Abstract
In this paper the quantum noise properties of phase-insensitive and phase-sensitive parametric processes are studied. Formulas for the field-quadrature and photon-number means and variances are derived, for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios associated with the direct and homodyne detection of optical signals. The consequences of the aforementioned formulas are described for frequency conversion, amplification, monitoring, and transmission through sequences of attenuators and amplifiers.
©2005 Optical Society of America
1. Introduction
Four-wave-mixing (FWM) in a fiber, driven by one or two pump waves, makes possible the parametric amplification (PA) of an optical signal [1, 2]. In this process, two pump photons are destroyed, and one signal and one idler photon are created: The signal wave is amplified and an idler wave is generated. Recent improvements in highly-nonlinear fibers have allowed parametric amplifiers (PAs) to produce high signal gains over broad signal-frequency bandwidths [3, 4]. Another FWM process [5], in which one pump and one signal photon are destroyed, and one pump and one idler photon are created, makes possible tunable frequency conversion (FC), without gain, over broad bandwidths [6, 7]. Because of these demonstrated functionalities, parametric devices (PDs) are candidates for a variety of signal-processing applications [8, 9, 10].
PAs can be used to amplify optical signals before, during or after transmission through a fiber link. Because the idlers produced by PA, or FC, are frequency-shifted copies of the signals, by detecting the idlers one can monitor the signals without disrupting the flow of information. Transmission of the idlers, rather than the signals, makes possible frequency-dependent routing. Because the idlers produced by PA are phase-conjugate images of the signals, transmission of these idlers makes possible the reduction of impairments caused by phenomena such as pulse distortion, pulse arrival-time and phase jitter, and inter-pulse FWM. In the FWM processes that underly these applications, the powers of the output signals (idlers) do not depend on the phases of the input signals. Such processes are said to be phase insensitive (PI). One can make FWM processes phase sensitive (PS) by providing input idlers that beat with the signals, or choosing the pump and signal frequencies in ways that allow the signals to beat with themselves. The benefits of PS amplification include the reduction of noise, and pulse arrival-time and phase jitter, and the suppression of the modulation instability.
In the aforementioned applications, signal amplification and idler generation are accompanied by the addition of quantum-mechanical uncertainty (noise). Because these applications involve the transmission of information, it is important to determine the noise properties of the underlying parametric processes. In his classic review of the quantum-mechanical limits on noise production by linear amplifiers [11], Caves described the structure of many-mode amplification processes and derived formulas for the field-quadrature uncertainties associated with two-mode processes. (A mode is a frequency component of the electromagnetic field.) In this paper formulas are derived for the field-quadrature and photon-number uncertainties associated with many-mode parametric processes. The consequences of these formulas are described, for several applications of current interest.
This paper is organized as follows: In Section 2 the significance and properties of the photon-annihilation and -creation operators are described. These operators are used to analyze direct detection (DD) and homodyne detection (HD), which allow one to measure the photon number and field quadratures, respectively, of a mode. In Section 3 the modal input-output relations are stated for a variety of parametric processes: Individual processes, such as PA, FC and attenuation, and composite processes, such as parametric monitoring (PM), and transmission through sequences of attenuators and PI or PS amplifiers. Although these processes are diverse, their input-output relations all have the same form. The consequences of these canonical relations are determined in Section 4. Formulas are derived for the field-quadrature and photon-number means, which one can measure, and their variances, which characterize the measurement uncertainties. In Sections 5.1–5.6, the results of Section 4 are used to derive formulas for the signal-to-noise ratios (SNRs) and noise figures (NFs) associated with FC, PI amplification, transmission through sequences of attenuators and PI amplifiers, PI monitoring, PS amplification, and transmission through sequences of attenuators and PS amplifiers, respectively. Finally, in Section 6 the main results of this paper are summarized.
2. Description and detection of light
The quantum-mechanical theory of light [12, 13] is formulated in terms of the photon-annihilation and -creation operators α_{j} and ${\alpha}_{j}^{\u2020}$ , where j denotes a mode and † denotes a hermitian conjugate. These dimensionless mode operators obey the boson commutation relations [α_{j} ;a_{k} ]=0 and [α_{j} ;${a}_{k}^{\u2020}$ ]=δ_{jk} .
The photon-number operator n_{j} =${a}_{j}^{\u2020}$ α_{j} . In DD a photodiode is used to measure the photon-number mean 〈n_{j} ,〉 where 〈 〉 denotes an expectation value. The uncertainty in such a measurement is the photon-number variance 〈${\delta n}_{j}^{2}$ 〉=〈${n}_{j}^{2}$ 〉-〈n_{j} 〉^{2}. For DD the SNR is 〈n_{j} 〉^{2}/〈${\delta n}_{j}^{2}$ 〉.
The first quadrature operator q_{j} (ϕ)=(α_{j}e ^{-iϕ}+${a}_{j}^{\u2020}$ e ^{iϕ} )/2, where ϕis a phase angle, and the second quadrature operator p_{j} (ϕ)=q_{j} (ϕ+π/2). If α_{j} were a complex number, rather than an operator, q_{j} and p_{j} would be the real and imaginary parts of that number, measured in a coordinate system rotated by ϕ radians relative to the reference system. The quadrature operators obey the commutation relation [q_{j} ; p_{j} ]=i/2, which shows that they are conjugate operators (apart from factors of 2^{1/2}). The mean of the first quadrature is denoted by q_{j} and the variance ${\delta q}_{j}^{2}$ = ${q}_{j}^{2}$ - q_{j} /^{2}. Similar definitions apply to the second quadrature. It follows from the Heisenberg uncertainty principle that ${\delta q}_{j}^{2}$ ${\delta p}_{j}^{2}$ ≥1/16. States for which the equality sign applies are called minimal-uncertainty states: Coherent states have symmetric (circular) probability clouds, whereas squeezed states have asymmetric (elliptical) probability clouds.
Suppose that two modes, j and k, are combined by a beam-splitter, which has two input ports and two output ports. Then the output modes (at position z) are related to the input modes (at position 0) by the equations
The commutation relations [a _{1};${a}_{1}^{\u2020}$]=1 and [a _{2};${a}_{2}^{\u2020}$]=1 require the beam-splitter transfer functions µ̄ and ν̄ to satisfy the (same) auxiliary condition |µ̄|^{2}+|ν̄|^{2}=1. This condition ensures that the total photon-number is conserved. For reference, the functions |ν̄|^{2}=T and |ν̄|^{2}=1-T are called the transmittance and reflectance, respectively. By using DD at the output ports, one can measure the photon numbers n_{j} and n_{k} , and the photon-number difference d_{jk} =n_{j} -n_{k} . It follows from Eqs. (1) and (2) that
If the transmittance and reflectance are equal, the beam-splitter is said to be balanced.
In HD the mode to be measured (j) is combined with a local-oscillator (LO) mode (l) of the same frequency, before detection. It is customary to assume that the LO is a coherent state, for which a_{l} |α_{l} 〈=α_{l} /α_{l} 〈, where the coherent-state parameter α_{l} is a complex number. Let $\stackrel{\u0304}{\mu}=\mid \stackrel{\u0304}{\mu}\mid {e}^{i{\varphi}_{\stackrel{\u0304}{\mu}}}$,$\stackrel{\u0304}{\nu}=\mid \nu \mid {e}^{i{\varphi}_{\stackrel{\u0304}{\nu}}}$, ${\alpha}_{l}=\mid {\alpha}_{l}\mid {e}^{i{\varphi}_{l}}$ and ${a}_{k}={a}_{l}{e}^{i{\varphi}_{l}}$. Then, for balanced HD, one can rewrite Eq. (3) in the simpler form
where the relative phase ϕ=ϕ_{l} -ϕ_{µ} ̄+ϕ_{ν} ̄. Because the LO phase is included explicitly in Eq. (4), the LO eigenvalue equation is a_{l} |α_{l} 〈=|α_{l} ‖α_{l} 〈. It follows from Eq. (4) and the boson commutation relations that
The subscript l in the last term in Eq. (6) indicates that it originates from the commutation relation [a_{l} ;${a}_{l}^{\u2020}$ ]=1. For typical cases, in which the LO is much stronger than the measured mode (|α_{l} |^{2}≫〈n_{j} 〉), the last term can be neglected. Thus, balanced HD measures the quadratures of mode j, with uncertainties that are characteristics of mode j, rather than the LO. For balanced HD the SNR is 〈d_{j} 〉^{2}/〈${\delta d}_{j}^{2}$ 〉≈〈q_{j} 〉^{2}/〈δq_{j} 〉^{2}. Henceforth, for simplicity, ϕ will be called the LO phase. The measurement of optical fields is discussed in detail in [14].
3. Examples of parametric processes
In this section several examples of multiple-mode parametric interactions are described. PA driven by two pump waves (1 and 2) involves four product waves that are coupled by three distinct FWM processes, as illustrated in Fig. 1. Suppose that the signal frequency ω _{1+}=ω _{1}+ω, where ω is the modulation frequency, and let γ denote a photon. Then the modulation interaction (MI) in which 2γ _{1}→γ _{1}_+γ _{1+} produces an idler with frequency ω _{1}=ω _{1}-ω, the phase-conjugation (PC) process in which γ _{1}+γ _{2} !γ _{1}++γ _{2} produces an idler with frequency ω _{2}_=ω _{2}-ω and the Bragg scattering (BS), or FC, process in which γ _{1+}+γ _{2}→γ _{1}+γ _{2+} produces an idler with frequency ω _{2+}=ω _{2} +ω. It is customary to use a classical model for the strong (constant-amplitude) pumps and a quantal model for the weak (variable-amplitude) products (sidebands). In this approach, each of the preceding FWM processes involves two (interaction-picture) mode operators (one for each sideband). MI is characterized by the input-output relation
The commutation relation [a _{1}+;${a}_{1+}^{\u2020}$]=1 requires the MI transfer functions µ and ν to satisfy the auxiliary equation |µ|^{2}-|ν|^{2}=1. This condition ensures that transformation (7) is unitary.
