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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 8 — Apr. 9, 2012
  • pp: 8367–8396
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Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt  »View Author Affiliations


Optics Express, Vol. 20, Issue 8, pp. 8367-8396 (2012)
http://dx.doi.org/10.1364/OE.20.008367


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Abstract

In this paper we consider frequency translation enabled by Bragg scattering, a four-wave mixing process. First we introduce the theoretical background of the Green function formalism and the Schmidt decomposition. Next the Green functions for the low-conversion regime are derived perturbatively in the frequency domain, using the methods developed for three-wave mixing, then transformed to the time domain. These results are also derived and verified using an alternative time-domain method, the results of which are more general. For the first time we include the effects of convecting pumps, a more realistic assumption, and show that separability and arbitrary reshaping is possible. This is confirmed numerically for Gaussian pumps as well as higher-order Hermite-Gaussian pumps.

© 2012 OSA

1. Introduction

As the computational needs of the world keep increasing, quantum information (QI) processing is of increasing interest [1

1. S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London) 437, 116–120 (2005). [CrossRef]

, 2

2. H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008). [CrossRef]

]. A fundamental process in QI is Hong-Ou-Mandel interference (HOM), in which two photons interfere through a quantum optical effect [3

3. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). [CrossRef] [PubMed]

]. Originally HOM interference was used to measure the delay between photons, but it has recently also been used in a scheme for quantum computation using linear optics [4

4. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001). [CrossRef]

, 5

5. I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005). [CrossRef] [PubMed]

].

For quantum key distribution and continuous variable teleportation it has been shown that inseparable three-mode entanglement is useful [6

6. A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004). [CrossRef]

]. This has been demonstrated in optical crystals using consecutive nonlinear optical interactions in resonance [6

6. A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004). [CrossRef]

9

9. R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (2005). [CrossRef] [PubMed]

]. Recently it has been demonstrated, theoretically and experimentally, that three-color tripartite entanglement is possible in an optical parametric oscillator across a wide frequency range [10

10. A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006). [CrossRef] [PubMed]

, 11

11. K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32, 695–697 (2007). [CrossRef] [PubMed]

].

QFC is also possible using non-degenerate four-wave mixing (FWM) in an optical-fiber [26

26. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]

]. It is in the form of Bragg scattering (BS), which is characterized by two strong pumps p and q that interact with two sidebands r and s such that πp +πsπq +πr. See Fig. 1 for the frequency locations of the four fields. This process has been used classically to allow FC (frequency conversion) over a wide frequency range [27

27. 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]

29

29. K. Uesaka, K. K. 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]

]. The advantages of BS are that it is tunable [30

30. 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]

], has low-noise transfer [31

31. A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006). [CrossRef] [PubMed]

], and allows for very distant FC (more than 200 nm) [32

32. 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]

, 33

33. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011). [CrossRef]

]. BS has also been used to FC single photons [34

34. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010). [PubMed]

].

Fig. 1 (a) The placement in the frequency spectrum of the two pumps p and q along with the sidebands s and i for frequency conversion in the near-conversion regime. ω0 is the zero-dispersion frequency. In this case the two pumps are closely spaced in frequency. (b) Illustration of frequency conversion in the far-conversion regime. Here the pumps are farther from each other. Arrows pointing up denote creation of photons and arrows pointing down destruction of photons.

One advantage of QFC using four-wave mixing in optical fibers is that the emitted photon wavepacket has a transverse distribution that is already mode-matched to existing transmission fibers. Also it allows for a very broad bandwidth of conversion as well as coupling from the visible to the telecom band and inter-telcom band conversion [26

26. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]

]. The quantum-noise properties of parametric amplification were considered in [35

35. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004). [CrossRef] [PubMed]

, 36

36. 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]

] and it has theoretically been shown that BS allows for noiseless QFC [26

26. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]

]. BS has also been shown theoretically to allow HOM-interference between photons of different colors [37

37. 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]

, 38

38. H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express 19, 17876–17907 (2011). [CrossRef] [PubMed]

].

2. General formalism of FC

Equations (1) and (2) also apply to quantum mechanical operators, where the classical fields Aj are replaced with the mode operators âj [26

26. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]

, 34

34. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010). [PubMed]

]. It is known that beam splitters do not add excess noise [43

43. R. Loudon, The Quantum Theory of Light, 3rd. ed. (Oxford University Press, 2000).

], and since FC by BS has mathematically equivalent input-output (IO) relations it does also not add excess noise [26

26. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]

]. The mode operators satisfy the boson commutation relations
[a^i(t),a^j(t)]=0and[a^i(t),a^j(t)]=δijδ(tt),
(3)
with i, j ∈ {r, s}, δij is the Kronecker delta and δ(tt′) is the Dirac delta function. The CMEs are valid in the so-called parametric approximation, in which the pumps are treated as strong continuous fields, and for which quantum fluctuations are ignored. The weak sidebands, however, are treated quantum mechanically.

Using the Green-function formalism, it is possible to write the solution of the CMEs in the IO form [37

37. 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]

, 38

38. H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express 19, 17876–17907 (2011). [CrossRef] [PubMed]

]
Aj(t,l)=kGjk(t,l;t,0)Ak(t,0)dt.
(4)
From this equation, the output of mode j at (t, l) is described by a function Gjk that represents the influence of the input-mode k at (t′, 0). In our example with two sidebands k ∈ {r, s}, Eq. (4) leads to
Ar(t)=Grr(t;t)Ar(t)dt+Grs(t;t)As(t)dt,
(5)
where the short notation Gjk(t;t′) = Gjk(t, l;t′, 0) has been introduced and with t and t′ as output and input times respectively. Similarly, As is described by the shape of itself and sideband r at the input of the fiber. Physically this means that Grr(t;t′) and Gss(t;t′) describe the influence on the output at time t, from the input of the field itself at time t′, see Fig. 2. In a similar way, the cross Green functions Grs(t;t′) and Gsr(t;t′) concern the influence on one sideband at time t, from the other sideband at the input at time t′.

Fig. 2 A characteristic diagram for the generation of an idler from pulsed pumps for βr = −βs. The solid diagonal lines show the influence from the signal at the input point (t′, 0) whereas the dashed lines show the domain that influences the output idler at (t, l) from the time-dependent pumps.

