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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 17 — Aug. 26, 2013
  • pp: 19437–19466
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Quadrature and number fluctuations produced by parametric devices driven by pulsed pumps

C. J. McKinstrie and M. E. Marhic  »View Author Affiliations


Optics Express, Vol. 21, Issue 17, pp. 19437-19466 (2013)
http://dx.doi.org/10.1364/OE.21.019437


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Abstract

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

© 2013 Optical Society of America

1. Introduction

In four-wave mixing (FWM), which occurs in third-order nonlinear media, one or two strong pump waves couple the evolution of weak signal and idler waves (sidebands). Parametric devices based on FWM in fibers provide a variety of useful functions for conventional communication systems. Examples include parametric amplification (PA), frequency conversion (FC), phase conjugation, buffering (delaying) and sampling [1

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

6

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

]. They can also generate photons for, and frequency convert photons in, quantum information experiments [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

12

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

]. In all of these applications, the noise properties of the underlying parametric processes are important. For example, in conventional systems noise degrades the information conveyed by the signals, whereas in quantum experiments noise can degrade the squeezing of particular quadratures or the purity of photon pair states. In early experiments relevant to conventional systems, the pumps were continuous waves (CWs), whereas the signals and idlers (sidebands) were CWs or pulses. The noise properties of such processes are known [13

13. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).

, 14

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

, 16

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

, 17

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

, 19

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

22

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

]. However, some recent conventional experiments and most quantum experiments involve pulsed pumps and sidebands. Such experiments are difficult to model and design.

In the undepleted-pump approximation, the coupled-mode equations (CMEs) for the sidebands (modes) are linear in the mode amplitudes (operators). Hence, the output operators depend linearly on the input operators (and possibly their hermitian conjugates). These dependencies are characterized by transfer (Green) functions. In some cases, one can determine the Green functions exactly, by solving the CMEs analytically, whereas in other cases one can only determine them approximately, by solving the CMEs numerically for a representative set of initial conditions.

In direct detection, the signal-to-noise ratio (SNR) is the square of the photon-number mean divided by the number variance, whereas in homodyne detection, the SNR is the square of the amplitude-quadrature mean divided by the quadrature variance. For each method of detection, the noise figure of a parametric process is the input SNR divided by the output SNR. This metric (which is a figure of demerit) is most useful for applications with many-photon signals.

Once the Green functions associated with a parametric process are known (exactly or approximately), the noise figures of the process can be determined (with no further approximations). The calculations are straightforward, but lengthy, and the final formulas involve products and time-integrals of the Green functions. Hence, it is useful to derive a complete set of formulas, for subsequent use in a wide variety of applications.

The aforementioned formulas allow one to calculate the noise figures of a particular process with specific inputs. However, they do not provide a complete physical understanding of the process or an efficient way to optimize it. Each of the aforementioned Green functions can be rewritten (decomposed) in terms of Schmidt coefficients and Schmidt modes (temporal eigenfunctions), which are physically significant [23

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

26

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

]. For example, in photon pair generation (PG) by pulsed pumps, the Schmidt modes are the temporal wave-packets of the output signal and idler photons, and the squares of the Schmidt coefficients are the probabilities with which photon pairs are produced [27

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

30

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

]. In FC by pulsed pumps, the Schmidt modes are the natural input and output shapes (wave-packets) of the signal and idler pulses (photons), and the squares of the Schmidt coefficients are the transmission and conversion efficiencies (probabilities) [31

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

34

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

]. In these examples, the optimization conditions can be stated concisely in terms of the Schmidt coefficients and modes, which reduces the optimization problem to the determination of how these quantities depend on the system parameters.

This report is organized as follows: In Sec. 2, the properties of the continuous-mode amplitude operators are stated, and used to review the direct and homodyne detection of signal pulses. In Sec. 3, the input–output (IO) equations for attenuation (modeled as two-continuous-mode beam splitting) and two-continuous-mode FC are stated, and used to determine the noise properties of these processes. Formulas are derived for the output quadrature and number probabilities (means, variances and correlations), which are valid for pumps and signals with arbitrary shapes. In Sec. 4, the IO equations for two-continuous-mode PA are stated and used to derive formulas for the aforementioned output quantities. The calculations for PA are more complicated than those for FC, so a procedure for minimizing the effort required to complete them is described. In Sec. 5, the Schmidt decomposition theorems for FC and PA are stated, and the formulas for the quadrature and number probabilities are rewritten in terms of the Schmidt coefficients and modes. The time-dependent formulas for the probability fluxes are complicated. However, the time-integrated formulas for the probabilities are simple and consistent with the reductions required by the decomposition theorems. Finally, in Sec. 6 the main results of this report are summarized. For completeness, the mathematical properties of the Green functions for FC and PA are reviewed in App. A, the reason why there is a simple relation between the quadrature-probability formulas and the signal–noise contributions to the number-probability formulas is explained in App. B, and formulas for the fluctuations produced by multiple-continuous-mode processes are derived in App. C.

2. Direct and homodyne detection

In quantum mechanics, light waves (modes) are described by the creation and destruction operators aj and aj, respectively, where † denotes a hermitian conjugate. These continuous-mode amplitude operators satisfy the boson commutation relations
[aj(t1),ak(t2)]=0,[aj(t1),ak(t2)]=δjkδ(t1t2),
(1)
where [,] denotes a commutator, and δjk and δ(t1t2) are the Kronecker and Dirac delta functions, respectively [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

].

In direct detection, photodiodes are used to measure the photon-number means 〈Nj〉, where 〈〉 denotes an expectation value. The uncertainties in such measurements are the number variances
δNj2=(NjNj)2=Nj2Nj2.
(2)
Also of interest are the number correlations
δNjδNk=(NjNj)(NkNk)=NjNkNjNk.
(3)
The number-flux operator
nj(t)=aj(t)aj(t)
(4)
and the associated number operator Nj = ∫nj(t)dt, where the limits of integration are −T/2 and T/2, and T is the measurement time. It follows from these definitions that
Nj=nj(t)dt,
(5)
NjNk=nj(t1)nk(t2)dt1dt2.
(6)
Thus, to model direct detection, one must calculate the second- and fourth-order amplitude moments 〈nj(t)〉 and 〈nj(t1)nk(t2)〉, respectively.

In many applications it is reasonable to approximate the input signal by a coherent state (CS), which is defined by the equations
aj(t)|αj=αj(t)|αj,αj|aj(t)=αj|αj*(t),
(7)
where |αj〉 is the state vector [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

]. For such a state, the mean (coherent) amplitude 〈aj(t)〉 = αj(t). By combining Eqs. (4) and (7), one finds that the second-order moment
nj(t)=aj(t)aj(t)=|αj(t)2.
(8)
The number flux is the squared modulus of the coherent amplitude. One also finds that the fourth-order moment
nj(t1)nj(t2)=aj(t1)aj(t1)aj(t2)aj(t2)=aj(t1)[aj(t2)aj(t1)+δ(t1t2)]aj(t2)=|αj(t1)|2|αj(t2)|2+αj*(t1)αj(t2)δ(t1t2).
(9)
The penultimate term in Eq. (9) is 〈nj(t1)〉〈n1(t2)〉. Hence, if one were to detect a CS (using a measurement time that is longer than the duration of the signal pulse), one would find that
Nj=|αj(t)|2dt,
(10)
δNj2=Nj.
(11)
Property (11) is characteristic of a CS. The vacuum state (VS) |0〉 is a special CS, for which αj(t) = 0 and δNj2=Nj=0. As stated earlier, the SNR is defined to be the square of the number mean divided by the number variance. Hence, for a CS the SNR equals 〈Nj〉, which is the standard result for a single discrete mode (or CW) [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

].

If one were to detect two CS (j and k), Eqs. (10) and (11) would apply to each measurement separately. In addition, the mixed fourth-order moment
nj(t1)nk(t2)=aj(t1)aj(t1)ak(t2)ak(t2)=|αj(t1)|2|αk(t2)|2,
(12)
because aj(t1) and ak(t2) commute. Hence,
δNjδNk=0.
(13)
The number fluctuations of two CS are uncorrelated.

