Abstract

We propose a scheme to achieve a uniform cross-phase modulation (XPM) for two nonclassical light pulses and study its application for quantum nondemolition measurements of the photon number in a pulse and for controlled phase gates in quantum information. We analyze the scheme by quantizing a common phenomenological model for classical XPM. Our analysis first treats the ideal case of equal XPM and pure unitary dynamics. This establishes the groundwork for more-complicated studies of nonunitary dynamics and difference in phase shifts between the two pulses where decohering effects severely affect the performance of the scheme.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593-2596 (1991).
    [CrossRef] [PubMed]
  4. M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959-2962 (1997).
    [CrossRef]
  5. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
    [CrossRef]
  6. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594-598 (1999).
    [CrossRef]
  7. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638-641 (2003).
    [CrossRef] [PubMed]
  8. H. Kang and Y. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
    [CrossRef] [PubMed]
  9. M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
    [CrossRef] [PubMed]
  10. D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split sublevels,” Phys. Rev. A 65, 033833 (2002).
    [CrossRef]
  11. A. B. Matsko, I. Novikova, G. R. Welch, and M. S. Zubairy, “Enhancement of Kerr nonlinearity by multiphoton coherence,” Opt. Lett. 28, 96-98 (2003).
    [CrossRef] [PubMed]
  12. C. Ottaviani, D. Vitali, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
    [CrossRef] [PubMed]
  13. D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
    [CrossRef]
  14. S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
    [CrossRef]
  15. Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in Rb87,” Phys. Rev. Lett. 97, 063901 (2006).
    [CrossRef] [PubMed]
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  17. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
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    [CrossRef]
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    [CrossRef] [PubMed]
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  21. J. H. Shapiro, “Single-photon Kerr nonlinearities do not help quantum computation,” Phys. Rev. A 73, 062305 (2006).
    [CrossRef]
  22. J. H. Shapiro and R. S. Bondurant, “Qubit degradation due to cross-phase-modulation photon-number measurement,” Phys. Rev. A 73, 022301 (2006).
    [CrossRef]
  23. J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
    [CrossRef]
  24. K. Koshino, “Transitional behavior between self-Kerr and cross-Kerr effects by two photons,” Phys. Rev. A 75, 063807 (2007).
    [CrossRef]
  25. P. Leung, T. Ralph, W. J. Munro, and K. Nemoto, “Spectral effects of fast response cross Kerr non-linearity on quantum gate,” arXiv:0810.2828v2 (2008).
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    [CrossRef] [PubMed]
  27. G. M. Gehring, R. W. Boyd, A. L. Gaeta, D. J. Gauthier, and A. E. Willner, “Fiber-based slow-light technologies,” J. Lightwave Technol. 26, 3752-3762 (2008).
    [CrossRef]
  28. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
    [CrossRef] [PubMed]
  29. P. Londero, V. Venkataraman, A. R. Bhagwat, A. D. Slepkov, and A. L. Gaeta, “Ultralow-power four-wave mixing with Rb in a hollow-core photonic band-gap fiber,” Phys. Rev. Lett. 103, 043602 (2009).
    [CrossRef] [PubMed]
  30. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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  33. T.-K. Chiang, N. Kagi, T. K. Fong, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in dispersive fibers: theoretical and experimental investigation of the impact of modulation frequency,” IEEE Photonics Technol. Lett. 6, 733-736 (1994).
    [CrossRef]
  34. F. M. Abbou, C. C. Hiew, H. T. Chuah, D. S. Ong, and A. Abid, “A detailed analysis of cross-phase modulation effects on OOK and dpsk optical WDM transmission systems in the presence of GVD, SPM, and ASE noise,” J. Russ. Laser Res. 29, 57-70 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  37. Answering the question whether the case v1V1(z)≠v2V2(z) of this phenomenological model describes a real physical system may require an ab-initio quantum description of a medium that supports XPM. This is a formidable task and beyond the aim of our work to propose schemes for QND measurements of the photon number and to generate a CPG.
  38. B. C. Sanders and G. J. Milburn, “Complementarity in a quantum nondemolition measurement,” Phys. Rev. A 39, 694-702 (1989).
    [CrossRef] [PubMed]
  39. T. Tyc and B. C. Sanders, “Operational formulation of homodyne detection,” J. Phys. A 37, 7341-7357 (2004).
    [CrossRef]
  40. S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
    [CrossRef]
  41. Strictly speaking, d denotes the initial distance between the pulses inside the medium.
  42. Usually the state in which the phase shift is acquired is taken to be |11〉, but we consider |10〉 to simplify the discussion. Both gates are related by a single-qubit NOT operation that acts on the second qubit.
  43. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245-2248 (1998).
    [CrossRef]
  44. G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys. 48, 119-130 (1976).
    [CrossRef]
  45. J. D. Jackson, Classical Electrodynamics, 3rd Ed. (Wiley, 1999).
  46. L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240-243 (1994).
    [CrossRef] [PubMed]
  47. For instance, for giant nonlinearities based on EIT we would have Δvt≫w because the light pulses would have a duration in the order of microseconds.