It is convenient to introduce the singular value (Schmidt) decomposition of the Green’s functions since it allows us to split the Green function into products of functions that depend only on the input and output times at the cost of an infinite sum [44

44. 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]

]. In general we write
G(t;t)=nvn(t)λn1/2un(t),
(6)
where the functions un and vn are the Schmidt modes normalized with respect to the integral of the absolute value squared. The normalized functions as well as the Schmidt coefficients λn1/2 are found from the integral eigenvalue equations
K1(t,t)vn(t)dt=λnvn(t),
(7)
K2(t,t)un(t)dt=λnun(t),
(8)
with the kernels K1(t,t′) ≡ ∫ G(t;t2)G*(t′;t2) dt2 and K2(t,t′) ≡ ∫ G(t1; t)G*(t1; t′) dt1. We remind the reader that the first argument of the Green function corresponds to the output and the second argument to the input. The physical interpretation of the Schmidt decomposition is thus that it takes the input mode un and converts it to the output mode vn with the probability λn [37

37. 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]

, 38

38. H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express 19, 17876–17907 (2011). [CrossRef] [PubMed]

]. In matrix notation, the preceding decomposition can be rewritten simply as [45

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

] G = VDU, with V and U being (different) unitary matrices and D a diagonal matrix containing the non-negative square roots of the eigenvalues of the non-negative matrix GG (which are equal to the eigenvalues of GG). Likewise V contains in its columns the eigenvectors of GG and the columns of U the eigenvectors of GG [46

46. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1990).

].

2.1. The frequency and time domains

Some aspects of the FC process are easier to model in the time domain, and others are easier to model in the frequency domain. Thus this section considers the relations between the two domains. The following analysis is based on the symmetric Fourier-transform
{A(t)}=(2π)1/2A(t)exp(iωt)dt,
(11)
1{A(ω)}=(2π)1/2A(ω)exp(iωt)dω.
(12)
The Fourier-transform of functions of two variables is defined in a similar way. Considering the input-output (IO) equation, Eq. (5), ignoring the self-band interactions, and taking the Fourier-transform with respect to ωr yields
Ar(ωr)=Grs(ωr;t)As(t)dt.
(13)
A product in the time-domain is the inverse Fourier-transform of a convolution in the frequency-domain, so
Ar(ωr)=(2π)1dtdωsdωGrs(ωr;ωsω)As(ω)exp(iωst)
(14)
We remind the reader of the relation, exp(±iωt)dt=2πδ(ω), from which the IO relation in the frequency-domain is obtained
Ar(ωr)=Grs(ωr;ωs)As(ωs)dωs.
(15)
Similarly, the signal Green function in the frequency domain is given as
As(ωs)=Gsr(ωs;ωr)Ar(ωr)dωr.
(16)
The Fourier transform of the Green function is given as
Grs(ωr;ωs)=(2π)1Grs(t;t)exp(iωrtiωst)dtdt,
(17)
Grs(t;t)=(2π)1Grs(ωr;ωs)exp(iωrt+iωst)dωrdωs.
(18)

Another interesting aspect is the Fourier-transform of the Schmidt-decomposed Green functions. Suppose that the time-domain Green function has the Schmidt-decomposition
Grs(t;t)=n=0λn1/2vn(t)un*(t),
(19)
and when using Eqs. (17) and (19) gives
Grs(ωr;ωs)=n=0λn1/2vn(t)exp(iωrt)(2π)1/2dt×[un(t)exp(iωsts)]*(2π)1/2dt,
(20)
where we have exchanged the order of the sum and the integral which is allowed since the sum by definition is convergent. This form shows that the Fourier transform of the Schmidt decomposition of the Green function involves only the Fourier-transform of the individual input and output modes
Grs(ωr;ωs)=n=0λn1/2vn(ωr)un*(ωs).
(21)
We also note from Parseval’s identity that if the eigenfunctions are normalized in the time-domain they are also normalized in the frequency-domain [47

47. M. P. des Chênes, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaire du second ordre, à coefficients constants,” Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assembleés. Sciences, mathématiques et physiques , 638–648 (1806). [PubMed]

]. The Schmidt coefficients, λn1/2 are naturally identical in the two domains. In the special case where un(t) are real the relation un*(ω)=u(ω) holds. A similar result exists when going from the frequency-domain to the time-domain. That is the Fourier transform of the Green function is the sum of products of the Fourier transform of the Schmidt modes. This is an important result that is used extensively through the remainder of this paper.

3. Stationary pumps

To gain physical insight we start by solving the FC problem in the low-conversion regime while assuming that the pumps do not convect relative to one another in the moving frame propagating at the average group slowness of the sidebands. This is a simplified model of FWM, since in most cases the pumps walk-off with respect to each other. However it is representative of TWM, because one pump is always stationary. One may simply choose the frame of reference to propagate with the pump. First we derive the equations in the frequency-domain, since this is the standard approach and afterwards we present an alternative derivation.

By Fourier transforming Eqs. (1) and (2) with respect to t and t′ one find that
zAr(ωr,z)=iβr(ωr)Ar(ωr,z)+i(2π)1/2γpq(ωrωs,z)As(ωs)dωs,
(22)
zAs(ωs,z)=iβs(ωs)As(ωs,z)+i(2π)1/2γpq*(ωsωr,z)Ar(ωr)dωr.
(23)
Introducing the transformed field Aj(ωj, z) = Bj(ωj, z) exp[iβj(ωj)z] simplifies the analysis. Furthermore it is assumed that βj(ωj)=βj(1)ωj, thus neglecting group velocity dispersion and higher-order effects, i.e. βj(3), βj(4), . . ., which is reasonable for a sufficiently short piece of fiber and a narrow pulse in the frequency domain. Throughout the remainder of this paper the simpler notation βj(1)=βj is used for the group slowness. These assumptions lead to the approximate solutions [38

38. H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express 19, 17876–17907 (2011). [CrossRef] [PubMed]

]
Br(ωr,l)i(2π)1/20lγpq(ωrωs,z)exp(izωsβs)×exp(izωrβr)Bs(ωs,0)dωsdz,
(24)
where l is the fiber length. A similar expression exists for the signal field. Using Eqs. (15) and (16), and inserting Bj = Aj exp[−iβj(ωj)z], one finds the Green functions
Grs(ωr;ωs)=i(2π)1/20lγpq(ωrωs,z)exp[iβr(lz)ωr+iβszωs]dz,
(25)
Gsr(ωs;ωr)=i(2π)1/20lγpq*(ωsωr,z)exp[iβs(lz)ωs+iβrzωr]dz.
(26)

3.1. Standard analysis

For many applications one is interested in separating the Green functions into functions depending on only one of the frequencies, as frequency entanglement is undesired for some quantum optical interference experiments [48

48. 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]

, 49

49. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

]. Separability has been studied extensively for photon-pair generation using three-wave mixing [48

48. 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]

, 49

49. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

], but not for FC. To achieve separability it is required that μ = 0 and, with the aforementioned assumption that αrs > 0, this leads to the separability requirement αrαs = −σ2. In the co-propagating case (where the group slownesses of the sidebands have the same sign) it is not possible to obtain separability since σ is real, whereas in the counter-propagating case (with different signs of the inverse group velocities) it is possible to obtain separability for one specific length of the fiber. For μ = 0, this leads to considerably simpler parameters
λ0=πγ02l2(αrαs)1/2αrs,
(38)
τr=(σ2+αr2)1/2,
(39)
τs=(σ2+αs2)1/2,
(40)
where λ0 is the only non-zero squared Schmidt coefficient. Notice that this Schmidt coefficient is indeed independent of the fiber length.

