Quantum entanglement in coupled lossy waveguides
Optics Express, Vol. 18, Issue 6, pp. 6241-6254 (2010)
http://dx.doi.org/10.1364/OE.18.006241
Acrobat PDF (215 KB)
Abstract
We investigate the viability of coupled waveguides as basic units of quantum circuits. We study entanglement when the waveguides are fed in by light produced by a down-converter working either in low gain limit or under large gain. We present explicit analytical results for the measure of entanglement in terms of the logarithmic negativity for a variety of input states. We also address the effect of loss on entanglement dynamics of waveguide modes. Our results indicate that the waveguide structures are reasonably robust against the effect of loss and thus quite appropriate for quantum architectures as well as for the study of coherent phenomena like random walks. Our analysis is based on realistic structures used currently.
© 2010 Optical Society of America
1. Introduction
1. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999). [CrossRef]
2. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature (London) 424, 817–823 (2003). [CrossRef]
3. S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A 79, 023811 (2009). [CrossRef]
4. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices,” Phys. Rev. Lett. 100, 170506 (2008). [CrossRef] [PubMed]
5. Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102, 253904 (2009). [CrossRef] [PubMed]
6. U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23, 1701–1703 (1998). [CrossRef]
7. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef]
8. R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot Effect in Waveguide Arrays,” Phys. Rev. Lett. 95, 053902 (2005). [CrossRef] [PubMed]
9. S. Longhi, “Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light,” Phys. Rev. Lett. 101, 193902 (2008). [CrossRef] [PubMed]
10. A. Rai, G. S. Agarwal, and J. H. H. Perk, “Transport and quantum walk of nonclassical light in coupled waveguides,” Phys. Rev. A 78, 042304 (2008). [CrossRef]
3. S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A 79, 023811 (2009). [CrossRef]
4. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices,” Phys. Rev. Lett. 100, 170506 (2008). [CrossRef] [PubMed]
5. Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102, 253904 (2009). [CrossRef] [PubMed]
10. A. Rai, G. S. Agarwal, and J. H. H. Perk, “Transport and quantum walk of nonclassical light in coupled waveguides,” Phys. Rev. A 78, 042304 (2008). [CrossRef]
4. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices,” Phys. Rev. Lett. 100, 170506 (2008). [CrossRef] [PubMed]
5. Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102, 253904 (2009). [CrossRef] [PubMed]
8. R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot Effect in Waveguide Arrays,” Phys. Rev. Lett. 95, 053902 (2005). [CrossRef] [PubMed]
9. S. Longhi, “Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light,” Phys. Rev. Lett. 101, 193902 (2008). [CrossRef] [PubMed]
11. A. Rai and G. S. Agarwal, “Possibility of coherent phenomena such as Bloch oscillations with single photons via W states,” Phys. Rev. A 79, 053849 (2009). [CrossRef]
12. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon Waveguide Quantum Circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed]
13. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photonics 3, 346–350 (2009). [CrossRef]
14. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733–2736 (2000). [CrossRef] [PubMed]
15. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404, 247–255 (2000). [CrossRef] [PubMed]
16. W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical ,” Rev. Mod. Phys. 75, 715–775 (2003). [CrossRef]
4. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices,” Phys. Rev. Lett. 100, 170506 (2008). [CrossRef] [PubMed]
17. B. Do, M. L. Stohler, S. Balasubramanian, D. S. Elliott, C. Eash, E. Fischbach, M. A. Fischbach, A. Mills, and B. Zwickl, “Experimental realization of a quantum quincunx by use of linear optical elements,” J. Opt. Soc. Am. B 22, 499–504 (2005). [CrossRef]
18. P. K. Pathak and G. S. Agarwal, “Quantum random walk of two photons in separable and entangled states,” Phys. Rev. A 75, 032351 (2007). [CrossRef]
