Schmidt decompositions of parametric processes III: simultaneous amplification and conversion

C. J. McKinstrie

doi:10.1364/OE.23.016949

1. Introduction

Parametric (wave-mixing) processes provide a variety of signal-processing functions required by classical communication systems [1–4] and quantum information experiments [5–8]. Classical processes are governed by coupled-mode equations (CMEs) of the form

d_{z} X = J X + K X^{*},

where d_z = d/dz is a space derivative, X = [x_j] is an n × 1 mode-amplitude vector, J and K are n × n coefficient matrices, and the superscript * denotes a complex conjugate. The self-action (-coupling) matrix J is anti-hermitian, whereas the cross-coupling matrix K is symmetric. Because Eq. (1) is linear in the amplitude vector and its conjugate, the solution of Eq. (1) can be written in the input–output (IO) form

X (z) = M (z) X (0) + N (z) X^{*} (0),

where M = [μ_jk] and N = [ν_jk] are transfer (Green) matrices.

Every complex matrix M has the singular value (Schmidt) decomposition M = VDU^†, where D is diagonal, U and V are unitary, and † denotes a hermitian conjugate [9]. The columns of U are the eigenvectors of M^†M, the columns of V are the eigenvectors of MM^† and the entries of D are the (common) non-negative eigenvalues of M^†M and MM^†. The laws of Hamiltonian mechanics [10], which are reviewed briefly in App. A, impose constraints on the aforementioned transfer matrices [11]. These constraints ensure that they have the simultaneous decompositions M = VD_μU^† and N = VD_νU^t, where the entries of the diagonal matrices (Schmidt coefficients) satisfy the auxiliary equations $μ_{j}^{2} - ν_{j}^{2} = 1$ and t denotes a transpose [12]. It follows from these decompositions that the columns of U(z) represent input Schmidt vectors, the columns of V(z) represent output vectors and the Schmidt-mode amplitudes x̄_j(0) and x̄_j(z), which are the components of X relative to the input and output Schmidt bases, respectively, satisfy the one-mode stretching (squeezing) equations

{\bar{x}}_{j} (z) = μ_{j} (z) {\bar{x}}_{j} (0) + ν_{j} (z) {\bar{x}}_{j}^{*} (0) .

Thus, every multiple-mode parametric process, no matter how complicated, can be decomposed into independent one-mode processes, the properties of which are known. If ν_j = 0 (μ_j = 1), the mode evolution is termed passive, because it corresponds to a basis change (U_j → V_j). Examples of passive transformations include phase shifting, beam splitting and frequency conversion (FC) by three-wave mixing (TWM) in a second-order nonlinear medium or four-wave mixing (FWM) in a third-order nonlinear medium. Conversely, if ν_j > 0 (μ_j > 1), the mode evolution is termed active, because the real quadrature (real part of x̄_j) is dilated by the factor μ_j + ν_j (stretched), whereas the imaginary quadrature is dilated by the reciprocal factor μ_j − ν_j (squeezed). Examples of active transformations include parametric amplification (PA) by TWM and FWM.

Although the preceding formalism was described for classical parametric processes, it applies also to quantum processes, in which the classical mode-amplitudes x_j are replaced by the quantum mode-operators x̂_j [10, 13, 14]. In the Heisenberg picture, the state vector is constant and the mode operators evolve (with distance). For systems governed by Hamiltonians that are second order (quadratic) in the mode operators (amplitudes), the Heisenberg equations for the operators are identical to the Hamilton equations for the amplitudes. Hence, the classical and quantum processes are described by the same transfer matrices (and decompositions thereof). In the Schrödinger picture, the mode operators are constant and the state vector evolves. For a single-mode system, the relation between the Heisenberg transfer functions and the Schrödinger output-state vector is known [13, 14]. Hence, if one can determine the Schmidt coefficients and vectors associated with the transfer matrices, one also can determine the quantum states.

The simultaneous Schmidt decomposition (of M and N) tells us how to interpret the (classical and quantum) IO equations. However, it does not tell us how to obtain them. In principle, one can solve the CMEs by rewriting them in the form

d_{z} Y = L Y,

where the augmented amplitude vector and coefficient matrix are

Y = [\begin{matrix} X \\ X^{*} \end{matrix}], L = [\begin{matrix} J & K \\ K^{*} & J^{*} \end{matrix}],

respectively. L has the adjoint (spectral) decomposition ED_λF^†, where D_λ is a diagonal matrix whose (nonzero) entries are the eigenvalues λ_j of L [9]. The columns of E are the eigenvectors of L and the columns of F are the eigenvectors of L^† (adjoint eigenvectors of L). Because Eq. (4) is linear in the (augmented) amplitude vector, its solution can be written in the IO form

Y (z) = T (z) Y (0),

where T is an augmented transfer matrix. If L is a constant matrix, then T(z) has the adjoint decomposition ED_τ(z)F^†, where D_τ is a diagonal matrix whose entries are the transmission factors τ_j(z) = exp(λ_jz). Notice that L and T share the same (adjoint) eigenvectors, which are constant.

The adjoint and Schmidt formalisms are well known. The simplest applications of these formalisms, to two-mode amplification and frequency conversion, are described in [15–18] and reviewed briefly in App. B. Both formalisms provide physical insight into the IO transformations. If the wavenumbers are matched, the eigenvectors and Schmidt vectors are identical, whereas if they are not matched, the components of the aforementioned vectors differ by phase factors. In these examples, which are simple, the adjoint and Schmidt decompositions are similar, and the associated formalisms are elegant, but nonessential, tools. What are missing from the literature are intermediate examples, which are complicated enough to require the use of these formalisms, and illustrate the differences between them, but straightforward enough to be tractable analytically.

One such example is simultaneous (wavenumber-matched) difference- and sum-frequency generation (DSFG), which occurs in a second-order nonlinear medium [19]. DFG is a TWM process in which π_p → π₁ + π₂, where π_j represents a photon with frequency ω_j, and SFG is a process in which π_p + π₁ → π₃ [20]. DFG amplifies the signal (1) and idler (2), whereas SFG transfers (converts) power from the signal to the idler (3), but both processes deplete the pump. The frequencies of the interacting waves are illustrated in Fig. 1. Similar processes occur in third-order nonlinear media [3, 21, 22].

Fig. 1 Frequency diagram for simultaneous difference- and sum-frequency generation. In the presence of a strong pump (p), a weak signal (1) interacts with a weak difference-frequency idler (2) and a weak sum-frquency idler (3).

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This paper is organized as follows: In Sec. 2, the three-mode equations for simultaneous PA and FC are written in matrix form [Eq. (4)]. The eigenvalues of the coefficient matrix, and the associated eigenvectors and adjoint eigenvectors, are determined and used to construct the augmented transfer matrix (T), the properties of which are described briefly. In Sec. 3, the Schmidt decompositions of the transfer matrices (M and N) are determined explicitly and contrasted to the adjoint decomposition of the augmented transfer matrix. In Sec. 4, it is shown that multiple-mode processes that involve unequal numbers of amplitudes and conjugate amplitudes always can be partitioned into passive processes (like phase shifting) and active processes (like amplification). In Sec. 5, the effects of stimulated Raman scattering (SRS) on PA and FC are discussed briefly. Finally, in Sec. 6, the main results of this paper are summarized.

