## Quadrature and number fluctuations produced by parametric devices driven by pulsed pumps |

Optics Express, Vol. 21, Issue 17, pp. 19437-19466 (2013)

http://dx.doi.org/10.1364/OE.21.019437

Acrobat PDF (1344 KB)

### Abstract

Two sets of formulas are derived for the field-quadrature and photon-number fluctuations (variances and correlations) produced by parametric amplifiers and frequency convertors that are driven by pulsed pumps and act on pulsed signals. The first set is based on the Green functions for the underlying parametric processes, whereas the second is based on the associated Schmidt coefficients and modes. These formulas facilitate the modeling and performance optimization of parametric devices used in a wide variety of applications.

© 2013 Optical Society of America

## 1. Introduction

1. M. E. Marhic, *Fiber Optical Parametric Amplifiers, Oscillators and Related
Devices* (Cambridge, 2007). [CrossRef]

6. S. Radic, “Parametric signal processing,”
IEEE J. Sel. Top. Quantum Electron. **18**, 670–680 (2012). [CrossRef]

12. M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single
photons,” Phys. Today **65**(11), 32–37
(2012). [CrossRef]

14. J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes
II,” Phys. Rev. **129**, 481–485 (1963). [CrossRef]

16. C. M. Caves, “Quantum limits on noise in linear
amplifiers,” Phys. Rev. D **26**, 1817–1839 (1982). [CrossRef]

17. R. Loudon, “Theory of noise accumulation in optical-amplifier
chains,” IEEE J. Quantum Electron. **21**, 766–773 (1985). [CrossRef]

19. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric
processes,” Opt. Express **13**, 4986–5012 (2005). [CrossRef] [PubMed]

22. M. E. Marhic, “Quantum-limited noise figures of networks of linear optical
elements,” J. Opt. Soc. Am. B **30**, 1462–1472 (2013). [CrossRef]

23. G. W. Stewart, “On the early history of the singular value
decomposition,” SIAM Rev. **35**, 551–566 (1993). [CrossRef]

26. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic
theory and simple examples,” Opt. Express **21**, 1374–1394 (2013). [CrossRef] [PubMed]

27. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert
space and entropy control,” Phys. Rev. Lett. **84**, 5304–5307 (2000). [CrossRef] [PubMed]

30. K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral
properties by spontaneous four-wave mixing in photonic-crystal fiber,”
Opt. Express **15**, 14870–14886 (2007). [CrossRef] [PubMed]

31. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different
color,” Opt. Commun. **283**, 747–752 (2010). [CrossRef]

34. C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and
pulse reshaping by four-wave mixing,” Phys. Rev. A **85**, 053829 (2012). [CrossRef]

## 2. Direct and homodyne detection

*a*, respectively, where † denotes a hermitian conjugate. These continuous-mode amplitude operators satisfy the boson commutation relations where [,] denotes a commutator, and

_{j}*δ*and

_{jk}*δ*(

*t*

_{1}−

*t*

_{2}) are the Kronecker and Dirac delta functions, respectively [7].

*N*〉, where 〈〉 denotes an expectation value. The uncertainties in such measurements are the number variances Also of interest are the number correlations The number-flux operator and the associated number operator

_{j}*N*=

_{j}*∫n*(

_{j}*t*)

*dt*, where the limits of integration are −

*T*/2 and

*T*/2, and

*T*is the measurement time. It follows from these definitions that Thus, to model direct detection, one must calculate the second- and fourth-order amplitude moments 〈

*n*(

_{j}*t*)〉 and 〈

*n*(

_{j}*t*

_{1})

*n*(

_{k}*t*

_{2})〉, respectively.

*α*〉 is the state vector [7]. For such a state, the mean (coherent) amplitude 〈

_{j}*a*(

_{j}*t*)〉 =

*α*(

_{j}*t*). By combining Eqs. (4) and (7), one finds that the second-order moment The number flux is the squared modulus of the coherent amplitude. One also finds that the fourth-order moment

*n*(

_{j}*t*

_{1})〉〈

*n*

_{1}(

*t*

_{2})〉. Hence, if one were to detect a CS (using a measurement time that is longer than the duration of the signal pulse), one would find that Property (11) is characteristic of a CS. The vacuum state (VS) |0〉 is a special CS, for which

*α*(

_{j}*t*) = 0 and

*N*〉, which is the standard result for a single discrete mode (or CW) [7].

