Abstract

We theoretically investigate a quantum nondemolition (QND) measurement with optical Kerr effect in an ultra-high-Q microtoroidal system. The analytical and numerical results predict that the present QND measurement scheme possesses a high sensitivity, which allows for detecting few photons or even single photons. Ultra-high-Q toroidal microcavity may provide a novel experimental platform to study quantum physics with nonlinear optics at low light levels.

© 2008 Optical Society of America

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  1. C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980).
    [CrossRef]
  2. F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002).
    [CrossRef]
  3. P. Grangier, J. -A. Levenson, and J.-P. Poizat, "Quantum nondemolition measurement in optics," Nature 396, 537-542 (1998).
    [CrossRef]
  4. J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992).
    [CrossRef]
  5. G. Leuchs, C. Silberhorn,K¨onig, P. K.  Lam, A. Sizmann, and N. Korolkova, in Quantum Information Theory with Continous Variables edited by S. L. Braunstein and A. K. Pati (Kluwer Academic, Dordrecht, 2002).
  6. 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]
  7. Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978).
    [CrossRef]
  8. V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum nondemolition measurements," Science 209, 547-557 (1980).
    [CrossRef] [PubMed]
  9. N. Imoto, H. A. Haus, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number via the optical Kerr effect," Phys. Rev. A 32, 2287-2292 (1985).
    [CrossRef] [PubMed]
  10. P. Alsing, G. J. Milburn, D. F. Walls, "Quantum nondemolition measurements in optical cavities," Phys. Rev. A 37, 2970-2978 (1988).
    [CrossRef] [PubMed]
  11. H. A. Haus and F. X. Kärtner, "Optical quantum nondemolition measurement and the Copenhagen Interpretation," Phys. Rev. A. 53, 3785-3791 (1986).
    [CrossRef]
  12. V. B. Braginsky and F. Ya. Khalili, "Quantum nondemolition measurements: the route from toys to tools," Rev. Mod. Phys. 68, 1-11 (1996).
    [CrossRef]
  13. J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998).
    [CrossRef]
  14. N. Imoto and S. Saito, "Quantum nondemolition measurement of photon number in a lossy optical Kerr medium", Phys. Rev. A 39, 675-682 (1989).
    [CrossRef] [PubMed]
  15. M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, "Nonideal quantum nondemolition measurements," Phys. Rev. A. 42, 2995-3005 (1990).
    [CrossRef] [PubMed]
  16. P. Grangier, J.M. Courty, and S. Reynaud, "Characterization of nonideal quantum nondemolition measurements," Opt. Commun. 89, 99-106 (1992).
    [CrossRef]
  17. M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, "Quantum nondemolition detection of optical quadrature amplitudes," Phys. Rev. Lett. 57, 2473-2476 (1986).
    [CrossRef] [PubMed]
  18. N. Imoto, S. Watkins, and Y. Sasaki, "A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number," Opt. Comm. 61, 159-163 (1987).
    [CrossRef]
  19. S. R. Friberg, S. Machida, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number of an optical soliton," Phys. Rev. Lett. 69, 3165-3168 (1992).
    [CrossRef] [PubMed]
  20. J. P. Poizat, P. Grangier, "Experimental realization of a quantum optical tap," Phys. Rev. Lett. 70, 271-274 (1993).
    [CrossRef] [PubMed]
  21. S. F. Pereira, Z. Y. Ou, and H. J. Kimble, "Back-action evading measurements for quantum non-demolition detection and quantum optical tapping," Phys. Rev. Lett. 72, 214-217 (1994).
    [CrossRef] [PubMed]
  22. K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, "Quantum non-demolition demonstration via repeated back-action evading measurements," Phys. Rev. Lett. 75, 3422-3425 (1995).
    [CrossRef] [PubMed]
  23. R. Bruckmeier, H. Hansen, and S. Schiller, "Repeated quantum non-demolition measurements of continuous optical waves," Phys. Rev. Lett. 79, 1463-1466 (1997).
    [CrossRef]
  24. K. Bencheick, C. Simonneau, and J. A. Levenson, "Cascaded amplifying quantum optical taps: A robust noiseless optical bus," Phys. Rev. Lett. 78, 34-37 (1997).
    [CrossRef]
  25. B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001).
    [CrossRef]
  26. J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
    [CrossRef]
  27. A. Kuzmich, L. Mandel, and N. P. Bigelow, "Generation of spin squeezing via continous quantum nondemolition measurement," Phys. Rev. Lett. 85, 1594-1597 (2000).
    [CrossRef] [PubMed]
  28. S. Peil, and G. Gabrielse, "Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states," Phys. Rev. Lett. 83, 1287-1290 (1999).
    [CrossRef]
  29. G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
    [CrossRef]
  30. C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
    [CrossRef] [PubMed]
  31. G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004).
    [CrossRef] [PubMed]
  32. Q2. I. Fushman and J. Vučković, "Analysis of a quantum dondemolition measurement scheme based on Kerr nonlinearity in photonic crystal waveguides," Opt. Express 15, 5559-5571 (2007).
    [CrossRef] [PubMed]
  33. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
    [CrossRef] [PubMed]
  34. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
    [CrossRef]
  35. M. Yamane and Y. Asahara, Glass for Photonics (Cambridge University Press, Cambridge, 2000).
    [CrossRef]
  36. S. X. Qian and R. K. Chang, "Multiorder stokes emission from micrometer-size droplets," Phys. Rev. Lett. 56, 926-929 (1986).
    [CrossRef] [PubMed]
  37. A. J. Campillo, J. D. Eversole, and H. B. Lin, "Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets," Phys. Rev. Lett. 67, 437-440 (1991).
    [CrossRef] [PubMed]
  38. F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).
  39. S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, "Stimulated raman scattering in a liquid-hydrogen droplet," Phys. Rev. A 61, 011803 (2000).
    [CrossRef]
  40. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621 (2002).
    [CrossRef] [PubMed]
  41. Q3. T. Carmon and K. J. Vahala, "Visible continuous emission from a silica microphonic device by third-harmonic generation," Nature Physics 3, 430-435 (2007).
    [CrossRef]
  42. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
    [CrossRef] [PubMed]
  43. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004).
    [CrossRef]
  44. T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behavior and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004).
    [CrossRef] [PubMed]
  45. D. F. Walls and G. J. Milburn, Quantum Optics(Springer-Verlag, Berlin Heidelberg, 1994)
  46. C.W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
    [CrossRef] [PubMed]
  47. S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, "Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation," Phys. Rev. A 66, 053809 (2002).
    [CrossRef]
  48. Mani Hossein-Zadeh, and Kerry J. Vahala "Free ultra-high-Q microtoroid: a tool for designing photonic devices," Opt. Express 15, 166-175 (2007).
    [CrossRef] [PubMed]