2. Adjoint decomposition of the transfer matrix

Simultaneous PA and FC is governed by the three-mode equations (TMEs)

d_{z} x_{1} = i δ_{1} x_{1} + i γ_{12} x_{2}^{*} + i γ_{13}^{*} x_{3},

d_{z} x_{2} = i δ_{2} x_{2} + i γ_{12} x_{1}^{*},

d_{z} x_{3} = i δ_{3} x_{3} + i γ_{13} x_{1},

where x₁, x₂ and x₃ are the (complex) amplitudes of the signal, and the PA and FC idlers, respectively, the δ_j are (real) wavenumber-mismatch coefficients, and γ₁₂ and γ₁₃ are (possibly complex) nonlinear coupling coefficients. By combining Eqs. (7)–(9) with their conjugates, one finds that

d_{z} ({| x_{1} |}^{2} - {| x_{2} |}^{2} + {| x_{3} |}^{2}) = 0 .

Equation (10), which is called the Manley–Rowe–Weiss (MRW) equation [23, 24], constrains the photon-flux evolution: For PA alone, signal and idler photons are produced in pairs, whereas for FC alone the total number of signal and idler photons is conserved. One can rewrite Eqs. (7)–(9) in the matrix form of Eq. (1) by defining the coefficient matrices

J = i [\begin{matrix} δ_{1} & 0 & γ_{13}^{*} \\ 0 & δ_{2} & 0 \\ γ_{13} & 0 & δ_{3} \end{matrix}], K = i [\begin{matrix} 0 & γ_{12} & 0 \\ γ_{12} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] .

Notice that J is anti-hermitian and K is symmetric, as stated earlier.

For the application described in [19], the signal frequency is of order 1 THz, whereas the pump and idler frequencies are about 200 THz. For such frequencies, DFG and SFG are not perfectly wavenumber-matched, but their mismatch-distance products (for GaAs waveguides shorter than 1 cm) are so small that one can consider them to be wavenumber-matched in practice. By assuming that δ_j = 0 and making the variable changes x₂ → ix₂e^iϕ₁₂ and x₃ → ix₃e^iϕ₁₃, where ϕ₁₂ = arg(γ₁₂) and ϕ₁₃ = arg(γ₁₃), one obtains the simplified TMEs

d_{z} x_{1} = γ_{12} x_{2}^{*} - γ_{13} x_{3}, d_{z} x_{2}^{*} = γ_{12} x_{1}, d_{z} x_{3} = γ_{13} x_{1},

in which γ₁₂ and γ₁₃ are real, and the physical difference between DFG (PA) and SFG (FC) is manifested by the − sign. One can rewrite Eqs. (12) in the matrix form of Eq. (4) by defining the amplitude vector

Y = {[x_{1}, x_{2}^{*}, x_{3}]}^{t}

and the coefficient matrix

L = [\begin{matrix} 0 & γ_{12} & - γ_{13} \\ γ_{12} & 0 & 0 \\ γ_{13} & 0 & 0 \end{matrix}] .

For Eqs. (12), no augmentation was actually necessary: x₁,

x_{2}^{*}

and x₃ are coupled to each other, but not to

x_{1}^{*}

, x₂ or

x_{3}^{*}

.

It is easy to verify that L has the eigenvalues

0, k, - k,

where

k = {(γ_{12}^{2} - γ_{13}^{2})}^{1 / 2}

, and the associated eigenvectors

[\begin{matrix} 0 \\ γ_{13} / k \\ γ_{12} / k \end{matrix}], \frac{1}{2^{1 / 2}} [\begin{matrix} 1 \\ γ_{12} / k \\ γ_{13} / k \end{matrix}], \frac{1}{2^{1 / 2}} [\begin{matrix} - 1 \\ γ_{12} / k \\ γ_{13} / k \end{matrix}] .

These eigenvectors are not mutually orthogonal. However, the adjoint matrix L^† has the same eigenvalues and the adjoint eigenvectors

[\begin{matrix} 0 \\ - γ_{13} / k \\ γ_{12} / k \end{matrix}], \frac{1}{2^{1 / 2}} [\begin{matrix} 1 \\ γ_{12} / k \\ - γ_{13} / k \end{matrix}], \frac{1}{2^{1 / 2}} [\begin{matrix} - 1 \\ γ_{12} / k \\ - γ_{13} / k \end{matrix}],

which are orthogonal to the eigenvectors. Let E be a matrix whose columns are eigenvectors of L, let Λ be the diagonal matrix whose entries are the eigenvalues of L and let F be a matrix whose columns are eigenvectors of L^†. Then, as stated earlier, L has the adjoint (spectral) decomposition EΛF^†. (Normalization of E and F is assumed.) The constituent matrices satisfy the equations F^†E = I (orthogonality) and EF^† = I (completeness).

The solution of Eq. (4) can be writen in the form of Eq. (6), where the transfer matrix T (z) = exp(Lz). By using the adjoint decomposition of L to calculate its exponential, one finds that the transfer matrix has the adjoint decomposition T (z) = ED_τ(z)F^†, where the entries of D_τ are exponentials of the eigenvalues of L. Thus, T has the same eigenvectors as L, which are independent of z, and the related eigenvalues

1, exp (k z), exp (- k z) .

For the nondegenerate case in which γ₁₂ ≠ γ₁₃, the transfer matrix

T (z) = [\begin{matrix} cosh (k z) & γ_{12} sinh (k z) / k & - γ_{13} sinh (k z) / k \\ γ_{12} sinh (k z) / k & [γ_{12}^{2} cosh (k z) - γ_{13}^{2}] / k^{2} & γ_{12} γ_{13} [1 - cosh (k z)] / k^{2} \\ γ_{13} sinh (k z) / k & γ_{12} γ_{13} [cosh (k z) - 1] / k^{2} & [γ_{12}^{2} - γ_{13}^{2} cosh (k z)] / k^{2} \end{matrix}] .

In this case the mode amplitudes grow exponentially with distance or oscillate sinusoidally, depending on whether the PA coefficient (γ₁₂) is larger or smaller than the FC coefficient (γ₁₃), respectively. For the degenerate case in which γ₁₂ = γ₁₃ = γ, the transfer matrix reduces to

T (z) = [\begin{matrix} 1 & γ z & - γ z \\ γ z & 1 + {(γ z)}^{2} / 2 & - {(γ z)}^{2} / 2 \\ γ z & {(γ z)}^{2} / 2 & 1 - {(γ z)}^{2} / 2 \end{matrix}] .

In this case the mode amplitudes grow linearly (quadratically) with distance. One can verify Eqs. (18) and (19) by solving Eqs. (12) directly.

