## Amplitude squeezing in a Mach-Zehnder fiber interferometer: Numerical analysis of experiments with microstructure fiber

Optics Express, Vol. 10, Issue 2, pp. 128-138 (2002)

http://dx.doi.org/10.1364/OE.10.000128

Acrobat PDF (493 KB)

### Abstract

We study a Mach-Zehnder nonlinear fiber interferometer for the generation of amplitude-squeezed light. Numerical simulations of experiments with microstructure fiber are performed using linearization of the quantum nonlinear Shrödinger equation. We include in our model the effect of distributed linear losses in the fiber.

© Optical Society of America

## 1 Introduction

1. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. **57**, 691 (1987). [CrossRef]

2. M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. **66**, 153 (1991). [CrossRef] [PubMed]

3. A. Sizmann and G. Leuchs, “The optical Kerr effect and quantum optics in fibers,” in *Progress in Optics XXXIX*, E. Wolf, ed. (Elsevier Scinece B.V.,1999). [CrossRef]

4. Ch. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fibre,” Phys. Rev. Lett. **86**, 4267 (2001). [CrossRef] [PubMed]

5. A. Furasawa, J.L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science282, 706 (1998); S. L. Braunstein and H. J. Kimble, “Dense coding for continuous variables,” Phys. Rev. A612302 (2000). [CrossRef]

2. M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. **66**, 153 (1991). [CrossRef] [PubMed]

6. S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. **81**, 2446 (1998). [CrossRef]

7. S. Spälter, M. Burk, U Ströner, A. Sizmann, and G. Leuchs, “Propagation of quantum properties of sub-picosecond solitons in a fiber,” Opt. Express **2**, 77 (1998). [CrossRef]

8. D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. **23**, 1390 (1998). [CrossRef]

9. D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. **24**, 948 (1999) [CrossRef]

6. S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. **81**, 2446 (1998). [CrossRef]

8. D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. **23**, 1390 (1998). [CrossRef]

6. S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. **81**, 2446 (1998). [CrossRef]

8. D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. **23**, 1390 (1998). [CrossRef]

10. M. Fiorentino, J. E. Sharping, P. Kumar, D. Levandovsky, and M. Vasilyev, “Soliton squeezing in a Mach-Zehnder fiber interferometer,” Phys. Rev. A **64**, 031801(R) (2001). [CrossRef]

10. M. Fiorentino, J. E. Sharping, P. Kumar, D. Levandovsky, and M. Vasilyev, “Soliton squeezing in a Mach-Zehnder fiber interferometer,” Phys. Rev. A **64**, 031801(R) (2001). [CrossRef]

12. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. **25**, 25 (2000). [CrossRef]

12. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. **25**, 25 (2000). [CrossRef]

13. X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggelton, S. G. Kosinsky, and R. S. Windeler, “Solitonself-phase shift in a short tapered air-silica microstructure fiber,” Opt. Lett. **26**, 358 (2001). [CrossRef]

14. J. E. Sharping, M. Fiorentino, and P. Kumar, “Four wave mixing in microstructure fibers,” Opt. Lett. **26**, 1048 (2001). [CrossRef]

15. F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,” Opt. Lett. **26**, 1558 (2001). [CrossRef]

## 2 Numerical Analysis, Lossless Fiber

*N*= 1), an analytical solution for the propagation of the quantum noise of the soliton is possible [16

16. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B **7**, 386 (1990). [CrossRef]

17. D. J. Kaup, “Perturbation theory for solitons in optical fibres,” Phys. Rev. A **42**, 5689 (1990). [CrossRef] [PubMed]

18. D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive loop mirror,” Opt. Lett. **24**, 89 (1999). [CrossRef]

18. D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive loop mirror,” Opt. Lett. **24**, 89 (1999). [CrossRef]

**81**, 2446 (1998). [CrossRef]

19. C. R. Doerr, M. Shirasaki, and F. I. Khatri, “Simulation of pulsed squeezing in optical fiber with chromatic dispersion,” J. Opt. Soc. Am. B, **11**, 142 (1994). [CrossRef]

10. M. Fiorentino, J. E. Sharping, P. Kumar, D. Levandovsky, and M. Vasilyev, “Soliton squeezing in a Mach-Zehnder fiber interferometer,” Phys. Rev. A **64**, 031801(R) (2001). [CrossRef]

