A Summation Relations

Theory of Multiwave Mixing within the Superconducting Kinetic-Inductance Traveling-Wave Amplifier

Abstract

We present a theory of parametric mixing within the coplanar waveguide (CPW) of a superconducting nonlinear kinetic-inductance traveling-wave (KIT) amplifier engineered with periodic dispersion loadings. This is done by first developing a metamaterial band theory of the dispersion-engineered KIT using a Floquet-Bloch construction and then applying it to the description of mixing of the nonlinear RF traveling waves. Our theory allows us to calculate signal gain vs. signal frequency in the presence of a frequency stop gap, based solely on loading design. We present results for both three-wave mixing (3WM), with applied DC bias, and four-wave mixing (4WM), without DC. Our theory predicts an intrinsic and deterministic origin to undulations of 4WM signal gain with signal frequency, apart from extrinsic sources, such as impedance mismatch, and shows that such undulations are absent from 3WM signal gain achievable with DC. Our theory is extensible to amplifiers based on Josephson junctions in a lumped LC transmission line (TWPA).

superconducting, amplifier, kinetic inductance, traveling wave, frequency dispersion, three-wave mixing, four-wave mixing, RF, microwave
pacs:
07.57.Kp,03.67.Lx,74.25.nn,85.25.Oj,85.25.Pb
\savesymbol

iint \restoresymbolWSSiint

I Introduction

Superconducting amplifiers with wide frequency bandwidth, high dynamic range, and low noise are used in both quantum computingWallraff et al. (2004) and photon-detectorDay et al. (2003) research. They have utility to measure large arrays of quantum-limited frequency-multiplexed microwave superconducting resonators, with recent strides made using nonlinear amplifiers based on Josephson junctions.Castellanos-Beltran et al. (2008); Bergeal et al. (2010); Spietz et al. (2010); Hatridge et al. (2011); Hover et al. (2012); Roch et al. (2012); Mutus et al. (2013); Macklin et al. (2015) In particular, a near-quantum-limited Josephson traveling-wave parametric amplifier (JTWPA) recently has been fabricated with a quantum efficiency of and a signal gain greater than 20 dB over a 3 GHz bandwidth.Macklin et al. (2015) Focus of the present discussion is the nonlinear kinetic-inductance traveling-wave (KIT) amplifier, first realized by Eom and co-workers.Eom et al. (2012) In the coplanar waveguide (CPW) of the KIT, degenerate four-wave mixing (4WM) can occur between RF input pump and signal, resulting in signal amplification and generation of an idler product. This parametric mixing of traveling RF waveforms is analogous to 4WM realizable in the optical frequency regime,Agrawal (2001); Agha et al. (2009); ?; ?; ?; ? as in recent nonlinear resonance experiments involving strongly pumped, high-Q optical microcavities made from nonlinear media.Kippenberg (2004); ?; ?; ? In this 4WM-mode of operation the KIT possesses signal gain of dB, a bandwidth of GHz (centered about the pump tone), and a dynamic range of the order of dB, comparable to microwave transistor amplifiers. Another promising mode of operation of the KIT is when a DC bias is applied: additional parametric three-wave mixing (3WM) may ensue, producing signal gain of dB over an exploitable bandwidth of GHz, centered about half the pump tone.Vissers et al. (2016) This latter 3WM gain is achieved with less pump input power than the 4WM mode due to DC biasing of the Kerr-like nonlinear kinetic inductance of the waveguide. Unwanted higher pump harmonics and shock waves, prevalent at higher pump powers, may be intentionally inhibited by proper CPW loading design, and additional dispersion loading may be engineered to customize the onset, magnitude, and bandwidth of the resulting signal gain. In this paper we present a quantifiable model of the dispersion engineering and parametric mixing of the KIT amplifier, taking into account DC biasing, showing how optimal amplifier design may be achieved.