In homodyne detection, which is illustrated in Fig. 1, a weak signal pulse is combined with a strong local oscillator (LO) pulse at a beam splitter, and photodiodes are used to measure the output number means and variances. The preceding analysis shows that homodyne detection allows one to measure the mean amplitude of the signal and the amplitude fluctuations. It is appropriate to define the quadrature-flux operator
qj(t)=[αp*(t)aj(t)+αp(t)aj(t)]/21/2,
(22)
where the LO amplitude αp(t) includes the phase factor ep implicitly and satisfies the normalization condition ∫ |αp(t)|2dt = 1. (In this report the subscripts p and q are used to denote LOs, because the subscripts i, j, k and l are used to denote weak signals participating in parametric processes.) By varying the LO phase, one can measure different quadratures of the signal. The associated quadrature operator Qj = ∫qj(t)dt. It follows from definition (22) that the quadrature-flux moments
qj(t)=[αp*(t)αj(t)+αp(t)αj*(t)]/21/2,
(23)
qj(t1)qj(t2)=[αp*(t1)aj(t1)+αp(t1)aj(t1)][αp*(t2)aj(t2)+αp(t2)aj(t2)]/2=[αp*(t1)αj(t1)+αp(t1)αj*(t1)][αp*(t2)αj(t2)+αp(t2)αj*(t2)]/2+αp*(t1)αp(t2)δ(t1t2)/2.
(24)
The penultimate term in Eq. (24) is 〈qj(t1)〉〈qj(t2)〉. Hence, if one were to detect a CS (using a measurement time that is longer than the signal and LO pulses), one would find that
Qj=[αp*(t)αj(t)+αp(t)αj*(t)]dt/21/2,
(25)
δQj2=1/2.
(26)
The quadrature mean (25) depends on the shapes and overlap of the signal and LO pulses, whereas the variance (26) does not depend on either pulse shape. The variance (26) has the minimal value allowed by the Heisenberg uncertainty principle. This result is characteristic of a CS [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

]. As stated earlier, the SNR is defined to be the square of the mean quadrature divided by the quadrature variance. If one uses the optimal LO [αp(t) ∝ αj(t), where the proportionality coefficient is real and positive], then 〈Qj〉 = 21/2Nj1/2 and the SNR equals 4〈Nj〉, which is the standard result for a single discrete mode [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

]. Notice that the optimal-LO condition is analogous to the matched-filter condition in detection theory [35

35. L. W. Couch, Digital and Analog Communication Systems, 6th ed. (Prentice Hall, 2000).

, 36

36. J. G. Proakis, Digital Communications, 4th Ed. (McGraw-Hill, 2001).

]

Fig. 1 In homodyne detection, a beam splitter (partially-reflecting mirror) is used to combine a signal mode (s) with a local-oscillator mode (l). Each output mode is a superposition of both input modes. In attenuation (modeled as beam splitting), a signal mode interacts with a loss-mode (l) at a virtual mirror. The input and output loss modes are inaccessible.

If one were to use two LOs (p and q) to detect two CS (j and k), Eqs. (25) and (26) would apply to each measurement separately. In addition, the mixed second-order moment
qj(t1)qk(t2)=[αp*(t1)aj(t1)+αp(t1)aj(t1)][αq*(t2)ak(t2)+αq(t2)ak(t2)]/2=[αp*(t1)αj(t1)+αp(t1)αj*(t1)][αq*(t2)αk(t2)+αq(t2)αk*(t2)]/2,
(27)
because aj(t1) and ak(t2) commute. Hence,
δQjδQk=0.
(28)
The quadrature fluctuations of two CS are uncorrelated.

3. Attenuation and frequency conversion

It is customary to model attenuation as two-mode beam splitting [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

]. In this process, which is illustrated in Fig. 1, the signal mode interacts with a scattered (loss) mode, which is inaccessible. The input loss mode is a VS, which contributes only fluctuations to the output signal. FC in a fiber [37

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

42

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

] is enabled by the nondegenerate FWM process called Bragg scattering (BS), in which a pump photon and a sideband (signal) photon are destroyed, and different pump and sideband (idler) photons are created (πp2 + πs1πp1 + πs2, where πj represents a photon with carrier frequency ωj). This process is illustrated in Fig. 2. The photon reaction used to describe BS is called the Manley–Rowe–Weiss (MRW) relation [43

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

, 44

44. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

]. Attenuation is a phase-insensitive process, which means that the output signal power does not depend of the input signal phase. In its standard configuration (nonzero input signal and zero input idler), BS is also phase-insensitive [37

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

, 39

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

, 40

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

]. However, in its alternative configuration (nonzero input signal and idler), BS is phase-sensitive [38

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

], which means that the output sideband powers depend on the input sideband phases.

Fig. 2 Frequency diagrams for (a) distant and (b) nearby Bragg scattering. Long arrows denote pumps (p1 and p2), whereas short arrows denote signal and idler sidebands (s1 and s2). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons. The directions of the arrows are reversible.

The IO equations for attenuation and FC can be written in the form
b1(t)=[g11(t,t)a1(t)+g12(t,t)a2(t)]dt,
(29)
b2(t)=[g21(t,t)a1(t)+g22(t,t)a2(t)]dt,
(30)
where aj are input operators, bj are output operators and gjk are Green functions. For attenuation, mode 1 is the signal and mode 2 is the loss mode [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

], whereas for FC, mode 1 is the signal and mode 2 is the idler [19

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

]. Notice that each output operator depends on both input operators, but not their conjugates. Because the IO transformation is unitary, the output operators satisfy the same commutation relations as the input operators [Eqs. (1)]. This fact imposes constraints on the Green functions (App. A), which facilitate the analyses of quadrature and number fluctuations, and ensure that the MRW relation is satisfied. Because Eqs. (29) and (30) depend symmetrically on the labels 1 and 2, the results in this section will be stated explicitly only for the signal.

To calculate the higher-order moments, one proceeds in the manner described in Sec. 2: One groups the operator moments into normally-ordered terms ( bj before bj) and anti-normally-ordered terms, and uses the commutation relations to rewrite the latter terms as normally-ordered terms plus extra terms, from which the fluctuations originate. The second-order moment
q1(t1)q1(t2)=[βp*(t1)b1(t1)+βp(t1)b1(t1)][βp*(t2)b1(t2)+βp(t2)b1(t2)]/2=[βp*(t1)β1(t1)+βp(t1)β1*(t1)][βp*(t2)β1(t2)+βp(t2)β1*(t2)]/2+βp*(t1)βp(t2)δ(t1t2)/2,
(33)
because b1(t1) and b1(t2) do not commute. In contrast, the mixed second-order moment
q1(t1)q2(t2)=[βp*(t1)b1(t1)+βp(t1)b1(t1)][βq*(t2)b2(t2)+βq(t2)b2(t2)]/2=[βp*(t1)β1(t1)+βp(t1)β1*(t1)][βq*(t2)β2(t2)+βq(t2)β2*(t2)]/2,
(34)
because b1(t1) and b2(t2) do commute. Hence,
δQ12=1/2,
(35)
δQ1δQ2=0.
(36)
The output quadrature variance is the same as the input variance [Eq. (26)] and the output quadrature fluctuations are uncorrelated.

We did the preceding calculations using only the properties of the output operators. If we had done them by using Eqs. (29) and (30) to rewrite the output operators in terms of the input operators, we would have encountered the terms
βp*(t1)βp(t2)[g11(t1,t)g11*(t2,t)+g12(t1,t)g12*(t2,t)]dt/2,
(37)
βp*(t1)βq(t2)[g11(t1,t)g21*(t2,t)+g12(t1,t)g22*(t2,t)]dt/2
(38)
in Eqs. (33) and (34), respectively. However, by using Eqs. (123) and (124), we would have obtained the stated results, βp*(t1)βp(t2)δ(t1t2)/2 and 0, respectively.

We did the preceding calculations using only the properties of the output operators. If we had done them by using Eqs. (29) and (30) to rewrite the output operators in terms of the input operators, we would have encountered the terms
β1*(t1)β1(t2)[g11(t1,t)g11*(t2,t)+g12(t1,t)g12*(t2,t)]dt,
(45)
β1*(t1)β2(t2)[g11(t1,t)g21*(t2,t)+g12(t1,t)g22*(t2,t)]dt
(46)
in Eqs. (41) and (42), respectively. However, by using Eqs. (123) and (124), we would have obtained the stated results, β1*(t1)β1(t2)δ(t1t2)/2 and 0, respectively. Notice that Eqs. (45) and (46) are just Eqs. (37) and (38), with the LO amplitudes βp(t) and βq(t) replaced by the sideband amplitudes β1(t) and β2(t), respectively, and the factors of 1/2 omitted.

4. Parametric amplification

PA [45

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

, 46

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

] and PG [47

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

52

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

] are enabled by a variety of FWM processes in fibers. Modulation interaction (MI) is the degenerate process in which two identical pump photons are destroyed and two different sideband (signal and idler) photons are created (2πp1πs1 +πs2), whereas inverse MI is the degenerate process in which two different pump photons are destroyed and two identical sideband (signal) photons are created (πp1 + πp2 → 2πs1). Phase conjugation (PC) is the nondegenerate process in which two different pump photons are destroyed and two different sideband photons are created (πp1 + πp2πs1 + πs2). These processes are illustrated in Fig. 3. The photon reactions used to describe them are called the MRW relations [43

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

, 44

44. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

]. Inverse MI is always phase-sensitive [53

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

55

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

]. In their standard configurations (nonzero input signal and zero input idler), MI and PC are phase-insensitive [1

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

,2

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]

], whereas in their alternative configurations (nonzero input signal and idler), MI and PC are phase-sensitive [56

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

58

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

]. The noise properties of PA depend on whether it is phase-sensitive or -insensitive [16

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

, 17

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

, 19

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

22

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

].