It is instructive to cast Eq. (31) in a slightly different way:
Grs(ωr;ωs)=in=0λn1/2vn(ωr)un*(ωs),
(41)
with vn(ωr) = ϕn(τrωr) exp(irωr/2) and un(ωs) = ϕn(τsωs) exp(−isωs/2). Since ωr and ωs correspond to the output and input frequencies respectively, we notice that τr and τs are characteristic time scales.

The time-domain Green function is found by Fourier transforming Eq. (30), see Appendix A for the details. The result is
Grs(t;t)=iγπσξ1/2|βrs|exp[(σ2+αs2)t¯22(σ2+αrαs)tt¯+(σ2+αr2)(t¯)22(σαrs)2]
(42)
where the retarded (or advanced) times are t̄ = tβrl/2 and t¯=t+βsl/2.

According to the inverse of Eq. (21) the Fourier transform of the Schmidt decomposition is simply the Fourier transform of the individual Schmidt modes and since 𝒡−1 {f()ei} = f[(tb)/a]/|a|, Eq. (41) becomes
Grs(t;t)=in=0λn1/2ϕn(tβrl/2τr)ϕn*(t+βsl/2τs),
(43)
where the Schmidt modes remain normalized and the square of the Schmidt coefficients are given in Eq. (33). This is the same decomposition we get by using Eqs. (104)(107) to decompose Eq. (42), thus confirming the results in Eqs. (19) and (21). For the case in which βr = −βs the input and output modes are shifted in phase by the factor jωj/2 corresponding to an interaction at the middle of the fiber which maximizes the interaction between the four fields [38

38. H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express 19, 17876–17907 (2011). [CrossRef] [PubMed]

].

3.2. Alternative analysis

The standard analysis in the frequency domain is based on two reasonable, but nonetheless restricting assumptions, i.e. similar Gaussian pumps and the approximation of the sinc-function with a Gaussian. We now present an alternative analysis that enables deriving the Green functions in the time-domain in the general case by interchanging the order of the frequency and length integrals. Finally we also present a simpler and more physical derivation.

Considering the Fourier transform of Eq. (25) (for brevity we only show a detailed derivation of Grs), which is
Grs(t;t)=i(2π)3/20lγpq(ωrωs,z)×exp[i(tβrl+βrz)ωr+i(t+βsz)ωs]dωrdωsdz.
(44)
Using the substitution ωr = ωrωs and the Fourier transform property 𝒡−1 {f(ω)eaiω} = f(ta) leads to
Grs(t;t)=i(2π)1/20lγpq(tβrl+βrz,z)exp[iωs(t+βszt+βrlβrz)]dωsdz.
(45)
Carrying out the second frequency integral gives a delta function in z, thus we find
Grs(t;t)=i/|βrs|γpq(βrtβs[tβrl]βrs,tt+βrlβrs)H(t+βrLt)H(ttβsL),
(46)
where H is the Heaviside step-function, which ensures causality. This result is valid for arbitrary pump shapes, and it does not approximate the way the system responds [beyond the perturbation theory used to derive Eq. (25)]. The argument of γpq is a complicated function of t and t′, but in the following section we present a simple physical derivation of it.

3.2.1. Time-domain collision analysis

Due to the simplicity of the Green function in the time-domain, Eq. (46), we find the Green functions directly, in the time domain, by using the method of characteristics [50

50. G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974), Chap. 2.

]. In the low-conversion regime, the presence of the idler has little effect on the signal. Hence, a signal impulse that enters the fiber at time t′ remains an impulse as it convects through the fiber. The part of the idler that exits the fiber at time t was generated by a collision of the idler pulse with the signal at the point (tc, zc), where t′ + βszc = tβr(lzc). Such a collision is illustrated in Fig. 3a. The collision distance and time are
zc=[t(tβrl)]/βrs,tc=[βrtβs(tβrl)]/βrs,
(47)
respectively. After the collision, the idler convects with constant amplitude. By integrating Eq. (1) across the collision region, which is infinitesimally thin, one finds that the cross Green function is given approximately by
Grs(t;t)[iγpq(zc,tc)/|βrs|]H(t+βrlt)H(ttβsl).
(48)

Fig. 3 Characteristic diagrams for (a) idler generation from a pulsed signal and (b) generation of a signal from a pulsed idler. The gray area shows the area of the high pump power region. The upward and downward diagonal lines are the characteristics of the idler and the signal respectively. The output idler (signal) at time t is generated by a collision with the signal (idler) occurring at the point c.

For signal generation by a pulsed idler, the collision distance and time are
zc=[(tβsl)t]/βrs,tc=[βr(tβsl)βst)]/βrs,
(49)
respectively, see Fig. 3b. By repeating the collision analysis described above, one obtains the cross Green function
Gsr(t,t)[iγpq*(zc,tc)/|βrs|]H(t+βrlt)H(ttβsl).
(50)

3.3. Comparing the time-domain and the frequency-domain results

The Green functions were found directly using the collision analysis, cf. Eq. (48). Notice that the collision time may be evaluated from
βrtβs(tβrl)=βr(t+βsl/2)βs(tβrl/2)=βrt¯βst¯,
(51)
where j are the same retarded times as used in Eq. (42). Second, notice that
H(ttβsl)H(t+βrlt)=rect[(tta)/(βrsl)]=rect[(t¯t¯)/(βrsl)],
(52)
where the average time is ta = t′ + (βr + βs)l/2 and the rectangle function [51

51. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

] is
rect(x)={1,if|x|<1/2;1/2,if x=±1/2;0,otherwise.
(53)
Thus, the Green function in Eq. (48) likewise only depends on the retarded times. The coupling function is defined as γpq(t)=γAp(t)Aq*(t) where both pump shapes are normalized. To compare the Green functions in Eqs. (42) and (48) we pick
Ap(t)Aq*(t)=exp[t2/(2σ2)]/(2πσ2)1/2.
(54)
In other words the Green function is of the form
G(t;t)=iγ(2π)1/2σ|βrs|exp[(βrt¯βst¯)22(βrsσ)2]rect(t¯t¯βrsl).
(55)
To compare this result with the frequency-domain result that was derived using the sinc approximation, Eq. (42), the rectangular function is approximated with the Gaussian
hexp[(t¯t¯)22w(βrsl/2)2],
(56)
where h and w are fitting parameters that will be determined later. For reference, h = 1 produces the correct peak height, whereas h = [2/(πw)]1/2 ≈ 1.28 produces the correct area. Aggregating these results the Green function is approximated by
Grs(t;t)iγh(2π)1/2σ|βrs|exp[(σ2+αs2)t2¯2(σ2+αrαs)tt¯+(σ2+αr2)2(t¯)22(σαrs)2],
(57)
where αj = w1/2βjl/2. Comparing Eqs. (42) and (57) they have the same general shape and we conclude that w = ξ and h = [2/(πw)]1/2 ≈ 1.28, which gives the same integral over the Gaussian and the rectangle function in the time domain. This shows that the effect of approximating the sinc with a Gaussian in the frequency-domain is equivalent to replacing the sharp boundaries from the rectangular function in the frequency-domain with a gradual effective boundary from the Gaussian. Physically this means that the Green function will allow effects from the input on the output from input and output times that are not allowed due to causality. Another issue is that in the limit of long fibers the rectangular function is unity for almost all times, so one has to pick h = 1 to get the best results.

Since Eq. (57) is of the canonical form for the Schmidt decomposition of a Gaussian, see Eq. (104) in Appendix A, we notice that the square of the lowest-order Schmidt coefficient and time scales (μj is used instead of τj here not to confuse it with the pump-width) are in the form
λ0=(γhl)2w4[(αr2+σ2)1/2(αs2+σ2)1/2+αrsσ],
(58)
μr=1(αrsσ)1/2(αs2+σ2αr2+σ2)1/4=1τr,
(59)
μs=1(αrsσ)1/2(αr2+σ2αs2+σ2)1/4=1τs,
(60)
where τj are the characteristic frequency-scales found in the frequency-domain Schmidt decomposition.

In the limit of short fibers βl/τ → 0 we find that
λ0(γhl)2w/(4σ2)=(γhl)2w/(2τ2),
(61)
μj1/(αrsσ)1/2=21/4/(αrsτ)1/2.
(62)
The square of the Schmidt coefficient scales quadratically with the length and the characteristic time-scales are the geometric means of αrs and σ, and increase with the square root of the fiber length. In this limit the separability coefficient (denoted t in the appendix) tends to
μ1αrs/σ,
(63)
which is close to unity, in other words this leads to a large number of non-zero Schmidt coefficients. In this limit of short fibers the sidebands experience approximately CW pumps when the pump-width is much larger than the sideband-width. Notice that the duration of the lowest-order Schmidt modes are much shorter than the pumps (it is the geometric mean of the transit time and the pump width). Also for signal-to-idler generation a pulse that is an arbitrary superposition of lower-order modes is converted without significant distortion, since the Schmidt coefficients decrease slowly as μ is close to unity.

For the other limit where βl/τ → ∞ we find
λ0(γh)2/|βrβs|,
(64)
μr(αs/αr)1/2/(αrsσ)1/2,
(65)
μs(αr/αs)1/2/(αrsσ)1/2.
(66)
The Schmidt coefficient tends to a constant, and the time-scales are now approximately the geometric mean of αrs and σ, and increase as the square root of the length. This is because the pumps overlap throughout the entire fiber. As discussed before; in this limit it is more reasonable to set h = 1 because the step-functions are almost equal to unity. The separability coefficient tends to
μ1σαrs/|αrαs|,
(67)
which is also close to unity. Thus, we would expect many Schmidt modes and therefore a non-separable Green function.

3.4. Numerical studies

Before we consider numerical studies of the various functions in this paper we discuss the natural dimensionless parameters to use. The efficiency of conversion is quantified by the dimensionless parameter γ̄ = γ/βrs, but this is not a parameter that is going to be varied since we consider the low-conversion efficiency limit which puts a natural limit on the conversion strength γ̄ ≪ 1. The natural unit to measure time in is in units of the pump-width and similarly for the length parameter it is natural to use the pump-width divided by β. For the remainder of the paper it is assumed that βr = β = −βs in the numerical studies, so βrs = 2β.

Fig. 4 Numerical studies of the Green function Grs. In all the plots β = 1, γ̄ = 0.1. The two pumps are normalized Gaussians with a root-mean-square width τ = 1. The white lines denote the cut-off due to the step-functions, but they only apply to the Green functions without the Gaussian approximation. For the approximate Green functions the height h = 1.28 was used. (a) The absolute value of the step-function Green function for βl/τ = 1, i.e. the non-separable case. (b) The Green function with the Gaussian approximation for βl/τ = 1. (c) The absolute value of Grs for βl/τ = 2.2768, the case where it is expected to be separable. The shape of the function implies that no separability is attainable. (d) The Gaussian approximation for the separable fiber-length, and it is indeed seen to be separable.
Fig. 5 (a) The Schmidt coefficients for the four plots in Fig. 4. Diamonds and crosses are for the separable and non-separable fiber length respectively with the Gaussian window whereas squares and open circles are the Schmidt coefficients with the rectangle window for the separable and non-separable length respectively. Notice that numerical study for the separable fiber-length, shows that the Gaussian approximation leads to separability, whereas the one with step-functions does not. (b) The first two Schmidt modes. The dashed curves are the Schmidt modes for the Gaussian Green function and the solid ones for the step-function window. The black curves are the zeroth-order modes and the blue curves the first-order modes. The normalized length of the fiber is one in this numerical study.

Fig. 6 (a) A plot of the Schmidt coefficients for a long fiber βl/τ = 10. The crosses and circles are the Schmidt coefficients for the Gaussian window for h = 1 and h = 1.28 respectively. The diamonds are for the rectangular window. (b) The first two Schmidt modes. The dashed curves are the Schmidt modes for the Gaussian window while the step-function Schmidt modes are solid. Again the black curves are the zeroth-order modes and the blue curves the first-order modes. Notice that since the Schmidt modes are normalized they are identical for both heights in the Gaussian approximation.

4. Convecting pumps

With the aforementioned assumption, the pumps are described by
Ap(z,t)=Fp[tβs(zzi)],
(68)
Aq(z,t)=Fq[tβr(zzi)],
(69)
where Fj are normalized shape-functions. The pumps intersect at the distance zi in the fiber. With this pump ansatz, our coupling function γpq in Grs depends on Ap and Aq which are functions of
tcrβs(zcrzi)=t+βszi,
(70)
tcrβr(zcrzi)=tβr(lzi),
(71)
respectively. Inserting these pumps in Eq. (46) leads to
Grs(t;t)=iγ¯Aq*[tβr(lzi)]Ap[t+βszi]×H(t+βrlt)H(ttβsl),
(72)
where γ̄ = γ/|βrs|. This Green function is naturally separable, and the input and output Schmidt modes are the shape functions of pumps p and q, respectively. Only the step-functions prevent complete separability, but for a sufficiently long fiber they are equal to unity for times of interest.