_{j}*α*(

_{p}*t*) includes the phase factor

*e*

^{iϕp}implicitly and satisfies the normalization condition ∫ |

*α*(

_{p}*t*)|

^{2}

*dt*= 1. (In this report the subscripts

*p*and

*q*are used to denote LOs, because the subscripts

*i*,

*j*,

*k*and

*l*are used to denote weak signals participating in parametric processes.) By varying the LO phase, one can measure different quadratures of the signal. The associated quadrature operator

*Q*= ∫

_{j}*q*(

_{j}*t*)

*dt*. It follows from definition (22) that the quadrature-flux moments

*q*(

_{j}*t*

_{1})〉〈

*q*(

_{j}*t*

_{2})〉. Hence, if one were to detect a CS (using a measurement time that is longer than the signal and LO pulses), one would find that The quadrature mean (25) depends on the shapes and overlap of the signal and LO pulses, whereas the variance (26) does not depend on either pulse shape. The variance (26) has the minimal value allowed by the Heisenberg uncertainty principle. This result is characteristic of a CS [7]. As stated earlier, the SNR is defined to be the square of the mean quadrature divided by the quadrature variance. If one uses the optimal LO [

*α*(

_{p}*t*) ∝

*α*(

_{j}*t*), where the proportionality coefficient is real and positive], then 〈

*Q*〉 = 2

_{j}^{1/2}〈

*N*〉

_{j}^{1/2}and the SNR equals 4〈

*N*〉, which is the standard result for a single discrete mode [7]. Notice that the optimal-LO condition is analogous to the matched-filter condition in detection theory [35, 36]

_{j}## 3. Attenuation and frequency conversion

37. K. Inoue, “Tunable and selective wavelength conversion using fiber
four-wave mixing with two pump lights,” IEEE Photon. Technol.
Lett. **6**, 1451–1453 (1994). [CrossRef]

42. A. S. Clark, S. Shahnia, M. J. Collins, C. Xiong, and B. J. Eggleton, “High-efficiency frequency conversion in the single-photon
regime,” Opt. Lett. **38**, 947–949 (2013). [CrossRef] [PubMed]

*π*

_{p2}+

*π*

_{s1}→

*π*

_{p1}+

*π*

_{s2}, where

*π*represents a photon with carrier frequency

_{j}*ω*). This process is illustrated in Fig. 2. The photon reaction used to describe BS is called the Manley–Rowe–Weiss (MRW) relation [43

_{j}43. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear
elements,” Proc. IRE **44**, 904–913 (1956). [CrossRef]

37. K. Inoue, “Tunable and selective wavelength conversion using fiber
four-wave mixing with two pump lights,” IEEE Photon. Technol.
Lett. **6**, 1451–1453 (1994). [CrossRef]

39. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within
entire C-band based on nondegenerate fiber four-wave mixing,” IEEE
Photon. Technol. Lett. **16**, 551–553 (2004). [CrossRef]

40. D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in
an optical fiber,” Opt. Express **14**, 8995–8999 (2006). [CrossRef] [PubMed]

38. K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear
dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top.
Quantum Electron. **8**, 560–568 (2002). [CrossRef]

*a*are input operators,

_{j}*b*are output operators and

_{j}*g*are Green functions. For attenuation, mode 1 is the signal and mode 2 is the loss mode [7], whereas for FC, mode 1 is the signal and mode 2 is the idler [19

_{jk}19. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric
processes,” Opt. Express **13**, 4986–5012 (2005). [CrossRef] [PubMed]

*β*(

_{p}*t*) and

*β*(

_{q}*t*) replaced by the sideband amplitudes

*β*

_{1}(

*t*) and

*β*

_{2}(

*t*), respectively, and the factors of 1/2 omitted.

## 4. Parametric amplification

45. T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric
amplifiers,” IEEE Photon. Technol. Lett. **19**, 650–652 (2007). [CrossRef]

46. J. M. Chavez Boggio, S. Moro, E. Myslivets, J. R. Windmiller, N. Alic, and S. Radic, “155-nm continuous-wave two-pump parametric
amplification,” IEEE Photon. Technol. Lett. **21**, 612–614 (2009). [CrossRef]

47. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum
communications,” IEEE Photon. Technol. Lett. **14**, 983–985 (2002). [CrossRef]

52. O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical
fibers,” Phys. Rev. Lett. **102**, 123603 (2009). [CrossRef] [PubMed]

*π*

_{p1}→

*π*

_{s1}+

*π*

_{s2}), whereas inverse MI is the degenerate process in which two different pump photons are destroyed and two identical sideband (signal) photons are created (

