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

C. J. McKinstrie; M. E. Marhic

doi:10.1364/OE.21.019437

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 [1–6]. They can also generate photons for, and frequency convert photons in, quantum information experiments [7–12]. 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, 14, 16, 17, 19–22]. 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 [23–26]. 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 [27–30]. 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) [31–34]. 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 $a_{j}^{†}$ and a_j, respectively, where † denotes a hermitian conjugate. These continuous-mode amplitude operators satisfy the boson commutation relations

[a_{j} (t_{1}), a_{k} (t_{2})] = 0, [a_{j} (t_{1}), a_{k}^{†} (t_{2})] = δ_{j k} δ (t_{1} - t_{2}),

where [,] denotes a commutator, and δ_jk and δ(t₁ − t₂) are the Kronecker and Dirac delta functions, respectively [7].

In direct detection, photodiodes are used to measure the photon-number means 〈N_j〉, where 〈〉 denotes an expectation value. The uncertainties in such measurements are the number variances

〈 δ N_{j}^{2} 〉 = 〈 {(N_{j} - 〈 N_{j} 〉)}^{2} 〉 = 〈 N_{j}^{2} 〉 - {〈 N_{j} 〉}^{2} .

Also of interest are the number correlations

〈 δ N_{j} δ N_{k} 〉 = 〈 (N_{j} - 〈 N_{j} 〉) (N_{k} - 〈 N_{k} 〉) 〉 = 〈 N_{j} N_{k} 〉 - 〈 N_{j} 〉 〈 N_{k} 〉 .

The number-flux operator

n_{j} (t) = a_{j}^{†} (t) a_{j} (t)

and the associated number operator N_j = ∫n_j(t)dt, where the limits of integration are −T/2 and T/2, and T is the measurement time. It follows from these definitions that

〈 N_{j} 〉 = \int 〈 n_{j} (t) 〉 d t,

〈 N_{j} N_{k} 〉 = \iint 〈 n_{j} (t_{1}) n_{k} (t_{2}) 〉 d t_{1} d t_{2} .

Thus, to model direct detection, one must calculate the second- and fourth-order amplitude moments 〈n_j(t)〉 and 〈n_j(t₁)n_k(t₂)〉, respectively.

In many applications it is reasonable to approximate the input signal by a coherent state (CS), which is defined by the equations

a_{j} (t) | α_{j} 〉 = α_{j} (t) | α_{j} 〉, 〈 α_{j} | a_{j}^{†} (t) = 〈 α_{j} | α_{j}^{*} (t),

where |α_j〉 is the state vector [7]. For such a state, the mean (coherent) amplitude 〈a_j(t)〉 = α_j(t). By combining Eqs. (4) and (7), one finds that the second-order moment

〈 n_{j} (t) 〉 = 〈 a_{j}^{†} (t) a_{j} (t) 〉 = {| α_{j} (t) 〉}^{2} .

The number flux is the squared modulus of the coherent amplitude. One also finds that the fourth-order moment

\begin{array}{l} 〈 n_{j} (t_{1}) n_{j} (t_{2}) 〉 & = & 〈 a_{j}^{†} (t_{1}) a_{j} (t_{1}) a_{j}^{†} (t_{2}) a_{j} (t_{2}) 〉 \\ = & 〈 a_{j}^{†} (t_{1}) [a_{j}^{†} (t_{2}) a_{j} (t_{1}) + δ (t_{1} - t_{2})] a_{j} (t_{2}) 〉 \\ = & {| α_{j} (t_{1}) |}^{2} {| α_{j} (t_{2}) |}^{2} + α_{j}^{*} (t_{1}) α_{j} (t_{2}) δ (t_{1} - t_{2}) . \end{array}

The penultimate term in Eq. (9) is 〈n_j(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

〈 N_{j} 〉 = \int {| α_{j} (t) |}^{2} d t,

〈 δ N_{j}^{2} 〉 = 〈 N_{j} 〉 .

Property (11) is characteristic of a CS. The vacuum state (VS) |0〉 is a special CS, for which α_j(t) = 0 and

〈 δ N_{j}^{2} 〉 = 〈 N_{j} 〉 = 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 〈N_j〉, which is the standard result for a single discrete mode (or CW) [7].

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

〈 n_{j} (t_{1}) n_{k} (t_{2}) 〉 = 〈 a_{j}^{†} (t_{1}) a_{j} (t_{1}) a_{k}^{†} (t_{2}) a_{k} (t_{2}) 〉 = {| α_{j} (t_{1}) |}^{2} {| α_{k} (t_{2}) |}^{2},

because a_j(t₁) and

a_{k}^{†} (t_{2})

commute. Hence,

〈 δ N_{j} δ N_{k} 〉 = 0 .

The number fluctuations of two CS are uncorrelated.

Now suppose that two modes (j and k), with the same carrier frequency, are combined by a beam splitter. Then the output operators (b_j and b_k) are related to the input operators (a_j and a_k) by the equations

b_{j} (t) = τ a_{j} (t) + ρ a_{k} (t),

b_{k} (t) = - ρ^{*} a_{j} (t) + τ^{*} a_{k} (t) .

The transmission and reflection coefficients satisfy the auxiliary equation |τ|² +|ρ|² = 1, which ensures that the total number of photons is conserved [7]. Each output mode is a superposition (combination) of both input modes. The output number-flux operators

n_{j} (t) = b_{j}^{†} (t) b_{j} (t)

and

n_{k} (t) = b_{k}^{†} (t) b_{k} (t)

, and the flux-difference operator n_jk(t) = n_j(t) − n_k(t). To limit the number of symbols, we will use the same notation for input and output quantities, wherever possible. (IO equations are obvious exceptions.) It is easy to show that

n_{j k} (t) = ({| τ |}^{2} - {| ρ |}^{2}) [a_{j}^{†} (t) a_{j} (t) - a_{k}^{†} (t) a_{k} (t)] + 2 [τ ρ^{*} a_{j} (t) a_{k}^{†} (t) + τ^{*} ρ a_{j}^{†} (t) a_{k} (t)] .

For a balanced beam splitter (|τ|² = |ρ|² = 1/2),

n_{j k} (t) = a_{j} (t) a_{k}^{†} (t) + a_{j}^{†} (t) a_{k} (t),

where the phase factor arg(ρτ^*) was absorbed into a_k. If the inputs are CS, then the output moments

〈 n_{j k} (t) 〉 = α_{j} (t) α_{k}^{*} (t) + α_{j}^{†} (t) α_{k} (t),

\begin{array}{l} 〈 n_{j k} (t_{1}) n_{j k} (t_{2}) 〉 & = & 〈 [a_{j} (t_{1}) a_{k}^{†} (t_{1}) + a_{j}^{†} (t_{1}) a_{k} (t_{1})] [a_{j} (t_{2}) a_{k}^{†} (t_{2}) + a_{j}^{†} (t_{2}) a_{k} (t_{2})] 〉 \\ = & [α_{j} (t_{1}) α_{k}^{*} (t_{1}) + α_{j}^{*} (t_{1}) α_{k} (t_{1})] [α_{j} (t_{2}) α_{k}^{*} (t_{2}) + α_{j}^{*} (t_{2}) α_{k} (t_{2})] \\ + α_{k}^{*} (t_{1}) α_{k} (t_{2}) δ_{j} (t_{1} - t_{2}) + α_{j}^{*} (t_{1}) α_{j} (t_{2}) δ_{k} (t_{1} - t_{2}), \end{array}

where the subscripts on the delta functions indicate their origin. (In abbreviated notation,

[a_{j}, a_{j}^{†}] = δ_{j}

) Hence, if one were to detect the number difference, one would find that

〈 N_{j k} 〉 = \int [α_{j} (t) α_{k}^{*} (t) + α_{j}^{*} (t) α_{k} (t)] d t,

〈 δ N_{j k}^{2} 〉 = 〈 N_{k} 〉 1_{j} + 〈 N_{j} 〉 1_{k},

where 1_j is the integral of δ_j. The mean of the number difference depends only on the input amplitudes and, for specified input numbers, is maximal when the input pulses have the same shape. The variance of the number difference depends on the input numbers and fluctuations of both modes (1_j and 1_k). However, if 〈N_k〉 ≫ 〈N_j〉, the variance depends only on the fluctuations of mode j. For this case, one would obtain the same result by treating the strong mode (k) classically and the weak mode (j) quantum-mechanically.

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

q_{j} (t) = [α_{p}^{*} (t) a_{j} (t) + α_{p} (t) a_{j}^{†} (t)] / 2^{1 / 2},

where the LO amplitude α_p(t) includes the phase factor e^iϕ_p implicitly and satisfies the normalization condition ∫ |α_p(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 Q_j = ∫q_j(t)dt. It follows from definition (22) that the quadrature-flux moments

〈 q_{j} (t) 〉 = [α_{p}^{*} (t) α_{j} (t) + α_{p} (t) α_{j}^{*} (t)] / 2^{1 / 2},

\begin{array}{l} 〈 q_{j} (t_{1}) q_{j} (t_{2}) 〉 & = & 〈 [α_{p}^{*} (t_{1}) a_{j} (t_{1}) + α_{p} (t_{1}) a_{j}^{†} (t_{1})] [α_{p}^{*} (t_{2}) a_{j} (t_{2}) + α_{p} (t_{2}) a_{j}^{†} (t_{2})] 〉 / 2 \\ = & [α_{p}^{*} (t_{1}) α_{j} (t_{1}) + α_{p} (t_{1}) α_{j}^{*} (t_{1})] [α_{p}^{*} (t_{2}) α_{j} (t_{2}) + α_{p} (t_{2}) α_{j}^{*} (t_{2})] / 2 \\ + α_{p}^{*} (t_{1}) α_{p} (t_{2}) δ (t_{1} - t_{2}) / 2 . \end{array}

The penultimate term in Eq. (24) is 〈q_j(t₁)〉〈q_j(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

〈 Q_{j} 〉 = \int [α_{p}^{*} (t) α_{j} (t) + α_{p} (t) α_{j}^{*} (t)] d t / 2^{1 / 2},

〈 δ Q_{j}^{2} 〉 = 1 / 2 .

The quadrature mean (25) depends on the shapes and overlap of the signal and LO pulses, whereas the variance (26) does not depend on either pulse shape. The variance (26) has the minimal value allowed by the Heisenberg uncertainty principle. This result is characteristic of a CS [7]. As stated earlier, the SNR is defined to be the square of the mean quadrature divided by the quadrature variance. If one uses the optimal LO [α_p(t) ∝ α_j(t), where the proportionality coefficient is real and positive], then 〈Q_j〉 = 2^1/2〈N_j〉^1/2 and the SNR equals 4〈N_j〉, which is the standard result for a single discrete mode [7]. Notice that the optimal-LO condition is analogous to the matched-filter condition in detection theory [35, 36]

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.

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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

\begin{array}{l} 〈 q_{j} (t_{1}) q_{k} (t_{2}) 〉 & = & 〈 [α_{p}^{*} (t_{1}) a_{j} (t_{1}) + α_{p} (t_{1}) a_{j}^{†} (t_{1})] [α_{q}^{*} (t_{2}) a_{k} (t_{2}) + α_{q} (t_{2}) a_{k}^{†} (t_{2})] 〉 / 2 \\ = & [α_{p}^{*} (t_{1}) α_{j} (t_{1}) + α_{p} (t_{1}) α_{j}^{*} (t_{1})] [α_{q}^{*} (t_{2}) α_{k} (t_{2}) + α_{q} (t_{2}) α_{k}^{*} (t_{2})] / 2, \end{array}

because a_j(t₁) and

a_{k}^{†} (t_{2})

commute. Hence,

〈 δ Q_{j} δ Q_{k} 〉 = 0 .

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]. 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 [37–42] 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 (π_p2 + π_s1 → π_p1 + π_s2, where π_j represents a photon with carrier frequency ω_j). This process is illustrated in Fig. 2. The photon reaction used to describe BS is called the Manley–Rowe–Weiss (MRW) relation [43, 44]. 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, 39, 40]. However, in its alternative configuration (nonzero input signal and idler), BS is phase-sensitive [38], 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.

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The IO equations for attenuation and FC can be written in the form

b_{1} (t) = \int [g_{11} (t, t^{'}) a_{1} (t^{'}) + g_{12} (t, t^{'}) a_{2} (t^{'})] {d t}^{'},

b_{2} (t) = \int [g_{21} (t, t^{'}) a_{1} (t^{'}) + g_{22} (t, t^{'}) a_{2} (t^{'})] {d t}^{'},

where a_j are input operators, b_j are output operators and g_jk are Green functions. For attenuation, mode 1 is the signal and mode 2 is the loss mode [7], whereas for FC, mode 1 is the signal and mode 2 is the idler [19]. 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.

