Abstract

Theoretical analysis is given of an experimental scheme that can perform individual photon operations such as the photon annihilation operation a, creation operation a, and commutation operation aaaa, utilizing atom–cavity field interactions and conditional measurements. In order for the scheme to perform the desired photon operation, the atom–cavity field interaction times are generally required to be sufficiently short that photon annihilation and/or creation are dominated by the one-half Rabi cycle process. Such short interaction times, however, lead inevitably to a low success probability of the scheme. It is shown that this problem of low success probability can be overcome by preparing the cavity field in a superposition of a small number (two) of Fock states and choosing the interaction times appropriately.

© 2010 Optical Society of America

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    [CrossRef]
  2. D. Meschede, “Radiating atoms in confined space: From spontaneous emission to micromasers,” Phys. Rep. 211, 201-250 (1992).
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  3. H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325-1382 (2006).
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  4. R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
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  7. L.-M. Duan, A. Kuzmich, and H. J. Kimble, “Cavity QED and quantum-information processing with “hot” trapped atoms,” Phys. Rev. A 67, 032305 (2003).
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  16. S.-Y. Lee, J. Park, S.-W. Ji, C. H. R. Ooi, and H.-W. Lee, “Nonclassicality generated by photon annihilation-then-creation and creation-then-annihilation operations,” J. Opt. Soc. Am. B 26, 1532-1537 (2009).
    [CrossRef]
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  21. A. S. Sørensen and K. Mølmer, “Measurement induced entanglement and quantum computation with atoms in optical cavities,” Phys. Rev. Lett. 91, 097905 (2003).
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  22. L.-M. Duan and H. J. Kimble, “Efficient engineering of multiatom entanglement through single-photon detections,” Phys. Rev. Lett. 90, 253601 (2003).
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  23. J. Lee, J. Park, S. M. Lee, H.-W. Lee, and A. H. Khosa, “Scalable cavity-QED-based scheme of generating entanglement of atoms and of cavity fields,” Phys. Rev. A 77, 032327 (2008).
    [CrossRef]
  24. L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, “Teleportation of an atomic state between two cavities using nonlocal microwave fields,” Phys. Rev. A 50, R895 (1994).
    [CrossRef] [PubMed]
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  26. S. M. Barnett and D. T. Pegg, “Optical state truncation,” Phys. Rev. A 60, 4965-4973 (1999).
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  27. M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A 59, 1658-1661 (1999).
    [CrossRef]
  28. G. M. D'Ariano, L. Maccone, M. G. A. Paris, and M. F. Sacchi, “Optical Fock-state synthesizer,” Phys. Rev. A 61, 053817 (2000).
    [CrossRef]
  29. M. Koniorczyk, Z. Kurucz, A. Gábris, and J. Janszky, “General optical state truncation and its teleportation,” Phys. Rev. A 62, 013802 (2000).
    [CrossRef]
  30. A. P. Lund and T. C. Ralph, “Nondeterministic gates for photonic single-rail quantum logic,” Phys. Rev. A 66, 032307 (2002).
    [CrossRef]
  31. A. Miranowicz, “Optical-state truncation and teleportation of qudits by conditional eight-port interferometry,” J. Opt. B: Quantum Semiclassical Opt. 7, 142-150 (2005).
    [CrossRef]
  32. A. Miranowicz, S. K. Özdemir, J. Bajer, M. Koashi, and N. Imoto, “Selective truncations of an optical state using projection synthesis,” J. Opt. Soc. Am. B 24, 379-383 (2007).
    [CrossRef]
  33. K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett. 88, 113601 (2002).
    [CrossRef] [PubMed]
  34. A. I. Lvovsky and J. Mlynek, “Quantum-optical catalysis: Generating nonclassical states of light by means of linear optics,” Phys. Rev. Lett. 88, 250401 (2002).
    [CrossRef] [PubMed]
  35. S. A. Babichev, J. Ries, and A. I. Lvovsky, “Quantum scissors: Teleportation of single-mode optical states by means of a nonlocal single photon,” Europhys. Lett. 64, 1 (2003).
    [CrossRef]

2009 (1)

2008 (3)

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing the quantum commutation rules through cavity QED,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

J. Lee, J. Park, S. M. Lee, H.-W. Lee, and A. H. Khosa, “Scalable cavity-QED-based scheme of generating entanglement of atoms and of cavity fields,” Phys. Rev. A 77, 032327 (2008).
[CrossRef]

2007 (3)

A. Miranowicz, S. K. Özdemir, J. Bajer, M. Koashi, and N. Imoto, “Selective truncations of an optical state using projection synthesis,” J. Opt. Soc. Am. B 24, 379-383 (2007).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890-1893 (2007).
[CrossRef] [PubMed]

2006 (1)

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325-1382 (2006).
[CrossRef]

2005 (3)

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Miranowicz, “Optical-state truncation and teleportation of qudits by conditional eight-port interferometry,” J. Opt. B: Quantum Semiclassical Opt. 7, 142-150 (2005).
[CrossRef]

2004 (1)

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660-662 (2004).
[CrossRef] [PubMed]

2003 (5)

L.-M. Duan, A. Kuzmich, and H. J. Kimble, “Cavity QED and quantum-information processing with “hot” trapped atoms,” Phys. Rev. A 67, 032305 (2003).
[CrossRef]

