## Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color |

Optics Express, Vol. 19, Issue 19, pp. 17876-17907 (2011)

http://dx.doi.org/10.1364/OE.19.017876

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### Abstract

We study quantum frequency translation and two-color photon interference enabled by the Bragg scattering four-wave mixing process in optical fiber. Using realistic model parameters, we computationally and analytically determine the Green function and Schmidt modes for cases with various pump-pulse lengths. These cases can be categorized as either “non-discriminatory” or “discriminatory” in regards to their propensity to exhibit high-efficiency translation or high-visibility two-photon interference for many different shapes of input wave packets or for only a few input wave packets, respectively. Also, for a particular case, the Schmidt mode set was found to be nearly equal to a Hermite-Gaussian function set. The methods and results also apply with little modification to frequency conversion by sum-frequency conversion in optical crystals.

© 2011 OSA

## 1. Introduction

3. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express **13**, 9131–9142 (2005). [CrossRef] [PubMed]

4. A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express **14**, 8989–8994 (2006). [CrossRef] [PubMed]

6. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” Photon. Technol. Lett. **23**, 109–111 (2011). [CrossRef]

2. S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature **437**, 116–120 (2005). [CrossRef] [PubMed]

7. J. M. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. **68**, 2153–2156 (1992). [CrossRef] [PubMed]

8. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. **105**, 093604 (2010). [CrossRef] [PubMed]

*ω*=

_{p}*ω*–

_{b}*ω*. The configuration for FWM with two pumps, denoted

_{g}*p*and

*q*, is that of dynamic Bragg scattering, in which for example, one photon is removed from pump

*p*and one is added to pump

*q*. At the same time, one photon is removed from the green signal and one is added to the blue signal so that

*ω*+

_{p}*ω*=

_{g}*ω*+

_{q}*ω*. This contrasts with optical parametric amplification, wherein photons are removed from the pump(s) and photons are added to both signals.

_{b}10. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. **283**, 747–752 (2010). [CrossRef]

11. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature **409**, 46–52 (2001). [CrossRef] [PubMed]

12. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. **79**, 135–174 (2007). [CrossRef]

13. N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. **101**, 130501 (2008). [CrossRef] [PubMed]

14. J. L. O’Brien, “Quantum computing over the rainbow,” Physics **1**, 23 (2008). [CrossRef]

15. A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express **19**, 13770–13778 (2010) [CrossRef]

16. J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. **8**, 478–491 (1990). [CrossRef]

17. M. E. Marhic, “Coherent optical CDMA networks,” J. Lightwave Technol. **11**, 854–864 (1993). [CrossRef]

18. B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper-engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. **13**, 065029 (2011). [CrossRef]

19. D. Kielpinski, A.F. Corney, and H.M. Wiseman, “Quantum optical waveform conversion” Phys. Rev Lett. **106**, 130501 (2011). [CrossRef] [PubMed]

3. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express **13**, 9131–9142 (2005). [CrossRef] [PubMed]

10. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. **283**, 747–752 (2010). [CrossRef]

3. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express **13**, 9131–9142 (2005). [CrossRef] [PubMed]

15. A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express **19**, 13770–13778 (2010) [CrossRef]

18. B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper-engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. **13**, 065029 (2011). [CrossRef]

19. D. Kielpinski, A.F. Corney, and H.M. Wiseman, “Quantum optical waveform conversion” Phys. Rev Lett. **106**, 130501 (2011). [CrossRef] [PubMed]

10. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. **283**, 747–752 (2010). [CrossRef]

## 2. Theory

4. A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express **14**, 8989–8994 (2006). [CrossRef] [PubMed]

8. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. **105**, 093604 (2010). [CrossRef] [PubMed]

21. K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. **6**, 1451–1453 (1994). [CrossRef]

*p*in Fig. 2, and one photon in the green field are annihilated while one photon from the other pump (denoted

*q*) and one photon in the blue field are created. The process conserves photon number in accordance with the Manley-Rowe relations [22] and can be summarized as

*γ*+

_{p}*γ*→

_{g}*γ*+

_{q}*γ*, where

_{b}*γ*represents a photon.

**13**, 9131–9142 (2005). [CrossRef] [PubMed]

*β*and

*κ*are parameters quantifying the effects of phase-mismatch and nonlinearity [3

**13**, 9131–9142 (2005). [CrossRef] [PubMed]

*a*

_{g(b)}(

*z*) as a function of position along the fiber

*z*for a fiber of length

*L*is given by where the “transfer” functions are given by where

*k*= (Δ

*β*

^{2}/4 + |

*κ*|

^{2})

^{1/2}. The transfer functions have the relation |

*μ*|

^{2}+ |

*ν*|

^{2}= 1, which ensures the process is unitary and conserves photon number. Mathematically, the BS translation process having frequency input and output “ports” is identical to the normal beam-splitter operation on degenerate monochromatic fields [23

23. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express **12**, 5037–5066 (2004). [CrossRef] [PubMed]

*ν*(

*L*)| = 1 for a medium of length

*L*. In this case the quantum states of the modes are completely swapped. For example, a single green photon is replaced by a single blue photon, that is, |1, 0〉 → |0, 1〉, where |

*i*,

*j*〉 denotes the state in which

*i*photons are in the green region and

*j*photons are in the blue region.

