## Schmidt decompositions of parametric processes I: Basic theory and simple examples |

Optics Express, Vol. 21, Issue 2, pp. 1374-1394 (2013)

http://dx.doi.org/10.1364/OE.21.001374

Acrobat PDF (940 KB)

### Abstract

Parametric devices based on four-wave mixing in fibers perform many signal-processing functions required by optical communication systems. In these devices, strong pumps drive weak signal and idler sidebands, which can have one or two polarization components, and one or many frequency components. The evolution of these components (modes) is governed by a system of coupled-mode equations. Schmidt decompositions of the associated transfer matrices determine the natural input and output mode vectors of such systems, and facilitate the optimization of device performance. In this paper, the basic properties of Schmidt decompositions are derived from first principles and are illustrated by two simple examples (one- and two-mode parametric amplification). In a forthcoming paper, several nontrivial examples relevant to current research (including four-mode parametric amplification) will be discussed.

© 2013 OSA

## 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]

10. 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*→

_{p}*π*+

_{s}*π*, where

_{i}*π*represents a photon with frequency

_{j}*ω*). Inverse MI is the degenerate process in which two photons from different pumps are destroyed and two signal photons are created (

_{j}*π*+

_{p}*π*→ 2

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

_{s}*π*+

_{p}*π*→

_{q}*π*+

_{s}*π*). The polarization properties of these processes are reviewed in [11

_{i}11. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. **28**, 883–894 (1992). [CrossRef]

14. C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express **14**, 8516–8534 (2006). [CrossRef] [PubMed]

*z*is distance,

*d*=

_{z}*d*/

*dz*,

*X*= [

*x*] is the vector of sideband amplitudes (modes),

_{j}*A*= [

*α*] and

_{jk}*B*= [

*β*] are coefficient matrices, and * denotes a complex conjugate. The entries of the amplitude vector could be the amplitudes of distinct monochromatic sidebands (continuous waves), or different frequency components of multichromatic sidebands (pulses), with one or two polarization components. For uniform fibers (media) the coupling coefficients are constants, whereas for nonuniform media they vary with distance. Because Eq. (1) is linear in the amplitude vector and its conjugate, the (explicit or implicit) solution of Eq. (1) can be written in the input–output (IO) form where

_{jk}*M*= [

*μ*] and

_{jk}*N*= [

*ν*] are transfer (Green) matrices.

_{jk}21. 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]

22. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A **64**, 063815 (2001). [CrossRef]

23. 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]

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

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

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

25. 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]

26. 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]

27. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans. Telecom. **10**, 585–595 (1999). [CrossRef]

28. C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” J. Sel. Top. Quantum Electron. **18**, 794–811 (2012). [CrossRef]

## 2. Simple examples of Schmidt decompositions

### 2.1. One-mode amplification

*x*is the mode amplitude,

*δ*is the (real) mismatch coefficient and

*γ*is the (complex) coupling coefficient [31

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

32. K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron. **14**, 648–658 (2008). [CrossRef]

*x*and

*x*

^{*}, so its solution can be written in the input–output (IO) form For the common case in which

*δ*and

*γ*are constants, the transfer (Green) functions and the characteristic wavenumber

*k*= (

*δ*

^{2}− |

*γ*|

^{2})

^{1/2}. If coupling is stronger than mismatch (|

*γ*| >

*δ*), the system is unstable. The transfer functions (7) satisfy the auxiliary equation They also have the interesting properties Furthermore, if

*γ*is real, then

*ν*

^{*}(

*z*) = −

*ν*(

*z*). These properties are not accidental.

*μ*= |

*μ*|

*e*

^{iϕμ}and

*ν*= |

*ν*|

*e*

^{iϕν}, and define the sum and difference phases

*ϕ*= (

_{s}*ϕ*+

_{μ}*ϕ*)/2 and

_{ν}*ϕ*= (

_{d}*ϕ*−

_{ν}*ϕ*)/2, respectively. Then Eq. (6) can be rewritten in the form where

_{μ}*u*=

*e*

^{iϕd}and

*v*=

*e*

^{iϕs}are phase factors (input and output phase references). If the signal phase

*ϕ*=

_{x}*ϕ*=

_{u}*ϕ*, the terms on the right side of Eq. (10) add constructively: The signal is said to be in-phase and is amplified (stretched) by the factor |

_{d}*μ*| + |

*ν*|. Conversely, if

*ϕ*=

_{x}*ϕ*+

_{d}*π*/2, the terms on the right side of Eq. (10) add destructively: The signal is said to be out-of-phase and is attenuated (squeezed) by the factor |

*μ*|+ |

*ν*| = 1/(|

*μ*|−|

*ν*|). If one were to measure the phase of the input signal relative to the aforementioned reference phase, one would say that the real quadrature is amplified and the imaginary quadrature is attenuated. Notice that Eq. (10) has the canonical form of Eq. (3).

