Spontaneous parametric downconversion in waveguides: What's loss got to do with it?

We derive frequency correlation and exit probability expressions for photons generated via spontaneous parametric downconversion (SPDC) in nonlinear waveguides that exhibit linear scattering loss. Such loss is included within a general Hamiltonian formalism by connecting waveguide modes to reservoir modes with a phenomenological coupling Hamiltonian, the parameters of which are later related to the usual loss coefficients. In the limit of a low probability of SPDC pair production, the presence of loss requires that we write the usual lossless generated pair state as a reduced density operator, and we find that this density operator is naturally composed of two photon, one photon, and zero photon contributions. The biphoton probability density, or joint spectral intensity (JSI), associated with the two-photon contribution is determined not only by a phase matching term, but also by a loss matching term. The relative size of the loss coefficients within this term lead to three qualitatively different regimes of SPDC JSIs. If either the pump or generated photon loss is much higher than the other, the side lobes of the phase matching squared sinc function are washed out. On the other hand, if pump and generated photon loss are appropriately balanced, the lossy JSI is identical to the lossless JSI. Finally, if the generated photon loss is frequency dependent, the shape of the JSI can be altered more severely, potentially leading to generated photons that are less frequency correlated though also produced less efficiently when compared to photons generated in low-loss waveguides.


I. INTRODUCTION
Waveguides are fundamental integrated optical components. Their performance is key to realizing increased miniaturization, stability, and scalability of both classical and quantum optical devices. In particular, as nonlinear quantum optics experiments continue to move from bulk crystal optics to chip-scale optics, waveguide losses will have a direct effect on nonlinear optical photon generation and manipulation. Indeed, loss mechanisms in waveguides have been well-investigated both theoretically and experimentally [1][2][3][4][5][6], a recent conclusion being that while propagation losses can certainly arise from material absorption and radiation associated with tight bends, the most significant source of loss in modern integrated waveguides is often scattering due to sidewall roughness inherent in fabrication processes [6].
Modern quantum-theoretical treatments of spontaneous photon generation typically proceed along one of two directions. In one school, the focus is on operator expectation values, and differential equations for these operators are developed in analogy with classical coupled mode equations (see e.g. Lin et al. [7]). In the other, Schrödinger picture states are followed from input to the nonlinear region, through a Hamiltonian interaction, to output while correctly accounting for material and modal dispersion (see e.g. Yang et al. [8]). Although the first approach has initially proven more amenable to extensions to include loss [9,10], the second places classical and quantum wave mixing processes within a consistent theoretical framework, making it easy to draw comparisons and develop new physical insights [11,12]. Therefore in this work we follow the second approach, extending a multiple-frequency mode Hamiltonian formalism of spontaneous parametric downconversion (SPDC) in waveguides [8] to include scattering loss in the nonlinear region.
We limit ourselves here to consideration of photon pair generation via SPDC, as this allows us to consider pump losses separate from generated photon losses; nonetheless, we expect many of the results presented here to carry over to photon pair generation via spontaneous four-wave mixing, a topic we intend to explore in detail in future work. While SPDC pair generation in waveguides has been studied in the past [13][14][15][16][17][18], the effects of scattering loss have rarely been included explicitly [19,20], and never within a multiple-frequency mode quantum state picture. Indeed, in the analysis of experimental results such loss is usually lumped in with detector efficiencies and losses associated with coupling on and off the chip. The theory that accompanies such an analysis models loss with the inclusion of asymmetric beam splitters [21] placed after the nonlinear region where the photons are generated, and thus any effects of photon generation in the presence of loss are missed.