*differing frequencies*will interfere with one another and give a perfect HOM dip under the appropriate circumstances [10

**283**, 747–752 (2010). [CrossRef]

*μ*|

^{2}= |

*ν*|

^{2}= 1/2, so that it is equally likely for the blue (green) photon to stay in the blue (green) mode or be translated to the green (blue) mode. Using the above equations it can be shown that in this case, for an input state consisting of one green photon and one blue photon, denoted by

*μ*|

^{2}and |

*ν*|

^{2}the |1, 1〉 component of the state cancels to zero.

*dωA*(0,

*ω*)

*a*

^{†}(0,

*ω*)|

*vac*〉, where

*A*(0,

*ω*) is the spectrum of the input wavepacket, is translated in the manner given by (9). The state can thus be expressed in terms of the output operator as

*g*” or “

*b*” subscript. For example, the creation operator for an green photon at a particular green frequency will be written as

**283**, 747–752 (2010). [CrossRef]

*G*refers to the evolution of an output creation operator in range

_{xy}*y*which came from range

*x*.

*G*function completely describes the evolution of arbitrary inputs, it is difficult to discover which input pulses give rise to desirable effects such as high-efficiency translation or good two-color HOM interference just by examining the

*G*function. However, by performing a Schmidt (singular-value) decomposition (SVD) [24

24. A. Ekert and P. L. Knight, “Entangled quantum systems and the schmidt decomposition,” Am. J. Phys. **63**, 415–423 (1995). [CrossRef]

*G*function or on the sub-matrices

*G*, the input wavefunctions necessary to achieve these effects can be found [10

_{xy}**283**, 747–752 (2010). [CrossRef]

*G*, which is not Hermitian and therefore does not admit a unitary decomposition. We write

_{gb}*G*in its SVD form as where the

_{gb}*ρ*are the real, positive singular values (the generalization of eigenvalues), and

_{n}*V*and

_{n}*w*are the

_{n}*n*th column vectors of the unitary “matrices”

*V*and

*w*describing the decomposition, respectively. The most useful way to interpret this decomposition is that a given green wavepacket

*V*will be translated to the blue wavepacket

_{n}*w*with probability

_{n}*V*that have associated singular values

_{n}*ρ*= 1 will translate over to the blue mode with 100 percent probability. The

_{n}*G*“matrix” describes an input green wavepacket scattering within the green spectral region at the output (no translation from green to blue, although the shape of the wavepacket may change). If the SVD of

_{gg}*G*is given by [10

_{gg}**283**, 747–752 (2010). [CrossRef]

*V*will scatter to an output green wavepacket

_{n}*υ*(the

_{n}*V*matrix here being the same as in the

*G*decomposition). The singular values satisfy the relation

_{gb}*G*and

_{bg}*G*have their own SVDs, given by where the

_{bb}*w*and

*υ*matrices, and the singular values

*τ*and

*ρ*, are the same as in (13) and (14). This highlights the interconnectedness between the various sub-Green functions in that there are only four unique Schmidt modes for each index

*n*that completely describe the process. Before treating exclusively single-photon inputs, we emphasize that the Green functions determined by our present approach can be used to treat arbitrary multi-photon inputs as well.

**283**, 747–752 (2010). [CrossRef]

*V*and

_{n}*W*will show perfect two-color HOM interference. Alternatively, this condition can be written as We refer to the quantity

_{n}*σ*= 2

_{n}*τ*as the HOM singular value, because it appears as such for a Schmidt decomposition kernel designed specifically to optimize the HOM interference. (See Eq. (37) of [10

_{n}ρ_{n}**283**, 747–752 (2010). [CrossRef]

*ψ*〉 must be normalized and the component states are orthonormal. Now, for input consisting of one green photon and one blue photon having input spectra of

*A*(0,

_{g}*ω*) and

_{g}*A*(0,

_{b}*ω*) respectively, the state is If the photons undergo the BS process, the input photon operators evolve according to (9). By switching the limits of integration and noting that

_{b}*X*and

*Y*could be either

*g*or

*b*and

*A*(

_{XY}*L*,

*ω*) is a component of the spectrum at the output of the fiber, the state can be written compactly as

_{Y}*C*|

_{ij}^{2}. For the |1, 1〉 state the coefficient is found by acting the aforementioned operators, with their accompanying functions, on the vacuum, yielding The first term,

*A*(

_{gg}*L*,

*ω*)

_{g}*A*(

_{bb}*L*,

*ω*), corresponds to the case where a photon initially in the green mode (i.e. wavepacket) scatters within the green mode at the output, and a photon initially in the blue mode scatters within the blue mode at the output. The second term,

_{b}*A*(

_{gb}*L*,

*ω*)

_{g}*A*(

_{bg}*L*,

*ω*), corresponds to the case where a photon initially in the green mode scatters to the blue mode at the output, and a photon initially in the blue mode scatters to the green mode at the output. Since these terms are being added before their modulus squared is taken, there is the chance that they could cancel if the terms are proportional and out of phase with one another. This is exactly what happens in the case in which ideal HOM interference occurs, leaving zero probability amplitude for the field to emerge in the |1, 1〉 state. The expression for the probability for this state to occur

_{b}*P*

_{11}is

**283**, 747–752 (2010). [CrossRef]

*P*

_{11}and the HOM singular values;

*P*

_{20}and

*P*

_{02}, the probabilities that two photons appear in one or the other output ports together. The expressions for these, and their derivation can be found in Appendix A.