S. A. Babichev, J. Ries, and A. I. Lvovsky, “Quantum scissors: Teleportation of single-mode optical states by means of a nonlocal single photon,” Europhys. Lett. 64, 1 (2003).
[CrossRef]

A. S. Sørensen and K. Mølmer, “Probabilistic generation of entanglement in optical cavities,” Phys. Rev. Lett. 90, 127903 (2003).
[CrossRef] [PubMed]

A. S. Sørensen and K. Mølmer, “Measurement induced entanglement and quantum computation with atoms in optical cavities,” Phys. Rev. Lett. 91, 097905 (2003).
[CrossRef] [PubMed]

L.-M. Duan and H. J. Kimble, “Efficient engineering of multiatom entanglement through single-photon detections,” Phys. Rev. Lett. 90, 253601 (2003).
[CrossRef] [PubMed]

2002 (4)

J. Hong and H.-W. Lee, “Quasideterministic generation of entangled atoms in a cavity,” Phys. Rev. Lett. 89, 237901 (2002).
[CrossRef] [PubMed]

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett. 88, 113601 (2002).
[CrossRef] [PubMed]

A. I. Lvovsky and J. Mlynek, “Quantum-optical catalysis: Generating nonclassical states of light by means of linear optics,” Phys. Rev. Lett. 88, 250401 (2002).
[CrossRef] [PubMed]

A. P. Lund and T. C. Ralph, “Nondeterministic gates for photonic single-rail quantum logic,” Phys. Rev. A 66, 032307 (2002).
[CrossRef]

2001 (1)

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565-582 (2001).
[CrossRef]

2000 (2)

G. M. D'Ariano, L. Maccone, M. G. A. Paris, and M. F. Sacchi, “Optical Fock-state synthesizer,” Phys. Rev. A 61, 053817 (2000).
[CrossRef]

M. Koniorczyk, Z. Kurucz, A. Gábris, and J. Janszky, “General optical state truncation and its teleportation,” Phys. Rev. A 62, 013802 (2000).
[CrossRef]

1999 (3)

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468-2475 (1999).
[CrossRef]

S. M. Barnett and D. T. Pegg, “Optical state truncation,” Phys. Rev. A 60, 4965-4973 (1999).
[CrossRef]

M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A 59, 1658-1661 (1999).
[CrossRef]

1998 (1)

D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical state truncation by projection synthesis,” Phys. Rev. Lett. 81, 1604-1606 (1998).
[CrossRef]

1997 (1)

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

1994 (1)

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, “Teleportation of an atomic state between two cavities using nonlocal microwave fields,” Phys. Rev. A 50, R895 (1994).
[CrossRef] [PubMed]

1992 (2)

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-poissonian statistics,” Phys. Rev. A 46, 485-488 (1992).
[CrossRef] [PubMed]

D. Meschede, “Radiating atoms in confined space: From spontaneous emission to micromasers,” Phys. Rep. 211, 201-250 (1992).
[CrossRef]

1991 (1)

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492-497 (1991).
[CrossRef] [PubMed]

1989 (1)

S. Haroche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24-30 (1989).
[CrossRef]

1963 (1)

E. Jaynes and F. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89-109 (1963).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-poissonian statistics,” Phys. Rev. A 46, 485-488 (1992).
[CrossRef] [PubMed]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492-497 (1991).
[CrossRef] [PubMed]

Al-Amri, M.

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing the quantum commutation rules through cavity QED,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

Babichev, S. A.

S. A. Babichev, J. Ries, and A. I. Lvovsky, “Quantum scissors: Teleportation of single-mode optical states by means of a nonlocal single photon,” Europhys. Lett. 64, 1 (2003).
[CrossRef]

Bajer, J.

Barnett, S. M.

S. M. Barnett and D. T. Pegg, “Optical state truncation,” Phys. Rev. A 60, 4965-4973 (1999).
[CrossRef]

D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical state truncation by projection synthesis,” Phys. Rev. Lett. 81, 1604-1606 (1998).
[CrossRef]

Becker, T.

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325-1382 (2006).
[CrossRef]

Beige, A.

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468-2475 (1999).
[CrossRef]

Bellini, M.

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890-1893 (2007).
[CrossRef] [PubMed]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660-662 (2004).
[CrossRef] [PubMed]

Birnbaum, K. M.

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
[CrossRef]

Boca, A.

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
[CrossRef]

Boozer, A. D.

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
[CrossRef]

Brune, M.

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565-582 (2001).
[CrossRef]

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, “Teleportation of an atomic state between two cavities using nonlocal microwave fields,” Phys. Rev. A 50, R895 (1994).
[CrossRef] [PubMed]

Clausen, J.

M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A 59, 1658-1661 (1999).
[CrossRef]

Cummings, F.

E. Jaynes and F. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89-109 (1963).
[CrossRef]

Dakna, M.

M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A 59, 1658-1661 (1999).
[CrossRef]

D'Ariano, G. M.

G. M. D'Ariano, L. Maccone, M. G. A. Paris, and M. F. Sacchi, “Optical Fock-state synthesizer,” Phys. Rev. A 61, 053817 (2000).
[CrossRef]

Davidovich, L.