20. M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B **7**, 30 (1990) [CrossRef]

*Û*for the envelope of the electric field in the fiber. Equation (1) is written in a retarded frame moving together with the pulse along

*z*, which is expressed in standard normalized-length units [21]. We linearize this equation by putting in Eq. (1), equation (2) yields a pair of coupled equations: the zeroth-order expansion represents the classical NLSE, describing the evolution of the envelope

*U*¯,

*U*¯ = (

*Û*) and u is the annihilation operator for the fluctuations, and keeping only terms up to first order in

*û*. Such linearization approximation is valid in the limit of fluctuations that are small compared with the mean values of the field. When substituted in Eq. (1), equation (2) yields a pair of coupled equations: the zeroth-order expansion represents the classical NLSE, describing the evolution of the envelope ¯,

*û*

*t*, and the propagation variable,

*z*. We use a split-step Fourier method [21], which is simple to implement with high resolution and precision. The idea underlying this method is illustrated schematically in Fig. 2 (a). The fiber is divided into small segments of length Δ

*z*, and an approximate solution to the equation is found by pretending that over Δ

*z*the nonlinearity [represented by the first term on the right side of Eq. (3)] and dispersion [represented by the second term on the right side of Eq. (3)] act independently. This approximation allows one to exactly solve each evolution step by matrix exponentiation, which can be efficiently implemented numerically. To further increase the precision of the solution, we use a symmetrized split-step method, where the dispersion is calculated at the mid plane of the segment and is preceded and followed by nonlinear propagation. Once the solution for Eq. (3) is obtained, one can solve Eq. (4) applying a discretization procedure as described by Doerr

*et alii*[19

19. C. R. Doerr, M. Shirasaki, and F. I. Khatri, “Simulation of pulsed squeezing in optical fiber with chromatic dispersion,” J. Opt. Soc. Am. B, **11**, 142 (1994). [CrossRef]

*M*temporal slices and each time slice is treated as an independent mode of the electromagnetic field. The modes’ operators evolve following Eq. (4). When Eq. (5) is substituted into Eq. (4) one obtains a set of coupled equations for

*û*and

*û*

^{†}. Without loss of generality one can express the mode

*u*

_{j}in

*z*as [19

19. C. R. Doerr, M. Shirasaki, and F. I. Khatri, “Simulation of pulsed squeezing in optical fiber with chromatic dispersion,” J. Opt. Soc. Am. B, **11**, 142 (1994). [CrossRef]

*μ*(

*z*) and

*ν*(

*z*) describe the evolution of the modes, and their off-diagonal elements represent cross-correlations between the various temporal slices of the pulse that develop due to the dispersion. Equation (6) is valid under the same limits that guarantee the validity of the linearization procedure [22

22. C. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D **26**, 1817 (1982). [CrossRef]

*ω*is the associated frequency variable. The coupled equations can then be solved using the symmetrized split-step Fourier method.

*U*¯′ and

*U*¯′′ , respectively, through equal lengths of fiber. Then, using these numerical solutions for the averaged envelopes, we propagate the associated noise operators,

*û*′ and

*û*″. Finally the two fields are mixed at the output beamsplitter (BS2) and the noise of the field emerging from one port of the interferometer is calculated, assuming the use of a direct-detection scheme. The two modes

*Û*′ and

*Û*″ propagating in the two arms of the interferometer can be linearized as in Eq. (2) and then expanded as in Eq. (6). The output mode will be

*Û*‴ =

*U*¯‴ +

*û*‴, where

*T*

_{2}is the transmittivity of the output beamsplitter (BS2),

*R*

_{2}= 1 -

*T*

_{2}is the corresponding reflectivity, and

*ϕ*is the phase difference between the two arms of the interferometer that can be changed at will. The low-frequency spectrum Φ