*π*

_{p1}+

*π*

_{p2}→ 2

*π*

_{s1}). Phase conjugation (PC) is the nondegenerate process in which two different pump photons are destroyed and two different sideband photons are created (

*π*

_{p1}+

*π*

_{p2}→

*π*

_{s1}+

*π*

_{s2}). These processes are illustrated in Fig. 3. The photon reactions used to describe them are called the MRW relations [43

43. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear
elements,” Proc. IRE **44**, 904–913 (1956). [CrossRef]

53. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a
fiber,” Opt. Express **12**, 4973–4979 (2004). [CrossRef] [PubMed]

55. R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of
binary phase-encoded signals,” IEEE J. Sel. Top. Quantum
Electron. **18**, 859–869 (2012). [CrossRef]

1. M. E. Marhic, *Fiber Optical Parametric Amplifiers, Oscillators and Related
Devices* (Cambridge, 2007). [CrossRef]

2. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their
applications,” IEEE J. Sel. Top. Quantum Electron. **8**, 506–520 (2002). [CrossRef]

56. R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate
phase-sensitive fiber-optic parametric amplifier with phase self-stabilized
input,” Opt. Express **13**, 10483–10493 (2005). [CrossRef] [PubMed]

58. Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical
amplifiers and their applications,” IEEE J. Sel. Top. Quantum
Electron **18**, 1016–1032 (2012). [CrossRef]

16. C. M. Caves, “Quantum limits on noise in linear
amplifiers,” Phys. Rev. D **26**, 1817–1839 (1982). [CrossRef]

17. R. Loudon, “Theory of noise accumulation in optical-amplifier
chains,” IEEE J. Quantum Electron. **21**, 766–773 (1985). [CrossRef]

19. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric
processes,” Opt. Express **13**, 4986–5012 (2005). [CrossRef] [PubMed]

22. M. E. Marhic, “Quantum-limited noise figures of networks of linear optical
elements,” J. Opt. Soc. Am. B **30**, 1462–1472 (2013). [CrossRef]

*g*is a Green function that couples

_{jj}*b*to

_{j}*a*(like operators), and

_{j}*g*is a Green function that couples

_{jk}*b*to

_{j}**13**, 4986–5012 (2005). [CrossRef] [PubMed]

*t*

_{1}and

*t*

_{2}denote the times at which the terms in brackets are evaluated. The anti-normally ordered terms in Eq. (51) are proportional to

*δq*

_{1}(

*t*

_{1})

*δq*

_{2}(

*t*

_{2})〉 = 〈

*δq*

_{2}(

*t*

_{2})

*δq*

_{1}(

*t*

_{1})〉. One can also use Eq. (142) to establish this symmetry directly.

*t*

_{1}and

*t*

_{2}. These formulas are so similar to Eqs. (53) and (56) that there is no point in stating them explicitly. However, by changing the order of integration (doing the

*t′*integral last), one obtains the alternative variance and correlation formulas respectively. [The first and second forms of Eq. (57) come from Eqs. (52) and (53), respectively, whereas Eq. (58) comes from Eq. (56).] By comparing Eqs. (57) and (58) to Eqs. (26) and (28), respectively, one finds that PA increases the variances of the sideband quadrature fluctuations (

*v*are all zero, and restoring the time integrals, one finds that

_{j}*v*are all zero, so the moment

_{j}*δn*

_{1}(

*t*

_{1})

*δn*

_{2}(

*t*

_{2})〉 = 〈

*δn*

_{2}(

*t*

_{2})

*δn*

_{1}(

*t*

_{1})〉.

*t*

_{1}and

*t*

_{2}. These formulas are so similar to Eqs. (67) and (71) that there is no need to state them explicitly. However, by changing the order of integration, one obtains the alternative variance and correlation formulas

*N*〉) and produces a correlation between them. It is shown in App. A that the presence of strong correlations is a consequence of the MRW relations. Notice that if the LO pulses have the same shapes as the output sideband pulses, the quadrature variance and correlation [Eqs. (57) and (58)] are proportional to the signal–noise contributions to the number variance and correlation, respectively. Notice also that the noise–noise contributions to the number variance and correlation are equal.