The output quadrature-flux operators are related to the output mode operators in the same way that the input quadrature-flux operators are related to the input mode operators [Eqs. (22)]. It follows from these definitions and the assumption that the inputs are CS, that the first-order output signal moment

〈 q_{1} (t) 〉 = [β_{p}^{*} (t) β_{1} (t) + β_{p} (t) β_{1}^{*} (t)] / 2^{1 / 2},

where the mean output signal amplitude β₁(t) is related to the mean input amplitudes α_j(t) by an equation similar to Eq. (29), and the LO amplitude is denoted by β_p(t), instead of α_p(t). Hence, if one were to detect the output signal, one would find that

〈 Q_{1} 〉 = \int [β_{p}^{*} (t) β_{1} (t) + β_{p} (t) β_{1}^{*} (t)] d t / 2^{1 / 2} .

One maximizes the detected signal quadrature by matching the LO pulse to the output signal pulse.

To calculate the higher-order moments, one proceeds in the manner described in Sec. 2: One groups the operator moments into normally-ordered terms ( $b_{j}^{†}$ before b_j) 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

\begin{array}{l} 〈 q_{1} (t_{1}) q_{1} (t_{2}) 〉 & = & 〈 [β_{p}^{*} (t_{1}) b_{1} (t_{1}) + β_{p} (t_{1}) b_{1}^{†} (t_{1})] [β_{p}^{*} (t_{2}) b_{1} (t_{2}) + β_{p} (t_{2}) b_{1}^{†} (t_{2})] 〉 / 2 \\ = & [β_{p}^{*} (t_{1}) β_{1} (t_{1}) + β_{p} (t_{1}) β_{1}^{*} (t_{1})] [β_{p}^{*} (t_{2}) β_{1} (t_{2}) + β_{p} (t_{2}) β_{1}^{*} (t_{2})] / 2 \\ + β_{p}^{*} (t_{1}) β_{p} (t_{2}) δ (t_{1} - t_{2}) / 2, \end{array}

because b₁(t₁) and

b_{1}^{†} (t_{2})

do not commute. In contrast, the mixed second-order moment

\begin{array}{l} 〈 q_{1} (t_{1}) q_{2} (t_{2}) 〉 & = & 〈 [β_{p}^{*} (t_{1}) b_{1} (t_{1}) + β_{p} (t_{1}) b_{1}^{†} (t_{1})] [β_{q}^{*} (t_{2}) b_{2} (t_{2}) + β_{q} (t_{2}) b_{2}^{†} (t_{2})] 〉 / 2 \\ = & [β_{p}^{*} (t_{1}) β_{1} (t_{1}) + β_{p} (t_{1}) β_{1}^{*} (t_{1})] [β_{q}^{*} (t_{2}) β_{2} (t_{2}) + β_{q} (t_{2}) β_{2}^{*} (t_{2})] / 2, \end{array}

because b₁(t₁) and

b_{2}^{†} (t_{2})

do commute. Hence,

〈 δ Q_{1}^{2} 〉 = 1 / 2,

〈 δ Q_{1} δ Q_{2} 〉 = 0 .

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

β_{p}^{*} (t_{1}) β_{p} (t_{2}) \int [g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) + g_{12} (t_{1}, t^{'}) g_{12}^{*} (t_{2}, t^{'})] {d t}^{'} / 2,

β_{p}^{*} (t_{1}) β_{q} (t_{2}) \int [g_{11} (t_{1}, t^{'}) g_{21}^{*} (t_{2}, t^{'}) + g_{12} (t_{1}, t^{'}) g_{22}^{*} (t_{2}, t^{'})] {d t}^{'} / 2

in Eqs. (33) and (34), respectively. However, by using Eqs. (123) and (124), we would have obtained the stated results,

β_{p}^{*} (t_{1}) β_{p} (t_{2}) δ (t_{1} - t_{2}) / 2

and 0, respectively.

The output number-flux operators are also related to the output mode operators in the same ways that the input number-flux operators are related to the input mode operators. It follows from these definitions that the second-order signal moment

〈 n_{1} (t) 〉 = 〈 b_{1}^{†} (t_{1}) b_{1} (t) 〉 = {| β_{1} (t) |}^{2} .

The output number flux is the squared modulus of the mean output amplitude, which depends implicitly on the mean input amplitudes, but not the input fluctuations. Hence, if one were to detect the output signal, one would find that

〈 N_{1} 〉 = \int {| β_{1} (t) |}^{2} d t .

The fourth-order moment

\begin{array}{l} 〈 n_{1} (t_{1}) n_{1} (t_{2}) 〉 & = & 〈 b_{1}^{†} (t_{1}) b_{1} (t_{1}) b_{1}^{†} (t_{2}) b_{1} (t_{2}) 〉 \\ = & {| β_{1} (t_{1}) |}^{2} {| β_{1} (t_{2}) |}^{2} + β_{1}^{*} (t_{1}) β_{1} (t_{2}) δ (t_{1} - t_{2}), \end{array}

because b₁(t₁) and

b_{1}^{†} (t_{2})

do not commute, whereas the mixed fourth-order moment

〈 n_{1} (t_{1}) n_{2} (t_{2}) 〉 = 〈 b_{1}^{†} (t_{1}) b_{1} (t_{1}) b_{2}^{†} (t_{2}) b_{2} (t_{2}) 〉 = {| β_{1} (t_{1}) |}^{2} {| β_{2} (t_{2}) |}^{2},

because b₁(t₁) and

b_{2}^{†} (t_{2})

do commute. Hence,

〈 δ N_{1}^{2} 〉 = 〈 N_{1} 〉,

〈 δ N_{1} δ N_{2} 〉 = 0 .

The output number variance and mean are related in the same way as the corresponding input quantities [Eq. (11)], and the output number fluctuations are uncorrelated. Results (35), (36), (43) and (44) reflect the facts that beam splitters and FCs transform pairs of input CS into pairs of different output CS [7, 19]. They do not add excess noise or correlate the (quadrature or number) fluctuations. It is shown in App. A that the absence of correlations is a consequence of the MRW relations.

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

β_{1}^{*} (t_{1}) β_{1} (t_{2}) \int [g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) + g_{12} (t_{1}, t^{'}) g_{12}^{*} (t_{2}, t^{'})] {d t}^{'},

β_{1}^{*} (t_{1}) β_{2} (t_{2}) \int [g_{11} (t_{1}, t^{'}) g_{21}^{*} (t_{2}, t^{'}) + g_{12} (t_{1}, t^{'}) g_{22}^{*} (t_{2}, t^{'})] {d t}^{'}

in Eqs. (41) and (42), respectively. However, by using Eqs. (123) and (124), we would have obtained the stated results,

β_{1}^{*} (t_{1}) β_{1} (t_{2}) δ (t_{1} - t_{2}) / 2

and 0, respectively. Notice that Eqs. (45) and (46) are just Eqs. (37) and (38), with the LO amplitudes β_p(t) and β_q(t) replaced by the sideband amplitudes β₁(t) and β₂(t), respectively, and the factors of 1/2 omitted.

4. Parametric amplification

PA [45, 46] and PG [47–52] 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 (2π_p1 → π_s1 +π_s2), whereas inverse MI is the degenerate process in which two different pump photons are destroyed and two identical sideband (signal) photons are created (π_p1 + π_p2 → 2π_s1). Phase conjugation (PC) is the nondegenerate process in which two different pump photons are destroyed and two different sideband photons are created (π_p1 + π_p2 → π_s1 + π_s2). These processes are illustrated in Fig. 3. The photon reactions used to describe them are called the MRW relations [43, 44]. Inverse MI is always phase-sensitive [53–55]. In their standard configurations (nonzero input signal and zero input idler), MI and PC are phase-insensitive [1,2], whereas in their alternative configurations (nonzero input signal and idler), MI and PC are phase-sensitive [56–58]. The noise properties of PA depend on whether it is phase-sensitive or -insensitive [16, 17, 19–22].

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.

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The IO equations for two-mode PA can be written in the form

b_{1} (t) = \int [g_{11} (t, t^{'}) a_{1} (t^{'}) + g_{12} (t, t^{'}) a_{2}^{†} (t^{'})] {d t}^{'},

b_{2} (t) = \int [g_{22} (t, t^{'}) a_{2} (t^{'}) + g_{21} (t, t^{'}) a_{1}^{†} (t^{'})] {d t}^{'},

where g_jj is a Green function that couples b_j to a_j (like operators), and g_jk is a Green function that couples b_j to

a_{k}^{†}

(unlike operators) [19]. 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.

If the inputs are CS, then the first-order output signal moment

〈 q_{1} (t) 〉 = [β_{p}^{*} (t) β_{1} (t) + β_{p} (t) β_{1}^{*} (t)] / 2^{1 / 2},

where the mean output signal amplitude β₁(t) is related to the mean input amplitudes α_j(t) by an equation similar to Eq. (47). Hence, if one were to detect the output signal, one would find that

〈 Q_{1} 〉 = \int [β_{p}^{*} (t) β_{1} (t) + β_{p} (t) β_{1}^{*} (t)] d t / 2^{1 / 2} .

One maximizes the detected signal quadrature by matching the LO pulse to the output signal pulse. Because Eqs. (47) and Eq. (48) depend symmetrically on the labels 1 and 2, the formulas for the corresponding idler quantities need not be stated explicitly.

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

\begin{array}{l} 〈 q_{1} (t_{1}) q_{1} (t_{2}) 〉 & = & 〈 {[β_{p}^{*} \int (g_{11} a_{1} + g_{12} a_{2}^{†}) {d t}^{'} + β_{p} \int (g_{11}^{*} a_{1}^{†} + g_{12}^{*} a_{2}) {d t}^{'}]}_{t_{1}} \\ \times {[β_{p}^{*} \int (g_{11} a_{1} + g_{12} a_{2}^{†}) {d t}^{″} + β_{p} \int (g_{11}^{*} a_{1}^{†} + g_{12}^{*} a_{2}) {d t}^{″}]}_{t_{2}} 〉 / 2, \end{array}

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

a_{1} a_{1}^{†}

and

a_{2} a_{2}^{†}

. By retaining only these terms, one finds that

\begin{array}{l} 〈 δ q_{1} (t_{1}) δ q_{1} (t_{2}) 〉 & = & β_{p}^{*} (t_{1}) β_{p} (t_{2}) \int g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) {d t}^{'} / 2 \\ + β_{p} (t_{1}) β_{p}^{*} (t_{2}) \int g_{12}^{*} (t_{1}, t^{'}) g_{12} (t_{2}, t^{'}) {d t}^{'} / 2 . \end{array}

By using Eq. (141), one can rewrite Eq. (52) in the alternative form

\begin{array}{l} 〈 δ q_{1} (t_{1}) δ q_{1} (t_{2}) 〉 & = & β_{p}^{*} (t_{1}) β_{q} (t_{2}) δ (t_{1} - t_{2}) / 2 \\ + β_{p}^{*} (t_{1}) β_{p} (t_{2}) \int g_{12} (t_{1}, t^{'}) g_{12}^{*} (t_{2}, t^{'}) {d t}^{'} / 2 \\ + β_{p} (t_{1}) β_{p}^{*} (t_{2}) \int g_{12}^{*} (t_{1}, t^{'}) g_{12} (t_{2}, t^{'}) {d t}^{'} / 2, \end{array}

which is manifestly real.