As we show, our approach correctly captures the full spectral structure of SPDC generated photons, including the effects of loss on photon frequency correlations, and enables a prediction of the quantum performance of nonlinear waveguides in the presence of loss. In particular, we show that the common representation of the biphoton probability density, or joint spectral intensity (JSI), as being composed of just a pump pulse spectrum term and a phase matching term, should also contain a loss matching term that can strongly modify its shape. We also show that the common way of quantifying device performance only in terms of photon pair exit probability should also consider the probability of accidental singles, from photon pairs that have lost one photon, exiting the waveguide. Put together, a picture of the trade-offs between frequency separability and photon pair to single photon exit probability emerges. Additionally, as we begin with a coherent state pump in the Schrödinger picture and follow its evolution through the device, our approach remains relatively straightforward and can easily be generalized to more complicated input states, additional nonlinear effects [22,23], and various integrated nonlinear structures beyond channel waveguides [24]. For a treatment of the simpler problem of including the effects of scattering loss following photon generation within our formalism, we refer the reader to Helt [25].
In Section II we introduce the general formalism, first reviewing how a calculation proceeds in the absence of loss, and then turning to differences that arise when scattering loss is included. In section III, working in the negligible multi-pair generation regime, we construct the reduced density operator associated with at most a single pair of photons exiting the lossy waveguide, first in wavevector-space and then switching to a frequency representation as well. This density operator is seen to naturally separate into the sum of a two-photon, a singlephoton, and a vacuum density operator. In Section IV, still in frequency space, we demonstrate the utility of our derived expressions. We first use the natural splitting of the total reduced density operator to calculate the probabilities with which two photons, one photon, or zero photons exit the waveguide. We then contrast a general lossy biphoton wave function with the usual biphoton wave function calculated in the absence of scattering loss, as well as the expression for the power generated in a corresponding lossy sum frequency generation (SFG) experiment. Finally, we consider biphoton probability densities in the three qualitatively different regimes: one in which either the pump or generated photon loss is much higher than the other, one in which pump and generated photon loss are appropriately balanced, and one in which generated photon loss is frequency dependent. For concreteness we calculate biphoton probability densities as well as the probabilities with which photon pairs and single photons exit the nonlinear device for a realistic Bragg reflection waveguide [26]. We conclude in Section V.

II. FROM INPUT PULSES TO OUTPUT PHOTONS
A. Summary of our formalism in the absence of scattering loss The formalism that we extend here was initially presented in an earlier work [8], and so we direct the reader there for additional details, providing just a summary here. It begins with linear and nonlinear Hamiltonians, which are built up from correctly normalized expansions of the full electric displacement and magnetic field operators in terms of the modes of interest of the linear problem. In particular, we assume that modes labeled by m = F for fundamental and m = SH for second harmonic have been solved for and write with all other commutators evaluating to zero. For simplicity, we have assumed that all generated photons are labeled by F, i.e. we have assumed type-I SPDC, though we note that generalizations are straightforward [27]. We assume that all pump and generated photons travel in the forward (positive k) direction, a valid approximation for typical dispersion relations [8], and therefore here and throughout all wavevector integrals are taken over the positive real axis. All of the nonlinear optics lives in the coupling term [8] where the nonlinearity has been assumed to exist between z = −L/2 and z = L/2, and n and χ 2 are, respectively, a typical effective index and second-order optical nonlinearity introduced solely for convenience. In particular, our final results depend on neither n nor χ 2 , as they cancel with counterparts in the definition of the effective area [8] A (k 1 , k 2 , k) with d i mk (x, y) the i-th component of the displacement field at wavenumber k, and n (x, y; ω mk ) the material refractive index at wavenumber k, both at waveguide crosssectional position (x, y). We have chosen the field amplitudes such that we can take the phase associated with the effective area to be zero.
