Abstract

Squeezed light generated by four-wave mixing near the Na atomic resonance is compared with a full quantum-theoretical model. Losses and dephasing that are due to spontaneous emission near the atomic resonance are included in this theoretical treatment. The experimental results show noise reduction greater than 1 dB below the standard quantum limits and are in good agreement with the quantum model at low pump intensities and large detuning. Theory and experiment show that limits to squeezed-light noise reduction resulting from spontaneous-emission losses can be largely avoided by using nondegenerate four-wave mixing and pump intensities near the atomic saturation values.

© 1987 Optical Society of America

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  1. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher and B. Yurke, in Frontiers in Optics, E. R. Pike and S. Sarkar, eds. (Hilger, Bristol, UK, 1986), pp. 1–41; R. E. Slusher, B. Yurke, and J. Mertz, “Experimental progress in intracavity generation of squeezed light using resonant atomic nonlinearities,” Opt. Acta (to be published).
    [Crossref] [PubMed]
  2. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
    [Crossref] [PubMed]
  3. L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [Crossref] [PubMed]
  4. M. W. Meada, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161 (1987).
    [Crossref]
  5. H. P. Yuen and J. H. Shapiro, Opt. Lett. 4, 334 (1979).
    [Crossref] [PubMed]
  6. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. A 31, 3512 (1985).
    [Crossref] [PubMed]
  7. M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985).
    [Crossref] [PubMed]
  8. T. Fu and M. Sargent, Opt. Lett. 4, 366 (1979).
    [Crossref] [PubMed]
  9. R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Marten, Phys. Rev. A 24, 411 (1981).
    [Crossref]
  10. M. D. Reid and D. F. Walls, Phys. Rev. A 34, 4929 (1986).
    [Crossref] [PubMed]
  11. H. Häken, Handbuch der Physik (Springer-Verlag, Berlin, 1970), Vol. XXV/2C.
  12. P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
    [Crossref]
  13. M. Sargent, D. A. Holm, and M. Zubairy, Phys. Rev. A 31, 3112 (1985).
    [Crossref]
  14. S. Stenholm, D. A. Holm, and N. Sargent, Phys. Rev. A 31, 3124 (1985).
    [Crossref] [PubMed]
  15. M. O. Scully and W. E. Lamb, Phys. Rev. 159, 208 (1967).
    [Crossref]
  16. D. A. Holm, M. Sargent, and B. A. Capron, Opt. Lett. 11, 443 (1986); D. A. Holm and M. Sargent, Phys. Rev. A 35, 2510 (1987).
    [Crossref] [PubMed]
  17. P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980).
    [Crossref]
  18. C. M. Savage and D. F. Walls, J. Opt. Soc. Am. B 4, 1514 (1987).
    [Crossref]
  19. B. Yurke, Phys. Rev. A 29, 408 (1984).
    [Crossref]
  20. M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1385 (1984).
    [Crossref]
  21. M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
    [Crossref] [PubMed]
  22. R. L. Abrams and R. C. Lind, Opt. Lett. 2, 94 (1973); errata 3, 205 (1978).
    [Crossref]
  23. B. Yurke, Phys. Rev. A 32, 300, 311 (1985).
    [Crossref] [PubMed]
  24. B. Yurke and R. E. Slusher, in Quantum Optics IV, J. D. Harvey and D. F. Walls, eds., Springer Proceedings in Physics (Springer-Verlag, New York, 1986).