Fig. 3 Frequency diagrams for (a) modulation interaction, (b) inverse modulation interaction, and (c) outer-band and (d) inner-band phase conjugation. Long arrows denote pumps (p1 and p2), whereas short arrows denote sidebands (s1 and s2). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons.

The IO equations for two-mode PA can be written in the form
b1(t)=[g11(t,t)a1(t)+g12(t,t)a2(t)]dt,
(47)
b2(t)=[g22(t,t)a2(t)+g21(t,t)a1(t)]dt,
(48)
where gjj is a Green function that couples bj to aj (like operators), and gjk is a Green function that couples bj to ak (unlike operators) [19

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

]. Once again, the output operators satisfy the same commutation relations as the input operators. This fact imposes constraints on the Green functions (App. A), which facilitate the analyses of the quadrature and number fluctuations, and ensure that the MRW relations are satisfied.

The calculations of the higher-order moments produced by PA are more complicated than those associated with FC, because normal ordering with respect to the output operators is not normal ordering with respect to the input operators [Eqs. (47) and (48)]. The second-order moment can be written in the abbreviated form
q1(t1)q1(t2)=[βp*(g11a1+g12a2)dt+βp(g11*a1+g12*a2)dt]t1×[βp*(g11a1+g12a2)dt+βp(g11*a1+g12*a2)dt]t2/2,
(51)
where the subscripts t1 and t2 denote the times at which the terms in brackets are evaluated. The anti-normally ordered terms in Eq. (51) are proportional to a1a1 and a2a2. By retaining only these terms, one finds that
δq1(t1)δq1(t2)=βp*(t1)βp(t2)g11(t1,t)g11*(t2,t)dt/2+βp(t1)βp*(t2)g12*(t1,t)g12(t2,t)dt/2.
(52)
By using Eq. (141), one can rewrite Eq. (52) in the alternative form
δq1(t1)δq1(t2)=βp*(t1)βq(t2)δ(t1t2)/2+βp*(t1)βp(t2)g12(t1,t)g12*(t2,t)dt/2+βp(t1)βp*(t2)g12*(t1,t)g12(t2,t)dt/2,
(53)
which is manifestly real.

The mixed second-order moment can be written in the abbreviated form
q1(t1)q2(t2)=[βp*(g11a1+g12a2)dt+βp(g11*a1+g12*a2)dt]t1×[βq*(g22a2+g21a1)dt+βq(g22*a2+g21*a1)dt]t2/2.
(54)
The anti-normally ordered terms in Eq. (54) are also proportional to a1a1 and a2a2. By retaining only these terms, one finds that
δq1(t1)δq2(t2)=βp*(t1)βq*(t2)g11(t1,t)g21(t2,t)dt/2+βp(t1)βq(t2)g12*(t1,t)g22*(t2,t)dt/2.
(55)
By using Eq. (142), one can rewrite Eq. (55) in the alternative form
δq1(t1)δq2(t2)=βp*(t1)βq*(t2)g11(t1,t)g21(t2,t)dt/2+βp(t1)βq(t2)g11*(t1,t)g21*(t2,t)dt/2,
(56)
which is also manifestly real. Because the output operators commute, 〈δq1(t1)δq2(t2)〉 = 〈δq2(t2)δq1(t1)〉. One can also use Eq. (142) to establish this symmetry directly.

One obtains formulas for the quadrature variance and correlation by integrating Eqs. (53) and (56), respectively, with respect to t1 and t2. These formulas are so similar to Eqs. (53) and (56) that there is no point in stating them explicitly. However, by changing the order of integration (doing the t′ integral last), one obtains the alternative variance and correlation formulas
δQ12=[|βp*(t)g11(t,t)dt|2+|βp*(t)g12(t,t)dt|2]dt/2=1/2+|βp*(t)g12(t,t)dt|2dt,
(57)
δQ1δQ2=Re[βp*(t1)g11(t1,t)dt1βq*(t2)g21(t2,t)dt2]dt,
(58)
respectively. [The first and second forms of Eq. (57) come from Eqs. (52) and (53), respectively, whereas Eq. (58) comes from Eq. (56).] By comparing Eqs. (57) and (58) to Eqs. (26) and (28), respectively, one finds that PA increases the variances of the sideband quadrature fluctuations ( δQj2>1/2), and produces a correlation between them. These results reflect the fact that PA of two input CS produces a two-mode squeezed (stretched) CS [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

]: Neither output pulse is squeezed or stretched by itself. Rather, certain quadratures of the sum and difference pulses are squeezed and stretched.

The second-order signal moment
n1(t)=[g11*(t,t)a1(t)+g12*(t,t)a2(t)]dt×[g11(t,t)a1(t)+g12(t,t)a2(t)]dt=|β1(t)|2+|g12(t,t)|2dt.
(59)
The number flux is the squared modulus of the mean output amplitude (which depends implicitly on the mean input amplitudes), supplemented by a term that represents the spontaneous amplification of input idler fluctuations. Hence, if one were to detect the output signal, one would find that
N1=|β1(t)|2dt+|g12(t,t)|2dtdt.
(60)
Notice that if the LO pulse has the same shape as the output signal pulse, the mean quadrature [Eq. (50)] is proportional to the square root of the coherent contribution to the mean number.

The fourth-order moment can be written in the abbreviated form
n1l(t1)n1r(t2)=[|β1|2+β1*g11v1+β1g12*v2+g12*v2(g11v1+g12v2)]t1×[|β1|2+β1*g12v2+β1g11*v1+(g11*v1+g12*v2)g12v2]t2,
(64)
in which all four time integrals are implicit. By using Eqs. (198) and (199), noting that the first-and third-order moments of vj are all zero, and restoring the time integrals, one finds that
n1(t1)n1(t2)=|β1(t1)β1(t2)|2+|β1(t1)|2|g12(t2,t)|2dt+|β1(t2)|2|g12(t1,t)|2dt+β1*(t1)β1(t2)g11(t1,t)g11*(t2,t)dt+β1(t1)β1*(t2)g12*(t1,t)g12(t2,t)dt+g11(t1,t)g11*(t2,t)dtg12*(t1,t)g12(t2,t)dt+|g12(t1,t)|2dt|g12(t2,t)|2dt.
(65)
It follows from Eqs. (59) and (65) that
δn1(t1)δn1(t2)=β1*(t1)β1(t2)g11(t1,t)g11*(t2,t)dt+β1(t1)β1*(t2)g12*(t1,t)g12(t2,t)dt+g11(t1,t)g11*(t2,t)dtg12*(t1,t)g12(t2,t)dt.
(66)
Just as PA modifies the quadrature variances [Eq. (52)], so also does it modify the number variances. The first two terms on the right side of Eq. (66) are called the signal–noise terms, whereas the third term is called the noise–noise term. By using Eq. (141), one can rewrite Eq. (66) in the alternative form
δn1(t1)δn1(t2)=2Re[β1*(t1)β1(t2)g12(t1,t)g12*(t2,t)dt]+β1*(t1)β1(t2)δ(t1t2)+|g12(t1,t)g12*(t2,t)dt|2+g12*(t1,t)g12(t2,t)dtδ(t1t2),
(67)
which is manifestly real.

The mixed fourth-order moment can be written in the abbreviated form
n1l(t1)n2r(t2)=[|β1|2+β1*g11v1+β1g12*v2+g12*v2(g11v1+g12v2)]t1×[|β2|2+β2*g21v1+β2g22*v2+(g22*v2+g21*v1)g21v1]t2.
(68)
Once again, the third-order moments of vj are all zero, so the moment
n1(t1)n2(t2)=|β1(t1)β2(t2)|2+|β1(t1)|2|g21(t2,t)|2dt+|β2(t2)|2|g12(t1,t)|2dt+β1*(t1)β2*(t2)g11(t1,t)g21(t2,t)dt+β1(t1)β2(t2)g12*(t1,t)g22*(t2,t)dt+g11(t1,t)g21(t2,t)dtg12*(t1,t)g22*(t2,t)dt+|g12(t1,t)|2dt|g21(t2,t)|2dt.
(69)
It follows from Eqs. (59) and (69) that
δn1(t1)δn2(t2)=β1*(t2)β2*(t2)g11(t1,t)g21(t2,t)dt+β1(t1)β2(t2)g12*(t1,t)g22*(t2,t)dt+g11(t1,t)g21(t2,t)dtg12*(t1,t)g22*(t2,t)dt.
(70)
Just as PA produces quadrature correlations [Eq. (55)], so also does it produce number correlations. By using Eq. (142), one can rewrite Eq. (70) in the alternative form
δn1(t1)δn2(t2)=2Re[β1*(t1)β2*(t2)g11(t1,t)g21(t2,t)dt]+|g11(t1,t)g21(t2,t)dt|2,
(71)
which is also manifestly real, and verify that 〈δn1(t1)δn2(t2)〉 = 〈δn2(t2)δn1(t1)〉.