*Y*= [

*x*,

*x*

^{*}]

*and the 2 × 2 coefficient matrix Notice that*

^{t}*L*is specified by three real parameters (

_{y}*δ*,

*γ*and

_{r}*γ*). The solution of Eq. (11) can be written in the IO form where the transfer (Green) matrix Two important results follow from Eqs. (12) and (14). First, tr(

_{i}*L*) = 0, so det(

_{y}*T*) = 1, and second,

_{y}*T*) = |

_{y}*μ*|

^{2}− |

*ν*|

^{2}= 1, so

*T*is defined by three real parameters (|

_{y}*ν*|,

*ϕ*and

_{μ}*ϕ*), the same number that specified

_{ν}*L*. Notice also that Hence,

_{y}*μ*(−

*z*) =

*μ*

^{*}(

*z*) and

*ν*(−

*z*) = −

*ν*(

*z*), as stated in Eqs. (9).

*T*are the eigenvectors of

_{y}*ϕ*and

_{s}*ϕ*were defined before Eq. (10). All three matrices in Eq. (17) depend on

_{d}*z*. The evolution equation (11) governs

*x*and

*x*

^{*}simultaneously, so it is natural that the associated transfer matrix describes stretching and squeezing simultaneously. Specifically, Eq. (17) shows that the stretching condition is 2

*ϕ*= 2

_{x}*ϕ*, whereas the squeezing condition is 2

_{d}*ϕ*= 2

_{x}*ϕ*+

_{d}*π*, and the associated Schmidt coefficients are reciprocals. The input Schmidt vectors, which are the natural inputs for one-mode amplification, correspond to in-phase and out-of-phase signals. These results are consistent with the discussion that follows Eq. (10).

*μ*→

*μ*

^{*}and

*ν*→ −

*ν*(which do not affect the moduli of the transfer functions) are equivalent to

*ϕ*→ −

_{μ}*ϕ*and

_{μ}*ϕ*→

_{ν}*ϕ*+

_{ν}*π*and, ultimately, to

*e*

^{iϕs}→

*ie*

^{iϕd}and

*e*

^{iϕd}→

*ie*

^{iϕs}. By making these replacements in decomposition (17), one obtains the inverse decomposition For reference, neither the forward decomposition (17), nor the backward decomposition (18), is unique.

### 2.2. Two-mode amplification

*x*is a mode amplitude and

_{j}*δ*is a (real) mismatch coefficient [1

_{j}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]

*δ*and

_{j}*γ*are constants, the two-mode transfer functions

*μ*

_{11}(

*z*) =

*e*(

*z*)

*μ*(

*z*),

*ν*

_{12}(

*z*) =

*e*(

*z*)

*ν*(

*z*),

*μ*

_{22}(

*z*) =

*e*

^{*}(

*z*)

*μ*(

*z*) and

*ν*

_{21}(

*z*) =

*e*

^{*}(

*z*)

*ν*(

*z*), where the one-mode transfer functions

*μ*(

*z*) and

*ν*(

*z*) are defined by Eqs. (7), with the mismatch coefficient

*δ*replaced by (

*δ*

_{1}+

*δ*

_{2})/2, and the phase factor

*e*(

*z*) = exp[

*i*(

*δ*

_{1}−

*δ*

_{2})

*z*/2] = exp(

*iϕ*). The two-mode transfer functions satisfy the auxiliary equations and have the interesting properties Furthermore, if

_{δ}*γ*is real, then

*δ*

_{1}=

*δ*

_{2}, solutions (20) reduce to solution (6).

*e*(

*z*), which affects the output signal phase, but does not affect the interference conditions. Hence, if

*ϕ*

_{1}+

*ϕ*

_{2}=

*ϕ*−

_{ν}*ϕ*= 2

_{μ}*ϕ*, the terms in the first of Eqs. (20) add constructively: The sidebands are said to be in-phase and (if their input amplitudes are equal) are stretched by the factor |

_{d}*μ*| + |

*ν*|. Conversely, if

*ϕ*

_{1}+

*ϕ*

_{2}=

*ϕ*−

_{ν}*ϕ*+

_{μ}*π*, the terms in the first of Eqs. (20) add destructively: The sidebands are said to be out-of-phase and (if their input amplitudes are equal) are squeezed by the factor |

*μ*| + |

*ν*| = 1/(|

*μ*| − |

*ν*|). The same interference conditions apply to the second of Eqs. (20).

*L*is specified by four real parameters (

_{x}*δ*

_{1},

*δ*

_{2},

*γ*and

_{r}*γ*). Notice also that Eq. (23) is closed (involves only

_{i}*x*

_{1}and

*L*and

_{x}*T*[Eqs. (24) and (26)] differ only slightly from those of

_{x}*L*and

_{y}*T*[Eqs. (12) and (14)]. Because tr(

_{y}*L*) =

_{x}*δ*

_{1}−

*δ*

_{2}≠ 0, det(

*T*) = exp[

_{x}*i*(

*δ*

_{1}−

*δ*

_{2})

*z*] ≠ 1. Nonetheless,

*T*(−

_{x}*z*) =

*T*

^{−1}(

*z*).