1987 (2)

1986 (4)

D. A. Holm, M. Sargent, and B. A. Capron, Opt. Lett. 11, 443 (1986); D. A. Holm and M. Sargent, Phys. Rev. A 35, 2510 (1987).
[Crossref] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 34, 4929 (1986).
[Crossref] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

1985 (7)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. A 31, 3512 (1985).
[Crossref] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher and B. Yurke, in Frontiers in Optics, E. R. Pike and S. Sarkar, eds. (Hilger, Bristol, UK, 1986), pp. 1–41; R. E. Slusher, B. Yurke, and J. Mertz, “Experimental progress in intracavity generation of squeezed light using resonant atomic nonlinearities,” Opt. Acta (to be published).
[Crossref] [PubMed]

M. Sargent, D. A. Holm, and M. Zubairy, Phys. Rev. A 31, 3112 (1985).
[Crossref]

S. Stenholm, D. A. Holm, and N. Sargent, Phys. Rev. A 31, 3124 (1985).
[Crossref] [PubMed]

M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

B. Yurke, Phys. Rev. A 32, 300, 311 (1985).
[Crossref] [PubMed]

1984 (2)

B. Yurke, Phys. Rev. A 29, 408 (1984).
[Crossref]

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1385 (1984).
[Crossref]

1981 (2)

P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[Crossref]

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Marten, Phys. Rev. A 24, 411 (1981).
[Crossref]

1980 (1)

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980).
[Crossref]

1979 (2)

1973 (1)

1967 (1)

M. O. Scully and W. E. Lamb, Phys. Rev. 159, 208 (1967).
[Crossref]

Abrams, R. L.

Boyd, R. W.

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Marten, Phys. Rev. A 24, 411 (1981).
[Crossref]

Capron, B. A.

Collett, M. J.

M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1385 (1984).
[Crossref]

DeVoe, R. G.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

Drummond, P. D.

P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[Crossref]

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980).
[Crossref]

Fu, T.

Gardiner, C. W.

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1385 (1984).
[Crossref]

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980).
[Crossref]

Häken, H.

H. Häken, Handbuch der Physik (Springer-Verlag, Berlin, 1970), Vol. XXV/2C.

Hall, J. L.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher and B. Yurke, in Frontiers in Optics, E. R. Pike and S. Sarkar, eds. (Hilger, Bristol, UK, 1986), pp. 1–41; R. E. Slusher, B. Yurke, and J. Mertz, “Experimental progress in intracavity generation of squeezed light using resonant atomic nonlinearities,” Opt. Acta (to be published).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. A 31, 3512 (1985).
[Crossref] [PubMed]

Holm, D. A.

D. A. Holm, M. Sargent, and B. A. Capron, Opt. Lett. 11, 443 (1986); D. A. Holm and M. Sargent, Phys. Rev. A 35, 2510 (1987).
[Crossref] [PubMed]

M. Sargent, D. A. Holm, and M. Zubairy, Phys. Rev. A 31, 3112 (1985).
[Crossref]

S. Stenholm, D. A. Holm, and N. Sargent, Phys. Rev. A 31, 3124 (1985).
[Crossref] [PubMed]

Kimble, H. J.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Kumar, P.

Lamb, W. E.

M. O. Scully and W. E. Lamb, Phys. Rev. 159, 208 (1967).
[Crossref]

Levenson, M. D.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

Lind, R. C.

Marten, D. J.

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Marten, Phys. Rev. A 24, 411 (1981).
[Crossref]

Meada, M. W.

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher and B. Yurke, in Frontiers in Optics, E. R. Pike and S. Sarkar, eds. (Hilger, Bristol, UK, 1986), pp. 1–41; R. E. Slusher, B. Yurke, and J. Mertz, “Experimental progress in intracavity generation of squeezed light using resonant atomic nonlinearities,” Opt. Acta (to be published).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. A 31, 3512 (1985).
[Crossref] [PubMed]

Narum, P.

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Marten, Phys. Rev. A 24, 411 (1981).
[Crossref]

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

Raymer, M. G.

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Marten, Phys. Rev. A 24, 411 (1981).
[Crossref]

Reid, M. D.

M. D. Reid and D. F. Walls, Phys. Rev. A 34, 4929 (1986).
[Crossref] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985).
[Crossref] [PubMed]

Sargent, M.

Sargent, N.

S. Stenholm, D. A. Holm, and N. Sargent, Phys. Rev. A 31, 3124 (1985).
[Crossref] [PubMed]

Savage, C. M.

Scully, M. O.

