Abstract

Two particles that are entangled with respect to continuous variables such as position and momentum exhibit a variety of nonclassical features. First, measurement of one particle projects the other particle into the state that is the complex conjugate of the state of the first particle; i.e., measurement of one particle projects the other particle into the time-reversed state. Second, continuous-variable entanglement can be used to implement a quantum magic bullet: When one particle manages to pass through a scattering potential, then, no matter how low the probability of this event, the second particle will also pass through a related scattering potential with probability 1. This phenomenon is investigated in terms of the original Einstein–Podolsky–Rosen state, and experimental realizations are suggested in terms of entangled photon states.

© 2002 Optical Society of America

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References

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  1. C. H. Bennett and P. W. Shor, “Quantum information theory,” IEEE Trans. Inf. Theory 44, 2724–2742 (1998).
    [CrossRef]
  2. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
    [CrossRef]
  3. A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
    [CrossRef] [PubMed]
  4. L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994).
    [CrossRef] [PubMed]
  5. S. L. Braunstein, “Error correction for continuous quantum variables,” Phys. Rev. Lett. 80, 4084–4087 (1998).
    [CrossRef]
  6. S. L. Braunstein, “Quantum error correction for communication with linear optics,” Nature 394, 47–49 (1998).
    [CrossRef]
  7. S. Lloyd and J.-J. E. Slotine, “Analog quantum error correction,” Phys. Rev. Lett. 80, 4088–4091 (1998).
    [CrossRef]
  8. S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
    [CrossRef]
  9. J. H. Shapiro, and K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
    [CrossRef]
  10. J. H. Shapiro and N. C. Wong, “An ultrabright narrowband source of polarization-entangled photon pairs,” J. Opt. B 2, L1–L4 (2000).
    [CrossRef]
  11. H. P. Yuen and J. H. Shapiro, “Optical Communication with two-photon coherent states. III. Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
    [CrossRef]
  12. M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
    [CrossRef] [PubMed]
  13. P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
    [CrossRef] [PubMed]
  14. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
    [CrossRef] [PubMed]
  15. Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
    [CrossRef]
  16. S. Reynaud, C. Fabre, and E. Giacobino, “Quantum fluctuations in a two-mode parametric oscillator,” J. Opt. Soc. Am. B 4, 1520–1524 (1987).
    [CrossRef]
  17. K. W. Leong, N. C. Wong, and J. H. Shapiro, “Nonclassical intensity correlation from a type-I phase-matched optical parametric oscillator,” Opt. Lett. 15, 1058–1060 (1990).
    [CrossRef] [PubMed]
  18. J. Teja and N. C. Wong, “Twin-beam generation in a triply resonant dual-cavity optical parametric oscillator,” Opt. Express 2, 65–71 (1998), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  19. J. H. Shapiro, “Long-distance high-fidelity teleportation using singlet states,” in Quantum Communication, Computing, and Measurement 3, O. Hirota and P. Tombesi, eds. (Kluwer, New York, 2001), pp. 367–374.
  20. 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]
  21. M. Vasilyev, S. K. Choi, P. Kumar, and G. M. D’Ariano, “Tomographic measurement of joint photon statistics of the twin-beam quantum state,” Phys. Rev. Lett. 84, 2354–2357 (2000).
    [CrossRef] [PubMed]
  22. S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long-distance, unconditional teleportation of atomic states via complete Bell state measurements,” Phys. Rev. Lett. 87, 167903 (2001).
    [CrossRef]

2001

S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long-distance, unconditional teleportation of atomic states via complete Bell state measurements,” Phys. Rev. Lett. 87, 167903 (2001).
[CrossRef]

2000

M. Vasilyev, S. K. Choi, P. Kumar, and G. M. D’Ariano, “Tomographic measurement of joint photon statistics of the twin-beam quantum state,” Phys. Rev. Lett. 84, 2354–2357 (2000).
[CrossRef] [PubMed]

J. H. Shapiro and N. C. Wong, “An ultrabright narrowband source of polarization-entangled photon pairs,” J. Opt. B 2, L1–L4 (2000).
[CrossRef]

1999

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[CrossRef]

1998

S. L. Braunstein, “Error correction for continuous quantum variables,” Phys. Rev. Lett. 80, 4084–4087 (1998).
[CrossRef]

S. L. Braunstein, “Quantum error correction for communication with linear optics,” Nature 394, 47–49 (1998).
[CrossRef]

S. Lloyd and J.-J. E. Slotine, “Analog quantum error correction,” Phys. Rev. Lett. 80, 4088–4091 (1998).
[CrossRef]

C. H. Bennett and P. W. Shor, “Quantum information theory,” IEEE Trans. Inf. Theory 44, 2724–2742 (1998).
[CrossRef]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

J. Teja and N. C. Wong, “Twin-beam generation in a triply resonant dual-cavity optical parametric oscillator,” Opt. Express 2, 65–71 (1998), http://www.opticsexpress.org.
[CrossRef] [PubMed]

1994

1992

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

1990

P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[CrossRef] [PubMed]

K. W. Leong, N. C. Wong, and J. H. Shapiro, “Nonclassical intensity correlation from a type-I phase-matched optical parametric oscillator,” Opt. Lett. 15, 1058–1060 (1990).
[CrossRef] [PubMed]

1988

M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[CrossRef] [PubMed]

1987

S. Reynaud, C. Fabre, and E. Giacobino, “Quantum fluctuations in a two-mode parametric oscillator,” J. Opt. Soc. Am. B 4, 1520–1524 (1987).
[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]

1980

H. P. Yuen and J. H. Shapiro, “Optical Communication with two-photon coherent states. III. Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

1935

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Bennett, C. H.