It is convenient to introduce the notation

T (z) = [\begin{matrix} μ_{11} (z) & ν_{12} (z) & μ_{13} (z) \\ ν_{21} (z) & μ_{22} (z) & ν_{23} (z) \\ μ_{31} (z) & ν_{32} (z) & μ_{33} (z) \end{matrix}],

where the transfer functions μ_jk connect like modes [x_j(z) to x_k(0)] and the transfer functions ν_jk connect unlike modes [x_j(z) to

x_{k}^{*} (0)

or

x_{j}^{*} (z)

to x_k(0)]. These functions are defined implicitly by Eqs. (18) and (19). Not only does the transfer matrix have an adjoint decomposition, it also has a Schmidt decomposition, which will be described in the next section.

3. Schmidt decomposition of the transfer matrix

Although the adjoint decomposition of the transfer matrix is useful, it is not canonical, because it mixes amplitudes and conjugate amplitudes. The IO equation can be written in the canonical form

B = M_{f} A + N_{f} A^{*},

where the input and output vectors A = X(0) and B = X(z), respectively, and the (forward) transfer matrices

M_{f} = [\begin{matrix} μ_{11} & 0 & μ_{13} \\ 0 & μ_{22} & 0 \\ μ_{31} & 0 & μ_{33} \end{matrix}], N_{f} = [\begin{matrix} 0 & ν_{12} & 0 \\ ν_{21} & 0 & ν_{23} \\ 0 & ν_{32} & 0 \end{matrix}]

depend implicitly on z. [The notations of Eqs. (20) and (22) are consistent because the transfer coefficients are real.] Hamiltonian evolution is symplectic, which means that the output vector B must satisfy the same Poisson-bracket relations as the input vector A [10]. By combining these relations (which are described in App. A) with Eq. (21), one obtains the auxiliary equations

M N^{t} - N M^{t} = 0, M M^{†} - N N^{†} = I,

where the subscript f was omitted for simplicity. Equations (23) impose constraints on the transfer matrices. For example, the diagonal elements of the second equation are

μ_{11}^{2} - ν_{12}^{2} + μ_{13}^{2} = 1, - ν_{21}^{2} + μ_{22}^{2} - ν_{23}^{2} = 1, μ_{31}^{2} - ν_{32}^{2} + μ_{33}^{2} = 1 .

These constraints (identities) apply to the rows of T [Eq. (20)].

For reference, the output–input equation can be written in the form

A = M_{b} B + N_{b} B^{†},

where the (backward) transfer matrices

M_{b} = M_{f}^{†}

and

N_{b} = - N_{f}^{t}

[11,15]. By repeating the procedure described above [applying Eqs. (23) to the backward matrices], one obtains the auxiliary equations

M^{†} N - N^{t} M^{*} = 0, M^{†} M - N^{t} N^{*} = I .

For example, the diagonal elements of the second of Eqs. (26) are

μ_{11}^{2} - ν_{21}^{2} + μ_{31}^{2} = 1, - ν_{12}^{2} + μ_{22}^{2} - ν_{32}^{2} = 1, μ_{13}^{2} - ν_{23}^{2} + μ_{33}^{2} = 1 .

These identities apply to elements in the columns of T [Eq. (20)] and are recognizable as the MRW equations (10) for the forward process. In retrospect, identities (24) are the MRW equations for the backward process.

As stated earlier, the Schmidt decomposition theorem asserts that the transfer matrices have the related decompositions M = VD_μU^† and N = VD_νU^t, where U and V are unitary, D_μ and D_ν are diagonal, and $D_{μ}^{2} - D_{ν}^{2} = I$ . The columns of U are the eigenvectors of M^†M and N^tN^*, and the columns of V are the eigenvectors of MM^† and NN^†. The entries of D_μ are the square roots of the (common) eigenvalues of M^†M and MM^†, whereas the entries of D_ν are the square roots of the (common) eigenvalues of N^tN^* and NN^†. In this case, the transfer matrices are real, so U and V are orthogonal (and real).

For the degenerate case, transfer matrix

N = [\begin{matrix} 0 & z & 0 \\ z & 0 & - z^{2} / 2 \\ 0 & z^{2} / 2 & 0 \end{matrix}],

where z is an abbreviation for γz. (According to the decomposition theorem, one can use either M or N to calculate the Schmidt coefficients and vectors. I chose N, because it has only two distinct nonzero elements). Routine calculations show that the Schmidt coefficients (diagonal entries of D_ν) are

0, z {(1 + z^{2} / 4)}^{1 / 2}, z {(1 + z^{2} / 4)}^{1 / 2},

the associated (canonical) input vectors are

\frac{1}{{(1 + z^{2} / 4)}^{1 / 2}} [\begin{matrix} z / 2 \\ 0 \\ 1 \end{matrix}], \frac{1}{{(1 + z^{2} / 4)}^{1 / 2}} [\begin{matrix} 1 \\ 0 \\ - z / 2 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}],

and the associated (canonical) output vectors are

\frac{1}{{(1 + z^{2} / 4)}^{1 / 2}} [\begin{matrix} - z / 2 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], \frac{1}{{(1 + z^{2} / 4)}^{1 / 2}} [\begin{matrix} 1 \\ 0 \\ z / 2 \end{matrix}] .

Notice that the Schmidt coefficients and both sets of Schmidt vectors depend on z. The other Schmidt coefficients (diagonal entries of D_μ) are

1, (1 + z^{2} / 2), (1 + z^{2} / 2) .

The input vectors of M are the same as the input vectors of N, but the output vectors are

\frac{1}{{(1 + z^{2} / 4)}^{1 / 2}} [\begin{matrix} - z / 2 \\ 0 \\ 1 \end{matrix}], \frac{1}{{(1 + z^{2} / 4)}^{1 / 2}} [\begin{matrix} 1 \\ 0 \\ z / 2 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] .

Notice that in Eqs. (31) and (33) the second and third vectors appear in opposite order.

Emboldened by the tractability of the degenerate case, I considered the nondegenerate case, in which the second transfer matrix can be written in the form

N = [\begin{matrix} 0 & s & 0 \\ s & 0 & - c \\ 0 & c & 0 \end{matrix}],

where s = γ₁₂ sinh(kz)/k and c = γ₁₂γ₁₃[cosh(kz) − 1]/k². Routine calculations show that the Schmidt coefficients (diagonal entries of D_ν) are

0, ν, ν,

where

ν = {(c^{2} + s^{2})}^{1 / 2} = {(ν_{21}^{2} + ν_{23}^{2})}^{1 / 2}

, the associated (canonical) input vectors are

\frac{1}{ν} [\begin{matrix} c \\ 0 \\ s \end{matrix}], \frac{1}{ν} [\begin{matrix} s \\ 0 \\ - c \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}],

and the associated (canonical) output vectors are

\frac{1}{ν} [\begin{matrix} - c \\ 0 \\ s \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], \frac{1}{ν} [\begin{matrix} s \\ 0 \\ c \end{matrix}] .

