## Abstract

In this paper we consider frequency translation enabled by Bragg scattering, a four-wave mixing process. First we introduce the theoretical background of the Green function formalism and the Schmidt decomposition. Next the Green functions for the low-conversion regime are derived perturbatively in the frequency domain, using the methods developed for three-wave mixing, then transformed to the time domain. These results are also derived and verified using an alternative time-domain method, the results of which are more general. For the first time we include the effects of convecting pumps, a more realistic assumption, and show that separability and arbitrary reshaping is possible. This is confirmed numerically for Gaussian pumps as well as higher-order Hermite-Gaussian pumps.

© 2012 Optical Society of America

## 1. Introduction

As the computational needs of the world keep increasing, quantum information (QI) processing is of increasing interest [1, 2]. A fundamental process in QI is Hong-Ou-Mandel interference (HOM), in which two photons interfere through a quantum optical effect [3]. Originally HOM interference was used to measure the delay between photons, but it has recently also been used in a scheme for quantum computation using linear optics [4, 5].

For quantum key distribution and continuous variable teleportation it has been shown that inseparable three-mode entanglement is useful [6]. This has been demonstrated in optical crystals using consecutive nonlinear optical interactions in resonance [6–9]. Recently it has been demonstrated, theoretically and experimentally, that three-color tripartite entanglement is possible in an optical parametric oscillator across a wide frequency range [10, 11].

A reliable and noise-free process for the translation (frequency conversion without
conjugation) of a quantum-state from one frequency to another is required for QI to
be able to send states from one quantum node to another [1, 2].
These states could be stored in quantum memories corresponding to wavelengths from
300–800 nm that need to be transmitted over a traditional optical link with
the low-loss windows in the range 1300–1600 nm [12–14]. It is important to note that this process does not violate the
no-cloning theorem since the original state is destroyed in the process
[15]. Quantum frequency
conversion (QFC) was first investigated using three-wave mixing (TWM) in optical
crystals where a strong pump p mitigates the conversion from the signal s to the
idler r, *i.e.*
*π*_{s} ↔
*π*_{p} +
*π*_{r} where
*π _{j}* represents a photon at frequency

*ω*,

_{j}*j*∈ {p,r,s}. The theoretical ground-work for QFC using TWM was presented in [16, 17] and first demonstrated experimentally in [18]. This process has been used for higher-efficiency single-photon detection using frequency up-conversion [19–21] and also for quantum networks using frequency down-conversion [22]. It has been demonstrated theoretically that TWM allows reshaping of pulses (

*i.e.*from a continuous-wave to a short pulse and vice versa) when using spectral phase modulation and propagation [13] or using dispersion engineering [23]. This is an important result since the states emitted from current quantum memory units have a pulse width many times larger than what is desired for traditional optical communication links [13, 14, 24, 25].

QFC is also possible using non-degenerate four-wave mixing (FWM) in an optical-fiber
[26]. It is in the form
of Bragg scattering (BS), which is characterized by two strong pumps p and q that
interact with two sidebands r and s such that *π*_{p}
+*π*_{s} ↔
*π*_{q}
+*π*_{r}. See Fig. 1 for the frequency locations of the four fields.
This process has been used classically to allow FC (frequency conversion) over a
wide frequency range [27–29]. The
advantages of BS are that it is tunable [30], has low-noise transfer [31], and allows for very distant FC (more than
200 nm) [32, 33]. BS has also been used to FC single photons
[34].

One advantage of QFC using four-wave mixing in optical fibers is that the emitted photon wavepacket has a transverse distribution that is already mode-matched to existing transmission fibers. Also it allows for a very broad bandwidth of conversion as well as coupling from the visible to the telecom band and inter-telcom band conversion [26]. The quantum-noise properties of parametric amplification were considered in [35, 36] and it has theoretically been shown that BS allows for noiseless QFC [26]. BS has also been shown theoretically to allow HOM-interference between photons of different colors [37, 38].

In this paper we describe the Green-function formalism for FC and the Schmidt
decomposition in Section 2. The advantage of the Schmidt decomposition is that it
allows for an easy interpretation of the results as it describes the natural modes
of the process and the conversion between them. In Section 3 FC is solved in the
perturbative regime, *i.e.* low energy-conversion efficiency. The
low-efficiency regime allows simple analytic solutions, including the Schmidt
decomposition, which provide baseline theoretical results to which to compare
higher-efficiency results typically obtained numerically, where it is found that up
to about 50 % efficiency the exact numerical results bear high similarity to
the perturbative ones [38].
The walk-off (pump convection) between the pumps is ignored. In Section 4 convection
of the pumps is included and the differences between the two models highlighted. In
both cases the results are decomposed and the Schmidt coefficients (mode-conversion
efficiencies) and the Schmidt modes (input and output modes) are related to physical
parameters: the pump energy, pump width, fiber length and the dispersion-induced
sideband walk-off. We find that re-shaping is indeed possible using BS, due to the
presence of the two pumps. Furthermore this process does not require dispersion
engineering or additional processing like what is required for TWM.

