Abstract

The difference between the two nonclassical photon states, i.e., the squeezed state and the number-phase minimum-uncertainty state, is discussed. Four different generation principles for number-phase minimum-uncertainty states (and number states) are described. They are the following: (1) unitary evolution, using self-phase modulation and interference; (2) nonunitary state reduction, using either quantum nondemolition photon-number measurement or parametrically amplified idler-wave photon counting; (3) negative-feedback oscillators, incorporating a correlated photon-pair generator; and (4) pump-amplitude noise-suppressed lasers. A number-phase minimum-uncertainty state with photon-number noise reduced to 1 dB (20%) below the standard quantum limit in a frequency range over 100 MHz was produced by the fourth-generation principle.

© 1987 Optical Society of America

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  11. Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 56, 1001 (1986).
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  43. G. Björk and Y. Yamamoto, “Generation and amplification of number states by nondegenerate parametric oscillators with idler measurement feedback,” Phys. Rev. A (to be published).
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    [CrossRef]
  45. S. M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1969).
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    [CrossRef] [PubMed]
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    [CrossRef]

1987 (5)

1986 (15)

J. H. Shapiro, M. C. Teich, B. E. A. Saleh, P. Kumar, and G. Saplakogln, Phys. Rev. Lett. 56, 1136 (1986).
[CrossRef] [PubMed]

Y. Yamamoto and N. Imoto, IEEE J. Quantum Electron. QE-22, 2032 (1986); O. Nilsson, Y. Yamamoto, and S. Machida, IEEE J. Quantum Electron. QE-22, 2043 (1986).
[CrossRef]

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, Phys. Rev. Lett. 57, 2473 (1986).
[CrossRef] [PubMed]

B. L. Schumaker, Phys. Rep. 135, 318 (1986).
[CrossRef]

Y. Yamamoto, S. Machida, and O. Nilsson, Phys. Rev. A 34, 4025 (1986).
[CrossRef] [PubMed]

M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974 (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. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986).
[CrossRef] [PubMed]

S. Machida and Y. Yamamoto, Opt. Commun. 57, 290 (1986); Y. Yamamoto, N. Imoto, and S. Machida, Phys. Rev. A 33, 3243 (1986); H. A. Haus and Y. Yamamoto, Phys. Rev. A 34, 270 (1986).
[CrossRef] [PubMed]

Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 56, 1001 (1986).
[CrossRef]

H. P. Yuen, Phys. Rev. Lett. 56, 2176 (1986).
[CrossRef] [PubMed]

J. Krause, M. O. Scully, and H. Walther, Phys. Rev. A 34, 2032 (1986).
[CrossRef] [PubMed]

P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

D. V. Avelin and K. K. Likharev, J. Low Temp. Phys. 62, 345 (1986).
[CrossRef]

1985 (6)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

J. G. Walker and E. Jakeman, Proc. Soc. Photo-Opt. Instrum. Eng. 492, 274 (1985); E. Jakeman and J. G. Rarity, Opt. Commun. 59, 219 (1986); J. G. Walker and E. Jakeman, Opt. Acta 32, 1303 (1985).
[CrossRef]

B. E. A. Saleh and M. C. Teich, Opt. Commun. 52, 429 (1985).
[CrossRef]

N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
[CrossRef] [PubMed]

M. Teich and B. E. A. Saleh, J. Opt. Soc. Am. B 2, 275 (1985).
[CrossRef]

C. W. Gardiner and M. J. Collet, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

1983 (3)

1982 (1)

C. Harder, J. Katz, S. Margalit, J. Shacharr, and A. Yariv, IEEE J. Quantum Electron. QE-18, 333 (1982).
[CrossRef]

1980 (2)

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, Science 209, 547 (1980).
[CrossRef] [PubMed]

C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmerman, Rev. Mod Phys. 52, 341 (1980).
[CrossRef]

1979 (1)

1976 (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

1971 (1)

D. Stoler, Phys. Rev. D 4, 1925 (1971).
[CrossRef]

1968 (2)

R. Jackiw, J. Math. Phys. 9, 339 (1968).
[CrossRef]

P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40, 411 (1968).
[CrossRef]

1967 (1)

H. Huang, Z. Phys. 206, 163 (1967).
[CrossRef]

1965 (1)

H. Takahashi, Adv. Commun. Syst. 1, 227 (1965).

1963 (1)

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

Avelin, D. V.

