Abstract

We demonstrate “simultaneous unbalanced shared local oscillator heterodyne interferometry” (SUSHI) a new method for minimally destructive, high signal-to-noise-ratio (SNR) dispersive detection of atomic spins. In SUSHI a dual-frequency probe laser interacts with atoms in one arm of a Mach–Zehnder interferometer, and then beats against a bright local oscillator (LO) beam traversing the other arm, resulting in two simultaneous, independent heterodyne measurements of the atom-induced phase shift. Measurement noise due to mechanical disturbances of beam paths is strongly rejected by the technique of active subtraction, in which antinoise is actively written onto the LO beam via an optical phase-locked loop. In SUSHI, technical noise due to phase, amplitude, and frequency fluctuations of the various laser fields is strongly rejected (i) for any mean phase bias between the interferometer arms, (ii) without the use of piezo actuated mirrors, and (iii) without signal balancing. We experimentally demonstrate an ultralow technical-noise-limited sensitivity of 51nrad/Hz over a measurement bandwidth of 60 Hz to 8 kHz using a 230 μW probe, and stay within 3dB of the standard quantum limit as probe power is reduced by more than 5 orders of magnitude to as low as 650 pW. SUSHI is therefore well suited to performing quantum nondemolition measurements for preparing spin-squeezed states and for high-SNR, truly continuous observations of ground-state Rabi flopping in cold atom ensembles.

© 2013 Optical Society of America

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  1. J. Hope and J. Close, “General limit to nondestructive optical detection of atoms,” Phys. Rev. A 71, 043822 (2005).
    [CrossRef]
  2. M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
    [CrossRef]
  3. W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, “Making, probing and understanding Bose–Einstein condensates,” eprint arXiv:cond-mat/9904034 (1999).
  4. Y. Liu, E. Gomez, S. Maxwell, L. Turner, and E. Tiesinga, “Number fluctuations and energy dissipation in sodium spinor condensates,” Phys. Rev. Lett. 102, 225301 (2009).
    [CrossRef]
  5. P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
    [CrossRef]
  6. P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
    [CrossRef]
  7. A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
    [CrossRef]
  8. D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
    [CrossRef]
  9. J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
    [CrossRef]
  10. N. Shiga and M. Takeuchi, “Locking the local oscillator phase to the atomic phase via weak measurement,” New J. Phys. 14, 023034 (2012).
    [CrossRef]
  11. H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983).
    [CrossRef]
  12. J. Shapiro and S. Wagner, “Phase and amplitude uncertainties in heterodyne detection,” IEEE J. Quantum Electron. 20, 803–813 (1984).
    [CrossRef]
  13. J. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237–250 (1985).
    [CrossRef]
  14. K. Leong and J. Shapiro, “Phase and amplitude uncertainties in multimode heterodyning,” Opt. Commun. 58, 73–77 (1986).
    [CrossRef]
  15. H. A. Haus, “From classical to quantum noise,” J. Opt. Soc. Am. B 12, 2019–2036 (1995).
    [CrossRef]
  16. G. D. Ariano and M. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022–3036 (1994).
    [CrossRef]
  17. Z. Ou, “Complementarity and fundamental limit in precision phase measurement,” Phys. Rev. Lett. 77, 2352–2355 (1996).
    [CrossRef]
  18. A. Luis and J. Peřina, “Optimum phase-shift estimation and the quantum description of the phase difference,” Phys. Rev. A 54, 4564–4570 (1996).
    [CrossRef]
  19. Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
    [CrossRef]
  20. T. Opatrný, M. Dakna, and D.-G. Welsch, “Number-phase uncertainty relations: verification by balanced homodyne measurement,” Phys. Rev. A 57, 2129–2133 (1998).
    [CrossRef]
  21. J. Lye, J. Hope, and J. Close, “Nondestructive dynamic detectors for Bose–Einstein condensates,” Phys. Rev. A 67, 043609 (2003).
    [CrossRef]
  22. G. Smith, S. Chaudhury, and P. Jessen, “Faraday spectroscopy in an optical lattice: a continuous probe of atom dynamics,” J. Opt. Soc. Am. B 5, 323–329 (2003).
  23. J. Lye, J. Hope, and J. Close, “Rapid real-time detection of cold atoms with minimal destruction,” Phys. Rev. A 69, 023601 (2004).
    [CrossRef]
  24. M. Kohnen, P. G. Petrov, R. A. Nyman, and E. A. Hinds, “Minimally destructive detection of magnetically trapped atoms using frequency-synthesized light,” New J. Phys. 13, 085006 (2011).
    [CrossRef]
  25. C. Figl, L. Longchambon, M. Jeppesen, M. Kruger, H. A. Bachor, N. P. Robins, and J. D. Close, “Demonstration and characterization of a detector for minimally destructive detection of Bose condensed atoms in real time,” Appl. Opt. 45, 3415–3419 (2006).
    [CrossRef]
  26. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).
  27. D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
    [CrossRef]
  28. R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Advances in Atomic, Molecular, and Optical Physics, Vol. 42 (Academic, 2000), pp. 95–170.
  29. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