*T*is determined by four real parameters (|

_{x}*ν*|,

*ϕ*,

_{μ}*ϕ*and

_{ν}*ϕ*), the same number that specified

_{δ}*L*. Notice also that Hence,

_{x}*ν*

_{12}(−

*z*) = −

*ν*

_{21}(

*z*), as stated in Eqs. (22).

*e*(

*z*) =

*e*

^{iϕδ}, so it follows from Eq. (17) that the forward transfer matrix has the Schmidt decomposition where

*ϕ*and

_{s}*ϕ*were defined after Eq. (9), and

_{d}*ϕ*was defined after Eq. (20). Suppose that the input sideband phases are measured relative to the reference phase

_{δ}*ϕ*. Then decomposition (29) implies that the combination

_{d}*ϕ*

_{1}+

*ϕ*

_{2}= 0, whereas for squeezing, the optimal phase condition is

*ϕ*

_{1}+

*ϕ*

_{2}=

*π*. These results are consistent with the results stated after Eq. (22). It follows from Eq. (18) that the backward transfer matrix has the Schmidt decomposition One can derive Eq. (30) from Eq. (29) by making the replacements

*e*

^{iϕs}→

*ie*

^{iϕd},

*e*

^{iϕd}→

*ie*

^{iϕs}and

*e*

^{iϕδ}→

*e*

^{−iϕδ}.

*M*to be diagonal and

*N*to be off-diagonal. Decompositions (33) and (34) are not quite in canonical form, because the elements of the diagonal matrices (

*μ*and

*ν*) are complex. However, by generalizing the derivation of Eq. (10), one obtains the Schmidt decompositions For reference, decompositions (29), (30), (35) and (36) are not unique.

*δ*= (

_{s}*δ*

_{1}+

*δ*

_{2})/2 and

*δ*= (

_{d}*δ*

_{1}−

*δ*

_{2})/2. For the special case in which

*δ*= 0, the sum (+) and difference (−) modes evolve independently: Each mode undergoes a one-mode parametric process that is governed by Eq. (4) or (6). Because the coupling coefficients in Eqs. (38) differ by a factor of −1, the input phases required for stretching differ by

_{d}*π*/2, as do the input phases required for squeezing. This method of analysis fails for the general case in which

*δ*≠ 0. However, the concept of a superposition mode remains useful.

_{d}*δ*, show that it is appropriate to measure the input sideband phases relative to the (common) reference phase

_{d}*ϕ*, as did decomposition (29). With this convention, the real quadrature of the sum mode and the imaginary quadrature of the difference mode are stretched. It follows from Eq. (4), in which the transfer functions are non-negative by construction, that the imaginary sum quadrature and the real difference quadrature are squeezed.

_{d}26. 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]

## 3. Basic theory of Schmidt decompositions

*m*coupled modes is governed by the Hamiltonian where

*X*is an

*m*× 1 mode-amplitude vector, and

*J*and

*K*are

*m*×

*m*coefficient matrices. In order for the first term on the right side of Eq. (39) to be real,

*J*must be Hermitian. The sum of the second and third terms is real by construction, and one can always write these terms in such a way that

*K*is symmetric. For reference,

*J*is specified by (up to)

*m*

^{2}real parameters, whereas

*K*is specified by

*m*(

*m*+ 1) real parameters. By applying the (complex) Hamilton equation to Hamiltonian (39), one obtains the CME (The complex Hamiltonian formalism is reviewed in the Appendix.) Equation (41) depends linearly on

*X*and

*X*

^{*}, so its solution can be written in the IO form where

*M*and

*N*are

*m*×

*m*transfer matrices. [Equation (41) is just Eq. (1), with

*A*=

*iJ*and

*B*=

*iK*, and Eq. (42) is just Eq. (2), repeated for convenience.] For the special case in which

*m*= 1,

*J*=

*δ*and

*K*=

*γ*, and Eq. (41) reduces to Eq. (5).

*m*× 1 mode vector and 2

*m*× 2

*m*coefficient matrix are respectively. The solution of Eq. (43) can be written in the IO form where

*T*is the 2

_{y}*m*× 2

*m*transmission matrix. If

*L*is a constant matrix, then Because Eq. (43) describes two copies of the same process (the original and its conjugate), the

_{y}*m*×

*m*blocks of

*T*are the transfer matrices

_{y}*M*and

*N*, which appeared in Eq. (42), and their conjugates [see Eq. (15)]. Clearly, Eqs. (43)–(46) are generalizations of Eqs. (11)–(14).

*X*is coupled to every component of

*X*and

*X*

^{*}. However, there are many important systems in which a subset of the components of

*X*(denoted by

*X*

_{1}and called the signal vector) is coupled to itself and a different subset of

*X*

^{*}(denoted by

*J*

_{1},

*J*

_{2}and

*K*are coefficient matrices.