_{j}*b*)]. The IO equation for this phase-sensitive process can be written in the form where the subscripts

*s*and

*c*denote coupling to similar and conjugate operators, respectively. One can obtain Eq. (74) from Eq. (47) or (48) by making the substitutions

*a*→

_{j}*a*,

*b*→

_{j}*b*,

*g*→

_{jj}*g*and

_{s}*g*→

_{jk}*g*. Because these IO equations are so similar, there is no need to describe the analysis of one-mode PA in detail. The quadrature flux and quadrature mean are specified by Eqs. (49) and (50), respectively, with the subscript 1 omitted. By following the procedure used to derive Eq. (52), one obtains the quadrature-flux moment Equation (75) contains two more terms than Eq. (52), because

_{c}*a*

_{1}commutes with

*a*does not commute with

*a*

^{†}. By using constraints analogous to Eqs. (141) and (142), one can show that the quadrature-flux moment is real. By integrating Eq. (75) with respect to

*t*

_{1}and

*t*

_{2}, one finds that the quadrature variance which is manifestly real and non-negative, but can be greater or less than 1/2 [15

15. H. P. Yuen, “Two-photon states of the radiation
field,” Phys. Rev. A **13**, 2226–2243 (1976). [CrossRef]

17. R. Loudon, “Theory of noise accumulation in optical-amplifier
chains,” IEEE J. Quantum Electron. **21**, 766–773 (1985). [CrossRef]

## 5. Schmidt decompositions of parametric processes

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

21. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by
parametric processes,” Opt. Express **18**, 19792–19823 (2010). [CrossRef] [PubMed]

59. S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam
splitter,” Opt. Commun. **62**, 139–145 (1987). [CrossRef]

60. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) summetry
and photon statistics,” Phys. Rev. A **40**, 1371–1384 (1989). [CrossRef] [PubMed]

*β*(

_{p}*t*) ∝

*v*(

_{j}*t*), in which case |

*β*| = 1 and

_{pj}*β*= 0 for

_{pk}*k*≠

*j*. It follows from Eq. (33) and the second of Eqs. (82) that the quadrature-flux moment One could obtain the same result by using Eq. (29) and the first of Eqs. (84) to reorder the

*t*

_{1}and

*t*

_{2}, one finds that the quadrature variance which is consistent with Eq. (35). The total quadrature variance is a weighted sum of discrete-mode contributions [Eq. (89)], which always equals 1/2, regardless of whether one selects a particular output mode or admits them all. Hence, one always maximizes the SNR by matching the LO pulse to the output signal pulse, regardless of whether the latter pulse is a single output mode or a superposition of such modes. Formulas for the number mean and variance will not be stated explicitly, because they can be deduced from the associated quadrature results, as noted above.

*t*

_{1}and

*t*

_{2}, one finds that The quadrature variance and correlation are weighted sums of discrete-mode contributions [Eqs. (102) and (103)].

*b*)]. The Green functions in Eq. (74) have the Schmidt decompositions [63

63. A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into
independent squeezers,” J. Mod. Opt. **54**, 721–733 (2007). [CrossRef]

64. M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave
phase-sensitive optical parametric amplifier,” J. Mod. Opt. **57**, 1908–1915 (2010). [CrossRef]

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

21. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by
parametric processes,” Opt. Express **18**, 19792–19823 (2010). [CrossRef] [PubMed]

22. M. E. Marhic, “Quantum-limited noise figures of networks of linear optical
elements,” J. Opt. Soc. Am. B **30**, 1462–1472 (2013). [CrossRef]

14. J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes
II,” Phys. Rev. **129**, 481–485 (1963). [CrossRef]

## 6. Summary

## Appendix A: Commutation relations and conservation laws

*h*are backward (adjoint) Green functions. It follows from Eqs. (125) and (126), and the commutation relations, that By combining Eqs. (29), (125) and (126), one obtains the identity

_{jk}31. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different
color,” Opt. Commun. **283**, 747–752 (2010). [CrossRef]

*a*

_{1}(

*t*) produces the same conditions. By combining Eqs. (127), (128) and (132), one finds that Thus, the Green functions for attenuation and FC satisfy constraints when integrated with respect to either (first or second) variable.

65. A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical
input-output theories of optical response,” Phys. Rev. A **43**, 3934–3938 (1991). [CrossRef] [PubMed]

66. C. J. McKinstrie, “Unitary and singular value decompositions of parametric
processes in fibers,” Opt. Commun. **282**, 583–593 (2009). [CrossRef]

*a*

_{1}(

*t*) produces the same conditions. By combining Eqs. (145), (146) and (150), one finds that Thus, the Green functions for PA also satisfy constraints when integrated with respect to either (first or second) variable.