2007 (4)

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

Q2. I. Fushman and J. Vučković, "Analysis of a quantum dondemolition measurement scheme based on Kerr nonlinearity in photonic crystal waveguides," Opt. Express 15, 5559-5571 (2007).
[CrossRef] [PubMed]

Q3. T. Carmon and K. J. Vahala, "Visible continuous emission from a silica microphonic device by third-harmonic generation," Nature Physics 3, 430-435 (2007).
[CrossRef]

Mani Hossein-Zadeh, and Kerry J. Vahala "Free ultra-high-Q microtoroid: a tool for designing photonic devices," Opt. Express 15, 166-175 (2007).
[CrossRef] [PubMed]

2005 (2)

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

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 (4)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004).
[CrossRef]

T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behavior and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004).
[CrossRef] [PubMed]

2003 (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

2002 (3)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621 (2002).
[CrossRef] [PubMed]

S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, "Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation," Phys. Rev. A 66, 053809 (2002).
[CrossRef]

F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002).
[CrossRef]

2001 (1)

B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001).
[CrossRef]

2000 (2)

A. Kuzmich, L. Mandel, and N. P. Bigelow, "Generation of spin squeezing via continous quantum nondemolition measurement," Phys. Rev. Lett. 85, 1594-1597 (2000).
[CrossRef] [PubMed]

S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, "Stimulated raman scattering in a liquid-hydrogen droplet," Phys. Rev. A 61, 011803 (2000).
[CrossRef]

1999 (2)

S. Peil, and G. Gabrielse, "Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states," Phys. Rev. Lett. 83, 1287-1290 (1999).
[CrossRef]

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

1998 (3)

P. Grangier, J. -A. Levenson, and J.-P. Poizat, "Quantum nondemolition measurement in optics," Nature 396, 537-542 (1998).
[CrossRef]

J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

1997 (3)

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
[CrossRef]

R. Bruckmeier, H. Hansen, and S. Schiller, "Repeated quantum non-demolition measurements of continuous optical waves," Phys. Rev. Lett. 79, 1463-1466 (1997).
[CrossRef]

K. Bencheick, C. Simonneau, and J. A. Levenson, "Cascaded amplifying quantum optical taps: A robust noiseless optical bus," Phys. Rev. Lett. 78, 34-37 (1997).
[CrossRef]

1996 (1)

V. B. Braginsky and F. Ya. Khalili, "Quantum nondemolition measurements: the route from toys to tools," Rev. Mod. Phys. 68, 1-11 (1996).
[CrossRef]

1995 (1)

K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, "Quantum non-demolition demonstration via repeated back-action evading measurements," Phys. Rev. Lett. 75, 3422-3425 (1995).
[CrossRef] [PubMed]