The other Green function was defined in Eq. (50) with the associated collision point defined in Eq. (49). Using the collision distance and time we find that
tcsβs(zcszi)=tβs(lzi),
(73)
tcsβr(zcszi)=t+βrzi,
(74)
and, hence, that
Gsr(t;t)=iγ¯Ap*[tβs(lzi)]Aq[t+βrzi]
(75)
×H(t+βrlt)H(ttβsl).
(76)
Results (72) and (76) have remarkable consequences. For a sufficiently long fiber we have in the case of signal-to-idler conversion that the natural input signal mode has the shape of pump p whereas the output idler mode attains the shape of pump q. The two modes are centered on βs(lzi) and βrzi respectively. For idler-to-signal conversion the natural input idler has the shape of pump q and the output idler attains the shape of pump p. In both cases the input and output Schmidt modes are timed to arrive at the intersection point of the pumps, since this will be the point of maximal interaction.

4.1. Gaussian pumps of equal width

Assuming two Gaussian pumps with the same width and zi = l/2, the Green function attains the form
Grs(t;t)=iγ¯πτexp[(tβrl/2)2+(t+βsl/2)22τ2]×H(t+βrlt)H(ttβsl).
(77)
The consequences of this result are illustrated in Fig. 7. We notice that this function is clearly separable for βl/τ = 3 and we confirm that the output mode, the first Schmidt mode for l = 3, indeed is a copy of pump q centered on βrl/2. It is difficult to conclude anything about the higher-order Schmidt modes, but a consequence of the Sturm comparison theorem, [52

52. G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd. ed. (McGraw-Hill, 1991).

], is that the eigenfunctions of a Sturm-Liouville problem have a monotonically increasing number of zeros. Since the Schmidt modes are eigenfunctions of the integral equations [cf. Eqs. (97) and (98)] we expect them to show the same behavior, which is confirmed by Fig. 7(d).

Fig. 7 Numerical studies of the Green function Grs when including the convecting pumps. In all the plots βr = 1 = −βs, γ̄ = 0.1. The numerical study uses Gaussian pumps with a root-mean square width τ = 1. In (a) and (b) white lines denote the cut-off due to the step-functions. (a) The absolute value of the Heaviside Green function for βl/τ = 1. (b) The Green function for βl/τ = 3. (c) A plot of the Schmidt coefficients for the two fiber lengths where the crosses are for the shorter length and the circles for βl/τ. For βl/τ = 3 the function is almost separable which is seen since we only have one dominating Schmidt mode. (d) The two first output Schmidt modes for the two lengths. The solid curves are for βl/τ = 1 and the dashed ones for βl/τ = 3. The lowest-order Schmidt mode is plotted in black and the next one in blue. For βl/τ = 3 the first output mode corresponds to the shape of pump q centered on βrl/2 like expected. The second output mode has another shape, but its Schmidt coefficient is almost zero, so this mode is negligible.

To check the hypothesis that the optimal interaction distance was half that of the fiber, a numerical study was performed of the square of the first two Schmidt coefficients λ0 and λ1 as a function of the interaction distance for two different fiber lengths. The result is seen in Figure 8 which confirms that the strongest frequency conversion is at zi = l/2. An interesting result is that for the short fiber the first two Schmidt coefficients have maxima at zi = l/2, which shows that the two lowest-order Schmidt modes have the maximal conversion there. This result was not replicated for the longer fiber, where the second Schmidt coefficient have minima at zi = l/2, but this is because the Green function is separable and changing zi moves the Green function in the (t,t′) plane leading to a cut-off due to the step-functions.

Fig. 8 The square of the first two Schmidt coefficients as a function of the interaction distance zi. The triangles and circles correspond to λ0 and λ1 respectively for βl/τ = 1. The dashed line is the square of the lowest-order Schmidt coefficient squared and the solid line the square of the next Schmidt coefficient for βl/τ = 3.

4.1.1. The Gaussian approximation

Comparing with Eq. (104) we note that the argument of Eq. (78) is of the same form, hence inserting from Eq. (78) and using Eq. (109) leads to
λ0=(γhl)2w2[αrs2+τ2+αrs(αrs2+2τ2)1/2],
(79)
μj=[αrs2(αrs2+2τ2)1/4]αrsτ,
(80)
μ=αrs2+τ2αrs(αrs2+2τ2)1/2τ2,
(81)
where again μj has been used instead of τj to avoid confusion.

In the limit of short fibers (βl/τ → 0) corresponding to ταrs we find
λ0(γhl)2w/(2τ2),
(82)
μj21/4/(αrsτ)1/2=1/(αrsσ)1/2,
(83)
μ1αrs21/2/τ=1αrs/σ.
(84)
Notice that these results are the same we found for the non-convecting case, Eqs. (61)(63), as expected since the effect of convection is imperceptible for short lengths. The square of the lowest-order Schmidt coefficient increases quadratically with length, a typical result from coherent scattering processes. The time-scale of the Schmidt modes is simply the geometric mean of αrs and σ, and increases as the square root of the length.

In the complementary limit in which βl/τ → ∞,
λ0(γh)2/βrs2,μj1/τ,μ0.
(85)
Thus the squared Schmidt coefficient tends to a constant (which is identical to the one found in the exact model for h = 1). Also the time-scale is simply the pump-width. Comparing with the non-convecting case, Eqs. (64)(67), the lowest-order Schmidt coefficient in both cases tends to a constant, but the time-scales differ as they now tend to a constant whereas the non-convecting ones grow with the square root of the fiber-length. Also the separability parameter tends to zero, such that the Green function is always separable for sufficiently long fibers, which was not the case for the non-convecting model that was only separable under the Gaussian approximation and under specific conditions. This contrasts to the non-convecting model because the interaction for the convecting case happens only during the collision, leading to separability and the drop-out of the length dependence.

These limits were tested numerically for the convecting case, see Fig. 9(a). The figure shows the square of the lowest-order Schmidt coefficient for the Green functions with the Gaussian-and step-function windows. It is seen that the Gaussian window over-estimates the value of the Schmidt coefficient for long fibers, but setting h = 1, such that the Gaussian has the same height as the rectangular function gives a better agreement. This is reasonable, as the rectangular window for large fiber-lengths is approximately unity. In the short fiber limit the two models disagree for h = 1, but one is free to choose w to obtain better accuracy since the long-fiber limit, Eq. (85) is independent of w. The width of the best-fit Gaussian in the time-domain was determined by matching the FWHM in the frequency-domain of the Gaussian and the sinc function. However, the FWHM of the inverse Fourier transform is not necessarily the same. By choosing the width such that the Gaussian and the rectangle have the same FWHM in the time-domain, one finds w = 0.7213. The square of the Schmidt coefficients with this width is seen in Fig. 9(b). Choosing this width results in a better fit in the short fiber limit, than the result of the traditional sinc approximation. By fitting w to the Schmidt coefficients with the rectangular window using Eq. (79), we were able to find a slightly better fit for intermediate fiber lengths for w ≈ 0.86, but at the cost of a worse fit for short fibers. Thus we conclude that the best fit is found for h = 1 and matching the FWHM in the time-domain.

