## Abstract

In a previous paper [Opt. Express **13**, 4986 (2005)], formulas were derived for the field-quadrature and photon-number variances produced by multiple-mode parametric processes. In this paper, formulas are derived for the quadrature and number correlations. The number formulas are used to analyze the properties of basic devices, such as two-mode amplifiers, attenuators and frequency convertors, and composite systems made from these devices, such as cascaded parametric amplifiers and communication links. Amplifiers generate idlers that are correlated with the amplified signals, or correlate pre-existing pairs of modes, whereas attenuators decorrelate pre-existing modes. Both types of device modify the signal-to-noise ratios (SNRs) of the modes on which they act. Amplifiers decrease or increase the mode SNRs, depending on whether they are operated in phase-insensitive (PI) or phase-sensitive (PS) manners, respectively, whereas attenuators always decrease these SNRs. Two-mode PS links are sequences of transmission fibers (attenuators) followed by two-mode PS amplifiers. Not only do these PS links have noise figures that are 6-dB lower than those of the corresponding PI links, they also produce idlers that are (almost) completely correlated with the signals. By detecting the signals and idlers, one can eliminate the effects of electronic noise in the detectors.

©2010 Optical Society of America

## 1. Introduction

Parametric devices based on four-wave mixing (FWM) in fibers perform a variety of signal-processing functions that are required by communication and sensing systems [1–4]. Because they should perform these functions without degrading the signals, their noise properties are important. In previous papers [5,6], detailed studies were made of the properties of multiple-mode parametric processes. Such processes occur when several two-mode processes occur simultaneously in the same device, or when several two-mode devices are concatenated. Formulas were derived for the field-quadrature variances associated with homodyne detection and the photon-number variances associated with direct detection, for devices of arbitrary complexity. These formulas are sufficient for many applications, which involve measurements of individual modes (signals).

However, there are other applications which require simultaneous measurements of two or more modes. For such applications, the correlations between the mode quadratures and numbers are important. An example is two-mode squeezing, in which neither the signal nor the idler mode is squeezed by itself. Instead, squeezing exists as a correlation between the modes [7, 8]. Although the variances of the signal and idler numbers are shot-noise limited, the variance of the number difference is not [9, 10].

This paper is organized as follows: In Sec. 2, formulas are derived for the quadrature and number correlations produced by multiple-mode processes. These formulas complete the results of [5, 6]. In Sec. 3, they are used to analyze the properties of basic devices, such as two-mode amplifiers, attenuators and frequency-convertors, and composite systems made from these devices, such as cascaded phase-sensitive (PS) amplifiers [11–14] and communication links [15–18]. Some consequences of these results are discussed in Sec. 4. (It is possible to read Sec. 4 after Sec. 2, and refer to Sec. 3 only as necessary.) Finally, in Sec. 5, the main results of this paper are summarized briefly.

## 2. General results

Multiple-mode parametric processes are governed by the input-output (IO) equations

where *a _{i}* is an input-mode operator,

*b*is an output-mode operator,

_{i}*μ*and

_{ik}*ν*are transfer coefficients, and † is a hermitian conjugate [5, 6]. The input modes satisfy the boson commutation relations [

_{ik}*a*,

_{i}*a*] = 0 and [

_{j}*a*,

_{i}*a*

^{†}

_{j}] =

*δ*, where [ , ] is a commutator and

_{ij}*δ*is the Kronecker delta function. The output modes satisfy similar commutation relations, which imply that

_{ij}The input quadrature operator
${p}_{i}=\frac{({a}_{i}{e}^{-i{\theta}_{i}}+{a}_{i}^{\u2020}{e}^{i{\theta}_{i}})}{{2}^{\frac{1}{2}}}$
, where *θ _{i}* is the phase of a local oscillator (LO), and the input number operator

*m*=

_{i}*a*

^{†}

_{i}

*a*. (In homodyne detection, a beam splitter is used to combine a signal with a LO, and the difference between the output numbers is proportional to the input quadrature of the signal.) If the inputs are independent coherent states (CS) with amplitudes 〈

_{i}*a*〉

_{i}*α*, where 〈 〉 is an expectation value, the input quadratures $\u3008{p}_{i}\u3009=\frac{({\alpha}_{i}{e}^{-i{\theta}_{i}}+{\alpha}_{i}^{*}{e}^{i{\theta}_{i}})}{{2}^{\frac{1}{2}}}$ and the input numbers 〈

_{i}*m*〉 = ∣

_{i}*α*∣

_{i}^{2}. [If some

*α*= 0, those inputs are vacuum states (VS).]

_{i}The output operators are defined in the same way as the input operators (*a _{i}*→

*b*,

_{i}*p*→

_{i}*q*and

_{i}*m*→

_{i}*n*). For CS inputs, Eqs. (1) imply that the output amplitudes (first-order moments)

_{i}In general, the output strengths ∣*β _{i}*∣

^{2}depend on the input phases

*ϕ*= arg(

_{k}*α*). The output quadratures (alternative first-order moments)

_{k}depend on both the input and LO phases.

There are two standard ways to calculate the higher-order output moments (quadrature products, numbers and number products). In the first method, one combines Eqs. (1) and calculates expectation values using the properties of CS (*a _{i}*∣

*α*〉 =

_{i}*α*∣

_{i}*α*〉). In the second method, one rewrites the mode operators as

_{i}where the auxiliary (noise) operators *ν _{i}* and

*w*also satisfy Eqs. (1) and the aforementioned commutation relations, and calculates expectation values using the properties of VS (

_{i}*ν*∣0〉=0). The second method will be used herein (because it is similar to the semi-classical method, which is familiar to communication engineers). It applies to CS inputs, but can be generalized to other inputs (such as squeezed CS).

_{i}#### 2.1. Quadrature fluctuations

The quadrature-deviation operator

It follows from this definition that the output quadrature correlation

By combining Eq. (8) with the noise moments

$$\u3008{w}_{i}^{\u2020}{w}_{j}\u3009={\Sigma}_{k}{\nu}_{\mathrm{ik}}^{*}{\nu}_{\mathrm{jk}},\u3008{w}_{i}^{\u2020}{w}_{j}^{\u2020}\u3009={\Sigma}_{k}{\nu}_{\mathrm{ik}}^{*}{\mu}_{\mathrm{jk}}^{*},$$

one finds that

When *i* = *j*, Eq. (10) reduces to Eq. (40) of [6], the right side of which is manifestly real. When *i* ≠ *j*, the right side of Eq. (10) involves summations of
${\mu}_{\mathrm{ik}}{\mu}_{\mathrm{jk}}^{*}{e}^{i({\theta}_{j}-{\theta}_{i})}$
,
${\mu}_{\mathrm{ik}}{v}_{\mathrm{jk}}{e}^{-i({\theta}_{i}+{\theta}_{j})}$
,
${v}_{\mathrm{ik}}^{*}{\mu}_{\mathrm{jk}}^{*}{e}^{i({\theta}_{i}+{\theta}_{j})}$
and
${v}_{\mathrm{ik}}^{*}{v}_{\mathrm{jk}}{e}^{i({\theta}_{i}-{\theta}_{j})}$
. Equation (2) implies that the sum of the second and third terms is real, whereas Eq. (3) implies that the sum of the first and fourth terms is real. Hence, the quadrature-correlation formula predicts real correlations (as it must do) and reduces to the known variance formula in the appropriate limit.

#### 2.2. Number fluctuations

The output number operator

The first term on the right side of Eq. (11) is the signal-signal term, the second and third terms are (collectively) the signal-noise term and the fourth term is the noise-noise term. It follows from Eq. (11) that the output number

where 〈*w*
^{†}
_{i}
*w _{i}*〉 = ∑

_{k}∣

*ν*∣

_{ik}^{2}[Eqs. (9)] is the number of noise photons.

The number-deviation operator

It follows from this definition, and the fact that the odd-order moments of *w*
^{(†)} have zero expectation values, that the output number correlation

The first term on the right side of Eq. (14) is the signal-noise term, and the second and third terms are (collectively) the noise-noise term. By comparing Eqs. (8) and (14), one finds that the signal-noise term

where *ρ _{j}* = ∣

*β*∣ and

_{j}*ϕ*= arg(

_{j}*β*) are the modulus and phase of the output amplitude, respectively. Equation (15) shows that the mode phase in direct (number) detection plays the role of the LO phase in homodyne (quadrature) detection. For many applications, the signal-noise terms are much larger than the noise-noise terms (and are much easier to calculate). By combining Eqs. (9) and (15), one obtains the alternative (explicit) formula

_{j}One can facilitate the evaluation of the fourth-order moment 〈*w*
^{†}
_{i}
*w _{i}w*

^{†}

*〉 by using the reduced noise operators*

_{j}w_{j}where l denotes an operator that acts (to the left) on the input ket-vector 〈0∣ and r denotes an operator that acts (to the right) on the input bra-vector ∣0〉. Notice that (*w*
^{†}
_{i}
*w _{i}*)

_{l}= (

*w*

^{†}

_{i}

*w*)

_{i}^{†}

_{r}(as it must do). By combining Eqs. (17) and (18), using the identities 〈

*ν*

_{k}ν_{l}ν^{†}

_{m}

*ν*

^{†}

_{n}〉 =

*δ*+

_{km}δ_{ln}*δ*and 〈

_{kn}δ_{lm}*ν*

_{k}ν^{†}

_{l}

*ν*

_{m}ν^{†}

_{n〉}=

*δ*, and collecting terms, one finds that the noise-noise term

_{kl}δ_{mn}It follows from Eqs. (16) and (19) that the output number correlation

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+{\Sigma}_{k}{\Sigma}_{l>k}({\mu}_{\mathrm{ik}}{\nu}_{\mathrm{il}}^{*}+{\mu}_{\mathrm{il}}{\nu}_{\mathrm{ik}}^{*})({\mu}_{\mathrm{jk}}^{*}{\nu}_{\mathrm{jl}}+{\mu}_{\mathrm{jl}}^{*}{\nu}_{\mathrm{jk}}).$$

When *i* = *j*, Eq. (20) reduces to Eq. (42) of [6]. When *i* ≠ *j*, the right side of Eq. (20) involves three complicated summations. Because the number deviations commute, 〈*δn _{i}δn_{j}*〉 = 〈

*δn*〉. It follows from this result, and the fact that interchanging

_{j}δn_{i}*i*and

*j*in these summations is equivalent to conjugating them, that the summations are real. (A similar argument could have been made in the context of quadrature correlations.) Hence, the number-correlation formula predicts real correlations (as it must do) and reduces to the known variance formula in the appropriate limit.