The mixed second-order moment can be written in the abbreviated form

\begin{array}{l} 〈 q_{1} (t_{1}) q_{2} (t_{2}) 〉 & = & 〈 {[β_{p}^{*} \int (g_{11} a_{1} + g_{12} a_{2}^{†}) {d t}^{'} + β_{p} \int (g_{11}^{*} a_{1}^{†} + g_{12}^{*} a_{2}) {d t}^{'}]}_{t_{1}} \\ \times {[β_{q}^{*} \int (g_{22} a_{2} + g_{21} a_{1}^{†}) {d t}^{″} + β_{q} \int (g_{22}^{*} a_{2}^{†} + g_{21}^{*} a_{1}) {d t}^{″}]}_{t_{2}} 〉 / 2 . \end{array}

The anti-normally ordered terms in Eq. (54) are also proportional to

a_{1} a_{1}^{†}

and

a_{2} a_{2}^{†}

. By retaining only these terms, one finds that

\begin{array}{l} 〈 δ q_{1} (t_{1}) δ q_{2} (t_{2}) 〉 & = & β_{p}^{*} (t_{1}) β_{q}^{*} (t_{2}) \int g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) {d t}^{'} / 2 \\ + β_{p} (t_{1}) β_{q} (t_{2}) \int g_{12}^{*} (t_{1}, t^{'}) g_{22}^{*} (t_{2}, t^{'}) {d t}^{'} / 2 . \end{array}

By using Eq. (142), one can rewrite Eq. (55) in the alternative form

\begin{array}{l} 〈 δ q_{1} (t_{1}) δ q_{2} (t_{2}) 〉 & = & β_{p}^{*} (t_{1}) β_{q}^{*} (t_{2}) \int g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) {d t}^{'} / 2 \\ + β_{p} (t_{1}) β_{q} (t_{2}) \int g_{11}^{*} (t_{1}, t^{'}) g_{21}^{*} (t_{2}, t^{'}) {d t}^{'} / 2, \end{array}

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

\begin{array}{l} 〈 δ Q_{1}^{2} 〉 & = & \int [{| \int β_{p}^{*} (t) g_{11} (t, t^{'}) d t |}^{2} + {| \int β_{p}^{*} (t) g_{12} (t, t^{'}) d t |}^{2}] {d t}^{'} / 2 \\ = & 1 / 2 + \int {| \int β_{p}^{*} (t) g_{12} (t, t^{'}) d t |}^{2} {d t}^{'}, \end{array}

〈 δ Q_{1} δ Q_{2} 〉 = \int Re [\int β_{p}^{*} (t_{1}) g_{11} (t_{1}, t^{'}) d t_{1} \int β_{q}^{*} (t_{2}) g_{21} (t_{2}, t^{'}) d t_{2}] {d t}^{'},

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 (

〈 δ Q_{j}^{2} 〉 > 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]: 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

\begin{array}{l} 〈 n_{1} (t) 〉 & = & 〈 \int [g_{11}^{*} (t, t^{'}) a_{1}^{†} (t^{'}) + g_{12}^{*} (t, t^{'}) a_{2} (t^{'})] {d t}^{'} \\ \times \int [g_{11} (t, t^{″}) a_{1} (t^{″}) + g_{12} (t, t^{″}) a_{2}^{†} (t^{″})] {d t}^{″} 〉 \\ = & {| β_{1} (t) |}^{2} + \int {| g_{12} (t, t^{'}) |}^{2} {d t}^{'} . \end{array}

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

〈 N_{1} 〉 = \int {| β_{1} (t) |}^{2} d t + \iint {| g_{12} (t, t^{'}) |}^{2} {d t}^{'} d t .

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 higher-order moments can be calculated directly, but it is tedious to do so, even for the simple two-mode process under consideration. To facilitate these calculations, one rewrites the input amplitude operators a_j(t) as the sums of the mean amplitudes α_j(t) = 〈a_j(t)〉 and the deviation (fluctuation) operators v_j(t) = a_j(t) − 〈a_j(t)〉. Notice that the input fluctuation operators obey the same commutation relations as the input amplitude operators [Eqs. (1)]. To evaluate the moments of v_j, one uses the VS limits of Eqs. (7). Similarly, one rewrites the output operators b_j(t) as sums of the mean amplitudes β_j(t) and the fluctuation operators w_j(t), which obey the aforementioned commutation relations. The output amplitudes and operators are related to the input amplitudes and operators by equations similar to Eqs. (47) and (48). In this notation, the number-flux operator

n_{1} (t) = {| β_{1} (t) |}^{2} + β_{1}^{*} (t) w_{1} (t) + β_{1} (t) w_{1}^{†} (t) + w_{1}^{†} (t) w_{1} (t) .

By combining Eq. (61) with the aforementioned IO equations, one can show that Eq. (61) is consistent with Eqs. (59) and (60). Because 〈

0_{j} | v_{j}^{†} (t) = 0

and v_j(t)|0_j〉 = 0, one can simplify the calculations further by defining the left and right number-flux operators

\begin{array}{l} n_{1 l} (t) & = & {| β_{1} (t) |}^{2} + β_{1}^{*} (t) \int g_{11} (t, t^{'}) v_{1} (t^{'}) {d t}^{'} + β_{1} (t) \int g_{12}^{*} (t, t^{'}) v_{2} (t^{'}) {d t}^{'} \\ + \int g_{12}^{*} (t, t^{'}) v_{2} (t^{'}) {d t}^{'} \int [g_{11} (t, t^{″}) v_{1} (t^{″}) + g_{12} (t, t^{″}) v_{2}^{†} (t^{″})] {d t}^{″}, \end{array}

\begin{array}{l} n_{1 r} (t) & = & {| β_{1} (t) |}^{2} + β_{1}^{*} (t) \int g_{12} (t, t^{'}) v_{2}^{†} (t^{'}) {d t}^{'} + β_{1} (t) \int g_{11}^{*} (t, t^{'}) v_{1}^{†} (t^{'}) {d t}^{'} \\ + \int [g_{11}^{*} (t, t^{'}) v_{1}^{†} (t^{'}) + g_{12}^{*} (t, t^{'}) v_{2} (t^{'})] {d t}^{'} \int g_{12} (t, t^{″}) v_{2}^{†} (t^{″}) {d t}^{″}, \end{array}

which act only on bra- and ket-vectors, respectively.

The fourth-order moment can be written in the abbreviated form

\begin{array}{l} 〈 n_{1 l} (t_{1}) n_{1 r} (t_{2}) 〉 & = & 〈 {[{| β_{1} |}^{2} + β_{1}^{*} g_{11} v_{1} + β_{1} g_{12}^{*} v_{2} + g_{12}^{*} v_{2} (g_{11} v_{1} + g_{12} v_{2}^{†})]}_{t_{1}} \\ \times {[{| β_{1} |}^{2} + β_{1}^{*} g_{12} v_{2}^{†} + β_{1} g_{11}^{*} v_{1}^{†} + (g_{11}^{*} v_{1}^{†} + g_{12}^{*} v_{2}) g_{12} v_{2}^{†}]}_{t_{2}} 〉, \end{array}

in which all four time integrals are implicit. By using Eqs. (198) and (199), noting that the first-and third-order moments of v_j are all zero, and restoring the time integrals, one finds that

\begin{array}{l} 〈 n_{1} (t_{1}) n_{1} (t_{2}) 〉 & = & {| β_{1} (t_{1}) β_{1} (t_{2}) |}^{2} + {| β_{1} (t_{1}) |}^{2} \int {| g_{12} (t_{2}, t^{'}) |}^{2} {d t}^{'} + {| β_{1} (t_{2}) |}^{2} \int {| g_{12} (t_{1}, t^{'}) |}^{2} {d t}^{'} \\ + β_{1}^{*} (t_{1}) β_{1} (t_{2}) \int g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) {d t}^{'} \\ + β_{1} (t_{1}) β_{1}^{*} (t_{2}) \int g_{12}^{*} (t_{1}, t^{'}) g_{12} (t_{2}, t^{'}) {d t}^{'} \\ + \int g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) {d t}^{'} \int g_{12}^{*} (t_{1}, t^{″}) g_{12} (t_{2}, t^{″}) {d t}^{″} \\ + \int {| g_{12} (t_{1}, t^{'}) |}^{2} {d t}^{'} \int {| g_{12} (t_{2}, t^{″}) |}^{2} {d t}^{″} . \end{array}

It follows from Eqs. (59) and (65) that

\begin{array}{l} 〈 δ n_{1} (t_{1}) δ n_{1} (t_{2}) 〉 & = & β_{1}^{*} (t_{1}) β_{1} (t_{2}) \int g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) {d t}^{'} \\ + β_{1} (t_{1}) β_{1}^{*} (t_{2}) \int g_{12}^{*} (t_{1}, t^{'}) g_{12} (t_{2}, t^{'}) {d t}^{'} \\ + \int g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) {d t}^{'} \int g_{12}^{*} (t_{1}, t^{″}) g_{12} (t_{2}, t^{″}) {d t}^{″} . \end{array}

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

\begin{array}{l} 〈 δ n_{1} (t_{1}) δ n_{1} (t_{2}) 〉 & = & 2 Re [β_{1}^{*} (t_{1}) β_{1} (t_{2}) \int g_{12} (t_{1}, t^{'}) g_{12}^{*} (t_{2}, t^{'}) {d t}^{'}] + β_{1}^{*} (t_{1}) β_{1} (t_{2}) δ (t_{1} - t_{2}) \\ + {| \int g_{12} (t_{1}, t^{'}) g_{12}^{*} (t_{2}, t^{'}) {d t}^{'} |}^{2} + \int g_{12}^{*} (t_{1}, t^{″}) g_{12} (t_{2}, t^{″}) {d t}^{″} δ (t_{1} - t_{2}), \end{array}

which is manifestly real.

The mixed fourth-order moment can be written in the abbreviated form

\begin{array}{l} 〈 n_{1 l} (t_{1}) n_{2 r} (t_{2}) 〉 & = & 〈 {[{| β_{1} |}^{2} + β_{1}^{*} g_{11} v_{1} + β_{1} g_{12}^{*} v_{2} + g_{12}^{*} v_{2} (g_{11} v_{1} + g_{12} v_{2}^{†})]}_{t_{1}} \\ \times {[{| β_{2} |}^{2} + β_{2}^{*} g_{21} v_{1}^{†} + β_{2} g_{22}^{*} v_{2}^{†} + (g_{22}^{*} v_{2}^{†} + g_{21}^{*} v_{1}) g_{21} v_{1}^{†}]}_{t_{2}} 〉 . \end{array}

Once again, the third-order moments of v_j are all zero, so the moment

\begin{array}{l} 〈 n_{1} (t_{1}) n_{2} (t_{2}) 〉 & = & {| β_{1} (t_{1}) β_{2} (t_{2}) |}^{2} + {| β_{1} (t_{1}) |}^{2} \int {| g_{21} (t_{2}, t^{'}) |}^{2} {d t}^{'} + {| β_{2} (t_{2}) |}^{2} \int {| g_{12} (t_{1}, t^{'}) |}^{2} {d t}^{'} \\ + β_{1}^{*} (t_{1}) β_{2}^{*} (t_{2}) \int g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) {d t}^{'} \\ + β_{1} (t_{1}) β_{2} (t_{2}) \int g_{12}^{*} (t_{1}, t^{'}) g_{22}^{*} (t_{2}, t^{'}) {d t}^{'} \\ + \int g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) {d t}^{'} \int g_{12}^{*} (t_{1}, t^{″}) g_{22}^{*} (t_{2}, t^{″}) {d t}^{″} \\ + \int {| g_{12} (t_{1}, t^{'}) |}^{2} {d t}^{'} \int {| g_{21} (t_{2}, t^{″}) |}^{2} {d t}^{″} . \end{array}

It follows from Eqs. (59) and (69) that

\begin{array}{l} 〈 δ n_{1} (t_{1}) δ n_{2} (t_{2}) 〉 & = & β_{1}^{*} (t_{2}) β_{2}^{*} (t_{2}) \int g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) {d t}^{'} \\ + β_{1} (t_{1}) β_{2} (t_{2}) \int g_{12}^{*} (t_{1}, t^{'}) g_{22}^{*} (t_{2}, t^{'}) {d t}^{'} \\ + \int g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) {d t}^{'} \int g_{12}^{*} (t_{1}, t^{″}) g_{22}^{*} (t_{2}, t^{″}) {d t}^{″} . \end{array}

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

\begin{array}{l} 〈 δ n_{1} (t_{1}) δ n_{2} (t_{2}) 〉 & = & 2 Re [β_{1}^{*} (t_{1}) β_{2}^{*} (t_{2}) \int g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) {d t}^{'}] \\ + {| \int g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) {d t}^{'} |}^{2}, \end{array}

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

\begin{array}{l} 〈 δ N_{1}^{2} 〉 & = & \int [{| \int β_{1}^{*} (t) g_{11} (t, t^{'}) d t |}^{2} + {| \int β_{1}^{*} (t) g_{12} (t, t^{'}) d t |}^{2}] {d t}^{'} \\ + \iint {| \int g_{11} (t, t^{'}) g_{12}^{*} (t, t^{″}) d t |}^{2} {d t}^{'} {d t}^{″} \\ = & \int {| β_{1} (t) |}^{2} d t + 2 \int {| \int β_{1}^{*} (t) g_{12} (t, t^{'}) d t |}^{2} {d t}^{'} \\ + \iint {| \int g_{12} (t, t^{'}) g_{12}^{*} (t, t^{″}) d t |}^{2} {d t}^{'} {d t}^{″} + \iint {| g_{12} (t, t^{'}) |}^{2} d t {d t}^{'}, \end{array}

\begin{array}{l} 〈 δ N_{1} δ N_{2} 〉 & = & 2 \int Re [\int β_{1}^{*} (t_{1}) g_{11} (t_{1}, t^{'}) d t_{1} \int β_{2}^{*} (t_{2}) g_{21} (t_{2}, t^{'}) d t_{2}] {d t}^{'} \\ + \iint {| \int g_{11} (t, t^{'}) g_{12}^{*} (t, t^{″}) d t |}^{2} {d t}^{'} {d t}^{″}, \end{array}

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 (

〈 δ N_{j}^{2} 〉

> 〈N_j〉) 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) = \int [g_{s} (t, t^{'}) a (t^{'}) + g_{c} (t, t^{'}) a^{†} (t^{'})] {d t}^{'},

where the subscripts s and c denote coupling to similar and conjugate operators, respectively. One can obtain Eq. (74) from Eq. (47) or (48) by making the substitutions a_j → a, b_j → b, g_jj → g_s and g_jk → g_c. 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

\begin{array}{l} 〈 δ q (t_{1}) δ q (t_{2}) 〉 & = & \int [β_{p}^{*} (t_{1}) g_{s} (t_{1}, t^{'}) + β_{p} (t_{1}) g_{c}^{*} (t_{1}, t^{'})] \\ \times [β_{p} (t_{2}) g_{s}^{*} (t_{2}, t^{'}) + β_{p}^{*} (t_{2}) g_{c} (t_{2}, t^{'})] {d t}^{'} . \end{array}

Equation (75) contains two more terms than Eq. (52), because a₁ commutes with

a_{2}^{†}

, 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

〈 δ Q^{2} 〉 = \int {| \int [β_{p}^{*} (t) g_{s} (t, t^{'}) + β_{p} (t) g_{c}^{*} (t, t^{'})] d t |}^{2} {d t}^{'} / 2,

which is manifestly real and non-negative, but can be greater or less than 1/2 [15–17].