_{0}of the current [19

**11**, 142 (1994). [CrossRef]

*θ*

_{j}= arg (

*U*¯‴

_{j}), and we have introduced the evolution matrices

*μ*

_{jk}′,

*ν*

_{jk}″,

*μ*

_{jk}′ and

*ν*

_{jk}″ for the modes

*û*′ and

*û*″ similar to those defined in Eq. (6). To evaluate the average in Eq. (9) we have assumed that the modes are initially in the vacuum state. This assumption is valid because in our experiment the measurements of photocurrent fluctuations are made in a frequency band where the laser source is shot-noise limited. The quantum-noise reduction (QNR) is then calculated as the ratio of the output photocurrent fluctuations as normalized to the shot noise [19

**11**, 142 (1994). [CrossRef]

## 3 Numerical Analysis: Lossy Fiber

*Û*with a reservoir of loss oscillators [23]. To evaluate the effect of this coupling on the overall evolution of the quantum noise in the fiber we introduce an approximate model schematically shown in Fig. 2(b). The fiber is divided into

*P*segments, where each segment is approximated by a piece of lossless fiber of length Δ

*z*followed by a beamsplitter. Propagation of noise through the lossless-fiber segment is described by Eq. (4), while the beamsplitter is introduced to account for the attenuation and the noise (including the back action of losses) accumulated in propagation through the segment. The

*k*-th beamsplitter has the twofold effect of attenuating the field transmitted by the

*k*-th lossless fiber segment and coupling in the noise through the modes

*j*), which we will assume to be in the vacuum state. Equation (6) can be generalized to the lossy case as

*μ*

^{(L)}and

*ν*

^{(L)}by multiplying the matrices resulting from lossless propagation in the

*k*-th segment by the factor (1 - ΓΔ

*z*). The ΓΔ

*z*factor on the second term in the right-hand side accounts for the back coupling of the beamsplitters. The propagation of the matrices

*ξ*

^{(k)}and

*η*

^{(k)}obey the same evolution equations as the matrices

*μ*

^{(L)}and

*ν*

^{(L)}, with the proviso that they propagate only through

*P*-

*k*segments of the fiber. The photocurrent spectral density can be rewritten as

**23**, 1390 (1998). [CrossRef]

9. D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. **24**, 948 (1999) [CrossRef]

*η*. The propagation length is

*L*= 4.3 soliton periods, the initial strong-pulse energy is that of a fundamental soliton, and the splitting ratio of BS1 is chosen to be 10% and that of BS2 to be 3.5% (i.e.,

*T*

_{1}=0.1 and

*T*

_{2}= 0.035). For comparison we have also included a plot of the effect that the same amount of total losses have on the QNR when the propagation in the fiber is lossless but the losses are lumped at the output of the interferometer. From the two curves in Fig. 3 it is evident that the degradation in QNR introduced by the distributed losses is always less than that caused by equal amount of lumped losses placed at the output of the interferometer. This result is consistent with that obtained via an analytical calculation in the case of squeezed-state generation by means of degenerate four-wave mixing [24

24. P. Kumar and J. Shapiro, “Squeezed-state generation via forward degenerate four-wave mixing,” Phys. Rev. A **30**, 1568 (1984). [CrossRef]

## 4 Experimental Setup

*sech*shaped and nearly Fourier transform limited (time-bandwidth product ≃ 0.4). We inject one arm of the interferometer with a strong pulse, propagating in the soliton regime, and the other with a weak, auxiliary pulse propagating in the dispersive regime. The total injected power and the input splitting ratio

*T*

_{1}of the interferometer are controlled by rotating a half-wave plate (HWP1) and a polarizing beamsplitter (PBS1). Since the pulses propagate with significantly different group velocities in the two polarization modes of the fiber, they are launched delayed with respect to each other so that they overlap at the fiber output. The relative delay is introduced by adding separate free-space propagation paths [

*s*(

*p*)-polarization reflects from M1 (M2)] for the two polarization modes in the interferometer. This arrangement also minimizes interaction between the two pulses, as they are temporally separated during most of the propagation distance in the fiber. In addition, a piezoelectric control on M1 allows fine tuning of the relative phase between the two pulses. A half-wave plate (HWP2) and a quarter-wave plate (QWP3) are used to inject the

*s*and

*p*polarized pulses from free space into the correct polarization modes of the fiber. At the output of the fiber, the two pulses are recombined using a half-wave plate (HWP3) and a polarizing beamsplitter (PBS3), which allows us to easily change the output splitting ratio

*T*

_{2}by turning HWP3. The combined pulse, reflected by PBS3, is reflected off another polarization beamsplitter (PBS4) in order to insure a high polarization purity while minimizing optical losses. The emerging pulse train is then analyzed for its noise characteristics with use of a balanced direct-detection apparatus.