By combining the formulas for 〈*δn*
^{2}
_{i}〉 and 〈*δn*
^{2}
_{j}〉, which follow from Eq. (20), with the formula for (*〈δn _{i}δn_{j}*〉 + 〈

*δn*〉)/2, which depends symmetrically on

_{j}δn_{i}*i*and

*j*, one finds that the differential variance

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+{\Sigma}_{k}{\Sigma}_{l>k}{\mid {\mu}_{\mathrm{ik}}{\nu}_{\mathrm{il}}^{*}+{\mu}_{\mathrm{il}}{\nu}_{\mathrm{ik}}^{*}-{\mu}_{\mathrm{jk}}{\nu}_{\mathrm{jl}}^{*}-{\mu}_{\mathrm{jl}}{\nu}_{\mathrm{jk}}^{*}\mid}^{2}$$

is non-negative (as it must be).

## 3. Applications

In this section, the consequences of Eq. (16) are determined for basic devices (two-mode amplifiers, attenuators and frequency convertors), and composite systems made from these devices (copiers, cascaded PS amplifiers and PS links). Results are stated for direct detection only, because in the aforementioned applications it is more common than homodyne detection. The results for homodyne detection are similar [Eq. (15)].

For the aforementioned devices (and many others), *μ _{ik}* and

*ν*are not nonzero simultaneously, so the equation for the output number variance can be rewritten in the compact form

_{ik}where *λ _{ik}* =

*μ*if

_{ik}*i*and

*k*are like (both odd or both even) and

*λ*=

_{ik}*ν*if

_{ik}*i*and

*k*are unlike (one odd and the other even). If

*i*and

*j*are like, the output number correlation

where *k* is like *i* (or *j*) and *l* is unlike. It follows from Eqs. (3) and (23) that

where *k* is arbitrary. Conversely, if *i* and *j* are unlike,

where *k* is like *i* and *l* is unlike. It follows from Eqs. (2) and (25) that

where *k* is arbitrary. Henceforth, the subscript sn will be omitted.

#### 3.1. Two-mode amplifier

A two-mode amplifier (Fig. 1) is governed by the IO equations

where mode 1 is the signal, mode 2 is the idler, and the transfer coefficients *μ* and *ν* satisfy the auxiliary equation ∣*μ*∣^{2} − ∣*ν*∣^{2} = 1 [7, 8]. It is convenient to define the phase-insensitive (PI) gain *G* = ∣*μ*∣^{2}, in which case ∣*ν*∣^{2} = *G*−1.

If both inputs are CS, Eqs. (27) and (28) imply that the output strengths

where the phase difference
$\theta ={\varphi}_{v}-{\varphi}_{\mu}-{\varphi}_{{\alpha}_{1}}-{\varphi}_{{\alpha}_{2}}$
. The signal and idler (sideband) gains are maximal when *θ* = 0 and minimal when *θ* = *π*. Notice that ∣*β*
_{1}∣^{2} − ∣*β*
_{2}∣^{2} = ∣*α*
_{1}∣^{2} − ∣*α*
_{2}∣^{2}. This relation is one of the Manley-Rowe-Weiss (MRW) equations [19, 20], and reflects the fact that sideband photons are produced in pairs.

Equations (22) and (26) imply that the output variances and correlation

respectively. On the right sides of Eqs. (31)–(33), the first terms stem from the input fluctuations, which combine incoherently, whereas the second terms stem from the input amplitudes, which combine coherently. By combining these equations, one finds that the differential variance

is a constant, which equals the sum of the variances of the input CS. The number difference is constant (because photons are produced in pairs), so its output variance equals its input variance, which equals the sum of the individual variances (because the inputs are independent).

One can also explain Eq. (34) in terms of superposition modes. By redefining the phases of the input and output modes in Eqs. (27) and (28), one can replace the transfer coefficients by their moduli. The (rephased) sum and difference modes *a*
_{±} = (*a*
_{1} ± *a*
_{2})/2^{1/2} satisfy the IO equations *b*
_{±} = ∣*μ*∣*a*
_{±} ± ∣*ν*∣*a*
^{†}
_{±}. In terms of these modes, the number-difference operator *n*
_{1} − *n*
_{2} = *b*
^{†}
_{+}
*b*
_{−}+*b*
^{†}
_{−}
*b*
_{+} = *a*
^{†}
_{+}
*a*
_{−}+*a*
^{†}
_{−}
*a*
_{+}, which is a constant operator. It is easy to verify that 〈(*δn*
_{1} − *δn*
_{2})^{2}〉 = 〈*a*
^{†}
_{+}
*a*
_{+}+*a*
^{†}
_{−}
*a*
_{−}〉 = ∣*α*
_{1}∣^{2} + ∣*α*
_{2}∣^{2}. Physically, the number difference is proportional to the product of the sum and difference amplitudes. The sum mode is stretched and the difference mode is squeezed (by the the same amount), so the product of their amplitudes is constant.

For the common case in which the input idler is a VS (*α*
_{2} = 0), the output amplitudes *β*
_{1} = *μα*
_{1} and *β*
_{2} = *να*
^{*}
_{1}. The output variances and correlation

and the differential variance

In the low-gain regime (*G*−1≪1), the signal variance is of order ∣*α*
_{1}∣^{2}. The idler variance and correlation are much smaller, because the idler is weak. Conversely, in the high-gain regime (*G* ≫ 1), the variances and correlation are all of order *G*
^{2}∣*α*
_{1}∣^{2}, which shows that the sidebands are strongly correlated. This correlation is also evidenced by the fact that the differential variance is much smaller than the individual variances (or that of a CS with strength *G*∣*α*
_{1}∣^{2}).

For the symmetric case in which the inputs are equal (*α*
_{1} = *α*
_{2} = *α*), the (common) output strength ∣*β*∣^{2} = *G*
_{θ}∣*α*∣^{2}, where *G _{θ}* = 2

*G*− 1 + 2[

*G*(

*G*− 1)]

^{1/2}cos

*θ*is the PS gain. Notice that the maximal PS gain is almost 4 times higher than the PI gain. The (common) output variance and correlation

For in-phase input (*θ* = 0) the output variance and correlation are both approximately 2*GG*
_{0}∣*α*∣^{2}, so the sidebands are strongly correlated. Conversely, for out-of-phase input (*θ* = *π*), the variance is approximately 2*GG _{π}*∣

*α*∣

^{2}, but the correlation is approximately −2

*GG*

_{π}∣

*α*∣

^{2}, so the sidebands are strongly anti-correlated. In the former case, the individual variances are much larger than ∣

*α*∣

^{2}and a positive correlation reduces the differential variance to 2∣

*α*∣

^{2}, whereas in the latter, the individual variances are smaller than ∣

*α*∣

^{2}and a negative correlation increases the differential variance to the aforementioned value.

A comparison of the one-and two-input results shows that the output strength is four times larger in the latter configuration than in the former, but the output variance is only two times larger, so the output signal-to-noise ratio is higher in the latter. (This result requires the inputs to be uncorrelated.)

#### 3.2. Two-mode attenuator or frequency convertor

A two-mode attenuator (Fig. 2) is governed by the IO equations

where mode 1 is the signal, mode 3 is the loss mode, and the transfer coefficients *τ* and *ρ* satisfy the auxiliary equation ∣*τ*∣^{2} + ∣*ρ*∣^{2} = 1 [7, 8]. It is convenient to define the transmission *T* = ∣*τ*∣^{2}, in which case ∣*ρ*∣^{2} = 1 − *T*. Equations (41) and (42) also govern a two-mode frequency convertor, in which mode 3 is the idler [5, 6, 21].

If both inputs are CS with nonzero amplitudes (frequency exchangers), Eqs. (41) and (42) imply that the output strengths

where the phase difference
$\theta ={\varphi}_{\rho}-{\varphi}_{\tau}+{\varphi}_{{\alpha}_{3}}-{\varphi}_{{\alpha}_{1}}$
. The conversion efficiency is minimal when *θ* = 0 and maximal when *θ* = *π*. Notice that ∣*β*
_{1}∣^{2} + ∣*β*
_{3}∣^{2} = ∣*α*
_{1}∣^{2} + ∣*α*
_{3}∣^{2}. This MRW relation reflects the fact that signal photons are converted to idler photons (or vice versa), but the total number of sideband photons is constant.

Equations (22) and (24) imply that the output variances and correlation

respectively. On the right side of Eq. (45), the first term stems from the input fluctuations, which combine incoherently (to the value 1), whereas the second term stems from the input amplitudes, which combine coherently. Equations (45) and (46) reflect the well-known fact that a frequency convertor (frequency-changing beam splitter [22]) converts two uncorrelated input CS to two (different) uncorrelated output CS. By combining these equations, one finds that the differential variance

is a constant, which equals the sum of the variances of the input CS. Because the output numbers are uncorrelated, the differential variance equals the total variance. But the total number is constant, so its output variance equals its input variance, which equals the sum of the individual variances (because the inputs are independent). For the common special case in which *α*
_{3} = 0 (attenuators or frequency convertors), ∣*β*
_{1}∣^{2} = *T*∣*α*
_{1}∣^{2} and ∣*β*
_{3}∣^{2} = (1 − *T*)∣*α*
_{1}∣^{2}.

#### 3.3. Amplifier followed by attenuators

For a two-mode amplifier followed by two attenuators in parallel (Fig. 3), the composite IO equations are

where modes 1 and 2 are the signal and idler, modes 3 and 4 are the loss modes of the signal and idler attenuators, respectively, *μ* and *ν* are the transfer coefficients of the amplifier, *τ*
_{1} and *ρ*
_{1} are the transfer coefficients of the signal attenuator, and *τ*
_{2} and *ρ*
_{2} are the transfer coefficients of the idler attenuator. The transfer coefficients satisfy auxiliary equations, which were stated in Secs. 3.1 and 3.2.

If both inputs are CS, Eqs. (48) and (49) imply that the output strengths

where the phase difference
$\theta ={\varphi}_{\nu}-{\varphi}_{\mu}-{\varphi}_{{\alpha}_{1}}-{\varphi}_{{\alpha}_{2}}$
. The sideband gains are maximal when *θ* = 0 and minimal when *θ* = *π*. Notice that the phase difference does not depend on the transmission coefficients and ∣*β*
_{1}/*τ*
_{1}∣^{2} − ∣*β*
_{2}/*τ*
_{2}∣^{2} = ∣*α*
_{1}∣^{2} − ∣*α*
_{2}∣^{2}.

Equations (22) and (26) imply that the output variances and correlation

respectively. The formula for the differential variance is not illuminating.