We frame evolution through the nonlinear waveguide in terms of 'asymptotic-in' and '-out' states, borrowing from scattering theory. Their introduction eliminates trivial linear evolution from our main calculation, as the asymptotic-in state is defined as the state evolved from t = t 0 , with energy localized at the beginning of the waveguide, to t = 0, at its center, according to only H L . Similarly, the asymptotic-out state is defined as the state at t = 0 that would evolve to t = t 1 , with energy localized at the end of the waveguide, if the evolution occurred according to the same linear Hamiltonian. The duration of the interaction is on the order of the length of the assumed nonlinear portion of waveguide, L, divided by the group velocity of the pump field, v SH , i.e. t 1 −t 0 ≈ L/v SH . However, as in scattering theory, it is common to take t 0 → −∞, t 1 → ∞. The calculation seeks the state of generated photons for an asymptotic-in coherent state with |z| 2 the average number of photons per pulse for a normalized pump pulse waveform φ P (k) and |vac = |vac F ⊗ |vac SH . In particular, we proceed by solving for the associated asymptotic-out state |ψ out = e iHLt1/ e −i(HL+HNL)(t1−t0)/ e −iHLt0/ |ψ in , (2.7) in the backwards Heisenberg picture [8], and seek an output state of the form (2.8) This form follows from our intuition that, for an undepleted pump, the SH mode will remain in a coherent state. Furthermore, any difference betweenā † SHk (t 0 ) and a † SHk can be identified with photon creation and, to first order, is naturally represented as a squeezing operation [8]. For a general barred operator O (t), which is seen to evolve backwards in time as |ψ in evolves forwards in time, one can show subject to the "final" condition with + H.c., (2.11) and S (k 1 , k 2 , k; t) = S (k 1 , k 2 , k) e i(ω Fk 1 +ω Fk 2 −ω SHk )t . In short, the problem of calculating the evolution of a state through the waveguide reduces to the integration of (2.9) from t = t 1 back to t = t 0 subject to (2.10) or solving differential equations for barred (backward Heisenberg) operators. The end result is that [8] |ψ out = exp ζC † is a two-photon creation operator characterized by the biphoton wave function (2.14) Note that it is symmetric Although (2.12) is already normalized, we are also free to choose C † II (t 0 ) |vac F to be normalized, which requires that we set Recalling (2.12), we see that this choice implies that in the limit |ζ| ≪ 1, |ζ| 2 can be thought of as the average number of generated photon pairs per pump pulse.

B. Photon generation in the presence of scattering loss
To include the effects of scattering loss in nonlinear waveguides within this formalism, we introduce a reservoir of radiation modes with the free Hamiltonian with operators that satisfy and that is coupled to pump and downconverted modes via the Hamiltonian (2.19) Here µ is a shorthand for all quantities necessary to specify reservoir modes in addition to m and k, and c mµk are waveguide-reservoir coupling terms. While it is possible in principle to solve for the radiation modes with which the b mµk are associated, and perform field overlap integrals to determine the c mµk , we do not attempt to do so here. We simply view the reservoir and coupling Hamiltonians as phenomenological entities, and later on in our calculation connect the waveguide-reservoir coupling terms to the usual experimentally determined loss coefficients. Although b Fµk and b SHµk come from the same field expansion, the division of the "total" reservoir into two parts through the sums in (2.17) and (2.19) is nevertheless justified. Following the rotating wave approximation used to simplify the coupling Hamiltonian, H C , the reservoir modes that couple strongly to the F waveguide modes, a Fk , are well-separated in frequency from the reservoir modes that couple strongly to the SH waveguide modes, a SHk . While this phenomenological model allows for light to couple both into and out of the guided waveguide modes via the reservoir, we eventually take the temperature of the reservoir to be zero so that no light can ever scatter back into the waveguide modes. This assumption, which is quite reasonable here as room temperature blackbody radiation at the frequencies of interest is negligible, is common in the study of open quantum optical systems and simplifies our calculations.