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, “Teleportation of an atomic state between two cavities using nonlocal microwave fields,” Phys. Rev. A 50, R895 (1994).
[CrossRef] [PubMed]

Duan, L.-M.

L.-M. Duan and H. J. Kimble, “Efficient engineering of multiatom entanglement through single-photon detections,” Phys. Rev. Lett. 90, 253601 (2003).
[CrossRef] [PubMed]

L.-M. Duan, A. Kuzmich, and H. J. Kimble, “Cavity QED and quantum-information processing with “hot” trapped atoms,” Phys. Rev. A 67, 032305 (2003).
[CrossRef]

Englert, B.-G.

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325-1382 (2006).
[CrossRef]

Gábris, A.

M. Koniorczyk, Z. Kurucz, A. Gábris, and J. Janszky, “General optical state truncation and its teleportation,” Phys. Rev. A 62, 013802 (2000).
[CrossRef]

Hagley, E.

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Haroche, S.

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565-582 (2001).
[CrossRef]

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, “Teleportation of an atomic state between two cavities using nonlocal microwave fields,” Phys. Rev. A 50, R895 (1994).
[CrossRef] [PubMed]

S. Haroche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24-30 (1989).
[CrossRef]

Hong, J.

J. Hong and H.-W. Lee, “Quasideterministic generation of entangled atoms in a cavity,” Phys. Rev. Lett. 89, 237901 (2002).
[CrossRef] [PubMed]

Huelga, S. F.

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468-2475 (1999).
[CrossRef]

Imoto, N.

Janszky, J.

M. Koniorczyk, Z. Kurucz, A. Gábris, and J. Janszky, “General optical state truncation and its teleportation,” Phys. Rev. A 62, 013802 (2000).
[CrossRef]

Jaynes, E.

E. Jaynes and F. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89-109 (1963).
[CrossRef]

Jeong, H.

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

Ji, S.-W.

Khosa, A. H.

J. Lee, J. Park, S. M. Lee, H.-W. Lee, and A. H. Khosa, “Scalable cavity-QED-based scheme of generating entanglement of atoms and of cavity fields,” Phys. Rev. A 77, 032327 (2008).
[CrossRef]

Kim, M. S.

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890-1893 (2007).
[CrossRef] [PubMed]

Kimble, H. J.

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
[CrossRef]

L.-M. Duan, A. Kuzmich, and H. J. Kimble, “Cavity QED and quantum-information processing with “hot” trapped atoms,” Phys. Rev. A 67, 032305 (2003).
[CrossRef]

L.-M. Duan and H. J. Kimble, “Efficient engineering of multiatom entanglement through single-photon detections,” Phys. Rev. Lett. 90, 253601 (2003).
[CrossRef] [PubMed]

Kleppner, D.

S. Haroche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24-30 (1989).
[CrossRef]

Knight, P. L.

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468-2475 (1999).
[CrossRef]

Knöll, L.

M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A 59, 1658-1661 (1999).
[CrossRef]

Koashi, M.

Koniorczyk, M.

M. Koniorczyk, Z. Kurucz, A. Gábris, and J. Janszky, “General optical state truncation and its teleportation,” Phys. Rev. A 62, 013802 (2000).
[CrossRef]

Kurucz, Z.

M. Koniorczyk, Z. Kurucz, A. Gábris, and J. Janszky, “General optical state truncation and its teleportation,” Phys. Rev. A 62, 013802 (2000).
[CrossRef]

Kuzmich, A.

L.-M. Duan, A. Kuzmich, and H. J. Kimble, “Cavity QED and quantum-information processing with “hot” trapped atoms,” Phys. Rev. A 67, 032305 (2003).
[CrossRef]

Lee, H.-W.

S.-Y. Lee, J. Park, S.-W. Ji, C. H. R. Ooi, and H.-W. Lee, “Nonclassicality generated by photon annihilation-then-creation and creation-then-annihilation operations,” J. Opt. Soc. Am. B 26, 1532-1537 (2009).
[CrossRef]

J. Lee, J. Park, S. M. Lee, H.-W. Lee, and A. H. Khosa, “Scalable cavity-QED-based scheme of generating entanglement of atoms and of cavity fields,” Phys. Rev. A 77, 032327 (2008).
[CrossRef]

J. Hong and H.-W. Lee, “Quasideterministic generation of entangled atoms in a cavity,” Phys. Rev. Lett. 89, 237901 (2002).
[CrossRef] [PubMed]

Lee, J.

J. Lee, J. Park, S. M. Lee, H.-W. Lee, and A. H. Khosa, “Scalable cavity-QED-based scheme of generating entanglement of atoms and of cavity fields,” Phys. Rev. A 77, 032327 (2008).
[CrossRef]

Lee, S. M.

J. Lee, J. Park, S. M. Lee, H.-W. Lee, and A. H. Khosa, “Scalable cavity-QED-based scheme of generating entanglement of atoms and of cavity fields,” Phys. Rev. A 77, 032327 (2008).
[CrossRef]

Lee, S.-Y.

Lund, A. P.

A. P. Lund and T. C. Ralph, “Nondeterministic gates for photonic single-rail quantum logic,” Phys. Rev. A 66, 032307 (2002).
[CrossRef]

Lundeen, J. S.