## 3. Numerical implementation

25. P. D. Drummond and C. W. Gardiner, “Generalized P-representations in quantum optics,” J. Phys. A **13**, 2353–2368 (1980). [CrossRef]

27. W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A , **73**, 063816 (2006). [CrossRef]

28. W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A , **73**, 063816 (2006). [CrossRef]

*E*(

*z*,

*t*) =

*A*(

*z*,

*t*)

*e*

^{i(β0z–ω0t)}, where

*A*(

*z*,

*t*) is a slowly varying envelope function,

*ω*

_{0}is the high frequency of the carrier wave, and

*β*

_{0}is the propagation constant at

*ω*

_{0}. In that form the evolution is governed by the nonlinear Schrödinger equation Although this is effective for single pulses, for electric fields made up of multiple, distinct pulses with significantly different frequency components, solving this form of the equation is unnecessarily computationally intensive and impractical. Instead, we solve four coupled equations, derived from the nonlinear Schrödinger equation, in which terms relating to dispersion, self-phase modulation, cross-phase, and the BS process were kept. Numerically, the frequency components of each field are associated with a frequency mesh pertaining only to that field. This drastically reduces the number of calculations needed to compute the solution in that it is unnecessary to resolve the large empty bandwidth in between fields. In this scheme, the total electric field is the sum of the four individual fields, each having a slowly varying amplitude function

*A*centered around a carrier wave of frequency

_{j}*ω*; where

_{j}*j*denotes

*p*,

*q*,

*g*or

*b*. In quantum theory,

*A*is proportional to an annihilation operator. Substituting this ansatz into the nonlinear Schrödinger equation leads to, with suppressed independent variables

_{j}*z*and

*t*, the four coupled BS equations (where

*∂*represents the partial derivative with respect to

_{z}*z*)

*β*is the dispersive mismatch given by Δ

*β*=

*β*+

_{p}*β*– (

_{g}*β*+

_{q}*β*) +

_{b}*γ*(

*P*–

_{q}*P*),

_{p}*P*and

_{q}*P*are the peak pump powers, and

_{p}*γ*is the nonlinear coefficient. The first term on the right hand side describes the effect of linear dispersion, the next two terms describe SPM and CPM respectively, and the last term describes the BS four-wave mixing process. The quantity

*β*(

_{p}*i∂*; Ω

_{T}_{0}) is given by where

*β*

^{(n)}(Ω

_{0}) is the

*nth*derivative of

*β*evaluated at Ω

_{0}, and

*T*=

*t –*

*β*

^{(1)}(

*ω*)

_{p}*z*. In this scheme all the fields are propagating in the arbitrary choice of the frame of reference of pump

*p*. With one frame of reference there need only be one universal time mesh.

*A*and

_{p}*A*by classical functions if these fields are in strong coherent states. We solve the first two equations for the pumps ignoring the small effects of the weak signal fields

_{q}*A*and

_{g}*A*, but retaining the SPM and CPM effects of the strong pumps. We then use these pump solutions in the second pair of equations for the weak signal fields. Because these equations are linear in the operators, we can treat them formally as classical equations. Then the quantum effects can be fully accounted for by using the commutation relations when calculating quantities such as

_{b}*P*

_{11}, as describe above.

*E*we approximately solve the operator equation

_{j}*∂*(

_{z}A_{j}*T*,

*z*) = (

*D̂*+

_{j}*N̂*)

_{j}*A*(

_{j}*T*,

*z*), where

*D̂*and

_{j}*N̂*are the dispersive and nonlinear differential operators, acting in the frequency and time domains respectively, which act on

_{j}*E*given in Eq. (25). The approximation is that

_{j}*A*(

*T*,

*z*) is taken to

*A*(

*T*,

*z*+ Δ

*z*) by where Δ

*z*=

*L*/

*N*and

*N*is the number of steps. The nonlinear component integral is found by use of a fourth-order fixed-step Runge-Kutta method.

*A*be an

_{in}*n*×

*n*matrix with each column equal to one time-domain vector from an orthogonal set,

*A*be an

_{out}*n*×

*n*matrix with each column equal to the output of the

*n*column of

^{th}*A*, and

_{in}*G*be the Green function. In this paper we use the Hermite-Gaussian (HG) functions as the orthogonal set, where the

*n*function is given by where the characteristic time of the basis set is set by the overall time window of the mesh. Due to the linear nature of the Green function formalism, as expressed in (9), the

^{th}*forward*Green function expressed in matrix notation is This is strictly true only if all of the output vectors of

*A*can be described as some linear combination of the orthogonal vectors of

_{out}*A*. In practice, an orthogonal vector of order

_{in}*j*will evolve to have components of both lower and higher order than

*j*. In the cases where the

*j*th mode evolves to include components of higher order than

*n*,

*G*won’t satisfy the unitary conditions of (11), which is often the case for the evolution of the very-high-order modes. In that case, only the subset of orthogonal input vectors that evolve into output that can be described by all

*n*input vectors can be considered “valid” output, as far as determining the Green function is concerned. Let

*A*

_{in,sub}be a

*m*×

*m*dimensional matrix that is the sub-matrix of

*A*where all the input vectors evolve to output vectors (described by the

_{in}*n*×

*m*matrix

*A*

_{out,sub}, where

*m*≦

*n*) that can be described as a linear combination of the vectors of

*A*. Then the Green function that always satisfies the unitary conditions is given by and will be a matrix of dimensions