2012 (1)

N. Shiga and M. Takeuchi, “Locking the local oscillator phase to the atomic phase via weak measurement,” New J. Phys. 14, 023034 (2012).
[CrossRef]

2011 (1)

M. Kohnen, P. G. Petrov, R. A. Nyman, and E. A. Hinds, “Minimally destructive detection of magnetically trapped atoms using frequency-synthesized light,” New J. Phys. 13, 085006 (2011).
[CrossRef]

2009 (2)

Y. Liu, E. Gomez, S. Maxwell, L. Turner, and E. Tiesinga, “Number fluctuations and energy dissipation in sodium spinor condensates,” Phys. Rev. Lett. 102, 225301 (2009).
[CrossRef]

J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
[CrossRef]

2008 (2)

P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
[CrossRef]

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

2006 (2)

2005 (2)

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

J. Hope and J. Close, “General limit to nondestructive optical detection of atoms,” Phys. Rev. A 71, 043822 (2005).
[CrossRef]

2004 (1)

J. Lye, J. Hope, and J. Close, “Rapid real-time detection of cold atoms with minimal destruction,” Phys. Rev. A 69, 023601 (2004).
[CrossRef]

2003 (2)

J. Lye, J. Hope, and J. Close, “Nondestructive dynamic detectors for Bose–Einstein condensates,” Phys. Rev. A 67, 043609 (2003).
[CrossRef]

G. Smith, S. Chaudhury, and P. Jessen, “Faraday spectroscopy in an optical lattice: a continuous probe of atom dynamics,” J. Opt. Soc. Am. B 5, 323–329 (2003).

1999 (1)

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
[CrossRef]

1998 (1)

T. Opatrný, M. Dakna, and D.-G. Welsch, “Number-phase uncertainty relations: verification by balanced homodyne measurement,” Phys. Rev. A 57, 2129–2133 (1998).
[CrossRef]

1997 (1)

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[CrossRef]

1996 (3)

Z. Ou, “Complementarity and fundamental limit in precision phase measurement,” Phys. Rev. Lett. 77, 2352–2355 (1996).
[CrossRef]

A. Luis and J. Peřina, “Optimum phase-shift estimation and the quantum description of the phase difference,” Phys. Rev. A 54, 4564–4570 (1996).
[CrossRef]

M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

1995 (1)

1994 (1)

G. D. Ariano and M. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022–3036 (1994).
[CrossRef]

1986 (1)

K. Leong and J. Shapiro, “Phase and amplitude uncertainties in multimode heterodyning,” Opt. Commun. 58, 73–77 (1986).
[CrossRef]

1985 (1)

J. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237–250 (1985).
[CrossRef]

1984 (1)

J. Shapiro and S. Wagner, “Phase and amplitude uncertainties in heterodyne detection,” IEEE J. Quantum Electron. 20, 803–813 (1984).
[CrossRef]

1983 (1)

Alzar, C. L. G.

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

Andrews, M.

M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Appel, J.

J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
[CrossRef]

P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
[CrossRef]

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

Ariano, G. D.

G. D. Ariano and M. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022–3036 (1994).
[CrossRef]

Bachor, H. A.

Bigelow, N. P.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Chan, V. W. S.

Chaudhury, S.

G. Smith, S. Chaudhury, and P. Jessen, “Faraday spectroscopy in an optical lattice: a continuous probe of atom dynamics,” J. Opt. Soc. Am. B 5, 323–329 (2003).

Close, J.