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett. 88, 113601 (2002).
[CrossRef] [PubMed]

Lvovsky, A. I.

S. A. Babichev, J. Ries, and A. I. Lvovsky, “Quantum scissors: Teleportation of single-mode optical states by means of a nonlocal single photon,” Europhys. Lett. 64, 1 (2003).
[CrossRef]

A. I. Lvovsky and J. Mlynek, “Quantum-optical catalysis: Generating nonclassical states of light by means of linear optics,” Phys. Rev. Lett. 88, 250401 (2002).
[CrossRef] [PubMed]

Maccone, L.

G. M. D'Ariano, L. Maccone, M. G. A. Paris, and M. F. Sacchi, “Optical Fock-state synthesizer,” Phys. Rev. A 61, 053817 (2000).
[CrossRef]

Maître, X.

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Meschede, D.

D. Meschede, “Radiating atoms in confined space: From spontaneous emission to micromasers,” Phys. Rep. 211, 201-250 (1992).
[CrossRef]

Miller, R.

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
[CrossRef]

Miranowicz, A.

A. Miranowicz, S. K. Özdemir, J. Bajer, M. Koashi, and N. Imoto, “Selective truncations of an optical state using projection synthesis,” J. Opt. Soc. Am. B 24, 379-383 (2007).
[CrossRef]

A. Miranowicz, “Optical-state truncation and teleportation of qudits by conditional eight-port interferometry,” J. Opt. B: Quantum Semiclassical Opt. 7, 142-150 (2005).
[CrossRef]

Mlynek, J.

A. I. Lvovsky and J. Mlynek, “Quantum-optical catalysis: Generating nonclassical states of light by means of linear optics,” Phys. Rev. Lett. 88, 250401 (2002).
[CrossRef] [PubMed]

Mølmer, K.

A. S. Sørensen and K. Mølmer, “Measurement induced entanglement and quantum computation with atoms in optical cavities,” Phys. Rev. Lett. 91, 097905 (2003).
[CrossRef] [PubMed]

A. S. Sørensen and K. Mølmer, “Probabilistic generation of entanglement in optical cavities,” Phys. Rev. Lett. 90, 127903 (2003).
[CrossRef] [PubMed]

Nogues, G.

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Northup, T. E.

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
[CrossRef]

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Özdemir, S. K.

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M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890-1893 (2007).
[CrossRef] [PubMed]

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G. M. D'Ariano, L. Maccone, M. G. A. Paris, and M. F. Sacchi, “Optical Fock-state synthesizer,” Phys. Rev. A 61, 053817 (2000).
[CrossRef]

Park, J.

S.-Y. Lee, J. Park, S.-W. Ji, C. H. R. Ooi, and H.-W. Lee, “Nonclassicality generated by photon annihilation-then-creation and creation-then-annihilation operations,” J. Opt. Soc. Am. B 26, 1532-1537 (2009).
[CrossRef]

J. Lee, J. Park, S. M. Lee, H.-W. Lee, and A. H. Khosa, “Scalable cavity-QED-based scheme of generating entanglement of atoms and of cavity fields,” Phys. Rev. A 77, 032327 (2008).
[CrossRef]

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S. M. Barnett and D. T. Pegg, “Optical state truncation,” Phys. Rev. A 60, 4965-4973 (1999).
[CrossRef]

D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical state truncation by projection synthesis,” Phys. Rev. Lett. 81, 1604-1606 (1998).
[CrossRef]

Phillips, L. S.

D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical state truncation by projection synthesis,” Phys. Rev. Lett. 81, 1604-1606 (1998).
[CrossRef]

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M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468-2475 (1999).
[CrossRef]

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J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565-582 (2001).
[CrossRef]

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, “Teleportation of an atomic state between two cavities using nonlocal microwave fields,” Phys. Rev. A 50, R895 (1994).
[CrossRef] [PubMed]

Ralph, T. C.

A. P. Lund and T. C. Ralph, “Nondeterministic gates for photonic single-rail quantum logic,” Phys. Rev. A 66, 032307 (2002).
[CrossRef]

Resch, K. J.

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett. 88, 113601 (2002).
[CrossRef] [PubMed]

Ries, J.

S. A. Babichev, J. Ries, and A. I. Lvovsky, “Quantum scissors: Teleportation of single-mode optical states by means of a nonlocal single photon,” Europhys. Lett. 64, 1 (2003).
[CrossRef]

Sacchi, M. F.

G. M. D'Ariano, L. Maccone, M. G. A. Paris, and M. F. Sacchi, “Optical Fock-state synthesizer,” Phys. Rev. A 61, 053817 (2000).
[CrossRef]

Sørensen, A. S.

A. S. Sørensen and K. Mølmer, “Measurement induced entanglement and quantum computation with atoms in optical cavities,” Phys. Rev. Lett. 91, 097905 (2003).
[CrossRef] [PubMed]

A. S. Sørensen and K. Mølmer, “Probabilistic generation of entanglement in optical cavities,” Phys. Rev. Lett. 90, 127903 (2003).
[CrossRef] [PubMed]

Steinberg, A. M.

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett. 88, 113601 (2002).
[CrossRef] [PubMed]

Sun, Q.