_{in}*n*×

*m*. In practice, the simplest way in which to determine the correct matrix subsets is by applying the unitary conditions (calculating

*G*·

*G*

^{†}) and seeing for what subset of vectors they hold. When that is determined,

8. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. **105**, 093604 (2010). [CrossRef] [PubMed]

31. P. S. J. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

32. G. K. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express **13**, 8662–8670 (2005). [CrossRef] [PubMed]

*n*(usually the index of bulk fused silica) and an effective cladding index

_{core}*n*, where

_{clad}*n*(

_{clad}*ω*) =

*f*+ (1 –

*f*)

*n*(

_{core}*ω*). Here

*f*is the air-filling fraction of the lattice region compromising the cladding, which along with the core radius

*a*, parameterizes the fiber. By measuring the modulation instability wavelength signals produced by a fiber as a function of pump wavelength, best-fit values for

*f*and

*a*can be determined. For a particular fiber that has been used in past experiments and is a good candidate for future experiments, this procedure has been carried out, resulting in parameter values of

*f*= 0.494 and

*a*= 0.72

*μ*m, with the resulting dispersion parameter

*D*shown in Fig. 3. This is the dispersion profile used for all numerical calculations in this paper.

*βL*/2), which defines the dispersive mismatch of the process. Figure 4 shows this sinc function for the fiber, central wavelengths, and length (20 m) used in the simulations. All fields were taken to be monochromatic, and the pumps where fixed at 808 nm and 845 nm. The “blue” signal field was varied from its central wavelength of approximately 649 nm (shown in frequency on the horizontal axis) while the “green” field was varied from its central wavelength of 673 nm to conserve the energy of the process (

*ω*=

_{s}*ω*+

_{p}*ω*–

_{i}*ω*). The full-width at half-maximum (FWHM) of the central sinc lobe is approximately 0.3 Trad/s. This number gives a rough estimate of the effective bandwidth of the BS process. As we will see, all else being equal, input signals that have spectral widths larger than this number do not translate as efficiently as signals that have spectral widths lower than this number.

_{q}## 4. Numerical results

*intensity*profiles had either a “long” FWHM duration of 1000 ps, making them quasi-CW for most input signal fields used, or a “short” FWHM duration of 70 ps. At the input to the fiber, the centers of the fields were placed at zero time. The central wavelengths for the pump fields

*p*and

*q*were 808 nm and 845 nm, respectively, placing them in the anomalous-dispersion region. The signals “green”

*g*and “blue”

*b*were at wavelengths 673 nm and 649 nm, respectively, placing them in the normal-dispersion region. The 845-nm and 649-nm fields had nearly equal group velocities (≅ 2.0044 × 10

^{8}m/s), and the 808-nm and 673-nm fields had nearly equal group velocities (≅ 2.0049 × 10

^{8}m/s). The nonlinear coefficient

*γ*was set to 100 W

^{−1}km

^{−1}for all calculations.

### 4.1. Frequency translation, long pulse

*V*matrix, as defined in (13), from an SVD of the

*G*Green function. This Green function corresponds to the case of translation from the green mode to the blue mode. As discussed previously, the columns of the

_{gb}*V*matrix represent the input Schmidt modes, so in this case the

*V*matrix represents the input signal field. That is, The horizontal axis is the Schmidt mode number

*n*, where order was determined by the magnitude of the corresponding singular value

*ρ*, largest to smallest, for the first 25 Schmidt modes. The vertical axis is the HG mode number

*j*of the Schmidt modes using the same HG basis used to find the Green function. The plotted quantity is

*V*matrix is mostly composed of the first HG function and almost completely described by the first four HG functions. Thus we can conclude that the temporal duration of the first Schmidt mode is close to the characteristic time of the optimal HG basis, 243 ps.

*w*matrix. The absolute value of this matrix, the column vectors of which correspond to output in the blue mode, is shown in Fig. 5(b). The axes are the same as those in part (a). Interestingly, the

*V*and

*w*matrices are qualitatively similar, although the

*w*matrix Schmidt modes tend to be dominated by either even or odd HG functions whereas the

*V*matrix Schmidt modes are composed of a more or less equal number of even and odd HG functions.

*ρ*

^{2}of the

*G*Green function for this case. The circles of the blue line correspond to the squared singular values

_{gb}*τ*

^{2}of the

*G*Green function. The

_{gg}*ρ*

^{2}describe the translation efficiencies of the input green Schmidt modes to the output blue Schmidt modes, whereas the

*τ*

^{2}describe the “non-translation” efficiencies of the input green Schmidt modes to the output green Schmidt modes. This corresponds to the transmission and possible reshaping of the green mode. For the Manley-Rowe relations to be satisfied, the square of these singular values must add to one, which they do numerically in this case (as well as in all other cases) to one part in one thousand. (The higher-order singular values are accurate to one part in one thousand, but the lower-order singular values are usually more accurate, on the order of one part in one million.) As is evident from the figure, many of the modes translate efficiently with the first several having efficiencies of over 90 percent. We call this configuration “non-discriminatory” in the sense that it effectively translates many different dissimilar modes efficiently.