1994 (1)

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, "Back-action evading measurements for quantum non-demolition detection and quantum optical tapping," Phys. Rev. Lett. 72, 214-217 (1994).
[CrossRef] [PubMed]

1993 (1)

J. P. Poizat, P. Grangier, "Experimental realization of a quantum optical tap," Phys. Rev. Lett. 70, 271-274 (1993).
[CrossRef] [PubMed]

1992 (3)

S. R. Friberg, S. Machida, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number of an optical soliton," Phys. Rev. Lett. 69, 3165-3168 (1992).
[CrossRef] [PubMed]

P. Grangier, J.M. Courty, and S. Reynaud, "Characterization of nonideal quantum nondemolition measurements," Opt. Commun. 89, 99-106 (1992).
[CrossRef]

J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992).
[CrossRef]

1991 (1)

A. J. Campillo, J. D. Eversole, and H. B. Lin, "Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets," Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

1990 (1)

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, "Nonideal quantum nondemolition measurements," Phys. Rev. A. 42, 2995-3005 (1990).
[CrossRef] [PubMed]

1989 (1)

N. Imoto and S. Saito, "Quantum nondemolition measurement of photon number in a lossy optical Kerr medium", Phys. Rev. A 39, 675-682 (1989).
[CrossRef] [PubMed]

1988 (1)

P. Alsing, G. J. Milburn, D. F. Walls, "Quantum nondemolition measurements in optical cavities," Phys. Rev. A 37, 2970-2978 (1988).
[CrossRef] [PubMed]

1987 (1)

N. Imoto, S. Watkins, and Y. Sasaki, "A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number," Opt. Comm. 61, 159-163 (1987).
[CrossRef]

1986 (3)

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, "Quantum nondemolition detection of optical quadrature amplitudes," Phys. Rev. Lett. 57, 2473-2476 (1986).
[CrossRef] [PubMed]

H. A. Haus and F. X. Kärtner, "Optical quantum nondemolition measurement and the Copenhagen Interpretation," Phys. Rev. A. 53, 3785-3791 (1986).
[CrossRef]

S. X. Qian and R. K. Chang, "Multiorder stokes emission from micrometer-size droplets," Phys. Rev. Lett. 56, 926-929 (1986).
[CrossRef] [PubMed]

1985 (2)

N. Imoto, H. A. Haus, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number via the optical Kerr effect," Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

C.W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

1980 (2)

C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980).
[CrossRef]

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

1978 (1)

Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978).
[CrossRef]

Alsing, P.

P. Alsing, G. J. Milburn, D. F. Walls, "Quantum nondemolition measurements in optical cavities," Phys. Rev. A 37, 2970-2978 (1988).
[CrossRef] [PubMed]

Andersen, U. L.

B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001).
[CrossRef]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Bachor, H. -A.

B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001).
[CrossRef]

Bartlett, S. D.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004).
[CrossRef] [PubMed]

Bencheick, K.

K. Bencheick, C. Simonneau, and J. A. Levenson, "Cascaded amplifying quantum optical taps: A robust noiseless optical bus," Phys. Rev. Lett. 78, 34-37 (1997).
[CrossRef]

Bencheikh, K.

K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, "Quantum non-demolition demonstration via repeated back-action evading measurements," Phys. Rev. Lett. 75, 3422-3425 (1995).
[CrossRef] [PubMed]

Bernu, J.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

Bigelow, N. P.

A. Kuzmich, L. Mandel, and N. P. Bigelow, "Generation of spin squeezing via continous quantum nondemolition measurement," Phys. Rev. Lett. 85, 1594-1597 (2000).
[CrossRef] [PubMed]

Braginsky, V. B.

V. B. Braginsky and F. Ya. Khalili, "Quantum nondemolition measurements: the route from toys to tools," Rev. Mod. Phys. 68, 1-11 (1996).
[CrossRef]

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Bruckmeier, R.

R. Bruckmeier, H. Hansen, and S. Schiller, "Repeated quantum non-demolition measurements of continuous optical waves," Phys. Rev. Lett. 79, 1463-1466 (1997).
[CrossRef]

Brune, M.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

Buchler, B.

F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002).
[CrossRef]

Buchler, B. C.

B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001).
[CrossRef]

Campillo, A. J.

A. J. Campillo, J. D. Eversole, and H. B. Lin, "Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets," Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

Carmon, T.

Q3. T. Carmon and K. J. Vahala, "Visible continuous emission from a silica microphonic device by third-harmonic generation," Nature Physics 3, 430-435 (2007).
[CrossRef]

T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behavior and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980).
[CrossRef]

Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978).
[CrossRef]

Chang, R. K.