Notice that one can deduce the output vectors (37) from the input vectors (36) by changing the sign of c, as implied by Eq. (34). Notice also that an input vector with nonzero 1- and 3-components is related to the output vector with nonzero 2-component, and vice versa. Lengthy, but straightforward, calculations show that the other Schmidt coefficients (diagonal entries of D_μ) are

1, μ, μ,

where μ = μ₂₂ = [γ₁₂ cosh(kz) − γ₁₃]/k². It is easy to verify that μ² − ν² = 1. The input vectors of M are the same as the input vectors of N, but the output vectors are

\frac{1}{ν} [\begin{matrix} - c \\ 0 \\ s \end{matrix}], \frac{1}{ν} [\begin{matrix} s \\ 0 \\ c \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] .

Once again, the order of the second and third vectors is reversed. The nondegenerate results reduce to the degenerate results in the limit γ₁₂ → γ₁₃.

Let u₀, u₁ and u₂ denote the input vectors in Eq. (30) or (36), and v₀, v₁ and v₂ denote the output vectors in Eq. (33) or (39). Then in both cases (degenerate and nondegenerate), the actions of M are summarized by the IO equations

M u_{0} = v_{0}, M u_{1} = μ v_{1}, M u_{2} = μ v_{2},

whereas the actions of N are summarized by the IO equations

N u_{0} = 0, N u_{1} = ν v_{2}, N u_{2} = ν v_{1} .

In Eqs. (41), N maps u₁ to v₂ and u₂ to v₁. By combining this fact with the degeneracy of the Schmidt coefficients, one finds that

M (u_{1} \pm u_{2}) = μ (v_{1} \pm v_{2}), N (u_{1} \pm u_{2}) = \pm ν (v_{1} \pm v_{2}) .

In Eqs. (42), N maps u₁ ± u₂ to v₁ ± v₂.

According to Eqs. (20) and (22), the transfer matrix T = M + N. By combining this definition with Eqs. (40)–(42), one finds that

T u_{0} = v_{0}, T (u_{1} \pm u_{2}) = (μ \pm ν) (v_{1} \pm v_{2}) .

Equations (43) describe the Schmidt decomposition of T: The Schmidt coefficients are 1, μ + ν and μ − ν, the associated input vectors are u₀, u₁ + u₂ and u₁ − u₂, respectively, and the associated output vectors are v₀, v₁ + v₂ and v₁ − v₂. (To normalize the second and third vectors, one divides them by 2^1/2.) Notice that the 0 mode is passive, the + mode is stretched and the − mode is squeezed [μ − ν = 1/(μ + ν)].

Although Eqs. (42) enabled the Schmidt decomposition of the transfer matrix, they are not in canonical form, because of the − sign in the second equation. One can rewrite them in canonical form by remembering that M acts on input vectors, whereas N acts on their conjugates. Hence,

M [i (u_{1} - u_{2})] = μ [i (v_{1} - v_{2})], N [- i (u_{1} - u_{2})] = ν [i (v_{1} - v_{2})] .

In Eqs. (44), both Schmidt coefficients are positive, as required.

To illustrate the preceding results, in Fig. 2 the relative powers of the Schmidt-vector components are plotted as functions of the interaction distance γz, where γ = (γ₁₂γ₁₃)^1/2, for the unstable case in which γ₁₂/γ = 1.1 and γ₁₃/γ = 0.9. Equations (36) and (39) show that the amplitude components of the input and output vectors differ by signs only, so the power components are the same for both vectors. The components of the passive Schmidt mode (u₀ and v₀) are displayed in Fig. 2(a), whereas the components for the active modes (u₁ ± u₂ and v₁ ± v₂) are displayed in Fig. 2(b). For each mode, the components evolve monotonically as the distance increases and tend to limits. The stretching parameter μ² is plotted as a function of distance in Fig. 3(a). This parameter increases monotonically as the distance increases.

Fig. 2 Relative powers of the Schmidt-vector components plotted as functions of the interaction distance γz in the unstable regime. The solid, dot-dashed and dashed curves represent components 1, 2 and 3, respectively. (a) Passive mode and (b) active modes.

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Fig. 3 Stretching parameter plotted as a function of the interaction distance γz in (a) the unstable regime and (b) the stable regime. Significant stretching occurs in both regimes.

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In Fig. 4 the relative powers of the Schmidt-vector components are plotted as functions of the interaction distance for the stable case in which γ₁₂/γ = 0.9 and γ₁₃/γ = 1.1. Once again, the power components of the input and output vectors are the same. The components of the passive Schmidt mode (u₀ and v₀) are displayed in Fig. 4(a), whereas the components of the active modes (u₁ ± u₂ and v₁ ± v₂) are displayed in Fig. 4(b). The stretching parameter μ² is plotted as a function of distance in Fig. 3(b). Although this parameter evolves periodically, its peak value is large. This behavior is similar to that of the constituent amplification process (γ₁₃ = 0), which is conditionally stable: If the coupling coefficient γ₁₂ is smaller than the mismatch coefficient |δ₁ + δ₂|/2, the idler power oscillates boundedly, as does the stretching parameter [21, 22].

Fig. 4 Relative powers of the Schmidt-vector components plotted as functions of the interaction distance γz in the stable regime. The solid, dot-dashed and dashed curves represent components 1, 2 and 3, respectively. (a) Passive mode and (b) active modes.

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In Figs. 2 and 4, the components of the input and output Schmidt vectors were plotted relative to the physical basis vectors. One can also plot their components relative to the eigenvectors: u_j(z) = ∑_kc_jk(z)e_k, where $c_{j k} (z) = f_{k}^{†} u_{j} (z)$ . In general, each input (and output) Schmidt vector is a linear combination of all three eigenvectors: There is no simple relation between the Schmidt vectors and the eigenvectors. (Some asymptotic results for the unstable case are described in App. C.)

Because the coefficient matrix in Eq. (4) is constant, the backward (inverse) transfer matrix T⁻¹(z) = T(−z). [See the text before Eq. (17).] Changing the sign of z in Eqs. (30) and (33), or Eqs. (36) and (39), interchanges the input and output vectors, and making the same change in Eq. (28), or Eq. (34), replaces ν by −ν in the second of Eqs. (42). Thus, the mode that was passive in the forward process remains passive in the backward process, whereas the mode that was stretched (squeezed) in the forward process is squeezed (stretched) in the backward process.

In the adjoint decomposition of the transfer matrix T, the output vectors are the same as the input vectors and their elements are constants. The eigenvalues are constants (degenerate case) or depend exponentially on z (nondegenerate case). In the Schmidt decomposition, the output vectors are different from the input vectors and their elements depend on z. The Schmidt coefficients also depend on z (both cases), in a nontrivial manner. Thus, the adjoint and Schmidt decompositions are fundamentally different. To the best of my knowledge, simultaneous PA and FC is the simplest physical example (process) that illustrates this difference.