Figure 1: (a) Schematic of KIT CPW of length depicting example dispersion loadings of pattern length , with non-loading width and pump wavelength . With no DC applied, pump and signal tones are input and an additional 4WM idler is observed on output. (b) Corresponding periodic unit cell of length with regions, representing the loading pattern of (a). Showcased is loaded region at distance from start of cell, of length , capacitance per unit length , and linear kinetic inductance per unit length .

KIT devices have been fabricated from superconducting TiN and NbTiN films on Si.Day et al. (2003); Mazin et al. (2006); ? These materials are used to construct a CPW of a meter and more length, , with width typically microns. As in Fig. 1 (a), engineered loadings of repeat length , where , are introduced that represent regions of increased width of the CPW–as much as 3 times wider than . Loadings are designed with , where is the wavelength of a propagating RF pump, to maximize destructive interference. If the pump propagates along the unloaded line with wavenumber and dispersion frequency , where is the group velocity and and are the unloaded inductance and capacitance per unit length, respectively, then one effect of the loadings is to open gaps in as a function of . Loadings designed in this way are often referred to as frequency stops, and their corresponding gaps are known as stop gaps, since a tone will not propagate down the waveguide if its frequency falls within a gap. For example, if pH/m and fF/m in the unloaded line, such that m/s, then loadings placed at intervals of m introduce a first stop gap at frequency GHz. Two additional loadings may be introduced between those of spacing , such that the spacing between nearest loadings becomes , as pictured in Fig. 1 (a). The lengths of these additional loadings can be made greater than the initially described loadings, as suggested in the figure. This has the effect of broadening stop gaps at every third gap, starting with the third stop gap at GHz. The advantage of this design is that if a strong pump tone is placed just above or below the first stop gap, at GHz, then the higher third harmonic of the pump, which is prevalent in the KIT amplifier when operated in a 4WM nonlinear regime, can be suppressed, increasing efficiency of parametric amplification.

Geometries such as a double spiral and a meandering line have been used to create very long KIT CPWs, on chips of area cm2. At low temperature, traveling RF waves, consisting of a strong pump of fixed frequency and a smaller-amplitude signal of adjustable frequency , input to one port of the CPW, undergo degenerate 4WM along the direction of the CPW due to the Kerr-like nonlinear kinetic inductance per unit length, , of the underlying superconducting film, viz.

(1)

where is the linear kinetic inductance per unit length, made dependent on to account for engineered loadings, is the total time-dependent electrical current of the mixing waveforms, and is a constant scaling factor.Day et al. (2003); Eom et al. (2012) The CPW may be modeled as a straight LC ladder circuit where the total current and voltage satisfy the equations

(2)
(3)

within the waveguide. The model assumes perfect impedance matching between end nodes of the CPW, although in practice high inductance of the thin lines can lead to mismatch, making it difficult to obtain a smooth transfer function, , on output. The solution of Eqs. (2) and (3) for traveling-wave boundary conditions produces output from the second port of the amplifier that includes the amplified signal () and the pump (), as well as a generated idler product of frequency . In the spectral output, and are equidistant from , as sketched in Fig. 1 (a), in accordance with energy conservation, i.e., . The output is analogous to the products of degenerate 4WM that are encountered in nonlinear optical fibers.Agrawal (2001)

i.1 Implications of a Periodic Loading Design

In 4WM of pump, signal, and idler within nonlinear optical fibers, the electromagnetic fields are described by plane waves. Signal gain arises on output by satisfying the three criteria of (i) energy conservation, i.e., ; (ii) linear momentum conservation, i.e., the respective plane-wave wavenumbers satisfy ; and (iii) overall phase matching of the constituent waveforms. In the last criterion, phase matching is tunable by adjusting the input power of the pump, which alters the extent of self-phase modulation of the waveforms, as well as the cross-phase modulation between them.Agrawal (2001) The RF pump, signal, and idler currents that propagate along the CPW of the KIT amplifier also must obey these same criteria in order to achieve signal gain. However, due to the engineered periodic loadings, which necessitate and in Eqs. (2) and (3), the RF waveforms of the KIT amplifier cannot be described by plane waves. This has particular implication for the definitions of both momentum conservation and phase matching, and thus, parametric mixing as a whole, within the KIT.

