Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime |
Optics Express, Vol. 20, Issue 8, pp. 8367-8396 (2012)
http://dx.doi.org/10.1364/OE.20.008367
Acrobat PDF (1894 KB)
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 OSA
1. Introduction
1. S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London) 437, 116–120 (2005). [CrossRef]
2. H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008). [CrossRef]
3. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). [CrossRef] [PubMed]
4. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001). [CrossRef]
5. I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005). [CrossRef] [PubMed]
6. A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004). [CrossRef]
6. A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004). [CrossRef]
9. R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (2005). [CrossRef] [PubMed]
10. A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006). [CrossRef] [PubMed]
11. K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32, 695–697 (2007). [CrossRef] [PubMed]
1. S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London) 437, 116–120 (2005). [CrossRef]
2. H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008). [CrossRef]
12. 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]
14. A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011). [CrossRef] [PubMed]
15. W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature (London) 299, 802–803 (1982). [CrossRef]
16. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1653 (1961). [CrossRef]
17. J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963). [CrossRef]
18. J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68, 2153–2156 (1992). [CrossRef] [PubMed]
21. R. V. Roussev, C. Langrock, J. R. Kurz, and M. M. Fejer, “Periodically poled lithium niobate waveguide sum-frequency generator for efficient single-photon detection at communication wavelengths,” Opt. Lett. 29, 1518–1520 (2004). [CrossRef] [PubMed]
22. Y. Ding and Z. Y. Ou, “Frequency downconversion for a quantum network,” Opt. Lett. 35, 2591–2593 (2010). [CrossRef] [PubMed]
13. D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106, 130501 (2011). [CrossRef] [PubMed]
23. 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]
13. D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106, 130501 (2011). [CrossRef] [PubMed]
14. A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011). [CrossRef] [PubMed]
24. C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London) 469, 508–511 (2011). [CrossRef]
25. E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London) 469, 512–515 (2011). [CrossRef]
26. 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]
27. 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]
29. K. Uesaka, K. K. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002). [CrossRef]
30. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004). [CrossRef]
31. 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]
32. 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]
33. 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,” IEEE Photon. Technol. Lett. 23, 109–111 (2011). [CrossRef]
34. 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). [PubMed]
26. 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]
35. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004). [CrossRef] [PubMed]
36. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005). [CrossRef] [PubMed]
26. 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]
37. 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]
38. 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). [CrossRef] [PubMed]
2. General formalism of FC
26. 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]
34. 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). [PubMed]
26. 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]
37. 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]
38. 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). [CrossRef] [PubMed]
44. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000). [CrossRef] [PubMed]
37. 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]
38. 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). [CrossRef] [PubMed]
45. C. J McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009). [CrossRef]
2.1. The frequency and time domains
47. M. P. des Chênes, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaire du second ordre, à coefficients constants,” Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assembleés. Sciences, mathématiques et physiques , 638–648 (1806). [PubMed]
3. Stationary pumps
38. 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). [CrossRef] [PubMed]
3.1. Standard analysis
48. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001). [CrossRef]
48. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001). [CrossRef]
48. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001). [CrossRef]
48. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001). [CrossRef]
38. 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). [CrossRef] [PubMed]
3.2. Alternative analysis
3.2.1. Time-domain collision analysis
3.3. Comparing the time-domain and the frequency-domain results
3.4. Numerical studies
4. Convecting pumps
34. 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). [PubMed]
38. 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). [CrossRef] [PubMed]
4.1. Gaussian pumps of equal width
4.1.1. The Gaussian approximation
4.2. Different Gaussian pumps
53. C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012). [CrossRef]
13. D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106, 130501 (2011). [CrossRef] [PubMed]
4.2.1. The Gaussian approximation
4.3. HG0/HG1 pumps
4.4. HG1/HG1 pumps
5. Conclusion
A. Appendix: Mehler identity and kernel decomposition
56. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen,” Mathematische Annalen 63, 433–476 (1907). [CrossRef]
58. C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012). [CrossRef]
Acknowledgments
References and links
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22. | Y. Ding and Z. Y. Ou, “Frequency downconversion for a quantum network,” Opt. Lett. 35, 2591–2593 (2010). [CrossRef] [PubMed] |
23. | 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] |
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25. | E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London) 469, 512–515 (2011). [CrossRef] |
26. | 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] |
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29. | K. Uesaka, K. K. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002). [CrossRef] |
30. | T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004). [CrossRef] |
31. | 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] |
32. | 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] |
33. | 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,” IEEE Photon. Technol. Lett. 23, 109–111 (2011). [CrossRef] |
34. | 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). [PubMed] |
35. | C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004). [CrossRef] [PubMed] |
36. | C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005). [CrossRef] [PubMed] |
37. | 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] |
38. | 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). [CrossRef] [PubMed] |
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45. | C. J McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009). [CrossRef] |
46. | R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1990). |
47. | M. P. des Chênes, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaire du second ordre, à coefficients constants,” Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assembleés. Sciences, mathématiques et physiques , 638–648 (1806). [PubMed] |
48. | W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001). [CrossRef] |
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58. | C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012). [CrossRef] |
OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(270.5585) Quantum optics : Quantum information and processing
ToC Category:
Nonlinear Optics
History
Original Manuscript: February 13, 2012
Revised Manuscript: March 16, 2012
Manuscript Accepted: March 16, 2012
Published: March 26, 2012
Citation
L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, "Quantum frequency translation by four-wave mixing in a fiber:
low-conversion regime," Opt. Express 20, 8367-8396 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8367
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References
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