4. Some generalizations

In the previous section, Schmidt decompositions were found explicitly for the transfer matrices M, N and T, and relations between the first two decompositions and the last one were established. However, the process considered was simple, in the sense that it involved only three modes, and the coefficient and transfer matrices were real, so it is natural to ask what aspects of these results are specific, and what are general.

It is convenient to group like modes together (1 with 3 and 2 by itself). By doing so, one obtains the canonical transfer matrices

M = [\begin{matrix} M_{11} & 0 \\ 0 & M_{22} \end{matrix}], N = [\begin{matrix} 0 & N_{12} \\ N_{21} & 0 \end{matrix}],

where the blocks

M_{11} = [\begin{matrix} μ_{11} & μ_{13} \\ μ_{31} & μ_{33} \end{matrix}], N_{12} = [\begin{matrix} ν_{12} \\ ν_{32} \end{matrix}], N_{21} = [ν_{21} ν_{23}], M_{22} = [μ_{22}] .

For generality, suppose that m odd modes interact with n even modes (conjugates), where m = n + d and d > 0. (If d < 0, conjugate the IO equations.) Then M₁₁ is an m × m matrix, N₁₂ is an m × n matrix, N₂₁ is an n × m matrix and M₂₂ is an n × n matrix. The symmetric case in which d = 0 was described in [16, 17], so my discussion of the asymmetric case will be brief, and will focus on the novel aspects of the latter case. By substituting decompositions (45) into the constraint equations (23) and (26), one finds that

M_{11} M_{11}^{†} - N_{12} N_{12}^{†} = I, M_{22} M_{22}^{†} - N_{21} N_{21}^{†} = I,

M_{11}^{†} M_{11} - N_{21}^{t} N_{21}^{*} = I, M_{22}^{†} M_{22} - N_{12}^{t} N_{12}^{*} = I .

According to the Schmidt decomposition theorem,

M_{11} = V_{11} D_{11} U_{11}^{†}

and

N_{12} = V_{12} D_{12} U_{12}^{t}

. D₁₁ is square and need not be discussed further. However, D₁₂ is not square (m × n). It can be written in either of the block forms [S₁₂, 0]^t or [0, S₁₂]^t, where S₁₂ is a square diagonal matrix (n × n). I prefer the latter form, because it corresponds to listing the passive modes first. M₂₂ and N₂₁ have similar decompositions, in which D₂₁ = [0, S₂₁], where S₂₁ is square. By substituting these decompositions in Eqs. (47), one finds that

V_{11} D_{11}^{2} V_{11}^{†} - V_{12} D_{12} D_{12}^{t} V_{12}^{†} = I, V_{22} D_{22}^{2} V_{22}^{†} - V_{21} D_{21} D_{21}^{t} V_{21}^{t} = I .

All the matrices in Eqs. (49) are hermitian. Because the unitary decomposition of a hermitian matrix is unique (apart from permutations), the output matrices V₁₁ = V₁₂ = V₁ and V₂₂ = V₂₁ = V₂. In the second of Eqs. (49),

D_{21} D_{21}^{t} = 0^{2} + S_{21}^{2}

, so the 0 matrices have no effect and

D_{22}^{2} - S_{21}^{2} = I

. However, in the first of Eqs. (49), the 0 matrices do have an effect. Specifically,

D_{11}^{2} = [\begin{matrix} I & 0 \\ 0 & I + S_{12}^{2} \end{matrix}] .

Thus, the first d odd modes have Schmidt coefficients that are equal to 1, which means that they are passive (do not evolve in a significant way). The remaining n odd modes have coefficients that are greater than (or equal to) 1, which means that they (can) interact with the n even modes. By substituting the aforementioned decompositions in Eqs. (48), one finds that the input matrices U₁₁ = U₂₁ = U₁ and U₂₂ = U₁₂ = U₂. One also finds that

D_{22}^{2} - S_{12}^{2} = I

and S₂₁ = S₁₂. By combining the preceding results, one finds that the canonical transfer matrices have the block decompositions

M = [\begin{matrix} V_{1} D_{11} U_{1}^{†} & 0 \\ 0 & V_{2} D_{22} U_{2}^{†} \end{matrix}], N = [\begin{matrix} 0 & V_{1} D_{12} U_{2}^{t} \\ V_{2} D_{21} U_{1}^{t} & 0 \end{matrix}],

where the diagonal matrices

D_{11} = [\begin{matrix} I & 0 \\ 0 & D_{μ} \end{matrix}], D_{12} = [\begin{matrix} 0 \\ D_{ν} \end{matrix}], D_{21} = [0 D_{ν}], D_{22} = D_{μ} .

In Eqs. (52), D_ν = S₁₂ and

D_{μ}^{2} - D_{ν}^{2} = I

.

Let U₀ be the m × d matrix whose columns are the input vectors of the passive odd modes, let U₁ be the m × n matrix whose columns are the input vectors of the active odd modes, divided by 2^1/2, and let U₂ be the n × n matrix whose columns are the input vectors of even modes (all of which are active), divided by 2^1/2. Furthermore, let V₀, V₁ and V₂ be matrices whose columns are the associated output vectors. Then it is easy to verify (by multiplication) that the canonical transfer matrices have the decompositions

M = [\begin{matrix} V_{0} & V_{1} & i V_{1} \\ 0 & V_{2} & - i V_{2} \end{matrix}] [\begin{matrix} I & 0 & 0 \\ 0 & D_{μ} & 0 \\ 0 & 0 & D_{μ} \end{matrix}] [\begin{matrix} U_{0}^{†} & 0 \\ U_{1}^{†} & U_{2}^{†} \\ - i U_{1}^{†} & i U_{2}^{†} \end{matrix}],

N = [\begin{matrix} V_{0} & V_{1} & i V_{1} \\ 0 & V_{2} & - i V_{2} \end{matrix}] [\begin{matrix} 0 & 0 & 0 \\ 0 & D_{ν} & 0 \\ 0 & 0 & D_{ν} \end{matrix}] [\begin{matrix} U_{0}^{t} & 0 \\ U_{1}^{t} & U_{2}^{t} \\ i U_{1}^{t} & - i U_{2}^{t} \end{matrix}] .

By combining decompositions (51), one finds that the transfer matrix has the block decomposition

T = [\begin{matrix} M_{11} & N_{12} \\ N_{21}^{*} & M_{22}^{*} \end{matrix}] = [\begin{matrix} V_{1} D_{11} U_{1}^{†} & V_{1} D_{12} U_{2}^{t} \\ V_{2}^{*} D_{21} U_{1}^{†} & V_{2}^{*} D_{22} U_{2}^{†} \end{matrix}],

which involves the U and V matrices defined between Eqs. (48) and (51). Alternatively, by combining decompositions (53) and (54), and adding conjugate signs in the appropriate places, one obtains the decomposition

T = [\begin{matrix} V_{0} & V_{1} & V_{1} \\ 0 & V_{2}^{*} & - V_{2}^{*} \end{matrix}] [\begin{matrix} I & 0 & 0 \\ 0 & D_{μ} + D_{ν} & 0 \\ 0 & 0 & D_{μ} - D_{ν} \end{matrix}] [\begin{matrix} U_{0}^{†} & 0 \\ U_{1}^{†} & U_{2}^{t} \\ U_{1}^{†} & - U_{2}^{t} \end{matrix}],

which involves the U and V matrices defined before Eq. (53). As explained in Sec. 3, one can deduce T⁻¹ from T by interchanging the input and output vectors, and changing the sign of D_ν.