One obtains formulas for the number variance and correlation by integrating Eqs. (67) and (71), respectively, with respect to t1 and t2. These formulas are so similar to Eqs. (67) and (71) that there is no need to state them explicitly. However, by changing the order of integration, one obtains the alternative variance and correlation formulas
δN12=[|β1*(t)g11(t,t)dt|2+|β1*(t)g12(t,t)dt|2]dt+|g11(t,t)g12*(t,t)dt|2dtdt=|β1(t)|2dt+2|β1*(t)g12(t,t)dt|2dt+|g12(t,t)g12*(t,t)dt|2dtdt+|g12(t,t)|2dtdt,
(72)
δN1δN2=2Re[β1*(t1)g11(t1,t)dt1β2*(t2)g21(t2,t)dt2]dt+|g11(t,t)g12*(t,t)dt|2dtdt,
(73)
respectively. [The first and second forms of Eq. (72) come from Eqs. (66) and (67) respectively, whereas the first term on the right side of Eq. (73) comes from Eq. (71), and the second term comes from Eqs. (70) and (152).] By comparing Eqs. (72) and (73) to Eqs. (11) and (13), respectively, one finds that PA increases the variances of the sideband number fluctuations ( δNj2> 〈Nj〉) and produces a correlation between them. It is shown in App. A that the presence of strong correlations is a consequence of the MRW relations. Notice that if the LO pulses have the same shapes as the output sideband pulses, the quadrature variance and correlation [Eqs. (57) and (58)] are proportional to the signal–noise contributions to the number variance and correlation, respectively. Notice also that the noise–noise contributions to the number variance and correlation are equal.

As stated earlier, one-mode PA is enabled by the inverse MI [Fig. 3(b)]. The IO equation for this phase-sensitive process can be written in the form
b(t)=[gs(t,t)a(t)+gc(t,t)a(t)]dt,
(74)
where the subscripts s and c denote coupling to similar and conjugate operators, respectively. One can obtain Eq. (74) from Eq. (47) or (48) by making the substitutions aja, bjb, gjjgs and gjkgc. Because these IO equations are so similar, there is no need to describe the analysis of one-mode PA in detail. The quadrature flux and quadrature mean are specified by Eqs. (49) and (50), respectively, with the subscript 1 omitted. By following the procedure used to derive Eq. (52), one obtains the quadrature-flux moment
δq(t1)δq(t2)=[βp*(t1)gs(t1,t)+βp(t1)gc*(t1,t)]×[βp(t2)gs*(t2,t)+βp*(t2)gc(t2,t)]dt.
(75)
Equation (75) contains two more terms than Eq. (52), because a1 commutes with a2, but a does not commute with a. By using constraints analogous to Eqs. (141) and (142), one can show that the quadrature-flux moment is real. By integrating Eq. (75) with respect to t1 and t2, one finds that the quadrature variance
δQ2=|[βp*(t)gs(t,t)+βp(t)gc*(t,t)]dt|2dt/2,
(76)
which is manifestly real and non-negative, but can be greater or less than 1/2 [15

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

17

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

].

The number flux and number mean are specified by Eqs. (59) and (60), respectively. Once again, the calculation of the number-flux moment is simplified by the use of left and right number-flux operators. The final result is
δn(t1)δn(t2)=[β*(t1)gs(t1,t)+β(t1)gc*(t1,t)]×[β(t2)gs(t2,t)+β*(t2)gc(t2,t)]dt+gs(t1,t)gs*(t2,t)dtgc*(t1,t)gc(t2,t)dt+gs(t1,t)gc(t2,t)dtgc*(t1,t)gs*(t2,t)dt.
(77)
Equation (77) contains more terms than Eq. (68), for the reason mentioned above. By using constraints analogous to Eqs. (141) and (142), one can show that the number-flux moment is real. By integrating Eq. (77) with respect to time, one finds that the number variance
δN2=|[β*(t)gs(t,t)+β(t)gc*(t,t)]dt|2dt+2|gs(t,t)gc*(t,t)dt|2dtdt,
(78)
which is also manifestly real and non-negative. Once again, if the LO pulse has the same shape as the output signal pulse, the quadrature variance (76) is proportional to the signal–noise contribution to the number variance. The reason for this (recurring) relation is explained in App. B.

5. Schmidt decompositions of parametric processes

The formulas derived in Sec. 3 and 4 allow one to determine the noise properties of attenuation, FC and PA for arbitrary input pulses. However, they do not provide a complete physical understanding of these processes, or a simple way to optimize them (especially if the Green functions were determined numerically). It is often useful to decompose parametric processes into their constituent sub-processes.

Each continuous input mode ai(t) has the decomposition
ai(t)=jaijuij(t),aij=uij*(t)ai(t)dt.
(84)
It follows from Eqs. (1) and (84) that the discrete input modes satisfy the commutation relations
[aik,ajl]=0,[aik,ajl]=δijδkl.
(85)
Each continuous output mode bi(t) has a similar decomposition. By substituting these decompositions into Eqs. (29) and (30), one obtains the discrete-mode IO equations
b1j=τja1j+ρja2j,
(86)
b2j=ρj*a1j+τj*a2j.
(87)
Equations (86) and (87) show that each discrete signal mode interacts with one discrete idler mode (and no other signal modes). They are similar in form to Eqs. (29) and (30), but simpler, because the operators no longer depend on time. They are identical to the IO equations for a two-discrete-mode beam splitter [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

]. By starting with the discrete-mode equations, one can show easily that
q1j=(β¯pj*β1j+β¯pjβ1j*)/21/2,
(88)
δq1j2=|β¯pj|2/2,
(89)
δq1jδq2j=0,
(90)
where β̄pj = ep is the LO phase factor. Because this factor has unit modulus, each discrete mode has a quadrature variance of 1/2. The associated signal and idler quadratures are uncor-related. It is also easy to show that
n1j=|β1j|2,
(91)
δn1j2=|β1j|2,
(92)
δn1jδn2j=0.
(93)
Each discrete mode has a number variance that equals the number mean, and the associated number quadratures are uncorrelated. Thus, the outputs are discrete-mode CS [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

]. Notice that one can deduce the number results from the associated quadrature results by replacing β̄pj with β1j and multiplying by powers of 21/2. In retrospect, the continuous-mode results, specifically Eqs. (32), (35) and (36), and Eqs. (40), (43) and (44), are simple generalizations of Eqs. (88)(93), respectively.

By decomposing the input and output modes in the manner of Eq. (84), one converts a two-continuous-mode process into an infinite family of two-discrete-mode processes. The advantages of doing so are that the discrete-mode processes are independent and their noise properties are known [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

, 19

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

, 21

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

]. The preceding analysis of FC was done in the Heisenberg picture, in which the state vector is constant and the mode operators evolve with distance, as do the classical mode amplitudes. Alternatively, one could analyze FC in the Schrödinger picture, in which the mode operators are constant and the state vector evolves. The effects of two-discrete-mode beam splitting (FC) on the state vector are also known [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

, 59

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

, 60

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

]. By using the decomposition described above, one can deduce the effects of two-continuous-mode FC on the state vector.

Instead of substituting decompositions (84) into the IO equations of Sec. 3, one could substitute them directly into the continuous-mode formulas. It follows from Eq. (31) that the quadrature flux
q1(t)=jk[βpj*v1j*(t)β1kv1k(t)+βpjv1j(t)β1k*v1k*(t)]/21/2.
(94)
There are two summations in Eq. (94) because the LO and signal pulses were both decomposed. By integrating Eq. (94) with respect to time, one finds that the mean quadrature
Q1=j(βpj*β1j+βpjβ1j*)/21/2,
(95)
which is consistent with Eq. (32). The total quadrature is a weighted sum of the discrete-mode quadratures and one can pick out (select) a particular mode by setting βp(t) ∝ vj(t), in which case |βpj| = 1 and βpk = 0 for kj. It follows from Eq. (33) and the second of Eqs. (82) that the quadrature-flux moment
δq1(t1)δq1(t2)=ijkβpl*v1i*(t1)v1j(t1)v1j*(t2)βpkv1k(t2)/2.
(96)
One could obtain the same result by using Eq. (29) and the first of Eqs. (84) to reorder the b1(t1)b1(t2) term in Eq. (33) directly. By integrating Eq. (96) with respect to t1 and t2, one finds that the quadrature variance
δQ12=j|βpj|2/2=1/2,
(97)
which is consistent with Eq. (35). The total quadrature variance is a weighted sum of discrete-mode contributions [Eq. (89)], which always equals 1/2, regardless of whether one selects a particular output mode or admits them all. Hence, one always maximizes the SNR by matching the LO pulse to the output signal pulse, regardless of whether the latter pulse is a single output mode or a superposition of such modes. Formulas for the number mean and variance will not be stated explicitly, because they can be deduced from the associated quadrature results, as noted above.