M. O. Scully and W. E. Lamb, Phys. Rev. 159, 208 (1967).
[Crossref]

Shapiro, J. H.

Shelby, R. M.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher and B. Yurke, in Frontiers in Optics, E. R. Pike and S. Sarkar, eds. (Hilger, Bristol, UK, 1986), pp. 1–41; R. E. Slusher, B. Yurke, and J. Mertz, “Experimental progress in intracavity generation of squeezed light using resonant atomic nonlinearities,” Opt. Acta (to be published).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. A 31, 3512 (1985).
[Crossref] [PubMed]

B. Yurke and R. E. Slusher, in Quantum Optics IV, J. D. Harvey and D. F. Walls, eds., Springer Proceedings in Physics (Springer-Verlag, New York, 1986).

Stenholm, S.

S. Stenholm, D. A. Holm, and N. Sargent, Phys. Rev. A 31, 3124 (1985).
[Crossref] [PubMed]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. A 31, 3512 (1985).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher and B. Yurke, in Frontiers in Optics, E. R. Pike and S. Sarkar, eds. (Hilger, Bristol, UK, 1986), pp. 1–41; R. E. Slusher, B. Yurke, and J. Mertz, “Experimental progress in intracavity generation of squeezed light using resonant atomic nonlinearities,” Opt. Acta (to be published).
[Crossref] [PubMed]

Walls, D. F.

C. M. Savage and D. F. Walls, J. Opt. Soc. Am. B 4, 1514 (1987).
[Crossref]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 34, 4929 (1986).
[Crossref] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985).
[Crossref] [PubMed]

M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[Crossref]

Wu, H.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Wu, L.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Yuen, H. P.

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher and B. Yurke, in Frontiers in Optics, E. R. Pike and S. Sarkar, eds. (Hilger, Bristol, UK, 1986), pp. 1–41; R. E. Slusher, B. Yurke, and J. Mertz, “Experimental progress in intracavity generation of squeezed light using resonant atomic nonlinearities,” Opt. Acta (to be published).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. A 31, 3512 (1985).
[Crossref] [PubMed]

B. Yurke, Phys. Rev. A 32, 300, 311 (1985).
[Crossref] [PubMed]

B. Yurke, Phys. Rev. A 29, 408 (1984).
[Crossref]

B. Yurke and R. E. Slusher, in Quantum Optics IV, J. D. Harvey and D. F. Walls, eds., Springer Proceedings in Physics (Springer-Verlag, New York, 1986).

Zubairy, M.

M. Sargent, D. A. Holm, and M. Zubairy, Phys. Rev. A 31, 3112 (1985).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980).
[Crossref]

Opt. Lett. (5)

Phys. Rev. (1)

M. O. Scully and W. E. Lamb, Phys. Rev. 159, 208 (1967).
[Crossref]

Phys. Rev. A (11)

B. Yurke, Phys. Rev. A 32, 300, 311 (1985).
[Crossref] [PubMed]

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Marten, Phys. Rev. A 24, 411 (1981).
[Crossref]

M. D. Reid and D. F. Walls, Phys. Rev. A 34, 4929 (1986).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. A 31, 3512 (1985).
[Crossref] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985).
[Crossref] [PubMed]

B. Yurke, Phys. Rev. A 29, 408 (1984).
[Crossref]

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1385 (1984).
[Crossref]

M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[Crossref]

M. Sargent, D. A. Holm, and M. Zubairy, Phys. Rev. A 31, 3112 (1985).
[Crossref]

S. Stenholm, D. A. Holm, and N. Sargent, Phys. Rev. A 31, 3124 (1985).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher and B. Yurke, in Frontiers in Optics, E. R. Pike and S. Sarkar, eds. (Hilger, Bristol, UK, 1986), pp. 1–41; R. E. Slusher, B. Yurke, and J. Mertz, “Experimental progress in intracavity generation of squeezed light using resonant atomic nonlinearities,” Opt. Acta (to be published).
[Crossref] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Other (2)

H. Häken, Handbuch der Physik (Springer-Verlag, Berlin, 1970), Vol. XXV/2C.

B. Yurke and R. E. Slusher, in Quantum Optics IV, J. D. Harvey and D. F. Walls, eds., Springer Proceedings in Physics (Springer-Verlag, New York, 1986).