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing the quantum commutation rules through cavity QED,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

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G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-poissonian statistics,” Phys. Rev. A 46, 485-488 (1992).
[CrossRef] [PubMed]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492-497 (1991).
[CrossRef] [PubMed]

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H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325-1382 (2006).
[CrossRef]

Viciani, S.

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660-662 (2004).
[CrossRef] [PubMed]

Walther, H.

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325-1382 (2006).
[CrossRef]

Welsch, D.-G.

M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A 59, 1658-1661 (1999).
[CrossRef]

Wunderlich, C.

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Zagury, N.

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, “Teleportation of an atomic state between two cavities using nonlocal microwave fields,” Phys. Rev. A 50, R895 (1994).
[CrossRef] [PubMed]

Zavatta, A.

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890-1893 (2007).
[CrossRef] [PubMed]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660-662 (2004).
[CrossRef] [PubMed]

Zubairy, M. S.

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing the quantum commutation rules through cavity QED,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

Europhys. Lett. (1)

S. A. Babichev, J. Ries, and A. I. Lvovsky, “Quantum scissors: Teleportation of single-mode optical states by means of a nonlocal single photon,” Europhys. Lett. 64, 1 (2003).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

A. Miranowicz, “Optical-state truncation and teleportation of qudits by conditional eight-port interferometry,” J. Opt. B: Quantum Semiclassical Opt. 7, 142-150 (2005).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. B (1)

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B 38, S551-S565 (2005).
[CrossRef]

Phys. Rep. (1)

D. Meschede, “Radiating atoms in confined space: From spontaneous emission to micromasers,” Phys. Rep. 211, 201-250 (1992).
[CrossRef]

Phys. Rev. A (14)

L.-M. Duan, A. Kuzmich, and H. J. Kimble, “Cavity QED and quantum-information processing with “hot” trapped atoms,” Phys. Rev. A 67, 032305 (2003).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492-497 (1991).
[CrossRef] [PubMed]

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-poissonian statistics,” Phys. Rev. A 46, 485-488 (1992).
[CrossRef] [PubMed]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468-2475 (1999).
[CrossRef]

J. Lee, J. Park, S. M. Lee, H.-W. Lee, and A. H. Khosa, “Scalable cavity-QED-based scheme of generating entanglement of atoms and of cavity fields,” Phys. Rev. A 77, 032327 (2008).
[CrossRef]

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, “Teleportation of an atomic state between two cavities using nonlocal microwave fields,” Phys. Rev. A 50, R895 (1994).
[CrossRef] [PubMed]

S. M. Barnett and D. T. Pegg, “Optical state truncation,” Phys. Rev. A 60, 4965-4973 (1999).
[CrossRef]

M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A 59, 1658-1661 (1999).
[CrossRef]

G. M. D'Ariano, L. Maccone, M. G. A. Paris, and M. F. Sacchi, “Optical Fock-state synthesizer,” Phys. Rev. A 61, 053817 (2000).
[CrossRef]

M. Koniorczyk, Z. Kurucz, A. Gábris, and J. Janszky, “General optical state truncation and its teleportation,” Phys. Rev. A 62, 013802 (2000).
[CrossRef]

A. P. Lund and T. C. Ralph, “Nondeterministic gates for photonic single-rail quantum logic,” Phys. Rev. A 66, 032307 (2002).
[CrossRef]

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing the quantum commutation rules through cavity QED,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

Phys. Rev. Lett. (9)

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett. 88, 113601 (2002).
[CrossRef] [PubMed]

A. I. Lvovsky and J. Mlynek, “Quantum-optical catalysis: Generating nonclassical states of light by means of linear optics,” Phys. Rev. Lett. 88, 250401 (2002).
[CrossRef] [PubMed]

D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical state truncation by projection synthesis,” Phys. Rev. Lett. 81, 1604-1606 (1998).
[CrossRef]

E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

J. Hong and H.-W. Lee, “Quasideterministic generation of entangled atoms in a cavity,” Phys. Rev. Lett. 89, 237901 (2002).
[CrossRef] [PubMed]

A. S. Sørensen and K. Mølmer, “Probabilistic generation of entanglement in optical cavities,” Phys. Rev. Lett. 90, 127903 (2003).
[CrossRef] [PubMed]

A. S. Sørensen and K. Mølmer, “Measurement induced entanglement and quantum computation with atoms in optical cavities,” Phys. Rev. Lett. 91, 097905 (2003).
[CrossRef] [PubMed]

L.-M. Duan and H. J. Kimble, “Efficient engineering of multiatom entanglement through single-photon detections,” Phys. Rev. Lett. 90, 253601 (2003).
[CrossRef] [PubMed]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

Phys. Today (1)

S. Haroche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24-30 (1989).
[CrossRef]

Proc. IEEE (1)

E. Jaynes and F. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89-109 (1963).
[CrossRef]

Rep. Prog. Phys. (1)

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325-1382 (2006).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565-582 (2001).
[CrossRef]

Science (2)

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660-662 (2004).
[CrossRef] [PubMed]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890-1893 (2007).
[CrossRef] [PubMed]

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Figures (1)

Fig. 1
Fig. 1

Experimental setup to realize the photon commutation operation a a a a . Atoms 1 and 3 are prepared in an entangled state 1 2 ( | e 1 | g 3 | g 1 | e 3 ) and Atom 2 in the excited state | e 2 . The three atoms enter the cavity C and interact with the cavity field one after another. The states of the three atoms after their interaction with the cavity are measured by the detectors D a and D b . The experiment succeeds if the atoms are measured to be in state | e 1 | g 2 | e 3 .