In summary, if m (odd) amplitudes interact with n conjugate (even) amplitudes (m > n) in a multiple-mode parametric process, the process can be decomposed into m − n passive processes (one-mode basis changes) and n active processes (two-mode stretching and squeezing). This property of parametric processes is reminiscent of the interaction between a two-level atom (with a two-dimensional basis) and a one-mode field (with an infinite-dimensional basis), in which only two Schmidt modes are required to describe the evolution of the field [25].

5. Discussion

In SRS, a strong pump wave interacts with weak frequency-downshifted (Stokes) and -upshifted (anti-Stokes) waves (optical modes) [20]. This interaction is mediated by a vibrational mode. If one allows all three modes to evolve self-consistently and assumes that the vibrational mode is weakly damped, the resulting equations are Hamiltonian and the Schmidt formalism applies to the transfer matrices M, N and T. This is true for discrete modes and continuous (spatial-temporal) modes [26]. However, if one makes the approximation that the vibrational mode follows the optical modes adiabatically, because it is strongly damped, gain terms (which are proportional to the pump power and inversely proportional to the damping rate) appear in the Stokes and anti-Stokes equations, which are no longer Hamiltonian. These equations can be solved analytically. The transfer matrix T has a Schmidt decomposition (because every complex matrix has one), but M and N are no longer guaranteed to have simultaneous Schmidt decompositions of the forms stated earlier.

SRS is strong when the (common) frequency difference between the pump and (anti-) Stokes waves equals a vibration frequency of the medium. In the context of DSFG, which occurs in a second-order nonlinear medium, the idlers (2 and 3) are the Stokes and anti-Stokes waves (Fig. 1). Because the aforementioned frequency difference also equals the signal frequency (ω₃ − ω_p = ω_p − ω₂ = ω₁), SRS is strong when the signal frequency equals a vibration frequency. In this case, the signal is strongly damped and DSFG is probably not useful. Conversely, if the signal is damped only weakly, DSFG is potentially useful and SRS is probably not significant. (For GaAs, the first vibrational resonance occurs for frequencies of about 8 THz, so signals with frequencies of order 1 THz are not strongly damped.) If three (or more) waves interact in a third-order nonlinear medium, the aforementioned frequency difference does not necessarily equal the frequency of any wave, so SRS can impact the performance of parametric devices that involve only weakly-damped waves [27].

6. Summary

Multiple-mode parametric processes have many important applications. In this paper, the interaction of three modes (one signal and two idlers) enabled by simultaneous amplification and frequency conversion was studied in detail. Such an interaction can occur in a second-or third-order nonlinear medium with suitable dispersion. The properties of this three-mode process are encapsulated by the transfer (Green) matrix, which relates arbitrary inputs to their associated outputs. This transfer matrix, and its adjoint (spectral) and Schmidt (singular value) decompositions, were determined analytically. In the former decomposition, the eigenvectors and adjoint eigenvectors are constants, and only the eigenvalues depend on distance. In the latter decomposition, the input and output Schmidt vectors, and the Schmidt coefficients (singular values), all depend on distance. These results showed that there is no simple relation between the eigenvectors and Schmidt vectors. (In the constituent two-mode amplification and conversion processes, the components of these vectors are related by phase factors.) To the best of my knowledge, the aforementioned three-mode process is the simplest physical process that illustrates this fundamental mathematical difference. The results also showed that one input mode is passive (experiences nothing more than a basis change), wheras the other two are active (participate in two-mode stretching and squeezing processes, in both the stable and unstable regimes). Further analysis showed that this feature is general: If m modes interact with n conjugate modes (m > n) in a multiple-mode parametric process, then there are m − n passive Schmidt modes and n pairs of active Schmidt modes (which are equivalent to 2n independent Schmidt modes). In summary, simultaneous amplification and frequency conversion is an interesting physical process, whose evolution illustrates several important features of multiple-mode processes.

Appendix A: Symplectic transformations

Consider a dynamical system [10], in which the evolution of each mode amplitude a_j is governed by the (complex) Hamilton equation

d_{z} a_{j} = i \partial H / \partial a_{j}^{*},

where the Hamiltonian H is a function of a_j and

a_{j}^{*}

. (When taking the partial derivative, a_j and

a_{j}^{*}

are treated as independent variables.) Equation (57) applies at all distances: Evolution preserves the form of the Hamilton equation.

Suppose that b_i is a function of a_j and $a_{j}^{*}$ . Then

\begin{array}{l} \frac{d b_{i}}{d z} & = & i \sum_{j} (\frac{\partial b_{i}}{\partial a_{j}} \frac{\partial H}{\partial a_{j}^{*}} - \frac{\partial b_{i}}{\partial a_{j}^{*}} \frac{\partial H}{\partial a_{j}}) \\ = & i \sum_{j} \frac{\partial b_{i}}{\partial a_{j}} (\frac{\partial H}{\partial b_{k}} \frac{\partial b_{k}}{\partial a_{j}^{*}} + \frac{\partial H}{\partial b_{k}^{*}} \frac{\partial b_{k}^{*}}{\partial a_{j}^{*}}) - i \sum_{j} \frac{\partial b_{i}}{\partial a_{j}^{*}} (\frac{\partial H}{\partial b_{k}} \frac{\partial b_{k}}{\partial a_{j}} + \frac{\partial H}{\partial b_{k}^{*}} \frac{\partial b_{k}^{*}}{\partial a_{j}}) \\ = & i \sum_{j} (\frac{\partial b_{i}}{\partial a_{j}} \frac{\partial b_{k}}{\partial a_{j}^{*}} - \frac{\partial b_{i}}{\partial a_{j}^{*}} \frac{\partial b_{k}}{\partial a_{j}}) \frac{\partial H}{\partial b_{k}} + i \sum_{j} (\frac{\partial b_{i}}{\partial a_{j}} \frac{\partial b_{k}^{*}}{\partial a_{j}^{*}} - \frac{\partial b_{i}}{\partial a_{j}^{*}} \frac{\partial b_{k}^{*}}{\partial a_{j}}) \frac{\partial H}{\partial b_{k}^{*}} . \end{array}

The right side of Eq. (58) has the canonical form

i \partial H / \partial b_{i}^{*}

if and only if

\sum_{j} (\frac{\partial b_{i}}{\partial a_{j}} \frac{\partial b_{k}}{\partial a_{j}^{*}} - \frac{\partial b_{i}}{\partial a_{j}^{*}} \frac{\partial b_{k}}{\partial a_{j}}) = 0, \sum_{j} (\frac{\partial b_{i}}{\partial a_{j}} \frac{\partial b_{k}^{*}}{\partial a_{j}^{*}} - \frac{\partial b_{i}}{\partial a_{j}^{*}} \frac{\partial b_{k}^{*}}{\partial a_{j}}) = δ_{i k} .