J. Hope and J. Close, “General limit to nondestructive optical detection of atoms,” Phys. Rev. A 71, 043822 (2005).
[CrossRef]

J. Lye, J. Hope, and J. Close, “Rapid real-time detection of cold atoms with minimal destruction,” Phys. Rev. A 69, 023601 (2004).
[CrossRef]

J. Lye, J. Hope, and J. Close, “Nondestructive dynamic detectors for Bose–Einstein condensates,” Phys. Rev. A 67, 043609 (2003).
[CrossRef]

Close, J. D.

Dakna, M.

T. Opatrný, M. Dakna, and D.-G. Welsch, “Number-phase uncertainty relations: verification by balanced homodyne measurement,” Phys. Rev. A 57, 2129–2133 (1998).
[CrossRef]

Durfee, D.

M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Durfee, D. S.

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, “Making, probing and understanding Bose–Einstein condensates,” eprint arXiv:cond-mat/9904034 (1999).

Ejnisman, R.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
[CrossRef]

Figl, C.

Garrido Alzar, C. L.

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

Gomez, E.

Y. Liu, E. Gomez, S. Maxwell, L. Turner, and E. Tiesinga, “Number fluctuations and energy dissipation in sodium spinor condensates,” Phys. Rev. Lett. 102, 225301 (2009).
[CrossRef]

Grimm, R.

R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Advances in Atomic, Molecular, and Optical Physics, Vol. 42 (Academic, 2000), pp. 95–170.

Haus, H. A.

Hinds, E. A.

M. Kohnen, P. G. Petrov, R. A. Nyman, and E. A. Hinds, “Minimally destructive detection of magnetically trapped atoms using frequency-synthesized light,” New J. Phys. 13, 085006 (2011).
[CrossRef]

Hoff, U. B.

J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
[CrossRef]

P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
[CrossRef]

Hope, J.

J. Hope and J. Close, “General limit to nondestructive optical detection of atoms,” Phys. Rev. A 71, 043822 (2005).
[CrossRef]

J. Lye, J. Hope, and J. Close, “Rapid real-time detection of cold atoms with minimal destruction,” Phys. Rev. A 69, 023601 (2004).
[CrossRef]

J. Lye, J. Hope, and J. Close, “Nondestructive dynamic detectors for Bose–Einstein condensates,” Phys. Rev. A 67, 043609 (2003).
[CrossRef]

Janis, J.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
[CrossRef]

Jeppesen, M.

Jessen, P.

G. Smith, S. Chaudhury, and P. Jessen, “Faraday spectroscopy in an optical lattice: a continuous probe of atom dynamics,” J. Opt. Soc. Am. B 5, 323–329 (2003).

Kasevich, M. A.

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

Ketterle, W.

M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, “Making, probing and understanding Bose–Einstein condensates,” eprint arXiv:cond-mat/9904034 (1999).

Kjaergaard, N.

J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
[CrossRef]

Kjærgaard, N.

P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
[CrossRef]

Kjaergaard, N.

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

Kohnen, M.

M. Kohnen, P. G. Petrov, R. A. Nyman, and E. A. Hinds, “Minimally destructive detection of magnetically trapped atoms using frequency-synthesized light,” New J. Phys. 13, 085006 (2011).
[CrossRef]

Kruger, M.

Kubasik, M.

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

Kurn, D.

M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Kuzmich, A.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
[CrossRef]

Leong, K.

K. Leong and J. Shapiro, “Phase and amplitude uncertainties in multimode heterodyning,” Opt. Commun. 58, 73–77 (1986).
[CrossRef]

Liu, Y.

Y. Liu, E. Gomez, S. Maxwell, L. Turner, and E. Tiesinga, “Number fluctuations and energy dissipation in sodium spinor condensates,” Phys. Rev. Lett. 102, 225301 (2009).
[CrossRef]

Longchambon, L.

Luis, A.

A. Luis and J. Peřina, “Optimum phase-shift estimation and the quantum description of the phase difference,” Phys. Rev. A 54, 4564–4570 (1996).
[CrossRef]

Lye, J.

J. Lye, J. Hope, and J. Close, “Rapid real-time detection of cold atoms with minimal destruction,” Phys. Rev. A 69, 023601 (2004).
[CrossRef]

J. Lye, J. Hope, and J. Close, “Nondestructive dynamic detectors for Bose–Einstein condensates,” Phys. Rev. A 67, 043609 (2003).
[CrossRef]

Mandel, L.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
[CrossRef]

Maxwell, S.