A quantifiable theory of KIT operation, which addresses the magnitude and bandwidth of parametric signal gain without introduction of ad hoc fitting parameters, must account for the reduced translational symmetry imposed by periodic loadings. For example, in the linear limit, where , with , a solution of the voltage and current of Eqs. (2) and (3) is properly formed using Floquet-Bloch functions, viz.

(4)

where and are Floquet-Bloch coefficients periodic in . This construction introduces a Bloch wavenumber , analogous to the wavenumber of a plane wave, but unique to the first of an infinite number of one-dimensional Brillouin zones. Any other wavenumber may be reduced to one within the first Brillouin zone by translation via a reciprocal lattice vector , where is an integer.Kittel (1976); ? In the KIT amplifier, momentum conservation between parametrically mixing waveforms is defined in terms of these Bloch wavenumbers, instead of their plane-wave counterparts. Additionally, the dispersion frequency , as a function of , forms one of a manifold of bands of dispersion frequencies that comprise the metamaterial band structure of the KIT amplifier. These engineered photonic bands are separated by the stop gaps we described earlier, and each band can have a distinctly different group velocity as a function of . As we shall see from our theory, the parametrically mixing waveforms may be described as superpositions of these band states, which has important consequences for how overall phase matching is defined and achieved within the KIT.

Application of the Floquet-Bloch equation to the study of elementary excitations in the bulk of solid materials is well known, where the arrangement of atoms on a periodic lattice dictates solutions of the form of Eq. (4). Introductory texts, such as those of Ref. (Kittel, 1976; ?), provide the reader with solutions for lattice vibrations (acoustic and optical phonons) and magnetic-moment precession (spin waves), to name a couple of examples. In particular, electronic states of the bulk formed in this way are the basis for determination of the electronic band structure of solids, and thus, account for the fundamental electronic properties of these materials. Similarly, the metamaterial bands engineered via loading design dictate the parametric behavior of the KIT amplifier, and resemble in principle the development of photonic crystals to manipulate light through the control of dispersion and formation of photonic band states.John (1987); ?; ?; ?; ?; ?

Thus, the periodic variations in CPW width depicted in Fig. 1(a), designed to create frequency stops, also control, in a more general sense, the dispersion of RF traveling waves as they propagate along the KIT. A similar concept has been developed for Josephson junctions in a lumped LC transmission line (TWPA) using resonator-based dispersion engineering.O’Brien et al. (2014); White et al. (2015) The engineered loadings are realized within our theory via the definitions we construct for both the periodic linear kinetic inductance per unit length, , and the periodic capacitance per unit length, , as these functions enter Eqs. (2) and (3).

Figure 1(b) shows a schematic of a single loading pattern, or unit cell, of length . The unit cell is representative of the repeated loadings of Fig. 1(a), and accounts for changes in inductance and capacitance attributable to variations in the width of the CPW. Specifically, within Fig 1(b), there are regions of different inductance and capacitance pairs, labeled by index , and denoted by and , respectively. To simplify matters, we confine our attention to unit cells of even symmetry, such that regions and of Fig. 1(b) possess the same loading sizes, and therefore, the same inductance and capacitance values. Thus, there exists a center region that may be defined as a non-loading region, with alternating loading and non-loading regions to either side, with the total number of regions always an odd number. We then model and of a unit cell () as

(5)

where and are the starting position and length of region , respectively, and is the conventional Heaviside step function. The definition in Eq. (5) may be extended to the entire length of the waveguide using and . To first approximation it is reasonable to model the waveguide as straight, with tens to thousands of repeated unit cells along the length.