S. X. Qian and R. K. Chang, "Multiorder stokes emission from micrometer-size droplets," Phys. Rev. Lett. 56, 926-929 (1986).
[CrossRef] [PubMed]

Collett, M. J.

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, "Nonideal quantum nondemolition measurements," Phys. Rev. A. 42, 2995-3005 (1990).
[CrossRef] [PubMed]

C.W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

Courty, J. M.

J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998).
[CrossRef]

J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992).
[CrossRef]

Courty, J.M.

P. Grangier, J.M. Courty, and S. Reynaud, "Characterization of nonideal quantum nondemolition measurements," Opt. Commun. 89, 99-106 (1992).
[CrossRef]

Delglise, S.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

Drever, R. W. P.

C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980).
[CrossRef]

Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978).
[CrossRef]

Eversole, J. D.

A. J. Campillo, J. D. Eversole, and H. B. Lin, "Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets," Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

Friberg, S. R.

S. R. Friberg, S. Machida, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number of an optical soliton," Phys. Rev. Lett. 69, 3165-3168 (1992).
[CrossRef] [PubMed]

Fushman, I.

Gabrielse, G.

S. Peil, and G. Gabrielse, "Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states," Phys. Rev. Lett. 83, 1287-1290 (1999).
[CrossRef]

Gardiner, C.W.

C.W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

Gleyzes, S.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

Goh, K. W.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

Grangier, P.

P. Grangier, J. -A. Levenson, and J.-P. Poizat, "Quantum nondemolition measurement in optics," Nature 396, 537-542 (1998).
[CrossRef]

K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, "Quantum non-demolition demonstration via repeated back-action evading measurements," Phys. Rev. Lett. 75, 3422-3425 (1995).
[CrossRef] [PubMed]

J. P. Poizat, P. Grangier, "Experimental realization of a quantum optical tap," Phys. Rev. Lett. 70, 271-274 (1993).
[CrossRef] [PubMed]

P. Grangier, J.M. Courty, and S. Reynaud, "Characterization of nonideal quantum nondemolition measurements," Opt. Commun. 89, 99-106 (1992).
[CrossRef]

J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992).
[CrossRef]

Grangier, Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
[CrossRef]

Grelu, Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
[CrossRef]

Guerlin, C.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

Hakuta, K.

S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, "Stimulated raman scattering in a liquid-hydrogen droplet," Phys. Rev. A 61, 011803 (2000).
[CrossRef]

Hansen, H.

R. Bruckmeier, H. Hansen, and S. Schiller, "Repeated quantum non-demolition measurements of continuous optical waves," Phys. Rev. Lett. 79, 1463-1466 (1997).
[CrossRef]

Hare, J.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Haroche, S.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Haus, H. A.

H. A. Haus and F. X. Kärtner, "Optical quantum nondemolition measurement and the Copenhagen Interpretation," Phys. Rev. A. 53, 3785-3791 (1986).
[CrossRef]

N. Imoto, H. A. Haus, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number via the optical Kerr effect," Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

Holland, M. J.

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, "Nonideal quantum nondemolition measurements," Phys. Rev. A. 42, 2995-3005 (1990).
[CrossRef] [PubMed]

Hossein-Zadeh, Mani

Ilchenko, V. S.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Imoto, N.

S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, "Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation," Phys. Rev. A 66, 053809 (2002).
[CrossRef]

N. Imoto and S. Saito, "Quantum nondemolition measurement of photon number in a lossy optical Kerr medium", Phys. Rev. A 39, 675-682 (1989).
[CrossRef] [PubMed]

N. Imoto, S. Watkins, and Y. Sasaki, "A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number," Opt. Comm. 61, 159-163 (1987).
[CrossRef]

N. Imoto, H. A. Haus, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number via the optical Kerr effect," Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

Kärtner, F. X.

H. A. Haus and F. X. Kärtner, "Optical quantum nondemolition measurement and the Copenhagen Interpretation," Phys. Rev. A. 53, 3785-3791 (1986).
[CrossRef]

Katsuragawa, M.

S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, "Stimulated raman scattering in a liquid-hydrogen droplet," Phys. Rev. A 61, 011803 (2000).
[CrossRef]

Khalili, F. Ya.

V. B. Braginsky and F. Ya. Khalili, "Quantum nondemolition measurements: the route from toys to tools," Rev. Mod. Phys. 68, 1-11 (1996).
[CrossRef]

Kimble, H. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, "Back-action evading measurements for quantum non-demolition detection and quantum optical tapping," Phys. Rev. Lett. 72, 214-217 (1994).
[CrossRef] [PubMed]

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621 (2002).
[CrossRef] [PubMed]

Koashi, M.

S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, "Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation," Phys. Rev. A 66, 053809 (2002).
[CrossRef]

König, F.

F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002).
[CrossRef]

Krönig, F.