Y. Liu, E. Gomez, S. Maxwell, L. Turner, and E. Tiesinga, “Number fluctuations and energy dissipation in sodium spinor condensates,” Phys. Rev. Lett. 102, 225301 (2009).
[CrossRef]

Metcalf, H. J.

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).

Mewes, M.

M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Mikkelsen, J. K.

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

Müller, J. H.

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

Nyman, R. A.

M. Kohnen, P. G. Petrov, R. A. Nyman, and E. A. Hinds, “Minimally destructive detection of magnetically trapped atoms using frequency-synthesized light,” New J. Phys. 13, 085006 (2011).
[CrossRef]

Oblak, D.

J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
[CrossRef]

P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
[CrossRef]

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

Opatrný, T.

T. Opatrný, M. Dakna, and D.-G. Welsch, “Number-phase uncertainty relations: verification by balanced homodyne measurement,” Phys. Rev. A 57, 2129–2133 (1998).
[CrossRef]

Ou, Z.

Z. Ou, “Complementarity and fundamental limit in precision phase measurement,” Phys. Rev. Lett. 77, 2352–2355 (1996).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[CrossRef]

Ovchinnikov, Y. B.

R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Advances in Atomic, Molecular, and Optical Physics, Vol. 42 (Academic, 2000), pp. 95–170.

Paris, M.

G. D. Ariano and M. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022–3036 (1994).
[CrossRef]

Perina, J.

A. Luis and J. Peřina, “Optimum phase-shift estimation and the quantum description of the phase difference,” Phys. Rev. A 54, 4564–4570 (1996).
[CrossRef]

Petrov, P. G.

M. Kohnen, P. G. Petrov, R. A. Nyman, and E. A. Hinds, “Minimally destructive detection of magnetically trapped atoms using frequency-synthesized light,” New J. Phys. 13, 085006 (2011).
[CrossRef]

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

Polzik, E. S.

J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
[CrossRef]

P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
[CrossRef]

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

Robins, N. P.

Saffman, M.

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

Shaham, Y. K.

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

Shapiro, J.

K. Leong and J. Shapiro, “Phase and amplitude uncertainties in multimode heterodyning,” Opt. Commun. 58, 73–77 (1986).
[CrossRef]

J. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237–250 (1985).
[CrossRef]

J. Shapiro and S. Wagner, “Phase and amplitude uncertainties in heterodyne detection,” IEEE J. Quantum Electron. 20, 803–813 (1984).
[CrossRef]

Shiga, N.

N. Shiga and M. Takeuchi, “Locking the local oscillator phase to the atomic phase via weak measurement,” New J. Phys. 14, 023034 (2012).
[CrossRef]

Smith, G.

G. Smith, S. Chaudhury, and P. Jessen, “Faraday spectroscopy in an optical lattice: a continuous probe of atom dynamics,” J. Opt. Soc. Am. B 5, 323–329 (2003).

Sørensen, J. L.

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

Stamper-Kurn, D. M.

W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, “Making, probing and understanding Bose–Einstein condensates,” eprint arXiv:cond-mat/9904034 (1999).

Takeuchi, M.

N. Shiga and M. Takeuchi, “Locking the local oscillator phase to the atomic phase via weak measurement,” New J. Phys. 14, 023034 (2012).
[CrossRef]

Tiesinga, E.

Y. Liu, E. Gomez, S. Maxwell, L. Turner, and E. Tiesinga, “Number fluctuations and energy dissipation in sodium spinor condensates,” Phys. Rev. Lett. 102, 225301 (2009).
[CrossRef]

Tittel, W.

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

Turner, L.

Y. Liu, E. Gomez, S. Maxwell, L. Turner, and E. Tiesinga, “Number fluctuations and energy dissipation in sodium spinor condensates,” Phys. Rev. Lett. 102, 225301 (2009).
[CrossRef]

van der Straten, P.

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).

van Druten, N.

M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Vershovski, A. K.

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

Wagner, S.

J. Shapiro and S. Wagner, “Phase and amplitude uncertainties in heterodyne detection,” IEEE J. Quantum Electron. 20, 803–813 (1984).
[CrossRef]

Weidemüller, M.

R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Advances in Atomic, Molecular, and Optical Physics, Vol. 42 (Academic, 2000), pp. 95–170.