Equations (102)

Equations on this page are rendered with MathJax. Learn more.

| ψ ( t ) = e i H t | ψ ( t = 0 ) = e i g t ( a σ + a σ + ) | ψ ( t = 0 ) .
| ψ ( t ) = | g C g ( t ) n c n | n i | e S g ( t ) n c n | n ,
C g ( t ) = cos ( g t a a ) = 1 ( g t ) 2 2 ! a a + ( g t ) 4 4 ! a a a a ,
S g ( t ) = a sin ( g t a a ) a a = g t a ( g t ) 3 3 ! a a a + ( g t ) 5 5 ! a a a a a .
| ψ ( t ) = | g n c n cos ( n g t ) | n i | e n c n sin ( n g t ) | n 1 .
| ψ ( t ) = | e C e ( t ) n c n | n i | g S e ( t ) n c n | n ,
C e ( t ) = cos ( g t a a ) = 1 ( g t ) 2 2 ! a a + ( g t ) 4 4 ! a a a a ,
S e ( t ) = a sin ( g t a a ) a a = g t a ( g t ) 3 3 ! a a a + ( g t ) 5 5 ! a a a a a .
| ψ ( t ) = | e n c n cos ( n + 1 g t ) | n i | g n c n sin ( n + 1 g t ) | n + 1 .
| ψ R ( t 1 ) = N { S g ( t 1 ) n c n | n } ,
P e ( t 1 ) = n | c n | 2 sin 2 ( n g t 1 ) .
| ψ R ( t 1 ) N { a n c n | n } .
P e ( t 1 ) n ¯ ( g t 1 ) 2
| ψ R ( t 1 ) = N { S e ( t 1 ) n c n | n } ,
P g ( t 1 ) = n | c n | 2 sin 2 ( n + 1 g t 1 ) .
| ψ R ( t 1 ) N { a n c n | n } .
P g ( t 1 ) n + 1 ¯ ( g t 1 ) 2 .
| ψ R ( t 1 , t 2 ) = N { S e ( t 2 ) S g ( t 1 ) n c n | n } ,
P e 1 g 2 = n | c n | 2 sin 2 ( n g t 1 ) sin 2 ( n g t 2 ) .
| ψ R ( t 1 , t 2 ) = N { a a n c n | n } ,
P e 1 g 2 n | c n | 2 n 2 ( g t 1 ) 2 ( g t 2 ) 2 = n 2 ¯ ( g t 1 ) 2 ( g t 2 ) 2
| ψ R ( t 1 , t 2 ) = N { S g ( t 2 ) S e ( t 1 ) n c n | n } ,
P g 1 e 2 = n | c n | 2 sin 2 ( n + 1 g t 1 ) sin 2 ( n + 1 g t 2 ) .
| ψ R ( t 1 , t 2 ) N { a a n c n | n } ,
P g 1 e 2 ( n + 1 ) 2 ¯ ( g t 1 ) 2 ( g t 2 ) 2 .
| ψ ( t = 0 ) = 1 2 ( | e 1 | e 2 | g 3 | g 1 | e 2 | e 3 ) n c n | n .
| ψ ( t 1 ) = N { | e 2 | g 3 C e ( t 1 ) n c n | n + i | e 2 | e 3 S g ( t 1 ) n c n | n } .
| ψ ( t 1 , t 2 ) = N { | g 3 S e ( t 2 ) C e ( t 1 ) n c n | n + i | e 3 S e ( t 2 ) S g ( t 1 ) n c n | n } .
| ψ R ( t 1 , t 2 , t 3 ) = N { S g ( t 3 ) S e ( t 2 ) C e ( t 1 ) n c n | n C e ( t 3 ) S e ( t 2 ) S g ( t 1 ) n c n | n } .
P e 1 g 2 e 3 = 1 2 n | c n | 2 [ sin ( n + 1 g t 3 ) sin ( n + 1 g t 2 ) cos ( n + 1 g t 1 ) cos ( n + 1 g t 3 ) sin ( n g t 2 ) sin ( n g t 1 ) ] 2 .
| ψ R ( t 1 , t 2 , t 3 ) N { ( a a a a ) n c n | n } .
P e 1 g 2 e 3 1 2 n | c n | 2 [ ( n + 1 ) n ] 2 ( g t ) 2 ( g t 2 ) 2 = 1 2 ( g t ) 2 ( g t 2 ) 2 .
| ψ R ( t = 0 ) = α 1 | N 1 + α 2 | N 2 ,
c n = α 1 δ n N 1 + α 2 δ n N 2 ,
| ψ R ( t 1 ) = N { n c n sin ( n g t 1 ) | n 1 } .
| ψ desired = N { a n c n | n } = N { n c n n | n 1 } .
sin ( n g t 1 ) n = const.
| ψ R ( t 1 ) = N { α 1 sin ( N 1 g t 1 ) | N 1 1 + α 2 sin ( N 2 g t 1 ) | N 2 1 } ,
| ψ desired = N { α 1 N 1 | N 1 1 + α 2 N 2 | N 2 1 } .