*J*

_{1}and

*J*

_{2}are Hermitian, whereas

*K*is arbitrary. For definiteness, suppose that

*X*

_{1}and

*X*

_{2}are

*n*× 1 vectors, where 2

*n*≤

*m*, so

*J*

_{1},

*J*

_{2}and

*K*are

*n*×

*n*matrices. Then

*J*

_{1}and

*J*

_{2}are each specified by (up to)

*n*

^{2}real parameters, whereas

*K*is specified by 2

*n*

^{2}real parameters. By applying the Hamilton equations to Hamiltonian (47), one obtains the CMEs The solutions of Eqs. (49) can be written in the IO forms where

*M*

_{11},

*N*

_{12},

*M*

_{22}and

*N*

_{21}are

*n*×

*n*transfer matrices. For the special case in which

*n*= 1,

*J*

_{1}=

*δ*

_{1},

*J*

_{2}=

*δ*

_{2}and

*K*=

*γ*, and Eqs. (49) reduce to Eqs. (19).

*n*× 1 mode vector and 2

*n*× 2

*n*coefficient matrix are respectively. The solution of Eq. (51) can be written in the IO form where

*T*is the 2

_{x}*n*× 2

*n*transmission matrix. If

*L*is a constant matrix, then The

_{x}*n*×

*n*blocks of

*T*are

_{x}*M*

_{11},

*N*

_{12},

*L*are not necessarily equal and the off-diagonal blocks are not necessarily symmetric, as are the corresponding blocks of

_{x}*L*[Eqs. (44) and (52)]. Hence, we will derive the properties of the special system and deduce the corresponding properties of the general system.

_{y}*c*= |

*x*

_{1}|

^{2}− |

*x*

_{2}|

^{2}is conserved [33

33. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE **44**, 904–913 (1956). [CrossRef]

*L*. By using this result, one can show that Eq. (59) is equivalent to the equations

_{x}*T*=

_{x}*VDU*

^{†}, where

*U*and

*V*are unitary, and

*D*is positive and diagonal. (Positivity is required because

*T*is invertible.) The columns of

_{x}*U*(input Schmidt vectors) are the eigenvectors of

*V*(output Schmidt vectors) are the eigenvectors of

*D*(Schmidt coefficients) are the square roots of the (common) eigenvalues of

*U*and

*V*by the same set of phase factors, and one can permute (reorder) the columns of

*U*and

*V*in the same way, without invalidating the decomposition. Notice that the input vectors of

*T*and the Schmidt coefficients of

_{x}*T*.

_{x}*A*, Hence, pre- and post-multiplying a matrix by

*S*does not change its eigenvalues. (Furthermore, if

*E*is an eigenvector of

*A*, then

*SE*is an eigenvector of

*SAS*.) For any invertible matrix

*A*and any other matrix

*B*, Hence, interchanging two matrices does not change the eigenvalues of their product. Equation (62) implies that

*λ*

^{−1}. (Furthermore, if

*E*is the eigenvector associated with

*λ*, then

*SE*is the eigenvector associated with

*λ*

^{−1}.) Because the Schmidt coefficients of

*T*are the square roots of these eigenvalues, they always occur in reciprocal pairs, as they did in Eqs. (17) and (29).

_{x}*M*

_{11},

*N*

_{12},

*N*

_{21}and

*M*

_{22}, which allow the decomposition to be determined. The former equation implies that By equating the blocks in Eq. (65), one finds that Notice that Eqs. (66) reduce to Eqs. (9) and (22) in the appropriate limits. The latter equation implies that

*V*

_{12}=

*V*

_{11}=

*V*

_{1}and

*D*

_{11}=

*D*

_{22}=

*D*and

_{μ}*D*

_{12}=

*D*

_{21}=

*D*, where

_{ν}35. 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]

*U*and

_{j}*V*by

_{j}*U*

_{j}e^{iϕj}and

*V*

_{j}e^{iϕj}, respectively, the diagonal blocks of the transfer matrix would be unaltered, whereas the off-diagonal blocks would be multiplied by

*e*

^{i(ϕ1+ϕ2)}and

*e*

^{−i(ϕ1+ϕ2)}. One can exploit this non-uniqueness to write some decompositions in particularly simple ways.

*n*sets of Schmidt modes in decomposition (73), there are two Schmidt coefficients (|

*μ*| and |

*ν*|) and four phase combinations (

*ϕ*

_{v}_{1}−

*ϕ*

_{u}_{1},

*ϕ*

_{v}_{1}+

*ϕ*

_{u}_{2}, −

*ϕ*

_{v}_{2}−

*ϕ*

_{u}_{1}and −

*ϕ*

_{v}_{2}+

*ϕ*

_{u}_{2}). However, only one of the coefficients is independent (|

*ν*|) and only three of the combinations are independent. If one defines the reference phase (

*ϕ*

_{v}_{1}−

*ϕ*

_{v}_{2}−

*ϕ*

_{u}_{1}+

*ϕ*

_{u}_{2})/2, then the first and fourth combinations have the relative phases ±(

*ϕ*

_{v}_{1}+

*ϕ*

_{v}_{2}−

*ϕ*

_{u}_{1}−

*ϕ*

_{u}_{2})/2, whereas the second and third combinations have the relative phases ±(