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

\begin{array}{l} 〈 δ n (t_{1}) δ n (t_{2}) 〉 & = & \int [β^{*} (t_{1}) g_{s} (t_{1}, t^{'}) + β (t_{1}) g_{c}^{*} (t_{1}, t^{'})] \\ \times [β (t_{2}) g_{s} (t_{2}, t^{'}) + β^{*} (t_{2}) g_{c} (t_{2}, t^{'})] {d t}^{'} \\ + \int g_{s} (t_{1}, t^{'}) g_{s}^{*} (t_{2}, t^{'}) {d t}^{'} \int g_{c}^{*} (t_{1}, t^{″}) g_{c} (t_{2}, t^{″}) {d t}^{″} \\ + \int g_{s} (t_{1}, t^{'}) g_{c} (t_{2}, t^{'}) {d t}^{'} \int g_{c}^{*} (t_{1}, t^{″}) g_{s}^{*} (t_{2}, t^{″}) {d t}^{″} . \end{array}

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

\begin{array}{l} 〈 δ N^{2} 〉 & = & \int {| \int [β^{*} (t) g_{s} (t, t^{'}) + β (t) g_{c}^{*} (t, t^{'})] d t |}^{2} {d t}^{'} \\ + 2 \iint {| \int g_{s} (t, t^{'}) g_{c}^{*} (t, t^{″}) d t |}^{2} {d t}^{'} {d t}^{″}, \end{array}

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.

Attenuation is modeled as beam splitting (Fig. 1) and FC is enabled by BS (Fig. 2). Each of the Green functions in Eqs. (29) and (30) has the Schmidt decomposition

g (t, t^{'}) = \sum_{j} v_{j} (t) σ_{j} u_{j}^{*} (t^{'}),

where u_j(t′) and v_j(t) are input and output Schmidt modes, respectively, and σ_j is a Schmidt coefficient [23, 25]. (In this section only, v denotes an output Schmidt mode rather than an input fluctuation operator.) The input and output modes are temporal eigenfunctions of integral operators with kernels

κ (t, t^{'}) = \int g^{*} (t^{″}, t) g (t^{″}, t^{'}) {d t}^{″},

λ (t, t^{'}) = \int g (t, t^{″}) g^{*} (t^{'}, t^{″}) {d t}^{″},

respectively, and the coefficients are square roots of the (common) eigenvalues of these kernels. The input modes satisfy the orthonormality and completeness conditions

\int u_{j}^{*} (t) u_{k} (t) d t = δ_{j k}, \sum_{j} u_{j} (t_{1}) u_{j}^{*} (t_{2}) = δ (t_{1} - t_{2}),

respectively, as do the output modes. Furthermore, because FC is a unitary process, the Green functions and their decompositions are related by the matrix equation

[\begin{array}{l} g_{11} (t, t^{'}) & g_{12} (t, t^{'}) \\ g_{21} (t, t^{'}) & g_{22} (t, t^{'}) \end{array}] = \sum_{j} [\begin{array}{l} v_{1 j} (t) τ_{j} u_{1 j}^{*} (t^{'}) & v_{1 j} (t) ρ_{j} u_{2 j}^{*} (t^{'}) \\ - v_{2 j} (t) ρ_{j}^{*} u_{1 j}^{*} (t^{'}) & v_{2 j} (t) τ_{j}^{*} u_{2 j}^{*} (t^{'}) \end{array}],

where each pair of Schmidt (transmission and conversion) coefficients satisfies the auxiliary equation |τ_j|² +|ρ_j|² = 1 [31–34]. (Notice that the phases of these coefficients can be absorbed into the modes, leaving the coefficients real and non-negative. We retained complex notation for the coefficients, because that is how they appear in most applications.) For the common case in which the transmission and reflection coefficients in the beam-splitter model of absorption are time-independent, the output modes equal the input modes, which are orthonormal, but in other respects are arbitrary.

Each continuous input mode a_i(t) has the decomposition

a_{i} (t) = \sum_{j} a_{i j} u_{i j} (t), a_{i j} = \int u_{i j}^{*} (t) a_{i} (t) d t .

It follows from Eqs. (1) and (84) that the discrete input modes satisfy the commutation relations

[a_{i k}, a_{j l}] = 0, [a_{i k}, a_{j l}^{†}] = δ_{i j} δ_{k l} .

Each continuous output mode b_i(t) has a similar decomposition. By substituting these decompositions into Eqs. (29) and (30), one obtains the discrete-mode IO equations

b_{1 j} = τ_{j} a_{1 j} + ρ_{j} a_{2 j},

b_{2 j} = - ρ_{j}^{*} a_{1 j} + τ_{j}^{*} a_{2 j} .

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]. By starting with the discrete-mode equations, one can show easily that

〈 q_{1 j} 〉 = ({\bar{β}}_{p j}^{*} β_{1 j} + {\bar{β}}_{p j} β_{1 j}^{*}) / 2^{1 / 2},

〈 δ q_{1 j}^{2} 〉 = {| {\bar{β}}_{p j} |}^{2} / 2,

〈 δ q_{1 j} δ q_{2 j} 〉 = 0,

where β̄_pj = e^iϕ_p 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

〈 n_{1 j} 〉 = {| β_{1 j} |}^{2},

〈 δ n_{1 j}^{2} 〉 = {| β_{1 j} |}^{2},

〈 δ n_{1 j} δ n_{2 j} 〉 = 0 .

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]. Notice that one can deduce the number results from the associated quadrature results by replacing β̄_pj with β₁_j and multiplying by powers of 2^1/2. 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, 19, 21]. 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, 59, 60]. 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

〈 q_{1} (t) 〉 = \sum_{j} \sum_{k} [β_{p j}^{*} v_{1 j}^{*} (t) β_{1 k} v_{1 k} (t) + β_{p j} v_{1 j} (t) β_{1 k}^{*} v_{1 k}^{*} (t)] / 2^{1 / 2} .

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

〈 Q_{1} 〉 = \sum_{j} (β_{p j}^{*} β_{1 j} + β_{p j} β_{1 j}^{*}) / 2^{1 / 2},

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 β_p(t) ∝ v_j(t), in which case |β_pj| = 1 and β_pk = 0 for k ≠ j. It follows from Eq. (33) and the second of Eqs. (82) that the quadrature-flux moment

〈 δ q_{1} (t_{1}) δ q_{1} (t_{2}) 〉 = \sum_{i} \sum_{j} \sum_{k} β_{p l}^{*} v_{1 i}^{*} (t_{1}) v_{1 j} (t_{1}) v_{1 j}^{*} (t_{2}) β_{p k} v_{1 k} (t_{2}) / 2 .

One could obtain the same result by using Eq. (29) and the first of Eqs. (84) to reorder the

b_{1} (t_{1}) b_{1}^{†} (t_{2})

term in Eq. (33) directly. By integrating Eq. (96) with respect to t₁ and t₂, one finds that the quadrature variance

〈 δ Q_{1}^{2} 〉 = \sum_{j} {| β_{p j} |}^{2} / 2 = 1 / 2,

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.

Two-mode PA is enabled by MI and PC (Fig. 3). The Green functions in Eqs. (47) and (48) also have Schmidt decompositions of the form (79), which are are related by the matrix equation

[\begin{array}{l} g_{11} (t, t^{'}) & g_{12} (t, t^{'}) \\ g_{21} (t, t^{'}) & g_{22} (t, t^{'}) \end{array}] = \sum_{j} [\begin{array}{l} v_{1 j} (t) μ_{j} u_{1 j}^{*} (t^{'}) & v_{1 j} (t) ν_{j} u_{2 j} (t^{'}) \\ v_{2 j} (t) ν_{j} u_{1 j} (t^{'}) & v_{2 j} (t) μ_{j} u_{2 j}^{*} (t^{'}) \end{array}],

where each pair of Schmidt coefficients satisfies the auxiliary equation |μ_j|² − |ν_j|² = 1. [24, 26]. (One can assume that these coefficients are real whenever it is convenient to do so.) By substituting decompositions of the form (84) into Eqs. (47) and (48), one obtains the discrete-mode IO equations

b_{1 j} = μ_{j} a_{1 j} + ν_{j} a_{2 j}^{†},

b_{2 j} = μ_{j} a_{2 j} + ν_{j} a_{1 j}^{†} .

Equations (99) and (100) show that each discrete signal mode interacts with one discrete idler mode (and no other signal modes). They are the IO equations for two-discrete-mode squeezing (stretching) [7]. By starting with these discrete-mode equations, one can show easily that

〈 q_{1 j} 〉 = ({\bar{β}}_{p j}^{*} β_{1 j} + {\bar{β}}_{p j} β_{1 j}^{*}) / 2^{1 / 2},

〈 δ q_{1 j}^{2} 〉 = {| {\bar{β}}_{p j} |}^{2} ({| μ_{j} |}^{2} + {| ν_{j} |}^{2}) / 2,

〈 δ q_{1 j} δ q_{2 j} 〉 = ({\bar{β}}_{p j}^{*} {\bar{β}}_{q j}^{*} μ_{j} ν_{j} + {\bar{β}}_{p j} {\bar{β}}_{q j} μ_{j}^{*} ν_{j}^{*}) / 2 .

Each discrete mode has a quadrature variance of (|μ_j|² + |ν_j|²)/2 = 1/2 + |ν_j|²: Coupling to the idler increases the quadrature variance of the signal (and vice versa). Furthermore, the signal and idler fluctuations are strongly correlated. It is also easy to show that

〈 n_{1 j} 〉 = {| β_{1 j} |}^{2} + {| ν_{j} |}^{2},

〈 δ n_{1 j}^{2} 〉 = {| β_{1 j} |}^{2} ({| μ_{j} |}^{2} + {| ν_{j} |}^{2}) + {| μ_{j} ν_{j} |}^{2},

〈 δ n_{1 j} δ n_{2 j} 〉 = β_{1 j}^{*} β_{2 j}^{*} μ_{j} ν_{j} + β_{1 j} β_{2 j} μ_{j}^{*} ν_{j}^{*} + {| μ_{j} ν_{j} |}^{2} .

Coupling to the idler adds noise photons to the signal (and vice versa). The output number variance is larger than the input variance, and the number fluctuations of the signal and idler are correlated, because sideband photons are produced in pairs. Notice that one can deduce the signal and signal–noise contributions to the number results from the corresponding quadrature results by replacing β̄_pj with β_1j and multiplying by powers of 2^1/2. In retrospect, the continuous-mode results, specifically Eqs. (50), (57) and (58), and Eqs. (60), (72) and (73), are nontrivial, but straightforward, generalizations of Eqs. (101)–(106), respectively. Once again, by decomposing the input and output modes in the manner of Eq. (84), one converts a two-continous-mode process into an infinite family of two-discrete-mode processes, the properties of which are known, in both the Heisenberg [7, 19–22] and Schrödinger [7, 61, 62] pictures.