D. V. Avelin and K. K. Likharev, J. Low Temp. Phys. 62, 345 (1986).
[CrossRef]

Björk, G.

G. Björk and Y. Yamamoto, “Generation and amplification of number states by nondegenerate parametric oscillators with idler measurement feedback,” Phys. Rev. A (to be published).

Bohm, D.

D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, N.J., 1951).

Braginsky, V. B.

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, Science 209, 547 (1980).
[CrossRef] [PubMed]

Buckingham, M. J.

M. J. Buckingham, Noise in Electronic Devices and Systems (Wiley, New York, 1983).

Carruthers, P.

P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40, 411 (1968).
[CrossRef]

Caves, C. M.

C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmerman, Rev. Mod Phys. 52, 341 (1980).
[CrossRef]

Chan, V. W. S.

Collet, M. J.

C. W. Gardiner and M. J. Collet, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

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]

Drever, R. W. P.

C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmerman, Rev. Mod Phys. 52, 341 (1980).
[CrossRef]

Filipowicz, P.

P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

Gardiner, C. W.

C. W. Gardiner and M. J. Collet, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

Glauber, R. J.

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

R. J. Glauber, Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965), p. 65.

Haken, H.

H. Haken, Encyclopedia of Physics (Springer-Verlag, Berlin, 1970), Vol. 25/2c.

Hall, J. L.

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

Harder, C.

C. Harder, J. Katz, S. Margalit, J. Shacharr, and A. Yariv, IEEE J. Quantum Electron. QE-18, 333 (1982).
[CrossRef]

Haus, H. A.

Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 56, 1001 (1986).
[CrossRef]

N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
[CrossRef] [PubMed]

Ho, S.-T.

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Hong, C. K.

C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986).
[CrossRef] [PubMed]

Huang, H.

H. Huang, Z. Phys. 206, 163 (1967).
[CrossRef]

Igeta, K.

K. Igeta and Y. Yamamoto, “Quantum mechanical optical computers,” tech. paper on optics and quantum electronics (Institute of Electronics and Communication Engineers of Japan, Tokyo, 1987.

Imoto, N.

M. Kitagawa, N. Imoto, and Y. Yamamoto, Phys. Rev. A 35, 5270 (1987).
[CrossRef] [PubMed]

Y. Yamamoto and N. Imoto, IEEE J. Quantum Electron. QE-22, 2032 (1986); O. Nilsson, Y. Yamamoto, and S. Machida, IEEE J. Quantum Electron. QE-22, 2043 (1986).
[CrossRef]

N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
[CrossRef] [PubMed]

N. Imoto and Y. Yamamoto, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), paper ThE4; N. Imoto, S. Watkins, and Y. Sasaki, Opt. Commun. 61, 159 (1987).
[CrossRef]

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

Jackiw, R.

R. Jackiw, J. Math. Phys. 9, 339 (1968).
[CrossRef]

Jakeman, E.

J. G. Walker and E. Jakeman, Proc. Soc. Photo-Opt. Instrum. Eng. 492, 274 (1985); E. Jakeman and J. G. Rarity, Opt. Commun. 59, 219 (1986); J. G. Walker and E. Jakeman, Opt. Acta 32, 1303 (1985).
[CrossRef]

Javanainen, J.

P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

Katz, J.

C. Harder, J. Katz, S. Margalit, J. Shacharr, and A. Yariv, IEEE J. Quantum Electron. QE-18, 333 (1982).
[CrossRef]

Kimble, H. J.

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

Kitagawa, M.

M. Kitagawa, N. Imoto, and Y. Yamamoto, Phys. Rev. A 35, 5270 (1987).
[CrossRef] [PubMed]

M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974 (1986).
[CrossRef] [PubMed]

Krause, J.