i.2 Parametric Multiwave Mixing within the KIT Amplifier

As mentioned, application of a DC bias, , to the KIT amplifier can induce 3WM processes, as well as additional 4WM processes.Vissers et al. (2016) We refer to this scenario as parametric multiwave mixing. Specifically, with , mixing of a pump of frequency and a signal of frequency produces three idlers of frequencies , , and . Figure 2(a) summarizes the six parametric scattering processes that occur with onset of . Only the degenerate 4WM process of (bright red), when , and 3WM process of (cyan), when , contribute to broadband signal gain since only these processes achieve momentum conservation: and , respectively, over a broad range of signal frequencies near the bottom of the amplifier dispersion-frequency manifold.

Figure 2: (a) Sketch of six parametric mixing processes of KIT with DC bias, involving pump (), signal (), and idler (, , ) frequencies. Degenerate 4WM (bright red) and topmost 3WM (cyan) produce broadband signal gain when and , respectively. Additional processes are: 4WM (dark red and orange) and 3WM (blue and purple). (b) Simple depiction of the low-lying frequency-dispersion bands of the KIT amplifier approximated as a CPW with no engineered loadings, where is the magnitude of the approximate group velocity, assumed the same in each band. (c) Total momentum of the six parametric processes of (a) as a function of when is at the first stop gap, as obtained using the approximate band structure of (b).

To see this for the case of , one can make a simple estimate of the total momentum of each of the six scattering processes as a function of , using the unloaded dispersion frequency . Conservation of energy for the six parametric processes is assumed as in Fig. 2(a). Recall that, within the loaded KIT, momentum conservation of parametrically mixing waveforms requires the use of Bloch wavenumbers. This necessitates folding into the first Brillouin zone so that every wavenumber is a reduced-zone wavenumber, as pictured in Fig. 2(b). In folding , we place the pump of tone at the first stop gap, such that its corresponding wavenumber is representative of the zone edge, i.e., , such that folding of is done with respect to . Thus, for example, the 4WM idler of frequency , matching to the second dispersion-frequency band, has the dispersion frequency , as illustrated in the figure. Because the simple approximate of Fig. 2(b) involves unloaded dispersion frequencies, there are technically no gaps in its manifold, and the magnitude of the group velocity is the same in each dispersion-frequency band. This is not the case in reality, but is useful for the present discussion.

For the case of the 3WM scattering process of , we have , , and , such that . Hence, assuming a pump tone placed near the first stop gap, momentum is conserved for this process across the range of signal frequencies that lie within the first dispersion-frequency band. As another example, consider the 4WM process of , where we note and . In particular, we have , and also . Thus, the total momentum in this case is . From Fig. 2(a) we have , , and , so alternatively we may write the total momentum as , which represents a straight line as a function of , with slope and intercept . The total momentum of the other four processes may be approximated similarly. Figure 2(c) summaries the approximate total momentum of the six parametric scattering processes as a function of signal frequency. Apart from incidental momentum conservation of the 4WM process of , at half the pump frequency, the 3WM process of is the source of signal amplification when , over the range of signal frequencies of the lowest-lying dispersion-frequency band. This exercise illustrates the importance of working within the reciprocal-lattice construct dictated by the underlying translational symmetry.

To understand the behavior of parametric multiwave mixing, let us define the current of the amplifier as , where is the total current of all RF waveforms: pump, signal, and three idler products. Substituting this expression into Eq. (1) we have a nonlinear kinetic inductance given by

(6)

where the term proportional to is associated with the three 3WM processes of Fig. 2(a). The three 4WM processes correspond to the term involving the square of . Noting the energy of kinetic inductance per unit length: , as RF input power increases from zero with DC bias applied, it is the 3WM processes that activate first, generating the three idler products. Only after RF input power is increased further, beyond a threshold of amplitude , do the three 4WM processes begin to dominate, with the two additional 4WM processes arising secondarily, after 3WM processes have generated the new idlers and . When and is situated just above the first stop gap, one finds the 3WM process of initiates broadband signal gain centered about half the pump frequency, i.e., .Vissers et al. (2016) In this case, as sketched in Fig. 2(c), the total momentum (cyan) is essentially zero, i.e., , for a range of signal frequencies about .