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Figures (13)

Fig. 1
Fig. 1

The variance V for the minimum-noise quadrature output from a four-wave mixing cavity is shown as a function of the normalized pump intensity I/Is, where Is is the saturation intensity at the pump frequency. This figure contrasts degenerate (δ = 0) and nondegenerate (δ = 60) squeezing with a cooperativity number C = 1000 and a normalized detuning from the atomic line Δ = −300. The dashed line represents the vacuum-fluctuation level.

Fig. 2
Fig. 2

Variance V for the minimum-noise quadrature as a function of the normalized pump intensity for nondegenerate four-wave mixing with δ = 60 and Δ = −250. The three curves are for cooperativity parameters C = 1500, 1000, and 750. The curves stop at the threshold for oscillation.

Fig. 3
Fig. 3

Variance V for the minimum-noise quadrature as a function of the normalized pump intensity for nondegenerate four-wave mixing with δ = 60 and Δ = −300. The three curves are for cooperativity values C = 1500, 1000, and 750. The C = 1500 curve terminates at the threshold for oscillation.

Fig. 4
Fig. 4

Variance V for the minimum-noise quadrature as a function of the normalized pump intensity for nondegenerate four-wave mixing with δ = 60 and Δ = −400. C designates the cooperativity values.

Fig. 5
Fig. 5

Variance V for the minimum-noise quadrature as a function of the normalized pump intensity for nondegenerate four-wave mixing with δ = 60 and Δ = −600. C designates the cooperativity values.

Fig. 6
Fig. 6

A schematic diagram of the present experimental arrangement for generating squeezed light by four-wave mixing near the Na atomic resonance. A cw ring dye laser is frequency and phase stabilized to an optical cavity formed by mirror SM (r = 0.999) and output mirror OSM (r = 0.98) in which squeezed light is generated. Stabilization is achieved by a locking scheme involving a phase-modulated beam generated by frequency shifting acousto-optic modulators (LOCK AOM’s) and an electro-optic modulator (LOCK EOM). This modulated signal is back-reflected from the squeezing cavity and detected at homodyne detector DF. The detected-error signal is mixed with the electro-optic modulator drive voltage and processed by feedback amplifiers (FB1 and FB2). It is then fed to a piezoelectric transducer (LOCK PZT) on mirror SM and to the stabilizing acousto-optic modulator (STAB AOM). The cavity parameters at the squeezing frequencies can be studied by the frequency-shifted (PROBE AOM’s) probe beams tuned near the cavity resonant frequencies. These are blocked by shutter S1 for measurements of squeezed-vacuum light in the cavity. The Na beam optical polarization is driven by a pump beam at frequency νp, which is backreflected by mirror PM for the four-wave mixing geometry. Output from the squeezing cavity is detected by using a balanced homodyne detector (BS5) and photodiode detectors (DA and DB). The local-oscillator beam (LO) can be phase shifted (ϕLO) by a piezoelectrically controlled mirror. The difference photocurrent from DA and DB is monitored on a spectrum analyzer (SA). BS1–BS4 are beam splitters.

Fig. 7
Fig. 7

Relative positions of the hyperfine-split Na D2 resonance, the pump frequency νp, the two-mode squeezing frequencies νp ± 4 νc, and the locking frequency are shown for the present experiments. The transmitted-to-incident light ratio outside the cavity, IT/Ii, is shown near the Na resonance. The pump light is tuned and locked (by the shifted beam at νp + 7 νc) at approximately 1.5 GHz above the weaker hyperfine line. Four-wave mixing generates conjugate photon pairs at cavity modes with frequencies near νp ± 4 νc. The cavity resonant frequencies are shown as the set of vertical lines spaced by 150 MHz to the right.