J. Krause, M. O. Scully, and H. Walther, Phys. Rev. A 34, 2032 (1986).
[CrossRef] [PubMed]

Kumar, P.

Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Levenson, M. D.

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, Phys. Rev. Lett. 57, 2473 (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]

Likharev, K. K.

D. V. Avelin and K. K. Likharev, J. Low Temp. Phys. 62, 345 (1986).
[CrossRef]

Machida, S.

Y. Yamamoto and S. Machida, Phys. Rev. A 35, 5114 (1987).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

S. Machida and Y. Yamamoto, Opt. Commun. 57, 290 (1986); Y. Yamamoto, N. Imoto, and S. Machida, Phys. Rev. A 33, 3243 (1986); H. A. Haus and Y. Yamamoto, Phys. Rev. A 34, 270 (1986).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Nilsson, Phys. Rev. A 34, 4025 (1986).
[CrossRef] [PubMed]

S. Machida and Y. Yamamoto, “Amplitude squeezing in semiconductor lasers,” tech. paper on optics and quantum electronics (Institute of Electronics and Communication Engineers of Japan, Tokyo, 1987).

Maeda, M. W.

Mandel, L.

C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986).
[CrossRef] [PubMed]

Margalit, S.

C. Harder, J. Katz, S. Margalit, J. Shacharr, and A. Yariv, IEEE J. Quantum Electron. QE-18, 333 (1982).
[CrossRef]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Messiah, A.

A. Messiah, Quantum Mechanics (McGraw-Hill, New York, 1961).

Meystre, P.

P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

Milburn, G. J.

G. J. Milburn and D. F. Walls, Phys. Rev. A 28, 2015 (1983).

Nieto, M. M.

P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40, 411 (1968).
[CrossRef]

Nilsson, O.

Y. Yamamoto, S. Machida, and O. Nilsson, Phys. Rev. A 34, 4025 (1986).
[CrossRef] [PubMed]

Pauli, W.

W. Pauli, Handbuch der Physik (Springer-Verlag, Berlin, 1958), Vol. V.

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]

Reid, M.

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, Phys. Rev. Lett. 57, 2473 (1986).
[CrossRef] [PubMed]

Saleh, B. E. A.

Sandberg, V. D.

C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmerman, Rev. Mod Phys. 52, 341 (1980).
[CrossRef]

Saplakogln, G.

J. H. Shapiro, M. C. Teich, B. E. A. Saleh, P. Kumar, and G. Saplakogln, Phys. Rev. Lett. 56, 1136 (1986).
[CrossRef] [PubMed]

Saplakoglu, G.

Sargent, M.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Schumaker, B. L.

B. L. Schumaker, Phys. Rep. 135, 318 (1986).
[CrossRef]

Scully, M. O.

J. Krause, M. O. Scully, and H. Walther, Phys. Rev. A 34, 2032 (1986).
[CrossRef] [PubMed]

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Shacharr, J.

C. Harder, J. Katz, S. Margalit, J. Shacharr, and A. Yariv, IEEE J. Quantum Electron. QE-18, 333 (1982).
[CrossRef]

Shapiro, J. H.

Shelby, R. M.

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, Phys. Rev. Lett. 57, 2473 (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]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Stoler, D.

D. Stoler, Phys. Rev. D 4, 1925 (1971).
[CrossRef]

Sze, S. M.

S. M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1969).

Takahashi, H.

H. Takahashi, Adv. Commun. Syst. 1, 227 (1965).

Teich, M.

Teich, M. C.

J. H. Shapiro, G. Saplakoglu, S.-T. Ho, P. Kumar, B. E. A. Saleh, and M. C. Teich, “Theory of light detection in the presence of feedback,” J. Opt. Soc. Am. B 4, 1604 (1987).
[CrossRef]

J. H. Shapiro, M. C. Teich, B. E. A. Saleh, P. Kumar, and G. Saplakogln, Phys. Rev. Lett. 56, 1136 (1986).
[CrossRef] [PubMed]

B. E. A. Saleh and M. C. Teich, Opt. Commun. 52, 429 (1985).
[CrossRef]

Thorne, K. S.