For the common case in which the idler is a VS (*α*
_{2} = 0) and the attenuators are identical (*τ*
_{1} = *τ*
_{2} = *τ*), the output strengths ∣*β*
_{1}∣^{2} = *TG*∣*α*
_{1}∣^{2} and ∣*β*
_{2}∣^{2} = *T*(*G*−1)∣*α*
_{1}∣^{2}, where the PI gain *G* and transmittance *T* were defined in Secs. 3.1 and 3.2, respectively. The system is balanced (∣*β*
_{1}∣^{2} = ∣*α*
_{1}∣^{2}) if *TG* = 1, in which case loss compensates PI gain. The output variances and correlation

If the reflection coefficient (*ρ*) were zero, Eqs. (55)–(57) would be just Eqs. (35)–(37), with *μ* and *ν* replaced by *τμ* and *τν*, respectively. However it is nonzero and the associated terms in Eqs. (55) and (56) represent the effects of loss-mode vacuum fluctuations (which do not contribute to the correlation). By combining Eqs. (55)–(57), one finds that the differential variance

There is no non-trivial condition under which Eq. (58) reduces to Eq. (38). If the system is balanced, 〈*δn*
^{2}
_{1}〉 = (3−2*T*)∣*α*
_{1}∣^{2}, 〈*δn*
^{2}
_{2}〉 = (3−2*T*)(1−*T*)∣*α*
_{1}∣^{2} and 〈*δn*
_{1}
*δn*
_{2}〉 = 2(1−*T*)∣*α*
_{1}∣^{2}. In the low-gain (low-loss) regime (1−*T* ≪ 1), the signal variance is of order ∣*α*
_{1}∣*2*, and the idler variance and correlation are much smaller than the signal variance: Only a weak idler is produced. Conversely, in the high-gain (high-loss) regime (*T* ≪ 1), both variances are approximately 3, whereas the correlation is approximately 2: The sidebands are both strong, but are not completely correlated. If the system is unbalanced (*TG* ≪ 1), 〈*δn*
^{2}
_{i}〉 ≈ 〈*n _{i}*〉 and 〈

*δn*

_{1}

*δn*

_{2}〉 ≪ 〈

*δn*

^{2}

_{i}〉: The signal and idler are (almost) independent CS. This system is the prototype of the copiers that precede realistic PS links (Sec. 3.6). The device described in this section is more flexible than the basic amplifier described in Sec. 3.1: By choosing the values of

*τ*

_{1}and

*τ*

_{2}judiciously, one can equalize the output strengths of the sidebands and control the degree of correlation between the sidebands.

#### 3.4. Attenuators followed by an amplifier

For two attenuators in parallel followed by a two-mode amplifier (Fig. 4), the composite IO equations are

The symbols in Eqs. (59) and (60) were defined in the previous subsection.

If both inputs are CS, Eqs. (59) and (60) imply that the output strengths

where the phase difference
$\theta ={\varphi}_{\nu}-{\varphi}_{\mu}-{\varphi}_{{\tau}_{1}}-{\varphi}_{{\tau}_{2}}-{\varphi}_{{\alpha}_{1}}-{\varphi}_{{\alpha}_{2}}$
. The sideband gains are maximal when *θ* = 0 and minimal when *θ* = *π*. Notice that ∣*β*
_{1}∣^{2} − ∣*β*
_{2}∣^{2} = ∣*τ*
_{1}
*α*
_{1}∣^{2} − ∣*τ*
_{2}
*α*
_{2}∣^{2}.

Equations (22) and (26) imply that the output variances and correlation

respectively. By combining Eqs. (63)–(65), one finds that the differential variance

Although Eqs. (59) and (60) involve four modes, whereas Eqs. (27) and (28) involve only two, Eqs. (61)–(66) are just Eqs. (29)–(34), with *α _{j}* replaced by

*τ*. This result reflects the fact that the attenuators convert input CS with amplitudes

_{j}α_{j}*α*to output CS with amplitudes

_{j}*τ*(Sec. 3.2), which are the inputs to the amplifier (Sec. 3.1).

_{j}α_{j}For the symmetric case in which *α*
_{1} = *α*
_{2} = *α* and *τ*
_{1} = *τ*
_{2} = *τ*, the (common) output strength ∣*β*∣^{2} = *G _{θ}T*∣

*α*∣

^{2}, where the PS gain

*G*and transmittance

_{θ}*T*were defined in Secs. 3.1 and 3.2, respectively. The system is balanced (∣

*β*∣

^{2}= ∣

*α*∣

^{2}) if

*G*

_{0}

*T*= 1, in which case in-phase gain compensates loss. This condition is satisfied when

*G*= (

*L*+ 2 + 1/

*L*)/4, where

*L*= 1/

*T*is the loss. The (common) output variance and correlation

If the system is balanced, 〈*δn _{i}*〉

^{2}= (

*L*+ 1/

*L*)∣

*α*∣

^{2}/2 and 〈

*δn*

_{1}

*δn*

_{2}〉 = (

*L*− 1/

*L*)∣

*α*∣

^{2}/2. In the low-loss regime (

*L*− 1 ≪ 1), the variance is approximately ∣

*α*∣

^{2}and the correlation is much smaller than the variance: Only a weak correlation is produced. Conversely, in the high-loss regime (

*L*≫ 1), the variance and correlation are both approximately

*L*∣

*α*∣

^{2}/2: A strong correlation is produced (even though the amplifier only restores the sideband strengths to their input value).

The system described in the preceding paragraph is the prototype of the stages in a PS link (Sec. 3.6). It is instructive to compare the properties of this system with those of a PI link [6, 15, 18], in which the input idler is a VS (*α*
_{2} = 0) and the output idler is discarded. For both types of link, Eq. (63) implies that the output signal variance 〈*δn*
^{2}
_{1}〉 = (2*G*
*−*
_{1})〈*n*
_{1}〉. However, a balanced PI link requires that *G* = *L*, whereas a balanced PS link requires only that *G* ≈ *L*/4: The higher efficiency of PS amplification (which requires *α*
_{2} ≠ 0) allows PS links to operate with smaller values of *G* than PI links, and amplify input fluctuations by smaller amounts.

#### 3.5. Cascaded phase-sensitive amplifier

A cascaded PS amplifier [13, 14] consists of a two-mode amplifier (which amplifies the input signal and generates an idler) followed by an optical processor (which controls the relative phase of the sidebands) and another two-mode amplifier (which provides PS amplification). The optical processor and connecting fibers are modeled as two attenuators in parallel. If the attenuator losses are comparable to the amplifier gain, the first two components produce a pair of sidebands whose amplitudes are comparable to the input signal amplitude. This part of the device is called a copier (Sec. 3.3). If the attenuator losses are equal, the sideband amplitudes differ slightly (because the signal and idler gains differ slightly). However, by choosing the losses judiciously, one can equalize the sideband amplitudes before the PS amplifier.

For a cascaded PS amplifier, the composite IO equations are

where modes 1 and 2 are the signal and idler, respectively, modes 3 and 4 are the loss modes of the attenuators, *μ _{c}* and

*ν*are the transfer coefficients of the first amplifier (copier),

_{c}*τ*and

_{j}*ρ*are the transfer coefficients of the attenuators, and

_{j}*μ*and

*ν*are the transfer coefficients of the second (PS) amplifier. The dependence of the cascaded PS amplifier on the phases of the input amplitude and transfer coefficients was studied in [23,24]. The results of this paper are based on the simplifying assumption that the amplitudes and coefficients are real, which is appropriate for in-phase (or out-of-phase) amplification.

If the copier is equalized (*τ*
_{1}
*μ _{c}* =

*τ*

_{2}

*ν*=

_{c}*σ*), the (common) output amplitude

*β*=

*σ*(

*μ*+

*ν*)

*α*. Equations (22) and (26) imply that the output variances and correlation

respectively. The difference between the variances is 2*τ*
^{2}
_{1}
*β*
^{2}, which is much smaller than the other contributions to the variances. Hence, both variances are approximately equal to the average variance, which involves the term (*μ*
^{2} + *ν*
^{2})(1 − *τ*
^{2}
_{1}). By combining Eqs. (71)–(73), one finds that the differential variance

Equation (74) is exact. In these results, *μ*
^{2} = *G* is the PI gain and (*μ* + *ν*)^{2} = *G*
_{0} is the inphase gain. These parameters are related by the identities *G*
_{0} = 2*G* − 1 + 2[*G*(*G* − 1)]^{1/2} and *G* = (*G*
_{0} + 2 + 1/*G*
_{0})/4.

By comparing Eqs. (71)–(73) with Eqs. (39) and (40), one finds that the *σ*-terms in the former equations represent the noise penalty (cost) associated with copying. (The *τ*
_{1}-terms are smaller than the *σ*-terms and can be omitted from this discussion.) If the copier is balanced (*σ* = 1), the amplitude *β* = (*μ* + *ν*)*α*, and the (common) variance and correlation

The (common) contribution of the copier to the variance and correlation (2*G*
_{0}
*β*
^{2}) is larger than that of the PS amplifier (≈ *G*
_{0}
*β*
^{2}/2). Although the variance and correlation are increased by the copier, the differential variance is not. For a balanced copier, Eq. (74) predicts that the differential variance is approximately 2*α*
^{2}, in agreement with Eq. (34).

In contrast, if the copier is symmetric (*τ*
_{1} = *τ*
_{2} = *τ _{c}*), the output amplitudes

*β*=

_{i}*τ*, where

_{c}γ_{i}α*γ*

_{1}= (

*μμ*+

_{c}*νν*) and

_{c}*γ*

_{2}= (

*μν*+

_{c}*νμ*) are the transfer coefficients associated with concatenated amplifiers, which satisfy the auxiliary equation

_{c}*γ*

^{2}

_{1}−

*γ*

^{2}

_{2}= 1. Equations (22) and (26) imply that the output variances and correlation

The only difference between the variances is their proportionality to *β*
^{2}
_{j}. By combining Eqs. (77) and (78), one finds that the differential variance

It is easy to verify that *μγ*
_{2} − *νγ*
_{1} = *ν _{c}*, which shows the equivalence of Eqs. (58) and (79), and the similarity between Eqs. (74) and (79). For a balanced, high-gain copier (

*τ*= 1 and

_{c}μ_{c}*μ*

^{2}

_{c}≫ 1) and a high-gain amplifier (

*μ*

^{2}≫ 1),

*β*

_{2}≈

*β*

_{1}= (

*μ*+

*ν*)

*α*, and the variances and correlation are approximately 5

*G*

_{0}

*β*

^{2}/2, in agreement with Eqs. (75) and (76). Just like an equalized copier, a symmetric copier increases the variances and correlation of the output sidebands (relative to those of an amplifier with CS inputs), but does not affect the differential variance significantly: The output sidebands are strongly correlated. If the second amplifier is absent (

*μ*= 1 and

*ν*= 0),

*γ*

_{1}=

*μ*,

_{c}*γ*

_{2}=

*ν*, and Eqs. (77) and (78) are consistent with Eqs. (55)–(57).