Fig. 8
Fig. 8

Noise levels corresponding to the rms photocurrent from the balanced homodyne detector are shown as a function of local-oscillator phase ϕLO. With the squeezing cavity blocked, the mean noise level indicated by the dashed line is obtained from the dotted trace. This noise level is primarily due to light noise and can be identified with the vacuum-fluctuation or shot-noise level. The photodetector amplifier noise is at −10 dB relative to the dashed line. The measured noise level increases +1.3 dB above and decreases −0.7 dB below the vacuum noise level as a function of ϕLO when the four-wave mixing output from the cavity is matched to the local-oscillator mode. The radio frequency is centered at 594.6 MHz with a bandwidth of 300 kHz. The video averaging bandwidth is 100 Hz. A theoretical model predicts squeezing of ±2 dB (solid curve) for an ideal measurement. For the experimental detection efficiency and amplifier noise this ideal behavior is degraded to the dashed–dotted curve. The pump frequency detuning for both theory and experiment is Δ = −400, the nondegenerate detuning is δ = 60, and the cooperativity parameter is C = 1000, corresponding to the model predictions in Fig. 4. The pump intensity is I/Is = 0.056.

Fig. 9
Fig. 9

Homodyne-detector noise level as a function of local-oscillator phase for a detuning Δ = −300 and δ = 60. The dashed line is the vacuum-fluctuation level plus a small amplifier noise component. The solid line represents data obtained with a pump intensity I/Is = 0.1 and cooperativity parameter C = 1000. The radio frequency is centered at 594.3 MHz, with a bandwidth of 300 kHz and an averaging bandwidth of 100 Hz.

Fig. 10
Fig. 10

A frequency scan of squeezed noise over a range of frequencies near the cavity resonance. The local-oscillator phase ϕLO is simultaneously swept. The experimental parameters are the same as in Fig. 9.

Fig. 11
Fig. 11

Homodyne-detector noise level as a function of local-oscillator phase for a detuning Δ ≃ −280 and δ = 60. The dashed line is the mean level of vacuum-fluctuation noise, with a small amplifier noise component. The center frequency is 594.3 MHz with a bandwidth of 300 kHz and an averaging bandwidth of 100 Hz. The oscillations in the solid data curve, with periods between 5 and 10 msec, may be due to phase jumps in the pump-laser frequency relative to the cavity frequency.

Fig. 12
Fig. 12

Homodyne-detector noise level as a function of local-oscillator, phase for a detuning of Δ = −280 and δ = 60. The parameters are similar to those in Fig. 11, except that the averaging bandwidth is 300 Hz. Again, there are oscillations in the solid data curve because of phase jumps in the pump frequency relative to the cavity frequency. In these data the phase stabilizes momentarily in a favorable ϕLO ~ π/2 region so that a large noise reduction (~ −1.5 dB) is recorded.

Fig. 13
Fig. 13

Homodyne-detector noise level as a function of local-oscillator phase for a detuning of Δ = −240 and δ = 60. The center frequency is 594.2 MHz, with a bandwidth of 300 kHz and an averaging bandwidth of 100 Hz.

Equations (24)