C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmerman, Rev. Mod Phys. 52, 341 (1980).
[CrossRef]

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, Science 209, 547 (1980).
[CrossRef] [PubMed]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

van der Ziel, A.

A. van der Ziel, Fluctuation Phenomena in Semiconductors (Butterworth, London, 1959).

von Neumann, J.

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1955).

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Figures (17)

Fig. 1
Fig. 1

Quasi-probability densities Q(α) and electric fields E(t). (a) CS, (b) SS, (c) quadrature-phase eigenstate of â1, (d) NUS, and (e) number state.

Fig. 2
Fig. 2

Four generation principles for NUS’s: (a) unitary transformation, (b) nonunitary state reduction by measurement, (c) measurement-feedback combination, (d) pump-noise-suppressed laser oscillator.

Fig. 3
Fig. 3

(a) Changes of Q(α) by self-phase modulation in the Kerr medium and by the interference at the output mirror. Q(α) indicated by dashed contours is shifted to solid contours by the interference. Contours are drawn for Q(α) = 0.75, 0.5, and 0.25 times its maximum value (+). (b) Uncertainties in photon-number (dashed curve) and phase (dashed–dotted curve) and their product (solid curve) for the Kerr-interferometer output state.

Fig. 4
Fig. 4

Photon-number QND measurement scheme.

Fig. 5
Fig. 5

Fano factor F (solid curve), sine uncertainty (dashed curve), square of cosine mean (dotted curve), and number-phase uncertainty product Pns (dashed–dotted curve) of the state after readout β2 = 0 numerically calculated from ρ ^ a ( meas ).

Fig. 6
Fig. 6

Parametric amplifier with idler photon counting.

Fig. 7
Fig. 7

Normalized photon-number mean and uncertainty (solid lines), sine uncertainty (dashed line), and uncertainty product PnS (dashed–dotted line) for the signal output state on condition that the idler output number has been measured to be nb.

Fig. 8
Fig. 8

(a) Experimental setup for photon-number fluctuation reduction by negative feedback. The in-loop current (terminal A) will show reduced noise below the SQL, whereas the out-loop photocurrent noise level (terminal B) will be above the SQL because of the quantum noise introduced by the 50–50 beam splitter (BS). (b) Measured photon statistics by analog photon counting.

Fig. 9
Fig. 9

(a) Basic configuration for NUS generation by using QND measurement and feedback. (b) An all-optical realization of QND measurement–feedback. DM’s, dichroic mirrors; BS, beam splitter.

Fig. 10
Fig. 10

(a) A nondegenerate optical parametric oscillator (OPO) with idler photon-flux measurement–feedback. The idler output photon flux is measured in a photon-counting detector and compared to some prescribed value Ni. The deviation is fed back to a pump-wave intensity modulator. (b) An all-optical realization of a nondegenerate OPO with idler photon-flux masurement–feedback. The asymmetric OPO cavity separates the idler beam (which has output to the left) from the signal (which has output to the right). The idler-beam photon flux is used to modulate the phase of the pump beam in the upper arm of a Mach–Zender interferometer using an optical Kerr medium, as described in Subsection 3.A. When the pump intensity and the lower-arm optical path length are adjusted according to the prescribed idler photon flux Ni, the interferometer will simultaneously perform idler flux measurement and pump intensity modulation.

Fig. 11
Fig. 11

The normalized in-phase (solid lines) and quadrature-phase (dotted line) signal fluctuation spectra as a function of the frequency normalized to the signal cavity decay rate γs. The idler and pump cavity decay rates are assumed to be much higher than γs, and the pump-beam intensity is much higher than the threshold level. As can be seen in the figure, the quadrature-phase fluctuations are independent of the feedback loop gain h. But below the cavity bandwidth the in-phase component fluctuations are decreased with increasing feedback gain. It should be noticed that the in-phase component is actually squeezed by 3 dB in the free-running condition (h ≡ 0). Above the signal bandwidth both quadrature spectra become flat and equal to the SQL. Also shown is the normalized spectral uncertainty product (dashed–dotted curve).

