## Field-quadrature and photon-number correlations produced by parametric processes |

Optics Express, Vol. 18, Issue 19, pp. 19792-19823 (2010)

http://dx.doi.org/10.1364/OE.18.019792

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### 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

1. J. Hansryd, P. A. Andrekson, M. Westland, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. **8**, 506–520 (2002). [CrossRef]

5. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express **12**, 5037–5066 (2004). [CrossRef] [PubMed]

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

7. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. **34**, 709–759 (1987). [CrossRef]

9. N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. **101**, 205–212 (1993). [CrossRef]

10. J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. **26**, 367–369 (2001). [CrossRef]

5. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express **12**, 5037–5066 (2004). [CrossRef] [PubMed]

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

11. R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. **17**, 1845–1847 (2005). [CrossRef]

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

## 2. General results

*a*is an input-mode operator,

_{i}*b*is an output-mode operator,

_{i}*μ*and

_{ik}*ν*are transfer coefficients, and † is a hermitian conjugate [5

_{ik}5. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express **12**, 5037–5066 (2004). [CrossRef] [PubMed]

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

*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}*θ*is the phase of a local oscillator (LO), and the input number operator

_{i}*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

_{i}*m*〉 = ∣

_{i}*α*∣

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

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

_{i}*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}*β*∣

_{i}^{2}depend on the input phases

*ϕ*= arg(

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

_{k}*a*∣

_{i}*α*〉 =

_{i}*α*∣

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

_{i}*ν*and

_{i}*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

*i*=

*j*, Eq. (10) reduces to Eq. (40) of [6

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

*i*≠

*j*, the right side of Eq. (10) involves summations of

### 2.2. Number fluctuations

*w*

^{(†)}have zero expectation values, that the output number correlation

*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}*w*

^{†}

_{i}

*w*

_{i}w^{†}

*〉 by using the reduced noise operators*

_{j}w_{j}*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}*i*=

*j*, Eq. (20) reduces to Eq. (42) of [6

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

*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.

*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

## 3. Applications

*μ*and

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

_{ik}*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

*k*is arbitrary. Conversely, if

*i*and

*j*are unlike,

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

### 3.1. Two-mode amplifier

*μ*and

*ν*satisfy the auxiliary equation ∣

*μ*∣

^{2}− ∣

*ν*∣

^{2}= 1 [7

7. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. **34**, 709–759 (1987). [CrossRef]

*G*= ∣

*μ*∣

^{2}, in which case ∣

*ν*∣

^{2}=

*G*−1.

*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

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

*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.

*α*

_{2}= 0), the output amplitudes

*β*

_{1}=

*μα*

_{1}and

*β*

_{2}=

*να*

^{*}

_{1}. The output variances and correlation

*α*

_{1}=

*α*

_{2}=

*α*), the (common) output strength ∣

*β*∣

^{2}=

*G*

_{θ}∣

*α*∣

^{2}, where

*G*= 2

_{θ}*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

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

*τ*and

*ρ*satisfy the auxiliary equation ∣

*τ*∣

^{2}+ ∣

*ρ*∣

^{2}= 1 [7

7. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. **34**, 709–759 (1987). [CrossRef]

*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

**12**, 5037–5066 (2004). [CrossRef] [PubMed]

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

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

*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.

### 3.3. Amplifier followed by attenuators

*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.

*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}.

*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

*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

### 3.4. Attenuators followed by an amplifier

*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}.

*α*replaced by

_{j}*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}*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

*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).

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

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

18. C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. **259**, 309–320 (2006). [CrossRef]

*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

14. J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express **18**, 4130–4137 (2010). [CrossRef] [PubMed]

*μ*and

_{c}*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

23. Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express **18**, 14820–14835 (2010). [CrossRef] [PubMed]

24. Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensative amplifiers,” Opt. Express **18**, 15426–15439 (2010). [CrossRef] [PubMed]

*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

*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

*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

*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).