Instead of substituting decompositions (84) into the IO equations of Sec. 4, one could substitute them directly into the continuous-mode formulas. The equations for the quadrature flux and mean are identical to Eqs. (94) and (95), respectively. It follows from Eqs. (52) and (55) that the quadrature-flux moments
δq1(t1)δq1(t2)=ijk[βpi*v1i*(t1)v1j(t1)|μj|2v1j*(t2)βpkv1k(t2)+βpiv1i(t1)v1j*(t1)|νj|2v1j(t2)βpk*v1k*(t2)]/2,
(107)
δq1(t1)δq2(t2)=ijk[βpi*v1i*(t1)v1j(t1)μjνjv2j(t2)βqk*v2k*(t2)+βpiv1i(t1)v1j*(t1)μj*νj*v2j*(t2)βqkv2k(t2)]/2.
(108)
By integrating Eqs. (107) and (108) with respect to t1 and t2, one finds that
δQ12=j|βpj|2(|μj|2+|νj|2)/2,
(109)
δQ1δQ2=j(βpj*βqj*μjνj+βpjβqjμj*νj*)/2.
(110)
The quadrature variance and correlation are weighted sums of discrete-mode contributions [Eqs. (102) and (103)].

The signal number flux and mean are
n1(t)=jkβ1j*v1j*(t)β1kv1k(t)+j|νj|2|v1j(t)|2,
(111)
N1=j(|β1j|2+|νj|2),
(112)
respectively. As noted previously, there is no need to state formulas for the signal–noise contributions to the number variance and correlation. However, in PA there are noise–noise contributions to the variance and correlation that have no counterpart in FC, so formulas for them must be derived. It follows from Eqs. (66), (70) and (84) that the number-flux moments
δn1(t1)δn1(t2)nn=[jv1j(t1)|μj|2v1j*(t2)][kv1k*(t1)|νk|2v1k(t2)],
(113)
δn1(t1)δn2(t2)nn=[jv1j(t1)μjνjv2j(t2)][kv1k*(t1)μk*νk*v2k*(t2)].
(114)
By integrating Eqs. (113) and (114) with respect to time, one finds that
δN12nn=j|μjνj|2=δN1δN2nn.
(115)
The noise–noise contributions to the signal variance and signal–idler correlation are identical because sideband photons are generated in pairs. Despite the complexity of the formulas for the quadrature- and number-flux moments, the final formulas for the quadrature and number moments do simplify in the manner predicted by the decomposed IO equations [(99) and (100)]. Notice that when one uses direct detection, one cannot separate the signal and noise contributions associated with the Schmidt modes of interest from those associated with the other Schmidt modes.

Recently, there has been considerable interest in photon generation by spontaneous FWM [30

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

,47

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

52

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

]. In such processes, the input signal and idler are VS (so only the noise–noise contributions are nonzero). It follows from Eqs. (111) and (114) that the mixed number-flux moment
n1(t1)n2(t2)nn=[jμjνjv1j(t1)v2j(t2)][kμk*νk*v1k*(t1)v2k*(t2)]+[j|νj|2|v1j(t1)|2][k|νk|2|v2k(t2)|2].
(116)
This moment is the joint probability (density) of detecting a signal photon at time t1 and an idler photon at time t2. In general, it is a complicated function of t1 and t2, and the detection (emission) times are correlated. However, if the joint probability is separable (can be written as the product of a function of t1 and function of t2), the emission times will be uncorrelated [28

28. 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]

]. This condition is satisfied if νj = 0 for j > 1, in which case
n1(t1)n2(t2)nn=(|μ1|2+|ν1|2)|ν1|2|v11(t1)v21(t2)|2.
(117)
The right side of Eq. (116) is separable in this case because both terms involve the same functions of t1 and t2 (squares of Schmidt modes). The emission probability (|μ1|2 + |ν1|2) |ν1|2 and the Schmidt modes v11(t) and v21(t) all depend on the gain parameter |ν1|2, which in turn depends on the pump power(s), and the fiber dispersion and length. In the low-gain regime a pair of signal and idler photons is produced, whereas in the high-gain regime a superposition of multiple-photon states is produced [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

]. This result is a straightforward generalization of the Grice separability condition [28

28. 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]

], which was derived for pair generation. It is also an example of an important physical condition that can be stated concisely in terms of Schmidt coefficients.

One-mode PA is enabled by the inverse MI [Fig. 3(b)]. The Green functions in Eq. (74) have the Schmidt decompositions [63

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

, 64

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

].
gs(t,t)=jvj(t)μjuj*(t),gc(t,t)=jvj(t)νjuj(t),
(118)
which are equivalent to the decompositions in the first row of Eq. (98). Because of the similarities between one- and two-mode PA, which were noted in Sec. 4, there is no need to describe the decomposition of the former process in detail. The quadrature-flux moment and quadrature variance are
δq(t1)δq(t2)=ijk[βpi*vi*(t1)μjvj(t1)+βpivi(t1)νj*vj*(t1)]×[μj*vj*(t2)βpkvk(t2)+νjvj(t2)βpk*vk*(t2)]/2,
(119)
δQ2=j|βpj*μj+βpjνj*|2/2,
(120)
respectively, and the number-flux moment and number variance are
δn(t1)δn(t2)=ijk[βi*vi*(t1)μjvj(t1)+βivi(t1)νj*vj*(t1)]×[μj*vj*(t2)βkvk(t2)+νjvj(t2)βk*vk*(t2)]+jk|μj|2vj(t1)vj*(t2)|νk|2vk*(t1)vk(t2)+jkμjνjvj(t1)vj(t2)μk*νk*vk*(t1)vk*(t2),
(121)
δN2=j|βj*μj+βjνj*|2+2j|μjνj|2,
(122)
respectively. The variances (120) and (122) are summations of contributions from the individual Schmidt modes, each of which is a standard discrete-mode contribution [7

7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).

, 19

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

]. In retrospect, the continuous-mode results, specifically Eqs. (76) and (78), are straightforward generalizations of these discrete-mode results.

Two further comments are in order. First, the results of Sec. 3, 4 and 5, for one- and two-mode parametric processes, can be generalized to multiple-mode processes (App. C). Results for discrete-mode processes are already known [19

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

, 21

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

, 22

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

] and, in view of the relations described in this section, it is a straightforward matter to generalize these results to continuous-mode processes. Second, because every multiple-continuous-mode process can be decomposed into a family of discrete-mode processes, and the quadrature fluctuations associated with discrete-mode processes are known to have Gaussian statistics [14

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

], the quadrature fluctuations associated with continuous-mode processes must also have Gaussian statistics [Eq. (84)].

6. Summary

In parametric devices based on four-wave mixing in fibers, one or two strong pump waves couple the evolution of weak signal and idler waves (sidebands). In the undepleted-pump approximation, the coupled-mode equations (CMEs) for the sidebands (modes) are linear in the mode operators (and possibly their hermitian conjugates). Hence, the output operators depend linearly on the input operators. These dependencies are characterized by transfer (Green) functions. In some cases, one can determine the Green functions exactly, by solving the CMEs analytically, whereas in other cases one can only determine them approximately, by solving the CMEs numerically for a representative set of initial conditions. Once the Green functions associated with a parametric process are known (exactly or approximately), the noise properties of the process can be determined. The calculations are straightforward, but lengthy, and the final formulas are complicated.

In this report, formulas were derived for the field-quadrature and photon-number means, variances and correlations, produced by two-mode frequency conversion (Sec. 3), and one- and two-mode parametric amplification (Sec. 4). These formulas are valid for pumps and sidebands with arbitrary input pulse shapes, and allow one to calculate the noise figures of the aforementioned processes, for direct and homodyne detection. They are nontrivial generalizations of formulas derived previously for continuous waves.

Although the aforementioned formulas determine the noise figures of a parametric process with specific inputs, they do not provide a complete physical understanding of the process or an efficient way to optimize it. Fortunately, each of the aforementioned Green functions can be rewritten (decomposed) in terms of Schmidt coefficients and Schmidt modes (temporal eigenfunctions), which are physically significant: The Schmidt modes are the natural input and output modes of a parametric process, and the Schmidt coefficients quantify the effects of the process on the input modes. For conversion, these coefficients are the transmission and conversion efficiencies, whereas for amplification, they describe the amounts by which the in-phase (out-of-phase) quadratures are stretched (squeezed).