C. H. Bennett and P. W. Shor, “Quantum information theory,” IEEE Trans. Inf. Theory 44, 2724–2742 (1998).
[CrossRef]

Braunstein, S. L.

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[CrossRef]

S. L. Braunstein, “Error correction for continuous quantum variables,” Phys. Rev. Lett. 80, 4084–4087 (1998).
[CrossRef]

S. L. Braunstein, “Quantum error correction for communication with linear optics,” Nature 394, 47–49 (1998).
[CrossRef]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Choi, S. K.

M. Vasilyev, S. K. Choi, P. Kumar, and G. M. D’Ariano, “Tomographic measurement of joint photon statistics of the twin-beam quantum state,” Phys. Rev. Lett. 84, 2354–2357 (2000).
[CrossRef] [PubMed]

D’Ariano, G. M.

M. Vasilyev, S. K. Choi, P. Kumar, and G. M. D’Ariano, “Tomographic measurement of joint photon statistics of the twin-beam quantum state,” Phys. Rev. Lett. 84, 2354–2357 (2000).
[CrossRef] [PubMed]

Drummond, P. D.

P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[CrossRef] [PubMed]

M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[CrossRef] [PubMed]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Fabre, C.

Fuchs, C. A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Furusawa, A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Giacobino, E.

Hemmer, P. R.

S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long-distance, unconditional teleportation of atomic states via complete Bell state measurements,” Phys. Rev. Lett. 87, 167903 (2001).
[CrossRef]

Hong, C. K.

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]

Kimble, H. J.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

Kumar, P.

M. Vasilyev, S. K. Choi, P. Kumar, and G. M. D’Ariano, “Tomographic measurement of joint photon statistics of the twin-beam quantum state,” Phys. Rev. Lett. 84, 2354–2357 (2000).
[CrossRef] [PubMed]

Leong, K. W.

Lloyd, S.

S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long-distance, unconditional teleportation of atomic states via complete Bell state measurements,” Phys. Rev. Lett. 87, 167903 (2001).
[CrossRef]

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[CrossRef]

S. Lloyd and J.-J. E. Slotine, “Analog quantum error correction,” Phys. Rev. Lett. 80, 4088–4091 (1998).
[CrossRef]

Mandel, L.

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]

Ou, Z. Y.

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

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]

Peng, K. C.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Pereira, S. F.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Polzik, E. S.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Reid, M. D.

P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[CrossRef] [PubMed]

M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[CrossRef] [PubMed]

Reynaud, S.

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Shahriar, M. S.

S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long-distance, unconditional teleportation of atomic states via complete Bell state measurements,” Phys. Rev. Lett. 87, 167903 (2001).
[CrossRef]

Shapiro, J. H.

S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long-distance, unconditional teleportation of atomic states via complete Bell state measurements,” Phys. Rev. Lett. 87, 167903 (2001).
[CrossRef]

J. H. Shapiro and N. C. Wong, “An ultrabright narrowband source of polarization-entangled photon pairs,” J. Opt. B 2, L1–L4 (2000).
[CrossRef]

J. H. Shapiro, and K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
[CrossRef]

K. W. Leong, N. C. Wong, and J. H. Shapiro, “Nonclassical intensity correlation from a type-I phase-matched optical parametric oscillator,” Opt. Lett. 15, 1058–1060 (1990).
[CrossRef] [PubMed]

H. P. Yuen and J. H. Shapiro, “Optical Communication with two-photon coherent states. III. Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

Shor, P. W.

C. H. Bennett and P. W. Shor, “Quantum information theory,” IEEE Trans. Inf. Theory 44, 2724–2742 (1998).
[CrossRef]

Slotine, J.-J. E.

S. Lloyd and J.-J. E. Slotine, “Analog quantum error correction,” Phys. Rev. Lett. 80, 4088–4091 (1998).
[CrossRef]

Sørensen, J. L.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Sun, K.-X.

Teja, J.

Vaidman, L.

L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994).
[CrossRef] [PubMed]

Vasilyev, M.

M. Vasilyev, S. K. Choi, P. Kumar, and G. M. D’Ariano, “Tomographic measurement of joint photon statistics of the twin-beam quantum state,” Phys. Rev. Lett. 84, 2354–2357 (2000).
[CrossRef] [PubMed]

Wong, N. C.

Yuen, H. P.