*ϕ*

_{v}_{1}+

*ϕ*

_{v}_{2}+

*ϕ*

_{u}_{1}+

*ϕ*

_{u}_{2})/2. Thus, if the Schmidt modes are known, only 4

*n*real parameters are required to specify the transfer matrix. [The other 4

*n*(

*n*− 1) parameters in the coefficient matrix specify the Schmidt modes.] For the special case in which

*J*

_{1}=

*J*

_{2}and

*K*=

*K*, the signal and idler equations are identical, so

^{t}*U*

_{1}=

*U*

_{2},

*V*

_{1}=

*V*

_{2}and the reference phase is 0. Thus, if the Schmidt modes are known, only 3

*n*real parameters are required to specify the transfer matrix. [The other 2

*n*(

*n*− 1) parameters in the coefficient matrix specify the Schmidt modes.] These results are consistent with Eqs. (15) and (27), which apply to cases in which

*n*= 1.

*E*

_{∓}=

*SE*

_{±}, as was predicted after Eq. (64). Hence, if

*E*

_{±}are the eigenvectors of

*SE*

_{±}=

*E*

_{∓}are the eigenvectors of

*U*,

_{j}*V*,

_{j}*D*and

_{μ}*D*depend implicitly on

_{ν}*z*. Let

*X*

_{1}=

*U*

_{1}

*X*̄

_{1}and

*X*

_{2}=

*U*

_{2}

*X*̄

_{2}, where

*X̄*

_{1}= [

*x*̄

_{1}

*]*

_{j}*and*

^{t}*X*̄

_{2}= [

*x*̄

_{2}

*]*

_{j}*. Then decomposition (78) implies that the (input) combinations*

^{t}*D*. However,

_{ν}*D*is non-negative by construction, so this empirical rule is not canonical.

_{ν}*T*=

_{x}*VDU*

^{†}, so the laws of matrix algebra require that

*vice versa*. It is also easy to verify that Decomposition (81) works by permuting the Schmidt vectors so that squeezed modes are stretched and

*vice versa*. These actions are equivalent ways to obtain the same result: In the (common) inversion formula

*P*matrices can act to the middle, or to the outsides.

*D*and

_{μ}*D*to remain positive. By applying them twice, one finds that

_{ν}*U*(

_{j}*z*) →

*iV*(−

_{j}*z*) →

*i*

^{2}

*U*(

_{j}*z*) and

*V*(

_{j}*z*) →

*iU*(−

_{j}*z*) →

*i*

^{2}

*V*(

_{j}*z*). These results are acceptable in the context of a Schmidt decomposition, because the signs (phases) of the Schmidt modes are not unique. By applying transformations (85) to the forward decomposition (78), one obtains a backward decomposition that is similar to decomposition (81): The first unitary matrix is multiplied by

*i*and the second is multiplied by −

*i*, so the decompositions are equivalent. Notice that transformations (85) are consistent with Eqs. (17) and (18), and Eqs. (29) and (30). In the former case

*u*

_{1}=

*u*

_{2}=

*e*

^{iϕd}and

*v*

_{1}=

*v*

_{2}=

*e*

^{iϕs}, whereas in the latter case

*u*

_{1}=

*u*

_{2}=

*e*

^{iϕd},

*v*

_{1}=

*e*

^{iϕs+iϕδ}and

*v*

_{2}=

*e*

^{iϕs−iϕδ}.

*M*

_{11}and

*N*

_{12}are the upper blocks of

*T*and

_{x}*N*

_{21}and

*M*

_{22}are the conjugates of the lower blocks. It is easy to verify that Relative to the basis vectors contained in

*U*

_{1}and

*U*

_{2}, the real quadratures of the sum modes and the imaginary quadratures of the difference modes are stretched. It follows from Eq. (4), in which the transfer functions are non-negative by construction, that the imaginary sum quadratures and the real difference quadratures are squeezed. These results are valid for the general case in which

*J*

_{1}≠

*J*

_{2}and

*K*≠

*K*. Notice that Eqs. (87) and (88) reduce to Eqs. (35) and (36) in the appropriate limit.

^{t}## 4. Unifying principles

*J*

_{2}is Hermitian, one can rewrite Hamiltonian (47) in the alternative form and by applying the alternative Hamilton equations to Hamiltonian (89), one can reproduce the aforementioned CMEs. Equations (89) and (90) are equivalent to the Hamiltonian and the single Hamilton equation where the mode vector and coefficient matrix are respectively, and the spin matrix

*S*was defined in Eq. (55). Notice that

*G*is Hermitian. (Consequently, if

*L*=

*SG*, then

*SL*=

*L*

^{†}

*S*.) Equations (91) and (92) are said to be in canonical form.

*X*′ =

*TX*, where

*T*is an arbitrary transformation (change-of-variables) matrix. Then, in component form, A transformation is said to be canonical (symplectic) if the equation for

*x*′

*has the same Hamiltonian form as the equation for*

_{i}*x*. Equation (94) shows that

_{i}*T*is symplectic if and only if Condition (95) can be rewritten in the matrix form If condition (96) is satisfied, then

*T*

^{−1}=

*ST*

^{†}

*S*and (

*T*

^{†})

^{−1}=

*STS*. For reference, the set of (nonsingular) matrices that satisfy condition (96) form a group with respect to multiplication.