_{c}The noise properties of cascaded PS amplifiers were measured experimentally [23, 24], and the results are consistent with the preceding analysis (and some straightforward extensions required by the experiment).

#### 3.6. Multiple-stage phase-sensitive link

Communication links are sequences of fibers (attenuators) and amplifiers. Links based on one-mode PS amplifiers were studied in [6, 15, 16], so in this section only links based on two-mode PS amplifiers (Sec. 3.1) are considered. Each stage in an idealized link consists of two attenuators in parallel, followed by an optical processor and a two-mode amplifier (Sec. 3.4), and both inputs are CS (Fig. 6 without the copier). Hence, one can determine the composite IO equations by iterating Eqs. (59) and (60).

For a link with two identical stages, the IO equations are

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\rho \tau \left(2\mu v\right){a}_{4}^{\u2020}+\left(\rho \mu \right){a}_{5}+\left(\rho v\right){a}_{6}^{\u2020},$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\rho \tau ({\mu}^{2}+{v}^{2}){a}_{4}+\left(\rho v\right){a}_{5}^{\u2020}+\left(\rho \mu \right){a}_{6},$$

where modes 1 and 2 are the signal and idler modes, respectively, modes 3 and 4 are the loss modes of stage 1, modes 5 and 6 are the loss modes of stage 2, *μ* and *ν* are the (common) transfer coefficients of the amplifiers, *τ* and *ρ* are the (common) transfer coefficients of the attenuators, and the transfer coefficients were assumed to be real. This simplification is sufficient to model a link in which the sidebands are in-phase with the transfer coefficients, which is the case of most interest. For a three-stage link,

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\rho \tau [({\mu}^{2}+{v}^{2}){a}_{5}+\left(2\mu v\right){a}_{6}^{\u2020}]+\rho (\mu {a}_{7}+{va}_{8}^{\u2020}),$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}+\rho \tau [\left(2\mu v\right){a}_{5}^{\u2020}+({\mu}^{2}+{v}^{2}){a}_{6}]+\rho (v{a}_{7}^{\u2020}+{\mu a}_{8}),$$

where modes 7 and 8 are the loss modes of stage 3. By continuing this sequence of equations, one finds that for an *n*-stage PS link, the composite IO equations are

where modes 2*r* + 1 and 2*r* + 2 are the loss modes of stage *r*. The polynomials *p _{n}* and

*q*are defined by the initial conditions

_{n}*p*

_{1}=

*μ*and

*q*

_{1}=

*ν*, together with the recursion relations

*p*

_{n+1}=

*μp*+

_{n}*νq*and

_{n}*q*

_{n+1}=

*μq*+

_{n}*νp*. (These polynomials should not be confused with the input and output quadratures of Sec. 2.) It is easy to verify that

_{n}*p*+

_{n}*q*= (

_{n}*μ*+

*ν*)

^{n}and

*p*−

_{n}*q*= (

_{n}*μ*−

*ν*)

^{n}, from which it follows that

If the input sidebands are CS with amplitudes *α _{i}*, the output amplitudes

If the input amplitudes are equal and in-phase with the transfer coefficients (positive), the (common) output amplitude *β* = *τ ^{n}*(

*μ*+

*ν*)

*. Hence, the balanced-link condition [*

^{n}α*τ*(

*μ*+

*ν*) = 1] does not depend on the number of stages in the link.

If the the inputs are equal and in-phase, Eqs. (22) and (26) imply that the (common) output variance and correlation

respectively. By combining Eqs. (86) and (87) with Eqs. (90) and (91), one obtains formulas that depend explicitly on the transfer coefficients of the constituent devices. The link is balanced if *G*
_{0} = *L*, where *G*
_{0} = (*μ* + *ν*)^{2} is the in-phase gain and *L* = 1/*τ*
^{2} is the loss of each stage. By making these substitutions in Eqs. (90) and (91), and doing the summations, one finds that

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.6em}{0ex}}=\frac{[1+n(L-1)+\frac{(1+\frac{1}{{L}^{2n-1}})}{(L+1)}]{\mid \alpha \mid}^{2}}{2},$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.6em}{0ex}}=\frac{[1+n(L-1)-\frac{(1+\frac{1}{{L}^{2n-1}})}{(L+1)}]{\mid \alpha \mid}^{2}}{2}.$$

In the first forms of Eqs. (92) and (93), the first pairs of terms stem from the signal and idler fluctuations, which are transmitted through the whole link, whereas the second pairs stem from the loss-mode (vacuum) fluctuations, which are added throughout the link. In the low-loss regime (*L* − 1 ≪ 1), 〈*δn*
^{2}
_{i}〉 ≈ ∣*α*∣^{2} and 〈*δn*
_{1}
*δn*
_{2}〉 ≈ 0: The outputs are (almost) independent CS, as were the inputs. Conversely, in the high-loss regime (*L* ≫ 1), 〈*δn*
^{2}
_{i}〉 ≈ [1 + *n*(*L* − 1)]∣*α*∣^{2}/2 ≈ 〈*δn*
_{1}
*δn*
_{2}〉: The output variance and correlation are determined primarily by the total loss *nL* and the outputs are (almost) completely correlated. At the inputs to the second and subsequent stages, the sidebands are strongly correlated. In each stage, the sidebands are diminished and decorrelated by the attenuators, then augmented and recorrelated by the PS amplifiers. (If the sidebands were not decorrelated at the inputs to the amplifiers, the factors of 1/2 would be absent from variance and correlation formulas.) For the special case in which *n* = 1, 〈*δn*
^{2}
_{i}〉 = (*L* + 1/*L*)∣*α*∣^{2}/2 and 〈*δn*
_{1}
*δn*
_{2}〉 = (*L* − 1/*L*)∣*α*∣^{2}/2. These results are consistent with Eqs. (67) and (68).

By combining Eqs. (90) and (91), one finds that the differential variance

For a balanced link,

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.7em}{0ex}}=\frac{2(1+\frac{1}{{L}^{2n-1}}){\mid \alpha \mid}^{2}}{(L+1)}.$$

In the first form of Eq. (95), the first term stems from sideband fluctuations and decreases rapidly as *L* increases (because the sideband fluctuations are strongly correlated), whereas the second term stems from vacuum fluctuations and decreases slowly as *L* increases (because uncorrelated vacuum fluctuations are added throughout the link). In the low-loss regime, the differential variance (≈ 2*α*
^{2}) has the value associated with two independent CS. In the high-loss regime, the differential variance (≈ 2*α*
^{2}/*L*) does not depend the number of stages in the link. For the special case in which *n* = 1, Eq. (95) reduces to Eq. (66).

Conventional communication systems are based on single-carrier-frequency signals. Hence, in realistic PS links, the signals must be copied (idlers must be generated) before they are transmitted. For a copier (Sec. 3.3) followed by an *n*-stage PS link (Fig. 6), the composite IO equations are

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\left({q}_{n}{\rho}_{2}\right){a}_{0}^{\u2020}]+\rho \underset{r=1}{\overset{n}{\Sigma}}{\tau}^{n-r}({p}_{n-r+1}{a}_{2r+1}+{q}_{n-r+1}{a}_{2r+2}^{\u2020}).$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\left({p}_{n}{\rho}_{2}\right){a}_{0}]+\rho \underset{r=1}{\overset{n}{\Sigma}}{\tau}^{n-r}({p}_{n-r+1}{a}_{2r+1}^{\u2020}+{q}_{n-r+1}{a}_{2r+2}),$$

where modes −1 and 0 are the loss modes of the copier attenuators, *μ _{c}* and

*ν*are the transfer coefficients of the copier amplifier,

_{c}*τ*and

_{j}*ρ*are the transfer coefficients of the copier attenuators, and the output superscripts (

_{j}*n*) were omitted. All the other symbols were defined above. The

*ρ*-terms in Eqs. (96) and (97) are identical to those in Eqs. (84) and (85), as are their contributions to the output variances and correlations. (These terms are associated with the loss modes of the link.) Hence, only the first four (copier) terms are retained the following analysis.

For an equalized copier (*τ*
_{1}
*μ _{c}* =

*τ*

_{2}

*ν*=

_{c}*σ*), the (common) output amplitude

*β*=

*στ*(

^{n}*p*+

_{n}*q*)

_{n}*α*. If the copier and link are balanced [

*σ*= 1 and (

*μ*+

*ν*)

*τ*= 1, respectively], so also is the combined system, in which case

*β*=

*α*. The contributions of the copier terms to the output variances and correlation are

respectively. The (normalized) signal and idler variances differ by the amount 2*τ*
^{2n}
*τ*
^{2}
_{1}, which is negligible, so both variances are approximately equal to the average variance, which involves the term (*p*
^{2}
_{n} + *q*
^{2}
_{n})(1 − *τ*
^{2}
_{1}). By combining Eqs. (98)–(100), one finds that the copier contributions to the differential variance are

Equation (101) is exact. For the case in which *n* = 1 and *τ* = 1, Eqs. (98)–(101) reduce to Eqs. (71)–(74), respectively.

By comparing Eqs. (90) and (91) with Eqs. (98)–(100), one finds that the copier increases the (normalized) variance and correlation by the amounts 2*σ*
^{2}
*τ*
^{2n}(*μ* + *ν*)^{2n} and −*τ*
^{2n}[(*μ* + *ν*)^{2n} + 1/(*μ* + *ν*)^{2n}]*τ*
^{2}
_{1}/2. For a balanced system, the second amount (≈ −*τ*
^{2}
_{1}/2) is negligible and the first amount (2) is much smaller than the other contributions (≈ *nL*/2). By comparing Eqs. (94) and (101), one finds that the copier has a negligible impact on the differential variance (because it produces correlated sideband photons).