*N*

^{2}in the case of PM fiber, which we attribute to an increasing temporal mismatch in the overlap between the soliton-like and the auxiliary pulses. We also point out that the lower QNR observed in the MF case cannot be explained by taking into account exclusively the distributed losses in the fiber.

## 5 Conclusions

## References and links

1. | R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. |

2. | M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. |

3. | A. Sizmann and G. Leuchs, “The optical Kerr effect and quantum optics in fibers,” in |

4. | Ch. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fibre,” Phys. Rev. Lett. |

5. | A. Furasawa, J.L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science282, 706 (1998); S. L. Braunstein and H. J. Kimble, “Dense coding for continuous variables,” Phys. Rev. A612302 (2000). [CrossRef] |

6. | S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. |

7. | S. Spälter, M. Burk, U Ströner, A. Sizmann, and G. Leuchs, “Propagation of quantum properties of sub-picosecond solitons in a fiber,” Opt. Express |

8. | D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. |

9. | D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. |

10. | M. Fiorentino, J. E. Sharping, P. Kumar, D. Levandovsky, and M. Vasilyev, “Soliton squeezing in a Mach-Zehnder fiber interferometer,” Phys. Rev. A |

11. | M. Fiorentino, J.E. Sharping, P. Kumar, A. Porzio, and R. Windeler, “Soliton squeezing in microstructure fiber,” submitted to Opt.Lett. |

12. | J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. |

13. | X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggelton, S. G. Kosinsky, and R. S. Windeler, “Solitonself-phase shift in a short tapered air-silica microstructure fiber,” Opt. Lett. |

14. | J. E. Sharping, M. Fiorentino, and P. Kumar, “Four wave mixing in microstructure fibers,” Opt. Lett. |

15. | F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,” Opt. Lett. |

16. | H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B |

17. | D. J. Kaup, “Perturbation theory for solitons in optical fibres,” Phys. Rev. A |

18. | D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive loop mirror,” Opt. Lett. |

19. | C. R. Doerr, M. Shirasaki, and F. I. Khatri, “Simulation of pulsed squeezing in optical fiber with chromatic dispersion,” J. Opt. Soc. Am. B, |

20. | M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B |

21. | G. P. Agrawal, |

22. | C. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D |

23. | W. H. Louisell, |

24. | P. Kumar and J. Shapiro, “Squeezed-state generation via forward degenerate four-wave mixing,” Phys. Rev. A |

25. | J. H. Shapiro and L. Boivin, “Raman-noise limit on squeezing in continuous-wave four-wave mixing,” Opt. Lett. |

26. | M. J. Werner, “Quantum soliton generation using an interferometer,” Phys. Rev. Lett. |

27. | K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, “Squeezing in a fiber interferometer with a gigahertz pump,” Opt. Lett. |

**OCIS Codes**

(060.2400) Fiber optics and optical communications : Fiber properties

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 21, 2001

Revised Manuscript: January 25, 2002

Published: January 28, 2002

**Citation**

Marco Fiorentino, Jay Sharping, Prem Kumar, and Alberto Porzio, "Amplitude squeezing in a Mach-Zehnder fiber interferometer: Numerical analysis of experiments with microstructure fiber," Opt. Express **10**, 128-138 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-2-128