2009

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef] [PubMed]

P. Londero, V. Venkataraman, A. R. Bhagwat, A. D. Slepkov, and A. L. Gaeta, “Ultralow-power four-wave mixing with Rb in a hollow-core photonic band-gap fiber,” Phys. Rev. Lett. 103, 043602 (2009).
[CrossRef] [PubMed]

2008

F. M. Abbou, C. C. Hiew, H. T. Chuah, D. S. Ong, and A. Abid, “A detailed analysis of cross-phase modulation effects on OOK and dpsk optical WDM transmission systems in the presence of GVD, SPM, and ASE noise,” J. Russ. Laser Res. 29, 57-70 (2008).
[CrossRef]

G. M. Gehring, R. W. Boyd, A. L. Gaeta, D. J. Gauthier, and A. E. Willner, “Fiber-based slow-light technologies,” J. Lightwave Technol. 26, 3752-3762 (2008).
[CrossRef]

2007

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
[CrossRef]

J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
[CrossRef]

K. Koshino, “Transitional behavior between self-Kerr and cross-Kerr effects by two photons,” Phys. Rev. A 75, 063807 (2007).
[CrossRef]

2006

J. H. Shapiro, “Single-photon Kerr nonlinearities do not help quantum computation,” Phys. Rev. A 73, 062305 (2006).
[CrossRef]

J. H. Shapiro and R. S. Bondurant, “Qubit degradation due to cross-phase-modulation photon-number measurement,” Phys. Rev. A 73, 022301 (2006).
[CrossRef]

S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
[CrossRef]

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in Rb87,” Phys. Rev. Lett. 97, 063901 (2006).
[CrossRef] [PubMed]

2005

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[CrossRef]

2004

K. Sanaka, T. Jennewein, J.-W. Pan, K. Resch, and A. Zeilinger, “Experimental nonlinear sign shift for linear optics quantum computation,” Phys. Rev. Lett. 92, 017902 (2004).
[CrossRef] [PubMed]

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
[CrossRef]

D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
[CrossRef]

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[CrossRef]

T. Tyc and B. C. Sanders, “Operational formulation of homodyne detection,” J. Phys. A 37, 7341-7357 (2004).
[CrossRef]

2003

A. B. Matsko, I. Novikova, G. R. Welch, and M. S. Zubairy, “Enhancement of Kerr nonlinearity by multiphoton coherence,” Opt. Lett. 28, 96-98 (2003).
[CrossRef] [PubMed]

C. Ottaviani, D. Vitali, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638-641 (2003).
[CrossRef] [PubMed]

H. Kang and Y. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

2002

D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split sublevels,” Phys. Rev. A 65, 033833 (2002).
[CrossRef]