For a symmetric copier (*τ*
_{1} = *τ*
_{2} = *τ _{c}*), the output amplitudes

*β*=

_{i}*τ*, where

^{n}τ_{c}γ_{i}α*γ*

_{1}= (

*p*+

_{n}μ_{c}*q*) and

_{n}ν_{c}*γ*

_{2}= (

*p*+

_{n}ν_{c}*q*) are the transfer coefficients associated with concatenated amplifiers, which satisfy the auxiliary equation stated in Sec. 3.5. By using the properties of the constituent transfer coefficients, one finds that

_{n}μ_{c}Equations (102) and (103) are extensions of Eqs. (86) and (87), respectively. If the system is balanced (*τ ^{n}τ_{c}γ*

_{1}= 1), the amplitudes

*β*

_{1}=

*α*and

*β*

_{2}=

*αγ*

_{2}/

*γ*

_{1}. The contributions of the copier terms to the output variances and correlation are

respectively. The only difference between the variances is their dependence on *β*
^{2}
_{j}. By combining Eqs. (104) and (105), one finds that the copier contributions to the differential variance are

where *p _{n}γ*

_{2}−

*q*

_{n}γ_{1}=

*ν*. For the case in which

_{c}*n*= 1 and

*τ*= 1, Eqs. (104)–(106) reduce to Eqs. (77)–(79), respectively.

By comparing Eqs. (90) and (91) with Eqs. (104) and (105), one finds that the copier increases the (normalized) variance and correlation by the amounts (*τ ^{n}τ_{c}*)

^{2}(

*γ*

^{2}

_{1}+

*γ*

^{2}

_{2}−

*p*

^{2}

_{n}−

*q*

^{2}

_{n}) and (

*τ*)

^{n}τ_{c}^{2}(2

*γ*

_{1}

*γ*

_{2}− 2

*p*), respectively. For a balanced system with a high-gain copier (

_{n}q_{n}*μ*

^{2}

_{c}≫ 1), the (common) copier contribution to the variance and correlation (≈ 2) is much smaller than the other contributions (≈

*nL*/2). By comparing Eqs. (94) and (106), one finds that the copier makes a negligible contribution to the differential variance. Thus, the performance of a copier and

*n*-stage PS link does not depend on whether the copier is equalized or symmetric and depends only weakly on the whether the copier is present.

For reference, if the link is balanced, Eqs. (102)–(105) imply that

where the in-phase copier gain *G*
_{c0} = 2*G _{c}* − 1 + 2[

*G*(

_{c}*G*− 1)]

_{c}^{1/2}.

## 4. Discussion

For direct detection, the signal-to-noise ratio of mode *i* is 〈*n _{i}*〉

^{2}/〈

*δn*

^{2}

_{i}〉 and the noise figure associated with mode

*i*is the input ratio of the signal divided by the output ratio of mode

*i*. The correlation coefficient of modes

*i*and

*j*is 〈

*δn*〉/(〈

_{i}δn_{j}*δn*

^{2}

_{i}〉〈

*δn*

^{2}

_{j}〉)

^{1/2}. For a two-mode PI amplifier, which has one CS and one VS input, Eqs. (35)–(37) imply that the sideband noise figures

where *G*
_{1} = *G* and *G*
_{2} = *G* − 1 are the PI signal and idler gains, respectively. The correlation coefficient

These quantities are plotted as functions of the PI gain parameter *G* in Fig. 7. As the gain increases, the signal noise figure increases monotonically and the idler noise figure decreases monotonically, to their (common) asymptotic value of 2 (3 dB). This factor of 2 arises because only the signal contributes coherent components to the outputs, whereas the signal and idler both contribute incoherent components (noise). The correlation coefficient tends to 1 rapidly as the gain increases (*C*
_{12} ≈ 1 − 1/8*G*
^{2}), so only a moderate gain is required to produce a strong correlation.

For a two-mode PS amplifier, which has two CS inputs of equal strength, Eqs. (39) and (40) imply that the (common) noise figure

where *G _{θ}* = 2G − 1 + 2[

*G*(

*G*− 1)]

^{1/2}cos

*θ*is the PS gain and

*θ*is the phase difference between the pumps and sidebands.

*G*≥ 1 unless cos

_{θ}*θ*< 0 and G < 1/(1 − cos

^{2}

*θ*). The correlation coefficient

where *C _{θ}* = 2[

*G*(

*G*− 1)]

^{1/2}+ (2

*G*− 1)cos

*θ*describes the phase dependence of the correlation.

*C*≥ 0 unless cos

_{θ}*θ*< 0 and

*G*< [1 + 1/(1 − cos

^{2}

*θ*)

^{1/2}]/2. If the inputs are in-phase with the pumps (

*θ*= 0),

*C*

_{0}= 2

*G*− 1 + 2[

*G*(

*G*− 1)]

^{1/2}=

*G*

_{0}, whereas if the inputs are out-of-phase (

*θ*=

*π*),

*C*

_{0}= −

*G*

_{0}. The noise figure and correlation coefficient are plotted as functions of the PI gain in Fig. 8. If

*θ*= 0, the noise figure tends rapidly to 1/2 (−3 dB) as the gain increases, because the coherent components of the sidebands increase twice as rapidly as the noise components. (Their gain factors are 4

*G*and 2

*G*, respectively.) This result requires the inputs to be independent (uncorrelated). The correlation coefficient tends rapidly to 1, so (once again) only a moderate gain is required to produce a strong correlation. In contrast, if

*θ*=

*π*, the noise figure increases, because the coherent components decrease while the noise components increase, and the correlation coefficient tends rapidly to −1. The case in which

*θ*=

*π*/2 is intermediate.

For a two-mode attenuator (frequency convertor), in which the input signal is a CS and the input loss mode (idler) is a VS, Eqs. (45) and (46) imply that the noise figures

where *T*
_{1} = *T* and *T*
_{3} = 1 − *T* are the signal and loss-mode (idler) transmissions, respectively. The correlation coefficient

for all values of the transmissions. These results reflect the fact that the outputs are also independent CS. The sideband noise figures are plotted as functions of the loss parameter *L* = 1/*T* in Fig. 9. *F*
_{1} equals the loss parameter. It exceeds 1 because the attenuator does not decrease the noise component of the output signal as much as the coherent component. (If it did, the properties of a strongly-attenuated signal would violate the Heisenberg uncertainty principle.) In the high-loss regime (*L* ≫ 1), the coherent component of the output loss mode (idler) is comparable to that of the input signal, so *F*
_{3} ≈ 1. In an attenuator the loss mode is inaccessible, whereas in a frequency convertor the idler is an accessible copy of the signal.

For a copier (PI amplifier followed by two attenuators in parallel), Eqs. (55)–(57) imply that the noise figures

and the correlation coefficient

Equations (115) and (116) are based on the simplifying assumption that the attenuators are identical. (One could also use attenuators with slightly different transmissions to equalize the output strengths of the sidebands. However, the performance of the copier does not depend sensitively on whether it is equalized or symmetric.) If the attenuators are absent (*T* = 1), Eqs. (115) and (116) reduce to Eqs. (109) and (110), respectively. Conversely, if the amplifier is absent (*G* = 1), the signal version of Eq. (115) reduces to the signal version of Eq. (113). For a balanced copier (*TG* = 1), *F*
_{1} ≈ 3 − 2*T* and *F*
_{2} ≈ (3 − 2*T*)/(1 − *T*) and *C*
_{12} = 2(1 − *T*)^{1/2}/(3 − 2*T*). These quantities are plotted as functions of gain in Fig. 10. As the gain increases, the signal noise figure increases, and the idler noise figure decreases, to their (common) asymptotic value of 3 (4.8 dB). The correlation coefficient starts to increase as the gain increases. However, its growth saturates rapidly as it approaches its asymptotic value of 2/3, because the noise added by the attenuators is uncorrelated. A balanced copier produces output sidebands that have comparable strengths and are partially correlated. For an unbalanced copier (*TG* ≪ 1), *F _{i}* ≈ 1/

*TG*and

_{i}*C*

_{12}≈ 2

*TG*. The output sidebands are much noisier than the input signal (as befits weak nearly-CS), and are only weakly correlated.

For the first stage of an idealized PS link (parallel attenuators followed by a two-mode PS amplifier), which has two CS inputs of equal strength, Eqs. (67) and (68) imply that the (common) noise figure

where *G _{θ}* was defined after Eq. (111). The correlation coefficient

where *C _{θ}* was defined after Eq. (112). The right side of Eq. (117) is just the right side of Eq. (111) divided by T, because the attenuators replace CS by diminished CS. Equation (118) is identical to Eq. (112), because the attenuators do not influence the correlation, which is produced by the amplifier. The noise figure and correlation coefficient are plotted as functions of loss in Fig. 11, for a balanced link (

*TG*

_{0}= 1). Results are also included for the associated PI link [6]. As the loss increases, the PS and PI noise figures both increase. However, the PS noise figure increases less rapidly and, for large values of loss, is about 6-dB lower. The PS correlation coefficient also increases less rapidly than its PI counterpart. The PI and PS formulas for these quantities have the same dependences on the gain. Differences between the displayed results exist only because the relations between the gain and loss are different for the two links (

*G*=

*L*and

*G*≈

*L*/4, respectively). This example and the preceding one show that the order of amplification and attenuation is important.

For a cascaded PS amplifier (PI amplifier followed by parallel attenuators and a PS amplifier), Eqs. (77) and (78) imply that the signal noise figure

where *H*
_{0} = *G*
_{2}
*G*
_{1} + (*G*
_{2} − 1)(*G*
_{1} − 1) + 2[*G*
_{2}(*G*
_{2} − 1)*G*
_{1}(*G*
_{1} − 1)]^{1/2} is the in-phase gain of both amplifiers, and *G*
_{1} and *G*
_{2} are the PI gains of the first and second amplifiers, respectively. The formula for the idler noise figure is similar. (In the denominator, *H*
_{0} is replaced by *H*
_{0} − 1.) The correlation coefficient

If the second amplifier is absent (*H*
_{0} = *G*
_{1}), Eqs. (119) and (120) reduce to Eqs. (115) and (116), respectively. Conversely, if the first amplifier is absent (*H*
_{0} = *G*
_{2}), Eq. (119) reduces to Eq. (109), adjusted for a diminished input signal, and Eq. (120) reduces to Eq. (110).