By decomposing the input and output pulses in terms of Schmidt modes (Sec. 5), one converts a (one-) two-continuous-mode process into an infinite family of (one-) two-discrete-mode processes. The advantages of doing so are that the discrete-mode processes are independent and their properties are known, in both the Heisenberg and Schrödinger pictures. Consequently, one can deduce the properties of complicated continuous-mode processes from the known properties of simple discrete-mode processes. One can also substitute the aforementioned output decompositions into the quadrature and number formulas of Secs. 3 and 4. By doing so, one obtains alternative formulas for the quadrature and number probabilities, to which each pair of Schmidt modes contributes independently (as predicted by the decomposition theorems). These formulas provide a natural bridge between the discrete- and continuous-mode models of parametric processes (and demonstrate the consistency of the direct and decomposition methods).

Appendix A: Commutation relations and conservation laws

Two-mode attenuation and FC are governed by the IO equations (29) and (30). Because the mode evolution is unitary, the output operators satisfy the same commutation relations as the input operators [Eqs. (1)]. Hence,
[b1(t1),b1(t2)]=[g11(t1,t)g11*(t2,t)+g12(t1,t)g12*(t2,t)]dt=δ(t1t2),
(123)
[b1(t1),b2(t2)]=[g11(t1,t)g21*(t2,t)+g12(t1,t)g22*(t2,t)]dt=0.
(124)
Equations (123) and (124) impose important constraints on the forward Green functions. Because Eqs. (29) and (30) depend symmetrically on the labels 1 and 2, similar constraints exist in which 1 ↔ 2. Most of the results of this appendix will be stated explicitly only for the signal. The corresponding results for the loss-mode or idler will be implied.

There also exist output–input (OI) equations, which can be written in the form
a1(t)=[h11(t,t)b1(t)+h12(t,t)b2(t)]dt,
(125)
a2(t)=[h21(t,t)b1(t)+h22(t,t)b2(t)]dt,
(126)
where hjk are backward (adjoint) Green functions. It follows from Eqs. (125) and (126), and the commutation relations, that
[a1(t1),a1(t2)]=[h11(t11,t)h11*(t2,t)+h12(t1,t)h12*(t2,t)]dt=δ(t1t2),
(127)
[a1(t1),a2(t2)]=[h11(t1,t)h21*(t2,t)+h12(t1,t)h22*(t2,t)]dt=0.
(128)
By combining Eqs. (29), (125) and (126), one obtains the identity
b1(t)=[g11(t,t)h11(t,t)+g12(t,t)h21(t,t)]b1(t)dtdt+[g11(t,t)h12(t,t)+g12(t,t)h22(t,t)]b2(t)dtdt,
(129)
from which follow the consistency conditions
[g11(t,t)h11(t,t)+g12(t,t)h21(t,t)]dt=δ(tt),
(130)
[g11(t,t)h12(t,t)+g12(t,t)h22(t,t)]dt=0.
(131)
By comparing Eqs. (123) and (124) with Eqs. (130) and (131), one obtains the reciprocity relations [31

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

]
h11(t1,t2)=g11*(t2,t1),h21(t1,t2)=g12*(t2,t1).
(132)
The identity for a1(t) produces the same conditions. By combining Eqs. (127), (128) and (132), one finds that
[g11*(t,t1)g11(t,t2)+g21*(t,t1)g21(t,t2)]dt=δ(t1t2),
(133)
[g11*(t,t1)g12(t,t2)+g21*(t,t1)g22(t,t2)]dt=0.
(134)
Thus, the Green functions for attenuation and FC satisfy constraints when integrated with respect to either (first or second) variable.

The constraint (auxiliary) equations have important physical consequences. By combining Eq. (29) with its conjugate, one finds that the signal number-flux density
n1(t)=[g11*(t,t)a1(t)g11(t,t)a1(t)+g11*(t,t)a1(t)g12(t,t)a2(t)+g12*(t,t)a2(t)g11(t,t)a1(t)+g12*(t,t)a2(t)g12(t,t)a2(t)]dtdt.
(135)
A similar formula exists for the idler number-flux density. By combining these formulas, one finds that the total-number flux
n1(t)+n2(t)=[g11*(t,t)g11(t,t)+g21*(t,t)g21(t,t)]a1(t)a1(t)dtdt+[g11*(t,t)g12(t,t)+g21*(t,t)g22(t,t)]a1(t)a2(t)dtdt+[g22*(t,t)g21(t,t)+g12*(t,t)g11(t,t)]a2(t)a1(t)dtdt+[g22*(t,t)g22(t,t)+g12*(t,t)g12(t,t)]a2(t)a2(t)dtdt.
(136)
The total-number flux is time dependent. However, by integrating Eq. (136) with respect to time, and using Eqs. (133) and (134), one finds that the total number
N1+N2=M1+M2,
(137)
where Mj and Nj denote input and output number operators, respectively. Thus, FC converts signal photons to idler photons and vice versa, but conserves the total number of photons, for arbitrary input pulses. Notice that Eq. (137) pertains to the total-number operator: Not only is the total-number mean conserved, so also are all the total-number moments (fluctuations). The first-moment equation, which involves the number means, is simply
N1+N2=M1+M2,
(138)
whereas the second-moment equation, which involves the variances and correlation, is
δN12+2δN1δN2+δN22=δM12+2δM1δM2+δM22.
(139)
If the inputs are (independent) CS, δMj2=Mj, 〈δM1δM2〉 = 0 and Eq. (139) reduces to
δN12+2δN1δN2+δN22=M1+M2.
(140)
It was shown in Sec. 3 that δN12+δN22=N1+N2. Because the total number is conserved, Eq. (140) implies that 〈δN1δN2〉 = 0: The output number fluctuations are uncorrelated.

Two-mode PA is governed by the IO equations (47) and (48). Because the mode evolution is unitary, the output operators satisfy the same commutation relations as the input operators. Hence,
[b1(t1),b1(t2)]=[g11(t1,t)g11*(t2,t)g12(t1,t)g12*(t2,t)]dt=δ(t1t2),
(141)
[b1(t1),b2(t2)]=[g11(t1,t)g21(t2,t)g12(t1,t)g22(t2,t)]dt=0.
(142)
Equations (141) and (142) impose important constraints on the Green functions. Because Eqs. (47) and (48) depend symmetrically on the labels 1 and 2, similar constraints exist in which 1 ↔ 2.

There also exist OI equations, which can be written in the form
a1(t)=[h11(t,t)b1(t)+h12(t,t)b2(t)]dt,
(143)
a2(t)=[h22(t,t)b2(t)+h21(t,t)b1(t)]dt.
(144)
It follows from Eqs. (143) and (144), and the commutation relations, that
[a1(t1),a1(t2)]=[h11(t1,t)h11*(t2,t)h12(t1,t)h12*(t2,t)]dt=δ(t1t2),
(145)
[a1(t1),a2(t2)]=[h11(t1,t)h21(t2,t)h12(t1,t)h22(t2,t)]dt=0.
(146)
By combining Eqs. (47), (143) and (144), one obtains the identity
b1(t)=[g11(t,t)h11(t,t)+g12(t,t)h21*(t,t)]b1(t)dtdt+[g11(t,t)h12(t,t)+g12(t,t)h22*(t,t)]b2(t)dtdt,
(147)
from which follow the consistency conditions
[g11(t,t)h11(t,t)+g12(t,t)h21*(t,t)]dt=δ(tt),
(148)
[g11(t,t)h12(t,t)+g12(t,t)h22*(t,t)]dt=0.
(149)
By comparing Eqs. (141) and (142) with Eqs. (148) and (149), one obtains the reciprocity relations [65

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

, 66

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

]
h11(t1,t2)=g11*(t2,t1),h21(t1,t2)=g12(t2,t1).
(150)
The identity for a1(t) produces the same conditions. By combining Eqs. (145), (146) and (150), one finds that
[g11*(t,t1)g11(t,t2)g21(t,t1)g21*(t,t2)]dt=δ(t1t2),
(151)
[g11*(t,t1)g12(t,t2)g21(t,t1)g22*(t,t2)]dt=0.
(152)
Thus, the Green functions for PA also satisfy constraints when integrated with respect to either (first or second) variable.