Welsch, D.-G.

T. Opatrný, M. Dakna, and D.-G. Welsch, “Number-phase uncertainty relations: verification by balanced homodyne measurement,” Phys. Rev. A 57, 2129–2133 (1998).
[CrossRef]

Windpassinger, P. J.

J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
[CrossRef]

P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
[CrossRef]

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Young, Y. E.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
[CrossRef]

Yuen, H. P.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

J. Shapiro and S. Wagner, “Phase and amplitude uncertainties in heterodyne detection,” IEEE J. Quantum Electron. 20, 803–813 (1984).
[CrossRef]

J. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237–250 (1985).
[CrossRef]

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

H. A. Haus, “From classical to quantum noise,” J. Opt. Soc. Am. B 12, 2019–2036 (1995).
[CrossRef]

G. Smith, S. Chaudhury, and P. Jessen, “Faraday spectroscopy in an optical lattice: a continuous probe of atom dynamics,” J. Opt. Soc. Am. B 5, 323–329 (2003).

New J. Phys. (3)

M. Kohnen, P. G. Petrov, R. A. Nyman, and E. A. Hinds, “Minimally destructive detection of magnetically trapped atoms using frequency-synthesized light,” New J. Phys. 13, 085006 (2011).
[CrossRef]

P. J. Windpassinger, D. Oblak, U. B. Hoff, J. Appel, N. Kjærgaard, and E. S. Polzik, “Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements,” New J. Phys. 10, 053032 (2008).
[CrossRef]

N. Shiga and M. Takeuchi, “Locking the local oscillator phase to the atomic phase via weak measurement,” New J. Phys. 14, 023034 (2012).
[CrossRef]

Opt. Commun. (1)

K. Leong and J. Shapiro, “Phase and amplitude uncertainties in multimode heterodyning,” Opt. Commun. 58, 73–77 (1986).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (9)

G. D. Ariano and M. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022–3036 (1994).
[CrossRef]

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999).
[CrossRef]

D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik, “Quantum-noise-limited interferometric measurement of atomic noise: towards spin squeezing on the cs clock transition,” Phys. Rev. A 71, 043807 (2005).
[CrossRef]

J. Hope and J. Close, “General limit to nondestructive optical detection of atoms,” Phys. Rev. A 71, 043822 (2005).
[CrossRef]

J. Lye, J. Hope, and J. Close, “Rapid real-time detection of cold atoms with minimal destruction,” Phys. Rev. A 69, 023601 (2004).
[CrossRef]

A. Luis and J. Peřina, “Optimum phase-shift estimation and the quantum description of the phase difference,” Phys. Rev. A 54, 4564–4570 (1996).
[CrossRef]

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[CrossRef]

T. Opatrný, M. Dakna, and D.-G. Welsch, “Number-phase uncertainty relations: verification by balanced homodyne measurement,” Phys. Rev. A 57, 2129–2133 (1998).
[CrossRef]

J. Lye, J. Hope, and J. Close, “Nondestructive dynamic detectors for Bose–Einstein condensates,” Phys. Rev. A 67, 043609 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

Y. Liu, E. Gomez, S. Maxwell, L. Turner, and E. Tiesinga, “Number fluctuations and energy dissipation in sodium spinor condensates,” Phys. Rev. Lett. 102, 225301 (2009).
[CrossRef]

P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjaergaard, and E. S. Polzik, “Nondestructive probing of Rabi oscillations on the cesium clock transition near the standard quantum limit,” Phys. Rev. Lett. 100, 103601 (2008).
[CrossRef]

Z. Ou, “Complementarity and fundamental limit in precision phase measurement,” Phys. Rev. Lett. 77, 2352–2355 (1996).
[CrossRef]

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjaergaard, and E. S. Polzik, “Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit,” Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965 (2009).
[CrossRef]

Science (1)

M. Andrews, M. Mewes, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Other (4)

W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, “Making, probing and understanding Bose–Einstein condensates,” eprint arXiv:cond-mat/9904034 (1999).

R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Advances in Atomic, Molecular, and Optical Physics, Vol. 42 (Academic, 2000), pp. 95–170.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).

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Figures (7)

Fig. 1.
Fig. 1.