## 2. General formalism of FC

Frequency conversion using four-wave mixing (FWM) includes the two pumps p and q as well as the two sidebands r and s, and comes in two different flavors: near and far conversion, see Fig. 1. This process is governed by the coupled-mode equations (CMEs) [26, 38]

*∂*and

_{z}*∂*are partial derivatives with respect to

_{t}*z*and

*t*respectively,

*β*

_{r}and

*β*

_{s}are the group slownesses (inverse group speeds) of the idler and the signal respectively, while

*A*

_{r}(

*t*,

*z*) and

*A*

_{s}(

*t*,

*z*) are the corresponding sideband amplitudes. Finally, ${\gamma}_{\text{pq}}=\gamma {A}_{\text{p}}(t-{\beta}_{\text{s}}z){A}_{\text{q}}^{*}(t-{\beta}_{\text{r}}z)$ which is based on the reasonable assumption that pump p co-propagates with sideband s and pump q co-propagates with sideband r [38]. In the case where the four fields are co-polarized and furthermore when the pumps fulfill the normalization condition ∫ |

*A*(

_{j}*t*)|

^{2}d

*t*= 1, we write

*γ*= 2

*γ*

_{K}(

*E*

_{p}

*E*

_{q})

^{1/2}where

*γ*

_{K}is the Kerr nonlinearity coefficient and

*E*,

_{j}*j*∈ {p,q}, are the pump energies. In the low-efficiency regime self-phase and cross-phase modulation are negligible. Intra-channel dispersion was also neglected which is a reasonable assumption for a recent experiment [34, 38] and a wide variety of related experiments. The case with different polarizations is considered in [35, 39–41]. The effect of spontaneous Raman scattering (SRS) is not modeled by Eqs. (1) and (2). The effect of SRS is minimized for very small or very large (≫ 13 THz) frequency shifts. Very large frequency shifts were demonstrated in [32, 33]. As shown in [42] SRS is weaker for cross-polarized signals than co-polarized signals, so utilizing vector BS might diminish the effect. In [34] SRS was minimized by having the pumps at longer wavelengths than the sidebands, which meant that SRS was observed but FC was still achievable.

Equations (1) and (2) also apply to quantum mechanical
operators, where the classical fields *A _{j}* are replaced
with the mode operators

*â*[26, 34]. It is known that beam splitters do not add excess noise [43], and since FC by BS has mathematically equivalent input-output (IO) relations it does also not add excess noise [26]. The mode operators satisfy the boson commutation relations

_{j}*i*,

*j*∈ {r, s},

*δ*is the Kronecker delta and

_{ij}*δ*(

*t*−

*t*′) is the Dirac delta function. The CMEs are valid in the so-called parametric approximation, in which the pumps are treated as strong continuous fields, and for which quantum fluctuations are ignored. The weak sidebands, however, are treated quantum mechanically.

Using the Green-function formalism, it is possible to write the solution of the CMEs in the IO form [37, 38]

*j*at (

*t*,

*l*) is described by a function

*G*that represents the influence of the input-mode

_{jk}*k*at (

*t*′, 0). In our example with two sidebands

*k*∈ {r, s}, Eq. (4) leads to

*G*(

_{jk}*t;t*′) =

*G*(

_{jk}*t*,

*l;t*′, 0) has been introduced and with

*t*and

*t*′ as output and input times respectively. Similarly,

*A*

_{s}is described by the shape of itself and sideband r at the input of the fiber. Physically this means that

*G*

_{rr}(

*t;t*′) and

*G*

_{ss}(

*t;t*′) describe the influence on the output at time

*t*, from the input of the field itself at time

*t*′, see Fig. 2. In a similar way, the cross Green functions

*G*

_{rs}(

*t;t*′) and

*G*

_{sr}(

*t;t*′) concern the influence on one sideband at time

*t*, from the other sideband at the input at time

*t*′.

It is convenient to introduce the singular value (Schmidt) decomposition of the Green’s functions since it allows us to split the Green function into products of functions that depend only on the input and output times at the cost of an infinite sum [44]. In general we write

where the functions*u*and

_{n}*v*are the Schmidt modes normalized with respect to the integral of the absolute value squared. The normalized functions as well as the Schmidt coefficients ${\lambda}_{n}^{1/2}$ are found from the integral eigenvalue equations

_{n}*K*

_{1}(

*t*,

*t*′) ≡ ∫

*G*(

*t;t*

_{2})

*G*

^{*}(

*t*′;

*t*

_{2}) d

*t*

_{2}and

*K*

_{2}(

*t*,

*t*′) ≡ ∫

*G*(

*t*

_{1};

*t*)

*G*

^{*}(

*t*

_{1};

*t*′) d

*t*

_{1}. We remind the reader that the first argument of the Green function corresponds to the output and the second argument to the input. The physical interpretation of the Schmidt decomposition is thus that it takes the input mode

*u*and converts it to the output mode

_{n}*v*with the probability

_{n}*λ*[37, 38]. In matrix notation, the preceding decomposition can be rewritten simply as [45]

_{n}*G*=

*VDU*

^{†}, with

*V*and

*U*being (different) unitary matrices and

*D*a diagonal matrix containing the non-negative square roots of the eigenvalues of the non-negative matrix

*GG*

^{†}(which are equal to the eigenvalues of

*G*

^{†}

*G*). Likewise

*V*contains in its columns the eigenvectors of

*GG*

^{†}and the columns of

*U*the eigenvectors of

*G*

^{†}

*G*[46].