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

\begin{array}{l} 〈 δ q_{1} (t_{1}) δ q_{1} (t_{2}) 〉 & = & \sum_{i} \sum_{j} \sum_{k} [β_{p i}^{*} v_{1 i}^{*} (t_{1}) v_{1 j} (t_{1}) {| μ_{j} |}^{2} v_{1 j}^{*} (t_{2}) β_{p k} v_{1 k} (t_{2}) \\ + β_{p i} v_{1 i} (t_{1}) v_{1 j}^{*} (t_{1}) {| ν_{j} |}^{2} v_{1 j} (t_{2}) β_{p k}^{*} v_{1 k}^{*} (t_{2})] / 2, \end{array}

\begin{array}{l} 〈 δ q_{1} (t_{1}) δ q_{2} (t_{2}) 〉 & = & \sum_{i} \sum_{j} \sum_{k} [β_{p i}^{*} v_{1 i}^{*} (t_{1}) v_{1 j} (t_{1}) μ_{j} ν_{j} v_{2 j} (t_{2}) β_{q k}^{*} v_{2 k}^{*} (t_{2}) \\ + β_{p i} v_{1 i} (t_{1}) v_{1 j}^{*} (t_{1}) μ_{j}^{*} ν_{j}^{*} v_{2 j}^{*} (t_{2}) β_{q k} v_{2 k} (t_{2})] / 2 . \end{array}

By integrating Eqs. (107) and (108) with respect to t₁ and t₂, one finds that

〈 δ Q_{1}^{2} 〉 = \sum_{j} {| β_{p j} |}^{2} ({| μ_{j} |}^{2} + {| ν_{j} |}^{2}) / 2,

〈 δ Q_{1} δ Q_{2} 〉 = \sum_{j} (β_{p j}^{*} β_{q j}^{*} μ_{j} ν_{j} + β_{p j} β_{q j} μ_{j}^{*} ν_{j}^{*}) / 2 .

The quadrature variance and correlation are weighted sums of discrete-mode contributions [Eqs. (102) and (103)].

The signal number flux and mean are

〈 n_{1} (t) 〉 = \sum_{j} \sum_{k} β_{1 j}^{*} v_{1 j}^{*} (t) β_{1 k} v_{1 k} (t) + \sum_{j} {| ν_{j} |}^{2} {| v_{1 j} (t) |}^{2},

〈 N_{1} 〉 = \sum_{j} ({| β_{1 j} |}^{2} + {| ν_{j} |}^{2}),

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

{〈 δ n_{1} (t_{1}) δ n_{1} (t_{2}) 〉}_{nn} = [\sum_{j} v_{1 j} (t_{1}) {| μ_{j} |}^{2} v_{1 j}^{*} (t_{2})] [\sum_{k} v_{1 k}^{*} (t_{1}) {| ν_{k} |}^{2} v_{1 k} (t_{2})],

{〈 δ n_{1} (t_{1}) δ n_{2} (t_{2}) 〉}_{nn} = [\sum_{j} v_{1 j} (t_{1}) μ_{j} ν_{j} v_{2 j} (t_{2})] [\sum_{k} v_{1 k}^{*} (t_{1}) μ_{k}^{*} ν_{k}^{*} v_{2 k}^{*} (t_{2})] .

By integrating Eqs. (113) and (114) with respect to time, one finds that

{〈 δ N_{1}^{2} 〉}_{nn} = \sum_{j} {| μ_{j} ν_{j} |}^{2} = {〈 δ N_{1} δ N_{2} 〉}_{nn} .

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 [30,47–52]. 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

\begin{array}{l} {〈 n_{1} (t_{1}) n_{2} (t_{2}) 〉}_{nn} & = & [\sum_{j} μ_{j} ν_{j} v_{1 j} (t_{1}) v_{2 j} (t_{2})] [\sum_{k} μ_{k}^{*} ν_{k}^{*} v_{1 k}^{*} (t_{1}) v_{2 k}^{*} (t_{2})] \\ + [\sum_{j} {| ν_{j} |}^{2} {| v_{1 j} (t_{1}) |}^{2}] [\sum_{k} {| ν_{k} |}^{2} {| v_{2 k} (t_{2}) |}^{2}] . \end{array}

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]. This condition is satisfied if ν_j = 0 for j > 1, in which case

{〈 n_{1} (t_{1}) n_{2} (t_{2}) 〉}_{nn} = ({| μ_{1} |}^{2} + {| ν_{1} |}^{2}) {| ν_{1} |}^{2} {| v_{11} (t_{1}) v_{21} (t_{2}) |}^{2} .

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]. This result is a straightforward generalization of the Grice separability condition [28], 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, 64].

g_{s} (t, t^{'}) = \sum_{j} v_{j} (t) μ_{j} u_{j}^{*} (t^{'}), g_{c} (t, t^{'}) = \sum_{j} v_{j} (t) ν_{j} u_{j} (t^{'}),

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

\begin{array}{l} 〈 δ q (t_{1}) δ q (t_{2}) 〉 & = & \sum_{i} \sum_{j} \sum_{k} [β_{p i}^{*} v_{i}^{*} (t_{1}) μ_{j} v_{j} (t_{1}) + β_{p i} v_{i} (t_{1}) ν_{j}^{*} v_{j}^{*} (t_{1})] \\ \times [μ_{j}^{*} v_{j}^{*} (t_{2}) β_{p k} v_{k} (t_{2}) + ν_{j} v_{j} (t_{2}) β_{p k}^{*} v_{k}^{*} (t_{2})] / 2, \end{array}

〈 δ Q^{2} 〉 = \sum_{j} {| β_{p j}^{*} μ_{j} + β_{p j} ν_{j}^{*} |}^{2} / 2,

respectively, and the number-flux moment and number variance are

\begin{array}{l} 〈 δ n (t_{1}) δ n (t_{2}) 〉 & = & \sum_{i} \sum_{j} \sum_{k} [β_{i}^{*} v_{i}^{*} (t_{1}) μ_{j} v_{j} (t_{1}) + β_{i} v_{i} (t_{1}) ν_{j}^{*} v_{j}^{*} (t_{1})] \\ \times [μ_{j}^{*} v_{j}^{*} (t_{2}) β_{k} v_{k} (t_{2}) + ν_{j} v_{j} (t_{2}) β_{k}^{*} v_{k}^{*} (t_{2})] \\ + \sum_{j} \sum_{k} {| μ_{j} |}^{2} v_{j} (t_{1}) v_{j}^{*} (t_{2}) {| ν_{k} |}^{2} v_{k}^{*} (t_{1}) v_{k} (t_{2}) \\ + \sum_{j} \sum_{k} μ_{j} ν_{j} v_{j} (t_{1}) v_{j} (t_{2}) μ_{k}^{*} ν_{k}^{*} v_{k}^{*} (t_{1}) v_{k}^{*} (t_{2}), \end{array}

〈 δ N^{2} 〉 = \sum_{j} {| β_{j}^{*} μ_{j} + β_{j} ν_{j}^{*} |}^{2} + 2 \sum_{j} {| μ_{j} ν_{j} |}^{2},

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, 19]. 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, 21, 22] 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], 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).

In summary, two sets of formulas were derived for the quadrature and number fluctuations produced by one- and two-continuous-mode parametric processes. The first set is based on Green functions, whereas the second is based on the associated Schmidt coefficients and modes. Related formulas for multiple-continuous-mode processes are derived in App. C. These formulas facilitate the modeling and performance optimization of parametric devices used in a wide variety of applications.

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,

[b_{1} (t_{1}), b_{1}^{†} (t_{2})] = \int [g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) + g_{12} (t_{1}, t^{'}) g_{12}^{*} (t_{2}, t^{'})] {d t}^{'} = δ (t_{1} - t_{2}),

[b_{1} (t_{1}), b_{2}^{†} (t_{2})] = \int [g_{11} (t_{1}, t^{'}) g_{21}^{*} (t_{2}, t^{'}) + g_{12} (t_{1}, t^{'}) g_{22}^{*} (t_{2}, t^{'})] {d t}^{'} = 0 .

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

a_{1} (t) = \int [h_{11} (t, t^{'}) b_{1} (t^{'}) + h_{12} (t, t^{'}) b_{2} (t^{'})] {d t}^{'},

a_{2} (t) = \int [h_{21} (t, t^{'}) b_{1} (t^{'}) + h_{22} (t, t^{'}) b_{2} (t^{'})] {d t}^{'},

where h_jk are backward (adjoint) Green functions. It follows from Eqs. (125) and (126), and the commutation relations, that

[a_{1} (t_{1}), a_{1}^{†} (t_{2})] = \int [h_{11} (t_{11}, t^{'}) h_{11}^{*} (t_{2}, t^{'}) + h_{12} (t_{1}, t^{'}) h_{12}^{*} (t_{2}, t^{'})] {d t}^{'} = δ (t_{1} - t_{2}),

[a_{1} (t_{1}), a_{2}^{†} (t_{2})] = \int [h_{11} (t_{1}, t^{'}) h_{21}^{*} (t_{2}, t^{'}) + h_{12} (t_{1}, t^{'}) h_{22}^{*} (t_{2}, t^{'})] {d t}^{'} = 0 .

By combining Eqs. (29), (125) and (126), one obtains the identity

\begin{array}{l} b_{1} (t) & = & \iint [g_{11} (t, t^{'}) h_{11} (t^{'}, t^{″}) + g_{12} (t, t^{'}) h_{21} (t^{'}, t^{″})] b_{1} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [g_{11} (t, t^{'}) h_{12} (t^{'}, t^{″}) + g_{12} (t, t^{'}) h_{22} (t^{'}, t^{″})] b_{2} (t^{″}) {d t}^{'} {d t}^{″}, \end{array}

from which follow the consistency conditions

\int [g_{11} (t, t^{'}) h_{11} (t^{'}, t^{″}) + g_{12} (t, t^{'}) h_{21} (t^{'}, t^{″})] {d t}^{'} = δ (t - t^{″}),

\int [g_{11} (t, t^{'}) h_{12} (t^{'}, t^{″}) + g_{12} (t, t^{'}) h_{22} (t^{'}, t^{″})] {d t}^{'} = 0 .

By comparing Eqs. (123) and (124) with Eqs. (130) and (131), one obtains the reciprocity relations [31]

h_{11} (t_{1}, t_{2}) = g_{11}^{*} (t_{2}, t_{1}), h_{21} (t_{1}, t_{2}) = g_{12}^{*} (t_{2}, t_{1}) .

The identity for a₁(t) produces the same conditions. By combining Eqs. (127), (128) and (132), one finds that

\int [g_{11}^{*} (t^{'}, t_{1}) g_{11} (t^{'}, t_{2}) + g_{21}^{*} (t^{'}, t_{1}) g_{21} (t^{'}, t_{2})] {d t}^{'} = δ (t_{1} - t_{2}),

\int [g_{11}^{*} (t^{'}, t_{1}) g_{12} (t^{'}, t_{2}) + g_{21}^{*} (t^{'}, t_{1}) g_{22} (t^{'}, t_{2})] {d t}^{'} = 0 .

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

\begin{array}{l} n_{1} (t) & = & \iint [g_{11}^{*} (t, t^{'}) a_{1}^{†} (t^{'}) g_{11} (t, t^{″}) a_{1} (t^{″}) + g_{11}^{*} (t, t^{'}) a_{1}^{†} (t^{'}) g_{12} (t, t^{″}) a_{2} (t^{″}) \\ + g_{12}^{*} (t, t^{'}) a_{2}^{†} (t^{'}) g_{11} (t, t^{″}) a_{1} (t^{″}) + g_{12}^{*} (t, t^{'}) a_{2}^{†} (t^{'}) g_{12} (t, t^{″}) a_{2} (t^{″})] {d t}^{'} {d t}^{″} . \end{array}

A similar formula exists for the idler number-flux density. By combining these formulas, one finds that the total-number flux

\begin{array}{l} n_{1} (t) + n_{2} (t) & = & \iint [g_{11}^{*} (t, t^{'}) g_{11} (t, t^{″}) + g_{21}^{*} (t, t^{'}) g_{21} (t, t^{″})] a_{1}^{†} (t^{'}) a_{1} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [g_{11}^{*} (t, t^{'}) g_{12} (t, t^{″}) + g_{21}^{*} (t, t^{'}) g_{22} (t, t^{″})] a_{1}^{†} (t^{'}) a_{2} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [g_{22}^{*} (t, t^{'}) g_{21} (t, t^{″}) + g_{12}^{*} (t, t^{'}) g_{11} (t, t^{″})] a_{2}^{†} (t^{'}) a_{1} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [g_{22}^{*} (t, t^{'}) g_{22} (t, t^{″}) + g_{12}^{*} (t, t^{'}) g_{12} (t, t^{″})] a_{2}^{†} (t^{'}) a_{2} (t^{″}) {d t}^{'} {d t}^{″} . \end{array}

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

N_{1} + N_{2} = M_{1} + M_{2},

where M_j and N_j 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

〈 N_{1} 〉 + 〈 N_{2} 〉 = 〈 M_{1} 〉 + 〈 M_{2} 〉,

whereas the second-moment equation, which involves the variances and correlation, is

〈 δ N_{1}^{2} 〉 + 2 〈 δ N_{1} δ N_{2} 〉 + 〈 δ N_{2}^{2} 〉 = 〈 δ M_{1}^{2} 〉 + 2 〈 δ M_{1} δ M_{2} 〉 + 〈 δ M_{2}^{2} 〉 .