Because the energy of 3WM contributions is one integer exponent less in the RF current amplitude than 4WM contributions, it takes less pump input power to operate the amplifier in this 3WM mode. However, if the input power is increased too high, the amplitude of traveling-wave current will increase beyond the threshold, allowing 4WM to dominate and thus wash out the effect of 3WM broadband gain. In particular, incidental momentum conservation of the 4WM process of , as approximated in Fig. 2(c), will become more prevalent for . Hence, the 3WM mode of the amplifier is limited to a range of RF input powers. Similarly, if the waveguide length is made too long then the increased run length of the traveling waves will also lead to current amplitude exceeding the 4WM threshold. Thus, when operated in 3WM mode, the signal gain of the KIT does not exhibit exponential growth with waveguide length; to the contrary, the amplifier has a waveguide length limitation.

i.3 Nonlinear Forward-Traveling Waves and the Floquet-Bloch Supermode

Periodic loadings create stop gaps in the dispersion-frequency spectrum of the KIT amplifier, but they also modify the group velocity of RF traveling waves as they propagate along the CPW. In our theory we assume the dispersive propagation is monochromatic, such that, for example, an RF signal of frequency injected into the amplifier can be matched to a specific dispersion frequency of the amplifier, as in Fig. 2(b), i.e., one has , where is the matching dispersion frequency, governed by the loading design, is the Bloch wavenumber of the signal as it propagates within the CPW, and is the index of the matching band. If the the signal is injected at sufficiently low power, i.e., small amplitude, and the waveguide length is not too long, then the traveling wave will retain a linear form throughout the waveguide, with current and voltage satisfying the Floquet-Bloch condition of Eq. (4). Since the coefficients and of Eq. (4) are periodic in the unit cell length , each may be expanded in a discrete Fourier series. Hence, in the linear limit of KIT operation, the voltage and current of the forward-traveling-wave signal assume the form

(7)

where and are Fourier coefficients.

On the other hand, if the input power is sufficiently high, or the waveguide length is long enough, then becomes nonlinear due to its dependence on the total current, and therefore Eq. (7) is no longer a viable solution of Eqs. (2) and (3). In this case, if the resulting nonlinear forward-traveling wave propagates adiabatically, then we may still assume the form of Eq. (7), except that the Fourier coefficients now take on a slowly varying dependence on , i.e., and . This nonlinear Floquet-Bloch forward-traveling-wave solution is then of the form

(8)

where the slowly varying coefficients satisfy the condition

(9)

Additionally, as we shall show in the development of the theory, the coefficients and can be further written as superpositions of the dispersion-frequency band states at . In this later expansion, the nonlinear Floquet-Bloch forward-traveling-wave solution of Eq. (8) may be referred to as a Floquet-Bloch supermode construction, and is not unlike the description of the plane-polarized electric field of nonlinear arrays of coupled optical waveguides.Mills and Trullinger (1987); ? The nonlinear response of these latter optical superlattices admit transverse-propagating soliton and gap-soliton solutions.Chen and Mills (1987a); ?; ? These collective supermode excitations, referred to as Floquet-Bloch solitons, have been demonstrated experimentally in optical waveguide arrays.Mandelik et al. (2003)

In the KIT amplifier, the parametric mixing of supermodes of pump, signal, and idler products makes for a complex description of overall phase matching, particularly as the band-superposition of each supermode evolves with increasing . Each band-component waveform of a given supermode corresponds to its own characteristic group velocity, which compounds the description of self-phase and cross-phase modulation between these components. In our theory we are able to account for this complex overall phase matching and obtain a quantifiable result for the magnitude and bandwidth of signal gain of the KIT amplifier directly from the loading design.