Simplified schematic of SUSHI detection. Thick lines represent optical paths (in fiber and/or free space), orange lines represent RF signal paths, and thin light black lines represent baseband signal paths. Padlock icons represent the electronic signal processing for OPLLs. Inset depicts detunings (not to scale) of the laser source and LSB, USB, and LO optical fields relative to an isolated atomic optical resonance. The dashed box denotes the set-point dither circuit used in bench-test measurements of the SNR for SUSHI; it can introduce a time-dependent phase modulation between the two probe beam colors, simulating the effect on the probe sidebands of an ensemble of atomic spins undergoing driven Rabi flopping.

Fig. 2.
Fig. 2.

SUSHI minimum detectable cloud phase shift versus total probe beam power. The dashed black line is the fundamental limit for heterodyne phase detection of an atom-cloud-induced phase shift, using a noiseless detector of quantum efficiency κ=0.6. The blue line is a no-free-parameters calculation of the theoretical sensitivity of a SUSHI measurement using a probe beam composed of two equal power sidebands totaling the shown power, a 2.4 mW LO beam, and a detector having dark noise SI2det=(17pA/Hz)2. The brown horizontal line at 37nrad/Hz is the sensitivity floor set by the εrpn=52nrad/Hz residual phase noise between the USB and LSB fields. (Section 2.C discusses why the former is smaller than the latter by the factor 1/2.) The black curve is the uncorrelated combination of the two sources of phase noise. Solid circles are our experimental determinations of the minimum detectable signal based on measurements of the phase-noise floor of the final SUSHI signal, as described in Section 4.

Fig. 3.
Fig. 3.

Plot of Eq. (20); the (per-atom) signal-to-noise root bandwidth product versus cross-sectional area of the cloud for a probe on the 780 nm D2 line of Rb87, in the two-level limit. The curves correspond to four different choices of (per-atom) spontaneous scattering rate. A quantum efficiency of κ=0.6 was assumed for the calculations.

Fig. 4.
Fig. 4.

Calculated SUSHI performance metrics for the internal bracketing detuning configuration on the Rb87 D1 line. A (main), plot of scaled phase shift versus inverse scattering rate. Blue curve is for sidebands bracketing the (F=2F=1) transition, as depicted in the upper right inset; red curve is for sidebands bracketing the (F=2F=2) transition, as depicted in the lower left inset. Black, dashed curve is a calculation valid for resolved fine structure but completely unresolved hyperfine structure, provided for reference. A (insets), LSB is the leftmost arrow, USB the rightmost. X is the total frequency interval between sidebands; a and b are the detuning of the LSB and USB, respectively, from the nearest allowed transition. B, plot of efficiency versus sideband splitting. Colors and points match those of A; black, dashed curve is η2l, the two-level atom result. C,D, same as A,B, but for atoms in the F=1 hyperfine ground state. Blue curves are for sidebands bracketing the F=1F=2 transition; red curve is for sidebands bracketing the F=1F=1 transition.

Fig. 5.
Fig. 5.

Similar to Fig. 4, but for the external bracketing configuration on the Rb87 D2 line, as described in the text. A, red curve is for unpolarized atoms are in the F=1 hyperfine ground state, LSB below the F=1F=0 transition and USB above the F=1F=2 transition, as depicted in the upper right inset. Blue curve is for unpolarized atoms in the F=2 hyperfine ground state, LSB below the F=2F=1 transition and USB above the F=2F=3 transition, as depicted in the lower left inset. Black, dashed curve is a calculation valid for resolved fine structure but completely unresolved hyperfine structure, provided for reference. B, red (blue) curve is for atoms in the F=1(F=2) ground state, respectively.

Fig. 6.
Fig. 6.

Example phase-noise spectra for SUSHI detection. Gain peaks in the OPLLs are observed around 10 kHz. Upper, blue (lower, black) trace is the observed noise floor for SUSHI detection with a 650 pW (230 μW) probe beam, respectively. The noise values used to infer the minimum detectable atomic phase signal of Fig. 2 are taken at 1.5 kHz from data sets like these, measured at various probe powers.

Fig. 7.
Fig. 7.

Minimum detectable phase shift for a single heterodyne dispersive phase measurement, versus probe beam power, for one- and two-arm configurations. Red lines are one-arm detection with f=1/2 (see text); blue lines are two-arm detection with Plo=2.4mW. Each color group shows curves for three κ=.6 photodetectors having dark noise levels of N.E.P.=3,17, and 100pW/Hz (lowest to highest curves on the graph). The noiseless detector limit for each type of measurement is shown in dashed lines of the same color. The top of the shaded region is the quantum limit for unit quantum efficiency, noiseless detection of a classical probe. A probe laser wavelength of 780 nm is assumed.