The properties of FC by BS is governed by the forward transformation matrix

*t*and the input time

*t*′. Furthermore ${\left|\tau \right|}_{n}^{2}$ is the transmission (nonconversion) probability and ${\left|\rho \right|}_{n}^{2}$ is the frequency conversion probability. We have the requirement |

*ρ*|

_{n}^{2}+ |

*τ*|

_{n}^{2}= 1 due to photon number conservation. The symmetry of this expansion is clear. Conversion from one sideband is described by the same input modes and, similarly, conversion to a sideband is described by identical output modes. In the case with only one non-zero Schmidt coefficient we obtain mode-filtering, since a lot of modes may be sent in, but only one is converted. Similarly for the case of many non-zero Schmidt coefficients we have mode non-filtering in the sense that the output is the sum of many inputs [38].

#### 2.1. The frequency and time domains

Some aspects of the FC process are easier to model in the time domain, and others are easier to model in the frequency domain. Thus this section considers the relations between the two domains. The following analysis is based on the symmetric Fourier-transform

*ω*

_{r}yields

Another interesting aspect is the Fourier-transform of the Schmidt-decomposed Green functions. Suppose that the time-domain Green function has the Schmidt-decomposition

*u*(

_{n}*t*) are real the relation ${u}_{n}^{*}(\omega )=u(-\omega )$ holds. A similar result exists when going from the frequency-domain to the time-domain. That is the Fourier transform of the Green function is the sum of products of the Fourier transform of the Schmidt modes. This is an important result that is used extensively through the remainder of this paper.

## 3. Stationary pumps

To gain physical insight we start by solving the FC problem in the low-conversion regime while assuming that the pumps do not convect relative to one another in the moving frame propagating at the average group slowness of the sidebands. This is a simplified model of FWM, since in most cases the pumps walk-off with respect to each other. However it is representative of TWM, because one pump is always stationary. One may simply choose the frame of reference to propagate with the pump. First we derive the equations in the frequency-domain, since this is the standard approach and afterwards we present an alternative derivation.

By Fourier transforming Eqs. (1) and
(2) with respect to
*t* and *t*′ one find that

*A*(

_{j}*ω*,

_{j}*z*) =

*B*(

_{j}*ω*,

_{j}*z*) exp[i

*β*(

_{j}*ω*)

_{j}*z*] simplifies the analysis. Furthermore it is assumed that ${\beta}_{j}({\omega}_{j})={\beta}_{j}^{(1)}{\omega}_{j}$, thus neglecting group velocity dispersion and higher-order effects,

*i.e.*${\beta}_{j}^{(3)}$, ${\beta}_{j}^{(4)}$, . . ., which is reasonable for a sufficiently short piece of fiber and a narrow pulse in the frequency domain. Throughout the remainder of this paper the simpler notation ${\beta}_{j}^{(1)}={\beta}_{j}$ is used for the group slowness. These assumptions lead to the approximate solutions [38]

*l*is the fiber length. A similar expression exists for the signal field. Using Eqs. (15) and (16), and inserting

*B*=

_{j}*A*exp[−i

_{j}*β*(

_{j}*ω*)

_{j}*z*], one finds the Green functions

#### 3.1. Standard analysis

The standard way to find the Green function is to consider a specific pump-shape
and to carry out the *z*-integral in the frequency-domain
[48]. A typical
pump-shape choice is two identical Gaussian pumps. It is assumed that the pumps
do not convect, thus we write them in the form

*τ*

_{0}is the root-mean-square width and where the pump-shapes are normalized. The Fourier transform of

*γ*

_{pq}(

*t*) is

*γ*(2

*π*)

^{−1}is denoted

*γ*

_{0}and includes the (2

*π*)

^{−1/2}in front of the Green function. Inserting this in the Green function and integrating with respect to

*z*leads to

*δ*

_{0}=

*β*

_{r}

*ω*

_{r}−

*β*

_{s}

*ω*

_{s}. The next step is to approximate the sinc with a Gaussian, sinc(

*x*) ≈ exp(−

*ξx*

^{2}/2) with

*ξ*≈ 0.3858. This value is chosen so that the sinc and the Gaussian have the same full-width-at-half-maximum (FWHM) [48]. All in all our Green function attains the form

*α*=

_{j}*ξ*

^{1/2}

*β*/2. To discuss whether the Green function is separable in its two frequencies the Schmidt decomposition is introduced in Eq. (108) of the Appendix. Since

_{j}l*λ*is real by definition we cast the Schmidt decomposition of the Green function under the Gaussian approximation in the following form:

_{n}*ϕ*is the

_{n}*n*th orthonormal Hermite polynomial

*H*is the Hermite polynomial of order

_{n}*n*. The square of the Schmidt coefficient (the dilation factor) is in the form

*α*

_{rs}=

*α*

_{r}−

*α*

_{s}is positive. The characteristic timescales are

For many applications one is interested in separating the Green functions into
functions depending on only one of the frequencies, as frequency entanglement is
undesired for some quantum optical interference experiments [48, 49]. Separability has been studied extensively for
photon-pair generation using three-wave mixing [48, 49], but not for FC. To achieve separability it is required that
*μ* = 0 and, with the aforementioned
assumption that *α*_{rs} > 0, this leads
to the separability requirement
*α*_{r}*α*_{s}
= −*σ*^{2}. In the co-propagating
case (where the group slownesses of the sidebands have the same sign) it is not
possible to obtain separability since *σ* is real,
whereas in the counter-propagating case (with different signs of the inverse
group velocities) it is possible to obtain separability for one specific length
of the fiber. For *μ* = 0, this leads to
considerably simpler parameters

*λ*

_{0}is the only non-zero squared Schmidt coefficient. Notice that this Schmidt coefficient is indeed independent of the fiber length.