If the inputs are (independent) CS,

〈 δ M_{j}^{2} 〉 = 〈 M_{j} 〉

, 〈δM₁δM₂〉 = 0 and Eq. (139) reduces to

〈 δ N_{1}^{2} 〉 + 2 〈 δ N_{1} δ N_{2} 〉 + 〈 δ N_{2}^{2} 〉 = 〈 M_{1} 〉 + 〈 M_{2} 〉 .

It was shown in Sec. 3 that

〈 δ N_{1}^{2} 〉 + 〈 δ N_{2}^{2} 〉 = 〈 N_{1} 〉 + 〈 N_{2} 〉

. 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,

[b_{1} (t_{1}), b_{1}^{†} (t_{2})] = \int [g_{11} (t_{1}, t^{'}) g_{11}^{*} (t_{2}, t^{'}) - g_{12} (t_{1}, t^{'}) g_{12}^{*} (t_{2}, t^{'})] {d t}^{'} = δ (t_{1} - t_{2}),

[b_{1} (t_{1}), b_{2} (t_{2})] = \int [g_{11} (t_{1}, t^{'}) g_{21} (t_{2}, t^{'}) - g_{12} (t_{1}, t^{'}) g_{22} (t_{2}, t^{'})] {d t}^{'} = 0 .

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

a_{1} (t) = \int [h_{11} (t, t^{'}) b_{1} (t^{'}) + h_{12} (t, t^{'}) b_{2}^{†} (t^{'})] {d t}^{'},

a_{2} (t) = \int [h_{22} (t, t^{'}) b_{2} (t^{'}) + h_{21} (t, t^{'}) b_{1}^{†} (t^{'})] {d t}^{'} .

It follows from Eqs. (143) and (144), and the commutation relations, that

[a_{1} (t_{1}), a_{1}^{†} (t_{2})] = \int [h_{11} (t_{1}, t^{'}) h_{11}^{*} (t_{2}, t^{'}) - h_{12} (t_{1}, t^{'}) h_{12}^{*} (t_{2}, t^{'})] {d t}^{'} = δ (t_{1} - t_{2}),

[a_{1} (t_{1}), a_{2} (t_{2})] = \int [h_{11} (t_{1}, t^{'}) h_{21} (t_{2}, t^{'}) - h_{12} (t_{1}, t^{'}) h_{22} (t_{2}, t^{'})] {d t}^{'} = 0 .

By combining Eqs. (47), (143) and (144), one obtains the identity

\begin{array}{l} b_{1} (t) & = & \iint [g_{11} (t, t^{'}) h_{11} (t^{'}, t^{″}) + g_{12} (t, t^{'}) h_{21}^{*} (t^{'}, t^{″})] b_{1} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [g_{11} (t, t^{'}) h_{12} (t^{'}, t^{″}) + g_{12} (t, t^{'}) h_{22}^{*} (t^{'}, t^{″})] b_{2}^{†} (t^{″}) {d t}^{'} {d t}^{″}, \end{array}

from which follow the consistency conditions

\int [g_{11} (t, t^{'}) h_{11} (t^{'}, t^{″}) + g_{12} (t, t^{'}) h_{21}^{*} (t^{'}, t^{″})] {d t}^{'} = δ (t - t^{″}),

\int [g_{11} (t, t^{'}) h_{12} (t^{'}, t^{″}) + g_{12} (t, t^{'}) h_{22}^{*} (t^{'}, t^{″})] {d t}^{'} = 0 .

By comparing Eqs. (141) and (142) with Eqs. (148) and (149), one obtains the reciprocity relations [65, 66]

h_{11} (t_{1}, t_{2}) = g_{11}^{*} (t_{2}, t_{1}), h_{21} (t_{1}, t_{2}) = - g_{12} (t_{2}, t_{1}) .

The identity for a₁(t) produces the same conditions. By combining Eqs. (145), (146) and (150), one finds that

\int [g_{11}^{*} (t^{'}, t_{1}) g_{11} (t^{'}, t_{2}) - g_{21} (t^{'}, t_{1}) g_{21}^{*} (t^{'}, t_{2})] {d t}^{'} = δ (t_{1} - t_{2}),

\int [g_{11}^{*} (t^{'}, t_{1}) g_{12} (t^{'}, t_{2}) - g_{21} (t^{'}, t_{1}) g_{22}^{*} (t^{'}, t_{2})] {d t}^{'} = 0 .

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

\begin{array}{l} n_{1} (t) & = & \iint [g_{11}^{*} (t, t^{'}) a_{1}^{†} (t^{'}) g_{11} (t, t^{″}) a_{1} (t^{″}) + g_{11}^{*} (t, t^{'}) a_{1}^{†} (t^{'}) g_{12} (t, t^{″}) a_{2}^{†} (t^{″}) \\ + g_{12}^{*} (t, t^{'}) a_{2} (t^{'}) g_{11} (t, t^{″}) a_{1} (t^{″}) + g_{12}^{*} (t, t^{'}) a_{2} (t^{'}) g_{12} (t, t^{″}) a_{2}^{†} (t^{″})] {d t}^{'} {d t}^{″}, \end{array}

\begin{array}{l} n_{2} (t) & = & \iint [g_{22}^{*} (t, t^{'}) a_{2}^{†} (t^{'}) g_{22} (t, t^{″}) a_{2} (t^{″}) + g_{22}^{*} (t, t^{'}) a_{2}^{†} (t^{'}) g_{21} (t, t^{″}) a_{1}^{†} (t^{″}) \\ + g_{21}^{*} (t, t^{'}) a_{1} (t^{'}) g_{22} (t, t^{″}) a_{2} (t^{″}) + g_{21}^{*} (t, t^{'}) a_{1} (t^{'}) g_{21} (t, t^{″}) a_{1}^{†} (t^{″})] {d t}^{'} {d t}^{″} . \end{array}

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

g_{11} a_{1} g_{12}^{*} a_{2}

and

g_{12} a_{2}^{†} g_{12}^{*} a_{2} + {| g_{12} |}^{2} δ

, respectively. Similarly, one can rewrite the second and fourth terms in Eq. (154) in the abbreviated forms

g_{21} a_{1}^{†} g_{22}^{*} a_{2}^{†}

and

g_{21} a_{1}^{†} g_{21}^{*} a_{1} + {| g_{21} |}^{2} δ

, respectively. By combining these results, one finds that the number-flux difference

\begin{array}{l} n_{1} (t) - n_{2} (t) & = & \iint [g_{11}^{*} (t, t^{'}) g_{11} (t, t^{″}) - g_{21} (t, t^{'}) g_{21}^{*} (t, t^{″})] a_{1}^{†} (t^{'}) a_{1} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [g_{11}^{*} (t, t^{'}) g_{12} (t, t^{″}) - g_{21} (t, t^{'}) g_{22}^{*} (t, t^{″})] a_{1}^{†} (t^{'}) a_{2}^{†} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [g_{11} (t, t^{'}) g_{12}^{*} (t, t^{″}) - g_{21}^{*} (t, t^{'}) g_{22} (t, t^{″})] a_{1} (t^{'}) a_{2} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [g_{12} (t, t^{'}) g_{12}^{*} (t, t^{″}) - g_{22}^{*} (t, t^{'}) g_{22} (t, t^{″})] a_{2}^{†} (t^{'}) a_{2} (t^{″}) {d t}^{'} {d t}^{″} \\ + \iint [{| g_{12} (t, t^{'}) |}^{2} - {| g_{21} (t, t^{'}) |}^{2}] {d t}^{'} . \end{array}

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

N_{1} - N_{2} = M_{1} - M_{2} + \iint [{| g_{12} (t, t^{'}) |}^{2} - {| g_{21} (t, t^{'}) |}^{2}] d t {d t}^{'} .

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

〈 N_{1} 〉 - 〈 N_{2} 〉 = 〈 M_{1} 〉 - 〈 M_{2} 〉,

whereas the second-moment equation, which involves the variances and correlation, is

〈 δ N_{1}^{2} 〉 - 2 〈 δ N_{1} δ N_{2} 〉 + 〈 δ N_{2}^{2} 〉 = 〈 δ M_{1}^{2} 〉 - 2 〈 δ M_{1} δ M_{2} 〉 + 〈 δ M_{2}^{2} 〉 .

If the inputs are (independent) CS, Eq. (158) reduces to

〈 δ N_{1}^{2} 〉 - 2 〈 δ N_{1} δ N_{2} 〉 + 〈 δ N_{2}^{2} 〉 = 〈 M_{1} 〉 + 〈 M_{2} 〉 .

In the high-gain regime (

〈 δ N_{j}^{2} 〉 > 〈 N_{j} 〉 ≫ 〈 M_{j} 〉

), 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, 44].

One-mode PA is governed by the IO equation (74). One can obtain this equation from Eq. (47) or (48) by making the substitutions a_j → a, b_j → b, g_jj → g_s and g_jk → g_c. 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 b_j(t) into its mean value 〈b_j(t)〉 = β_j(t), which depends only on the input signals, and its deviation b_j(t) − 〈b_j(t)〉 = w_j(t), which depends only on the input fluctuations. By using these definitions, one finds that the mean and deviation of each quadrature flux are

〈 q_{j} (t) 〉 = [β_{p}^{*} (t) β_{j} (t) + β_{p} (t) β_{j}^{*} (t)] / 2^{1 / 2},

δ q_{j} (t) = [β_{p}^{*} (t) w_{j} (t) + β_{p} (t) w_{j}^{†} (t)] / 2^{1 / 2},

respectively, where β_p(t) is the (normalized) amplitude of the LO used to detect mode j. To calculate quadrature correlations, one evaluates quadrature-flux moments of the form

〈 δ q_{j} (t_{1}) δ q_{k} (t_{2}) 〉 = 〈 [β_{p}^{*} (t_{1}) w_{j} (t_{1}) + β_{p} (t_{1}) w_{j}^{†} (t_{1})] [β_{q}^{*} (t_{2}) w_{k} (t_{2}) + β_{q} (t_{2}) w_{k}^{†} (t_{2})] 〉 / 2,

where β_q(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

〈 n_{j} (t) 〉 = {| β_{j} (t) |}^{2} + 〈 w_{j}^{†} (t) w_{j} (t) 〉,

δ n_{j} (t) = β_{j}^{*} (t) w_{j} (t) + β_{j} (t) w_{j}^{†} (t) + w_{j}^{†} (t) w_{j} (t) - 〈 w_{j}^{†} (t) w_{j} (t) 〉,

respectively. To calculate number correlations, one evaluates number-flux moments of the form

\begin{array}{l} 〈 δ n_{j} (t_{1}) δ n_{k} (t_{2}) 〉 & = & 〈 [β_{j}^{*} (t_{1}) w_{j} (t_{1}) + β_{j} (t_{1}) w_{j}^{†} (t_{1}) + w_{j}^{†} (t_{1}) w_{j} (t_{1}) - 〈 w_{j}^{†} (t_{1}) w_{j} (t_{1}) 〉] \\ \times 〈 [β_{k}^{*} (t_{2}) w_{k} (t_{2}) + β_{k} (t_{2}) w_{k}^{†} (t_{2}) + w_{k}^{†} (t_{2}) w_{k} (t_{2}) - 〈 w_{k}^{†} (t_{2}) w_{k} (t_{2}) 〉] 〉, \end{array}

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

\begin{array}{l} 〈 δ n_{j} (t_{1}) δ n_{k} (t_{2}) 〉 & = & 〈 [β_{j}^{*} (t_{1}) w_{j} (t_{1}) + β_{j} (t_{1}) w_{j}^{†} (t_{1})] [β_{k}^{*} (t_{2}) w_{k} (t_{2}) + β_{k} (t_{2}) w_{k}^{†} (t_{2})] 〉 \\ + 〈 w_{j}^{†} (t_{1}) w_{j} (t_{1}) w_{k}^{†} (t_{2}) w_{k} (t_{2}) 〉 - 〈 w_{j}^{†} (t_{1}) w_{j} (t_{1}) 〉 〈 w_{k}^{†} (t_{2}) w_{k} (t_{2}) 〉 . \end{array}

The right side of Eq. (166) contains both signal–noise terms, which depend on the mean signals and the fluctuations (second-order moments of w_j and w_k), 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

{〈 δ n_{j} (t_{1}) δ n_{k} (t_{2}) 〉}_{sn} = 〈 [β_{j}^{*} (t_{1}) w_{j} (t_{1}) + β_{j} (t_{1}) w_{j}^{†} (t_{1})] [β_{k}^{*} (t_{2}) w_{k} (t_{2}) + β_{k} (t_{2}) w_{k}^{†} (t_{2})] 〉 .