sin ( N 1 g t 1 ) N 1 = sin ( N 2 g t 1 ) N 2 .
| ψ R ( t = 0 ) = α 1 | 1 + α 2 | 2 .
sin ( g t 1 ) = sin ( 2 g t 1 ) 2 .
P e ( t 1 ) = | α 1 | 2 sin 2 ( g t 1 ) + | α 2 | 2 sin 2 ( 2 g t 1 ) = ( | α 1 | 2 + 2 | α 2 | 2 ) sin 2 g t 1 = n ¯ sin 2 g t 1 0.34 n ¯ .
sin ( n + 1 g t 1 ) n + 1 = const.
sin ( N 1 + 1 g t 1 ) N 1 + 1 = sin ( N 2 + 1 g t 1 ) N 2 + 1 .
| ψ R ( t = 0 ) = α 1 | 0 + α 2 | 1 .
| ψ R ( t 1 , t 2 ) = N { n c n sin ( n g t 2 ) sin ( n g t 1 ) | n } .
| ψ desired ( t 1 , t 2 ) = N { a a n c n | n } = N { n c n n | n } .
sin ( n g t 2 ) sin ( n g t 1 ) n = const.
| ψ R ( t 1 , t 2 ) = N { α 1 sin ( N 1 g t 2 ) sin ( N 1 g t 1 ) | N 1 + α 2 sin ( N 2 g t 2 ) sin ( N 2 g t 1 ) | N 2 } ,
| ψ desired = N { α 1 N 1 | N 1 + α 2 N 2 | N 2 } .
sin ( N 1 g t 2 ) sin ( N 1 g t 1 ) N 1 = sin ( N 2 g t 2 ) sin ( N 2 g t 1 ) N 2 .
P e 1 g 2 = | α 1 | 2 sin 2 ( N 1 g t 1 ) sin 2 ( N 1 g t 2 ) + | α 2 | 2 sin 2 ( N 2 g t 1 ) sin 2 ( N 2 g t 2 ) .
sin ( g t 2 ) sin ( g t 1 ) = sin ( 2 g t 2 ) sin ( 2 g t 1 ) 2 .
P e 1 g 2 = | α 1 | 2 sin 2 ( g t 1 ) sin 2 ( g t 2 ) + | α 2 | 2 sin 2 ( 2 g t 1 ) sin 2 ( 2 g t 2 ) = ( | α 1 | 2 + 4 | α 2 | 2 ) sin 2 ( g t 1 ) sin 2 ( g t 2 ) 0.11 ( | α 1 | 2 + 4 | α 2 | 2 ) .
sin ( n + 1 g t 2 ) sin ( n + 1 g t 1 ) n + 1 = const.
| ψ R ( t 1 , t 2 ) = N { α 1 sin ( N 1 + 1 g t 2 ) sin ( N 1 + 1 g t 1 ) | N 1 + α 2 sin ( N 2 + 1 g t 2 ) sin ( N 2 + 1 g t 1 ) | N 2 } ,
| ψ desired = N { α 1 ( N 1 + 1 ) | N 1 + α 2 ( N 2 + 1 ) | N 2 } .
sin ( N 1 + 1 g t 2 ) sin ( N 1 + 1 g t 1 ) N 1 + 1 = sin ( N 2 + 1 g t 2 ) sin ( N 2 + 1 g t 1 ) N 2 + 1 ,
P g 1 e 2 = | α 1 | 2 sin 2 ( N 1 + 1 g t 1 ) sin 2 ( N 1 + 1 g t 2 ) + | α 2 | 2 sin 2 ( N 2 + 1 g t 1 ) sin 2 ( N 2 + 1 g t 2 ) .
1 2 ( | e 1 | g 3 + e i ϕ | g 1 | e 3 ) | e 2 ,
| ψ R ( t 1 , t 2 , t 3 ) = N { n c n [ sin ( n + 1 g t 3 ) sin ( n + 1 g t 2 ) cos ( n + 1 g t 1 ) + e i ϕ cos ( n + 1 g t 3 ) sin ( n g t 2 ) sin ( n g t 1 ) ] | n } .
| ψ desired = ( a a + e i ϕ a a ) n c n | n = N { n c n [ ( n + 1 ) + e i ϕ n ] | n } .
sin ( n + 1 g t 3 ) sin ( n + 1 g t 2 ) cos ( n + 1 g t 1 ) n + 1 = cos ( n + 1 g t 3 ) sin ( n g t 2 ) sin ( n g t 1 ) n
sin ( N 1 + 1 g t 3 ) sin ( N 1 + 1 g t 2 ) cos ( N 1 + 1 g t 1 ) N 1 + 1 = cos ( N 1 + 1 g t 3 ) sin ( N 1 g t 2 ) sin ( N 1 g t 1 ) N 1 = sin ( N 2 + 1 g t 3 ) sin ( N 2 + 1 g t 2 ) cos ( N 2 + 1 g t 1 ) N 2 + 1 = cos ( N 2 + 1 g t 3 ) sin ( N 2 g t 2 ) sin ( N 2 g t 1 ) N 2 .
sin ( g t 3 ) sin ( g t 2 ) cos ( g t 1 ) = sin ( N 2 + 1 g t 3 ) sin ( N 2 + 1 g t 2 ) cos ( N 2 + 1 g t 1 ) N 2 + 1 = cos ( N 2 + 1 g t 3 ) sin ( N 2 g t 2 ) sin ( N 2 g t 1 ) N 2 .