A similar transformation characterizes the generation of the 1-idler. For reference, the functions |µ|^{2}=G and |ν|^{2}=G-1 are called the signal and idler gain, respectively. In quantum optics, transformation (7) is called a two-mode squeezing transformation [15].
PC is characterized by the input-output relation
where the PC transfer functions µ and ν also satisfy the auxiliary equation |µ|^{2}-|ν|^{2}=1. A similar relation characterizes the generation of the 2- idler. Notice that Eq. (8) has the same form as Eq. (7).
BS is characterized by the input-output relation
The BS transfer functions µ̄ and ν̄ satisfy the auxiliary condition |µ̄|^{2}+|ν̄|^{2}=1, which ensures that transformation (9) is unitary. A similar transformation characterizes the generation of the 2+ idler. For reference, the BS relations have the same form as the beam-splitter relations (1) and (2). Formulas for the MI, PC and BS transfer functions are stated in [16].
In a two-pump PA the aforementioned two-mode processes occur simultaneously. If one relabels the 1-, 1+, 2- and 2+ modes as 1, 2, 3 and 4, respectively, one can write the four-mode input-output relation in the form
where the four-mode transfer functions satisfy the auxiliary equation -|µ _{21}|^{2}+|µ _{22}|^{2}-|ν _{23}|^{2}+|µ _{24}|^{2}=1. Similar relations characterize the idler-generation processes. Formulas for the four-mode transfer functions are stated in [16].
The preceding example shows how p pumps couple the evolution of (at least) 2p sidebands. Although one can increase the number of coupled modes by increasing the number of pumps, the extent to which one does this is limited in practice (by fiber dispersion). One can couple many more modes by concatenation [17]. Consider a communication link that consists of s stages, in each of which fiber attenuation (loss) is followed by PA (gain). The architecture of a typical stage (r) is illustrated in Fig. 2.
The loss process at the beginning of stage r is characterized by the input-output relation
where mode 1 is the signal, mode 2r+1 is the scattered mode associated with the loss mechanism, ${z}_{r-1}^{\u2033}$ denotes the end of stage r-1 (beginning of stage r) and ${z}_{r}^{\prime}$ denotes the end of the fiber in stage r. The attenuator transfer functions µ̄ and ν̄ satisfy the auxiliary equation |µ̄|^{2}+|ν̄|^{2}=1. A similar relation characterizes the generation of the scattered mode, but is of lesser interest. For reference, the input-output relations for attenuators are identical to the relations for beam-splitters and frequency converters.
The gain process at the end of stage r is characterized by the input-output relation
where mode 2r is the idler, ${z}_{r}^{\u2033}$ denotes the end of stage r and |µ|^{2}-|ν|^{2}=1. A similar relation characterizes the idler-generation process, but is of lesser interest. Equation (12) has the same form as Eqs. (7) and (8).
By combining Eqs. (11) and (12), one obtains the composite input-output relation
$$+{\mu}_{12r+1}({z}_{r}^{\u2033},{z\u2033}_{r-1}){a}_{2r+1}\left({z\u2033}_{r-1}\right),$$
where the composite transfer functions µ _{11}=µµ̄, ν _{12r}=ν and µ _{12r+1}=µν̄. By using the auxiliary equations for the loss and gain processes, one can show that |µ _{11}|^{2}-|ν _{12r}|^{2}+|µ _{12r+1}|^{2}=1: The product of unitary transformations is also unitary. By iterating Eq. (13), one finds that signal transmission through the entire link is characterized by the many-mode input-output relation
For simplicity, in Eq. (14) the output point ${z}_{s}^{\u2033}$ was denoted by z_{s} and the input points of the idlers and scattered modes were denoted by 0. (One can formalize this relabeling of the input points by using step functions to extend the domains of the transfer functions.)
The idlers generated by PAs are frequency-shifted copies of the signals, and carry the same information. Consequently, by detecting only the generated idlers, one can access the information carried by the signals without disrupting the flow of information. In a real parametric monitor, coupling losses precede and follow the PA. Consequently, the input-output relation for the monitor [17] has the same form as the relation for a two-stage link, in which the second PA is absent (G _{2}=1).
The aforementioned amplification processes (MI and PC) were PI, because the idler was generated within the amplifier, and the idler phase is determined by the phases of the pump(s) and signal. These processes become PS in the presence of an externally-generated idler (second signal) against which the (first) signal can beat [2, 18]. One can produce such an idler by using BS prior to MI or PC [18, 19].
As stated earlier, the PC process in which γ _{1}+γ _{2}→γ _{1}+γ _{2-} involves waves whose frequencies satisfy the matching condition ω _{1}+ω _{2}=ω _{1}+ω _{2}. Equation (8) shows that the signal operator a _{1+} is coupled to the idler operator ${a}_{2-}^{\u2020}$. For the special case in which ω _{1}+ω _{2}=2ω _{1+}, which is illustrated in Fig. 3, the idler coincides with the signal and a _{1+} is coupled to ${a}_{1+}^{\u2020}$ [18, 19]. If the pump frequencies are sufficiently far apart that dispersion prevents the 1- and 2+ idlers from interacting strongly with the 1+/2- signal, one can characterize this degenerate PC process by the input-output relation
where the one-mode transfer functions µ and ν satisfy the auxiliary equation |µ|^{2}-|ν|^{2}=1. Formulas for these transfer functions are stated in [18]. In quantum optics, transformation (15) is called a one-mode squeezing transformation [13, 15]. It is intrinsically PS.
Conversely, if the pump frequencies are not far apart, one must account for the idlers. As stated earlier, the MI of pump 1 couples a _{1+} to ${a}_{1-}^{\u2020}$. The BS process in which γ _{1}+γ _{1+}→γ _{1}+γ _{2} also couples a _{1+} to a _{1-}. Likewise, the MI of pump 2, in which 2γ _{2}→γ _{1+} +γ _{2+}, couples a _{1+} to ${a}_{2+}^{\u2020}$ and the BS process in which γ _{1+} +γ _{2}→γ _{1}+γ _{2+} couples a _{1+} to a _{2+}. If one relabels the 1-, 1+/2- and 2+ modes as 1, 2 and 3, respectively, one can write the three-mode input-output relation in the form
$$+{\nu}_{22}\left(z\right){a}_{2}^{\u2020}\left(0\right)+{\mu}_{23}\left(z\right){a}_{3}\left(0\right)+{\nu}_{23}\left(0\right){a}_{3}^{\u2020}\left(0\right),$$
where the three-mode transfer functions satisfy the auxiliary equation-|ν _{21}|^{2}+|µ _{21}|^{2}-|ν ^{22}|^{2}+|µ _{22}|^{2}-|ν _{23}|^{2}+|µ _{23}|^{2}=1. The relations that characterize the idler-generation processes are similar, but not identical (a _{1} is coupled to ${a}_{3}^{\u2020}$ by PC, but is not coupled to a _{3}, and a _{3} is coupled to ${a}_{1}^{\u2020}$ by PC, but is not coupled to a _{1}).
Consider a communication link that consists of s stages, in each of which fiber loss is compensated by PS gain (produced by degenerate PC, for example). The architecture of a typical stage (r) is illustrated in Fig. 4. (The branching and rejoining of the signal line in the amplifier symbolizes the interaction of the signal with itself, rather than an idler.) The loss process at the beginning of stage r is characterized by the input-output relation
which is equivalent to relation (11), and the gain process at the end of stage r is characterized by the input-output relation
which is equivalent to relation (15). By combining Eqs. (17) and (18), one obtains the composite input-output relation
$$+{\mu}_{1r+1}({z\u2033}_{r},{z\u2033}_{r-1}){a}_{r+1}\left({z\u2033}_{r-1}\right)+{\nu}_{1r+1}({z}_{r}^{\u2033},{z\u2033}_{r-1}){a}_{r+1}^{\u2020}\left({z\u2033}_{r-1}\right),$$
where the composite transfer functions µ _{11}=µµ̄, ν _{11}=νµ̄*, µ _{1r+1}=µν̄ and ν _{1r+1}=νν̄*. By using the auxiliary equations for the loss and gain processes, one can show that |µ _{11}|^{2}-|ν _{11}|^{2}+ |µ _{1r+1}|^{2}-|ν _{1r+1}|^{2}=1: Once again, the composite transformation is unitary. By iterating Eq. (19), one finds that signal transmission through the entire link is characterized by the many-mode input-output relation
$$+\sum _{r=1}^{s}\left[{\mu}_{1r+1}\left({z}_{s}\right){a}_{r+1}\left(0\right)+{\nu}_{1r+1}\left({z}_{s}\right){a}_{r+1}^{\u2020}\left(0\right)\right],$$
where the output point ${z}_{s}^{\u2033}$ was denoted by z_{s} and the input points of the scattered modes were denoted by 0.