Fig. 9 Plots of the square of the lowest-order Schmidt coefficient as a function of the length of the fiber. (a) For w = ξ = 0.3858, the value giving the same FWHM in the frequency-domain. The solid line is the square of the analytic lowest-order Schmidt coefficient, Eq. (79) with h = [2/()]1/2 ≈ 1.28. The crosses are the numerically found values for the Gaussian approximated Green function, which agrees with the analytic value. The open circles are also found numerically, but are for the convecting Green function with step-function window. Finally the dots is Eq. (79) with h = 1. (b) The same curves as the left panel but for h = 1 and w = 0.7213 which is the width giving the same FWHM in the time-domain.

4.2. Different Gaussian pumps

Fig. 10 Plots of the Green function Grs with different pump widths in this case τq = τp/2. In (a) βl/τp = 1 whereas (b) considers βl/τp = 3. The function is elongated in the t′ direction since pump p has the largest width. (c) shows the Schmidt coefficients where open circles are for the longer fiber-length. The long fiber Green function is still separable even though the pumps have different widths. In (d) the two lowest-order Schmidt modes for the two lengths is plotted. The black curves are the lowest-order Schmidt mode and the blue ones the next one. Dashed curves are for the longer fiber.

The lowest-order Schmidt coefficient was also considered for various aspect ratios, see Fig. 11(a). As the aspect ratio increases for a constant length the coefficient falls off since the value of the Green-function at the cut-off points increases, and it is therefore less separable. However, if the length increases with the aspect ratio it is possible to achieve separability over a wide range of aspect ratios. This is expected since in this case where pump q is wide compared to p and thus sideband r is wide and s narrow, for a long enough fiber the two short pulses propagate past the longer ones and hence experience a full collision [53

53. C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012). [CrossRef]

].

Fig. 11 Plot of the lowest-order Schmidt coefficient as a function of the aspect ratio. The crosses are for βl/τp = 2 and the open circles for the variable length 2τq/τp. (b) The Schmidt coefficients for a pulse with a very narrow aspect ratio (τq = τp/100) for two different fiber lengths βl/τp = 0.5 (crosses) and 3 (open circles).

For quantum communication it is of interest to convert the states emitted from a quantum memory unit to a shape suitable for transmission in an optical communication system. This might include reshaping the pulse width by a factor of 100 [13

13. D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106, 130501 (2011). [CrossRef] [PubMed]

]. To check whether such a reshaping is possible within the perturbative framework, a numerical study with τq = τp/100 was carried out with the Schmidt coefficients in Fig. 11(b). This definitely shows that the Green function for the longer fiber lengths is separable, but this is a natural extension of the discussion in Fig. 10(c).

4.2.1. The Gaussian approximation

In a similar way as the analysis for the Green function with two identical Gaussian pumps we are interested in investigating the effect of the Gaussian approximation. Two different pump-widths corresponds to replacing τ with (τpτq)1/2, and τq and τp in front of and t¯ respectively, in Eq. (78). For the limit where the aspect ratio tends to infinity or in other words τp → 0 and τq → ∞ we find that
λ0(γh/βrs)2(αrs/τq),μr1/αrs,μs1/τp.
(87)
Notice that the two time-scales are no longer identical, as expected, and they are determined by the smaller of either αrs or τj. The square of the Schmidt coefficient differs from γ̄2 by the factor αrs/τq.

These results were simulated, see Fig. 12(a). In general, the Gaussian window with h = 1 underestimates the lowest-order Schmidt coefficient, but for large aspect ratios the lower window height approximates the right result. Also the lowest-order Schmidt coefficients fall off since the length of the fiber was held constant. In the right panel the square of the lowest-order Schmidt coefficient is plotted for w = 0.7213. In this case the lower window height does give a better approximation, but it is only moderately accurate. This is because the Green function has a large value at the cut-off which makes the Gaussian approximation a less accurate fit.

Fig. 12 Plots of the limits of the square of the lowest-order Schmidt coefficient as a function of the aspect ratio. (a) This figure shows convecting pumps where βl/τp = 2, w = 0.3858 and the open circles are for the rectangular windows and filled circles and crosses are for the Gaussian window with h = 1 and h = 1.28 respectively. (b) The same functions as the left panel but for the the width yielding the same FWHM in the time-domain, w = 0.7213.

4.3. HG0/HG1 pumps

Fig. 13 Numerical study of the Green function Grs with pump p as HG0 and pump q as HG1. In all the cases γ̄ = 0.1. (a) A plot of the Green function for βl/τ = 1. (b) The Green function for the long fiber interaction, βl/τ = 4. (c) The first 10 Schmidt coefficients for the two fiber lengths, where the open circles are for the longer case. The longer one is clearly separable. (d) The first two output Schmidt modes for each of the numerical studies, again black is for the lowest-order mode and blue for the next one, while dashed curves are the ones for the longer fiber. The long fiber first-order Schmidt mode is a HG1 centered on βl/τ = 2 like expected, whereas the one for the shorter fiber is slightly distorted.

4.4. HG1/HG1 pumps

The final case considered is two identical HG1 pumps. Due to the larger width of the HG1 pumps, we consider βl/τ = 5 to ensure separability. The result is seen in Fig. 14. From Fig. 14(b) and (c) it is clear that the Green function is separable for a sufficiently long fiber, which is expected. Considering Eq. (72), we expect the output and input modes for the separable case simply to be copies of the two pumps. This is indeed confirmed from Fig. 14(d) where the HG1 pumps coincide with the lowest-order input and output Schmidt mode (only the output mode has been plotted here since it was indistinguishable from the input mode). Again the shorter fiber leads to a slightly distorted HG1 mode. With this study we showed that FC is possible for relatively short fibers and more complicated pump-shapes.

Fig. 14 Plots of the Green function Grs where both pumps are HG1 functions and γ̄ = 0.1. (a) A contour plot of the Green function for the fiber length βl/τ = 1. (b) The contour plot, for βl/τ = 5. (c) The computed first 10 Schmidt coefficients for the two fiber lengths, open circles are again for the longer fiber which is clearly separable. (d) A plot of the lowest-order output Schmidt mode as well as the theoretical HG1 modes. The black curves are for the shorter fiber and the dashed curves are the numerically found Schmidt modes. The input modes have not been plotted since they coincided (as expected) with the output modes.