If the gains are high (*G*
_{1} and *G*
_{2} ≫ 1), *H*
_{0} ≈ 4*G*
_{1}
*G*
_{2}, the (common) noise figure *F* ≈ [8*G*
_{2}
*TG*
_{1} + 2*G*
_{2}(1 − *T*)]/4*G*
_{2}
*TG*
_{1} ≥ 2 and the correlation coefficient *C*
_{12} ≈ 1. The outputs have comparable strengths and are strongly correlated, for all values of the gain ratios and transmission. For a low-loss copier (1 − *T* ≪ 1), *F* ≈ 2 (3.0 dB), so the cascaded PS amplifier has the same noise figure as a PI amplifier (but different PS gain). For a balanced copier (*TG*
_{1} = 1) with significant loss (*T* ≪ 1), *F* ≈ 2.5 (4.0 dB), which is 1.0 dB higher than the noise figure of a PI amplifier, but is 0.8 dB lower than that of the constituent copier. The second (PS) amplifier reduces the sideband noise, even though its inputs (which are the copier outputs) are partially correlated. The sideband noise figures and correlation coefficient are plotted as functions of the PI gain of the second amplifier in Fig. 12. Notice that only moderate values of *G*
_{2} are required to establish the properties of the cascaded PS amplifier:With the exception of the net gain *TH*
_{0}, the properties of this device depend only weakly on *G*
_{2}. For an unbalanced copier with high loss (*TG*
_{1} ≪ 1), *F* ≈ 1/2*TG*
_{1}, which is much higher than the noise figure of a PI amplifier, but is lower than that of the copier by a factor of 2 (3.0 dB). The second amplifier achieves its maximal noise reduction because its inputs are uncorrelated. The noise figure and correlation coefficient are plotted as functions of loss in Fig. 13. Results are also included for the copier alone. For small values of loss, the noise figures of both devices are approximately 3 dB, which is appropriate for high-gain amplifiers. As the loss increases, the noise figure of the cascaded PS amplifier increases more slowly than that of the copier. This result demonstrates (again) the beneficial effects of PS amplification. For the cascaded PS amplifier, the correlation coefficient is (almost) independent of loss, because its last constituent device is an amplifier. In contrast, for the copier alone, the correlation decreases as the loss increases.

For an idealized *n*-stage PS link (sequence of parallel attenuators and two-mode PS amplifiers), which has two CS inputs of equal strength, Eqs. (92) and (93) imply that the (common) noise figure

where *L* is the stage loss, and the correlation coefficient

Equations (121) and (122) are based on the (realistic) assumptions that the loss is high (*L* ≫ 1) and the link is balanced (*G*
_{0} = *L*, where *G*
_{0} is the in-phase stage gain). To be precise, for high losses *F* ≈ *nL*/2 and *C*
_{12} ≈ 1 − 2/*nL*
^{2}. The extra terms in Eqs. (121) and (122) increase their ranges of accuracy. The noise figure and correlation coefficient are plotted as functions of the stage loss in Fig. 14. Notice that the approximate results [Eqs. (121) and (122)] agree well with the exact results [Eqs. (92) and (93)], even for moderate values of loss. The asymptotic (high-loss) noise figure of an idealized PS link (*nL*/2) is smaller than that of a PI link (2*nL*) by a factor of 4 (6.0 dB). For both types of link, the noise figure is 2*nG*, where *G* is the PI stage gain. However, a balanced PI link requires that *G* = *L*, whereas a balanced PS link requires that *G* ≈ *L*/4: The higher efficiency of PS amplification allows PS links to operate with smaller values of *G* than PI links, and produce less noise. In each stage, the sidebands are diminished and decorrelated by the attenuators, then augmented and recorrelated by the amplifiers. The correlation evolution is nearly periodic, so the correlation coefficient depends only weakly on the number of stages.

A realistic *n*-stage PS link (copier followed by a sequence of parallel attenuators and PS amplifiers) has one CS input and one VS input (because the copier generates the idler that is required by the PS link). As stated above, the noise properties of a copier depend only weakly on whether it is equalized or symmetric. For a symmetric copier and a balanced link, Eqs. (92), (93), (107) and (108) imply that the (common) noise figure

where *G*
_{c0} = 2*G _{c}* − 1 + 2[

*G*(

_{c}*G*− 1)]

_{c}^{1/2}is the in-phase copier gain,

*G*is the PI copier gain and

_{c}*T*is the copier transmission. The correlation coefficient

_{c}For a balanced high-gain copier (*T _{c}G_{c}* = 1 and

*G*≫ 1, so

_{c}*G*

_{c0}≈ 4

*G*),

_{c}*F*≈ [5 +

*n*(

*L*− 1)]/2 and

*C*

_{12}≈ 1 − 2/(

*L*+ 1)[5 +

*n*(

*L*− 1)]. The noise figure and correlation coefficient are plotted as functions of the stage loss in Fig. 15. Once again, the approximate results [Eqs. (123) and (124)] agree well with the exact results [Eqs. (107) and (108)]. For small values of loss the effects of the copier are significant: The noise figure is 3 (rather than 1) and the correlation coefficient is 2/3 (rather than 0). In contrast, for large values of loss the copier effects are minor: The sidebands are still strongly correlated (

*C*

_{12}≈ 1 − 2/

*nL*

^{2}) and the copier contribution to the noise figure (2) is much less than the link contribution (

*nL*/2). Thus, one can use a copier to generate the idler that is required by a PS link, without sacrificing the 6-dB advantage of the link (relative to a PI link). This positive result was obtained because the moderate amount of noise produced by the copier is swamped by the large amount of noise produced by the link (loss and gain).

A cascaded PS attenuator is a cascaded PS amplifier with more loss than gain. For such a device, it was shown above that *F* ≈ 1/2*TG*
_{1}. If one were to split the total transmission *T* into two parts, the first *T _{c}* associated with copying and the second 1/

*L*associated with transmission, and relabel

_{t}*G*

_{1}as

*G*, one would find that

_{c}*F*≈

*L*/2

_{t}*T*, which is just Eq. (123) with

_{c}G_{c}*nL*replaced by

*L*. Furthermore, Eqs. (120) and (124) both imply that

_{t}*C*

_{12}≈ 1. Thus, the noise properties of a cascaded PS attenuator, which is straightforward to construct, mimic those of a realistic PS link, which is difficult to construct.

A common feature of devices that use two-mode amplifiers is the strong correlation between the output signal and idler. By measuring both sidebands, one can subtract the effects of electrical noise in the detectors [25] and improve the performances of these devices.

## 5. Summary

In this paper, formulas were derived for the field-quadrature and photon-number variances and correlations produced by multiple-mode parametric processes. These formulas were used to analyze the properties of basic devices, such as two-mode amplifiers, attenuators and frequency convertors, and composite systems made from these devices, such as cascaded parametric amplifiers and communication links. For these systems (and many others), the general formulas for the variances and correlations [Eqs. (16) and (19)] simplify significantly [Eq. (22) and Eq. (24) or (26)].

Two-mode amplifiers with one coherent-state (CS) input (signal) and one vacuum-state (VS) input are phase insensitive (PI), so the output signal powers do not depend on the input signal phases. These amplifiers generate idlers that are correlated with the amplified signals [Eqs. (35)–(37)]. The noise-figure of a device is the input signal-to-noise ratio divided by the output ratio. PI amplifiers have (high-gain) noise figures of 3 dB [Eq. (109)], because they add amplified VS fluctuations to the signals. In contrast, amplifiers with two CS inputs are phase sensitive (PS). These amplifiers correlate their input modes [Eqs. (39) and (40)] and have (in-phase) noise figures of −3 dB [Eq. (111)], because they combine the input amplitudes coherently, but only combine the CS fluctuations incoherently. (This remarkable performance is only possible if the inputs are uncorrelated.) Two-mode attenuators with one CS input and one VS input produce two uncorrelated CS outputs [Eqs. (45) and (46)]. The noise figure of an attenuator equals its loss factor [Eq. (113)]. Two attenuators acting in parallel on correlated modes (such as those produced by amplifiers) decorrelate the modes.

To operate in a PS manner, a two-mode amplifier requires two nonzero inputs. However, current communication systems are based on one-carrier-frequency signals. A standard way to produce the second input (copy the signal) is to use a two-mode amplifier, which amplifies the signal and generates an idler of comparable strength, and two attenuators, which can equalize the strengths of the output sidebands, reduce them to the level of the input signal and control their degree of correlation (as required). The noise figure of a balanced (zero-net-gain) copier is 4.8 dB [Eq. (115)].

A cascaded PS amplifier is a PI amplifier (which copies the signal as described above), followed by an optical processor (which controls the relative phase of the sidebands) and a PS amplifier (which combines the sidebands). Like its constituent amplifiers, this composite amplifier produces correlated sidebands [Eqs. (75) and (76) or Eqs. (77) and (78)]. The noise figure of a cascaded PS amplifier is 4 dB [Eq. (119)], which is 0.8-dB lower than that of a copier. However, it is 1-dB higher than that of a PI amplifier, because the second amplifier cannot compensate completely the noise added by the first amplifier and the processor (connecting fibers and splices). PS signal processing is obtained at only a moderate (noise) cost.

Two-mode PS links are sequences of transmission fibers (attenuators) followed by optical processors and two-mode PS amplifiers. An idealized link has two CS input sidebands. As these sidebands propagate through the link, they are periodically diminished and decorrelated by the attenuators, then augmented and re-correlated by the amplifiers [Eqs. (90) and (91)]. The noise figure of this PS link is proportional to its total loss, but is 6-dB lower than that of the corresponding PI link [Eqs. (121) and (123)]. Furthermore, the output sidebands are (almost) completely correlated (because the last element in the link is an amplifier), so one can eliminate the effects of electronic noise by detecting both sidebands. A realistic link requires only one nonzero input, because a copier placed before the first attenuator provides the required second input. Analyses show that the presence of the copier and the way in which it is configured (equalized or symmetric) have only minor effects on the noise properties of the link [Eqs. (98)–(100), and Eqs. (104) and (105)], because the (correlated) copier noise is swamped by the (uncorrelated) attenuator noise in the link. Hence, the predicted 6-dB noise advantage of a two-mode PS link is realizable.

In this paper, the number variances and correlations (moments) produced by parametric devices were described in detail. The relation between the amplitude and number moments is described in Appendix A, which also contains more examples of correlations affecting the performances of parametric devices. The main results of this paper were obtained by retaining the signal-noise contributions to the variances and correlations, and omitting the noise-noise contributions, which are usually smaller. For completeness, the latter contributions are calculated in Appendix B.

## Appendix A: Effects of correlations

If the noise-only contributions to the number variances and correlations are neglected, these second-order number moments depend only on the second-order amplitude moments (which include the amplitude correlations). Input amplitude correlations (produced by prior parametric processes) affect the output number moments and, hence, the noise properties of the current parametric process.