4. Properties of parametric processes
The modal input-output relations associated with the parametric processes described in Section 3 are all of the form [11]
where the transfer functions µ_{jk} and ν_{jk} couple the output annihilation operator of mode j to the input annihilation and creation operators of mode k, respectively. The boson commutation relations, which are valid for all distances, imply that
Suppose that the input signal is a coherent state with displacement (amplitude) α_{i} (0). Then the input state |α_{i} 〈=D(α_{i} )|0〈, where D(α_{i} ) is the displacement operator [13] and |0‹ is the vacuum state. It is customary to use the vacuum state as a virtual input state and incorporate the displacement required to produce the actual input state in the input-output relations associated with the process. By using the identity D ^{†}(α_{i} )α_{j}D(α_{i} )=α_{j} +α_{i}δ_{ij} , one can rewrite the input-output relation (21) in the form
where the complex number
is the output amplitude of mode j and the operator
describes the effects of vacuum fluctuations on mode j. Because the v_{j} operators only differ from the α_{j} operators by complex numbers, the v_{j} operators also must satisfy the boson commutation relations. The quadrature and number means and variances of mode j depend on the first-, second- and fourth-order moments of α_{j} . The first-order moment is Eq. (25), from which it follows that
$$+{\left({v}_{j}^{\u2020}{v}_{j}\right)}^{2}+{2\mid {\alpha}_{j}\mid}^{2}\left({\alpha}_{j}{v}_{j}^{\u2020}+{\alpha}_{j}^{*}{v}_{j}\right)+2{\mid {\alpha}_{j}\mid}^{2}{v}_{j}^{\u2020}{v}_{j}$$
$$+\left({\alpha}_{j}{v}_{j}^{\u2020}+{\alpha}_{j}^{*}{v}_{j}\right){v}_{j}^{\u2020}{v}_{j}+{v}_{j}^{\u2020}{v}_{j}\left({\alpha}_{j}{v}_{j}^{\u2020}+{\alpha}_{j}^{*}{v}_{j}\right).$$
Thus, to calculate the moments of α_{j} , one must first calculate the moments of v_{j} .
It follows from Eq. (27) that
where |1 _{k} 〈 represents a state with 1 photon in mode k and no photons in the other modes. Because different number states are orthogonal, Eqs. (32) and (33) imply that the first-order moments 〈v_{j} 〉 and 〈${v}_{j}^{\u2020}$ 〉 are both zero. It follows from Eqs. (27), (32) and (33) that
where |2 _{k} 〉 represents a state with 2 photons in mode k and no photons in the other modes, and |1 _{k} 1 _{l} 〉 represents a state with 1 photon in mode k, one photon in mode l and no photons in the other modes. The second-order moments are all nonzero. Notice that 〈v_{j} ${v}_{j}^{\u2020}$ 〉-〈${v}_{j}^{\u2020}$ v_{j} 〉=1, as it must. Because each state in Eqs. (34)–(37) differs from the vacuum state by zero or two raising operations, each state in the third-order moment equations must differ from the vacuum state by one or three raising operations. Consequently, the third-order moments must all be zero. Because the operator ${v}_{j}^{\u2020}$ v_{j} is hermitian, one can deduce the fourth-order moment 〈(${v}_{j}^{\u2020}$ v_{j} )^{2}〉 from Eq. (36) and the identity ∑ _{k} ∑_{l≠k} µ* _{jl} ν _{jk} |1 _{k} 1 _{l} 〉=∑ _{k} ∑_{l>k}(µ* _{jk} ν_{jl} +µ* _{jl} ν_{jk} )|1 _{k} 1 _{l} 〉.
By using the preceding results, one finds that the quadrature mean and variance are respectively. The quadrature mean depends on the output amplitude, which depends on the phases of the input signal and the transfer functions, and the LO phase. In contrast, the quadrature variance depends on the transfer functions and the LO phase, but not on the output amplitude. Let λ_{jk} =µ_{jk}e ^{-iϕ}+ν* _{jk}e^{iϕ} . Then Eq. (39) can be rewritten in the compact form
The number mean and variance are
$$+2\sum _{k}{\mid {\mu}_{jk}{\nu}_{jk}\mid}^{2}+\sum _{k}\sum _{l>k}{\mid {\mu}_{jk}^{*}{\nu}_{jl}+{\mu}_{jl}^{*}{\nu}_{jk}\mid}^{2},$$
respectively. The number mean depends implicity on the phases of the input signal and the transfer functions. However, the number of noise photons ∑| _{k} |ν_{jk} |^{2}=∑ _{k} |µ_{jk} |^{2}-1 depends only on the magnitudes of the transfer coefficients. In contrast, the number variance depends on the phases of the output amplitude and the transfer coefficients. Let ${\alpha}_{j}=\mid {\alpha}_{j}\mid {e}^{i{\varphi}_{j}}$ and ${\lambda}_{jk}^{\text{'}}={\nu}_{jk}{e}^{-i{\varphi}_{j}}+{e}_{jk}^{*}{e}^{i{\varphi}_{j}}$. (The definition of ${\lambda}_{\mathit{\text{jk}}}^{\prime}$ differs slightly from the definition of λ_{jk} .) Then Eq. (42) can be rewritten in the compact form In the context of our model, which was described at the beginning of Section 3, Eqs. (38)–(43) are exact.
If there is more than one (coherent) input signal, the input state |{α_{i} }〉=∏ _{i}D(α_{i} )|0〉. The output amplitude is defined by the equation
which is a generalization of Eq. (26). The vacuum operator is still defined by Eq. (27). Hence, the quadrature and number means and variances are given by Eqs. (38)–(43), together with the modified amplitude equation (44). Other input states are considered briefly in Appendix A.
5. Selected applications
As stated in Section 2, the SNR associated with HD is 〈q_{j} 〉^{2}/〈${\delta q}_{j}^{2}$ 〉, whereas the SNR associated with DD is 〈n_{j} 〉^{2}/〈${\delta n}_{j}^{2}$ 〉. For a coherent input signal with state-parameter (amplitude) ${\alpha}_{i}=\mid {\alpha}_{i}\mid {e}^{i{\varphi}_{i}}$, the quadrature mean 〈q_{i} 〉=|α_{i} |cos(ϕ_{i} -ϕ), where ϕ_{i} and ϕ are the input-signal and LO phases, respectively. The quadrature mean attains its maximal value |a_{i} | when ϕ=ϕ_{i} , whereas the quadrature variance 〈${\delta q}_{i}^{2}$ 〉=1/4 for any value of ϕ. The photon-number mean 〈n_{i} 〉=|α_{i} |^{2} and variance 〈${\delta n}_{i}^{2}$ 〉=|α_{i} |^{2}. Hence, for HD the maximal input-signal SNR
whereas for DD the input-signal SNR
Although HD is more complicated than DD, it is more sensitive by a factor of 4 (6 dB).
The effects of parametric processes on the transmitted signal and generated idler(s) were determined in Section 4. It follows from Eqs. (38) and (40) that for HD the output SNR
where ${\alpha}_{j}=\mid {\alpha}_{j}\mid {e}^{i{\varphi}_{j}}$ is the output amplitude of mode j and λ_{jk} was defined before Eq. (40). As the following sections demonstrate, the SNR is often maximal when ϕ=ϕ_{j} . For these cases
It follows from Eqs. (41) and (43) that for DD the SNRs
where ${\lambda}_{\mathit{\text{jk}}}^{\prime}$ was defined before Eq. (43). For a many-photon input signal (|α_{i} |^{2}≫1), the stimulated terms in Eq. (48), which depend on the output strength |α_{j} |^{2}, are usually much larger that the spontaneous terms, which do not. For the usual cases
For each output mode (signal or idler) and detection method (HD or DD), the NF of the process is the SNR of the input signal divided by the SNR of the output mode.
5.1 Frequency conversion
The noise properties of PI convertors (or attenuators) and PI amplifiers were reviewed recently [16]. In Sections 5.1 and 5.2 these properties are described briefly, because they determine the properties of PI links, and contrast with the properties of PS devices and links.
Consider two-mode FC, which is made possible by BS (Fig. 1). Let the signal and idler modes be labeled 1 and 2, respectively, and suppose that the input signal is a coherent state with amplitude α_{1}(0). Then it follows from the results of Section 4 that the output quantities α_{j} (z)=µ_{j} _{1} α _{1}(0), 〈q_{j} (z)〉=|α_{j} (z)|, 〈${\delta q}_{j}^{2}$ (z)〉=1/4, 〈n_{j} (z)〉=|α_{j} (z)|^{2} and 〈${\delta n}_{j}^{2}$ (z)〉=|α_{j} (z)|^{2}, where µ _{11}=µ̄, µ _{21}=-ν̄* and the LO phase ϕ=arg[α_{j} (z)] is optimal. This standard FC process is PI, because the output photon-numbers (powers) of the signal and idler do not depend on the phase of the input signal.
The preceding results reflect the fact that the transmitted signal and generated idler are also coherent states. Hence, Eqs. (45) and (46) apply to HD and DD, respectively, with α_{i} replaced by α_{j} (z): HD is still more sensitive than DD by a factor of 4. The SNRs of the transmitted signal are reduced by the common factor 1=|µ̄|^{2}, whereas the SNRs of the generated idler are lower than those of the input signal by the common factor 1=|ν̄|^{2}. Because the SNRs associated with HD and DD are reduced by the same factors, the NFs associated with HD and DD are equal. The common NFs
where the transmittance T=|µ̄|^{2} is a periodic function of distance [16]. The signal and idler NFs are plotted as functions of the transmittance in Fig. 5. If T=1, F _{1}=1 (0 dB): The signal is transmitted perfectly and no idler is generated. Conversely, if T=0, F _{2}=1 (0 dB): No signal is transmitted and the generated idler is a perfect copy of the input signal.