2001

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

2000

M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

1999

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594-598 (1999).
[CrossRef]

1998

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245-2248 (1998).
[CrossRef]

1997

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 52(6), 36-42 (1997).
[CrossRef]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959-2962 (1997).
[CrossRef]

1996

1995

I. L. Chuang and Y. Yamamoto, “Simple quantum computer,” Phys. Rev. A 52, 3489-3496 (1995).
[CrossRef] [PubMed]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

1994

T.-K. Chiang, N. Kagi, T. K. Fong, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in dispersive fibers: theoretical and experimental investigation of the impact of modulation frequency,” IEEE Photonics Technol. Lett. 6, 733-736 (1994).
[CrossRef]

L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

1993

1992

1991

K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

1989

B. C. Sanders and G. J. Milburn, “Complementarity in a quantum nondemolition measurement,” Phys. Rev. A 39, 694-702 (1989).
[CrossRef] [PubMed]

1976

G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys. 48, 119-130 (1976).
[CrossRef]

Abbou, F. M.

F. M. Abbou, C. C. Hiew, H. T. Chuah, D. S. Ong, and A. Abid, “A detailed analysis of cross-phase modulation effects on OOK and dpsk optical WDM transmission systems in the presence of GVD, SPM, and ASE noise,” J. Russ. Laser Res. 29, 57-70 (2008).
[CrossRef]

Abid, A.

F. M. Abbou, C. C. Hiew, H. T. Chuah, D. S. Ong, and A. Abid, “A detailed analysis of cross-phase modulation effects on OOK and dpsk optical WDM transmission systems in the presence of GVD, SPM, and ASE noise,” J. Russ. Laser Res. 29, 57-70 (2008).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th Ed., (Academic, 2007).

Anderson, D. Z.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Artoni, M.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[CrossRef]

Bajcsy, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef] [PubMed]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638-641 (2003).
[CrossRef] [PubMed]

Balic, V.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef] [PubMed]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594-598 (1999).
[CrossRef]

Bhagwat, A. R.

P. Londero, V. Venkataraman, A. R. Bhagwat, A. D. Slepkov, and A. L. Gaeta, “Ultralow-power four-wave mixing with Rb in a hollow-core photonic band-gap fiber,” Phys. Rev. Lett. 103, 043602 (2009).
[CrossRef] [PubMed]

Boivin, L.

L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

Boller, K.-J.

K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

Bondurant, R. S.

J. H. Shapiro and R. S. Bondurant, “Qubit degradation due to cross-phase-modulation photon-number measurement,” Phys. Rev. A 73, 022301 (2006).
[CrossRef]

Boyd, R. W.

Cataliotti, F.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[CrossRef]

Chang, D. E.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
[CrossRef]

Chiang, T.-K.

T.-K. Chiang, N. Kagi, T. K. Fong, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in dispersive fibers: theoretical and experimental investigation of the impact of modulation frequency,” IEEE Photonics Technol. Lett. 6, 733-736 (1994).
[CrossRef]

Chuah, H. T.

F. M. Abbou, C. C. Hiew, H. T. Chuah, D. S. Ong, and A. Abid, “A detailed analysis of cross-phase modulation effects on OOK and dpsk optical WDM transmission systems in the presence of GVD, SPM, and ASE noise,” J. Russ. Laser Res. 29, 57-70 (2008).
[CrossRef]

Chuang, I. L.

I. L. Chuang and Y. Yamamoto, “Simple quantum computer,” Phys. Rev. A 52, 3489-3496 (1995).
[CrossRef] [PubMed]

Corbalan, R.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[CrossRef]

Cornell, E. A.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Di Giuseppe, G.

S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594-598 (1999).
[CrossRef]

Fleischhauer, M.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959-2962 (1997).
[CrossRef]

Fong, T. K.