Let *a* be a mode operator, which satisfies the commutation relation [*a,a*
^{†}] = 1, and define the mode amplitude *α* = 〈*a*〉, where 〈 〉 is an expectation value. Then the noise operator *ν* = *a* − 〈*a*〉 satisfies the commutation relation [*ν*,*ν*
^{†}] = 1 and has zero mean. The number operator *m* = *a*
^{†}
*a*, which can be rewritten in terms of the noise operator as

Hence, the number mean

The squared number

where noise-only terms were neglected. The number variance 〈*δm*
^{2}〉 = 〈*m*
^{2}〉 − 〈*m*〉^{2}. By combining Eqs. (126) and (127), one finds that the number variance

The number variance depends on all four of the second-order self-moments, two of which are independent. (The fourth moment is the conjugate of the first, and the third differs from the second by 1.) If the mode is associated with a CS, the noise operator is a VS operator, so 〈*δm*
^{2}〉 = ∣*α*∣^{2} = 〈*m*〉, which is the standard result.

The applications considered in this paper involve two (or more) modes (sidebands), which are called the signal (1) and idler (2). Equations (125)–(128) apply to each mode separately. By combining the signal and idler versions of Eq. (125), one finds that the number product

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+{\alpha}_{1}^{*}{\alpha}_{2}^{*}\u3008{v}_{1}{v}_{2}\u3009+{\alpha}_{1}^{*}{\alpha}_{2}\u3008{v}_{1}{v}_{2}^{\u2020}\u3009+{\alpha}_{1}{\alpha}_{2}^{*}\u3008{v}_{1}^{\u2020}{v}_{2}\u3009+{\alpha}_{1}{\alpha}_{2}\u3008{v}_{1}^{\u2020}{v}_{2}^{\u2020}\u3009.$$

The number correlation 〈*δm*
_{1}
*δm*
_{2}〉 = 〈*m*
_{1}
*m*
_{2}〉 − 〈*m*
_{1}〉〈*m*
_{2}〉. By combining the signal and idler versions of Eqs. (126) with Eq. (129), one finds that the number correlation

The number correlation depends on all four of the second-order cross-moments (correlations), two of which are independent. (The fourth moment is the conjugate of the first, and the third is the conjugate of the second, because *ν*
_{1} and *ν*
_{2} commute.) If the sidebands are independent CS, 〈*δm*
_{1}
*δm*
_{2}〉 = 0.

As the modes propagate through a device, the amplitudes *α _{i}* and noise operators

*ν*evolve in the same way as the mode operators

_{i}*a*. (Specific IO equations for

_{i}*a*were stated in Sec. 3.) Equations (128) and (130) apply wherever one chooses to measure the sideband variances and correlation. Henceforth, the approximation signs will be omitted.

_{i}To illustrate how correlations develop in practice, some examples will be considered. The operation of a two-mode amplifier is governed by Eqs. (27) and (28). By combining these equations, one obtains the second-order IO equations

where *ν _{i}* and

*w*are input and output noise operators, respectively. Each of the output moments depends on input self- and cross-moments, so input correlations affect the outputs and, hence, the noise figures of two-mode amplifiers. Notice that the six moment equations decouple into two sets of three equations. The first set involves

_{i}*w*

^{2}

_{1},

*w*

^{2}

_{2}and

*w*

_{1}

*w*

^{†}

_{2}, whereas the second involves

*w*

^{†}

_{1}

*w*

_{1},

*w*

^{†}

_{2}

*w*

_{2}and

*w*

_{1}

*w*

_{2}. Equation (132) and its idler counterpart imply that

*w*

^{†}

_{1}

*w*

_{1}−

*w*

^{†}

_{2}

*w*

_{2}=

*ν*

^{†}

_{1}

*ν*

_{1}−

*ν*

^{†}

_{2}

*ν*

_{2}. Regardless of the input conditions, noise photons are produced in pairs [19, 20].

If the inputs are CS, the first set of output moments are zero, whereas the other output moments

By combining Eqs. (128), (130) and (135), one obtains the number variances and correlation

respectively, where the *β _{i}* are output amplitudes. Equations (136) and (137) are consistent with Eqs. (31)–(33).

If two amplifiers (also labeled 1 and 2) are concatenated, the composite amplifier is governed by IO equations of the forms (27) and (28), in which the composite transfer coefficients *$\overline{\mu}$* = *μ*
_{2}
*μ*
_{1} + *ν*
_{2}
*ν*
^{*}
_{1} and *$\overline{\nu}$* = *μ*
_{2}
*ν*
_{1} + *ν*
_{2}
*μ*
^{*}
_{1}. These coefficients have the squared moduli

which depend on the relative phase
$\theta ={\varphi}_{{\mu}_{2}}-{\varphi}_{{\nu}_{2}}+{\varphi}_{{\mu}_{1}}+{\varphi}_{{\nu}_{1}}$
, and satisfy the auxiliary equation ∣*$\overline{\mu}$*∣^{2} − ∣*$\overline{\nu}$*∣^{2} = 1. One can control the relative phase by varying the pump phases or imposing phase shifts on the sidebands between the amplifiers. If the amplifiers combine constructively (*θ* = 0), ∣*$\overline{\mu}$*∣ = ∣*μ*
_{2}
*μ*
_{1}∣ + ∣*ν*
_{2}
*ν*
_{1}∣ and ∣*$\overline{\nu}$*∣ = ∣*μ*
_{2}
*ν*
_{1}∣ + ∣*ν*
_{2}
*μ*
_{1}∣, whereas if they combine destructively (*θ* = *π*), ∣*$\overline{\mu}$*∣ = ∣*μ*
_{2}
*μ*
_{1}∣ − ∣*ν*
_{2}
*ν*
_{1}∣ and ∣*$\overline{\nu}$*∣ = ∣*μ*
_{2}
*ν*
_{1}∣ − ∣*ν*
_{2}
*μ*
_{1}∣. The noise figure of the composite amplifier is determined solely by the transmission coefficient ∣*$\overline{\nu}$*∣ (or ∣*$\overline{\mu}$*∣). Suppose that the amplifiers have equal and high PI gains. If the amplifiers combine constructively, ∣*$\overline{\mu}$*∣ = ∣*μ*∣^{2} + ∣*ν*∣^{2} ≫ 1 and ∣*$\overline{\nu}$*∣ = 2∣*μν*∣ ≫ 1: The noise figures of the first amplifier and the composite amplifier both are 3 dB. This result implies that the effective noise figure of the second amplifier is 0 dB, which is not possible if the inputs to the second amplifier are uncorrelated. (The effective noise figure is the input SNR divided by the output SNR, without the proviso that the input is a CS.) Conversely, if the amplifiers combine destructively, ∣*$\overline{\mu}$*∣ = 1 and ∣*$\overline{\nu}$*∣ = 0: The noise added by the first amplifier is removed by the second amplifier, which also is not possible if the inputs to the second amplifier are uncorrelated.

According to Eqs. (135), the nonzero moments produced by the first amplifier are ∣*ν*
_{1}∣^{2} and *μ*
_{1}
*ν*
_{1}. By combining these results with Eq. (132), one finds that the nonzero output moment

which equals ∣*$\overline{\nu}$*∣^{2} [Eq. (139)]. Thus, to predict correctly the properties of the composite amplifier, one is required to account for the correlation between the inputs to the second amplifier. If this correlation were absent, Eqs. (128) and (132) would underestimate the output variance in the constructive regime (by a factor of 2) and overestimate it in the destructive regime (because both nonzero contributions would be positive).

The operation of a two-mode attenuator (frequency convertor) is governed by Eqs. (41) and (42). By combining these equations, one obtains the second-order IO equations

where mode 3 is the loss mode (idler). Once again, each of the output moments depends on the input self- and cross-moments, so input correlations affect the outputs and, hence, the noise figures of two-mode attenuators (frequency convertors). Notice that the six moment equations decouple into two sets of three equations. The first set involves *w*
^{2}
_{1}, *w*
^{2}
_{3} and *w*
_{1}
*w*
_{3}, whereas the second involves *w*
^{†}
_{1}
*w*
_{1}, *w*
^{†}
_{3}
*w*
_{3} and *w*
_{1}
*w*
^{†}
_{3}. (This decomposition differs from the previous one.) Equation (142) and its idler counterpart imply that *w*
^{†}
_{1}
*w*
_{1} + *w*
^{†}
_{3}
*w*
_{3} = *ν*
^{†}
_{1}
*ν*
_{1} + *ν*
^{†}
_{3}
*v*
_{3}. Regardless of the input conditions, the total number of noise photons is conserved [19, 20].

If both inputs are CS, all six normally-ordered output moments are zero. The only nonzero moments are the anti-normally-ordered moments 〈*w _{i}w*

^{†}

_{i}〉 = 1. By combining these results with Eqs. (128) and (130), one obtains the number variances and correlation

respectively. Equations (145) and (146) reflect the fact that input CS are converted into different output CS. They are consistent with Eqs. (45) and (46).

If the input signal is an amplified CS (and the input loss mode is a VS), the nonzero input moment 〈*ν*
^{†}
_{1}
*ν*
_{1}〉 = ∣*ν*∣^{2}. By combining this result with Eqs. (141)–(144), one obtains the nonzero output moments

In contrast to the previous case, the outputs contain noise photons and are correlated. By combining Eqs. (128), (130) and (147), one obtains the number variances and correlation

respectively. By combining Eqs. (148)–(150), one finds that

The right side of Eq. (151) is the variance of the input signal: Because the attenuator preserves the number of photons, the total output variance must equal the input variance, which would not be possible if the outputs were uncorrelated. (The variance of the input loss mode is zero.)

Now consider a parametric amplifier followed by a frequency convertor that operates on the same sidebands (1 and 2). By combining Eqs. (27) and (28) with Eqs. (41) and (42), one obtains the composite IO equations

For simplicity, suppose that *τ* = 1/2^{1/2} = *ρ*. Then Eqs. (152) and (153) can be rewritten in the simple forms

respectively, where the superposition modes *a*
_{±} = (*a*
_{1} ± *a*
_{2})/2^{1/2}. If the input signal and idler are independent CS with amplitudes *α*
_{1} and 0, respectively, the superposition modes are independent CS with (common) amplitude *α*
_{1}/2^{1/2}. Equations (154) and (155) show that the outputs of the composite device are independent squeezed CS with amplitudes (∓*μα*
_{1} + *να*
^{*}
_{1})/2^{1/2} [26]: The frequency convertor decorrelates the amplifier outputs. Squeezed CS are discussed in [6–8]. Their output numbers and variances depend on the phases of the input amplitudes and transfer coefficients.

As stated previously, the nonzero moments produced by the amplifier are 〈*ν*
^{†}
_{i}
*ν _{i}*〉 = ∣

*ν*∣

^{2}and 〈

*ν*

_{1}

*ν*

_{2}〉 =

*μν*. By combining these results with Eqs. (141)–(144), one finds that the nonzero moments produced by the composite device are

where the + and − signs apply to modes 1 and 2, respectively. The frequency convertor destroys the cross-moment 〈*ν*
_{1}
*ν*
_{2}〉 and creates the self-moments 〈*w*
^{2}
_{i}〉, which would not be possible if the inputs were uncorrelated. It is these self-moments that make the noise properties of the device PS. By combining Eqs. (128), (130) and (156), one obtains the number variances and correlation

respectively. Equation (157) is consistent with Eq. (16) and Eq. (158) confirms the statement that the outputs are uncorrelated.