If both modes have nonzero input amplitudes, the output modes are coherent states with amplitudes α_{j} (z)=µ _{j1} α _{1}(0)+µ _{j2} α _{2}(0), where µ _{12}=ν̄ and µ _{22}=µ̄*. Let $\stackrel{\u0304}{\mu}=\mid \stackrel{\u0304}{\mu}\mid {e}^{i{\varphi}_{\stackrel{\u0304}{\mu}}}$ and $\stackrel{\u0304}{\nu}=\mid \stackrel{\u0304}{\nu}\mid {e}^{i{\varphi}_{\stackrel{\u0304}{\nu}}}$. Then the output photon-numbers
where the relative phase ξ=ϕ_{ν} ̄-ϕ_{µ} ̄+ϕ _{2}(0)-ϕ _{1}(0). On the right sides of Eqs. (53) and (54), α_{j} is an abbreviation for α_{j} (0). This alternative FC process is PS, because the output numbers of both modes depend on the phases of the input modes and transfer functions. However, Eqs. (45) and (46) still apply to HD and DD, respectively, with α_{i} replaced by the PS α_{j} (z), and the NFs associated with HD and DD are still equal. In Fig. 6 the signal and idler NFs are plotted as functions of the relative phase for the case in which |α _{1}|=|α _{2}| and T=0:5. Because of constructive interference, which allows one of the output amplitudes to be larger that the corresponding input amplitude, the NFs of the FC process can be less than 1 (0 dB). One could also define NFs based on the total input number, which would be greater than (or equal to) 1.
5.2 Phase-insensitive parametric amplification
Consider two-mode PA, which is made possible by MI or PC (Fig. 1). Let the signal and idler modes be labeled 1 and 2, respectively, and suppose that the input signal is a coherent state with amplitude α _{1}(0). Then it follows from the results of Section 4 that the output quantities α _{1}(z)=µ _{11} α _{1}(0), µ _{2}(z)=ν _{21} a _{1}* (0), 〈q_{j} (z)〉=|α_{j} (z)|, 〈${\delta q}_{j}^{2}$ (z)〉=(|µ_{jj} |^{2}+|ν_{jk} |^{2})/4,〈n_{j} (z)〉=|α_{j} (z)|^{2}+|ν_{jk} |^{2} and 〈${\delta n}_{j}^{2}$ (z)=|α_{j} (z)|^{2}(|µ_{jj} |^{2}+|ν_{jk} |^{2})+|µ_{jj} ν_{jk} |^{2}, where µ _{11}=µ=µ _{22}, ν _{12}=ν=ν _{21}, k≠j and the LO phase is optimal. This standard PA process is PI, because the output photon-numbers of the signal and idler do not depend on the phase of the input signal.
The preceding results show that the amplified signal and generated idler are not coherent states. For HD the SNRs
where the gain G=|µ|^{2} is a monotonically-increasing function of distance [16] and the input photon-number 〈n _{1}〉=|α _{1}(0)|^{2}. By combining Eqs. (45), (55) and (56), one finds that the NFs
For DD the SNRs
Formulas (59) and (60) are valid for input signals with arbitrary numbers. However, current communication systems use many-photon input-signals, for which 〈n _{1}〉≫1. In this limit, the SNR formulas simplify and the NFs
Hence, for many-photon signals, the NFs associated with HD are the same as those associated with DD. The signal and idler NFs are plotted as functions of the gain in Fig. 7. In the high-gain limit (G≫1) the signal and idler NFs are both about 2 (3 dB). The degradation in signal quality is caused by the coupling of the signal to the (amplified) vacuum fluctuations associated with the idler.
Now consider four-mode PA driven by two pumps (Fig. 1), and suppose that the input signal is a coherent state with amplitude a_{i} (0). Then it follows from the results of Section 4 that the output quantities α_{j} (z)=k_{ji} [(1-σ _{ji} )α_{i} (0)+σ_{ji} α_{i} * (0)], 〈q_{j} (z)〉=|α_{j} (z)|, 〈${\delta q}_{j}^{2}$ (z)i=∑ _{k} |k_{jk} |^{2}/4, 〈n_{j} (z)〉=|α_{j} (z)|^{2}+∑ _{k} |k_{jk} |^{2}σ _{jk} and 〈${\delta n}_{j}^{2}$ (z)〉=|α_{j} (z)|^{2}∑ _{k} |k_{jk} |^{2}+∑ _{k} ∑_{l>k}|k_{jk} k_{jl} |^{2}σ _{kl} , where k_{jk} =µ_{jk} if j and k are both odd, or even, k_{jk} =ν_{jk} if one of j and k is odd and the other is even, σ_{jk} =0 if j and k are both odd, or even, σ_{jk} =1 if one of j and k is odd and the other is even, and the LO phase is optimal. The formulas for the photon-number mean and variance are consistent with Eqs. (122) and (123) of [16], which were derived by a different method.
For HD the SNRs and NFs
where 〈n _{1}〉=|α_{i} (0)|^{2}. For DD the SNRs
In the many-photon limit (〈n_{i} 〉≫1), the SNR formulas simplify and the NFs
Hence, for many-photon signals, the NFs associated with HD are the same as those associated with DD.
The consequences of Eqs. (64) and (66) were described in detail in [16]. As a general rule, the signal and idler noise-levels increase in proportion to the number of modes that interact strongly. The interaction strengths (transfer functions |k_{jk} |^{2}) depend on the physical parameters associated with the pumps and signal, and the fiber. In some applications, such as optical switching [10], the pump frequencies are tuned in such a way that the output powers of the signals and idlers are comparable. The transfer functions |k_{jk} |^{2} are also comparable , and (in fibers with random birefringence) the signal and idler NFs are closer to 6 dB than 0 dB (two-mode FC) or 3 dB (two-mode PA): Extra frequency diversity comes at the price of extra noise. However, if the pump frequencies are tuned in ways such that the PA or FC bandwidths are maximized, the signal is coupled strongly to the primary idler (which is generated by PC or BS, respectively), and is coupled weakly to the (other) secondary idlers. The signal and primary-idler NFs are only slightly higher than the NFs associated with the limiting two-mode processes: PA with signal and primary-idler NFs of about 3 dB, and FC with a primary-idler NF of about 0 dB, are possible [16]. PA with more than one input signal will be discussed in Section 5.5.
5.3 Transmission through a phase-insensitive link
Consider the transmission of a signal through a one-stage PI link (Fig. 2). Let the signal, idler and scattered modes be labeled 1, 2 and 3, respectively, and suppose that the input signal is a coherent state with amplitude a1(0). Then it follows from the results of Section 4 that the output quantities α _{1}(z)=µ _{11} α _{1}(0), 〈q _{1}(z)〉=|α _{1}(z)|, 〈${\delta q}_{1}^{2}$(z)〉=(|µ _{11}|^{2}+|ν _{12}|^{2}+|µ _{13}|^{2})/4, 〈n _{1}(z)〉=|α _{1}(z)|^{2}+|ν _{12}|^{2} and 〈${\delta n}_{1}^{2}$(z)〉=|α _{1}(z)|^{2}(|µ _{11}|^{2}+|ν _{12}|^{2}+|µ _{13}|^{2}) +|µ _{11} ν _{12}|^{2}+|ν _{12} µ _{13}|^{2}, where µ _{11}=µ µ̄, ν _{12}=ν, µ _{13}=µν̄ and the LO phase is optimal.
For HD the SNR and NF
where the gain G=|µ|^{2}, the attenuation (transmittance) T=|µ̄|^{2} and the input photon-number 〈n _{1}〉=|a _{1}(0)|^{2}. Formula (67) has a simple physical interpretation: Attenuation transforms a coherent state with number 〈n _{1}〉 into a coherent state with number T〈n _{1}〉, which is amplified subsequently. Hence, Eq. (67) is like Eq. (55), with a modified input number. For DD the SNR
Equation (69) is like Eq. (59), with the input number 〈n _{1}〉 replaced by Thn _{1}i. For many-photon signals (T〈n _{1}〉≫1), the NF
Hence, for many-photon signals, the NF associated with HD is the same as that associated with DD. For a balanced link (GT=1), F _{1}=2G-1. In the high-gain limit (G≫1) the NF is about 2G.
Now consider the transmission of a signal through a link with s stages (Fig. 2). Let the input mode (1) be a coherent state with amplitude α _{1}(0). Then it follows from the results of Section 4 that the output quantities α _{1}(z)=k _{1} α _{1}(0), 〈q _{1}(z)〉=|α _{1}(z)|, 〈${\delta q}_{1}^{2}$ (z)〉=∑ _{k} |k_{k} |^{2}/4, 〈n _{1}(z)〉=|α _{1}(z)|^{2}+∑ _{k} |k_{k} |^{2}σ_{1k} and 〈${\delta n}_{1}^{2}$(z)〉=|α _{1}(z)|^{2}+∑ _{k} |k_{k} |^{2}+∑ _{k} ∑_{l>k}|k_{k}k_{l} |^{2}σ _{kl} , where k_{k} =µ _{1k} if k is odd, k_{k} =ν _{1k} if k is even, σ _{jk} =0 if j and k are both odd, or even, σ _{jk} =1 if one of j and k is odd and the other is even, and the LO phase is optimal. The formulas for the photon-number mean and variance are consistent with Eqs. (122) and (123) of [16], which were derived by a different method, for processes that occur simultaneously, rather than sequentially. If every stage in the link is identical, µ _{11}=(µµ̄) ^{s} , ν _{12} _{r} =(µµ̄)^{s-r} ν and µ _{12r+1}=(µµ̄)^{s-r} µν̄ (as shown in Appendix B). For a balanced link |µ _{11}|^{2}=1, |ν _{12r}|^{2}=G-1 and |µ _{12r+1}|^{2}=G-1.
By combining the preceding results, one finds that for HD the SNR and NF
Equations (71) and (72) are consistent with the results of [20], which were obtained by a different method. In Eq. (72) one factor of s(G-1) comes from the |ν _{12r}|^{2} terms, whereas the other comes from the |α _{12r+1}|^{2} terms. Because the latter terms are only nonzero if ν̄ is nonzero (|µ̄|<1), one can conclude that attenuation and amplification both degrade the signal quality. For DD the SNR
Equation (73) is consistent with Eqs. (47) and (48) of [17], and the results of [20], which were obtained by different methods. For many-photon signals [〈n _{1}〉≫s(G-1)], the NF
Hence, for many-photon signals, the NF associated with HD is the same as that associated with DD. In the high-gain limit the NF is about 2sG.