J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998).
[CrossRef]

Kuhr, S.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

Kuzmich, A.

A. Kuzmich, L. Mandel, and N. P. Bigelow, "Generation of spin squeezing via continous quantum nondemolition measurement," Phys. Rev. Lett. 85, 1594-1597 (2000).
[CrossRef] [PubMed]

Lam, P. K.

B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001).
[CrossRef]

Lefevre-Seguin, V.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Leuchs, G.

F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002).
[CrossRef]

J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998).
[CrossRef]

Levenson, J. A.

K. Bencheick, C. Simonneau, and J. A. Levenson, "Cascaded amplifying quantum optical taps: A robust noiseless optical bus," Phys. Rev. Lett. 78, 34-37 (1997).
[CrossRef]

K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, "Quantum non-demolition demonstration via repeated back-action evading measurements," Phys. Rev. Lett. 75, 3422-3425 (1995).
[CrossRef] [PubMed]

Levenson, J. -A.

P. Grangier, J. -A. Levenson, and J.-P. Poizat, "Quantum nondemolition measurement in optics," Nature 396, 537-542 (1998).
[CrossRef]

Levenson, M. D.

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, "Nonideal quantum nondemolition measurements," Phys. Rev. A. 42, 2995-3005 (1990).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, "Quantum nondemolition detection of optical quadrature amplitudes," Phys. Rev. Lett. 57, 2473-2476 (1986).
[CrossRef] [PubMed]

Lin, H. B.

A. J. Campillo, J. D. Eversole, and H. B. Lin, "Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets," Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

Lopez, O.

K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, "Quantum non-demolition demonstration via repeated back-action evading measurements," Phys. Rev. Lett. 75, 3422-3425 (1995).
[CrossRef] [PubMed]

Machida, S.

S. R. Friberg, S. Machida, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number of an optical soliton," Phys. Rev. Lett. 69, 3165-3168 (1992).
[CrossRef] [PubMed]

Mandel, L.

A. Kuzmich, L. Mandel, and N. P. Bigelow, "Generation of spin squeezing via continous quantum nondemolition measurement," Phys. Rev. Lett. 85, 1594-1597 (2000).
[CrossRef] [PubMed]

Milburn, G. J.

P. Alsing, G. J. Milburn, D. F. Walls, "Quantum nondemolition measurements in optical cavities," Phys. Rev. A 37, 2970-2978 (1988).
[CrossRef] [PubMed]

Miranowicz, A.

S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, "Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation," Phys. Rev. A 66, 053809 (2002).
[CrossRef]

Munro, W. J.

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]

Nemoto, K.

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]

Nogues, G.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

O’Brien, J. L.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004).
[CrossRef] [PubMed]

Osnaghi, S.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

Ou, Z. Y.

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, "Back-action evading measurements for quantum non-demolition detection and quantum optical tapping," Phys. Rev. Lett. 72, 214-217 (1994).
[CrossRef] [PubMed]

Ozdemir, S. K.

S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, "Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation," Phys. Rev. A 66, 053809 (2002).
[CrossRef]

Peil, S.

S. Peil, and G. Gabrielse, "Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states," Phys. Rev. Lett. 83, 1287-1290 (1999).
[CrossRef]

Pereira, S. F.

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, "Back-action evading measurements for quantum non-demolition detection and quantum optical tapping," Phys. Rev. Lett. 72, 214-217 (1994).
[CrossRef] [PubMed]

Poizat, J. P.

J. P. Poizat, P. Grangier, "Experimental realization of a quantum optical tap," Phys. Rev. Lett. 70, 271-274 (1993).
[CrossRef] [PubMed]

Poizat, J.-P.

P. Grangier, J. -A. Levenson, and J.-P. Poizat, "Quantum nondemolition measurement in optics," Nature 396, 537-542 (1998).
[CrossRef]

Poizat, J.-Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
[CrossRef]

Pryde, G. J.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004).
[CrossRef] [PubMed]

Qian, S. X.

S. X. Qian and R. K. Chang, "Multiorder stokes emission from micrometer-size droplets," Phys. Rev. Lett. 56, 926-929 (1986).
[CrossRef] [PubMed]

Raimond, J. M.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Ralph, T. C.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004).
[CrossRef] [PubMed]

B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001).
[CrossRef]

Rauschenbeutel, A.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

Rechtenwald, T.

F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002).
[CrossRef]

Reid, M.

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, "Quantum nondemolition detection of optical quadrature amplitudes," Phys. Rev. Lett. 57, 2473-2476 (1986).
[CrossRef] [PubMed]

Reynaud, S.