*τ*

_{1}=

*τ*

_{2}=

*τ*), the output amplitudes

_{c}*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

*β*

^{2}

_{j}. By combining Eqs. (77) and (78), one finds that the differential variance

23. Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express **18**, 14820–14835 (2010). [CrossRef] [PubMed]

24. Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensative amplifiers,” Opt. Express **18**, 15426–15439 (2010). [CrossRef] [PubMed]

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

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

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

16. R. E. Slusher and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. **8**, 466–477 (1990). [CrossRef]

*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,

*n*-stage PS link, the composite IO equations are

*r*+ 1 and 2

*r*+ 2 are the loss modes of stage

*r*. The polynomials

*p*and

_{n}*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

*α*, the output amplitudes

_{i}*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.

*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

*n*-stage PS link (Fig. 6), the composite IO equations are

*μ*and

_{c}*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.

*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

*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

*n*= 1 and

*τ*= 1, Eqs. (98)–(101) reduce to Eqs. (71)–(74), respectively.

*τ*

_{1}=

*τ*

_{2}=

*τ*), the output amplitudes

_{c}*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}*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

*β*

^{2}

_{j}. By combining Eqs. (104) and (105), one finds that the copier contributions to the differential variance are

*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.

*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.

*G*

_{c0}= 2

*G*− 1 + 2[

_{c}*G*(

_{c}*G*− 1)]

_{c}^{1/2}.

## 4. Discussion

*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

*G*

_{1}=

*G*and

*G*

_{2}=

*G*− 1 are the PI signal and idler gains, respectively. The correlation coefficient

*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.

*G*= 2G − 1 + 2[

_{θ}*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

*T*

_{1}=

*T*and

*T*

_{3}= 1 −

*T*are the signal and loss-mode (idler) transmissions, respectively. The correlation coefficient

*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.

*G*was defined after Eq. (111). The correlation coefficient

_{θ}*C*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 (

_{θ}*TG*

_{0}= 1). Results are also included for the associated PI link [6

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

*G*=

*L*and

*G*≈

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

*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

*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).

*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

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

*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

*G*

_{c0}= 2

*G*− 1 + 2[

_{c}*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}*F*≈ 1/2

*TG*

_{1}. If one were to split the total transmission

*T*into two parts, the first

*T*associated with copying and the second 1/

_{c}*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.

25. L. A. Krivitsky, U. L. Andersen, R. Dong, A. Huck, C. Wittmann, and G. Leuchs, “Electronic noise-free measurements of squeezed light,” Opt. Lett. **33**, 2395–2397 (2008). [CrossRef] [PubMed]

## 5. Summary

## Appendix A: Effects of correlations

*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

*δm*

^{2}〉 = 〈

*m*

^{2}〉 − 〈

*m*〉

^{2}. By combining Eqs. (126) and (127), one finds that the number variance

*δm*

^{2}〉 = ∣

*α*∣

^{2}= 〈

*m*〉, which is the standard result.

*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

*ν*

_{1}and

*ν*

_{2}commute.) If the sidebands are independent CS, 〈

*δm*

_{1}

*δm*

_{2}〉 = 0.

*α*and noise operators

_{i}*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}*ν*and

_{i}*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

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

*β*are output amplitudes. Equations (136) and (137) are consistent with Eqs. (31)–(33).

_{i}*=*μ ¯

*μ*

_{2}

*μ*

_{1}+

*ν*

_{2}

*ν*

^{*}

_{1}and

*=*ν ¯

*μ*

_{2}

*ν*

_{1}+

*ν*

_{2}

*μ*

^{*}

_{1}. These coefficients have the squared moduli

*ν*

_{1}∣

^{2}and

*μ*

_{1}

*ν*

_{1}. By combining these results with Eq. (132), one finds that the nonzero output moment

*∣*ν ¯

^{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).

*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

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

*w*

_{i}w^{†}

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

*ν*

^{†}

_{1}

*ν*

_{1}〉 = ∣

*ν*∣

^{2}. By combining this result with Eqs. (141)–(144), one obtains the nonzero output moments

*τ*= 1/2

^{1/2}=

*ρ*. Then Eqs. (152) and (153) can be rewritten in the simple forms

*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

26. C. J. McKinstrie, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. **257**, 146–163 (2006). [CrossRef]

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

*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

## Appendix B: Noise-noise contributions

**13**, 4986–5012 (2005). [CrossRef] [PubMed]

*i*=

*j*, Eq. (160) reduces to Eq. (159), as it must do. For the applications considered herein,

*μ*and

_{ik}*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

*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

*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

*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

*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

*ν*

_{14}= 0 =

*ν*

_{23}. It follows from these facts and Eqs. (166)–(168) that

*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

18. C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. **259**, 309–320 (2006). [CrossRef]

*T*tend to zero, the noise-only contributions to the variances tend to zero as the first power of

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

_{j}*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}).