*t′*and

*t″*are dummy variables, one can rewrite the third and fourth terms in Eq. (153) in the abbreviated forms

*g*

_{12}(

*t*,

*t′*)|

^{2}does not equal |

*g*

_{21}(

*t*,

*t′*)|

^{2}. However, by using the Schmidt decompositions stated in Sec. 5, one can show that the integrals of these functions are equal, so the last term in Eq. (156) is zero. Thus, PA generates signal and idler photons in pairs, for arbitrary input pulses. Notice that Eq. (156) pertains to the number-difference operator: Not only is the number-difference mean conserved, so also are all the number-difference moments (fluctuations). The first-moment equation, which involves the number means, is simply whereas the second-moment equation, which involves the variances and correlation, is If the inputs are (independent) CS, Eq. (158) reduces to In the high-gain regime (

43. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear
elements,” Proc. IRE **44**, 904–913 (1956). [CrossRef]

*a*→

_{j}*a*,

*b*→

_{j}*b*,

*g*→

_{jj}*g*and

_{s}*g*→

_{jk}*g*. Like their two-mode counterparts, the one-mode transfer functions satisfy constraints, which one can deduce by making the same substitutions in Eqs. (141) and (142), and similar substitutions in Eqs. (145) and (146), and reciprocity relations, which one can deduce from Eqs. (150).

_{c}## Appendix B: Quadrature and number fluctuations

*b*(

_{j}*t*) into its mean value 〈

*b*(

_{j}*t*)〉 =

*β*(

_{j}*t*), which depends only on the input signals, and its deviation

*b*(

_{j}*t*) − 〈

*b*(

_{j}*t*)〉 =

*w*(

_{j}*t*), which depends only on the input fluctuations. By using these definitions, one finds that the mean and deviation of each quadrature flux are respectively, where

*β*(

_{p}*t*) is the (normalized) amplitude of the LO used to detect mode

*j*. To calculate quadrature correlations, one evaluates quadrature-flux moments of the form where

*β*(

_{q}*t*) is the amplitude of the LO used to detect mode

*k*, then integrates with respect to

*t*

_{1}and

*t*

_{2}.

*w*and

_{j}*w*), and noise–noise terms, which depend only on the fluctuations (fourth-order moments). For signals that contain many photons, one can neglect the latter terms, in which case If the LOs used to detect the mode quadratures have the same shapes as the output pulses (which is the optimal situation), the quadrature-flux moments (162) are proportional to the signal–noise contributions to the number-flux moments (167). Notice that this result is kinematic, in the sense that it depends only on the definitions of the quadrature- and number-flux moments, and not on the details of the parametric processes (kinetics) that produced them. It is a generalization of the discrete-mode result stated in [21

_{k}21. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by
parametric processes,” Opt. Express **18**, 19792–19823 (2010). [CrossRef] [PubMed]

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

## Appendix C: Multiple-mode processes

*a*and

_{k}*b*are input and output mode-amplitude operators, respectively,

_{i}*μ*is a forward transfer (Green) function that couples

_{ik}*b*to

_{i}*a*(like operators),

_{k}*ν*is a forward Green function that couples

_{ik}*b*to

_{i}*b*was coupled to

_{i}*a*or

_{k}*μ̄*and

_{ik}*ν̄*are backward Green functions. These Green functions satisfy constraints with the same forms as Eqs. (169) and (170). By combining Eqs. (168) and (171), one obtains the identity

_{ik}*a*produces equivalent results. Notice that Eqs. (169), (170) and (175)–(177) reduce to the corresponding equations of App. A in the appropriate limits.

_{i}*δq*and

_{i}*δq*commute, as do

_{j}*δQ*and

_{i}*δQ*), these properties are not evident in Eqs. (184) and (185). It is convenient to rewrite Eq. (184) in the abbreviated form in which summation with respect to

_{j}*k*, integration with respect to

*t′*and the factor of 1/2 are implicit, and the subscripts 1 and 2 represent the times

*t*

_{1}and

*t*

_{2}, respectively. Consider the first and fourth terms on the right side of Eq. (186). By applying Eq. (170) twice and reordering, one finds that

*i*and

*j*(and

*p*and

*q*) interchanged. Thus, the second-order moments are real and do not depend on the order in which the deviation operators are written: 〈

*δq*(

_{i}*t*

_{1})

*δq*(

_{j}*t*

_{2})〉 = 〈

*δq*(

_{j}*t*

_{2})

*δq*(

_{i}*t*

_{1})〉 and 〈

*δQ*〉 = 〈

_{i}δQ_{j}*δQ*〉. When

_{j}δQ_{i}*i*=

*j*, Eq. (185) specifies the quadrature variance which is manifestly real and non-negative.