It is instructive to cast Eq. (31) in a slightly different way:

*v*(

_{n}*ω*

_{r}) =

*ϕ*(

_{n}*τ*

_{r}

*ω*

_{r}) exp(i

*lβ*

_{r}

*ω*

_{r}/2) and

*u*(

_{n}*ω*

_{s}) =

*ϕ*(

_{n}*τ*

_{s}

*ω*

_{s}) exp(−i

*lβ*

_{s}

*ω*

_{s}/2). Since

*ω*

_{r}and

*ω*

_{s}correspond to the output and input frequencies respectively, we notice that

*τ*

_{r}and

*τ*

_{s}are characteristic time scales.

The time-domain Green function is found by Fourier transforming Eq. (30), see Appendix A for the details. The result is

*t*̄ =

*t*−

*β*

_{r}

*l*/2 and $\overline{{t}^{\prime}}={t}^{\prime}+{\beta}_{\text{s}}l/2$.

According to the inverse of Eq.
(21) the Fourier transform of the Schmidt decomposition is simply the
Fourier transform of the individual Schmidt modes and since
^{−1}
{*f*(*aω*)e^{ibω}}
= *f*[(*t* −
*b*)/*a*]/|*a*|, Eq. (41) becomes

*β*

_{r}= −

*β*

_{s}the input and output modes are shifted in phase by the factor

*lβ*/2 corresponding to an interaction at the middle of the fiber which maximizes the interaction between the four fields [38].

_{j}ω_{j}#### 3.2. Alternative analysis

The standard analysis in the frequency domain is based on two reasonable, but
nonetheless restricting assumptions, *i.e.* similar Gaussian
pumps and the approximation of the sinc-function with a Gaussian. We now present
an alternative analysis that enables deriving the Green functions in the
time-domain in the general case by interchanging the order of the frequency and
length integrals. Finally we also present a simpler and more physical
derivation.

Considering the Fourier transform of Eq.
(25) (for brevity we only show a detailed derivation of
*G*_{rs}), which is

*ω*′

_{r}=

*ω*

_{r}−

*ω*

_{s}and the Fourier transform property

^{−1}{

*f*(

*ω*)e

^{a}^{i}

*} =*

^{ω}*f*(

*t*−

*a*) leads to

*z*, thus we find

*γ*

_{pq}is a complicated function of

*t*and

*t*′, but in the following section we present a simple physical derivation of it.

### 3.2.1. Time-domain collision analysis

Due to the simplicity of the Green function in the time-domain, Eq. (46), we find the Green
functions directly, in the time domain, by using the method of
characteristics [50].
In the low-conversion regime, the presence of the idler has little effect on
the signal. Hence, a signal impulse that enters the fiber at time
*t*′ remains an impulse as it convects through
the fiber. The part of the idler that exits the fiber at time
*t* was generated by a collision of the idler pulse with
the signal at the point (*t*_{c},
*z*_{c}), where *t*′
+
*β*_{s}*z*_{c}
= *t* −
*β*_{r}(*l* −
*z*_{c}). Such a collision is illustrated in
Fig. 3a. The collision distance
and time are

For signal generation by a pulsed idler, the collision distance and time are

#### 3.3. Comparing the time-domain and the frequency-domain results

The Green functions were found directly using the collision analysis,
*cf.*
Eq. (48). Notice that the
collision time may be evaluated from

*t̄*are the same retarded times as used in Eq. (42). Second, notice that

_{j}*t*

_{a}=

*t*′ + (

*β*

_{r}+

*β*

_{s})

*l*/2 and the rectangle function [51] is

*h*and

*w*are fitting parameters that will be determined later. For reference,

*h*= 1 produces the correct peak height, whereas

*h*= [2/(

*πw*)]

^{1/2}≈ 1.28 produces the correct area. Aggregating these results the Green function is approximated by

*α*=

_{j}*w*

^{1/2}

*β*/2. Comparing Eqs. (42) and (57) they have the same general shape and we conclude that

_{j}l*w*=

*ξ*and

*h*= [2/(

*πw*)]

^{1/2}≈ 1.28, which gives the same integral over the Gaussian and the rectangle function in the time domain. This shows that the effect of approximating the sinc with a Gaussian in the frequency-domain is equivalent to replacing the sharp boundaries from the rectangular function in the frequency-domain with a gradual effective boundary from the Gaussian. Physically this means that the Green function will allow effects from the input on the output from input and output times that are not allowed due to causality. Another issue is that in the limit of long fibers the rectangular function is unity for almost all times, so one has to pick

*h*= 1 to get the best results.

Since Eq. (57) is of the canonical
form for the Schmidt decomposition of a Gaussian, see Eq. (104) in Appendix A, we notice that the square of the lowest-order
Schmidt coefficient and time scales (*μ _{j}* is
used instead of

*τ*here not to confuse it with the pump-width) are in the form

_{j}*τ*are the characteristic frequency-scales found in the frequency-domain Schmidt decomposition.

_{j}In the limit of short fibers
*βl*/*τ* → 0 we find
that

*α*

_{rs}and

*σ*, and increase with the square root of the fiber length. In this limit the separability coefficient (denoted

*t*in the appendix) tends to which is close to unity, in other words this leads to a large number of non-zero Schmidt coefficients. In this limit of short fibers the sidebands experience approximately CW pumps when the pump-width is much larger than the sideband-width. Notice that the duration of the lowest-order Schmidt modes are much shorter than the pumps (it is the geometric mean of the transit time and the pump width). Also for signal-to-idler generation a pulse that is an arbitrary superposition of lower-order modes is converted without significant distortion, since the Schmidt coefficients decrease slowly as

*μ*is close to unity.