Fig. 12
Fig. 12

(a) Amplitude noise spectra for various pump levels, r = I/Ith − 1, in an ordinary laser with shot-noise-limited pump-amplitude fluctuation. The origins for its SQL limited output at r = ∞ is indicated. (b) Amplitude and phase noise spectra in a pump-noise-suppressed laser.

Fig. 13
Fig. 13

Noise-equivalent circuit of a semiconductor-laser diode.

Fig. 14
Fig. 14

Amplitude noise spectra as a function of normalized pump level r and source resistance, Rs.

Fig. 15
Fig. 15

Experimental setup for measuring the amplitude noise spectrum. HWP, half-wavelength plate; PBS, polarization beam splitter; ATT, optical attenuator; D1, D2, detectors.

Fig. 16
Fig. 16

(a) Photocurrent spectrum for the bias-circuit-driven laser shown in the inset. A high source impedance is realized only at the LC-circuit resonant frequency fr. (b) Photocurrent spectrum normalized by SQL at a pump level of 3.9 times threshold. The bias circuit features a high source impedance (750 Ω) near 400 MHz. LD, laser diode; PD, photodiode.

Fig. 17
Fig. 17

Theoretical (solid line) and experimental (open circles) amplitude noise levels versus pump level for Rs = 750 Ω. The dotted line is the theoretical amplitude noise level for a laser with a shot-noise-limited pump fluctuation.

Equations (113)