Conditions (59) are referred to as the symplectic conditions and, if they are satisfied, the transformation from a_j to b_i (change of variables) is termed symplectic.

If the transformation is linear, then

b_{i} = \sum_{j} (μ_{i j} a_{j} + ν_{i j} a_{j}^{*}),

where μ_ij and ν_ij are transfer coefficients. By combining Eqs. (59) and (60), one obtains the linear symplectic conditions

\sum_{j} (μ_{i j} ν_{k j} - ν_{i j} μ_{k j}) = 0, \sum_{j} (μ_{i j} μ_{k j}^{*} - ν_{i j} ν_{k j}^{*}) = δ_{i k} .

Not only do Eqs. (61) constrain variable changes, they also constrain mode evolutions in linear systems (for which the Hamiltonians are quadratic functions of a_j and

a_{j}^{*}

). As noted above, evolution preserves the form of the Hamilton equation. Hence, if a_j is an (input) amplitude at distance z and b_i is an (output) amplitude at distance z + l, where l is a constant, the transfer coefficients μ_ij(l) and ν_ij(l) associated with the solutions of Eqs. (57) must satisfy conditions (61). Let M = [μ_ij] and N = [ν_ij] be transfer matrices. Then conditions (61) can be rewritten in the matrix forms

M N^{t} - N M^{t} = 0, M M^{†} - N N^{†} = I .

Equations (62) were used in Secs. 3 and 4 to determine the Schmidt decompositions of the transfer matrices.

In Hamiltonian mechanics, the (complex) Poisson bracket (PB)

{u, v}_{a} = \sum_{j} (\frac{\partial u}{\partial a_{j}} \frac{\partial v}{\partial a_{j}^{*}} - \frac{\partial u}{\partial a_{j}^{*}} \frac{\partial v}{\partial a_{j}}) .

By replacing u with a_i, and v with a_k and

a_{k}^{*}

, one obtains the fundamental PB relations

{a_{i}, a_{k}}_{a} = 0, {a_{i}, a_{k}^{*}}_{a} = δ_{i k},

respectively. In PB notation, the symplectic conditions (59) are

{b_{i}, b_{k}}_{a} = 0, {b_{i}, b_{k}^{*}}_{a} = δ_{i k} .

It can be shown that {u, v}_a = {u, v}_b if and only if conditions (65) are satisfied. Specific examples of this result are

{b_{i}, b_{k}^{(*)}}_{a} = {b_{i}, b_{k}^{(*)}}_{b}

. Thus, if a transformation is symplectic (like evolution), the input and output amplitudes satisfy the fundamental PB relations (in which the numerators and denominators involve the same variables).

Suppose that u is a function of a_j and $a_{j}^{*}$ , which depend on z. Then it follows from the first of Eqs. (58) that

d_{z} u = i {u, H} .

In particular, if u = a_j, Eq. (66) is an alternative to Eq. (57): The Hamilton equation can be rewritten in terms of a PB, which maintins its form under evolution (as it must do).

To make the transition from classical to quantum mechanics [10], one replaces the mode amplitudes a_j by the mode operators â_j, which satisfy the commutation relations

[{\hat{a}}_{i}, {\hat{a}}_{k}] = 0, [{\hat{a}}_{i}, {\hat{a}}_{k}^{†}] = δ_{i k},

where the commutator [a, b] = ab − ba. The mode operators evolve according to the Heisenberg equation

d_{z} {\hat{a}}_{j} = i [{\hat{a}}_{j}, \hat{H}],

where the Hamiltonian operator

\hat{H} = H ({\hat{a}}_{j}, {\hat{a}}_{j}^{†})

. Notice the similarities between the symplectic conditions (64) and the commutation relations (67), and the Hamilton equation (66) and the Heisenberg equation (68). For systems with quadratic Hamiltonians, the Hamilton and Heisenberg equations are linear in the mode amplitudes and operators, respectively, and have identical coefficients. Hence, their solutions are specified by the same transfer matrices.

Appendix B: Two-mode processes

In this appendix the transfer matrices associated with the limiting two-mode processes (parametric amplification and frequency conversion) are considered briefly. First, consider wavenumber-matched amplification. The two-mode equations can be written in the form of Eq. (4), in which the amplitude vector and coefficient matrix are

Y = [\begin{matrix} x_{1} \\ x_{2}^{*} \end{matrix}], L = [\begin{matrix} 0 & γ \\ γ & 0 \end{matrix}],

respectively, where γ is an abbreviation for γ₁₂. The coefficient matrix is hermitian. It has the real eigenvalues ±γ, which occur as a positive and negative pair. The associated eigenvectors are

\frac{1}{2^{1 / 2}} [\begin{matrix} 1 \\ 1 \end{matrix}], \frac{1}{2^{1 / 2}} [\begin{matrix} 1 \\ - 1 \end{matrix}] .

Hence, the two-mode transfer matrix has the unitary (adjoint) decomposition

T = \frac{1}{2^{1 / 2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] [\begin{matrix} τ_{+} & 0 \\ 0 & τ_{-} \end{matrix}] \frac{1}{2^{1 / 2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}],

where the eigenvalues τ_±(z) = exp(±γz). By multiplying the matrices in Eq. (71), or solving the first and second of Eqs. (12) directly, one obtains the transfer matrix

T = [\begin{matrix} μ & ν \\ ν & μ \end{matrix}],

where the transfer functions μ(z) = cosh(γz) and ν(z) = sinh(γz) satisfy the auxiliary equation μ² − ν² = 1, which is the second of Eqs. (62).

By following the procedure described in the text, one finds that the transfer matrix also has the Schmidt decomposition

T = \frac{1}{2^{1 / 2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] [\begin{matrix} μ + ν & 0 \\ 0 & μ - ν \end{matrix}] \frac{1}{2^{1 / 2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] .

In this example, the Schmidt coefficients equal the (z-dependent) eigenvaulues. The input and output Schmidt vectors are the same, and equal the (z-independent) eigenvectors. One eigen-mode (Schmidt mode) is stretched, whereas the other mode is squeezed (by the same amount). Both of these processes can be considered active (in the sense that the output differs significantly from the input). In the presence of wavenumber mismatch, the Schmidt vectors differ from the eigenvectors only by phase factors [15,16]. These factors complicate the mathematics of two-mode amplification, but not the physics.