*V*matrix) of the green (673 nm) mode. The input amplitudes of the pump fields are also shown as a black dashed line. As could be guessed from Fig. 5(a), the modes look very much like HG functions, although there is some asymmetry and delay in both time and frequency. These effects are likely the results of pump evolution, which includes convection, dispersion and CPM. Figure 7(a) shows that first few Schmidt modes are much narrower in time than the pump fields. Looking at Fig. 7(b), which shows the phases in time, all the input Schmidt modes, disregarding the

*π*phase jumps, are distinctly parabolic with positive curvature. For the phase convention we use, this shape implies the Schmidt modes at the fiber’s input are down-chirped (frequency decreases in time), meaning that when they propagate through the medium, which for the signals is normally-dispersive, they will temporally compress. That is, they are dispersion pre-compensated. CPM also chirps the signals in the same way as linear dispersion. Figure 7(c) shows that the pump fields are nearly monochromatic as compared to the Schmidt modes, although all modes are well within the translation phase-matching bandwidth of the fiber.

### 4.2. Frequency translation, short pulse

*V*matrix (input green Schmidt modes) from the SVD of the

*G*Green function, while Fig. 8(b) shows the absolute value of the

_{gb}*w*matrix (output Schmidt modes). The horizontal and vertical axes have the same meaning as they did in the previous case.

*V*matrix). The input amplitudes of the pump fields are also shown as a black dashed line. Qualitatively, the amplitudes for this case are similar to the amplitudes in the former case; although asymmetric, they are roughly HG-like. Quantitatively, these amplitudes are much broader spectrally than in the former case. The phases in time are essentially parabolas with positive curvature much larger than that of the previous case, as expected because the bandwidths are much larger.

### 4.3. HOM interference, long pulse

*V*matrix, as defined in (13), from an SVD of the

*G*Green function. This Green function corresponds to the case of translation from the green mode to the blue mode. Figure 13(b) shows the absolute value of the

_{gb}*w*matrix. The horizontal and vertical axes are the same as they were the previous cases. Surprisingly, both the

*V*and

*w*matrices are nearly diagonal, meaning that the input Schmidt mode set and the HG basis functions are nearly identical. In Appendix B we present a theoretical model explaining this result. The long pump pulse of 1000 ps enables many modes to be translated with nearly similar efficiencies, as can be seen from 13(c). For this reason we describe this case as “non-discriminatory” with respect to its two-color HOM interference properties, since many different types of modes lead to good HOM interference.

*P*

_{11}for the first three input Schmidt modes as a function of fiber length. We emphasize that a value

*P*

_{11}=0 means that two green or two blue photons exit the fiber, but never one green and one blue. This is the HOM interference effect as it applies to interference of photons of different colors [10

**283**, 747–752 (2010). [CrossRef]

*V*matrix). The input amplitudes of the pump fields are also shown as a black dashed line. Since the

*V*matrix is nearly diagonal, the Schmidt mode amplitudes are almost quantitatively identical to the HG basis functions, which stands in contrast to the case in section 4.1 which had the same duration pulse length but twice the pump power. Similarly to the previous cases, the phase in time is parabolic with positive curvature, showing that these inputs are also down-chirped.

### 4.4. HOM interference, short pulse

*V*matrix from the SVD of the

*G*Green function, while Fig. 16(b) shows the absolute value of the

_{gb}*w*matrix. The horizontal and vertical axes are the same as they were in the previous cases.

*w*and

*V*matrices are nearly diagonal for low Schmidt mode number, but unlike the previous case, they quickly become seemingly random combinations of HG functions for higher-order Schmidt modes. The reason for this is the same as it was in the other case having short pump pulses; there are many Schmidt modes that have low singular values, and practically any combination of higher-order HG functions will have a low, but non-zero singular value. As shown in Fig. 16(c), only the first few Schmidt modes have a singular value meaningfully above zero and will therefore have a more ordered mode structure.

*P*

_{11}for the first three input Schmidt modes as a function of fiber length. Figure 17(a) shows that, in contrast to the previous case, the HOM singular values decrease rapidly with Schmidt mode number, making this case highly “discriminatory” in regards to its HOM interference properties; only a few Schmidt modes exhibit HOM singular values significantly above zero.

*V*matrix). The input amplitudes of the pump fields are also shown as a black dashed line. Similarly to the previous case, these first few Schmidt modes are qualitatively similar to the HG basis functions, although they exhibit more asymmetry relative to their width than do the Schmidt modes in the previous case. As can be seen in Fig. 18 (c), the frequency amplitudes have widths that are comparable to (and, for the second and third mode, somewhat larger than) the translation bandwidth, which contributes to the fact that the higher-order modes experience a low translation efficiency. The phase plots are similar to what they were in the previous cases; the phases in time are mostly parabolic with positive curvature, implying the inputs are down-chirped.

*P*

_{11}value achieved along the fiber is plotted as a function of input signal pulse FWHM duration. For short signal durations the minimum

*P*

_{11}value hardly changes from 1, which is the initial value (

*z*= 0) of

*P*

_{11}for all modes, but for long signal durations the minimum

*P*

_{11}value drops to nearly zero. This is most likely due to the short signals having large spectral bandwidths and the long signals having small spectral bandwidths as compared to the translation bandwidth. This follows because pulses with large relative bandwidth experience poor translation as compared to pulses with small relative bandwidths. In Fig. 11(b), the length at which the minimal

*P*

_{11}value occurs is plotted as a function of input signal duration. For short signal widths, the optimal length is the same as the fiber length. This is likely due to the difficulty translating signals with large bandwidths; it takes the entire length of the fiber to translate the small amount of the signal that is within the translation phase-matching bandwidth. This dynamic changes once the signal width is on the order of the phase-matching bandwidth, at which point the BS process is no longer limited by bandwidth but by the shape and power of the pumps, hence the longer optimal lengths for longer signal pulses.