*T*(

*z*) is the transfer matrix for the system, which satisfies the evolution equation together with the input condition

*T*(0) =

*I*. Then because

*G*is Hermitian. Hence, Equation (99) implies that Hence, the MRW variable

*X*

^{†}

*SX*is conserved. Equation (99) also implies that

*T*

^{−1}=

*ST*

^{†}

*S*and (

*T*

^{†})

^{−1}=

*STS*.

*X*evolves. One can prove this statement directly. Alternatively, by multiplying the identity

*T*

^{†}

*ST*=

*S*by

*S*(

*T*

^{†})

^{−1}on the left and

*T*

^{−1}

*S*on the right, one can show that

*S*= (

*TS*)

*S*(

*ST*

^{†}) =

*TST*

^{†}. Hence, the transfer matrix satisfies the symplectic condition, which is equivalent to the MRW condition (99). Notice that the proofs of the preceding results were based on the assumption that

*T*is a linear transformation, but not on the assumption that

*G*is a constant: The results remain valid when

*G*is a function of

*z*.

*T*

^{−1}=

*ST*

^{†}

*S*. For example, if (

*T*

^{†}

*T*)

*E*=

*λE*, where

*λ*≠ 0, then Thus, not only do the Schmidt coefficients occur in reciprocal pairs, but there is also a simple relation between the associated Schmidt vectors. The Schmidt decomposition (73) owes its form to the constraints imposed on the blocks of the transfer matrix [Eqs. (67) and (68)], which are just (

*ST*

^{†}

*S*)

*T*=

*I*and

*T*(

*ST*

^{†}

*S*) =

*I*. Decompositions (78), (79), (81), (87) and (88) all follow directly from decomposition (73). Furthermore, if

*T*=

*VDU*

^{†}, then It is always true that the input (output) vectors of

*T*

^{†}are the output (input) vectors of

*T*. Equations (101) and (102) show that the stretched modes for the forward transformation are the squeezed modes for the backward transformation. These relationships guarantee that the combined transformation is the identity transformation.

*m*,C) is the general linear group, whose members are

*m*×

*m*complex matrices. SL(

*m*,C) is the special linear group of degree

*m*, whose members are unimodular (have determinant 1).

*m*) is the unitary group, whose members are

*m*×

*m*unitary matrices (which are complex by definition). The actions of these matrices preserve the quadratic form

*X*

^{†}

*X*, where

*X*is an arbitrary

*m*× 1 vector. SU(

*m*) is the special unitary group of degree

*m*, whose members have determinant 1. It is sometimes called the unimodular unitary group. These groups occur in models of conservative phenomena, in which

*X*

^{†}

*X*is the total power (or energy). For example,

*U*(2) and

*SU*(2) underly polarization rotation, beam splitting, directional coupling and (stable) frequency conversion.

*n*,

*n*) is the pseudo-unitary group, whose members are 2

*n*× 2

*n*complex matrices. The actions of these matrices preserve the quadratic form

*X*

^{†}

*SX*, where

*X*is an arbitrary 2

*n*× 1 vector. (This group has a subgroup of diagonal matrices that are unitary, but most of its members are nonunitary.) In the context of parametric amplification,

*X*

^{†}

*SX*is the MRW variable and

*T*is a member of U(

_{x}*n*,

*n*). SU(

*n*,

*n*) is the special pseudo-unitary group of degree 2

*n*, whose members have determinant 1. (This group also has a unitary subgroup.) In the aforementioned context,

*T*is a member of SU(

_{y}*n*,

*n*).

*T*is also a member of the symplectic group Sp(2

_{y}*n*), whose members satisfy condition (96) and have determinant 1. (In the Appendix, it is shown that this definition of the symplectic group is equivalent to the standard definition, which involves a different auxiliary matrix.) The mathematical properties of continuous groups are described in [39

39. D. H. Sattinger and O. L. Weaver, *Lie groups and Algebras with Applications to Physics, Geometry and Mechanics* (Springer, 1986). [CrossRef]

41. H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nou. Rev. Opt. **4**, 37–41 (1973). [CrossRef]

44. C. C. Gerry, “Remarks on the use of group theory in quantum optics,” Opt. Express **8**, 76–85 (2001). [CrossRef] [PubMed]

## 5. Summary

*n*modes (

*n*is arbitrary). The coefficient matrix that appears in the coupled-mode equation (51) has a symmetry property [Eq. (56)] that constrains the associated transfer matrix [Eq. (59)]. This constraint links the decompositions of the signal and idler blocks, and allows the decomposition of the forward transfer matrix to be determined [Eqs. (73) and (78)]. The transfer matrix involves 4

*n*mode vectors, for the input and output signal and idler, which are equivalent to 2

*n*input and output Schmidt vectors (combinations of the signal and idler vectors). In addition to these vectors, the transfer matrix involves (up to) 4

*n*real parameters (

*n*Schmidt coefficients and 3

*n*phase factors). This number of parameters is much smaller than the number required to specify the aforementioned coefficient matrix, which is of order

*n*

^{2}. If the forward matrix and decomposition are known, so also are the backward matrix and decomposition [Eqs. (79) and (80)]: One obtains the latter entities from the former by interchanging the input and output vectors, and interchanging the stretching and squeezing factors.