For the other limit where *βl*/*τ*
→ ∞ we find

*α*

_{rs}and

*σ*, and increase as the square root of the length. This is because the pumps overlap throughout the entire fiber. As discussed before; in this limit it is more reasonable to set

*h*= 1 because the step-functions are almost equal to unity. The separability coefficient tends to which is also close to unity. Thus, we would expect many Schmidt modes and therefore a non-separable Green function.

#### 3.4. Numerical studies

Before we consider numerical studies of the various functions in this paper we
discuss the natural dimensionless parameters to use. The efficiency of
conversion is quantified by the dimensionless parameter
*γ*̄ =
*γ*/*β*_{rs}, but this
is not a parameter that is going to be varied since we consider the
low-conversion efficiency limit which puts a natural limit on the conversion
strength *γ*̄ ≪ 1. The natural unit to
measure time in is in units of the pump-width and similarly for the length
parameter it is natural to use the pump-width divided by
*β*. For the remainder of the paper it is assumed
that *β*_{r} = *β*
= −*β*_{s} in the numerical
studies, so *β*_{rs} =
2*β*.

To better understand the implications of the step-functions and the Gaussian
approximation several numerical studies were performed, see Fig. 4. In Figs.
4(a) and 4(b) the Green
function is plotted in the time-domain with and without the Gaussian
approximation for a short fiber
*βl*/*τ* = 1 and for
the Gaussian approximation we choose *h* = 1.28. In this
limit the Gaussian approximation gives a qualitative answer, but it is only
moderately accurate. In Figs. 4(c) and
4(d) the two Green functions are
plotted with the normalized fiber-length set such that it should be separable.
We remind the reader of the criterion for separability is
*α*_{r}*α*_{s}
= −*σ*^{2}, thus giving
*βl*/*τ* =
(2/*w*)^{1/2} ≈ 2.2768. By looking at the
contour-plots it is clear that the Green function with the Gaussian
approximation is clearly separable whereas the step-function Green function is
not, as it is rotated with respect to the frequency axes. This hypothesis is
confirmed in Fig. 5(a) where the Schmidt
coefficients for the four Green functions are plotted. The Schmidt coefficients
are determined by using the analytic form of the Green function for which the
Schmidt decomposition is performed numerically. It gives qualitatively the same
Schmidt coefficients for the non-separable fiber-length. For the separable
fiber-length we have only one non-zero Schmidt coefficient with the Gaussian
approximation, but several non-zero coefficients for the rectangular window. In
Fig. 5(b) the first two Schmidt modes
are compared for the two Green functions for the non-separable fiber-length.
Again the qualitative behavior is the same, but it is only moderately accurate.
In conclusion, the time-domain collision analysis does highlight new physics
compared to the standard sinc/Gaussian approximation. The standard criteria for
separability was seen to be an artifact from the approximation of the
step-function with a Gaussian. Indeed for the non-convecting pumps the Green
function is never completely separable for Gaussian pumps since they depend only
on ${t}_{c}^{2}$ which includes both the input and the output
times. However, the first Schmidt coefficient is 36 times larger than the next
one, so the Green function is approximately separable.

Previously it was discussed that there was a discrepancy in the choice of the
height of the approximating Gaussian in the case of short and long fibers. In
Fig. 6(a) the first ten Schmidt
coefficients are plotted for *h* = 1 and
*h* = 1.28 for the Gaussian windows as well as the
step-index window for a long fiber
*βl*/*τ*. Notice that none of
the Green functions are separable, as discussed in the previous paragraph.
Furthermore, neither of the Gaussian Green functions accurately represents the
Green function with the rectangular window. This is because the Green function
is constant along the *t* − *t*′
contours, such that the Green function always has a finite value at the cut-off
lines. Figure 6(b) shows the two
lowest-order Schmidt modes for this long fiber. Notice that since the Schmidt
modes are normalized they are identical for the two different heights for the
Gaussian window. In this case there is a large discrepancy between the Schmidt
modes of the two different windows, thus reaffirming that replacing the
rectangular function with the Gaussian is not always insignificant.

## 4. Convecting pumps

The results presented in the previous section gave physical insight into FC by BS. To improve the accuracy of the model we have to include the walk-off between the pumps since they have different group slownesses. As seen from Fig. 1, we treat the common case that the pumps and sidebands are placed pairwise symmetrically around the zero-dispersion frequency, which means that for closely and moderately spaced pumps and sidebands, pump p co-propagate with the signal and likewise pump q co-propagates with the idler, because the group-slowness is symmetrical around the zero-dispersion frequency. This approximation is valid for a wide range of experimental parameters and throughout the paper we assume that this is the case [34, 38]. Even when this assumption is not completely accurate, Eq. (46) is still valid, so it is easy to determine the accuracy of this approximation.

With the aforementioned assumption, the pumps are described by

where*F*are normalized shape-functions. The pumps intersect at the distance

_{j}*z*

_{i}in the fiber. With this pump ansatz, our coupling function

*γ*

_{pq}in

*G*

_{rs}depends on

*A*

_{p}and

*A*

_{q}which are functions of

*γ*̄ =

*γ*/|

*β*

_{rs}|. This Green function is naturally separable, and the input and output Schmidt modes are the shape functions of pumps p and q, respectively. Only the step-functions prevent complete separability, but for a sufficiently long fiber they are equal to unity for times of interest.