P. Grangier, J.M. Courty, and S. Reynaud, "Characterization of nonideal quantum nondemolition measurements," Opt. Commun. 89, 99-106 (1992).
[CrossRef]

J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992).
[CrossRef]

Roch, J. F.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992).
[CrossRef]

Roch, J.-F.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
[CrossRef]

Roger, G.

J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992).
[CrossRef]

Saito, S.

N. Imoto and S. Saito, "Quantum nondemolition measurement of photon number in a lossy optical Kerr medium", Phys. Rev. A 39, 675-682 (1989).
[CrossRef] [PubMed]

Sandberg, V. D.

C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980).
[CrossRef]

Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978).
[CrossRef]

Sasaki, Y.

N. Imoto, S. Watkins, and Y. Sasaki, "A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number," Opt. Comm. 61, 159-163 (1987).
[CrossRef]

Sayrin, C.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

Schiller, S.

R. Bruckmeier, H. Hansen, and S. Schiller, "Repeated quantum non-demolition measurements of continuous optical waves," Phys. Rev. Lett. 79, 1463-1466 (1997).
[CrossRef]

Shelby, R. M.

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, "Quantum nondemolition detection of optical quadrature amplitudes," Phys. Rev. Lett. 57, 2473-2476 (1986).
[CrossRef] [PubMed]

Simonneau, C.

K. Bencheick, C. Simonneau, and J. A. Levenson, "Cascaded amplifying quantum optical taps: A robust noiseless optical bus," Phys. Rev. Lett. 78, 34-37 (1997).
[CrossRef]

Sinatra, A.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
[CrossRef]

Sizmann, A.

J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998).
[CrossRef]

Sizmann, S.

F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002).
[CrossRef]

Sp¨alter, S.

J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998).
[CrossRef]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621 (2002).
[CrossRef] [PubMed]

Spiller, T. P.

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]

Suzuki, M.

S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, "Stimulated raman scattering in a liquid-hydrogen droplet," Phys. Rev. A 61, 011803 (2000).
[CrossRef]

Thorne, K, S.

C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980).
[CrossRef]

Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978).
[CrossRef]

Thorne, K. S.

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Treussart, F.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Uetake, S.

S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, "Stimulated raman scattering in a liquid-hydrogen droplet," Phys. Rev. A 61, 011803 (2000).
[CrossRef]

Vahala, K. J.

Q3. T. Carmon and K. J. Vahala, "Visible continuous emission from a silica microphonic device by third-harmonic generation," Nature Physics 3, 430-435 (2007).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004).
[CrossRef]

T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behavior and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621 (2002).
[CrossRef] [PubMed]

Vahala, Kerry J.

Vigneron, K.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
[CrossRef]

Vorontsov, Y. I.

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Vuckovic, J.

Walls, D. F.

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, "Nonideal quantum nondemolition measurements," Phys. Rev. A. 42, 2995-3005 (1990).
[CrossRef] [PubMed]

P. Alsing, G. J. Milburn, D. F. Walls, "Quantum nondemolition measurements in optical cavities," Phys. Rev. A 37, 2970-2978 (1988).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, "Quantum nondemolition detection of optical quadrature amplitudes," Phys. Rev. Lett. 57, 2473-2476 (1986).
[CrossRef] [PubMed]

Watkins, S.

N. Imoto, S. Watkins, and Y. Sasaki, "A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number," Opt. Comm. 61, 159-163 (1987).
[CrossRef]

White, A. G.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004).
[CrossRef] [PubMed]

Wilcut, E.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

Yamamoto, Y.

S. R. Friberg, S. Machida, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number of an optical soliton," Phys. Rev. Lett. 69, 3165-3168 (1992).
[CrossRef] [PubMed]

N. Imoto, H. A. Haus, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number via the optical Kerr effect," Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

Yang, L.

Zimmermann, M.

C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980).
[CrossRef]

Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978).
[CrossRef]

Appl. Phys. B (1)

J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992).
[CrossRef]

Appl. Phys. Lett. (1)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004).
[CrossRef]

Eur. Phys. J. D (1)

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Nature (5)

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621 (2002).
[CrossRef] [PubMed]

P. Grangier, J. -A. Levenson, and J.-P. Poizat, "Quantum nondemolition measurement in optics," Nature 396, 537-542 (1998).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Nature Physics (1)

Q3. T. Carmon and K. J. Vahala, "Visible continuous emission from a silica microphonic device by third-harmonic generation," Nature Physics 3, 430-435 (2007).
[CrossRef]

New J. Phys. (1)

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]

Opt. Comm. (1)

N. Imoto, S. Watkins, and Y. Sasaki, "A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number," Opt. Comm. 61, 159-163 (1987).
[CrossRef]

Opt. Commun. (1)

P. Grangier, J.M. Courty, and S. Reynaud, "Characterization of nonideal quantum nondemolition measurements," Opt. Commun. 89, 99-106 (1992).
[CrossRef]