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[ O ^ 1 , O ^ 2 ] = i O ^ 3 Δ O ^ 1 2 Δ O ^ 2 2 1 4 O ^ 1 , O ^ 2 2 = 1 4 O ^ 3 2 ,
O ^ ( r ) = e r O ^ 1 + i e - r O ^ 2 ,
Δ O ^ 1 2 = Δ O ^ 1 2 r = 0 e - 2 r Δ O ^ 2 2 = Δ O ^ 2 2 r = 0 e 2 r }             ( minimum - uncertainty product ) .
n ^ SS = a ^ 1 2 + a ^ 2 2 + sinh 2 ( r ) ( r ) .
O ^ 1 = n ^ a ^ a ^ ,             O ^ 2 = S ^ 1 2 i [ ( n ^ + 1 ) - 1 / 2 a ^ - a ^ ( n ^ + 1 ) - 1 / 2 ]
O ^ 3 = C ^ ½ [ ( n ^ + 1 ) - 1 / 2 a ^ + a ^ ( n ^ + 1 ) - 1 / 2 ] ,
Q ( α ) α ρ ^ α = α ψ 2 ,
H I = 2 ( χ a ^ 2 + χ * a ^ 2 ) .
χ = { χ ( 2 ) , parametric amplification χ ( 3 ) 2 , four - photon mixing ,
ψ out = U 2 ( L ) ψ in ,
U 2 ( L ) = exp [ i L 2 v ( χ * a ^ 2 + ξ a ^ 2 ] ,
a ^ out U 2 a ^ in U 2 = a ^ in cosh ( r ) - a ^ in e i θ sinh ( r ) ,
( Δ a ^ out , 1 ) 2 = ( Δ a ^ in , 1 ) 2 e - 2 r
( Δ a ^ out , 2 ) 2 = ( Δ a ^ in , 2 ) 2 e 2 r .
ρ out ψ out out ψ = U 2 α in α in U 2 .
H I = χ a ^ 2 a ^ 2 .
U 4 ( L ) = exp [ i χ L v ( a ^ 2 a ^ 2 ) ] = exp [ i γ 2 n ^ ( n ^ - 1 ) ] ,
a ^ out U 4 a ^ in U 4 = e i γ n ^ a ^ in .
D ( ξ ) = exp ( ξ a ^ - ξ * a ^ ) ,
a ^ out D a ^ out D = a ^ out + ξ ,
U T D ( ξ ) U 4 ( L ) = exp ( ξ a ^ - ξ * a ^ ) exp [ i γ 2 n ^ ( n ^ - 1 ) ] .
a ^ out U T a ^ in U T = e i γ n ^ a ^ in + ξ .
γ 1 = 1 2 n ^ out - 2 / 3 .
( Δ n ^ out ) 2 min = n ^ out 1 / 3 ,
H I = 2 ( χ a ^ + a ^ - + χ * a ^ + a ^ - ) ,
ρ ^ 0 = α 0 a α 0 β 0 a b b β 0 .
H I = χ n ^ a n ^ b ,
U = exp ( i μ n ^ a n ^ b ) ,
ρ ^ a b U ρ ^ 0 U = exp ( i μ n ^ a n ^ b ) ρ ^ 0 exp ( - i μ n ^ a n ^ b ) .
Q a ( α ) α ρ ^ a α a ^ a = n P b ( n ) exp ( - α - α 0 e i μ n 2 )
P a ( n ) n ρ ^ a n a a = exp ( - n a ) n a n / n !             ( Poisson ) ,
ρ ^ a Tr b ρ ^ a b = n P b ( n ) α 0 e i μ n α 0 e i μ n .
ρ ^ a ( meas ) = N Tr b ( β 2 b b β 2 ρ ^ a b ) ,
P a ( meas ) ( n ) n ρ ^ a ( meas ) n a a = P a ( n ) G ( n ) ,
G ( n ) = N b β 2 β 0 e i μ n b 2 = N × exp { - 2 [ β 0 sin ( μ n - n a sin μ ) - β 2 ] 2 } .
μ π / α 0 .
G ( n ) N exp { - 2 [ β 0 μ ( n - n a ) - β 2 ] 2 } ,
P a ( meas ) ( n ) N exp [ - ( n - n a - 2 β 2 ) 2 / 2 ( Δ n ^ ) 2 ] .
( Δ n ^ ) 2 = ( 1 n a + 4 μ 2 n b ) - 1 .
( Δ S ^ ) 2 1 4 n a + [ 1 - exp ( - 2 μ 2 n b ) ] / 2.
C ^ 2 exp ( - μ 2 n b ) .