At this point, there are many ways that one could proceed. One could derive an expression for the evolution of the reduced density operator describing the generated photons. Written as a differential equation, this is known as the Master Equation, and can be put into Lindblad form [28]. Alternatively, one could work with quasiprobability distribution function representations for the same density operator [29], and arrive at a Fokker-Planck type equation [30,31]. However, we find that the approach that most clearly captures the physics and lends itself to integration with a Schrödinger state picture approach is a quantum Langevin formalism in which equations of motion are derived for waveguide operators in terms of "fluctuating force" reservoir operators, which are later traced out of the appropriate density operator [31], and it is this procedure that we follow here.
We imagine the same generalized asymptotic-in coherent state as above (2.6) incident on a lossy nonlinear waveguide, and also work in the backwards Heisenberg picture as above, the only difference being that we now include the reservoir and coupling Hamiltonians in our calculation. With H L → H L + H R in (2.7), we find that the asymptotic-out state can be written  (2.22) subject to the "final" condition The vacuum ket now encompasses the Hilbert space of the reservoir in addition to the F and SH waveguide operator spaces, i.e. |vac = |vac F ⊗ |vac SH ⊗ |vac R . Writing the state of the reservoir in this way is justified in the zero-temperature limit, where the reservoir density operator with T the temperature and k B the Boltzmann constant, becomes the pure state The method presented here has eliminated trivial linear evolution according to both H L as well as H R and, in addition to nonlinear effects from H NL , our backwards Heisenberg equation now includes the effects of coupling to the reservoir. Explicitly, the differential equations for the barred reservoir operators, which follow from (2.21) are 27) which can then be substituted in the differential equations for the barred waveguide operators a mk (t). Treating the nonlinear term containing S (k 1 , k 2 , k; t) as a perturbation, we are interested in the first-order solution for the waveguide operator in our output state a SHk (t). This operator has the first-order equation with zeroth-order equations for both F and SH barred waveguide operators where we have used (2.27). We remind readers that c mµk are coupling coefficients, not operators, and that ω mk and Ω mµk are, respectively, energies associated with waveguide mode and radiation mode (reservoir) operators.
To evaluate these integrals we note that, as a first approximation, the guided modes labeled by a particular m and k couple equally to all reservoir modes of the same m and k regardless of µ. That is, for a fixed m and k, there are so many degrees of freedom represented by µ that scattering into each is equally likely. Physically, this is because we assume that waveguide roughness at different positions is approximately uncorrelated and the scattering spectrum is flat over the µ of interest, an approximation certainly accurate down to a few nanometers [6]. With this in mind, we approximate |c mµk | 2 ≈ C mk as independent of µ and write dµ = (dµ/dΩ mµk ) dΩ mµk , also approximating the density of states dµ/dΩ mµk ≈ D mk as independent of µ so that (2.29) can be cast into the form of a quantum mechanical Langevin equation where the loss rate β mk = C mk πD mk > 0 and fluctuating force operator The zeroth-order solution for a † mk (t) is then (2.32) It is easy to verify that the fluctuating force operators satisfy (recall (2.18)) does not decay in time, as quantum mechanics requires. In fact, the commutation relation is the same as that for a † mk (2.3). We remark that these zerothorder equations are just multimode generalizations of the well-known single-mode quantum Langevin equation [31], da † /dt = −βa † + F † (t), with F (t) , F † (t ′ ) = 2βδ (t − t ′ ). The difference in sign of the loss rate β between the two differential equations (recall (2.30)) arises because in the usual Heisenberg picture systems operators evolve forwards in time, whereas in the backwards Heisenberg picture here the system operators evolve backwards in time such that their associated Schrödinger picture state correctly evolves forwards in time. The first-order solution for a † SHk (t) follows immediately from (2.28), and we see that to this order the asymptotic-out state (2.20) can be written |vac SH is the initial coherent state (2.6), having decayed exponentially in time from t 0 to t 1 and is a two-photon creation operator characterized by the total biphoton wave function (2.36) The two-photon creation operator and biphoton wave function calculated in the presence of loss are key results. They represent generalizations of (2.13) and (2.14) to include the effects of scattering loss for photons generated in nonlinear waveguides. However, much more can be learned in the frequency representation, and although we have progressed from the c mµk to β mk , the β mk are still not in the form of the standard attenuation coefficients α mk , usually expressed as inverse lengths.