Opt. Express (3)

Phys. Rev. A (9)

C.W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, "Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation," Phys. Rev. A 66, 053809 (2002).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, "Stimulated raman scattering in a liquid-hydrogen droplet," Phys. Rev. A 61, 011803 (2000).
[CrossRef]

J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998).
[CrossRef]

N. Imoto and S. Saito, "Quantum nondemolition measurement of photon number in a lossy optical Kerr medium", Phys. Rev. A 39, 675-682 (1989).
[CrossRef] [PubMed]

F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002).
[CrossRef]

N. Imoto, H. A. Haus, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number via the optical Kerr effect," Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

P. Alsing, G. J. Milburn, D. F. Walls, "Quantum nondemolition measurements in optical cavities," Phys. Rev. A 37, 2970-2978 (1988).
[CrossRef] [PubMed]

Phys. Rev. A. (3)

H. A. Haus and F. X. Kärtner, "Optical quantum nondemolition measurement and the Copenhagen Interpretation," Phys. Rev. A. 53, 3785-3791 (1986).
[CrossRef]

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, "Nonideal quantum nondemolition measurements," Phys. Rev. A. 42, 2995-3005 (1990).
[CrossRef] [PubMed]

B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001).
[CrossRef]

Phys. Rev. Lett. (15)

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997).
[CrossRef]

A. Kuzmich, L. Mandel, and N. P. Bigelow, "Generation of spin squeezing via continous quantum nondemolition measurement," Phys. Rev. Lett. 85, 1594-1597 (2000).
[CrossRef] [PubMed]

S. Peil, and G. Gabrielse, "Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states," Phys. Rev. Lett. 83, 1287-1290 (1999).
[CrossRef]

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, "Quantum nondemolition detection of optical quadrature amplitudes," Phys. Rev. Lett. 57, 2473-2476 (1986).
[CrossRef] [PubMed]

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004).
[CrossRef] [PubMed]

S. X. Qian and R. K. Chang, "Multiorder stokes emission from micrometer-size droplets," Phys. Rev. Lett. 56, 926-929 (1986).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H. B. Lin, "Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets," Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978).
[CrossRef]

S. R. Friberg, S. Machida, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number of an optical soliton," Phys. Rev. Lett. 69, 3165-3168 (1992).
[CrossRef] [PubMed]

J. P. Poizat, P. Grangier, "Experimental realization of a quantum optical tap," Phys. Rev. Lett. 70, 271-274 (1993).
[CrossRef] [PubMed]

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, "Back-action evading measurements for quantum non-demolition detection and quantum optical tapping," Phys. Rev. Lett. 72, 214-217 (1994).
[CrossRef] [PubMed]

K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, "Quantum non-demolition demonstration via repeated back-action evading measurements," Phys. Rev. Lett. 75, 3422-3425 (1995).
[CrossRef] [PubMed]

R. Bruckmeier, H. Hansen, and S. Schiller, "Repeated quantum non-demolition measurements of continuous optical waves," Phys. Rev. Lett. 79, 1463-1466 (1997).
[CrossRef]

K. Bencheick, C. Simonneau, and J. A. Levenson, "Cascaded amplifying quantum optical taps: A robust noiseless optical bus," Phys. Rev. Lett. 78, 34-37 (1997).
[CrossRef]

Rev. Mod. Phys. (2)

V. B. Braginsky and F. Ya. Khalili, "Quantum nondemolition measurements: the route from toys to tools," Rev. Mod. Phys. 68, 1-11 (1996).
[CrossRef]

C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980).
[CrossRef]

Science (1)

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Other (3)

G. Leuchs, C. Silberhorn,K¨onig, P. K.  Lam, A. Sizmann, and N. Korolkova, in Quantum Information Theory with Continous Variables edited by S. L. Braunstein and A. K. Pati (Kluwer Academic, Dordrecht, 2002).

M. Yamane and Y. Asahara, Glass for Photonics (Cambridge University Press, Cambridge, 2000).
[CrossRef]

D. F. Walls and G. J. Milburn, Quantum Optics(Springer-Verlag, Berlin Heidelberg, 1994)

Cited By

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Figures (3)

Fig. 1
Fig. 1

Schematic of QND measurement with an ultra-high-Q toroidal microcavity. BS1 and BS2 are beamspitters with reflection coefficients r and 1/2, respectively, while M1 and M2 are mirrors with unit reflectivity for the probe field and unit transmissivity for the signal wave. Signal and probe waves are introduced into the same fiber waveguide and then coupled to the microcavity. The change in the phase of the probe field is finally detected by balanced homodyne detection using two photon detectors D1 and D2.