19. A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shors Quantum Factoring Algorithm on a Photonic Chip,” Science 325, 1221 (2009). [CrossRef] [PubMed]
12. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon Waveguide Quantum Circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed]
13. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photonics 3, 346–350 (2009). [CrossRef]
19. A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shors Quantum Factoring Algorithm on a Photonic Chip,” Science 325, 1221 (2009). [CrossRef] [PubMed]
2. The model
21. W. K. Lai, V. Bužek, and P. L. Knight, “Nonclassical fields in a linear directional coupler ,” Phys. Rev. A 43, 6323–6336 (1991). [CrossRef] [PubMed]
3. Evolution of entanglement for input states at single photon level
25. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002). [CrossRef]
3.1. Separable photon number state as an input
5. Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102, 253904 (2009). [CrossRef] [PubMed]
12. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon Waveguide Quantum Circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed]
26. 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]
5. Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102, 253904 (2009). [CrossRef] [PubMed]
3.2. NOON state as an input
14. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733–2736 (2000). [CrossRef] [PubMed]
5. Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102, 253904 (2009). [CrossRef] [PubMed]
4. Evolution of entanglement for squeezed input states
4.1. Separable two mode squeezed state as an input
30. R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems,”Phys. Rev. Lett. 84, 2726–2729 (2000). [CrossRef] [PubMed]
25. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002). [CrossRef]
31. G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004). [CrossRef]
31. G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004). [CrossRef]
4.2. Entangled two mode squeezed state as an input
5. Lossy waveguides
5.1. Effect of loss on entanglement for photon number states
22. R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of Discrete Quadratic Solitons ,” Phys. Rev. Lett. 93, 113902 (2004). [CrossRef] [PubMed]
23. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002). [CrossRef]
22. R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of Discrete Quadratic Solitons ,” Phys. Rev. Lett. 93, 113902 (2004). [CrossRef] [PubMed]
23. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002). [CrossRef]
13. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photonics 3, 346–350 (2009). [CrossRef]
12. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon Waveguide Quantum Circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed]
5.2. Effect of loss on entanglement for squeezed input states
33. Sumanta Das and G. S. Agarwal, “Bright and dark periods in the entanglement dynamics of interacting qubits in contact with the environment,” J. Phys. B 42, 141003 (2009). [CrossRef]
34. Sumanta Das and G. S. Agarwal, “Decoherence effects in interacting qubits under the influence of various environments,” J. Phys. B 42, 205502 (2009). [CrossRef]
3. S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A 79, 023811 (2009). [CrossRef]
35. S. Longhi, “Transfer of light waves in optical waveguides via a continuum,” Phys. Rev. A 78, 013815 (2008). [CrossRef]
6. Conclusion
References and links
1. | R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999). [CrossRef] |
2. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature (London) 424, 817–823 (2003). [CrossRef] |
3. | S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A 79, 023811 (2009). [CrossRef] |
4. | H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices,” Phys. Rev. Lett. 100, 170506 (2008). [CrossRef] [PubMed] |
5. | Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102, 253904 (2009). [CrossRef] [PubMed] |
6. | U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23, 1701–1703 (1998). [CrossRef] |
7. | T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef] |
8. | R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot Effect in Waveguide Arrays,” Phys. Rev. Lett. 95, 053902 (2005). [CrossRef] [PubMed] |
9. | S. Longhi, “Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light,” Phys. Rev. Lett. 101, 193902 (2008). [CrossRef] [PubMed] |