P n S ( Δ n ^ ) 2 ( Δ S ^ ) 2 C ^ 2 1 4 sinh ( μ 2 n b ) / ( μ 2 n b ) .
ρ ^ 0 = α 0 a α 0 0 a b b 0 .
H 1 = 2 ( χ a ^ b ^ + χ * a ^ b ^ ) ,
U = exp [ i η ( a ^ b + a ^ b ^ ) ] = 1 cosh η exp [ i ( tanh η ) a ^ b ] exp [ - ln ( cosh η ) a ^ a ^ ] × exp [ - ln ( cosh η ) b ^ b ^ ] exp [ i ( tanh η ) a ^ b ^ ] ,
ρ ^ a b U ρ ^ 0 U = 1 cosh 2 η exp [ i ( tanh η ) a ^ b ] × exp [ - ln ( cosh η ) a ^ a ^ ] × ρ ^ 0 exp [ - ln ( cosh η ) a ^ a ^ ] × exp [ - i ( tanh η ) a ^ b ^ ] .
n ^ a α 0 2 cosh 2 η ,
Δ n ^ a 2 n ^ a ( 2 sinh 2 η + 1 ) ,
Δ S ^ a 2 1 4 n ^ a ( 2 sin 2 η + 1 ) .
n ^ b α 0 2 sinh 2 η .
ρ ^ a ( meas ) = N Tr b ( n b b n b ρ ^ a b ) b ,
P a ( meas ) ( n ) = { N exp ( - α 0 tanh η 2 ) ( n n b ) ( tanh 2 η ) n b cosh 2 η exp ( - α 0 cosh η 2 ) α 0 cosh η 2 ( n - n b ) ( n - n b ) ! ( for n n b ) 0 ( for n < n b ) ,
d d t A ^ = - ½ [ γ e + γ o - ω n r 2 ( χ ˜ i - i χ ˜ r ) ] A ^ + G ^ ( t ) + γ e f ^ ( t ) .
G ^ 1 ( t ) G ^ 1 ( s ) = G ^ 2 ( t ) G ^ 2 ( s ) = δ ( t - s ) 1 4 [ γ 0 + ( E ˜ c v + E ˜ v c ) ]
f ^ 1 ( t ) f ^ 1 ( s ) = f ^ 2 ( t ) f ^ 2 ( s ) = δ ( t - s ) 1 4 .
d d t N ˜ c = P ˜ - N ˜ c τ sp - ( E ˜ c v - E ˜ v c ) n ^ - E ˜ c v + F ˜ c ( t ) .
F ˜ c ( t ) F ˜ c ( s ) = δ ( t - s ) [ P ˜ + N ˜ c τ sp + ( E ˜ c v + E ˜ v c ) n ˜ ] .
F ˜ c ( t ) G ^ 1 ( s ) = - δ ( t - s ) 1 2 n ^ 1 / 2 ( E ˜ c v + E ˜ v c ) .
r ^ ( Ω ) = - f ^ ( Ω ) + γ e A ^ ( Ω ) .
Δ N ^ ( meas ) = Δ N ^ - Δ ϕ ^ p μ = 2 r ^ Δ r ^ - Δ ϕ ^ p μ ,
P ˜ = P ˜ - o h ( θ ) [ 2 r ^ Δ r ^ ( t - θ ) - Δ ϕ ^ p ( t - θ ) μ ] d θ .
Δ ψ ^ = Δ ψ ^ 0 - μ Δ n ^ p ,
S Δ r ^ 1 ( Ω ) = 1 2 γ e 2 + Ω 2 + H ( Ω ) 2 γ e 2 2 μ 2 r ^ 2 S Δ ϕ ^ p ( Ω ) γ e 2 1 + H ( Ω ) 2 + Ω 2 H ( Ω ) ) S Δ ϕ ^ p ( Ω ) 4 μ 2 r ^ 2
S Δ r ^ 2 ( Ω ) = γ e 2 Ω 2 + 1 2 + μ 2 S Δ n ^ p ( Ω ) r ^ 2 ,
S Δ N ^ ( Ω ) 4 r ^ 2 S Δ r ^ ( Ω ) = S Δ ϕ p ( Ω ) μ 2 .
S Δ N ^ ( Ω ) S Δ ψ ^ ( Ω ) = S Δ n ^ p ( Ω ) S Δ ϕ ^ p ( Ω ) ,
[ r ^ ( Ω ) , r ^ ( Ω ) ] = δ ( Ω - Ω ) ( Schwartz inequality ) S Δ r ^ 1 ( Ω ) S Δ r ^ 2 ( Ω ) ¼ S Δ N ^ ( Ω ) S Δ ψ ^ ( Ω ) 1 , } .
n ^ s ( t ) - n ^ i ( t ) = n ^ s ( 0 ) - n ^ i ( 0 ) ,
H = ω s a ^ s a ^ s + ω i a ^ i a ^ i + ω p a ^ p a ^ p + i χ ( a ^ s a ^ i a ^ p - a ^ s a ^ i a ^ p ) + i E ^ p [ a ^ p exp ( - i ω p t ) - a ^ p exp ( i ω p t ) ] + i ( γ s a ^ s f ^ s + γ i a ^ i f ^ i + γ p a ^ p f ^ p - H . c ) .