The constraint equations have important physical consequences. By combining Eq. (47) and (48) with their conjugates, one finds that the sideband number-flux densities
n1(t)=[g11*(t,t)a1(t)g11(t,t)a1(t)+g11*(t,t)a1(t)g12(t,t)a2(t)+g12*(t,t)a2(t)g11(t,t)a1(t)+g12*(t,t)a2(t)g12(t,t)a2(t)]dtdt,
(153)
n2(t)=[g22*(t,t)a2(t)g22(t,t)a2(t)+g22*(t,t)a2(t)g21(t,t)a1(t)+g21*(t,t)a1(t)g22(t,t)a2(t)+g21*(t,t)a1(t)g21(t,t)a1(t)]dtdt.
(154)
By using the commutation relations and the fact that t′ and t″ are dummy variables, one can rewrite the third and fourth terms in Eq. (153) in the abbreviated forms g11a1g12*a2 and g12a2g12*a2+|g12|2δ, respectively. Similarly, one can rewrite the second and fourth terms in Eq. (154) in the abbreviated forms g21a1g22*a2 and g21a1g21*a1+|g21|2δ, respectively. By combining these results, one finds that the number-flux difference
n1(t)n2(t)=[g11*(t,t)g11(t,t)g21(t,t)g21*(t,t)]a1(t)a1(t)dtdt+[g11*(t,t)g12(t,t)g21(t,t)g22*(t,t)]a1(t)a2(t)dtdt+[g11(t,t)g12*(t,t)g21*(t,t)g22(t,t)]a1(t)a2(t)dtdt+[g12(t,t)g12*(t,t)g22*(t,t)g22(t,t)]a2(t)a2(t)dtdt+[|g12(t,t)|2|g21(t,t)|2]dt.
(155)
The number-flux difference is time dependent. However, by integrating Eq. (155) with respect to time, and using Eqs. (151) and (152), one finds that the number difference
N1N2=M1M2+[|g12(t,t)|2|g21(t,t)|2]dtdt.
(156)
In general, |g12(t, t′)|2 does not equal |g21(t, t′)|2. However, by using the Schmidt decompositions stated in Sec. 5, one can show that the integrals of these functions are equal, so the last term in Eq. (156) is zero. Thus, PA generates signal and idler photons in pairs, for arbitrary input pulses. Notice that Eq. (156) pertains to the number-difference operator: Not only is the number-difference mean conserved, so also are all the number-difference moments (fluctuations). The first-moment equation, which involves the number means, is simply
N1N2=M1M2,
(157)
whereas the second-moment equation, which involves the variances and correlation, is
δN122δN1δN2+δN22=δM122δM1δM2+δM22.
(158)
If the inputs are (independent) CS, Eq. (158) reduces to
δN122δN1δN2+δN22=M1+M2.
(159)
In the high-gain regime ( δNj2>NjMj), the correlation is approximately equal to the average of the variances. Furthermore, if the inputs are VS, the correlation equals the (common) variance for arbitrary gains, which reflects the fact that photons are generated in pairs. For reference, Eqs. (138) and (157) are called the MRW equations [43

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

, 44

44. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

].

One-mode PA is governed by the IO equation (74). One can obtain this equation from Eq. (47) or (48) by making the substitutions aja, bjb, gjjgs and gjkgc. Like their two-mode counterparts, the one-mode transfer functions satisfy constraints, which one can deduce by making the same substitutions in Eqs. (141) and (142), and similar substitutions in Eqs. (145) and (146), and reciprocity relations, which one can deduce from Eqs. (150).

Appendix B: Quadrature and number fluctuations

As demonstrated in Sec. 4, when one calculates quadrature and number correlations (including variances), it is convenient to separate each output mode operator bj(t) into its mean value 〈bj(t)〉 = βj(t), which depends only on the input signals, and its deviation bj(t) − 〈bj(t)〉 = wj(t), which depends only on the input fluctuations. By using these definitions, one finds that the mean and deviation of each quadrature flux are
qj(t)=[βp*(t)βj(t)+βp(t)βj*(t)]/21/2,
(160)
δqj(t)=[βp*(t)wj(t)+βp(t)wj(t)]/21/2,
(161)
respectively, where βp(t) is the (normalized) amplitude of the LO used to detect mode j. To calculate quadrature correlations, one evaluates quadrature-flux moments of the form
δqj(t1)δqk(t2)=[βp*(t1)wj(t1)+βp(t1)wj(t1)][βq*(t2)wk(t2)+βq(t2)wk(t2)]/2,
(162)
where βq(t) is the amplitude of the LO used to detect mode k, then integrates with respect to t1 and t2.

By proceeding in a similar way, one finds that the mean and deviation of each number flux are
nj(t)=|βj(t)|2+wj(t)wj(t),
(163)
δnj(t)=βj*(t)wj(t)+βj(t)wj(t)+wj(t)wj(t)wj(t)wj(t),
(164)
respectively. To calculate number correlations, one evaluates number-flux moments of the form
δnj(t1)δnk(t2)=[βj*(t1)wj(t1)+βj(t1)wj(t1)+wj(t1)wj(t1)wj(t1)wj(t1)]×[βk*(t2)wk(t2)+βk(t2)wk(t2)+wk(t2)wk(t2)wk(t2)wk(t2)],
(165)
then integrates with respect to time. Because the odd moments of the fluctuations have zero means, one can rewrite Eq. (165) in the simpler form
δnj(t1)δnk(t2)=[βj*(t1)wj(t1)+βj(t1)wj(t1)][βk*(t2)wk(t2)+βk(t2)wk(t2)]+wj(t1)wj(t1)wk(t2)wk(t2)wj(t1)wj(t1)wk(t2)wk(t2).
(166)
The right side of Eq. (166) contains both signal–noise terms, which depend on the mean signals and the fluctuations (second-order moments of wj and wk), and noise–noise terms, which depend only on the fluctuations (fourth-order moments). For signals that contain many photons, one can neglect the latter terms, in which case
δnj(t1)δnk(t2)sn=[βj*(t1)wj(t1)+βj(t1)wj(t1)][βk*(t2)wk(t2)+βk(t2)wk(t2)].
(167)
If the LOs used to detect the mode quadratures have the same shapes as the output pulses (which is the optimal situation), the quadrature-flux moments (162) are proportional to the signal–noise contributions to the number-flux moments (167). Notice that this result is kinematic, in the sense that it depends only on the definitions of the quadrature- and number-flux moments, and not on the details of the parametric processes (kinetics) that produced them. It is a generalization of the discrete-mode result stated in [21

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

]. For specific examples of the relation between the quadrature and number fluctuations, see Eqs. (35), (36), (43) and (44), and Eqs. (57), (58), (72) and (73).

As stated earlier, the noise figure of a parametric process is the input SNR divided by the output SNR. If the shapes and phases of the LOs are chosen optimally, the relation between formulas (162) and (167) ensures that the homodyne SNRs are larger than the direct SNRs by a (common) factor of 4 at the input and output. Hence, the direct and homodyne noise figures are equal. This result is a generalization of the discrete-mode result stated in [19

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

].

Appendix C: Multiple-mode processes

In Secs. 3 and 4, formulas were derived for the quadrature and number fluctuations produced by one- and two-mode PA, and two-mode FC. In this appendix, these formulas are generalized for processes that involve arbitrary numbers of continuous modes. Because the physics and applications of the one- and two-mode formulas were described in the main text, this appendix is restricted to an efficient mathematical derivation of the multiple-mode formulas.

The input–output (IO) equations for multiple-mode parametric processes can be written in the form
bi(t)=kdt[μik(t,t)ak(t)+νik(t,t)ak(t)],
(168)
where ak and bi are input and output mode-amplitude operators, respectively, μik is a forward transfer (Green) function that couples bi to ak (like operators), νik is a forward Green function that couples bi to ak (unlike operators) and † denotes a hermitian conjugate. (In this appendix, different symbols are used for the Green functions that couple like and unlike operators. Such a distinction was unnecessary in the text, because bi was coupled to ak or ak, but not both.) The input operators satisfy the commutation relations (1). Because the mode evolution is unitary, the output operators satisfy similar commutation relations, which impose constraints on the Green functions. Specifically,
[bi(t1),bj(t2)]=kdt[μik(t1,t)μjk*(t2,t)νik(t1,t)νjk*(t2,t)]=δijδ(t1t2),
(169)
[bi(t1),bj(t2)]=kdt[μik(t1,t)νjk(t2,t)νik(t1,t)μjk(t2,t)]=0.
(170)

There also exist output–input (OI) equations, which can be written in the form
ai(t)=kdt[μ¯ik(t,t)bk(t)+ν¯ik(t,t)bk(t)],
(171)
where μ̄ik and ν̄ik are backward Green functions. These Green functions satisfy constraints with the same forms as Eqs. (169) and (170). By combining Eqs. (168) and (171), one obtains the identity
bi(t)=jkdtdt{[μij(t,t)μ¯jk(t,t)+νij(t,t)ν¯jk*(t,t)]bk(t)+[μij(t,t)ν¯jk(t,t)+νij(t,t)μ¯jk*(t,t)]bk(t)},
(172)
from which follow the consistency conditions
jdt[μij(t,t)μ¯jk(t,t)+νij(t,t)ν¯jk*(t,t)]=δikδ(tt),
(173)
jdt[μij(t,t)ν¯jk(t,t)+νij(t,t)μ¯jk*(t,t)]=0.
(174)
By combining Eqs. (173) and (174) with Eqs. (169) and (170), one obtains the reciprocity relations
μ¯ij(t1,t2)=μji*(t2,t1),ν¯ij(t1,t2)=νji(t2,t1).
(175)
By combining the constraints imposed on the backward Green functions with the reciprocity relations, one finds that
kdt[μki*(t,t1)μkj(t,t2)νki(t,t1)νkj*(t,t2)]=δijδ(t1t2),
(176)
kdt[μki*(t,t1)νkj(t,t2)νki(t,t1)μkj*(t,t2)]=0.
(177)
Thus, the Green functions satisfy constraints when integrated with respect to either (first or second) variable. The identity for ai produces equivalent results. Notice that Eqs. (169), (170) and (175)(177) reduce to the corresponding equations of App. A in the appropriate limits.