10. | A. Rai, G. S. Agarwal, and J. H. H. Perk, “Transport and quantum walk of nonclassical light in coupled waveguides,” Phys. Rev. A 78, 042304 (2008). [CrossRef] |
11. | A. Rai and G. S. Agarwal, “Possibility of coherent phenomena such as Bloch oscillations with single photons via W states,” Phys. Rev. A 79, 053849 (2009). [CrossRef] |
12. | A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon Waveguide Quantum Circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed] |
13. | J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photonics 3, 346–350 (2009). [CrossRef] |
14. | A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733–2736 (2000). [CrossRef] [PubMed] |
15. | C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404, 247–255 (2000). [CrossRef] [PubMed] |
16. | W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical ,” Rev. Mod. Phys. 75, 715–775 (2003). [CrossRef] |
17. | B. Do, M. L. Stohler, S. Balasubramanian, D. S. Elliott, C. Eash, E. Fischbach, M. A. Fischbach, A. Mills, and B. Zwickl, “Experimental realization of a quantum quincunx by use of linear optical elements,” J. Opt. Soc. Am. B 22, 499–504 (2005). [CrossRef] |
18. | P. K. Pathak and G. S. Agarwal, “Quantum random walk of two photons in separable and entangled states,” Phys. Rev. A 75, 032351 (2007). [CrossRef] |
19. | A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shors Quantum Factoring Algorithm on a Photonic Chip,” Science 325, 1221 (2009). [CrossRef] [PubMed] |
20. | B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Edition, (Wiley, New York2007), p. 319. |
21. | W. K. Lai, V. Bužek, and P. L. Knight, “Nonclassical fields in a linear directional coupler ,” Phys. Rev. A 43, 6323–6336 (1991). [CrossRef] [PubMed] |
22. | R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of Discrete Quadratic Solitons ,” Phys. Rev. Lett. 93, 113902 (2004). [CrossRef] [PubMed] |
23. | U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002). [CrossRef] |
24. | K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, “Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices,” Elec-trophoresis 25, 3788–3795 (2004). [CrossRef] |
25. | G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002). [CrossRef] |
26. | 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] |
27. | G. S. Agarwal, “Entropy, the Wigner Distribution Function, and the Approach to Equilibrium of a System of Coupled Harmonic Oscillators ,” Phys. Rev. A 3, 828–831 (1971). [CrossRef] |
28. | P. J. Dodd and J. J. Halliwell, “Disentanglement and decoherence by open system dynamics ,” Phys. Rev. A 69, 052105 (2004). [CrossRef] |
29. | L. -M . Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability Criterion for Continuous Variable Systems,” Phys. Rev. Lett. 84, 2722–2725 (2000). [CrossRef] [PubMed] |
30. | R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems,”Phys. Rev. Lett. 84, 2726–2729 (2000). [CrossRef] [PubMed] |
31. | G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004). [CrossRef] |
32. | S.M. Barnett and P. Radmore, Methods in Theoretical Quantum Optics, (Oxford University Press, 2002), p. 168. |
33. | Sumanta Das and G. S. Agarwal, “Bright and dark periods in the entanglement dynamics of interacting qubits in contact with the environment,” J. Phys. B 42, 141003 (2009). [CrossRef] |
34. | Sumanta Das and G. S. Agarwal, “Decoherence effects in interacting qubits under the influence of various environments,” J. Phys. B 42, 205502 (2009). [CrossRef] |
35. | S. Longhi, “Transfer of light waves in optical waveguides via a continuum,” Phys. Rev. A 78, 013815 (2008). [CrossRef] |
OCIS Codes
(230.7370) Optical devices : Waveguides
(250.5300) Optoelectronics : Photonic integrated circuits
(270.5585) Quantum optics : Quantum information and processing
ToC Category:
Quantum Optics
History
Original Manuscript: December 21, 2009
Manuscript Accepted: February 18, 2010
Published: March 12, 2010
Citation
Amit Rai, Sumanta Das, and Girish Agarwal, "Quantum entanglement in coupled lossy waveguides," Opt. Express 18, 6241-6254 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6241
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References
- R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999). [CrossRef]