T.-K. Chiang, N. Kagi, T. K. Fong, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in dispersive fibers: theoretical and experimental investigation of the impact of modulation frequency,” IEEE Photonics Technol. Lett. 6, 733-736 (1994).
[CrossRef]

Gaeta, A. L.

P. Londero, V. Venkataraman, A. R. Bhagwat, A. D. Slepkov, and A. L. Gaeta, “Ultralow-power four-wave mixing with Rb in a hollow-core photonic band-gap fiber,” Phys. Rev. Lett. 103, 043602 (2009).
[CrossRef] [PubMed]

G. M. Gehring, R. W. Boyd, A. L. Gaeta, D. J. Gauthier, and A. E. Willner, “Fiber-based slow-light technologies,” J. Lightwave Technol. 26, 3752-3762 (2008).
[CrossRef]

Gauthier, D. J.

Gehring, G. M.

Hafezi, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef] [PubMed]

Harris, S. E.

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594-598 (1999).
[CrossRef]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 52(6), 36-42 (1997).
[CrossRef]

K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594-598 (1999).
[CrossRef]

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

Haus, H. A.

L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

Hemmer, P. R.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
[CrossRef]

Hiew, C. C.

F. M. Abbou, C. C. Hiew, H. T. Chuah, D. S. Ong, and A. Abid, “A detailed analysis of cross-phase modulation effects on OOK and dpsk optical WDM transmission systems in the presence of GVD, SPM, and ASE noise,” J. Russ. Laser Res. 29, 57-70 (2008).
[CrossRef]

Hofferberth, S.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
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M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
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D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
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M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638-641 (2003).
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M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
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M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959-2962 (1997).
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D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
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T.-K. Chiang, N. Kagi, T. K. Fong, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in dispersive fibers: theoretical and experimental investigation of the impact of modulation frequency,” IEEE Photonics Technol. Lett. 6, 733-736 (1994).
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F. M. Abbou, C. C. Hiew, H. T. Chuah, D. S. Ong, and A. Abid, “A detailed analysis of cross-phase modulation effects on OOK and dpsk optical WDM transmission systems in the presence of GVD, SPM, and ASE noise,” J. Russ. Laser Res. 29, 57-70 (2008).
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S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
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S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
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C. Ottaviani, D. Vitali, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
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K. Sanaka, T. Jennewein, J.-W. Pan, K. Resch, and A. Zeilinger, “Experimental nonlinear sign shift for linear optics quantum computation,” Phys. Rev. Lett. 92, 017902 (2004).
[CrossRef] [PubMed]

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D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
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D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split sublevels,” Phys. Rev. A 65, 033833 (2002).
[CrossRef]

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M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
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S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
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M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
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K. Sanaka, T. Jennewein, J.-W. Pan, K. Resch, and A. Zeilinger, “Experimental nonlinear sign shift for linear optics quantum computation,” Phys. Rev. Lett. 92, 017902 (2004).
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K. Sanaka, T. Jennewein, J.-W. Pan, K. Resch, and A. Zeilinger, “Experimental nonlinear sign shift for linear optics quantum computation,” Phys. Rev. Lett. 92, 017902 (2004).
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W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
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S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
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S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
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S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
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S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
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M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
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M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
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Phys. Rev. A

J. H. Shapiro, “Single-photon Kerr nonlinearities do not help quantum computation,” Phys. Rev. A 73, 062305 (2006).
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[CrossRef]

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[CrossRef]

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
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M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef] [PubMed]

P. Londero, V. Venkataraman, A. R. Bhagwat, A. D. Slepkov, and A. L. Gaeta, “Ultralow-power four-wave mixing with Rb in a hollow-core photonic band-gap fiber,” Phys. Rev. Lett. 103, 043602 (2009).
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M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
[CrossRef]

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in Rb87,” Phys. Rev. Lett. 97, 063901 (2006).
[CrossRef] [PubMed]

K. Sanaka, T. Jennewein, J.-W. Pan, K. Resch, and A. Zeilinger, “Experimental nonlinear sign shift for linear optics quantum computation,” Phys. Rev. Lett. 92, 017902 (2004).
[CrossRef] [PubMed]

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[CrossRef]

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Phys. Today

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P. Leung, T. Ralph, W. J. Munro, and K. Nemoto, “Spectral effects of fast response cross Kerr non-linearity on quantum gate,” arXiv:0810.2828v2 (2008).