The other Green function was defined in Eq. (50) with the associated collision point defined in Eq. (49). Using the collision distance and time we find that

*β*

_{s}(

*l*−

*z*

_{i}) and

*β*

_{r}

*z*

_{i}respectively. For idler-to-signal conversion the natural input idler has the shape of pump q and the output idler attains the shape of pump p. In both cases the input and output Schmidt modes are timed to arrive at the intersection point of the pumps, since this will be the point of maximal interaction.

#### 4.1. Gaussian pumps of equal width

Assuming two Gaussian pumps with the same width and
*z*_{i} = *l*/2, the Green
function attains the form

*βl*/

*τ*= 3 and we confirm that the output mode, the first Schmidt mode for

*l*= 3, indeed is a copy of pump q centered on

*β*

_{r}

*l*/2. It is difficult to conclude anything about the higher-order Schmidt modes, but a consequence of the Sturm comparison theorem, [52], is that the eigenfunctions of a Sturm-Liouville problem have a monotonically increasing number of zeros. Since the Schmidt modes are eigenfunctions of the integral equations [

*cf.*Eqs. (97) and (98)] we expect them to show the same behavior, which is confirmed by Fig. 7(d).

To check the hypothesis that the optimal interaction distance was half that of
the fiber, a numerical study was performed of the square of the first two
Schmidt coefficients *λ*_{0} and
*λ*_{1} as a function of the interaction
distance for two different fiber lengths. The result is seen in Figure 8 which confirms that the strongest frequency
conversion is at *z*_{i} = *l*/2.
An interesting result is that for the short fiber the first two Schmidt
coefficients have maxima at *z*_{i} =
*l*/2, which shows that the two lowest-order Schmidt modes
have the maximal conversion there. This result was not replicated for the longer
fiber, where the second Schmidt coefficient have minima at
*z*_{i} = *l*/2, but this is
because the Green function is separable and changing
*z*_{i} moves the Green function in the
(*t*,*t*′) plane leading to a cut-off
due to the step-functions.

### 4.1.1. The Gaussian approximation

To compare the results related to convecting Gaussian pumps with the ones found in the non-convecting case, once again the step-functions are approximated by a best-fit Gaussian, yielding

*t̄*and $\overline{{t}^{\prime}}$ are the delayed and advanced output and input times respectively,

*h*is the height fitting parameter, and

*ξ*= 0.3858 is the constant that gives the same full-width half-maximum for the approximation of the sinc with a Gaussian. This expression clearly shows that the Green function is separable in the limit of long fibers since

*α*

_{rs}∝

*l*and the cross-term is proportional to $1/{\alpha}_{\text{rs}}^{2}$.

Comparing with Eq. (104) we note that the argument of Eq. (78) is of the same form, hence inserting from Eq. (78) and using Eq. (109) leads to

*μ*has been used instead of

_{j}*τ*to avoid confusion.

_{j}In the limit of short fibers
(*βl*/*τ* → 0)
corresponding to *τ* ≫
*α*_{rs} we find

*α*

_{rs}and

*σ*, and increases as the square root of the length.

In the complementary limit in which
*βl*/*τ* →
∞,

*h*= 1). Also the time-scale is simply the pump-width. Comparing with the non-convecting case, Eqs. (64)–(67), the lowest-order Schmidt coefficient in both cases tends to a constant, but the time-scales differ as they now tend to a constant whereas the non-convecting ones grow with the square root of the fiber-length. Also the separability parameter tends to zero, such that the Green function is always separable for sufficiently long fibers, which was not the case for the non-convecting model that was only separable under the Gaussian approximation and under specific conditions. This contrasts to the non-convecting model because the interaction for the convecting case happens only during the collision, leading to separability and the drop-out of the length dependence.

These limits were tested numerically for the convecting case, see Fig. 9(a). The figure shows the square
of the lowest-order Schmidt coefficient for the Green functions with the
Gaussian-and step-function windows. It is seen that the Gaussian window
over-estimates the value of the Schmidt coefficient for long fibers, but
setting *h* = 1, such that the Gaussian has the same
height as the rectangular function gives a better agreement. This is
reasonable, as the rectangular window for large fiber-lengths is
approximately unity. In the short fiber limit the two models disagree for
*h* = 1, but one is free to choose
*w* to obtain better accuracy since the long-fiber limit,
Eq. (85) is independent
of *w*. The width of the best-fit Gaussian in the time-domain
was determined by matching the FWHM in the frequency-domain of the Gaussian
and the sinc function. However, the FWHM of the inverse Fourier transform is
not necessarily the same. By choosing the width such that the Gaussian and
the rectangle have the same FWHM in the time-domain, one finds
*w* = 0.7213. The square of the Schmidt
coefficients with this width is seen in Fig.
9(b). Choosing this width results in a better fit in the short
fiber limit, than the result of the traditional sinc approximation. By
fitting *w* to the Schmidt coefficients with the rectangular
window using Eq. (79), we
were able to find a slightly better fit for intermediate fiber lengths for
*w* ≈ 0.86, but at the cost of a worse fit for
short fibers. Thus we conclude that the best fit is found for
*h* = 1 and matching the FWHM in the
time-domain.