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]. 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].

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

b_{i} (t) = \sum_{k} \int {d t}^{'} [μ_{i k} (t, t^{'}) a_{k} (t^{'}) + ν_{i k} (t, t^{'}) a_{k}^{†} (t^{'})],

where a_k and b_i are input and output mode-amplitude operators, respectively, μ_ik is a forward transfer (Green) function that couples b_i to a_k (like operators), ν_ik is a forward Green function that couples b_i to

a_{k}^{†}

(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 b_i was coupled to a_k or

a_{k}^{†}

, 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,

[b_{i} (t_{1}), b_{j}^{†} (t_{2})] = \sum_{k} \int {d t}^{'} [μ_{i k} (t_{1}, t^{'}) μ_{j k}^{*} (t_{2}, t^{'}) - ν_{i k} (t_{1}, t^{'}) ν_{j k}^{*} (t_{2}, t^{'})] = δ_{i j} δ (t_{1} - t_{2}),

[b_{i} (t_{1}), b_{j} (t_{2})] = \sum_{k} \int {d t}^{'} [μ_{i k} (t_{1}, t^{'}) ν_{j k} (t_{2}, t^{'}) - ν_{i k} (t_{1}, t^{'}) μ_{j k} (t_{2}, t^{'})] = 0 .

There also exist output–input (OI) equations, which can be written in the form

a_{i} (t) = \sum_{k} \int {d t}^{'} [{\bar{μ}}_{i k} (t, t^{'}) b_{k} (t^{'}) + {\bar{ν}}_{i k} (t, t^{'}) b_{k}^{†} (t^{'})],

where μ̄_ik and ν̄_ik 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

\begin{array}{l} b_{i} (t) & = & \sum_{j} \sum_{k} \int {d t}^{'} \int {d t}^{″} {[μ_{i j} (t, t^{'}) {\bar{μ}}_{j k} (t^{'}, t^{″}) + ν_{i j} (t, t^{'}) {\bar{ν}}_{j k}^{*} (t^{'}, t^{″})] b_{k} (t^{″}) \\ + [μ_{i j} (t, t^{'}) {\bar{ν}}_{j k} (t^{'}, t^{″}) + ν_{i j} (t, t^{'}) {\bar{μ}}_{j k}^{*} (t^{'}, t^{″})] b_{k}^{†} (t^{″})}, \end{array}

from which follow the consistency conditions

\sum_{j} \int {d t}^{'} [μ_{i j} (t, t^{'}) {\bar{μ}}_{j k} (t^{'}, t^{″}) + ν_{i j} (t, t^{'}) {\bar{ν}}_{j k}^{*} (t^{'}, t^{″})] = δ_{i k} δ (t - t^{″}),

\sum_{j} \int {d t}^{'} [μ_{i j} (t, t^{'}) {\bar{ν}}_{j k} (t^{'}, t^{″}) + ν_{i j} (t, t^{'}) {\bar{μ}}_{j k}^{*} (t^{'}, t^{″})] = 0 .

By combining Eqs. (173) and (174) with Eqs. (169) and (170), one obtains the reciprocity relations

{\bar{μ}}_{i j} (t_{1}, t_{2}) = μ_{j i}^{*} (t_{2}, t_{1}), {\bar{ν}}_{i j} (t_{1}, t_{2}) = - ν_{j i} (t_{2}, t_{1}) .

By combining the constraints imposed on the backward Green functions with the reciprocity relations, one finds that

\sum_{k} \int {d t}^{'} [μ_{k i}^{*} (t^{'}, t_{1}) μ_{k j} (t^{'}, t_{2}) - ν_{k i} (t^{'}, t_{1}) ν_{k j}^{*} (t^{'}, t_{2})] = δ_{i j} δ (t_{1} - t_{2}),

\sum_{k} \int {d t}^{'} [μ_{k i}^{*} (t^{'}, t_{1}) ν_{k j} (t^{'}, t_{2}) - ν_{k i} (t^{'}, t_{1}) μ_{k j}^{*} (t^{'}, t_{2})] = 0 .

Thus, the Green functions satisfy constraints when integrated with respect to either (first or second) variable. The identity for a_i produces equivalent results. Notice that Eqs. (169), (170) and (175)–(177) reduce to the corresponding equations of App. A in the appropriate limits.

It is convenient to separate each output mode operator into its mean value 〈b_i〉 = β_i, which depends only on the input means α_k, and its deviation b_i − 〈b_i〉 = w_i, which depends only on the input deviations v_k. The output means (deviations) are related to the input means (deviations) by equations similar to Eq. (168), and the input and output deviations all satisfy the boson commutation relations. By using the preceding definitions and assuming that the inputs are coherent states, one finds that the mean and deviation of each quadrature flux [Eq. (22)] are

〈 q_{i} (t) 〉 = [β_{p}^{*} (t) β_{i} (t) + β_{p} (t) β_{i}^{*} (t)] / 2^{1 / 2},

δ q_{i} (t) = [β_{p}^{*} (t) w_{i} (t) + β_{p} (t) w_{i}^{†} (t)] / 2^{1 / 2},

respectively, where β_p(t) is the (normalized) amplitude of the LO used to detect mode i. Hence, the mean quadrature

〈 Q_{i} 〉 = \int d t [β_{p}^{*} (t) β_{i} (t) + β_{p} (t) β_{i}^{*} (t)] / 2^{1 / 2} .

By using the IO equation for w_i to rewrite Eq. (179) in terms of v_k, one finds that

\begin{array}{l} δ q_{i} (t) & = & \sum_{k} \int {d t}^{'} {β_{p}^{*} (t) [μ_{i k} (t, t^{'}) v_{k} (t^{'}) + ν_{i k} (t, t^{'}) v_{k}^{†} (t^{'})] \\ + β_{p} (t) [μ_{i k}^{*} (t, t^{'}) v_{k}^{†} (t^{'}) + ν_{i k}^{*} (t, t^{'}) v_{k} (t^{'})]} / 2^{1 / 2} . \end{array}

Equation (181) has four terms on its right side, so if one were to use it to calculate a second-order quadrature moment (variance or correlation), one would need to evaluate sixteen terms. However, because 〈

0_{k} | v_{k}^{†} = 0

and v_k|0_k〉 = 0, one can simplify the calculation by defining the left and right quadrature-flux operators

{[δ q_{i} (t)]}_{l} = \sum_{k} \int {d t}^{'} [β_{p}^{*} (t) μ_{i k} (t, t^{'}) + β_{p} (t) ν_{i k}^{*} (t, t^{'})] v_{k} (t^{'}) / 2^{1 / 2},

{[δ q_{i} (t)]}_{r} = \sum_{k} \int {d t}^{'} [β_{p}^{*} (t) ν_{i k} (t, t^{'}) + β_{p} (t) μ_{i k}^{*} (t, t^{'})] v_{k}^{†} (t^{'}) / 2^{1 / 2},

which act only on bra- and ket-vectors, respectively. Notice that δq_i(t) = [δq_i(t)]_l + [δq_i(t)]_r and

{[δ q_{i} (t)]}_{r} = {[δ q_{i} (t)]}_{1}^{†}

. By using these definitions, one finds that the quadrature-flux moment

\begin{array}{l} 〈 δ q_{i} (t_{1}) δ q_{j} (t_{2}) 〉 & = & \sum_{k} \int {d t}^{'} [β_{p}^{*} (t_{1}) μ_{i k} (t_{1}, t^{'}) + β_{p} (t_{1}) ν_{i k}^{*} (t_{1}, t^{'})] \\ \times [β_{q}^{*} (t_{2}) ν_{j k} (t_{2}, t^{'}) + β_{q} (t_{2}) μ_{j k}^{*} (t_{2}, t^{'})] / 2, \end{array}

where β_q is the amplitude of the LO used to detect mode j. By integrating Eq. (184) with respect to t₁ and t₂, one finds that the quadrature correlation

\begin{array}{l} 〈 δ Q_{i} δ Q_{j} 〉 & = & \sum_{k} \int {d t}^{'} \int d t_{1} [β_{p}^{*} (t_{1}) μ_{i k} (t_{1}, t^{'}) + β_{p} (t_{1}) ν_{i k}^{*} (t_{1}, t^{'})] \\ \times \int d t_{2} [β_{q}^{*} (t_{2}) ν_{j k} (t_{2}, t^{'}) + β_{q} (t_{2}) μ_{j k}^{*} (t_{2}, t^{'})] / 2 . \end{array}

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 δq_i and δq_j commute, as do δQ_i and δQ_j), these properties are not evident in Eqs. (184) and (185). It is convenient to rewrite Eq. (184) in the abbreviated form

\begin{array}{l} 〈 δ q_{i} (t_{1}) δ q_{j} (t_{2}) 〉 & \propto & {(β_{p}^{*} μ_{i k})}_{1} {(β_{q}^{*} ν_{j k})}_{2} + {(β_{p}^{*} μ_{i k})}_{1} {(β_{q} μ_{j k}^{*})}_{2} \\ + {(β_{p} ν_{i k}^{*})}_{1} {(β_{q}^{*} ν_{j k})}_{2} + {(β_{p} ν_{i k}^{*})}_{1} {(β_{q} μ_{j k}^{*})}_{2}, \end{array}

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

\begin{array}{l} {(β_{p}^{*} μ_{i k})}_{1} {(β_{q}^{*} ν_{j k})}_{2} + {(β_{p} ν_{i k}^{*})}_{1} {(β_{q} μ_{j k}^{*})}_{2} & = & {(β_{p}^{*} μ_{i k})}_{1} {(β_{q}^{*} ν_{j k})}_{2} + {(β_{p} μ_{i k}^{*})}_{1} {(β_{q} ν_{j k}^{*})}_{2} \\ = & {(β_{p}^{*} ν_{i k})}_{1} {(β_{q}^{*} μ_{j k})}_{2} + {(β_{p} μ_{i k}^{*})}_{1} {(β_{q} ν_{j k}^{*})}_{2} \\ = & {(β_{q}^{*} μ_{j k})}_{2} {(β_{p}^{*} ν_{i k})}_{1} + {(β_{q} ν_{j k}^{*})}_{2} {(β_{p} μ_{i k}^{*})}_{1} . \end{array}

Now consider the second and third terms in Eq. (186). By applying Eq. (169) twice and reordering, one finds that

\begin{array}{l} {(β_{p}^{*} μ_{i k})}_{1} {(β_{q} μ_{j k}^{*})}_{2} + {(β_{p} ν_{i k}^{*})}_{1} {(β_{q}^{*} ν_{j k})}_{2} & = & {(β_{p}^{*} ν_{i k})}_{1} {(β_{q} ν_{j k}^{*})}_{2} + {(β_{p} ν_{i k}^{*})}_{1} {(β_{q}^{*} ν_{j k})}_{2} + {({| β_{p} |}^{2})}_{1} δ_{i j} δ_{12} \\ = & {(β_{p}^{*} ν_{i k})}_{1} {(β_{q} ν_{j k}^{*})}_{2} + {(β_{p} μ_{i k}^{*})}_{1} {(β_{q}^{*} μ_{j k})}_{2} \\ = & {(β_{q}^{*} μ_{j k})}_{2} {(β_{p} μ_{i k}^{*})}_{1} + {(β_{q} ν_{j k}^{*})}_{2} {(β_{p}^{*} ν_{i k})}_{1} . \end{array}

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: 〈δq_i(t₁)δq_j(t₂)〉 = 〈δq_j(t₂)δq_i(t₁)〉 and 〈δQ_iδQ_j〉 = 〈δQ_jδQ_i〉. When i = j, Eq. (185) specifies the quadrature variance

〈 δ Q_{i}^{2} 〉 = \sum_{k} \int {d t}^{'} {| \int d t [β_{p}^{*} (t) μ_{i k} (t, t^{'}) + β_{p} (t) ν_{i k}^{*} (t, t^{'})] |}^{2} / 2,

which is manifestly real and non-negative.

The mean and deviation of each number flux are

〈 n_{i} (t) 〉 = {| β_{i} (t) |}^{2} + 〈 w_{i}^{†} (t) w_{i} (t) 〉,

δ n_{i} (t) = β_{p}^{*} (t) w_{i} (t) + β_{p} (t) w_{i}^{†} (t) + w_{i}^{†} (t) w_{i} (t) - 〈 w_{i}^{†} (t) w_{i} (t) 〉,

respectively. By integrating Eq. (190) with respect to time and using the aforementioned properties of vacuum states, one finds that the number mean

〈 N_{i} 〉 = \int d t {| β_{i} (t) |}^{2} + \sum_{k} \int d t {\int {d t}^{'} | ν_{i k} (t, t^{'}) |}^{2} .