5. Conclusion

The collision method was generalized to also include convecting pumps. The Schmidt decomposition was used to find the natural modes of the problem and obtain important limits that allowed us to compare the stationary and the convecting models. In the short-fiber limit the predictions of the two models agree. It was also shown that convecting pumps allow for separable Green functions for sufficiently long fibers. This is in contrast to the stationary result that is only separable for one specific length. Additionally, we showed that it is possible to obtain arbitrary reshaping of a signal by a proper selection of the pump pulses. This was confirmed for simple Gaussian pumps and was also shown to be possible for two Gaussian pumps with very different widths. Finally higher-order Hermite-Gaussian shapes were also seen to allow for separability and reshaping. Preliminary numerical results show that reshaping also occurs in the high-conversion regime.

These results show that frequency conversion by four-wave mixing is a valuable resource for quantum information systems, as an convenient and reliable source for reshaping and frequency conversion, both of which are paramount for these systems to be used in practice. The low-conversion analysis will be extended to the high-conversion regime in future work.

A. Appendix: Mehler identity and kernel decomposition

The Schmidt decomposition theorem [56

56. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen,” Mathematische Annalen 63, 433–476 (1907). [CrossRef]

, 57

57. B. L. Moiseiwitsch, Integral Equations (Dover, 2005).

] states that a complex kernel K(x, y) may be written as the series
K(x,y)=n=0λn1/2vn(x)un*(y).
(94)
The (non-negative) singular values λn are the (common) eigenvalues of the integral equations
λu(x)=Luu(x),
(95)
λv(x)=Lvv(x),
(96)
where the hermitian (and non-negative) kernels are
Lu(x,x)=K*(x,x)K(x,x)dx,
(97)
Lv(x,x)=K(x,x)K*(x,x)dx,
(98)
and the associated eigenfunctions u and v satisfy the orthonormality relations
um(x)un*(x)dx=δmn=vm(x)vn*(x)dx.
(99)
If K is real and symmetric, u and v are real and Lu = Lv.

Suppose that
K(x,y)=exp[(1+t2)(x2+y2)2(1t2)+2txy(1t2)].
(100)
Then the (common) hermitian kernel
L(x,y)=[π(1t2)(1+t2)]1/2exp[(1+t4)(x2+y2)2(1t4)+2t2xy(1t4)].
(101)
It follows from Eqs. (92) and (101) that
L(x,y)=π(1t2)n=0t2nψn(x)ψn(y).
(102)
Hence, the eigenfunctions that appear in decomposition (94) are the Hermite functions ψn(x), and the eigenvalues
λn=π(1t2)t2n.
(103)
These results also follow directly from Eqs. (92) and (100).

For asymmetrically-pumped FC,
K(ωr,ωs)=exp[(aωr2+2bωrωs+cωs2)/2],
(104)
where a, c and acb2 are all non-negative. One can rewrite Eq. (104) in the form of Eq. (100) by defining
t=[(ac)1/2(acb2)1/2]/b,
(105)
x = τrωr and y = τsωs, where
τr=[a(acb2)/c]1/4,
(106)
τs=[c(acb2)/a]1/4.
(107)
[The choice of root in Eq. (105) is determined by the requirement that t → 0 as b → 0]. The result is
K(ωr,ωs)=n=0λn1/2ψn(τrωr)ψn(τsωs)=n=0(λn/τrτs)1/2τr1/2ψn(τrωr)τs1/2ψn(τsωs),
(108)
where the singular values are specified by Eqs. (103) and (105), and the eigenfunctions τj1/2ψn(τjωj) are normalized. If b is positive, t is negative and vice versa. However, the Hermite functions Eq. (93) do not depend on t and the singular values depend only on t2, so Eqs. (103) and (108) omit sign information as written. One can restore this information by replacing t with |t| and multiplying τr or τs by sc = −sign(c). This change is equivalent to changing the sign of x or y in Eq. (100). By combining Eqs. (105)(107), one can show that
π(1t2)τrτs=2π(ac)1/2+(acb2)1/2,
(109)
This term appears in the formula for the Schmidt coefficients associated with normalized eigenfunctions.

The kernel was decomposed in the frequency-domain and the Schmidt modes were Fourier-transformed to the time-domain. Alternatively one can inverse-transform the kernel directly to the time-domain and then decompose it. To be able to compare the result in the time-domain with what we obtained in the frequency domain we consider the generalized Gaussian in the frequency-domain
F(ωr,ωs)=exp[(aωr2+2bωrωs+cωs2)/2].
(115)
By doing the requisite integrals explicitly, one can show that
F(t,t)=exp[ctr2+2btrts+ats22(acb2)]/(acb2)1/2,
(116)
which is valid for a and c positive and acb2 > 0. Equations (115) and (116) comprise a specific example of the general transform relation
exp(XtMX/2)exp(KtM1K/2)/[det(M)]1/2,
(117)
M is a symmetric matrix, X and K are column vectors, and the superscript t denotes a transpose. Formula (117) is a standard result. It is proved in Appendix A of [58

58. C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012). [CrossRef]

].

Comparing with Eqs. (105)(107) the time-domain kernel has an analytic Schmidt decomposition as it is in the canonical form. It has the same Schmidt-coefficients but the characteristic time-scales are 1/τr and 1/τs respectively. The time-domain Schmidt-modes are indeed Hermite functions, as stated previously.

Acknowledgments

MR was supported by the National Science Foundation, EPDT.

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I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005). [CrossRef] [PubMed]

6.

A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004). [CrossRef]

7.

A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253–256 and 582 (2004).

8.

O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004). [CrossRef]

9.

R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (2005). [CrossRef] [PubMed]

10.

A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006). [CrossRef] [PubMed]

11.

K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32, 695–697 (2007). [CrossRef] [PubMed]

12.

W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A 73, 063816 (2006). [CrossRef]

13.

D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106, 130501 (2011). [CrossRef] [PubMed]

14.

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011). [CrossRef] [PubMed]

15.

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature (London) 299, 802–803 (1982). [CrossRef]

16.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1653 (1961). [CrossRef]

17.

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]

18.

J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68, 2153–2156 (1992). [CrossRef] [PubMed]

19.

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency up-conversion,” J. Mod. Opt. 51, 1433–1445 (2004).

20.

M. A. Albota and F. N. C. Wong, “Efficient single-photon counting at 1.55 μm by means of frequency upconversion,” Opt. Lett. 29, 1449–1451 (2004). [CrossRef] [PubMed]

21.