5.4 Phase-insensitive parametric monitoring.
Consider the transmission of a signal by a PI monitor. As stated in Section 3, the input-output relation for a monitor, with coupling losses before and after the PA, has the same form as the relation for a two-stage link, in which the second PA is absent (Fig. 2). Let the signal and idler modes be labeled 1 and 2, respectively, and let the scattered modes be labeled 3 and 5. Then it follows from the results of Section 4 (and Appendix B) that α _{1}(z)=k _{1} α _{1}(0), 〈q _{1}(z)〉=|α _{1}(z)|, 〈${\delta q}_{1}^{2}$(z)〉=(|k _{1}|^{2}+|k _{2}|^{2}+|k _{3}|^{2}+|k _{5}|^{2})=4, 〈n _{1}(z)〉=|α _{1}(z)|^{2}+|k _{2}|^{2} and 〈${\delta n}_{1}^{2}$(z)〉=|α _{1}(z)|^{2}(|k _{1}|^{2}+|k _{2}|^{2}+|k _{3}|^{2}+|k _{5}|^{2})+|k _{1} k _{2}|^{2}+|k _{2} k _{3}|^{2}+|k _{2} k _{5}|^{2}, where k _{1}=µ̄ _{f}µµ̄〉 _{i} , k _{2}=µ̄ _{f} ν, k _{3}=µ̄ _{f}µ ν̄ _{f} 〉, k _{5}=ν̄ _{f} and the LO phase is optimal. The subscripts i and f denote the initial loss (in stage 1) and the final loss (in stage 2), respectively.
For HD the SNR and NF
where the gain G=|µ|^{2}, the transmittances T_{i} =|µ̄ _{i} |^{2} and T_{f} =|µ̄ _{f} |^{2}, and the input photon-number 〈n _{1}〉=|α _{1}(0)|^{2}. For DD the SNR
$$+{T}_{f}\left(G-1\right)\left[1+{T}_{f}\left(G-1\right)\right]\}.$$
In the many-photon limit (T_{i} 〈n _{1}〉≫1), the SNR formula simplifies and the NF
The results of Sections 5.1–5.4 show that the NFs associated with HD and DD are always equal for PI processes with one many-photon input signal.
Equations (76) and (78) quantify the effects on the signal of a monitor that is adjacent to the transmitter. Coupling losses degrade the performance of adjacent monitors because they impose a lower bound on the NF of the signal-monitoring process. For example, if the monitor is balanced (T_{f}GT_{i} =1), F _{1}=1+2(1=T_{i} -T_{f} )>1. In contrast, coupling losses have little effect on the performance of distant monitors, because the SNR degradations associated with them are insignificant compared to the SNR degradations associated with the fiber losses in typical links [17].
5.5 Phase-sensitive parametric amplification
The noise properties of PS frequency down-convertors were reviewed in [13, 15] and the noise properties of PS amplifiers were reviewed in [18, 19]. In Section 5.5 the latter properties are described briefly, because they determine the properties of PS links, and contrast with the properties of PI devices and links.
Consider the degenerate PC process in which a signal is amplified by two pumps, whose frequencies differ from the signal frequency by equal and opposite amounts (Fig. 3). This frequency condition allows the signal to interact with itself, rather than an idler. Let the signal mode be labeled 1. Then it follows from the results of Section 4 that α _{1}(z)=µα _{1}(0)+να*_{1}(0), 〉q _{1}(z)〉=|α _{1}(z)|cos[ϕ _{1}(z)-ϕ], 〈${\delta q}_{1}^{2}$(z)〉=|λ|^{2}/4, 〈ν _{1}(z)〉=|α _{1}(z)|^{2}+|ν|^{2} and 〈δ${n}_{1}^{2}$(z)〉=|α _{1}(z)|^{2}|λ ^{′}|^{2}+2|µν|^{2}, where ϕ _{1}(z)=arg[α _{1}(z)] is the output phase, ϕ is the LO phase, λ=µe ^{-iϕ}+ν*e ^{iϕ} and $\lambda \text{'}=\mu {e}^{-i{\varphi}_{1}}+{\nu}^{*}{e}^{i{\varphi}_{1}}$. Let $\mu =\mid \mu \mid {e}^{i{\varphi}_{\mu}}$ and $\nu =\mid \nu \mid {e}^{i{\varphi}_{\nu}}$. Then the output strength and phase
where the PI gain G=|µ|^{2} is a monotonically-increasing function of distance [18] and the relative phase ξ=ϕ_{ν} -ϕ_{µ} -2ϕ _{1}(0). It is convenient to define the PS gain-function H(ξ)=2G-1+2[G(G-1)]^{1/2} cosξ. The output photon-number
where 〈n _{1}〈=|α _{1}(0)|^{2}, and the output quadrature- and number-variances
where the relative phases η=ϕ_{v} +ϕ_{µ} -2ϕ and ζ=ϕ_{v} +ϕ_{µ} -2ϕ _{1}(z). Notice that ξ-η=2[ϕ-ϕ _{1}(z)]. It follows from Eq. (80) and the definition of ζ that
The output strength, phase, number and number-variance are all controlled by the relative phase ϕ_{ν} -ϕ_{µ} -2ϕ _{1}(0), which depends on the coupling phases ϕ_{µ} and ν_{v} , and the input phase ϕ _{1}(0).
In contrast, the output quadrature-variance is controlled by the relative phase ϕ_{ν} +ϕ_{µ} -2ϕ, which depends on the coupling phases and the LO phase ϕ, but not the input phase. These phase dependences were illustrated in [19].
whereas for DD the SNR
In the many-photon limit (〈n _{1}≫1), the SNR formula simplifies and the NF
As noted after Eq. (83), one can obtain z from η by replacing the LO phase with the output phase, in which case formula (86) reduces to formula (88): One can consider DD as self-homodyning detection.
Formulas for the transfer functions µ and ν were stated in [18, 19]. For typical processes ϕ_{µ} =0, ϕ_{v} is related to the pump phases, and the relative phases ξ=ϕ_{ν} -2ϕ _{1}(0), η=ϕ_{ϕ} ν-2ϕ and ζ=ϕ_{ν} -2ϕ _{1}(z). If one measures the signal and LO phases relative to the reference phase ϕ_{ν} =2, one can refer to ξ, η and ζ as the input, LO and output phases, respectively. One can vary the input phase freely. If ξ=0 the output strength is maximal and, in the high-gain limit (G≫1), is about 4G〈n _{1}〉. In contrast, if ξ=π the output strength is minimal and, in the high-gain limit, is about 〈n _{1}〉=4G. For each choice of ξ, one can vary η independently. In contrast, Eq. (84) shows that ζ is a function of ξ: One cannot vary ξ and ζ independently.
In Fig. 8 the NF associated with HD is plotted as a function of ξ and η, for the case in which G=10. The region of Fig. 8a in which the contours merge (η≈π) is magnified in Fig. 8b.
With the exception of this narrow region, the NF contours are almost vertical: The NF is almost independent of η. This behavior has a simple explanation: When a signal is subject to PS amplification, its in-phase component is amplified, whereas its out-of-phase component is attenuated. Hence, in the high-gain regime, the output signal is predominantly in-phase (ζ≈0 or 2π). For the same reason, the in-phase axis of the probability cloud that characterizes the (initially-coherent) signal fluctuations is stretched, whereas the out-of-phase axis is squeezed. As one varies the LO phase (quadrature-measurement axis), the measured signal-strength decreases at almost the same rate as the (squared) width of the probability cloud. Hence, the NF is almost independent of η. For most values of η, the NF is maximal when ξ≈π, because the signal strength is minimal. In Fig. 9 the NF associated with HD is plotted as a function ξ, and as a function of η, for cases in which G=10.
In Fig. 10 the NF associated with DD is plotted as a function of ξ, for the case in which G=10 and 〈n _{1}〉=100. The approximate formula (88) predicts the NF accurately, provided that the signal is amplified significantly (ξ≠π).
Now consider two-mode PA, which is made possible by MI or PC (Fig. 1). If both modes have nonzero input amplitudes, the output amplitudes α_{j} (z)=µ_{jj}α_{j} (0)+ν_{jk}α* _{k} (0), where µ_{jj} =µ, ν_{jk} =ν and k≠j. It follows from the results of Section 4 that 〈q_{j} (z)〈=|α_{j} (z)|cos[ϕ_{j} (z)-ϕ], 〈${\xi q}_{j}^{2}$ (z)〉=(2G-1)/4, 〈n_{j} (z)〉=|α_{j} (z)|^{2}+G-1 and 〈${\delta n}_{j}^{2}$ (z)〉=|α_{j} (z)^{2}(2G-1)+G(G-1), where the PI gain G=|µ|^{2}. This alternative PA process is PS [2], because the output photon-numbers of both modes depend on the phases of the input modes and transfer functions. The quadrature mean attains its maximal value |α_{j} (z)| when ϕ=ϕ_{j} (z), whereas the quadrature variance does not depend on ϕ. The output strengths
where the relative phase ξ=ϕ_{ν} -ϕ_{µ} -ϕ _{2}(0)-ϕ _{1}(0). On the right sides of Eqs. (89) and (90), α_{j} is an abbreviation for α_{j} (0).
For HD the SNRs and NFs
whereas for DD the SNRs
In the many-photon limit (|α_{j} (z)|^{2}≫G-1), the SNR formulas simplify and the NFs
Hence, for many-photon inputs, the NFs associated with HD are the same as those associated with DD.