*b*is coupled. The first two terms on the right side of Eq. (191) are (collectively) the signal–noise contribution to the number deviation, whereas the last two terms are the noise–noise contribution. The former contribution to the number deviation has the same form as the quadrature deviation (179), with

_{i}*β*replaced by

_{p}*β*and the factor of 1/2

_{j}^{1/2}omitted. Hence, the derivations of the signal–noise contributions to the number-flux moment and number correlation need not be described. The final results are

*δn*(

_{i}*t*

_{1})

*δn*(

_{j}*t*

_{2})〉

_{sn}= 〈

*δn*(

_{j}*t*

_{2})

*δn*(

_{i}*t*

_{1})〉

_{sn}and 〈

*δN*〉

_{i}δN_{j}_{sn}= 〈

*δN*〉

_{j}δN_{i}_{sn}. When

*i*=

*j*, Eq. (194) specifies the number variance which is manifestly real and non-negative.

*k*,

*l*) = (

*m*,

*n*), where

*m*and

*n*are integers, there is another term with (

*k*,

*l*) = (

*n*,

*m*). By using this fact to rewrite the summations, one can show that

*v*and

_{k}*k*and

*l*, and reordered the terms on the right side of Eq. (201), to make it resemble previous results, which will be mentioned shortly. By integrating Eq. (201) with respect to time, one obtains the number correlation

*t′*and

*t″*are implicit. Consider the first term on the right side of Eq. (203). By applying Eq. (169) twice and reordering, one finds that

*k*,

*l*) = (

*m*,

*n*), there is another term with (

*k*,

*l*) = (

*n*,

*m*), which is the conjugate of the first term. Thus, the fourth-order moments are real. The third forms of the right sides of Eqs. (204) and (205) are just the left sides, with

*i*and

*j*interchanged. Thus, the fourth-order moments do not depend on the order in which the deviation operators are written: 〈

*δn*(

_{i}*t*

_{1})

*δn*(

_{j}*t*

_{2})〉

_{nn}= 〈

*δn*(

_{j}*t*

_{2})

*δn*(

_{i}*t*

_{1})〉

_{nn}and 〈

*δN*〉

_{i}δN_{j}_{nn}= 〈

*δN*〉

_{j}δN_{i}_{nn}.

**18**, 19792–19823 (2010). [CrossRef] [PubMed]

*i*=

*j*, Eq. (206) reduces to which is consistent with Eq. (43) of [19

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

*u*and

_{n}*v*are input and output Schmidt modes (temporal eigenfunctions), respectively, and

_{n}*σ*is a Schmidt coefficient [25]. In general, the Schmidt coefficients and modes depend on both

_{n}*i*and

*j*. However, for systems in which the output modes depend only on

*i*and the input modes depend only on

*j*, one can use these modes as basis functions for the input and output pulses (Sec. 5). By doing so, one converts a multiple-continuous-mode process into a set of independent multiple-discrete-mode processes, each of which is governed by IO equations that are identical to those considered in [19

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

**18**, 19792–19823 (2010). [CrossRef] [PubMed]

**30**, 1462–1472 (2013). [CrossRef]

24. S. L. Braunstein, “Squeezing as an irreducible
resource,” Phys. Rev. A **71**, 055801 (2005). [CrossRef]

26. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic
theory and simple examples,” Opt. Express **21**, 1374–1394 (2013). [CrossRef] [PubMed]

*M*(

*t*,

*t′*) = [

*μ*(

_{ij}*t*,

*t′*)] and

*N*(

*t*,

*t′*) = [

*ν*(

_{ij}*t*,

*t′*)], which couple the output vector

*B*(

*t*) = [

*b*(

_{i}*t*)] to the input vector

*A*(

*t′*) = [

*a*(

_{j}*t′*)] and its conjugate, respectively, to determine the input and output Schmidt supermodes (vector temporal eigenfunctions), and the associated Schmidt coefficients.

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**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(190.2620) Nonlinear optics : Harmonic generation and mixing

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

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 6, 2013

Revised Manuscript: July 16, 2013

Manuscript Accepted: July 22, 2013

Published: August 12, 2013

**Citation**

C. J. McKinstrie and M. E. Marhic, "Quadrature and number fluctuations produced by parametric devices driven by pulsed
pumps," Opt. Express **21**, 19437-19466 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-19437

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### References

- M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2007). [CrossRef]
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