P e 1 g 2 e 3 = 1 2 { | α 1 | 2 [ sin ( N 1 + 1 g t 3 ) sin ( N 1 + 1 g t 2 ) cos ( N 1 + 1 g t 1 ) cos ( N 1 + 1 g t 3 ) sin ( N 1 g t 2 ) sin ( N 1 g t 1 ) ] 2 + | α 2 | 2 [ sin ( N 2 + 1 g t 3 ) sin ( N 2 + 1 g t 2 ) cos ( N 2 + 1 g t 1 ) cos ( N 2 + 1 g t 3 ) sin ( N 2 g t 2 ) sin ( N 2 g t 1 ) ] 2 } .
P e 1 g 2 e 3 1 2 ( g t ) 2 ( g t 2 ) 2
sin ( N 1 + 1 g t 2 ) N 1 + 1 = sin ( N 1 g t 2 ) N 1 = sin ( N 2 + 1 g t 2 ) N 2 + 1 = sin ( N 2 g t 2 ) N 2 ,
sin ( g t 2 ) = sin ( N 2 + 1 g t 2 ) N 2 + 1 = sin ( N 2 g t 2 ) N 2 .
sin ( g t 2 ) = sin ( 2 g t 2 ) 2 .
P e 1 g 2 e 3 1 2 [ sin 2 g t 2 ] ( g t ) 2 = 0.17 ( g t ) 2 .
sin ( 2 g t 2 ) 2 = sin ( g t 2 ) = sin ( 3 g t 2 ) 3 .
P e 1 g 2 e 3 = 1 2 [ sin 2 ( g t 2 ) ] ( g t ) 2 = 0.13 ( g t ) 2 .
sin ( g t 3 ) sin ( g t 2 ) cos ( g t 1 ) = sin ( 2 g t 3 ) sin ( 2 g t 2 ) cos ( 2 g t 1 ) 2 = cos ( 2 g t 3 ) sin ( g t 2 ) sin ( g t 1 ) .
P e 1 g 2 e 3 = 1 2 sin 2 ( g t 3 ) sin 2 ( g t 2 ) cos 2 ( g t 1 ) 0.014 ,
| ψ in = ( n c n | n a ) | 0 b ,
| ψ out = n c n k = 0 n n ! ( n k ) ! k ! r k t n k | n k a | k b ,
| ψ a out = N { n c n n r t n 1 | n 1 a } .
P = n | c n | 2 ( n r t n 1 ) 2 .
| ψ desired = N { a n c n | n a } = N { n c n n | n 1 a } .
t 1 , r 0 .
P n ¯ r 2
| ψ in = ( α 1 | N 1 a + α 2 | N 2 a ) | 0 b .
| ψ a out = N { α 1 N 1 r t N 1 1 | N 1 1 a + α 2 N 2 r t N 2 1 | N 2 1 a } .
| ψ desired = N { a ( α 1 | N 1 a + α 2 | N 2 a ) } = N { α 1 N 1 | N 1 1 a + α 2 N 2 | N 2 a } .
| ψ out = n c n 1 cosh n + 1 g k = 0 tanh k g ( n + k ) ! n ! k ! | n + k a | k b ,
| ψ a out = N { n c n tanh g cosh n + 1 g n + 1 | n + 1 a } .
P = n | c n | 2 tanh 2 g cosh 2 ( n + 1 ) g ( n + 1 ) .
| ψ desired = N { a n c n | n } = N { n c n n + 1 | n + 1 } .
g 0 ,
P n + 1 ¯ sinh 2 g n + 1 ¯ g 2 ,
| ψ a in = α 1 | N 1 a + α 2 | N 2 a .
| ψ a out = N { α 1 tanh g cosh N 1 + 1 g N 1 + 1 | N 1 + 1 a + α 2 tanh g cosh N 2 + 1 g N 2 + 1 | N 2 + 1 a } .
| ψ desired = N { a ( α 1 | N 1 a + α 2 | N 2 a ) } = N { α 1 N 1 + 1 | N 1 + 1 a + α 2 N 2 + 1 | N 2 + 1 a } .
| ψ a out = N { n c n n r t n 1 tanh g cosh n g n | n a } = N { n c n n t n 1 cosh n g | n a } .
| ψ desired = N { a a n c n | n a } = N { n c n n | n a } ,
| ψ a out = N { n c n ( n + 1 ) t n cosh n + 1 g | n n } ,
| ψ desired = N { a a n c n | n a } = N { n c n ( n + 1 ) | n + 1 a } .
| ψ a out = N { n c n [ ( n + 1 ) t n r t n cosh n + 1 g n r t n 1 t n cosh n g ] | n a } ,
| ψ desired = N { ( a a a a ) n c n | n a } = N { n c n [ ( n + 1 ) n ] | n a } .
r t n 1 t n cosh n g = t n r t n cosh n + 1 g = const.

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