J. D. Jackson, Classical Electrodynamics, 3rd Ed. (Wiley, 1999).

Strictly speaking, d denotes the initial distance between the pulses inside the medium.

Usually the state in which the phase shift is acquired is taken to be |11〉, but we consider |10〉 to simplify the discussion. Both gates are related by a single-qubit NOT operation that acts on the second qubit.

G. P. Agrawal, Nonlinear Fiber Optics, 4th Ed., (Academic, 2007).

Answering the question whether the case v1V1(z)≠v2V2(z) of this phenomenological model describes a real physical system may require an ab-initio quantum description of a medium that supports XPM. This is a formidable task and beyond the aim of our work to propose schemes for QND measurements of the photon number and to generate a CPG.

For instance, for giant nonlinearities based on EIT we would have Δvt≫w because the light pulses would have a duration in the order of microseconds.

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Figures (4)

Fig. 1
Fig. 1

Scheme to achieve uniform XPM: two light pulses ψ 1 , ψ 2 travel at different group velocities v 1 > v 2 through an XPM medium such that ψ 1 can overtake ψ 2 . d denotes the initial distance between the two pulses.

Fig. 2
Fig. 2

Construction of a CPG for two time-bin qubits using uniform XPM.

Fig. 3
Fig. 3

Concurrence as a function of the XPM phase shift for minimal decoherence (dots) and the decoherence model presented in Appendix D (solid curve).

Fig. 4
Fig. 4

The function θ i ( z 1 , z 2 ) for a Gaussian potential V i ( z ) of width r.

Equations (63)