Formulas for the transfer functions µ and ν were stated in [16]. For typical processes ϕ_{µ} =0, fn is related to the pump phase(s) and the relative phase ξ=ϕ_{ν} -ϕ _{2}(0)-ϕ _{1}(0). One can vary the relative phase freely. Suppose that the input modes have equal strengths (|α _{1}(0)|=|α|=|α _{2}(0)j). Then, if ξ=0 the output strengths are maximal and, in the high-gain limit (G≫1), are about 4G|α|^{2}. In contrast, if ξ=π the output strengths are minimal and, in the high-gain limit, are about |α|^{2}/4G.
In Fig. 11 the NF associated with DD is plotted as a function of the relative phase for the case in which G=10 and |α _{1}(0)|^{2}=100=|α _{2}(0)|^{2} (and the signal and idler NFs are equal). Because of constructive interference, the common NF of this alternative PA process can be less than 1 (0 dB). The approximate formula (94) predicts the NF accurately, provided that the input modes are amplified significantly (ξ≠π). The dashed curve in Fig. 11 also represents the common NF associated with HD. The noise properties of four-mode PA with two nonzero input amplitudes are similar.
The degenerate PC process produces an output signal whose strength depends on the input phase ϕ _{1}(0), and whose quadrature mean and variance both depend on the LO phase ϕ. These properties typify a one-mode squeezing transformation [13, 15]. The PA process with two input modes also produces output modes whose strengths depend on the input phases ϕ _{1}(0) and ϕ _{2}(0), and whose quadrature means depend on ϕ. However, the quadrature variances of the output modes are independent of ϕ: Neither mode is squeezed by itself. Although the output modes are not squeezed, they are correlated: By using the FC process to combine them, one can produce new output modes that are squeezed [18, 19]. These properties typify a two-mode squeezing transformation [15].
5.6 Transmission through a phase-sensitive link
Consider the transmission of a signal through a one-stage PS link (Fig. 4). Let the signal and scattered modes be labelled 1 and 2, respectively, and suppose that the input signal is a coherent state with amplitude α _{1}(0). Then it follows from the results of Section 4 that the output quantities α _{1}(z)=µ _{11} α _{1}(0) +ν _{11} α*_{1}(0), 〈q _{1}(z)〉=|α _{1}(z)|cos[ϕ _{1}(z)-ϕ], 〈${\delta q}_{1}^{2}$(z)〉=(|λ _{11}|^{2}+|λ _{12}|^{2})/4, 〈ν _{1}(z)〉=|α _{1}(z)|^{2}+|ν _{11}|^{2}+|ν _{12}|^{2}, 〈${\delta \nu}_{1}^{2}$(z)〉=|α _{1}(z)|^{2}(|${\lambda}_{12}^{\prime}$|^{2}+|${\lambda}_{12}^{\prime}$|^{2})+^{2}(|µ _{11} ν _{11}|^{2}+|µ _{12} ν _{12}|^{2})+|µ*_{11} ν _{12}+µ*_{12} ν _{11}|^{2}, where µ _{11}=µµ̄, ν _{11}=νµ̄*,µ _{12}=µν̄ and ν _{12}=νν̄*,ϕ _{1}(z)=arg[α _{1}(z)] is the output phase, ϕ is the LO phase, λ _{1j}=µ _{1} j^{e} ^{-iϕ}+ν*_{1j} e ^{iϕ} and ${\lambda}_{1j}^{\text{'}}={\mu}_{1j}{e}^{-i{\varphi}_{1}}+{\nu}_{1j}^{*}{e}^{i{\varphi}_{1}}$. Let $\mu =\mid \mu \mid {e}^{i{\varphi}_{\mu}}$, $\nu =\mid \nu \mid {e}^{i{\varphi}_{\nu}}$,$\stackrel{\u0304}{\mu}=\mid \stackrel{\u0304}{\mu}\mid {e}^{i{\varphi}_{\stackrel{\u0304}{\mu}}}$ and $\stackrel{\u0304}{\nu}=\mid \stackrel{\u0304}{\nu}\mid {e}^{i{\varphi}_{\stackrel{\u0304}{\nu}}}$. Then the output strength and phase
where the PI gain G=|µ|^{2}, the attenuation (transmittance) T=|µ̄|^{2} and the relative phase ξ=ϕ_{ν} -ϕ_{µ} -2[ϕ _{1}(0)+ϕ_{µ} ̄]. It is convenient to define the PS gain-function H(ξ)=2G-1+2[G(G-1)]^{1/2} cosξ. The output photon-number
where the input number 〈n _{1}〉=|α _{1}(0)|^{2}, and the output quadrature- and number-variances
where the relative phases η=ϕ_{ν} +ϕ_{µ} -2ϕ and ζ=ϕ_{ν} +ϕ_{µ} -2ϕ _{1}(z). It follows from Eq. (96) that x and z are related by Eq. (84). These results have a simple physical interpretation: Attenuation transforms a coherent state with number 〈n _{1}〉 and phase ϕ _{1} into a coherent state with number T〈n _{1}〉 and phase ϕ _{1}+ϕ_{µ} ̄. Hence, Eqs. (95)–(99) are like Eqs. (79)–(83), with modified input number and phase. The output quantities do not depend on ϕ_{v} ̄, because the vacuum fluctuations associated with the scattered mode have indefinite phase.
For HD the SNR and NF
whereas for DD the SNR
In the many-photon limit (〈n _{1}≫1), the SNR formula simplifies and the NF
Formulas for the transfer functions µ and ν were stated in [18, 19]. For typical processes ϕ_{µ} =0, ϕ_{ν} is related to the pump phases, and the relative phases ξ=ϕ_{ν} -2[ϕ _{1}(0)+ϕ_{µ} ̄], η=ϕ_{ν} -2ϕ, and ζ=ϕ_{ν} -2ϕ _{1}(z). If one measures the signal and LO phases relative to the reference phase ϕ_{ν} =2, one can refer to ξ as the input phase (shifted by ϕ_{µ} ̄), and one can refer to η and ζ as the LO and output phases, respectively. One can vary ξ and η freely. The PS gain attains its maximal value (|µ|+|ν|)^{2} when the input phase has its optimal value ξ=0. For this value of x the output phase ζ=0 and the optimal value of the LO phase η=0 (Fig. 5b), in which case the NFs associated with HD and DD are equal [Eqs. (101) and (103)]. It follows from these observations that F _{1}=1/T. For a balanced stage [H(0)T=1] the PI gain G=(1+T)^{2}/4T, from which it follows that |µ|^{2}≈1/4T+1/2 and |ν|^{2}≈1=4T-1/2. Because the PI gains required for balanced PI and PS stages (Section 5.3) differ by a factor of 4, one can facilitate comparisons of PI and PS links by writing the NFs in terms of their common loss L=1/T. For a one-stage PI link F _{1}=2L-1 [Eqs. (68) and (70)], whereas for a PS link F _{1}=L. In the high-loss limit (L≫1) the NF of a one-stage PS link is lower than that of a PI link by a factor of 2 (3 dB).
Now consider a PS link with s stages (Fig. 4). Let the input mode (1) be a coherent state with amplitude α _{1}(0). Then the output quantities are given by Eqs. (26), (38), (40), (41) and (43). Because these formulas are complicated, we illustrate their consequences for a simple case. The preceding analysis of a one-stage link showed that the quadrature and number variances do not depend on the phase shift ϕ_{µ} ̄ imposed on the signal by the fiber, or the phase ϕ_{v} ̄. Furthermore, the optimal value of the input phase ξ=0, in which case the amplitude contributions µα _{1}(0) and νa*_{1} (0) add constructively, the output phase ζ=0, and the optimal value of the LO phase η=0. Consequently, in our discussion of a many-stage link, we assume, without significant loss of generality, that µ̄, ν̄, µ and ν are all real, and that ξ=0 and η=0. Under these conditions, and the assumption that every stage in the link is identical, µ _{11}=µ̄ ^{s} ps, ν _{11}=µ̄ ^{s} qs, µ _{1r+1}=µ̄ ^{s-r} ν̄ p _{s+1-r} and ν _{1r+1}=µ̄ ^{s-r} ν̄ q _{s+1-r}, where p_{s} and q_{s} are polynomial functions of µ and ν with the property p_{s} +q_{s} =(µ+ν) ^{s} (as shown in Appendix C). For a balanced link [µ̄ (µ+ν)=1], |µ _{11}+ν _{11}|^{2}=1 and |µ _{1r+1}+ν _{1r+1}|^{2}=L-1.
By combining Eqs. (26), (38) and (40) with the preceding formulas for |µ _{1k}+ν _{1k}|^{2}, one finds that for HD the SNR and NF
Equations (104) and (105) are consistent with the results of [20], which were obtained by a different method. By combining Eqs. (41) and (43) with the formulas for µ _{1k} and ν _{1k} (as detailed in Appendix C), one finds that for DD the SNR
Equation (73) shows that, in a PI link, the spontaneous contributions to the number mean and variance grow as sL and (sL)^{2}, respectively. Equation (106) shows that, in a PS link, the spontaneous contributions grow in the same ways. However, in a PS link the coefficients are smaller by factors of 4 and 8, respectively. For many-photon signals [|α _{1}(0)|^{2}≫s(L-1)], the SNR formula simplifies and the NF
Hence, for many-photon signals, the NF associated with DD is the same as that associated with HD. In the high-loss limit, the NF of a PS link is lower than that of a PI link by a factor of 2 (3 dB). The difference is small (compared to 2s) because loss adds PI uncertainty at each stage.