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### References

- R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, ?Broad-band parametric deamplification of quantum noise in an optical fiber,? Phys. Rev. Lett. 57, 691 (1987). [CrossRef]
- M. Rosenbluh and R. M. Shelby, ?Squeezed optical solitons,? Phys. Rev. Lett. 66, 153 (1991). [CrossRef] [PubMed]
- A. Sizmann and G. Leuchs, ?The optical Kerr effect and quantum optics in fibers,? in Progress in Optics XXXIX, E. Wolf, ed. (Elsevier Scinece B.V., 1999). [CrossRef]
- Ch. Silberhorn , P. K. Lam, O. Wei?, F. Koenig, N. Korolkova, and G. Leuchs, ?Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fibre,? Phys. Rev. Lett. 86, 4267 (2001). [CrossRef] [PubMed]
- A. Furasawa, J.L. Soren sen ,S. L. Braun stein , C. A. Fuchs, H. J. Kimble, and E. S. Polzik, ?Unconditional quantum teleportation,? Science 282, 706 (1998); S. L. Braunstein and H. J. Kimble, ?Dense coding for continuous variables,? Phys. Rev. A 61 2302 (2000). [CrossRef]
- S. Schmitt, J. Ficker, M. Wol., F. K?onig, A. Sizmann, and G. Leuchs, ?Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,? Phys. Rev. Lett. 81, 2446 (1998). [CrossRef]
- S. Spalter, M. Burk, U Stro?ner, A. Sizmann, and G. Leuchs, ?Propagation of quantum properties of sub-picosecond solitons in a fiber,? Opt. Express 2, 77 (1998). [CrossRef]
- D. Krylov and K. Bergman, ?Amplitude-squeezed solitons from an asymmetric fiber interferometer,? Opt. Lett. 23, 1390 (1998). [CrossRef]
- D. Levandovsky, M. Vasilyev, and P. Kumar, ?Amplitude squeezing of light by means of a phasesensitive fiber parametric amplifier,? Opt. Lett. 24, 948 (1999) [CrossRef]
- M. Fiorentino, J. E. Sharping, P. Kumar, D. Levandovsky, and M. Vasilyev, ?Soliton squeezing in a Mach-Zehnder fiber interferometer,? Phys. Rev. A 64, 031801(R) (2001). [CrossRef]
- M. Fiorentino, J.E. Sharping, P. Kumar, A. Porzio, and R. Windeler, ?Soliton squeezing in microstructure fiber,? submitted to Opt. Lett.
- J. K. Ranka, R. S. Windeler, and A. J. Stentz, ?Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,? Opt. Lett. 25, 25 (2000). [CrossRef]
- X. Liu, C. Xu,W. H. Kn ox, J. K. Chan dalia, B. J. Eggelton , S. G. Kosin sky, and R. S.Windeler, ?Soliton self-phase shift in a short tapered air-silica microstructure fiber,? Opt. Lett. 26, 358 (2001). [CrossRef]
- J. E. Sharping, M. Fiorentino and P. Kumar, ?Four wave mixing in microstructure fibers,? Opt. Lett. 26, 1048 (2001). [CrossRef]
- F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, ?Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,? Opt. Lett. 26, 1558 (2001). [CrossRef]
- H. A. Haus and Y. Lai, ?Quantum theory of soliton squeezing: a linearized approach,? J. Opt. Soc. Am. B 7, 386 (1990) [CrossRef]
- D. J. Kaup, ?Perturbationtheor y for solitons in optical fibres,? Phys. Rev. A 42, 5689 (1990). [CrossRef] [PubMed]
- D. Levandovsky, M. Vasilyev, and P. Kumar, ?Soliton squeezing in a highly transmissive loop mirror,? Opt. Lett. 24, 89 (1999). [CrossRef]
- C. R. Doerr, M. Shirasaki, and F. I. Khatri, ?Simulation of pulsed squeezing in optical fiber with chromatic dispersion,? J. Opt. Soc. Am. B, 11, 142 (1994). [CrossRef]
- M. Shirasaki and H. A. Haus, ?Squeezing of pulses in a nonlinear interferometer,? J. Opt. Soc. Am. B 7, 30 (1990) [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).
- C. Caves, ?Quantum limits on noise in linear amplifiers,? Phys. Rev. D 26, 1817 (1982). [CrossRef]
- W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
- P. Kumar and J. Shapiro, ?Squeezed-state generation via forward degenerate four-wave mixing,? Phys. Rev. A 30, 1568 (1984). [CrossRef]
- J. H. Shapiro and L. Boivin, ?Raman-noise limit on squeezing in continuous-wave four-wave mixing,? Opt. Lett. 20, 925 (1990). [CrossRef]
- M. J. Werner, ?Quantum soliton generation using an interferometer,? Phys. Rev. Lett. 81, 4132 (1998). [CrossRef]
- K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, ?Squeezing in a fiber interferometer with a gigahertz pump,? Opt. Lett. 19, 290 (1994). [CrossRef] [PubMed]

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