*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

*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.

*n*

_{1}〉 = 〈

*c*

_{1e}

*c*

^{†}

_{1e}〉 and the squared number

*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

## Acknowledgments

## References and links

1. | J. Hansryd, P. A. Andrekson, M. Westland, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. |

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

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

4. | S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. |

5. | C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express |

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

7. | R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. |

8. | S. M. Barnett and P. M. Radmore, |

9. | N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. |

10. | J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. |

11. | R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. |

12. | R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, and P. Kumar, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express |

13. | C. Lundström, J. Kakande, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Experimental comparison of gain and saturation characteristics of a parametric amplifier in phase-sensitive and phase-insensitive mode,” ECOC 2009, paper 1.1.1. |

14. | J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express |

15. | R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” J. Quantum Electron. |

16. | R. E. Slusher and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. |

17. | M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express |

18. | C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. |

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

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

21. | C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express |

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

23. | Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express |

24. | Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensative amplifiers,” Opt. Express |

25. | L. A. Krivitsky, U. L. Andersen, R. Dong, A. Huck, C. Wittmann, and G. Leuchs, “Electronic noise-free measurements of squeezed light,” Opt. Lett. |

26. | C. J. McKinstrie, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. |

**OCIS Codes**

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

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

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: June 9, 2010

Revised Manuscript: August 26, 2010

Manuscript Accepted: August 28, 2010

Published: September 1, 2010

**Citation**

C. J. McKinstrie, M. Karlsson, and Z. Tong, "Field-quadrature and photon-number correlations produced by parametric processes," Opt. Express **18**, 19792-19823 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19792

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### References

- J. Hansryd, P. A. Andrekson, M. Westland, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002). [CrossRef]
- S. Radic, and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron,” E 88-C, 859–869 (2005). [CrossRef]
- P. A. Andrekson, and M. Westlund, “Nonlinear optical fiber based all-optical waveform sampling,” Laser Photon. Rev. 1, 231–248 (2007). [CrossRef]
- S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2, 489–513 (2008). [CrossRef]
- C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004). [CrossRef] [PubMed]
- C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005). [CrossRef] [PubMed]
- R. Loudon, and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987). [CrossRef]
- S. M. Barnett, and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, 1997).
- N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. 101, 205–212 (1993). [CrossRef]
- J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001). [CrossRef]
- R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005). [CrossRef]
- R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, and P. Kumar, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005). [CrossRef] [PubMed]
- C. Lundström, J. Kakande, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani and D. J. Richardson, “Experimental comparison of gain and saturation characteristics of a parametric amplifier in phase-sensitive and phase-insensitive mode,” ECOC 2009, paper 1.1.1.
- J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18, 4130–4137 (2010). [CrossRef] [PubMed]
- R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” J. Quantum Electron. 21, 766–773 (1985). [CrossRef]
- R. E. Slusher, and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990). [CrossRef]
- M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005). [CrossRef] [PubMed]
- C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259, 309–320 (2006). [CrossRef]
- J. M. Manley, and H. E. Rowe, “Some general properties of nonlinear elements–Part I. General energy relations,” Proc. IRE 44, 904–913 (1956). [CrossRef]
- M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).
- C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]
- M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010). [CrossRef]
- Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010). [CrossRef] [PubMed]
- Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensative amplifiers,” Opt. Express 18, 15426–15439 (2010). [CrossRef] [PubMed]
- L. A. Krivitsky, U. L. Andersen, R. Dong, A. Huck, C. Wittmann, and G. Leuchs, “Electronic noise-free measurements of squeezed light,” Opt. Lett. 33, 2395–2397 (2008). [CrossRef] [PubMed]
- C. J. McKinstrie, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. 257, 146–163 (2006). [CrossRef]

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