*P*

_{11}, shown in part (a), for short signal inputs behaves similarly to the previous long-pumps case: a sharp increase in

*P*

_{11}for very short signal widths due to the signal having larger translation phase-matching bandwidth than the fiber. Unlike the previous case,

*P*

_{11}reaches a minimum around 40 ps and begins to slowly increase for wider input signal widths. The increase for wider signal widths is likely due to signals beginning or evolving to be wider than the pumps, so that only part of the signals wavepacket could be translated. The length at which the minimum

*P*

_{11}occurs is shown in part (b). This is a qualitatively similar result to the previous long-pumps cases. For short signals the optimal length is equal to the length of fiber (fixed at 20 m), due to the excessive bandwidth of the signal in relation to the phase-matching bandwidth. But when the signal durations are of the same order as the pumps, the optimal length is not as constrained by the phase-matching bandwidth in the same manner, and becomes somewhat shorter than the length of the fiber. But when the signals become as long as or longer than the pump, the overlap between all the fields in time significantly decreases, leading to less translation and raising the value of

*P*

_{11}. In this regime the fiber length also limits the translation, hence the reason the optimal length is equal to the length of the fiber. This suggests that were the fiber longer the value of

*P*

_{11}would be lower, which indeed is the case.

## 5. Analytic derivation of Schmidt modes

*V*and

_{n}*W*, the output modes

_{n}*v*and

_{n}*w*, and the Schmidt coefficients

_{n}*ρ*and

_{n}*G*and

_{gg}*G*can be deduced from the off-diagonal Green functions

_{bb}*G*and

_{gb}*G*(and vice versa). In Appendix B we rewrite the third and fourth lines of (25) in a frame moving with the average group speed of the signals. We include the effects of signal convection and second-order dispersion, but do not include the effects of time-dependent cross-phase modulation, which were discussed qualitatively after Fig. 7. In this frame the dispersion relations for the green and blue signal fields

_{bg}*β*

_{1}is negative, so the green mode is faster than the blue mode. For the parameters in our simulations, the temporal walk-off of pulses is significant, but pulse broadening by linear group-velocity dispersion is negligible. Therefore, we set

*β*

_{2}= 0 here for the simplest analytical solution. We assume the pumps have identical temporal intensity profiles, proportional to exp(–

*t*

^{2}/2

*σ*

^{2}), where

*σ*is a width parameter. We then find that the perturbative solution for the signals (to first order in the coupling parameter

*γ*) involves Green functions that can be expressed in SVD form, as are those in (13)–(16). In this perturbation theory the Schmidt modes are independent of the coupling parameter

*γ*and pump peak intensities, but of course this will break down at higher values of

*γ*or pump peak intensities. Notably, we find that the Schmidt modes are given by HG functions in this regime. The explicit forms of the input green and blue Schmidt modes are, respectively: where the

*ψ*(

_{n}*t*

_{0}

*ω*) are the HG functions defined in (26), and the temporal scale factor is

_{g}*t*

_{0}= (0.621

*σβ*

_{1}

*L*)

^{1/2}. Likewise, the output green and blue Schmidt modes are, respectively: Note that the spectral phases in these mode solutions correspond to temporal delays or advances, which in this simple model result in both green and blue modes maximally overlapping with the pump pulses at the midpoint (

*z*=

*L*/2) of the medium. In our simulations

*β*

_{1}is negative, so the green input mode is delayed and the blue input mode is advanced (although note that in our simulations the two pumps walk off from each other, which is not the case for the analytic solution given here).

## 6. Discussion and conclusions

**13**, 9131–9142 (2005). [CrossRef] [PubMed]

**283**, 747–752 (2010). [CrossRef]

**13**, 9131–9142 (2005). [CrossRef] [PubMed]

**283**, 747–752 (2010). [CrossRef]

16. J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. **8**, 478–491 (1990). [CrossRef]

17. M. E. Marhic, “Coherent optical CDMA networks,” J. Lightwave Technol. **11**, 854–864 (1993). [CrossRef]

15. A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express **19**, 13770–13778 (2010) [CrossRef]

**19**, 13770–13778 (2010) [CrossRef]

## Appendix A: probabilities *P*_{20} and *P*_{02}

*ω*′

*to*

_{b}*ω*was for example). This ambiguity can be avoided by calculating

_{b}*P*

_{20}directly as the product of the appropriate bra and ket vectors (similarly for

*P*

_{02}). From (20) the term corresponding to the creation of two green photons leads to an expression for

*P*

_{20}of

*a*(

*ω*),

*a*

^{†}(

*ω*′)] =

*δ*(

*ω*–

*ω*′). Hence the commutation relation to be used on the innermost operators is This breaks the expression into two parts, and leads to the expression, employing dummy variable notation, The expression for

*P*

_{02}is the same when

*A*→

_{gg}*A*and

_{bb}*A*→

_{gb}*A*. The two terms of (38) have distinct physical interpretations. The first term, derived in part from the delta function of (37), is essentially the overlap of the two spectral functions

_{bg}*A*and

_{gg}*A*, and accounts for cases in which the two green photons are created in the same mode. This purely quantum mechanical term arises from the bosonic nature of photons, the operators of which obey the above-mentioned commutation relation. The second term accounts for the cases in which the two green fields are purely classical, as if these fields did not obey the quantum bosonic commutation relations.