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( 1 v i t + z ) E i ( z , t ) = i γ i ( I i ( z , t ) + 2 I 3 i ( z , t ) ) E i ( z , t ) ,
( 1 v i t + z ) E i ( z , t ) = i d z I 3 i ( z , t ) V i ( z z ) E i ( z , t ) .
[ E i ( z , t ) , E j ( z , t ) ] = ω i 2 ε 0 A δ i j δ ( z z ) η δ i j δ ( z z ) ,
a i [ u ] = 1 η d z ψ i * ( z u ) E i ( z , 0 ) .
E i ( z , t ) = U 3 i ( z , t ) E i ( z v i t , 0 ) ,
U i ( z , t ) = exp ( i I i ( z , 0 ) v 3 i v i v 3 i θ 3 i ( z z v 3 i t , [ v i v 3 i ] t ) d z ) ,
θ i ( z 1 , z 2 ) 0 z 2 d z V i ( z 1 z ) .
| ψ = 1 n ! D 1 ( α ) ( a 2 ) n | 0 .
D 1 ( α ) E 1 ( z , 0 ) D 1 ( α ) = E 1 ( z , 0 ) + α η ψ 1 ( z ) .
ψ | E 1 ( z , t ) | ψ = α η n ! ψ 1 ( z v 1 t ) 0 | a 2 n U 2 ( z , t ) ( a 2 ) n | 0 = α η n ! ψ 1 ( z v 1 t ) 0 | a 2 n ( a ̃ 2 [ 0 ; z , t ] ) n | 0 ,
a ̃ i [ u ; z , t ] 1 η d z ψ i * ( z u ) E i ( z , 0 ) × e i η v 3 i v i v 3 i θ 3 i ( z z v 3 i t , ( v i v 3 i ) t ) .
0 | a 2 n ( a ̃ 2 [ 0 ; z , t ] ) n | 0 = n ! [ a 2 , a ̃ 2 [ 0 ; z , t ] ] n ,
E 1 ( z , t ) = α η ψ 1 ( z v 1 t ) × ( d z | ψ 2 ( z ) | 2 e i η v 1 Δ v θ 1 ( z z + v 1 t , Δ v t ) ) n ,
V i d z V i ( z ) .
E 1 ( v 1 t , t ) = α η ψ 1 ( 0 ) e i n φ 1 ,
φ i η v i V i Δ v .
U CPG | k l = e i k ( 1 l ) β | k l , k , l = 0 , 1 .
| i j ( t ) = a 1 [ δ z i ( 1 ) + v 1 t ] a 2 [ δ z j ( 2 ) + v 2 t ] | 0 ,
ρ i j ; k l ( t ) = X k l * X i j ,
X i j = c i j d z 1 d z 2 | ψ 1 ( z 1 ) ψ 2 ( z 2 + d ) | 2 × e i η v 2 Δ v θ 2 ( z 2 z 1 + d + δ z j ( 2 ) δ z i ( 1 ) , Δ v t ) .
X i j = c i j e i i ( 1 j ) φ 2 ,
ρ i j ; k l ( t ) = c i j c k l * e i ( i ( 1 j ) k ( 1 l ) ) φ 2 .
E 2 ( z , t ) E 1 ( z , t ) = E 1 ( z , t ) E 2 ( z , t ) e i Φ ( z z , t ) ,
Φ ( z z , t ) η Δ v ( v 1 θ 1 ( z z + Δ v t , Δ v t ) v 2 θ 2 ( z z + Δ v t , Δ v t ) ) ,
( 1 v i t + z ) E i ( z , t ) = i ( d z I 3 i ( z , t ) V i ( z z ) + m i ( z , t ) ) E i ( z , t ) .
[ m i ( z , t ) , m j ( z , t ) ] = i η ( V i ( z z ) v j V j ( z z ) v i ) δ ( t t ) .
E i ( z , t ) = R i ( z , t ) U 3 i ( z , t ) E i ( z v i t , 0 ) ,
R i ( z , t ) = exp ( i v i 0 t d t m i ( z v i ( t t ) , t ) ) .
R 1 ( z , t ) R 2 ( z , t ) R 1 ( z , t ) = R 2 ( z , t ) e i Φ ( z z , t ) ,