Second, consider wavenumber-matched conversion. The two-mode equations can be written in the form of Eq. (4), in which

Y = [\begin{matrix} x_{1} \\ x_{3} \end{matrix}], L = [\begin{matrix} 0 & - γ \\ γ & 0 \end{matrix}],

where γ is an abbreviation for γ₁₃. The coefficient matrix is anti-hermitian (differs from a hermitian matrix by a factor of i). It has the imaginary eigenvalues ±iγ, which occur as a positive and negative pair. The associated eigenvectors are

\frac{1}{2^{1 / 2}} [\begin{matrix} 1 \\ - i \end{matrix}], \frac{1}{2^{1 / 2}} [\begin{matrix} 1 \\ i \end{matrix}] .

Hence, the two-mode transfer matrix has the unitary (adjoint) decomposition

T = \frac{1}{2^{1 / 2}} [\begin{matrix} 1 & 1 \\ - i & i \end{matrix}] [\begin{matrix} τ_{+} & 0 \\ 0 & τ_{-} \end{matrix}] \frac{1}{2^{1 / 2}} [\begin{matrix} 1 & i \\ 1 & - i \end{matrix}],

where the eigenvalues τ_±(z) = exp(±iγz) have modulus 1. In the eigenbasis, the conversion process appears as phase shifts. By multiplying the matrices in Eq. (76), or solving the first and third of Eqs. (12) directly, one obtains the transfer matrix

T = [\begin{matrix} τ & - ρ \\ ρ & τ \end{matrix}],

where the transfer coefficients τ(z) = cos(γz) and ρ(z) = sin(γz) satisfy the auxiliary equation τ² + ρ² = 1, which is also the second of Eqs. (62). The eigenvalues should not be confused with the transfer coefficients, because the former quantities have subscripts, whereas the latter do not. In the physical basis, the process appears as a rotation (which is a form of basis change).

In this example T^†T = TT^† = I, so the Schmidt vectors are not unique. It is easy to verify that the transfer matrix has the complex familty of Schmidt decompositions

T = \frac{1}{2^{1 / 2}} [\begin{matrix} e_{o} & e_{o}^{*} \\ - i e_{o} & i e_{o}^{*} \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \frac{1}{2^{1 / 2}} [\begin{matrix} e_{i} & i e_{i} \\ e_{i}^{*} & - i e_{i}^{*} \end{matrix}],

where e_o = exp(iθ_o), e_i = exp(iθ_i), and θ_i and θ_o are any two angles that sum to γz. The Schmidt coefficients are the moduli of the (z-dependent) eigenvaulues, and the input and output Schmidt vectors differ from the eigenvectors by (z-dependent) phase factors. Because the coefficients are degenerate, the transfer matrix also has the real family of decompositions

T = \frac{1}{2^{1 / 2}} [\begin{matrix} c_{o} & - s_{o} \\ s_{o} & c_{o} \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \frac{1}{2^{1 / 2}} [\begin{matrix} c_{i} & - s_{i} \\ s_{i} & c_{i} \end{matrix}],

where c_j = cosθ_j and s_j = sinθ_j. In every Schmidt basis, the process appears as a basis change. The transfer of power from one physical mode to another is useful in some applications. However, because this process can be recast as a pair of power-preserving phase shifts or a basis change (relabelling), it can be considered passive (in the sense that the output does not differ significantly from the input). In the presence of wavenumber mismatch, the Schmidt vectors still differ from the eigenvectors only by phase factors [15, 17]. These factors complicate the mathematics of two-mode conversion, but not the physics.

In the preceding wavenumber-matched examples, the coefficient matrices are hermitian or anti-hermitian. For such coefficient matrices, the Schmidt coefficients of the transfer matrices are the moduli of the eigenvalues, and Schmidt vectors exist that differ from the eigenvectors only by phase factors.

Appendix C: Asymptotic results

It was shown in Secs. 2 and 3 that there is no simple relation between the eigenvectors of the transfer matrix (whose elements are constants) and the Schmidt vectors (whose elements depend on z). However, in the unstable regime (γ₁₂ > γ₁₃), there are asymptotic relations between the largest eigenvalue, and its associated eigenvector and adjoint eigenvector, and the largest Schmidt coefficient, and its associated input and output vectors.

Let τ_j denote an eigenvalue of T, and let e_j and f_j be the associated eigenvector and adjoint eigenvector, respectively. Then the adjoint decomposition can be written in the alternative form $T (z) = \sum_{j} e_{j} τ_{j} (z) f_{j}^{†}$ . Now suppose that one eigenvalue (+) dominates as z → ∞. Then

T (z) \to e_{+} τ_{+} (z) f_{+}^{†} .

Similarly, let μ₊ and ν₊ be the dominant Schmidt coefficients, and u₊ and v₊ be the associated input and output Schmidt vectors. Then

T (z) \to v_{+} (z) [μ_{+} (z) + ν_{+} (z)] u_{+}^{†} (z) .

By comparing expressions (80) and (81), one finds that u₊ and v₊ can differ from f₊ and e₊, respectively, by at most scalar functions of z. Furthermore, since u and v are normalized, these functions must be constants or (z-dependent) phase factors. It follows from these observations that the stretching factor μ₊ + ν₊ can differ from the eigenvalue τ₊ by at most a constant or phase factor.

For the problem under consideration, these predictions can be verified. It follows from Eqs. (36), (39) and (43) that, in the limit of large z,

u_{+} ~ \frac{1}{2^{1 / 2} n_{+}} [\begin{matrix} 1 \\ γ_{12} / k \\ - γ_{13} / k \end{matrix}], v_{+} ~ \frac{1}{2^{1 / 2} n_{+}} [\begin{matrix} 1 \\ γ_{12} / k \\ γ_{13} / k \end{matrix}],

where the normalization constant n₊ = γ₁₂/k. By comparing Eqs. (15), (16) and (82), one finds that u₊ = f₊/n₊ and v₊ = e₊/n₊. Hence,

(μ_{+} + ν_{+}) / τ_{+} = n_{+}^{2}

, which is constant. In Fig. 5 the ratio of the stretching factor [Eqs. (35) and (38)] and eigenvalue [Eq. (17)] is plotted as a function of the interaction distance γz, where γ = (γ₁₂γ₁₃)^1/2, for the case in which γ₁₂/γ = 1.1 and γ₁₃/γ = 0.9. For these parameters, k/γ = 0.63 and

n_{+}^{2} = 3.02

. The numerical results are consistent with the theoretical prediction.

Fig. 5 Ratio of the stretching factor and eigenvalue plotted as a function of the interaction distance γz. This ratio tends to a constant as the distance increases.

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Acknowledgments

I thank J. M. Dailey for bringing this problem to my attention.

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Schmidt decompositions of parametric processes III: simultaneous amplification and conversion

Abstract

1. Introduction

2. Adjoint decomposition of the transfer matrix

3. Schmidt decomposition of the transfer matrix

4. Some generalizations

5. Discussion

6. Summary

Appendix A: Symplectic transformations

Appendix B: Two-mode processes

Appendix C: Asymptotic results

Acknowledgments

References and links

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Figures (5)

Equations (82)

Optics Express