 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. 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]
- 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]
- 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. 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]
- C. M. Caves, "Quantum-mechanical radiation-pressure fluctuations in an interferometer," Phys. Rev. Lett. 45, 75-79 (1980). [CrossRef]
- R. Loudon, The quantum theory of light (Oxford University Press, 1983), 2nd ed.
- R. Loudon, "Theory of noise accumulation in linear optical-amplifier chains," IEEE J. Quantum Electron. 21, 766-773 (1985). [CrossRef]
- 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]

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