Noise photons are generated by each conjugate mode

a_{k}^{†}

to which b_i 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 β_p replaced by β_j and the factor of 1/2^1/2 omitted. Hence, the derivations of the signal–noise contributions to the number-flux moment and number correlation need not be described. The final results are

\begin{array}{l} {〈 δ n_{i} (t_{1}) δ n_{j} (t_{2}) 〉}_{sn} & = & \sum_{k} \int {d t}^{'} [β_{i}^{*} (t_{1}) μ_{i k} (t_{1}, t^{'}) + β_{i} (t_{1}) ν_{i k}^{*} (t_{1}, t^{'})] \\ \times [β_{j}^{*} (t_{2}) ν_{j k} (t_{2}, t^{'}) + β_{j} (t_{2}) μ_{j k}^{*} (t_{2}, t^{'})], \end{array}

\begin{array}{l} {〈 δ N_{i} δ N_{j} 〉}_{sn} & = & \sum_{k} \int {d t}^{'} \int d t_{1} [β_{i}^{*} (t_{1}) μ_{i k} (t_{1}, t^{'}) + β_{i} (t_{1}) ν_{i k}^{*} (t_{1}, t^{'})] \\ \times \int d t_{2} [β_{j}^{*} (t_{2}) ν_{j k} (t_{2}, t^{'}) + β_{j} (t_{2}) μ_{j k}^{*} (t_{2}, t^{'})] . \end{array}

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: 〈δn_i(t₁)δn_j(t₂)〉_sn = 〈δn_j(t₂)δn_i(t₁)〉_sn and 〈δN_iδN_j〉_sn = 〈δN_jδN_i〉_sn. When i = j, Eq. (194) specifies the number variance

{〈 δ N_{i}^{2} 〉}_{sn} = \sum_{k} \int {d t}^{'} {| \int d t [β_{i}^{*} (t) μ_{i k} (t, t^{'}) + β_{i} (t) ν_{i k}^{*} (t, t^{'})] |}^{2},

which is manifestly real and non-negative.

It only remains to calculate the noise–noise contributions to the fourth-order moments. One can simplify the calculations by defining the left and right number-flux operators

\begin{array}{l} {[w_{i}^{†} (t) w_{i} (t)]}_{1} & = & \sum_{k} \sum_{l} \int {d t}^{'} \int {d t}^{″} ν_{i k}^{*} (t, t^{'}) v_{k} (t^{'}) [μ_{i l} (t, t^{″}) v_{l} (t^{″}) + ν_{i l} (t, t^{″}) v_{l}^{†} (t^{″})] \\ = & \sum_{k} \sum_{l} \int {d t}^{'} \int {d t}^{″} [ν_{i k}^{*} (t, t^{'}) μ_{i l} (t, t^{″}) v_{k} (t^{'}) v_{l} (t^{″}) + ν_{i k}^{*} (t, t^{'}) ν_{i l} (t, t^{″}) v_{k} (t^{'}) v_{l}^{†} (t^{″})], \end{array}

\begin{array}{l} {[w_{i}^{†} (t) w_{i} (t)]}_{r} & = & \sum_{k} \sum_{l} \int {d t}^{'} \int {d t}^{″} [μ_{i k}^{*} (t, t^{'}) v_{k}^{†} (t^{'}) + ν_{i k}^{*} (t, t^{'}) v_{k} (t^{'})] ν_{i l} (t, t^{″}) v_{l}^{†} (t^{″}) \\ = & \sum_{k} \sum_{l} \int {d t}^{'} \int {d t}^{″} [μ_{i k}^{*} (t, t^{'}) ν_{i l} (t, t^{″}) v_{k}^{†} (t^{'}) v_{l}^{†} (t^{″}) + ν_{i k}^{*} (t, t^{'}) ν_{i l} (t, t^{″}) v_{k} (t^{'}) v_{l}^{†} (t^{″})], \end{array}

respectively. For each term in Eq. (196) or (197) with (k, l) = (m, n), where m and n are integers, there is another term with (k, l) = (n, m). By using this fact to rewrite the summations, one can show that

{[w_{i}^{†} (t) w_{i} (t)]}_{r} = {[w_{i}^{†} (t) w_{i} (t)]}_{l}^{†}

. In the equation for

〈 w_{i}^{†} (t_{1}) w_{i} (t_{1}) w_{j}^{†} (t_{2}) w_{j} (t_{2}) 〉

, the terms with odd powers of v_k and

v_{l}^{†}

have zero expectation values, whereas the terms with even powers have nonzero values. One can evaluate the latter terms by using the identities

〈 v_{k} (t_{k}) v_{l}^{†} (t_{l}) v_{m} (t_{m}) v_{n}^{†} (t_{n}) 〉 = δ_{k l} δ (t_{k} - t_{l}) δ_{m n} δ (t_{m} - t_{n}),

\begin{array}{l} 〈 v_{k} (t_{k}) v_{l} (t_{l}) v_{m}^{†} (t_{m}) v_{n}^{†} (t_{n}) 〉 & = & δ_{k m} δ (t_{k} - t_{m}) δ_{\ln} δ (t_{l} - t_{n}) \\ + δ_{k n} δ (t_{k} - t_{n}) δ_{l m} δ (t_{l} - t_{m}) . \end{array}

The calculation is straightforward and the final result is

\begin{array}{l} 〈 w_{i}^{†} (t_{1}) w_{i} (t_{1}) w_{j}^{†} (t_{2}) w_{j} (t_{2}) 〉 & = & \sum_{k} \sum_{l} \int {d t}^{'} \int {d t}^{″} [{| ν_{i k} (t_{1}, t^{'}) |}^{2} {| ν_{j l} (t_{2}, t^{″}) |}^{2} \\ + ν_{i k}^{*} (t_{1}, t^{'}) μ_{j k}^{*} (t_{2}, t^{'}) μ_{i l} (t_{1}, t^{″}) ν_{j l} (t_{2}, t^{″}) \\ + ν_{i k}^{*} (t_{1}, t^{'}) ν_{j k} (t_{2}, t^{'}) μ_{i l} (t_{1}, t^{″}) μ_{j l}^{*} (t_{2}, t^{″})] . \end{array}

To model photon-multiplet generation (Sec. 5), one must retain all three terms on the right side of Eq. (200). Howevever, to calculate variances and correlations, one can omit the first term, which equals

〈 w_{i}^{†} (t_{1}) w_{i} (t_{1}) 〉 〈 w_{j}^{†} (t_{2}) w_{j} (t_{2}) 〉

. Thus, the number-flux moment

\begin{array}{l} {〈 δ n_{i} (t_{1}) δ n_{j} (t_{2}) 〉}_{nn} & = & \sum_{k} \sum_{l} \int {d t}^{'} \int {d t}^{″} [μ_{i k} (t_{1}, t^{'}) μ_{j k}^{*} (t_{2}, t^{'}) ν_{i l}^{*} (t_{1}, t^{″}) ν_{j l} (t_{2}, t^{″}) \\ + μ_{i k} (t_{1}, t^{'}) ν_{j k} (t_{2}, t^{'}) ν_{i l}^{*} (t_{1}, t^{″}) μ_{j l}^{*} (t_{2}, t^{″})] . \end{array}

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

\begin{array}{l} {〈 δ N_{i} δ N_{j} 〉}_{nn} & = & \sum_{k} \sum_{l} \int {d t}^{'} \int {d t}^{″} [\int d t_{1} μ_{i k} (t_{1}, t^{'}) ν_{i l}^{*} (t_{1}, t^{″}) \int d t_{2} μ_{j k}^{*} (t_{2}, t^{'}) ν_{j l} (t_{2}, t^{″}) \\ + \int d t_{1} μ_{i k} (t_{1}, t^{'}) ν_{i l}^{*} (t_{1}, t^{″}) \int d t_{2} ν_{j k} (t_{2}, t^{'}) μ_{j l}^{*} (t_{2}, t^{″})] . \end{array}

Equation (201) can be rewritten in the abbreviated form

{〈 δ n_{i} (t_{1}) δ n_{j} (t_{2}) 〉}_{nn} \propto \sum_{k} \sum_{l} [{(μ_{i k})}_{1} {(μ_{j k}^{*})}_{2} {(ν_{i l}^{*})}_{1} {(ν_{j l})}_{2} + {(μ_{i k})}_{1} {(ν_{j k})}_{2} {(ν_{i l}^{*})}_{1} {(μ_{j l}^{*})}_{2}],

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

\begin{array}{l} \sum_{k} \sum_{l} {(μ_{i k})}_{1} {(μ_{j k}^{*})}_{2} {(ν_{i l}^{*})}_{1} {(ν_{j l})}_{2} & = & \sum_{k} \sum_{l} {(ν_{i k})}_{1} {(ν_{j k}^{*})}_{2} {(ν_{i l}^{*})}_{1} {(ν_{j l})}_{2} + δ_{i j} \sum_{l} {({| ν_{i l} |}^{2})}_{1} \\ = & \sum_{k} \sum_{l} {(ν_{i k})}_{1} {(ν_{j k}^{*})}_{2} {(μ_{i l}^{*})}_{1} {(μ_{j l})}_{2} \\ - δ_{i j} \sum_{k} {({| ν_{i k} |}^{2})}_{1} + δ_{i j} \sum_{l} {({| ν_{i l} |}^{2})}_{1} \\ = & \sum_{k} \sum_{l} {(μ_{j l})}_{2} {(μ_{i l}^{*})}_{1} {(ν_{j k}^{*})}_{2} {(ν_{i k})}_{1} . \end{array}

Now consider the second term in Eq. (203). By applying Eq. (170) twice and reordering, one finds that

\begin{array}{l} \sum_{k} \sum_{l} {(μ_{i k})}_{1} {(ν_{j k})}_{2} {(ν_{i l}^{*})}_{1} {(μ_{j l}^{*})}_{2} & = & \sum_{k} \sum_{l} {(μ_{i k})}_{1} {(ν_{j k})}_{2} {(μ_{i l}^{*})}_{1} {(ν_{j l}^{*})}_{2} \\ = & \sum_{k} \sum_{l} {(ν_{i k})}_{1} {(μ_{j k})}_{2} {(μ_{i l}^{*})}_{1} {(ν_{j l}^{*})}_{2} \\ = & \sum_{k} \sum_{l} {(μ_{j k})}_{2} {(ν_{i k})}_{1} {(ν_{j l}^{*})}_{2} {(μ_{i l}^{*})}_{1} . \end{array}

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: 〈δn_i(t₁)δn_j(t₂)〉_nn = 〈δn_j(t₂)δn_i(t₁)〉_nn and 〈δN_iδN_j〉_nn = 〈δN_jδN_i〉_nn.

Notice that Eq. (202) can be rewritten in the form

{〈 δ N_{i} δ N_{j} 〉}_{nn} \propto 2 \sum_{k} {(μ_{i k} ν_{i k}^{*})}_{1} {(μ_{j k}^{*} ν_{j k})}_{2} + \sum_{k} \sum_{l > k} {(μ_{i k} ν_{i l}^{*} + μ_{i l} ν_{i k}^{*})}_{1} {(μ_{j k}^{*} ν_{j l} + μ_{j l}^{*} ν_{j k})}_{2},

in which the time integrals are implicit and which is consistent with Eq. (20) of [21]. When i = j, Eq. (206) reduces to

{〈 δ N_{i}^{2} 〉}_{nn} \propto 2 \sum_{k} {| μ_{i k} ν_{i k}^{*} |}^{2} + \sum_{k} \sum_{l > k} {| μ_{i k} ν_{i l}^{*} + μ_{i l} ν_{i k}^{*} |}^{2},

which is consistent with Eq. (43) of [19]. 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

\sum_{n} v_{n} (t) σ_{n} u_{n}^{*} (t^{'})

, where u_n and v_n are input and output Schmidt modes (temporal eigenfunctions), respectively, and σ_n is a Schmidt coefficient [25]. 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, 21, 22]. 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, 26] to the Green matrices M(t, t′) = [μ_ij(t, t′)] and N(t, t′) = [ν_ij(t, t′)], which couple the output vector B(t) = [b_i(t)] to the input vector A(t′) = [a_j(t′)] and its conjugate, respectively, to determine the input and output Schmidt supermodes (vector temporal eigenfunctions), and the associated Schmidt coefficients.

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

Abstract

1. Introduction

2. Direct and homodyne detection

3. Attenuation and frequency conversion

4. Parametric amplification

5. Schmidt decompositions of parametric processes

6. Summary

Appendix A: Commutation relations and conservation laws

Appendix B: Quadrature and number fluctuations

Appendix C: Multiple-mode processes

References and links

Cited By

Figures (3)

Equations (207)

Optics Express