Note that, just as seen for its non-lossy counterpart above (2.14), the biphoton wave function here (2.36) is symmetric (2.37) We normalize it as well implying that in the limit |ζ| ≪ 1, |ζ| 2 can still be thought of as the average number of generated photon pairs per pump pulse. As the inclusion of loss in our formalism has introduced fluctuating force operators with a different time dependence to the system operators, the time integral has moved from within the biphoton wave function in the lossless expression (2.14), to a more explicit part of the two-photon creation operator (2.35). However, it is easy to write the lossless two-photon creation operator with the time integral in the same position, and indeed we note that if there were no loss, i.e. no coupling to the reservoirs c mµk = 0, then β mk = 0, F mk (t) = 0, and (2.35) would be the same as (2.13).

III. THE DENSITY OPERATOR REPRESENTATION OF GENERATED PHOTON PAIRS
We now turn to the statistical properties of the generated photon states by moving to a density operator picture. We imagine that we are in a pulsed pump regime where the probability of generating a pair of photons is low enough, |ζ| ≪ 1, that we may approximate Anticipating the eventual trace over reservoir operators, we can write this in density operator form as where (recall (2.25)) and While vacuum-pair correlations, or "cross terms", in (3.3) are present, we note that they would be present even if one constructed a density operator in the low probability of pair production regime in a lossless calculation. There (recall (2.12)) one also has 1 + ζC † II (t 0 ) |vac F , and often focuses on the normalized two-photon state C † II (t 0 ) |vac F , produced with probability |ζ| 2 . As our interest in this work is in photon pairs, we focus on the corresponding piece of our density operator produced with probability |ζ| 2 in all that follows.

A. k-space expressions
We construct the reduced density operator describing the state of generated photons by tracing over the reservoir operators in ρ II at zero temperature where we have used (2.37). Thus our state of generated photons naturally separates into where we have dropped the mode label F on the vacuum state for notational convenience, leaving We recognize ρ 2 as representing a pure state of two photons with photon creation operators multiplied by decaying exponentials and correlations determined by φ (k 1 , k 2 ; τ ). The middle term ρ 1 is not pure, and represents a mixed state of single photons, while the final term ρ 0 is simply a number that sits in front of the vacuum density operator |vac vac|. More will be said about these terms in the next section.

B. ω-space expressions
The density operator derived at the end of the previous section (3.10) is best understood in the frequency representation. To get there, as in [8], we take and where the derivatives have been introduced to ensure normalization in frequency space. Furthermore, we approximate the effective area A and group velocities v m as being constant over the frequency ranges of interest [8], and take t 0 = −t 1 . While earlier work took , doing so here would result in scattering loss occurring for all times. Thus, as mentioned above, as a first approximation we take t 0 = −L/ (2v SH ), where L is the length of the waveguide, and v SH is the group velocity of the pump field. As shown in the Appendix (A31), the usual attenuation coefficients α m (ω) are related to our β m (ω) via α m (ω) = 2β m (ω) /v m . This equality combined with our choice of interaction duration leads to ratios of group velocities, r ≡ v F /v SH , appearing in expressions written in terms of α m (ω). Recalling (2.4) and performing the temporal integrals in (3.7)-(3.9), the two-photon term can be written as the one-photon term as  The three different wave functions differ only in the argument of a hyperbolic sine term and

A. Exit probabilities
In the lossless case, in the limit of a low probability of pair production, the state that is generated with probability |ζ| 2 per pump pulse contains two photons that also exit the waveguide with probability |ζ| 2 . Working in the same regime in our current calculation, however, for a ρ gen that is generated with probability |ζ| 2 , there is only a probability Tr F [ρ 2 ] = dω 1 dω 2 |φ 2 (ω 1 , ω 2 )| 2 e −[αF(ω1)+αF(ω2)]Lr/2 ≡ P 2 , (4.1) that both photons of the pair exit the waveguide. There is also a probability that only one photon of the pair exits, and a probability that neither exits. The physics of these probabilities is that with increasing device length the probability of two photons exiting the device (4.1) always decreases. As there is an additional α F (ω) in the decaying exponential coefficient in front of the negative |φ 2 (ω 1 , ω 2 )| 2 in (4.2) compared to the decaying exponential coefficient in front of the positive |φ 1 (ω 1 , ω 2 )| 2 , the probability of a single photon exiting the device will first increase and then decrease as L increases from zero. Finally, as the length tends to infinity P 0 tends to 1. Its form ensures that the three probabilities sum to one regardless of device length, as the trace of a density operator must: We note that the unit trace of our generated photon density operator can also be confirmed in k-space, using (3.10). To first order in the nonlinearity, which we identify with the undepleted pump approximation, and in the limit of a low probability of pair production, these results are exact and enable many calculations, which we now explore.