H. P. Yuen and J. H. Shapiro, “Optical Communication with two-photon coherent states. III. Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

Appl. Phys. B

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

IEEE Trans. Inf. Theory

H. P. Yuen and J. H. Shapiro, “Optical Communication with two-photon coherent states. III. Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

C. H. Bennett and P. W. Shor, “Quantum information theory,” IEEE Trans. Inf. Theory 44, 2724–2742 (1998).
[CrossRef]

J. Opt. B

J. H. Shapiro and N. C. Wong, “An ultrabright narrowband source of polarization-entangled photon pairs,” J. Opt. B 2, L1–L4 (2000).
[CrossRef]

J. Opt. Soc. Am. B

Nature

S. L. Braunstein, “Quantum error correction for communication with linear optics,” Nature 394, 47–49 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Phys. Rev. A

L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994).
[CrossRef] [PubMed]

P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[CrossRef] [PubMed]

S. L. Braunstein, “Error correction for continuous quantum variables,” Phys. Rev. Lett. 80, 4084–4087 (1998).
[CrossRef]

S. Lloyd and J.-J. E. Slotine, “Analog quantum error correction,” Phys. Rev. Lett. 80, 4088–4091 (1998).
[CrossRef]

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[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]

M. Vasilyev, S. K. Choi, P. Kumar, and G. M. D’Ariano, “Tomographic measurement of joint photon statistics of the twin-beam quantum state,” Phys. Rev. Lett. 84, 2354–2357 (2000).
[CrossRef] [PubMed]

S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long-distance, unconditional teleportation of atomic states via complete Bell state measurements,” Phys. Rev. Lett. 87, 167903 (2001).
[CrossRef]

Science

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Other

J. H. Shapiro, “Long-distance high-fidelity teleportation using singlet states,” in Quantum Communication, Computing, and Measurement 3, O. Hirota and P. Tombesi, eds. (Kluwer, New York, 2001), pp. 367–374.

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Figures (4)

Fig. 1
Fig. 1

Schematic of a type II phase-matched doubly resonant OPA.

Fig. 2
Fig. 2

(a) Shot-noise level and (b) signal-minus-idler intensity difference from a KTP OPO.

Fig. 3
Fig. 3

Normalized photocount-difference variance, σn2 versus the ratio of measurement-cavity linewidth to source-cavity linewidth, Γc/Γ, for 1% OPA pumping (G2=0.01) and various values of the normalized detuning, Δω/Γ.

Fig. 4
Fig. 4

Schematic of filter-penetration optical magic bullets: BW, bandwidth.

Equations (23)

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

|ψEPR=- |x1|x2dx=- |p1|-p2dp.
|ψEPR=|ϕz1|ϕz2*dz.
|ψ(t)EPR exp(-iHˆt/)|ϕz1 exp(-iHˆt/)|ϕz2*dz=|ϕz(t)1|ϕz(-t)2*dz.
S(n)(ω)-AˆS(t+τ)AˆS(t)exp(-iωτ)dτ-AˆS(t+τ)AˆI(t)exp(-iωτ)dτ=2G1-G2-(ω/Γ)2-2iω/Γ2,
S(p)(ω)-AˆS(t+τ)AˆI(t)exp(-iωτ)dτ=2G[1+G2+(ω/Γ)2]|1-G2-(ω/Γ)2-2iω/Γ|2.
|ψSI=n=0N¯n(N¯+1)n+11/2|nS|nI,
|ψSI=-- ψ(αS1, αI1)|αS1S|αI1IdαS1dαI1,
ψ(αS1, αI1)exp[-(1+2N¯)αS12+4N¯(N¯+1)αS1αI1-(1+2N¯)αI12]/π/2.
ψ(αS1, αI1)=exp-αS121+2N¯[π(1+2N¯)/2]1/4exp-(1+2N¯)αI1-[4N¯(N¯+1)]1/21+2N¯αS12[π/2(1+2N¯)]1/4.
1(πN¯)1/4δ(αI1-αS1),N¯.
aˆSn0T EˆS(t)exp[i(ωS+2πn/T)t]T dt,
aˆIn0T EˆI(t)exp[i(ωI-2πn/T)t]T dt.
|ψSI=n ψn|1Sn|1In
|ψI=n ψnϕn*|1Inn|ψn|2|ϕn|21/2.
|ψInϕn*|1In;
aˆS(Tc)=aˆS(0)exp[-(Γc+iΔω)Tc]+0Tc dt2Γc exp[-(Γc+iΔω)(Tc-t)+iωSt]EˆS(t),
aˆI(Tc)=aˆI(0)exp[-(Γc-iΔω)Tc]+0Tc dt2Γc exp[-(Γc-iΔω)(Tc-t)+iωIt]EˆI(t),
|HS(ω)|2=11+[(ω-ωS-Δω)/ωc]2K,
|HI(ω)|2=11+[(ω-ωI+Δω)/ωc]2K.
|ψSI=n k=0Pk(n)|kSn|kIn,
Pk(n)[S(n)(2πn/T)]k[S(n)(2πn/T)+1]k+1,k=0, 1, 2 ,
ρˆj=n k=0 Pk(n)|kjnjnk|,j=S, I.
ρˆSI=n k=0 Pk(n)|kSnSnk||kInInk|.

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