## Appendix: Real and complex Hamiltonian systems

*H*(

*q*,

*p*), where

*q*and

*p*are conjugate variables. The Hamilton equations (for

*z*-evolution) are By defining the vector variable

*X*= [

*x*

_{1},

*x*

_{2}]

*= [*

^{t}*p*,

*q*]

*, one can rewrite Eqs. (103) in the matrix form where the auxiliary matrix and the derivative of*

^{t}*H*is taken componentwise. Notice that

*J*

^{2}= −

*I*.

*α*̄,

*β*̄ and

*γ*̄ are real constants (parameters). By applying the Hamilton equations (103) to Hamiltonian (106), one obtains the linear evolution equations Alternatively, one can rewrite Hamiltonian (106) in the compact form where the coefficient matrix Notice that

*G*is symmetric. By applying the Hamilton equation (104) to Hamiltonian (108), one obtains the matrix evolution equation which is equivalent to the component equations (107).

*X*′ =

*TX*, where

*T*is an arbitrary transformation (change-of-variables) matrix. Then, in component form, Hence, the transformation is symplectic if and only if This condition can be rewritten in the matrix form If condition (113) is satisfied, then

*T*

^{−1}= −

*JT*and (

^{t}J*T*)

^{t}^{−1}= −

*JTJ*.

*T*(

*z*) is the transfer matrix for the system, which satisfies the evolution equation together with the input condition

*T*(0) =

*I*. Then because

*J*= −

^{t}*J*,

*J*

^{2}= −

*I*and

*G*=

*G*. Hence, Conditions (113) and (116) are equivalent, and require that det(

^{t}*T*) = ±1. However,

*G*is symmetric, so tr(

*JG*) = 0 and det(

*T*) = 1. Hence,

*T*is a member of Sp(2,R), the three-parameter group whose members are symplectic 2 × 2 matrices with determinant 1. [This group is isomororphic to SL(2,R) and SU(1,1).]

*H*(

*a*,

*a*

^{*}) =

*H*[

*q*(

*a*,

*a*

^{*}),

*p*(

*a*,

*a*

^{*})], where

*a*and

*a*

^{*}are treated as independent variables. By combining Eqs. (103), (117) and (118), one obtains the complex Hamilton equations By defining the vector variable

*X*= [

*a*,

*a*]

^{*}*, one can rewrite Eqs. (119) in the matrix form where the auxiliary matrix*

^{t}*S*was defined in Eq. (55) and the derivative of

*H*is taken componentwise. Equation (120) is equivalent to Eq. (104).

*δ*,

*γ*and

_{r}*γ*). By applying the Hamilton equations (119) to Hamiltonian (121), one obtains the linear evolution equations The first of Eqs. (122) is just Eq. (5), which describes one-mode squeezing. One can reconcile the real and complex descriptions of this process by setting Alternatively, one can rewrite Hamiltonian (121) in the compact form where the coefficient matrix Notice that

_{i}*G*is Hermitian. By applying the Hamilton equation (120) to Hamiltonian (124) componentwise, and reassembling the results, one obtains the matrix evolution equation However, it is easier to rewrite Hamiltonian (124) without the factor of 2, and differentiate it with respect to

*X*

^{†}, treated as a single (vector) variable. This approach was taken in the main text.

*X*= [

_{r}*p*,

*q*]

*and*

^{t}*X*= [

_{c}*a*,

*a*

^{*}]

*. Then*

^{t}*X*=

_{c}*UX*, where the unitary matrix It follows from these definitions that if

_{r}*X*(

_{r}*z*) =

*T*(

_{r}*z*)

*X*(0), then

_{r}*X*(

_{c}*z*) =

*UT*(

_{r}*z*)

*U*

^{†}

*X*(0), so

_{c}*T*(

_{c}*z*) =

*UT*(

_{r}*z*)

*U*

^{†}. By multiplying Eq. (113) by

*U*on the left and

*U*

^{†}on the right, one finds that But

*UJU*

^{†}=

*iS*, so Eq. (128) is just Eq. (96). Thus, the real and complex symplectic identities are equivalent, so the groups formed by

*T*and

_{r}*T*[Sp(2,R) and SU(1,1)] are isomorphic. This equivalence extends to systems of 2

_{c}*n*variables.