_{gb}## Appendix B: derivation of the approximate Schmidt modes

*V*and

_{n}*W*, the output modes

_{n}*v*and

_{n}*w*, and the Schmidt coefficients

_{n}*ρ*and

_{n}*G*and

_{gg}*G*can be deduced from the off-diagonal Green functions

_{bb}*G*and

_{gb}*G*(and vice versa). In this appendix, the off-diagonal Green functions and the associated Schmidt modes will be determined approximately, for low conversion efficiencies.

_{bg}*ω*is an envelope frequency,

_{j}*(*γ ¯

*z*,

*ω*) is the Fourier transform of the coupling term

*π*. Equations (39) and (40) are valid in a frame moving with the average group speed of the signals (so ±

*β*

_{1}are the relative group slownesses). They include the effects of convection and second-order dispersion, but do not include the effect of time-dependent cross-phase modulation (which chirps the signals).

*A*(

_{j}*z*,

*ω*) =

_{j}*B*(

_{j}*z*,

*ω*) exp[

_{j}*ik*(

_{j}*ω*)

_{j}*z*]. Then the transformed amplitudes

*B*obey the transformed equations In general, the

_{j}*B*-equations are no simpler than the

*A*-equations, because they depend explicitly on

*z*. However, in the low-conversion-efficiency regime, one can replace the mode amplitudes on the right sides by the input amplitudes. In this regime, where

*κ*(

*z*,

*ω*,

_{g}*ω*) =

_{b}*(*γ ¯

*z*,

*ω*–

_{g}*ω*) exp[

_{b}*ik*(

_{b}*ω*)

_{b}*z – ik*(

_{g}*ω*)

_{g}*z*] is the kernel in (41). Note that

*γ*

^{*}(

*z*,

*ω*) = [

*γ*(

*z*, –

*ω*)]*, so

*κ*(

*z*,

*ω*,

_{b}*ω*) =

_{g}*κ**(

*z*,

*ω*,

_{g}*ω*).

_{b}*A*=

_{p}A_{q}*p*

_{0}exp(−

*t*

^{2}/

*τ*

^{2}), where

*p*

_{0}is the peak pump power and

*τ*is the (Gaussian) pump width. In this case, where the width parameter

*σ*=

*τ*/2

^{1/2}. Define the wavenumber-mismatch function

*δ*(

*ω*,

_{g}*ω*) =

_{b}*k*(

_{g}*ω*) –

_{g}*k*(

_{b}*ω*). Then By combining the preceding results, and using the identity

_{b}*ω*terms in the first exponential produce delays (advances) in the time domain, whereas the

_{j}*x*) ≈ exp(−

*cx*

^{2}/2), where

*c*= 0.3858, and omitting dispersion (setting

*β*

_{2}= 0), which is negligible for the parameters of our simulations, one finds that where the walk-off parameter

*β*=

*c*

^{1/2}

*β*

_{1}

*L*/2.

33. F. G. Mehler, “Über die entwicklung einer funktion von beliebig vielen variablen nach Laplaceschen functionen höherer ordnung,” J. Reine Angew. Math. **66**, 161–176 (1866). [CrossRef]

*ψ*were defined in (27). By comparing (48) and (49), one finds that

_{n}*μ*= (

*σ*

*–*

*β*)/(

*σ*+

*β*),

*x*=

*t*and

_{g}ω_{g}*y*=

*t*, where

_{b}ω_{b}*t*= (2

_{g}*βσ*)

^{1/2}=

*t*. Hence, kernel (48) has the singular value (Schmidt) decomposition where the Schmidt coefficients

_{b}*λ*=

_{n}*γp*

_{0}

*L*[(

*σ*/

*β*)(1 –

*μ*

^{2})]

^{1/2}|

*μ*|

*and the (normalized) Schmidt modes*

^{n}*μ*= 0 (

*σ*=

*β*), in which case

*λ*

_{0}=

*γp*

_{0}

*L*and

*t*= 2

_{j}^{1/2}

*σ*=

*τ*. In this case, only one mode will be partially translated; the others will be unaffected, creating a mode-selective filter, similar to that proposed in [15

**19**, 13770–13778 (2010) [CrossRef]

*H*(

_{gb}*ω*,

_{g}*ω*) be the Green function that describes the effect on the output transformed amplitude

_{b}*B*(

_{g}*L*,

*ω*) of the input transformed amplitude

_{g}*B*(0,

_{b}*ω*). Then (43) implies that

_{b}*H*(

_{gb}*ω*,

_{g}*ω*) =

_{b}*iK*(

*ω*,

_{g}*ω*). The associated (forward) Green function for the original amplitudes,

_{b}*A*(

_{g}*L*,

*ω*) and

_{g}*A*(0,

_{b}*ω*), is

_{b}*J*(

_{gb}*ω*,

_{g}*ω*) =

_{b}*H*(

_{gb}*ω*,

_{g}*ω*) exp(

_{b}*iβ*

_{1}

*Lω*). Hence, from which it follows that the output (green) and input (blue) Schmidt mode functions are given by respectively. Likewise, (44) implies that the transformed Green function