B. General expressions
Having moved to a frequency representation and performed temporal integrals enables a comparison between the biphoton wave function associated with the twophoton term and the biphoton wave function that results in the absence of scattering loss. Rewriting (3.14) as where as the biphoton wave function associated with the twophoton contribution to the density matrix describing SPDC in a lossy nonlinear waveguide. Comparing this biphoton wave function with the biphoton wave function that results in a lossless nonlinear waveguide (2.14), two new features can be noticed. The first, seen on the final line of (4.8), are exponential decay terms associated with the scattering loss of pump and generated photons as they traverse the waveguide. The second, and perhaps less expected, is that the hyperbolic sine term responsible for energy conservation and often approximated as a Dirac delta function [8] sinh (4.10) now contains loss coefficients as well. It is these coefficients that lead to the "loss-matching" term alluded to in the introduction and, in fact, their inclusion might have been predicted from a classical sum frequency generation (SFG) calculation that includes scattering loss in the continuous wave (CW) limit. Approximating the frequency integral over (3.21) as (cf. (4.10)) we find where P SFG SH is the generated power, P SFG S the 'signal' pump power, P SFG I the 'idler' pump power, and (4.14) Note how the exponential, sine, and hyperbolic sine terms are essentially the same in both equations.

C. Biphoton probability densities
Here we plot the biphoton probability density |Φ 2 (ω 1 , ω 2 )| 2 associated with coincidence counts (4.12) for three qualitatively distinct cases. As an example, we consider the effects of including loss in the calculation of previously predicted biphoton probability densities in Bragg reflection waveguides [26]. We Taylor expand The relevant parameters are v SH = 74.3 µm/ps, v F = 89.8 µm/ps, Λ SH = 2.92 × 10 −6 ps 2 /µm, Λ F = 7.07 × 10 −7 ps 2 /µm. Furthermore, we take L = 2 mm, and a Gaussian pump pulse waveform with an intensity full width at half maximum (FWHM) in time of T = 2 ln (2)/∆ = 20 fs, and ω P = 2πc/775 nm. If losses are frequency independent and α SH v SH ≈ 2α F v F the shape of the biphoton probability density is exactly as in the absence of loss: the exponential attenuation factor alters the number of generated photons, but not the photon pair frequency correlations (see Fig. 1a) If instead frequency independent losses are such that α SH v SH ≫ 2α F v F the biphoton probability density is near Lorentzian 18) a shape that we plot in Fig. 1b for α SH = 40 cm −1 , α F = 2 cm −1 [33]. (The large difference in the size of these loss coefficients is due to the different guiding mechanisms employed at the second harmonic frequency versus the fundamental frequency. In this Bragg reflection waveguide, the F mode is guided by total internal reflection, while the SH mode is guided by Bragg reflections.) Using (4.1) and (4.2), we see that compared to the lossless case, when all photon pairs generated per pump pulse |ζ| 2 exit the waveguide, here only 0.44 |ζ| 2 do as pairs, while 0.49 |ζ| 2 do as singles. Additionally, the associated Schmidt number [34], K , characterizing the frequency correlations of the biphoton wave functions, has been reduced from 582 to 168 as we have moved moved from balanced losses to large SH losses and the side lobes of the phase matching sinc function contribution to the biphoton probability density have become