## Appendix B: Noise-noise contributions

The results of this paper were obtained by retaining the signal-noise contributions to the quantities of interest and omitting the noise-noise contributions. It was shown in [6] and Sec. 2.2 that the noise-noise contributions to the output variances and correlations are

respectively. If *i* = *j*, Eq. (160) reduces to Eq. (159), as it must do. For the applications considered herein, *μ _{ik}* and

*ν*are not nonzero simultaneously. For such applications, Eqs. (159) and (160) can be simplified. The first terms on the right sides are zero. Consider the second term on the right side of Eq. (159). If

_{ik}*k*is like

*i*(both odd or even) and

*l*is unlike

*i*(one odd and the other even), the contribution to the summation is ∣

*μ*∣

_{ik}ν_{il}^{2}with

*k*<

*l*. Conversely, if

*k*is unlike and

*l*is like, the contribution is ∣

*μ*∣

_{il}ν_{ik}^{2}with

*l*>

*k*, which is equivalent to the first type of contribution with

*k*>

*l*. Hence, the output variance

where *k* is like *i*, *l* is unlike *i* and the subscript nn was omitted. The right side of Eq. (161) is manifestly real. Now consider the second term on the right side of Eq. (160). If *i* and *j* are like, the contributions are *μ _{ik}ν*

^{*}

_{il}

*μ*

^{*}

_{jk}

*ν*=

_{jl}*μ*

_{ik}μ^{*}

_{jk}ν^{*}

*with*

_{il}ν_{jl}*k*<

*l*or

*μ*

_{il}ν^{*}

_{ik}μ^{*}

*=*

_{jl}ν_{jk}*μ*

_{il}μ^{*}

_{jl}ν^{*}

*with*

_{ik}ν_{jk}*l*>

*k*. Hence, the output correlation

where *k* is like and *l* is unlike. Equation (3) ensures that the right side of Eq. (162) is real. If *i* = *j*, Eq. (162) reduces to Eq. (161), as it must do. Conversely, if *i* and *j* are unlike, the contributions are *μ _{ik}ν*

^{*}

_{il}μ^{*}

*=*

_{jl}ν_{jk}*μ*

_{ik}ν_{jk}μ^{*}

_{jl}ν^{*}

_{il}with

*k*<

*l*or

*μ*

_{il}ν^{*}

_{ik}μ^{*}

*=*

_{jk}ν_{jl}*μ*

_{il}ν_{jl}μ^{*}

_{jk}ν^{*}

_{ik}with

*l*>

*k*. Hence, the output correlation

where *k* is like *i* and *l* is unlike *i*. Equation (2) ensures that the right side of Eq. (163) is real.

It is instructive to consider some examples. Two-mode amplification is governed by Eqs. (27) and (28), in which the signal operator *a*
_{1} is coupled to the conjugate of the idler operator *a*
^{†}
_{2}. The nonzero transfer coefficients *μ*
_{11} = *μ* = *μ*
_{22} and *ν*
_{12} = *ν* = *ν*
_{21}, where ∣*μ*∣^{2} = *G* is the PI gain and ∣*ν*∣^{2} = *G* − 1. It follows from these facts, and Eqs. (161) and (163), that

Because the amplitude fluctuations associated with a CS are the same as those associated with a VS, the noise-only contributions to the sideband variances are equal. The correlation equals the (common) variance because sideband photons are created in pairs (by the destruction of pump photons).

Two-mode attenuation (frequency conversion) is governed by Eqs. (41) and (42), in which the signal operator *a*
_{1} is coupled to the loss-mode (idler) operator *a*
_{3}. The nonzero transfer coefficients *μ*
_{11} = *τ* = *μ*
^{*}
_{33} and *μ*
_{13} = *ρ* = − *μ*
^{*}
_{31}. It follows from these facts, and Eqs. (161) and (162), that

These results reflect the fact that attenuation (frequency conversion) is a stable process. Unlike amplification, which is an unstable process, there is no mechanism to convert input vacuum fluctuations into output number fluctuations. (In frequency conversion, signal and pump photons are destroyed, and idler and other pump photons are created.)

Now consider four-mode processes, of which amplification followed by attenuation (Sec. 3.3), attenuation followed by amplification (Sec. 3.4) and cascaded PS amplification (Sec. 3.5) are examples. It follows from Eqs. (161)–(163) that

Equations (166)–(169) are stated explicitly because they apply to all of the aforementioned examples. It is easy to verify that they are consistent with Eqs. (159) and (160).

Amplification followed by an attenuation is governed by Eqs. (48) and (49), in which *ν*
_{14} = 0 = *ν*
_{23}. It follows from these facts and Eqs. (166)–(168) that

where *G* is the PI gain, and *T*
_{1} and *T*
_{2} are the signal and idler transmissions, respectively. Equations (170) and (171) are consistent with Eq. (52) of [18], which were obtained by a different method. As the transmissions *T _{j}* tend to zero, the noise-only contributions to the variances tend to zero as the first power of

*T*, whereas the noise-only contributions to the correlation tends to zero as the second power. Hence, attenuation decorrelates the sideband fluctuations completely.

_{j}Attenuation followed by amplification is governed by Eqs. (59) and (60). It follows from Eqs. (166)–(168) that

Equations (173) are identical to Eqs. (164) because attenuators replace CS and VS by diminished CS and identical VS, respectively, all of which have the same amplitude fluctuations, so prior attenuation does not change the noise-only contributions to the output variances and correlation.

Cascaded PS amplification is governed by Eqs. (69) and (70), from which it follows that

In Eqs. (174)–(176), the *T* terms represent fluctuations that were transmitted through both amplifiers (*H*
_{0}), whereas the 1 − *T* terms represent fluctuations that were added by the attenuators and transmitted through the second amplifier (*G*
_{2}).

A noteworthy feature of the preceding equations is the occurrence of the signal-like terms *H*
_{0} and *G*
_{2}, the idler-like terms *H*
_{0} − 1 and *G*
_{2} − 1 and the (symmetric) product terms [*H*
_{0}(*H*
_{0} − 1)]^{1/2} and [*G*
_{2}(*G*
_{2} − 1)]^{1/2}. One can explain why these terms occur by writing the sideband IO equations in the compact forms

where the effective-input-mode operators *c*
_{1o} = *μ*
_{11}
*a*
_{1} + *μ*
_{13}
*a*
_{3}, *c*
_{1e} = *ν*
^{*}
_{12}
*a*
_{2} + *ν*
***
_{14}
*a*
_{4}, *c*
_{2o} = *ν*
^{*}
_{21}
*a*
_{1} + *ν*
***
_{23}
*a*
_{3} and *c*
_{2e} = *μ*
_{22}
*a*
_{2} + *μ*
_{24}
*a*
_{4}. These effective-mode operators satisfy the commutation relations listed in Table 1. As an example of how to read the table, the entry ∣*μ*
_{11}∣^{2} + ∣*μ*
_{13}∣^{2} in the first row and first column is the value of [*c*
_{1o},*c*
^{†}
_{1o}]. All commutators of the form [*x,y*] are zero. The effective-mode operators also have the property that [*x,y*
^{†}] = 〈*xy*
^{†}〉, where the expectation value is associated with a (four-mode) vacuum state.

Equation (177) converts a four-mode process into a two-effective-mode process, the properties of which are well known. By proceeding in the standard way, one finds that the signal number 〈*n*
_{1}〉 = 〈*c*
_{1e}
*c*
^{†}
_{1e}〉 and the squared number

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.8em}{0ex}}=\u3008{c}_{1o}{c}_{1o}^{\u2020}\u3009\phantom{\rule{.2em}{0ex}}\u3008{c}_{1e}{c}_{1e}^{\u2020}\u3009+{\u3008{c}_{1e}{c}_{1e}^{\u2020}\u3009}^{2}.$$

Terms that are obviously zero (because they involve 〈0∣*c*
^{†} or *c*∣0〉) were omitted from the first version of Eq. (178). One obtains the second version from the first by applying the effective-mode commutation relations described above. By combining the formulas for the first and second powers of the signal number, one finds that the signal variance

Equation (179) is consistent with Eq. (166). The IO equations (177) are related by the interchanges 1 ↔ 2 and o ↔ *e*. Hence, the idler number 〈*n*
_{2}〉 = 〈*c*
_{2o}
*c*
^{†}
_{2o}〉 and the idler variance

Equation (180) is consistent with Eq. (167). By proceeding in a similar way, one finds that the number product

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}=\u3008{c}_{1o}{c}_{2o}^{\u2020}\u3009\u3008{c}_{1e}{c}_{2e}^{\u2020}\u3009+\u3008{c}_{1e}{c}_{1e}^{\u2020}\u3009\u3008{c}_{2o}{c}_{2o}^{\u2020}\u3009,$$

from which it follows that the correlation

Equation (182) is consistent with Eq. (168).

In the context of a cascaded PS amplifier, the only modes of interest are the unlike modes 1 and 2, so the preceding analysis is sufficient. However, in other applications, such as four-mode amplification [5], the properties of modes 3 and 4 also of interest. The IO equation for mode 3 is related to the first of Eqs. (177) by the interchange 1 ↔ 3, so the variance of mode 3 and the correlation between modes 2 and 3 follow from Eqs. (179) and (182), respectively. For the like modes 1 and 3, a separate calculation is required. By following the procedure described above, and using the commutation relations listed in Table 2, one finds that the number product

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}=\u3008{c}_{1o}{c}_{3o}^{\u2020}\u3009\phantom{\rule{.2em}{0ex}}\u3008{c}_{1e}{c}_{3e}^{\u2020}\u3009+\u3008{c}_{1e}{c}_{1e}^{\u2020}\u3009\u3008{c}_{3e}{c}_{3e}^{\u2020}\u3009.$$

Hence, the correlation

Equation (184) is consistent with Eq. (169). One can deduce the properties of mode 4 from those of mode 2. Thus, the variances and correlations produced by four-mode (and multiple-mode) processes are determined by the commutators of the odd and even effective-mode operators, as are the variances and correlations produced by the prototypical two-mode processes of amplification (Sec. 3.1) and frequency conversion (Sec. 3.2).

## Acknowledgments

CJM acknowledges useful discussions with S. Radic and M. G. Raymer.

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