_{g}*H*(

_{bg}*ω*,

_{b}*ω*) =

_{g}*iK*

^{*}(

*ω*,

_{g}*ω*), and the original Green function

_{b}*J*(

_{bg}*ω*,

_{b}*ω*) =

_{g}*H*(

_{bg}*ω*,

_{b}*ω*) exp(−

_{g}*iβ*

_{1}

*Lω*). Hence, from which it follows that the output (blue) and input (green) Schmidt mode functions are given by respectively. To obtain these, we needed to account for the distinction between forward and backward Green functions, and note that here we are propagating

_{b}*A*, which corresponds to the quantum operator

*a*, not

*a*

^{†}. If

*β*

_{1}is positive, the green mode is slower than the blue mode. The preceding results show that the green input is advanced and the blue input is delayed, so their collision is centered on the midpoint of the fiber. This location maximizes the distance over which the signals interact with the peaks of the pump pulses. In our simulations

*β*

_{1}is negative, so the green input is delayed and blue input is advanced.

## Acknowledgments

## References and links

1. | A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency upconversion,” J. Mod. Opt. |

2. | S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature |

3. | C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express |

4. | A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express |

5. | D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express |

6. | H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” Photon. Technol. Lett. |

7. | J. M. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. |

8. | H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. |

9. | M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. |

10. | M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. |

11. | E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature |

12. | P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. |

13. | N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. |

14. | J. L. O’Brien, “Quantum computing over the rainbow,” Physics |

15. | A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express |

16. | J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. |

17. | M. E. Marhic, “Coherent optical CDMA networks,” J. Lightwave Technol. |

18. | B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper-engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. |

19. | D. Kielpinski, A.F. Corney, and H.M. Wiseman, “Quantum optical waveform conversion” Phys. Rev Lett. |

20. | H. J. McGuinness, “The creation and frequency translation of single-photon states of light in optical fiber,” Ph.D. thesis, University of Oregon, Eugene, Oregon (2011). |

21. | K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. |

22. | R. W. Boyd, |

23. | C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express |

24. | A. Ekert and P. L. Knight, “Entangled quantum systems and the schmidt decomposition,” Am. J. Phys. |

25. | P. D. Drummond and C. W. Gardiner, “Generalized P-representations in quantum optics,” J. Phys. A |

26. | C. W. Gardiner, |

27. | W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A , |

28. | W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A , |

29. | G. P. Agrawal, |

30. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

31. | P. S. J. Russell, “Photonic crystal fibers,” Science |

32. | G. K. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express |

33. | F. G. Mehler, “Über die entwicklung einer funktion von beliebig vielen variablen nach Laplaceschen functionen höherer ordnung,” J. Reine Angew. Math. |

34. | P. M. Morse and H. Feschbach, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: July 13, 2011

Revised Manuscript: July 14, 2011

Manuscript Accepted: July 14, 2011

Published: August 29, 2011

**Citation**

H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, "Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color," Opt. Express **19**, 17876-17907 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-17876

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### References

- A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency upconversion,” J. Mod. Opt. 51, 1433–1445 (2004).
- S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005). [CrossRef] [PubMed]
- C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]
- A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006). [CrossRef] [PubMed]
- D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006). [CrossRef] [PubMed]
- H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” Photon. Technol. Lett. 23, 109–111 (2011). [CrossRef]
- J. M. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68, 2153–2156 (1992). [CrossRef] [PubMed]
- H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010). [CrossRef] [PubMed]
- M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010). [CrossRef]
- M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010). [CrossRef]
- E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]
- P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]
- N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008). [CrossRef] [PubMed]
- J. L. O’Brien, “Quantum computing over the rainbow,” Physics 1, 23 (2008). [CrossRef]
- A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2010) [CrossRef]
- J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. 8, 478–491 (1990). [CrossRef]
- M. E. Marhic, “Coherent optical CDMA networks,” J. Lightwave Technol. 11, 854–864 (1993). [CrossRef]
- B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper-engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011). [CrossRef]
- D. Kielpinski, A.F. Corney, and H.M. Wiseman, “Quantum optical waveform conversion” Phys. Rev Lett. 106, 130501 (2011). [CrossRef] [PubMed]
- H. J. McGuinness, “The creation and frequency translation of single-photon states of light in optical fiber,” Ph.D. thesis, University of Oregon, Eugene, Oregon (2011).
- K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994). [CrossRef]
- R. W. Boyd, Nonlinear Optics , 3rd ed. (Elsevier Science, 2008).
- C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004). [CrossRef] [PubMed]
- A. Ekert and P. L. Knight, “Entangled quantum systems and the schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995). [CrossRef]
- P. D. Drummond and C. W. Gardiner, “Generalized P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980). [CrossRef]
- C. W. Gardiner, Quantum Noise (Springer, 1992).
- W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A , 73, 063816 (2006). [CrossRef]
- W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A , 73, 063816 (2006). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics , 4th ed. (Elsevier Science, 2008).
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran , 2nd ed. (Cambridge University Press, 1992).
- P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]
- G. K. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express 13, 8662–8670 (2005). [CrossRef] [PubMed]
- F. G. Mehler, “Über die entwicklung einer funktion von beliebig vielen variablen nach Laplaceschen functionen höherer ordnung,” J. Reine Angew. Math. 66, 161–176 (1866). [CrossRef]
- P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

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