Equations (42)

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

SNRB=KΔϕ2κPprobe,
LSB(D1)=t2Elsbcos[ωlsbt+εrpn],
USB(D1)=t2Eusbcos[ωusbt],
Δϕusb(t)=Δϕlsb(t)Δϕcl(t).
LSB(D2)=dr2t3Elsbcos[ωlsbt+εshΔϕcl(t)+εrpn],
USB(D2)=dr2t3Eusbcos[ωusbt+εsh+Δϕcl(t)].
LO(D2)=r3Elocos[ωlot+εlo].
Itot=ρ(LO(D2)+LSB(D2)+USB(D2))2=IRF+IDC,
IRF=Ilolsb+Ilousb+Ilsbusb,
Ilolsb=Ilolsb0cos[εlo+εsh+εrpnΔϕclt(ωlo+ωlsb)],
Ilousb=Ilousb0cos[εlo+εsh+Δϕclt(ωlo+ωusb)].
Vpfd(ϕlolsb)=mlsb[εcomΔϕcl(t)+εrpn+εhet(Ilolsb0)],
Vpfd(ϕlousb)=musb[εcom+Δϕcl(t)+εhet(Ilousb0)],
VSUSHI=Vpfd(ϕlolsb)Vpfd(ϕlousb)=m[2Δϕcl+εrpn+εhet(Ilolsb0)+εhet(Ilousb0)]VSUSHI+εSUSHI,
LO(D2,locked)=r3Elocos[ωlot+εshΔϕcl(t)+εhet(Ilolsb0)],
VSUSHI=Vpfd(ϕlousb)=musb[2Δϕcl+εrpn+εhet(Ilolsb0)+εhet(Ilousb0)],
SNR(VSUSHI)=22×Δϕclκ2Plo2PxsbB2κPlo+2κPxsb+SI2det+SI2rpn.
Δϕmin,cl,SUSHI=(Δϕmin,cl,SUSHI)2+Sϕ2rpn
SI2rpn=Sϕ2rpnκ2Plo2Pxsb,
[SNR(VSUSHI)]PloB=Δϕcl2κPprobe.
[Δϕcl,min,SUSHI]Plo=B2κPprobe.
ΔϕSUSHI(2l)=Δϕcl,lsbΔϕcl,usb=Nσ04A(1δlsb+1δusb)=Nσ02Aδ,
Γsc(2l)=1(t3)2×σ04A(Plsbδlsb2+Pusbδusb2)=Pprobeσ0(t3)24Aδ2,
[SNR(VSUSHI)]Plo(2l)NB=2κσ0Γsc(2l)A
Δϕ˜SUSHIANΔϕSUSHI,
Γ˜scAPprobeΓsc=APprobe/ωΓsc,
Δϕ˜SUSHI(Dn,F)=σ0FT.S.(Dn,F,F)×(δlsb(Dn,F,F)1+4[δlsb(Dn,F,F)]2δusb(Dn,F,F)1+4[δusb(Dn,F,F)]2),
Γ˜sc(Dn,F)=1(t3)2×σ0FT.S.(Dn,F,F)×(11+4[δlsb(Dn,F,F)]2+11+4[δusb(Dn,F,F)]2).
SNRB=Δϕ2κP12κP22κP1+2κP2+SI2det
SNRB=(KK)Δϕ2κPprobe.
K1arm=f(1f),
K2arm=(1+PprobePlo)1/2.
K1arm=(1+SI2det2Pprobe)1/2,
K2arm=(1+SI2det2Plo+2Pprobe)1/2.
E(x=,t)=E0exp[iωt+iRe(κ)Im(κ)]
κ=ωc1+nαε0ωc(1+nα2ε0),
α=6πε0c3Γ/ω03ω02ω2i(ω3/ω02)Γ.
Re(α)=12πε0c3ω3δ1+4δ2,
Im(α)=12δRe(α).
E(x=,t)=E0exp[iωt+iω(/c)iΔϕA],
Δϕncolσ0δ1+4δ2
A12ncolσ011+4δ2

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