washed out. Lastly, we consider the ability of losses to further reduce the frequency correlations of a biphoton probability density having a reasonably small associated K to begin with. For this we consider the same structure as above, with a pump pulse duration of T = 2 ps and a quadratic frequency dependent loss model. When losses are balanced the biphoton probability density looks as in Fig. 1c., whereas for the quadratic loss profile, α F (ω) = 1.77 × 10 10 (1/2 − ω/ω P ) 2 cm −1 , with α SH (ω) = 0, it takes the shape shown in Fig. 1d. For this loss profile, the generation bandwidth has been greatly reduced, much as when strong filters are applied to achieve nearly frequency uncorrelated photons, and as such only 0.0077 |ζ| 2 exit the waveguide as pairs, while 0.011 |ζ| 2 do as singles. However note that the interesting loss feature has led to a biphoton wave function that is naturally nearly frequency uncorrelated, with an associated Schmidt number of only 1.29 compared to the balanced loss case of 76.3.

V. CONCLUSIONS
In conclusion, we have presented a formalism capable of handling SPDC and linear scattering loss concurrently in a waveguide. A density operator describing the state of generated photons was calculated and shown to be composed of a two photon, a single photon, and a vacuum piece, with traces that sum to 1 corresponding to the probabilities of two photons, one photon, or zero photons exiting the device. In general, the biphoton wave function associated with the two photon contribution in a lossy waveguide was shown to additionally exhibit both exponential decay terms and a loss-matching hyperbolic sine term compared to the usual lossless biphoton wave function. By looking at biphoton probability densities in three different regimes we demonstrated that it is possible to tailor losses such that they do not affect the shape of the biphoton probability density whatsoever, that losses can wash out the side lobes of the phase matching sinc function contribution to the biphoton probability density, and also that losses might possibly be controlled to engineer desired biphoton probability densities.
the first of which has solution h mµ (z, t) = h mµ (z, 0) e −iΩmµt −i t 0 g m (z, τ ) c * mµ e −iΩmµ(t−τ ) dτ, (A15) assuming that the interaction is switched on at t = 0 (i.e. light enters the waveguide at t = 0). Following arguments presented in the main text, we approximate approximate |c mµ | 2 ≈ C m as independent of µ and write dµ = (dµ/dΩ mµ ) dΩ mµ , also approximating the density of states dµ/dΩ mµ ≈ D m , and substitute (A15) into the equations for the g m , yielding where we have defined and the fixed k version of our loss rate above as β m = C m πD m . We note that the commutation relation for this real-space fluctuation operator is exactly what might be expected from its k space analogue (2.33) r m (z, t) , r † m ′ (z ′ , t ′ ) = 2β m δ mm ′ δ (t − t ′ ) δ (z − z ′ ) (A20) We then put and write the equations above in terms of new operators such that G † m G m = P m has units of power: We work in the undepleted pump approximation, in the limit of stationary fields, where the time derivatives vanish, and the limit of strong pumps G S,I ≫ G SH , leaving Tracing over reservoir operators at zero temperature, we find that R m R = 0. Writing G m R = G m for nota-tional simplicity, and recognizing we arrive at coupled mode equations where we have assumed v S ≈ v I ≈ v F , ω S ≈ ω I ≈ ω SH /2 ≡ ω P /2, and used (4.14). For a waveguide extending from z = −L/2 to z = L/2 we find (A37)