## References and links

1. | M. E. Marhic, |

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. |

3. | J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron. |

4. | S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron. |

5. | P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev. |

6. | S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron. |

7. | R. Loudon, |

8. | M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. |

9. | M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express |

10. | 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. |

11. | K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. |

12. | C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express |

13. | M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optic parametric amplifiers with lineary or circularly polarized waves,” J. Opt. Soc. Am. B |

14. | C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express |

15. | G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Rev. |

16. | G. J. Gbur, |

17. | A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A |

18. | S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A |

19. | H. P. Yuen, “Two-photon states of the radiation field,” Phys. Rev. A |

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

21. | C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett. |

22. | W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A |

23. | M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. |

24. | C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A |

25. | C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express |

26. | 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. |

27. | E. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans. Telecom. |

28. | C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” J. Sel. Top. Quantum Electron. |

29. | C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. |

30. | D. A. Edwards, J. D. Fehribach, R. O. Moore, and C. J. McKinstrie, “An application of matrix theory to the evolution of coupled modes,” to appear in SIAM Rev. |

31. | C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express |

32. | K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron. |

33. | J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE |

34. | M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE |

35. | A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. |

36. | C. J. McKinstrie, M. G. Raymer, and H. J. McGuinness, “Spatial-temporal evolution of asymmetrically-pumped phase conjugation I: General formalism,” Alcatel-Lucent ITD-09-48636Q, available upon request. |

37. | H. Goldstein, |

38. | V. I. Arnold, |

39. | D. H. Sattinger and O. L. Weaver, |

40. | M. Hamermesh, |

41. | H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nou. Rev. Opt. |

42. | Y. S. Kim and M. E. Noz, “Illustrative examples of the symplectic group,” Am. J. Phys. |

43. | A. Mufti, H. A. Schmitt, and M. Sargent, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys. |

44. | C. C. Gerry, “Remarks on the use of group theory in quantum optics,” Opt. Express |

**OCIS Codes**

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

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

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 11, 2012

Revised Manuscript: December 5, 2012

Manuscript Accepted: December 30, 2012

Published: January 14, 2013

**Citation**

C. J. McKinstrie and M. Karlsson, "Schmidt decompositions of parametric processes I: Basic theory and simple examples," Opt. Express **21**, 1374-1394 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1374

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### References

- M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2007). [CrossRef]
- 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]
- J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron.E88-C, 327–334 (2005). [CrossRef]
- S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.E88-C, 859–869 (2005). [CrossRef]
- P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev.1, 231–248 (2007). [CrossRef]
- S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron.18, 670–680 (2012). [CrossRef]
- R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford, 2000).
- 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]
- M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express17, 4670–4676 (2009). [CrossRef] [PubMed]
- 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]
- K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron.28, 883–894 (1992). [CrossRef]
- C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express12, 2033–2055 (2004). [CrossRef] [PubMed]
- M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optic parametric amplifiers with lineary or circularly polarized waves,” J. Opt. Soc. Am. B20, 2425–2433 (2003). [CrossRef]
- C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express14, 8516–8534 (2006). [CrossRef] [PubMed]
- G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Rev.35, 551–566 (1993). [CrossRef]
- G. J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge, 2011), Sec. 5.4.
- A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A43, 3934–3938 (1991). [CrossRef] [PubMed]
- S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A71, 055801 (2005). [CrossRef]
- H. P. Yuen, “Two-photon states of the radiation field,” Phys. Rev. A13, 2226–2243 (1976). [CrossRef]
- C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D26, 1817–1839 (1982). [CrossRef]
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- W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001). [CrossRef]
- 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]
- C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A85, 053829 (2012). [CrossRef]
- C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005). [CrossRef] [PubMed]
- 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]
- E. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans. Telecom.10, 585–595 (1999). [CrossRef]
- C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” J. Sel. Top. Quantum Electron.18, 794–811 (2012). [CrossRef]
- C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282, 583–593 (2009). [CrossRef]
- D. A. Edwards, J. D. Fehribach, R. O. Moore, and C. J. McKinstrie, “An application of matrix theory to the evolution of coupled modes,” to appear in SIAM Rev.
- C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 4973–4979 (2004). [CrossRef] [PubMed]
- K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron.14, 648–658 (2008). [CrossRef]
- J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE44, 904–913 (1956). [CrossRef]
- M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE45, 1012–1013 (1957).
- 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]
- C. J. McKinstrie, M. G. Raymer, and H. J. McGuinness, “Spatial-temporal evolution of asymmetrically-pumped phase conjugation I: General formalism,” Alcatel-Lucent ITD-09-48636Q, available upon request.
- H. Goldstein, Classical Mechanics, 2nd Ed. (Addison-Wesley, 1980).
- V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd Ed. (Springer, 2000).
- D. H. Sattinger and O. L. Weaver, Lie groups and Algebras with Applications to Physics, Geometry and Mechanics (Springer, 1986). [CrossRef]
- M. Hamermesh, Group Theory and its Application to Physical Problems (Dover, 1989).
- H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nou. Rev. Opt.4, 37–41 (1973). [CrossRef]
- Y. S. Kim and M. E. Noz, “Illustrative examples of the symplectic group,” Am. J. Phys.51, 368–375 (1983). [CrossRef]
- A. Mufti, H. A. Schmitt, and M. Sargent, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys.61, 729–733 (1993). [CrossRef]
- C. C. Gerry, “Remarks on the use of group theory in quantum optics,” Opt. Express8, 76–85 (2001). [CrossRef] [PubMed]

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