Fig. 2
Fig. 2

(a) Photon number difference 〈n12〉 versus the intracavity average signal photon number 〈ns〉. The solid and dashed lines represent the precise numerical calculation and approximately analytical solution, respectively. (b) The transmission of the signal field versus 〈ns〉. Parameters used in the calculations: r = 1/2, κj,1 = 50κj,0, κs = κp, Q0 = 4×108, wavelength λ = 1550 nm, third-order nonlinear susceptibility χ(3) = 2.2 × 10−22 m2/V2, quantization volume V = 50 μm3, and 〈nu〉 = 1013.

Fig. 3
Fig. 3

Dependence of observed variance ( Δ n s obs ) 2 on (a) the probe photon number which is linearly proportional to 〈uu〉, and (b) the intrinsic quality factor Q0 for the signal prepared as coherent and Fock states. The dashed and dotted lines are for coherent states with mean photon numbers 〈ns〉 = 1 and 〈ns〉 = 10, respectively. The curves are obtained from Eq. 25 using the parameters given in Fig.2. For (b) we used 〈nu〉 = 1014. Lower bound of observed variance for a coherent state with 〈ns〉 is determined by its intrinsic noise 〈ns〉 when Q0 ∼ 108 and 〈nu〉 ∼ 1014. For a Fock state, it is determined by Q0, 〈nu〉 and K, hence can be made arbitrarily small by proper choice of these parameters.

Equations (27)

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d a j d t = i [ a j , H ] κ j 2 a j κ j , 1 a j in ,
a j out = a j in + κ j , 1 a j .
H = H o + H int , H o = h ¯ ω sc a s a s + h ¯ ω pc a p a p , H int = K [ ( a s a s ) 2 + ( a p a p ) 2 + 4 ε a s a s a p a p ] ,
d a s d t = [ i ( δ s + K + 2 K n s + 4 K n p ) + κ s 2 ] a s κ s , 1 a s in ,
d a p d t = [ i ( δ p + K + 2 K n p + 4 K n s ) + κ p 2 ] a p κ p , 1 a p in ,
[ i ( δ s + K + 2 K N s + 4 K N p ) + κ s 2 ] A s + κ s , 1 A s in e i ( ϕ s φ s ) = 0 ,
[ i ( δ p + K + 2 K N p + 4 K N s ) + κ p 2 ] A p + κ p , 1 A p in e i ( ϕ p φ p ) = 0 ,
A j out = A j in + κ j , 1 A j e i ( φ j ϕ j ) ,
[ i ( δ p + K + 2 K N p ) + κ p 2 ] A p = κ p , 1 A p in e i ( ϕ p φ p ) .
N p 3 + δ p + K K c 0 N p 2 + κ p 2 + 4 ( δ p + K ) 2 16 K 2 c 1 N p κ p , 1 4 K 2 N p in = 0
δ p + K ( 1 + 2 N p ) = 0.
N p in = N p 4 κ p 2 κ p , 1 ,
| N p in | > κ p 3 8 3 K κ p , 1 .
d a j d t = ( i 2 μ j K n s + κ j 2 ) a j κ j , 1 a j in ,
a j ( ω ) = 2 κ j , 1 κ j i 2 ( ω 2 μ j K n s ) a j in ( ω ) ,
G j ( ω ) = ( 1 2 κ j , 1 κ j i 2 ω ) + i 8 μ j κ j , 1 K n s ( κ j i 2 ω ) 2 .
G j e i 8 μ j ( K / κ j ) n s ,
a j = a j out = a j in e i 8 μ j ( K / κ j ) n s
a p in = i r u + 1 r v , a L O = 1 r u + i r v ,
n s obj d 2 d 2 d 1 d 1 2 ζ p r ( 1 r ) n u = n 21 2 ζ p r ( 1 r ) n u ,
n 21 = i r ( 1 r ) ( u u v v ) ( t p * t p ) r ( u v v u ) ( t p * t p ) ( t p u v + t p * v u )
n s obs = u u v v n u n s + ir ( u v v u ) r ( 1 r ) n u n s t p u v + t p * v u 2 ζ p r ( 1 r ) n u
n s obs = n s = n 21 2 ζ p r ( 1 r ) n u
( Δ n s obs ) 2 = ( Δ n u ) 2 n u n u 2 n s 2 + ( 1 + 1 ( 1 r ) n u ) ( Δ n s ) 2 + 1 4 ζ p 2 r ( 1 r ) n u
( Δ n s obs ) 2 = ( Δ n s ) 2 + 1 4 ζ p 2 r ( 1 r ) n u .
( Δ n s obs ) 2 = 1 4 ζ p 2 r ( 1 r ) n u 1 4 ζ p 2 n p in
( Δ n s obs ) 2 = n s + 1 4 ζ p 2 r ( 1 r ) n u n s + 1 4 ζ p 2 n p in .

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