6. Summary
In this paper the quantum noise properties of parametric processes were studied. Formulas for the field-quadrature and photon-number means and variances were derived [Eqs. (38), (39), (41) and (42)], for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios (SNRs) associated with the direct detection (DD) and homodyne detection (HD) of optical signals. The consequences of the aforementioned formulas were described for frequency conversion (FC), parametric amplification (PA), monitoring, and transmission through communication links (sequences of attenuators and PAs).
In fiber optics, a process is termed phase-sensitive (PS) if the power of the output signal depends on the phase of the input signal. Otherwise, it is termed phase-insensitive (PI). In quantum optics, the term PS (PI) is used to describe a process in which the quadrature variance of the output signal depends (does not depend) on the phase of the local oscillator (LO) used to measure it. To avoid confusion, we term such a process isotropic (anisotropic).
All of the PI processes considered herein are isotropic (one-input-mode FC and PA, PI monitoring and transmission through a PI link), as are the processes that are made PS by the presence of a second input mode (two-input-mode FC and PA). For an isotropic process, the SNR associated with HD is maximal when the LO phase equals the relevant output phase (signal or idler), in which case the SNR is given by Eq. (48), with λ _{jk} =µ_{jk} or ν* _{jk} . For a many-photon signal, the spontaneous contributions to the photon-number mean and variance are negligible, in which case the SNR associated with DD is given by Eq. (50), with ${\lambda}_{\mathit{\text{jk}}}^{\prime}$ =µ_{jk} or ν* _{jk} . Hence, for isotropic processes with coherent many-photon input signals, the input and output SNRs associated with HD are larger than those associated with DD by a factor of 4 (6 dB). The noise figure (NF) of a parametric process is the SNR of the input signal divided by the SNR of the relevant output mode (signal or idler). For isotropic processes with many-photon signals, the NFs associated with HD and DD are equal. Two of the PS processes considered herein (PS amplification and transmission through a PS link) are anisotropic. In the high-gain limit, the signal SNR associated with HD is maximal when the LO phase equals the signal phase, in which case it is given by Eq. (48), with λ_{jk} =${\lambda}_{\mathit{\text{jk}}}^{\prime}$ . The SNR depends only weakly on the LO phase. For a many-photon signal, the spontaneous contributions to the photon-number mean and variance are negligible, in which case the SNR associated with DD is given by Eq. (50). Hence, for anisotropic processes with coherent many-photon input signals and high gains, the input and output SNRs associated with HD are larger than those associated with DD by a factor of 4 (6 dB), so the NFs associated with HD and DD are equal. These results allow the term NF to be used unambiguously. The preceding statements were based on one-time measurements of signals. Many-time measurements of signals [14], and joint measurements of signals and correlated idlers [15], were not considered.
For one-input-mode FC (Section 5.1), the signal NF varies from 1 (0 dB) to ∞, whereas the idler NF varies from ∞ to 1: It is possible to generate an idler that is a perfect copy of the input signal. For two-mode FC (Section 5.1), constructive interference allows the standard NFs to be less than 1. If the input numbers are equal, the minimal NFs are 1/2 (-3 dB). However, if one were to define alternative NFs based on the total input number, the minimal NFs would be 1. These results reflect the fact that the FC images of coherent states are also coherent states.
For one-input-mode PA (Section 5.2), the signal NF varies from 1 (0 dB) to 2 (3 dB), whereas the idler NF varies from ∞ to 2. In the high-gain limit, both NFs are about 2. The degradation in signal quality is caused by the coupling of the signal to the (amplified) vacuum fluctuations associated with the idler. For two-input-mode PA (Section 5.5), constructive interference allows the standard signal and idler NFs to be less than 1. If the input numbers are equal, the minimal NFs are 1/2 (-3 dB). However, if one were to define alternative NFs based on the total input number, the minimal NFs would be 1 (0 dB). Many-mode PI processes (Section 5.2) were discussed in [16]. In general, the presence of extra idlers increases the signal and idler NFs (relative to those of two-mode FC and PA).
In fiber communication systems the effects of signal attenuation (loss) must be compensated (balanced) by amplification (gain). Consider a link that consists of s identical stages with loss L. For a balanced PI link (Section 5.3), the signal NF is approximately equal to 2sL. One factor of sL comes from the amplifiers, whereas the other comes from the attenuators: Both types of component degrade the signal quality.
Monitors based on PAs generate high-quality idlers, which can be detected without disrupting the flow of information. In a typical monitor, coupling losses precede and follow the PA. For a balanced monitor that is adjacent to the transmitter (Section 5.4), the signal NF is 1+2(L_{i} -1/L_{f} )>1 (0 dB), where L_{i} and L_{f} are the initial and final losses, respectively. Coupling losses have little effect on the performance of distant monitors, because the SNR degradations associated with them are insignificant compared to those associated with the losses in typical links [17].
PS amplification in a fiber occurs when the signal frequency is the average of the two pump frequencies. This frequency condition allows the signal to interact with itself, rather than an idler: No vacuum fluctuations are coupled to the signal. For in-phase amplification (Section 5.5), the signal NF is 1 (0 dB), because the mean input-signal and the in-phase component of the (coherent) input-signal fluctuations are amplified by the same amount. An out-of-phase signal is attenuated.
For transmission through a balanced PS link (Section 5.6), the signal NF is approximately equal to sL. Although the PS amplifiers do not degrade the signal quality, the PI attenuators do, so the NF of a PS link is lower than that of PI link by a factor of only 2 (3 dB).
In this paper a formalism was developed to determine the noise properties of parametric processes; those of current interest (described above) and those yet to be imagined. The results of this paper apply to interactions between discrete modes. Interactions between modes with continuous frequency spectra will be described elsewhere.
Appendix A: Arbitrary input states
In Section 4 formulas for the quadrature and number means and variances were derived for coherent input states. In this appendix formulas are derived for arbitrary input states. First, suppose that there is only a single input mode, which, without loss of generality, is labeled 1. Then one can rewrite the input-output relation (21) in the form
where the output operator
describes the effects of the input mode and the output operator
describes the effects of vacuum fluctuations.
To determine the quadrature and number means and variances, one must first calculate the lower-order moments of the output operator. The first-order moment is Eq. (108), the second-order moments are
and the fourth-order moment is
$$+{\left({v}_{j}^{\u2020}{v}_{j}\right)}^{2}+{u}_{j}^{\u2020}{u}_{j}\left({u}_{j}{v}_{j}^{\u2020}+{u}_{j}^{\u2020}{v}_{j}\right)+\left({u}_{j}{v}_{j}^{\u2020}+{u}_{j}^{\u2020}{v}_{j}\right){u}_{j}^{\u2020}{u}_{j}$$
$$+2{u}_{j}^{\u2020}{u}_{j}{v}_{j}^{\u2020}{v}_{j}+\left({u}_{j}{v}_{j}^{\u2020}+{u}_{j}^{\u2020}{v}_{j}\right){v}_{j}^{\u2020}{v}_{j}+{v}_{j}^{\u2020}{v}_{j}\left({u}_{j}{v}_{j}^{\u2020}+{u}_{j}^{\u2020}{v}_{j}\right).$$
The moments of v_{j} were evaluated in Section 4. Because the (expectation values of the) first-and third-order moments of v_{j} are zero, the first-order moment of α_{j} is determined solely by u_{j} , the contributions of u_{j} and v_{j} to the second-order moments add independently, and the fourth, fifth, seventh and eighth terms on the right side of Eq. (114) are zero. It follows from Eq. (108), Eqs. (111)–(113) and these observations, that the quadrature mean and variance
where 〈δq ^{2}(u_{j} )〉=〈q ^{2}(u_{j} )〉-〈q(u_{j} )〉^{2} is the variance contribution from u_{j} and
where λ_{jk} =µ_{jk}e ^{-iϕ}+ν* _{jk}e^{iϕ} , is the other variance contribution. It follows from Eqs. (113) and (114), and the preceding observations, that the number mean and variance
$$+{u}_{j}^{\u2020}{u}_{j}{v}_{j}{v}_{j}^{\u2020}+{\left({u}_{j}^{\u2020}\right)}^{2}{v}_{j}^{2}\u3009+\u3008\delta {n}^{2}\left({v}_{j}\right)\u3009,$$
where 〈δn ^{2}(u_{j} )〉=〈n ^{2}(u_{j} )〉-〈n(u_{j} )〉^{2} is the variance contribution from u_{j} , the second-order moments
and the other variance contribution
Further progress requires the specification of the input state. In the absence of such a specification, all one can do is demonstrate that the preceding results are consistent with the results of Section 4. If the input state 1 is a coherent state, Eqs. (38)–(43) apply to the moments of u_{j} , with k=1. It follows from this observation that
where α_{j} =µ_{j} _{1} α _{1}+ν _{j1} α*_{1} and λ _{j1}=µ _{j1} e ^{-iϕ}+ν*_{j1} e ^{iϕ}. Equation (125) is identical to Eq. (38). By combining Eqs. (116), (117) and (126), one obtains Eq. (40). It also follows from the preceding observation that
where ${\lambda}_{j1}^{\text{'}}={\mu}_{j1}{e}^{-i{\varphi}_{j}}+{\nu}_{j1}^{*}{e}^{i{\varphi}_{j}}$ and ϕ_{j} =arg(α_{j} ). By combining Eqs. (118), (122) and (127), one obtains Eq. (41). It follows from Eqs. (119)–(124) and Eq. (128) that
$$+\left({\alpha}_{j}^{2}+{\mu}_{j1}{\nu}_{j1}\right)\sum _{k>1}{\left({\mu}_{jk}{\nu}_{jk}\right)}^{*}+\left({\mid {\alpha}_{j}\mid}^{2}+{\mid {\mu}_{j1}\mid}^{2}\right)\sum _{k>1}{\mid {\nu}_{jk}\mid}^{2}$$
$$+\left({\mid {\alpha}_{j}\mid}^{2}+{\mid {\nu}_{j1}\mid}^{2}\right)\sum _{k>1}{\mid {\mu}_{jk}\mid}^{2}+{\left({\alpha}_{j}^{2}+{\mu}_{j1}{\nu}_{j1}\right)}^{*}\sum _{k>1}{\mu}_{jk}{\nu}_{jk}$$
$$+2\sum _{k>1}{\mid {\mu}_{jk}{\nu}_{jk}\mid}^{2}+\sum _{k>1}\sum _{l>k}{\mid {\mu}_{jk}^{*}{\nu}_{jl}+{\mu}_{jl}^{*}{\nu}_{jk}\mid}^{2}.$$
one can show that Eq. (129) is equivalent to Eq. (43).