In the preceding remarks we introduced several important concepts:

  1. KIT amplifiers are engineered with periodic loadings to create frequency stops to inhibit higher pump harmonics, as well as to modify dispersion characteristics of RF traveling waves. The loading design is incorporated into our theory.

  2. Periodic loadings reduce the translational symmetry of the amplifier, which necessitates introduction of a band structure of dispersion frequencies and the Bloch wavenumber of a first Brillouin zone, as in Fig 2(b). In our theory the band structure contains the stop gaps of the loading design, as well as group velocity that may vary from Brillouin zone center to Brillouin zone edge, as well as from band to band.

  3. The criteria for parametric amplification is altered by the reduced translational symmetry: momentum conservation must be expressed in terms of Bloch wavenumbers and overall phase matching between parametrically mixing waveforms must include dispersion-frequency bands with varying group velocities. Our theory incorporates these modified criteria.

  4. In our theory nonlinear, parametrically-mixing RF forward-traveling waves may be expressed as Floquet-Bloch supermodes constructed from slowly-varying superpositions of dispersion-frequency band states. As these dispersive forward-traveling waves propagate along the CPW, the evolution of the components of their respective superpositions defines the phase matching between them.

  5. In this way our theory allows us to calculate the magnitude and bandwidth of parametric signal gain directly from the loading design, without need to introduce ad hoc fitting parameters.

In what follows we first derive the metamaterial band theory of the KIT amplifier and use it as a basis for the theory of parametric multiwave mixing of nonlinear traveling waves. We then present results of calculations using a specific even-symmetry loading design for a KIT amplifier. The parameters of our model are those of Eq. (5), i.e., the set of , , , and , which determine the band structure. With band structure calculated, we show the dispersion of a single nonlinear forward-traveling wave as it propagates down the CPW of the KIT. We then present calculations of the signal gain of the KIT as a function of signal frequency, both without and with DC bias. We conclude with remarks about the extensiblility of our theory to other ladder-type, equivalent-circuit models of nonlinear traveling-wave parametric amplifiers.

Ii Theory

ii.1 Band Theory of the KIT Amplifier

We consider a KIT amplifier with engineered dispersion loadings as in Fig. 1(a), modeled as a straight LC ladder-type transmission line of total length . The variable measures position along the length of the CPW. A unit cell of length of the loading design is sketched in Fig. 1(b) and expressed via Eq. (5). We first confine our attention to the linear limit of , and with we look for band solutions of Eqs. (2) and (3) using the Floquet-Bloch construction of Eq. (4). The current and voltage defined in this way satisfy periodic boundary conditions where is the Bloch wavenumber and is the band frequency.

Like and , the Bloch amplitudes , of Eq. (4) are periodic in with periodicity . Thus, we have four periodic functions that may be expanded in a discrete Fourier series, i.e., each may be transformed in the manner

(10)
(11)

where . We also introduce discrete Fourier transforms and , which are elements of matrix inverses corresponding to and , respectively. These may be written explicitly as

(12)
(13)

from which one may easily show and .

If we now substitute Eq. (4) for and in Eqs. (2) and (3), make use of the discrete Fourier transform pair of Eqs. (10) and (11) for each of our four periodic functions, and apply the matrix inverses of Eqs. (12) and (13), we may decouple voltage and current in the transform space, obtaining the result

(14)
(15)

where we have introduced a non-Hermitian dispersion matrix with elements given by

(16)

Diagonalization of the matrix of Eq. (16) produces the metamaterial band structure of the KIT amplifier.

Since the above dispersion matrix is non-Hermitian, we must introduce left, , and right, , eigenvectors and express the diagonalization formally as

(17)
(18)

Here refers to a band index, of which there are an infinite number at each value of . For orthonormality and completeness we define

(19)
(20)

which presupposes the existence of a non-unitary similarity transformation diagonalizing Eq. (16). Several notable properties of the dispersion matrix, which are easily verified, are

(21)
(22)

These relations are consistent with reciprocity of linear waveform propagation in either direction of the waveguide.