R 2 ( z , t ) R 1 ( z , t ) R 2 ( z , t ) = R 1 ( z , t ) e i Φ ( z z , t ) .
X i j = c i j e i i ( 1 j ) φ 2 d z 1 d z 2 | ψ 1 ( z 1 ) ψ 2 ( z 2 + d ) | 2 R 1 ( z 1 + v 1 t + δ z i ( 1 ) , t ) R 2 ( z 2 + d + v 2 t + δ z j ( 2 ) , t ) .
ρ i j ; k l ( t ) = Tr E ( ρ E ( 0 ) X k l X i j ) ,
X i j = c i j e i i ( 1 j ) φ 2 R 1 ( v 1 t + δ z i ( 1 ) , t ) R 2 ( d + v 2 t + δ z j ( 2 ) , t ) .
ρ i j ; k l ( t ) = c i j c k l * e i ( i ( 1 j ) k ( 1 l ) ) φ 2 N i j ; k l ( t ) ,
N i j ; k l ( t ) R 2 ( d + v 2 t + δ z l ( 2 ) , t ) R 1 ( v 1 t + δ z k ( 1 ) , t ) R 1 ( v 1 t + δ z i ( 1 ) , t ) R 2 ( d + v 2 t + δ z j ( 2 ) , t ) .
N = ( 1 c 2 c 1 e i ( φ 1 φ 2 ) N 00 ; 11 c 2 * 1 N 01 ; 10 c 1 c 1 * e i ( φ 1 φ 2 ) N 01 ; 10 * 1 c 2 N 00 ; 11 * c 1 * c 2 * 1 ) ,
c n R n ( v n t + δ z , t ) R 1 ( v n t , t ) .
R i ( z , t ) = exp ( η 4 v i v 3 i Δ v t V ( 0 ) ) .
R i ( z , t ) exp ( φ 1 φ 2 4 π v i v 3 i Δ v t w ) .
U i E i ( z , 0 ) U i = E i ( z , 0 ) exp ( i η v 3 i v i v 3 i θ 3 i ( z z v 3 i t , ( v i v 3 i ) t ) ) ,
U i E i ( z , 0 ) U i = E i ( z , 0 ) exp ( i η v 3 i v i v 3 i θ 3 i ( z z v 3 i t , ( v i v 3 i ) t ) ) .
[ a i [ u ] , E i ( z , 0 ) ] = η ψ i * ( z u ) ,
U i ( z , t ) a i [ u ] U i ( z , t ) = a ̃ i [ u ; z , t ] ,
θ i ( z 1 , z 2 ) ( θ ( z 1 ) θ ( z 2 z 1 ) θ ( z 1 ) θ ( z 1 z 2 ) ) V i ,
ρ ( 0 ) = i , j , k , l c k l * c i j | i j ( 0 ) k l ( 0 ) | .
i j ( t ) | U ( t ) = 0 | U ( t ) a 1 [ δ z i ( 1 ) + v 1 t ] a 2 [ δ z j ( 2 ) + v 2 t ] U ( t ) .
i j ( t ) | U ( t ) = 0 | a 1 [ δ z i ( 1 ) + v 1 t , t ] a 2 [ δ z j ( 2 ) + v 2 t , t ] ,
a i [ z , t ] 1 η d z ψ i * ( z u ) E i ( z , t ) .
ρ i j ; k l ( t ) = ψ ( 0 ) | a 2 [ δ z l ( 2 ) + v 2 t , t ] a 1 [ δ z k ( 1 ) + v 1 t , t ] | 0 × 0 | a 1 [ δ z i ( 1 ) + v 1 t , t ] a 2 [ δ z j ( 2 ) + v 2 t , t ] | ψ ( 0 ) .
X i j 0 | a 1 [ δ z i ( 1 ) + v 1 t , t ] a 2 [ δ z j ( 2 ) + v 2 t , t ] | ψ ( 0 ) = 1 η d z 1 d z 2 ψ 1 * ( z 1 v 1 t δ z i ( 1 ) ) ψ 2 * ( z 2 v 2 t δ z j ( 2 ) ) 0 | U 2 ( z 1 , t ) E 1 ( z 1 v 1 t , 0 ) U 1 ( z 2 , t ) E 2 ( z 2 v 2 t , 0 ) | ψ ( 0 ) = 1 η d z 1 d z 2 ψ 1 * ( z 1 v 1 t δ z i ( 1 ) ) ψ 2 * ( z 2 v 2 t δ z j ( 2 ) ) 0 | U 1 ( z 2 , t ) E 1 ( z 1 v 1 t , 0 ) U 1 ( z 2 , t ) E 2 ( z 2 v 2 t , 0 ) | ψ ( 0 ) = 1 η d z 1 d z 2 ψ 1 * ( z 1 ) ψ 2 * ( z 2 + d ) e i η v 2 Δ v θ 2 ( z 2 z 1 + d + δ z j ( 2 ) δ z i ( 1 ) , Δ v t ) 0 | E 1 ( z 1 + δ z i ( 1 ) , 0 ) E 2 ( z 2 + d + δ z j ( 2 ) , 0 ) | ψ ( 0 ) .