#### 4.2. Different Gaussian pumps

The next investigation considers Gaussian pumps with different widths. This leads to the Green function

*z*

_{i}=

*l*/2. This case is investigated in Fig. 10 for

*τ*

_{q}=

*τ*

_{p}/2. In (a) and (b) the Green function is plotted for

*βl*/

*τ*

_{p}= 1 and 3 respectively. The function is clearly elongated in the

*t*′ direction because pump p is twice as broad. Fig. 10(c) shows the Schmidt coefficients for the two cases and for the longer fiber the function is definitely still separable in spite of the different pump widths. This agrees with the hypothesis since it is only the step-functions that prevent separability and as long as the ridge is wider than the pump-widths the Schmidt decomposition should only contain one term. The shorter fiber has a slightly larger lowest-order Schmidt coefficient compared to Fig. 10(c) resulting in a larger degree of separability because the higher-order Schmidt coefficients are smaller. This is because the pulse is narrower in one direction and thus less of the Green function is exposed to the sharp boundaries. In Fig. 10(d) the two lowest output modes for the different fiber lengths are shown. As expected the lowest-order mode for the separable state is simply a copy of pump q centered on

*βl*/

*τ*= 1/2 [

*cf.*Eq. (72)], whereas the lowest-order mode for the shorter fiber is distorted, because the step-functions are not negligible in this case.

The lowest-order Schmidt coefficient was also considered for various aspect ratios, see Fig. 11(a). As the aspect ratio increases for a constant length the coefficient falls off since the value of the Green-function at the cut-off points increases, and it is therefore less separable. However, if the length increases with the aspect ratio it is possible to achieve separability over a wide range of aspect ratios. This is expected since in this case where pump q is wide compared to p and thus sideband r is wide and s narrow, for a long enough fiber the two short pulses propagate past the longer ones and hence experience a full collision [53].

For quantum communication it is of interest to convert the states emitted from a
quantum memory unit to a shape suitable for transmission in an optical
communication system. This might include reshaping the pulse width by a factor
of 100 [13]. To check
whether such a reshaping is possible within the perturbative framework, a
numerical study with *τ*_{q} =
*τ*_{p}/100 was carried out with the Schmidt
coefficients in Fig. 11(b). This
definitely shows that the Green function for the longer fiber lengths is
separable, but this is a natural extension of the discussion in Fig. 10(c).

### 4.2.1. The Gaussian approximation

In a similar way as the analysis for the Green function with two identical
Gaussian pumps we are interested in investigating the effect of the Gaussian
approximation. Two different pump-widths corresponds to replacing
*τ* with
(*τ*_{p}*τ*_{q})^{1/2},
and *τ*_{q} and
*τ*_{p} in front of
*t̄* and $\overline{{t}^{\prime}}$ respectively, in Eq. (78). For the limit where
the aspect ratio tends to infinity or in other words
*τ*_{p} → 0 and
*τ*_{q} → ∞ we find that

*α*

_{rs}or

*τ*. The square of the Schmidt coefficient differs from

_{j}*γ*̄

^{2}by the factor

*α*

_{rs}/

*τ*

_{q}.

These results were simulated, see Fig.
12(a). In general, the Gaussian window with *h*
= 1 underestimates the lowest-order Schmidt coefficient, but for
large aspect ratios the lower window height approximates the right result.
Also the lowest-order Schmidt coefficients fall off since the length of the
fiber was held constant. In the right panel the square of the lowest-order
Schmidt coefficient is plotted for *w* = 0.7213. In
this case the lower window height does give a better approximation, but it
is only moderately accurate. This is because the Green function has a large
value at the cut-off which makes the Gaussian approximation a less accurate
fit.

#### 4.3. HG0/HG1 pumps

Next we consider the case with a Hermite-Gaussian (HG) temporal shape of the pump
of zeroth order (HG0, a Gaussian) for pump p, and a HG pump of first order, HG1,
for pump q. The results are plotted in Fig.
13. Since the HG1 profile is slightly wider a longer fiber was used
for the long-fiber case (*βl*/*τ*
= 4). Again separability is indeed possible for the long fiber as seen
from the fact that the second coefficient is almost zero in Fig. 13(c). From Eq. (72) it is expected that the output mode corresponds to
that of pump q centered on *l*/2 for a sufficiently long fiber
where the step-functions are negligible. Since pump q in this case is HG1 it is
expected that the output mode will also be HG1 which is confirmed for
*βl*/*τ* = 4 in Fig. 13(d) the zeroth-order output mode is
a HG1 centered on *l*/2. The zeroth-order mode for the shorter
fiber is a distorted HG1 function centered also on *l*/2.

#### 4.4. HG1/HG1 pumps

The final case considered is two identical HG1 pumps. Due to the larger width of
the HG1 pumps, we consider
*βl*/*τ* = 5 to ensure
separability. The result is seen in Fig.
14. From Fig. 14(b) and (c) it
is clear that the Green function is separable for a sufficiently long fiber,
which is expected. Considering Eq.
(72), we expect the output and input modes for the separable case
simply to be copies of the two pumps. This is indeed confirmed from Fig. 14(d) where the HG1 pumps coincide
with the lowest-order input and output Schmidt mode (only the output mode has
been plotted here since it was indistinguishable from the input mode). Again the
shorter fiber leads to a slightly distorted HG1 mode. With this study we showed
that FC is possible for relatively short fibers and more complicated
pump-shapes.