Second, suppose that there are multiple input modes, which are labeled by the subscript i. Then one can rewrite the input-output relation in the form of Eq. (108), in which the output operators
The quadrature and number means and variances are still given by Eqs. (115), (116), (118) and (119), in which the moments and variance contribution of v_{j} are given by truncated versions of Eqs. (120)–(124). Once again, further progress requires the specification of the input states.
Appendix B: Phase-insensitive link
Consider the transmission of a signal through a PI link (Fig. 2). For a one-stage link, Eq. (13) states that
It follows from Eqs. (13) and (134) that, for a two-stage link,
By iterating Eq. (13) s times, one finds that
Thus, µ _{11}=(µµ̄) ^{s} , ν _{12r}=(µµ̄) ^{s-r} ν and µ _{12r+1}=(µµ̄)^{s-r} µν̄, as stated in Section 5.3.
To derive the SNRs (71) and (73), one must evaluate the sums ∑ _{k} |k_{k} |^{2} and ∑ _{k} ∑_{l>k}|k_{k}k_{l} |^{2}σ _{kl} , where σ _{kl} =0 if k and l are both odd, or even, and σ _{kl} =1 if one of k and l is odd and the other is even. For a balanced link, |k _{1}|^{2}=1 and |k_{k} |^{2}=G-1 if k>1. It follows from these facts that the first sum ∑ _{k} |k_{k} |^{2}=1+2s(G-1), as stated in Eqs. (71) and (73). The second sum ∑ _{k} ∑_{l>k}|k_{k}k_{l} |^{2}σ _{kl} is evaluated as follows: For stage 0 (k=1) there are s contributions of the form |k _{1} k_{l} |^{2}=1(G-1), which sum to s(G-1). For stage 1 (k=2 and 3) there are s contributions of the form |k _{2} k_{l} |^{2}=(G-1)^{2} and s-1 contributions of the form |k _{3} k_{l} |^{2}=(G-1)^{2}, which sum to (2s-1)(G-1)^{2}. For stage 2 (k=4 and 5) there are s-1 contributions of the form |k _{4} k_{l} |^{2}=(G-1)^{2} and s-2 contributions of the form |k _{5} k_{l} |^{2}=(G-1)^{2}, which sum to (2s-3)(G-1)^{2}. By continuing this counting process, one finds that the second sum is s(G-1)+${\sum}_{r=1}^{s}$[2(s-r)+1](G-1)^{2}=s(G-1)[1+s(G-1)], as stated in Eq. (73).
Appendix C: Phase-sensitive link
Consider the transmission of a signal through a PS link. For a one-stage link, Eq. (19) states that
For the case considered in the text, µ, ν, µ̄ and ν̄ are real. It follows from Eqs. (19) and (137) that, for a two-stage link,
$$+\stackrel{\u0304}{\mu}\stackrel{\u0304}{\nu}\left(2\mu \nu \right){a}_{2}^{\u2020}\left({z}_{0}^{\u2033}\right)+\stackrel{\u0304}{\nu}\mu {a}_{3}\left({z}_{1}^{"}\right)+\stackrel{\u0304}{\nu}\nu {a}_{3}^{\u2020}\left({z}_{1}^{"}\right).$$
By iterating Eq. (19) s times, one finds that
$$\times \left[{p}_{s+1-r}(\mu ,\nu ){a}_{r+1}\left({z\u2033}_{r-1}\right)+{q}_{s+1-r}(\mu ,\nu ){a}_{r+1}^{\u2020}\left({z\u2033}_{r-1}\right)\right],$$
where the polynomials p_{s} and q_{s} are defined recursively: p _{1}=µ, q _{1}=ν, p _{s+1}=µ p _{s} +νq_{s} and q _{s+1}=µq_{s} +νp_{s} . Thus, µ _{11}=µ̄ ^{s} p_{s} , ν _{11}=µ̄ ^{s} q_{s} , µ _{1r+1}=µ̄ ^{s-r} ν̄ p _{s+1-r} and ν _{1r+1}=µ̄ ^{s-r} ν̄ q _{s+1-r}, as stated in Section 5.6. It follows from the preceding definitions that p_{s} ±q_{s} =(µ±ν) ^{s} , µ _{11}±ν _{11}=[µ̄ (µ±ν)] ^{s} and µ _{1r+1}±ν _{1r+1}=[µ̄ (µ±ν)]^{s-r} ν̄ (µ±ν), where 1·r ·s. For a balanced link [µ̄ (µ+ν)=1], µ=L ^{1/2}(1+T)/2 and ν=L ^{1/2}(1-T)/2, where T=|µ̄ |^{2} and L=1/T. It follows from the preceding results that µ _{11}=(1+T^{s} )/2, ν _{11}=(1-T^{s} )/2, µ _{1r+1}=(L-1)^{1/2}(1+T ^{s+1-r})/2 and ν _{1r+1}=(L-1)^{1/2}(1-T ^{s+1-r})/2.
To derive the SNRs (104) and (106), one must evaluate the sums ∑ _{k} |λ _{1k}|^{2} and ∑ _{k} |${\lambda}_{1k}^{\prime}$|^{2}, where λ _{1k}=µ _{1k} e ^{-iϕ}+ν*_{1k} e ^{iϕ} and ${\lambda \text{'}}_{1k}={\mu}_{1k}{e}^{-i{\varphi}_{1}}+{\nu}_{1k}^{*}{e}^{i{\varphi}_{1}}$. For the in-phase quadrature, ϕ=0. It follows from the results of the preceding paragraph that |λ _{11}|^{2}=1 and |λ _{1r+1}|^{2}=L-1. Thus, ${\sum}_{r=0}^{s}$|λ _{1r+1}(0)|^{2}=1+s(L-1), as stated in Eq. (104): The in-phase sum (quadrature variance) increases monotonically as the number of stages increases. For the case considered, in which ϕ _{1}(z)=0, ${\lambda}_{1r+1}^{\prime}$=λ _{1r+1}, so ${\sum}_{r=0}^{s}$|${\lambda}_{1r+1}^{\prime}$|^{2}=1+s(L-1), as stated in Eq. (106). For the out-of-phase quadrature ϕ=π/2. It follows from the results of the preceding paragraph that |λ _{11}|^{2}=T ^{2s} and |λ _{1r+1}|^{2}=T(1-T)T ^{2s-2r}. Thus, ${\sum}_{r=0}^{s}$|λ _{1r+1}(π/2)|^{2}=(T+T ^{2s})/(1+T): As the number of stages increases, the out-of-phase sum (quadrature variance) tends quickly to its limit T/(1+T). It follows from Eq. (40), the definition of λ _{1k}, and the assumption that µ _{1k} and ν _{1k} are real, that 〈${\delta q}_{1}^{2}$(ϕ)〉=〈${\delta q}_{1}^{2}$(0)〉cos^{2} ϕ+〈${\delta q}_{1}^{2}$(π/2)sin^{2} ϕ. Because the in-phase variance is much larger then the out-of-phase variance, 〈${\delta q}_{1}^{2}$(ϕ)〉≈〈${\delta q}_{1}^{2}$(0)〉cos^{2} ϕ. Recall that 〈q _{1}(ϕ)〉^{2}=|α _{1}(0)|^{2} cos^{2} ϕ. By combining these results, one finds that the SNR associated with HD depends weakly on ϕ, and in-phase measurement (ϕ=0) is optimal.
To derive the SNR (106), one must evaluate the sums ∑ _{k} |ν _{1k}|^{2}, ∑ _{k} |µ _{1k} ν _{1k}|^{2} and ∑ _{k} ∑_{l>k} |µ _{1k} ν _{1l}+µ _{1l} ν _{1k}|^{2}. The first sum
For a long link (L≫1 and s≫1), ∑ _{k} |ν _{1k}|^{2}≈[s(L-1)-1]=4. The second sum
$$+{T}^{4}\left(1-{T}^{4s}\right)\u2044\left(1-{T}^{4}\right)\left]\right\}\u204416.$$
For a long link, ∑ _{k} |µ _{1k} ν _{1k}|^{2}≈[s(L-1)2-1]=16. It is convenient to split the third sum into two parts. The first part
$$+{T}^{2s+2}\left(1-{T}^{2s}\right)\u2044\left(1-{T}^{2}\right)]\u20444.$$
For a long link, ∑_{l>1}|µ _{11} ν _{1l}+µ _{1l} ν _{11}|^{2}≈s(L-1)/4. The second part involves contributions of the form
$$+{T}^{2s+4-2q}\left(1-{T}^{2s-2q}\right)\u2044\left(1-{T}^{2}\right)]\u20444,$$
where 1≤q≤s-1. It is not difficult to sum these contributions. For a long link, ∑ _{k} >1∑_{l>k} |µ _{1k} ν _{1l}+µ _{1l} ν _{1k}|^{2}≈s(s-1)(L-1)^{2}=8. By combining the preceding results, one finds that
as stated in Eq. (106).
Acknowledgment
We acknowledge a useful discussion with M. V. Vasilyev.
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