Although the preceding second-order moments are real by construction and do not depend on the order in which the deviation operators are written (because δqi and δqj commute, as do δQi and δQj), these properties are not evident in Eqs. (184) and (185). It is convenient to rewrite Eq. (184) in the abbreviated form
δqi(t1)δqj(t2)(βp*μik)1(βq*νjk)2+(βp*μik)1(βqμjk*)2+(βpνik*)1(βq*νjk)2+(βpνik*)1(βqμjk*)2,
(186)
in which summation with respect to k, integration with respect to t′ and the factor of 1/2 are implicit, and the subscripts 1 and 2 represent the times t1 and t2, respectively. Consider the first and fourth terms on the right side of Eq. (186). By applying Eq. (170) twice and reordering, one finds that
(βp*μik)1(βq*νjk)2+(βpνik*)1(βqμjk*)2=(βp*μik)1(βq*νjk)2+(βpμik*)1(βqνjk*)2=(βp*νik)1(βq*μjk)2+(βpμik*)1(βqνjk*)2=(βq*μjk)2(βp*νik)1+(βqνjk*)2(βpμik*)1.
(187)
Now consider the second and third terms in Eq. (186). By applying Eq. (169) twice and reordering, one finds that
(βp*μik)1(βqμjk*)2+(βpνik*)1(βq*νjk)2=(βp*νik)1(βqνjk*)2+(βpνik*)1(βq*νjk)2+(|βp|2)1δijδ12=(βp*νik)1(βqνjk*)2+(βpμik*)1(βq*μjk)2=(βq*μjk)2(βpμik*)1+(βqνjk*)2(βp*νik)1.
(188)
The first forms of the right sides of Eqs. (187) and (188) are manifestly real, and the third forms of the right sides are just the left sides, with i and j (and p and q) interchanged. Thus, the second-order moments are real and do not depend on the order in which the deviation operators are written: 〈δqi(t1)δqj(t2)〉 = 〈δqj(t2)δqi(t1)〉 and 〈δQiδQj〉 = 〈δQjδQi〉. When i = j, Eq. (185) specifies the quadrature variance
δQi2=kdt|dt[βp*(t)μik(t,t)+βp(t)νik*(t,t)]|2/2,
(189)
which is manifestly real and non-negative.

The mean and deviation of each number flux are
ni(t)=|βi(t)|2+wi(t)wi(t),
(190)
δni(t)=βp*(t)wi(t)+βp(t)wi(t)+wi(t)wi(t)wi(t)wi(t),
(191)
respectively. By integrating Eq. (190) with respect to time and using the aforementioned properties of vacuum states, one finds that the number mean
Ni=dt|βi(t)|2+kdtdt|νik(t,t)|2.
(192)
Noise photons are generated by each conjugate mode ak to which bi is coupled. The first two terms on the right side of Eq. (191) are (collectively) the signal–noise contribution to the number deviation, whereas the last two terms are the noise–noise contribution. The former contribution to the number deviation has the same form as the quadrature deviation (179), with βp replaced by βj and the factor of 1/21/2 omitted. Hence, the derivations of the signal–noise contributions to the number-flux moment and number correlation need not be described. The final results are
δni(t1)δnj(t2)sn=kdt[βi*(t1)μik(t1,t)+βi(t1)νik*(t1,t)]×[βj*(t2)νjk(t2,t)+βj(t2)μjk*(t2,t)],
(193)
δNiδNjsn=kdtdt1[βi*(t1)μik(t1,t)+βi(t1)νik*(t1,t)]×dt2[βj*(t2)νjk(t2,t)+βj(t2)μjk*(t2,t)].
(194)
It is easy to show that both fourth-order moments are real and neither depends on the order in which the deviation operators are written: 〈δni(t1)δnj(t2)〉sn = 〈δnj(t2)δni(t1)〉sn and 〈δNiδNjsn = 〈δNjδNisn. When i = j, Eq. (194) specifies the number variance
δNi2sn=kdt|dt[βi*(t)μik(t,t)+βi(t)νik*(t,t)]|2,
(195)
which is manifestly real and non-negative.

Equation (201) can be rewritten in the abbreviated form
δni(t1)δnj(t2)nnkl[(μik)1(μjk*)2(νil*)1(νjl)2+(μik)1(νjk)2(νil*)1(μjl*)2],
(203)
in which integrations with respect to t′ and t″ are implicit. Consider the first term on the right side of Eq. (203). By applying Eq. (169) twice and reordering, one finds that
kl(μik)1(μjk*)2(νil*)1(νjl)2=kl(νik)1(νjk*)2(νil*)1(νjl)2+δijl(|νil|2)1=kl(νik)1(νjk*)2(μil*)1(μjl)2δijk(|νik|2)1+δijl(|νil|2)1=kl(μjl)2(μil*)1(νjk*)2(νik)1.
(204)
Now consider the second term in Eq. (203). By applying Eq. (170) twice and reordering, one finds that
kl(μik)1(νjk)2(νil*)1(μjl*)2=kl(μik)1(νjk)2(μil*)1(νjl*)2=kl(νik)1(μjk)2(μil*)1(νjl*)2=kl(μjk)2(νik)1(νjl*)2(μil*)1.
(205)
The first forms of the right sides of Eqs. (204) and (205) are not manifestly real. However, for each term with (k, l) = (m, n), there is another term with (k, l) = (n, m), which is the conjugate of the first term. Thus, the fourth-order moments are real. The third forms of the right sides of Eqs. (204) and (205) are just the left sides, with i and j interchanged. Thus, the fourth-order moments do not depend on the order in which the deviation operators are written: 〈δni(t1)δnj(t2)〉nn = 〈δnj(t2)δni(t1)〉nn and 〈δNiδNjnn = 〈δNjδNinn.

Notice that Eq. (202) can be rewritten in the form
δNiδNjnn2k(μikνik*)1(μjk*νjk)2+kl>k(μikνil*+μilνik*)1(μjk*νjl+μjl*νjk)2,
(206)
in which the time integrals are implicit and which is consistent with Eq. (20) of [21

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

]. When i = j, Eq. (206) reduces to
δNi2nn2k|μikνik*|2+kl>k|μikνil*+μilνik*|2,
(207)
which is consistent with Eq. (43) of [19

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

]. The structural similarity between the continuous-mode equations (206) and (207), and the aforementioned discrete-mode equations, is striking. For some systems, there is even a direct correspondence. Each of the Green functions in Eqs. (168) has a Schmidt decomposition of the form nvn(t)σnun*(t), where un and vn are input and output Schmidt modes (temporal eigenfunctions), respectively, and σn is a Schmidt coefficient [25

25. G. J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge, 2011), Sec. 5.4.

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

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

, 21

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

, 22

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

]. To the best of our knowledge, such a simplification is possible for some, but not all, systems. One can always apply the Schmidt decomposition theorem [24

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

, 26

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

] to the Green matrices M(t, t′) = [μij(t, t′)] and N(t, t′) = [νij(t, t′)], which couple the output vector B(t) = [bi(t)] to the input vector A(t′) = [aj(t′)] and its conjugate, respectively, to determine the input and output Schmidt supermodes (vector temporal eigenfunctions), and the associated Schmidt coefficients.

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OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(270.6570) Quantum optics : Squeezed states

ToC Category:
Nonlinear Optics

History
Original Manuscript: May 6, 2013
Revised Manuscript: July 16, 2013
Manuscript Accepted: July 22, 2013
Published: August 12, 2013

Citation
C. J. McKinstrie and M. E. Marhic, "Quadrature and number fluctuations produced by parametric devices driven by pulsed pumps," Opt. Express 21, 19437-19466 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-19437


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References

  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]
  3. J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron.E88-C, 327–334 (2005). [CrossRef]
  4. C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News18(3), 34–40 (2007). [CrossRef]
  5. P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev.1, 231–248 (2007). [CrossRef]
  6. S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron.18, 670–680 (2012). [CrossRef]
  7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Cambridge, 2000).
  8. I. A. Walmsley and M. G. Raymer, “Toward quantum information processing with photons,” Science307, 1733–1734 (2005). [CrossRef] [PubMed]
  9. S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature437, 116–120 (2005). [CrossRef] [PubMed]

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