R. V. Roussev, C. Langrock, J. R. Kurz, and M. M. Fejer, “Periodically poled lithium niobate waveguide sum-frequency generator for efficient single-photon detection at communication wavelengths,” Opt. Lett. 29, 1518–1520 (2004). [CrossRef] [PubMed]

22.

Y. Ding and Z. Y. Ou, “Frequency downconversion for a quantum network,” Opt. Lett. 35, 2591–2593 (2010). [CrossRef] [PubMed]

23.

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011). [CrossRef]

24.

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London) 469, 508–511 (2011). [CrossRef]

25.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London) 469, 512–515 (2011). [CrossRef]

26.

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]

27.

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]

28.

M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, “Widely tunable spectrum translation and wavelength exchange by four-wave mixing in optical fibers,” Opt. Lett. 21, 1906–1908 (1996). [CrossRef] [PubMed]

29.

K. Uesaka, K. K. 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]

30.

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]

31.

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006). [CrossRef] [PubMed]

32.

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]

33.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011). [CrossRef]

34.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010). [PubMed]

35.

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004). [CrossRef] [PubMed]

36.

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]

37.

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]

38.

H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express 19, 17876–17907 (2011). [CrossRef] [PubMed]

39.

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

40.

M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003). [CrossRef]

41.

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

42.

X. Li, P. L. Voss, J. Chen, K. F. Lee, and P. Kumar, “Measurement of co- and cross-polarized Raman spectra in silica fiber for small detunings,” Opt. Express 13, 2236–2244 (2005). [CrossRef] [PubMed]

43.

R. Loudon, The Quantum Theory of Light, 3rd. ed. (Oxford University Press, 2000).

44.

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]

45.

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

46.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1990).

47.

M. P. des Chênes, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaire du second ordre, à coefficients constants,” Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assembleés. Sciences, mathématiques et physiques , 638–648 (1806). [PubMed]

48.

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]

49.

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

50.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974), Chap. 2.

51.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

52.

G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd. ed. (McGraw-Hill, 1991).

53.

C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012). [CrossRef]

54.

F. G. Mehler, “Über die Entwicklung einer Funktion von beliebig vielen Variablen nach Laplaceshen Functionen höherer Ordnung,” Journal für die reine und angewandte Mathematik , 161–176 (1866). [CrossRef] [PubMed]

55.

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 781 and 786.

56.

E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen,” Mathematische Annalen 63, 433–476 (1907). [CrossRef]

57.

B. L. Moiseiwitsch, Integral Equations (Dover, 2005).

58.

C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012). [CrossRef]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 13, 2012
Revised Manuscript: March 16, 2012
Manuscript Accepted: March 16, 2012
Published: March 26, 2012

Citation
L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, "Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime," Opt. Express 20, 8367-8396 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8367


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References

  1. S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London) 437, 116–120 (2005). [CrossRef]
  2. H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008). [CrossRef]
  3. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). [CrossRef] [PubMed]
  4. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001). [CrossRef]
  5. I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005). [CrossRef] [PubMed]
  6. A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004). [CrossRef]
  7. A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253–256 and 582 (2004).
  8. O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004). [CrossRef]
  9. R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (2005). [CrossRef] [PubMed]
  10. A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006). [CrossRef] [PubMed]
  11. K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32, 695–697 (2007). [CrossRef] [PubMed]
  12. W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A 73, 063816 (2006). [CrossRef]
  13. D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106, 130501 (2011). [CrossRef] [PubMed]
  14. A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011). [CrossRef] [PubMed]
  15. W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature (London) 299, 802–803 (1982). [CrossRef]
  16. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1653 (1961). [CrossRef]
  17. 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]
  18. J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68, 2153–2156 (1992). [CrossRef] [PubMed]
  19. A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency up-conversion,” J. Mod. Opt. 51, 1433–1445 (2004).
  20. M. A. Albota and F. N. C. Wong, “Efficient single-photon counting at 1.55 μm by means of frequency upconversion,” Opt. Lett. 29, 1449–1451 (2004). [CrossRef] [PubMed]
  21. R. V. Roussev, C. Langrock, J. R. Kurz, and M. M. Fejer, “Periodically poled lithium niobate waveguide sum-frequency generator for efficient single-photon detection at communication wavelengths,” Opt. Lett. 29, 1518–1520 (2004). [CrossRef] [PubMed]
  22. Y. Ding and Z. Y. Ou, “Frequency downconversion for a quantum network,” Opt. Lett. 35, 2591–2593 (2010). [CrossRef] [PubMed]
  23. B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011). [CrossRef]
  24. C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London) 469, 508–511 (2011). [CrossRef]
  25. E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London) 469, 512–515 (2011). [CrossRef]
  26. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]
  27. 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]
  28. M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, “Widely tunable spectrum translation and wavelength exchange by four-wave mixing in optical fibers,” Opt. Lett. 21, 1906–1908 (1996). [CrossRef] [PubMed]
  29. K. Uesaka, K. K. 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]
  30. 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]
  31. A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006). [CrossRef] [PubMed]
  32. 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]
  33. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011). [CrossRef]
  34. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010). [PubMed]
  35. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004). [CrossRef] [PubMed]
  36. 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]
  37. 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]
  38. H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express 19, 17876–17907 (2011). [CrossRef] [PubMed]
  39. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992). [CrossRef]
  40. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003). [CrossRef]
  41. C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (2006). [CrossRef] [PubMed]
  42. X. Li, P. L. Voss, J. Chen, K. F. Lee, and P. Kumar, “Measurement of co- and cross-polarized Raman spectra in silica fiber for small detunings,” Opt. Express 13, 2236–2244 (2005). [CrossRef] [PubMed]
  43. R. Loudon, The Quantum Theory of Light, 3rd. ed. (Oxford University Press, 2000).
  44. 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]
  45. C. J McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009). [CrossRef]
  46. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1990).
  47. M. P. des Chênes, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaire du second ordre, à coefficients constants,” Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assembleés. Sciences, mathématiques et physiques, 638–648 (1806). [PubMed]
  48. 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]
  49. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).
  50. G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974), Chap. 2.
  51. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  52. G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd. ed. (McGraw-Hill, 1991).
  53. C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012). [CrossRef]
  54. F. G. Mehler, “Über die Entwicklung einer Funktion von beliebig vielen Variablen nach Laplaceshen Functionen höherer Ordnung,” Journal für die reine und angewandte Mathematik, 161–176 (1866). [CrossRef] [PubMed]
  55. P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 781 and 786.
  56. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen,” Mathematische Annalen 63, 433–476 (1907). [CrossRef]
  57. B. L. Moiseiwitsch, Integral Equations (Dover, 2005).
  58. C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012). [CrossRef]

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