Appendix B shows how one may apply Eq. (5) to Eq. (16) to obtain a useful formulation of for arbitrary loading design of even symmetry. The result is the real-valued dispersion matrix element of Eq. (81), which we may write as

(23)

where we have introduced coefficients

(24)

Note that is just Eq. (24) with , interchanged with , . The generality of Eq. (23) allows us to explore the band structure of a wide range of loading designs of even symmetry.

For the remainder of our discussion we adopt a specific convention for labeling the bands of the KIT amplifier. Figure 3 provides a comparison of reduced-zone scheme () and extended-zone scheme () representations of KIT dispersion frequencies, using the no-load limit of constant and to illustrate how the extended-zone dispersion curve is mapped to bands. The dashed vertical blue lines represent Brillouin zone boundaries defined by reciprocal lattice vectors . The black (blue) line segments of the reduced zone, corresponding to specific positive (negative) indexes , map to the extended-zone dispersion curve of (). Our convention is to label the bands such that in the reduced zone we have the ascending order , for , and , for . In this way the mapping of extended to reduced zone follows as , for any loading design. The red and green dashed lines show several examples of mapping, involving the wavenumbers of the reduced zone. This labeling convention is useful to the understanding of the effect of a loading design on parametric mixing.

Figure 3: (a) Comparison of reduced-zone scheme () and extended-zone scheme () representations of KIT dispersion frequencies, as described in the text. (b) The five types of multichannel parametric scattering associated with multiwave mixing within the KIT amplifier, also discussed in the text.

ii.2 Theory of Multiwave Mixing within the KIT Amplifier

If dispersion loadings of the KIT are engineered to inhibit formation of pump harmonics and shock waves then we may confine our solution of Eqs. (2) and (3) to forward-traveling RF waves, labeled by index , consisting solely of the pump (), signal (), and idlers (), with corresponding frequencies , as depicted in Fig. 2(a). Inside the waveguide we approximate voltage and current of the -th forward-traveling wave in terms of a Floquet-Bloch-like function, as in the example of Eq. (8) for the case of , again introducing a Bloch wavenumber, defined as . Assuming monochromatic dispersion, is determined by matching frequency to a specific band of the KIT, i.e., , as in Fig. 2(b). Like Eq. (8), we allow the Fourier-like coefficients of each traveling wave to assume a slowly-varying dependence on . Specifically, we may write

(25)

where the assumption of a slowly-varying-amplitude may be expressed as

(26)

as in Eq. (9). Allowing for a DC bias the full solution, including all mixing waveforms and DC, may be expressed as

(27)
(28)

Equations (27) and (28) may be substituted into Eqs. (2) and (3) to obtain an expression of the current coefficients decoupled from those of the voltage, . The steps of derivation are similar to those we outlined for the linear limit, with the caveat that we may also leverage the canonical transformation implied by Eqs. (19) and (20). For example, the current coefficients may be expanded as

(29)

where the dimensionless current amplitudes also are slowly varying in , in the manner of Eq. (26). Equation (29) expresses the fact that, although an initial boundary condition may be imposed on the slowly-varying amplitude at , the effect of dispersion will cause the waveform to hybridize with other states of as it evolves in along the waveguide. These assumptions regarding the form of our solution hold as long as the traveling wave propagates adiabatically along the waveguide.

With application of the canonical transformation, solution of the current , in particular, follows by solving an infinite set of coupled differential equations involving amplitudes , expressed in terms of eigenvalues and vectors of our band theory. Appendix C gives a general derivation of the coupled equations. Our focus will be restricted to dynamics of the six mixing processes of Fig. 2(a). These involve the phases

(30)
(31)
(32)
(33)
(34)
(35)

which measure the extent of momentum conservation for each process, respectively. Hence, defining , we have directly from Eq. (92) the coupled subset of pump, signal, and idler amplitude equations given by

(36)
(37)