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003). [CrossRef]
- S. Longhi, "Optical analog of population trapping in the continuum: Classical and quantum interference effects," Phys. Rev. A 79, 023811 (2009). [CrossRef]
- H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008). [CrossRef] [PubMed]
- Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, "Quantum and Classical Correlations in Waveguide Lattices," Phys. Rev. Lett. 102, 253904 (2009). [CrossRef] [PubMed]
- U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998). [CrossRef]
- T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999). [CrossRef]
- R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005). [CrossRef] [PubMed]
- S. Longhi, "Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light," Phys. Rev. Lett. 101, 193902 (2008). [CrossRef] [PubMed]
- A. Rai, G. S. Agarwal, and J. H. H. Perk, "Transport and quantum walk of nonclassical light in coupled waveguides," Phys. Rev. A 78, 042304 (2008). [CrossRef]
- A. Rai and G. S. Agarwal, "Possibility of coherent phenomena such as Bloch oscillations with single photons via W states," Phys. Rev. A 79, 053849 (2009). [CrossRef]
- A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008). [CrossRef] [PubMed]
- J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits," Nature Photonics 3, 346-350 (2009). [CrossRef]
- A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000). [CrossRef] [PubMed]
- C. H. Bennett and D. P. DiVincenzo, "Quantum information and computation," Nature 404, 247-255 (2000). [CrossRef] [PubMed]
- W. H. Zurek, "Decoherence, einselection, and the quantum origins of the classical, " Rev. Mod. Phys. 75, 715-775 (2003). [CrossRef]
- B. Do, M. L. Stohler, S. Balasubramanian, D. S. Elliott, C. Eash, E. Fischbach, M. A. Fischbach, A. Mills, and B. Zwickl, "Experimental realization of a quantum quincunx by use of linear optical elements," J. Opt. Soc. Am. B 22, 499-504 (2005). [CrossRef]
- P. K. Pathak and G. S. Agarwal, "Quantum random walk of two photons in separable and entangled states," Phys. Rev. A 75, 032351 (2007). [CrossRef]
- A. Politi, J. C. F. Matthews, and J. L. O’Brien, "Shors Quantum Factoring Algorithm on a Photonic Chip," Science 325, 1221 (2009). [CrossRef] [PubMed]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Edition, (Wiley, New York 2007), p. 319.
- W. K. Lai, V. Buˇzek, and P. L. Knight, "Nonclassical fields in a linear directional coupler, " Phys. Rev. A 43, 6323-6336 (1991). [CrossRef] [PubMed]
- R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004). [CrossRef] [PubMed]
- U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties," J. Opt. Soc. Am. B 19, 2637- 2644 (2002). [CrossRef]
- K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004). [CrossRef]
- G. Vidal and R. F. Werner, "Computable measure of entanglement," Phys. Rev. A 65, 032314 (2002). [CrossRef]
- 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]
- G. S. Agarwal, "Entropy, the Wigner Distribution Function, and the Approach to Equilibrium of a System of Coupled Harmonic Oscillators, " Phys. Rev. A 3, 828-831 (1971). [CrossRef]
- P. J. Dodd and J. J. Halliwell, "Disentanglement and decoherence by open system dynamics, " Phys. Rev. A 69, 052105 (2004). [CrossRef]
- L. -M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2722-2725 (2000). [CrossRef] [PubMed]
- R. Simon, "Peres-Horodecki Separability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2726- 2729 (2000). [CrossRef] [PubMed]
- G. Adesso, A. Serafini, and F. Illuminati, "Extremal entanglement and mixedness in continuous variable systems," Phys. Rev. A 70, 022318 (2004). [CrossRef]
- S. M. Barnett and P. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, 2002), p. 168.
- Sumanta Das and G. S. Agarwal, "Bright and dark periods in the entanglement dynamics of interacting qubits in contact with the environment," J. Phys. B 42, 141003 (2009). [CrossRef]
- Sumanta Das and G. S. Agarwal, "Decoherence effects in interacting qubits under the influence of various environments," J. Phys. B 42, 205502 (2009). [CrossRef]
- S. Longhi, "Transfer of light waves in optical waveguides via a continuum," Phys. Rev. A 78, 013815 (2008). [CrossRef]
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