## 5. Conclusion

In this paper we considered quantum-state preserving frequency conversion in both the frequency- and the time-domain using a perturbative analysis that is valid for low conversion efficiencies. The theoretical foundation was discussed and the Green function formalism introduced. The Schmidt decomposition was used as a useful tool to discuss separability and the temporal modes comprising the Green functions. The Schmidt modes of the Green function are the natural input and output modes of the process. Next the Green functions were obtained in the frequency-domain in the low-conversion limit for stationary pumps that do not convect with respect to each other. The stationary model is a simplification of four-wave mixing but it is realistic for three-wave mixing where there is only one pump. The results were obtained using a standard analysis that assumes Gaussian pumps and the approximation of the system’s sinc-function response by a best-fit Gaussian. The standard result was inverse Fourier transformed to the time-domain. Using the time-domain collision method, the solution was found in the time-domain for arbitrary pump-shapes and the effects of the assumptions made in the standard analysis were discussed. It was shown that for relatively stationary pumps complete separability is never possible for Gaussian pumps, and the conditions found using the standard model are artifacts of the sinc/Gaussian approximation. However the Green functions are close to being separable, and the predictions of the standard theory are reasonable in practice.

The collision method was generalized to also include convecting pumps. The Schmidt decomposition was used to find the natural modes of the problem and obtain important limits that allowed us to compare the stationary and the convecting models. In the short-fiber limit the predictions of the two models agree. It was also shown that convecting pumps allow for separable Green functions for sufficiently long fibers. This is in contrast to the stationary result that is only separable for one specific length. Additionally, we showed that it is possible to obtain arbitrary reshaping of a signal by a proper selection of the pump pulses. This was confirmed for simple Gaussian pumps and was also shown to be possible for two Gaussian pumps with very different widths. Finally higher-order Hermite-Gaussian shapes were also seen to allow for separability and reshaping. Preliminary numerical results show that reshaping also occurs in the high-conversion regime.

These results show that frequency conversion by four-wave mixing is a valuable resource for quantum information systems, as an convenient and reliable source for reshaping and frequency conversion, both of which are paramount for these systems to be used in practice. The low-conversion analysis will be extended to the high-conversion regime in future work.

## A. Appendix: Mehler identity and kernel decomposition

Decompositions of Gaussian kernels are made possible by the Mehler identity [54, 55]

*x*

^{2}+

*y*

^{2})/2], one obtains the related identity

The Schmidt decomposition theorem [56, 57] states that a
complex kernel *K*(*x*, *y*) may be
written as the series

*λ*are the (common) eigenvalues of the integral equations where the hermitian (and non-negative) kernels are

_{n}*u*and

*v*satisfy the orthonormality relations

*K*is real and symmetric,

*u*and

*v*are real and

*L*=

_{u}*L*.

_{v}Suppose that

*ψ*(

_{n}*x*), and the eigenvalues These results also follow directly from Eqs. (92) and (100).

For asymmetrically-pumped FC,

*a*,

*c*and

*ac*−

*b*

^{2}are all non-negative. One can rewrite Eq. (104) in the form of Eq. (100) by defining

*x*=

*τ*

_{r}

*ω*

_{r}and

*y*=

*τ*

_{s}

*ω*

_{s}, where [The choice of root in Eq. (105) is determined by the requirement that

*t*→ 0 as

*b*→ 0]. The result is

*b*is positive,

*t*is negative and vice versa. However, the Hermite functions Eq. (93) do not depend on

*t*and the singular values depend only on

*t*

^{2}, so Eqs. (103) and (108) omit sign information as written. One can restore this information by replacing

*t*with |

*t*| and multiplying

*τ*or

_{r}*τ*by

_{s}*s*= −sign(

_{c}*c*). This change is equivalent to changing the sign of

*x*or

*y*in Eq. (100). By combining Eqs. (105)–(107), one can show that

For reference, the Fourier transform of a Hermite function is also a Hermite
function. This result is a consequence of the fact that the Hermite functions
are eigenstates of the harmonic-oscillator Hamiltonian, which is symmetric with
respect to the position and momentum operators. One can prove this by
multiplying the Hermite generating function [55] by
exp(−*x*^{2}/2), to obtain the identity

*t*in Eq. (113), one finds that

^{n}*ψ*was defined in Eq. (93). Hence, if the sideband wavepackets can be expressed as sums of frequency-domain Hermite functions, they can be expressed as related sums of time-domain Hermite functions.

_{n}The kernel was decomposed in the frequency-domain and the Schmidt modes were Fourier-transformed to the time-domain. Alternatively one can inverse-transform the kernel directly to the time-domain and then decompose it. To be able to compare the result in the time-domain with what we obtained in the frequency domain we consider the generalized Gaussian in the frequency-domain

*a*and

*c*positive and

*ac*−

*b*

^{2}> 0. Equations (115) and (116) comprise a specific example of the general transform relation

*M*is a symmetric matrix,

*X*and

*K*are column vectors, and the superscript

*t*denotes a transpose. Formula (117) is a standard result. It is proved in Appendix A of [58].

Comparing with Eqs.
(105)–(107)
the time-domain kernel has an analytic Schmidt decomposition as it is in the
canonical form. It has the same Schmidt-coefficients but the characteristic
time-scales are 1/*τ*_{r} and
1/*τ*_{s} respectively. The time-domain
Schmidt-modes are indeed Hermite functions, as stated previously.

## Acknowledgments

MR was supported by the National Science Foundation, EPDT.

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