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

H = χ ( E 2 a 2 a 3 + E * 2 a 2 a 3 ) ,
X ϕ = a 2 e - i ϕ + a 3 e i ϕ .
V ( X ϕ ) = ½ ( X ϕ X ϕ + X ϕ X ϕ ) - X ϕ X ϕ .
V ( X ϕ ) = e - r , V ( X ϕ + π / 2 ) = e r ,
r = E 2 χ t
H = μ = 0 4 H μ , H 0 = j = 1 3 ω j a j a j + i = 1 N ω 0 σ z i , H 1 = i g i = 1 N { σ i [ a 1 exp ( - i k 1 r i ) + a 2 exp ( - i k 2 r i ) + a 3 exp ( - i k 3 r i ) } + h . c . ] , H 2 = i [ a 1 E exp ( - i ω p t ) - a 1 E * exp ( i ω p t ) ] , H 3 = i = 1 N ( σ 1 Γ + σ 1 Γ ) , H 4 = j = 1 3 ( a j Γ c + a j Γ c ) ,
α ˙ = E exp ( - i ω p t ) - ( κ + i ω 1 ) α + q ν + Γ α , ν ˙ = ( γ + i ω 0 ) ν + g α D + Γ ν , D ˙ = γ ( D + N ) - 2 g ( ν α + ν α ) + Γ D ,
Γ ν ( t ) Γ ν ( t ) = 2 g α ν δ ( t - t ) , Γ D ( t ) Γ D ( t ) = [ 2 γ ( D + N ) - κ g ( ν α + ν α ) ] δ ( t - t ) ,
α = α 1 exp ( - i ω p t ) + α 2 exp [ - i ( ω p - ) t ] + α 3 exp [ - i ( ω p + ) t ] , ν = ν 1 exp ( - i ω p t ) + ν 2 exp [ - i ( ω p - ) t ] + ν 3 exp [ - i ( ω p + ) t ] , D = D 1 + D 2 e - i t + D 2 e i t .
α ˙ 1 = E - κ ( 1 + i θ ) α 1 - 2 C κ α 1 ( 1 + i Δ ) ( 1 + i 1 + Δ 2 ) + Γ α 1 ( t ) , α ˙ 2 = κ [ 1 + i θ + γ ( δ ) ] α 2 + κ χ ( δ ) α 3 exp ( 2 i ϕ 0 ) + Γ α 2 ( t ) , α ˙ 3 = - κ [ 1 + i θ + γ ( - δ ) ] α 3 + κ χ ( - δ ) α 2 exp ( 2 i ϕ 0 ) + Γ α 3 ( t ) ,
Γ α 2 ( t ) Γ α 3 ( t ) = κ R exp ( 2 i ϕ 0 ) δ ( t - t ) , Γ α 2 ( t ) Γ α 3 ( t ) = κ R * exp ( - 2 i ϕ 0 ) δ ( t - t ) , Γ α 2 ( t ) Γ α 2 ( t ) = Γ α 3 ( t ) Γ α 3 ( t ) = κ L Λ δ ( t - t ) .
α ˙ 2 = i χ α 3 + Γ α 2 ( t ) , α ˙ 3 = i χ α 2 + Γ α 3 ( t ) ,
Γ α 2 ( t ) Γ α 3 ( t ) = - i χ δ ( t - t ) .
X ϕ = a 2 OUT exp ( - i ϕ LO ) + a 3 OUT exp ( i LO ) ,
d d t α = - A α + D 1 / 2 f ( t ) ,
Γ i ( t ) Γ j ( t ) = D i j δ ( t - t ) .
α 2 ss = α 2 ss = α 3 ss = α 3 ss = 0.
S i j ( ω , δ ) = - e - i ω t α i ( t ) α j ( 0 ) d t , = [ ( A + i ω J ) - 1 D ( A - i ω J ) - 1 ] i j .
V ( X ϕ , δ ) = 1 + 2 κ [ S 12 ( 0 , δ ) + S 34 ( 0 , δ ) - 2 S 13 ( 0 , δ ) ] ,
P s = { [ η ( V sin 2 ϕ LO + V M cos 2 ϕ LO ) + ( 1 - η ) ] P 0 + P A } ,
α ( Δ ) = α 0 [ 1 + Δ 2 ( 1 + 4 I / I s ) 3 / 2 ) ]
n ( Δ ) = 1 - c α 0 ω [ Δ ( 1 + 2 I / I s ) 1 + Δ 2 ( 1 + 4 I / I s ) 3 / 2 ] ,
ν i = N i c 2 n ( Δ ) L ,
ν i + 1 - ν i c 2 L + c 2 π α 0 [ Δ i + 1 ( 1 + 2 I / I s ) 1 + Δ i + 1 2 ( 1 + 4 I / I s ) 3 / 2 - Δ i ( 1 + 2 I / I s ) 1 + Δ i 2 ( 1 + 4 I / I s ) 3 / 2 ] .

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