0 | E 1 ( z 1 + δ z i ( 1 ) , 0 ) E 2 ( z 2 + d + δ z j ( 2 ) , 0 ) | ψ ( 0 ) = k , l c k l 0 | E 1 ( z 1 + δ z i ( 1 ) , 0 ) E 2 ( z 2 + d + δ z j ( 2 ) , 0 ) a 1 [ δ z k ( 1 ) ] a 2 [ δ z l ( 2 ) ] | 0 = η k , l c k l ψ 1 ( z 1 + δ z i ( 1 ) δ z k ( 1 ) ) ψ 2 ( z 2 + d + δ z j ( 2 ) δ z l ( 2 ) ) .
0 | E 1 ( z 1 + δ z i ( 1 ) , 0 ) E 2 ( z 2 + d + δ z j ( 2 ) , 0 ) | ψ ( 0 ) = η c i j ψ 1 ( z 1 ) ψ 2 ( z 2 + d ) ,
ρ i j ; k l ( t ) = Tr E ( i j ( t ) | U ( t ) ρ ( 0 ) ρ E ( 0 ) U ( t ) | k l ( t ) ) ,
X i j 0 | a 1 [ δ z i ( 1 ) + v 1 t , t ] a 2 [ δ z j ( 2 ) + v 2 t , t ] | ψ ( 0 ) = 1 η d z 1 d z 2 R 1 ( z 1 , t ) R 2 ( z 2 , t ) ψ 1 * ( z 1 v 1 t δ z i ( 1 ) ) ψ 2 * ( z 2 v 2 t δ z j ( 2 ) ) e i η v 2 Δ v θ 2 ( z 2 z 1 + Δ v t , Δ v t ) 0 | E 1 ( z 1 v 1 t , 0 ) E 2 ( z 2 v 2 t , 0 ) | ψ ( 0 ) .
X i j = c i j d z 1 d z 2 | ψ 1 ( z 1 ) ψ 2 ( z 2 + d ) | 2 R 1 ( z 1 + v 1 t + δ z i ( 1 ) , t ) R 2 ( z 2 + d + v 2 t + δ z j ( 2 ) , t ) × e i η v 2 Δ v θ 2 ( z 2 z 1 + d + δ z j ( 2 ) δ z i ( 1 ) , Δ v t ) ,
N i j ; i l ( t ) = R 2 ( d + v 2 t + δ z l ( 2 ) , t ) R 2 ( d + v 2 t + δ z j ( 2 ) , t ) .
N i j ; k j ( t ) = R 2 ( d + v 2 t + δ z j ( 2 ) , t ) R 1 ( v 1 t + δ z k ( 1 ) , t ) R 1 ( v 1 t + δ z i ( 1 ) , t ) R 2 ( d + v 2 t + δ z j ( 2 ) , t ) = R 1 ( v 1 t + δ z k ( 1 ) , t ) R 1 ( v 1 t + δ z i ( 1 ) , t ) e i Φ ( d Δ v t + δ z j ( 2 ) δ z k ( 1 ) , t ) i Φ ( d Δ v t + δ z j ( 2 ) δ z i ( 1 ) , t ) = R 1 ( v 1 t + δ z k ( 1 ) , t ) R 1 ( v 1 t + δ z i ( 1 ) , t ) × e i ( φ 1 φ 2 ) ( 1 j ) ( k i ) .
m 1 ( z , t ) = M ( z , t ) + M ( z , t ) ,
m 2 ( z , t ) = i M ( z , t ) + i M ( z , t ) ,
M ( z , t ) = η Δ v 4 π v 1 v 2 0 d ω ( i B ( z , ω ) e i ω t + C ( z , ω ) e i ω t ) ,
[ B ( z , ω ) , B ( z , ω ) ] = [ C ( z , ω ) , C ( z , ω ) ] = V ( z z ) δ ( ω ω ) .
c i = R i ( z , t ) 2 ,
N 00 ; 11 = N 01 , 10 = e i 2 ( φ 1 φ 2 ) c 1 c 2 .

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