d d t a ^ s = - ( i ω s + γ s 2 ) a ^ s + χ a ^ i a ^ p + γ s f ^ s ,
d d t a ^ i = - ( i ω i + γ i 2 ) a ^ i + χ a ^ s a ^ p + γ i f ^ i ,
d d t a ^ p = - ( i ω p + γ p 2 ) a ^ p - χ a ^ s a ^ i + E ^ p exp ( - i ω p t ) + γ p f ^ p .
a ^ s A ^ s exp ( - i ω s t ) = ( A s + Δ A ^ 1 s + i Δ A ^ 2 s ) exp ( - i ω s t ) ,
γ s 2 A s = χ A i * A p ,
γ i 2 A i = χ A s * A p ,
γ p 2 A p = E ^ p - χ A s A i ,
d d t Δ A ^ 1 s = - γ s 2 Δ A ^ 1 s + χ A p Δ A ^ 1 i + χ A i Δ A ^ 1 p + γ s f ^ 1 s ,
d d t Δ A ^ 2 s = - γ s 2 Δ A ^ 2 s - χ A p Δ A ^ 2 i + χ A i Δ A ^ 2 p + γ s f ^ 2 s ,
d d t Δ A ^ 1 i = - γ i 2 Δ A ^ 1 i + χ A p Δ A ^ 1 s + χ A s Δ A ^ 1 p + γ i f ^ 1 i ,
d d t Δ A ^ 2 i = - γ i 2 Δ A ^ 2 i - χ A p Δ A ^ 2 s + χ A s Δ A ^ 2 p + γ i f ^ 2 i ,
d d t Δ A ^ 1 p = - γ p 2 Δ A ^ 1 p - 0 h ( θ ) [ E ^ p ( t - θ ) - E ^ p ] d θ - χ A ^ s Δ A ^ 1 i - χ A i Δ A ^ 1 s + γ p f ^ 1 p ,
d d t Δ A ^ 2 p = - γ p 2 Δ A ^ 2 p - χ A s Δ A ^ 2 i - χ A i Δ A ^ 2 s + γ p f ^ 2 p .
n ^ s A s 2 = γ i / γ s 1 χ ( E ^ p - γ i γ s γ p 4 χ ) ,
n ^ i A i 2 = γ s γ i n ^ s ,
n ^ p A p 2 = γ s γ i 4 χ 2 .
Δ N ^ i 2 r ^ i Δ r ^ 1 i = 2 γ i A i ( - f ^ 1 i + γ i Δ A ^ 1 i ) ,
E ^ p ( t - θ ) - E ^ p = 2 γ i A i Δ A ^ 1 i ( t - θ ) - 2 γ i A i f ^ 1 i ( t - θ ) .
Δ r ^ 1 s ( Ω ) = - f ^ 1 s ( Ω ) + γ s Δ A ^ 1 s ( Ω )
Δ r ^ 2 s ( Ω ) = - f ^ 2 s ( Ω ) + γ s Δ A ^ 1 s ( Ω ) .
S Δ N ^ s ( Ω ) = 4 r ^ s 2 S Δ r ^ 1 s ( Ω ) .
S Δ r ^ 1 s ( Ω ) = 1 2 Ω 2 Ω 2 + γ s 2 .
S Δ r ^ 2 s ( Ω ) = 1 2 Ω 2 + 2 γ s 2 Ω 2 .
Ω π / τ .
S Δ r ^ ( Ω ) = { γ e A 3 2 [ p ˜ + N ˜ c τ sp + ( E ˜ c v + E ˜ v c ) n ˜ ] + γ e ( A 1 2 + Ω 2 ) 1 4 [ ( γ 0 + γ e ) + ( E ˜ c v + E ˜ c v ) ] + 1 4 [ γ e A 1 - A 2 A 3 - Ω 2 ) 2 + Ω 2 ( γ e + A 1 ) 2 ] + γ e A 1 A 3 ( E ˜ c v + E ˜ v c ) n ˜ 1 / 2 } × [ ( A 2 2 A 3 3 + Ω 2 ) 2 + A 1 2 Ω 2 ] - 1 .
A 1 = 1 τ sp + 1 τ st ,
A 2 = 2 ( γ e + γ 0 ) n ^ 1 / 2 ,
A 3 = 1 2 n ˜ 1 / 2 τ st ,
S Δ r ^ ( Ω ) = 1 2 Ω 2 Ω 2 + γ e 2
S Δ ψ ^ ( Ω ) = 1 r ^ 2 ( 1 2 + γ e 2 Ω 2 ) ,
R ( d I d V ) - 1 = m V T q N c ( 1 τ sp + 1 τ st ) - 1
C d ( q N e ) d V = q N m V T .
S i ( Ω ) ~ 2 q 2 [ N c τ c + ( E ˜ c v + E ˜ v c ) n ˜ ] ~ 2 q I + 4 q 2 E ˜ v c n ˜ .
S V s ( Ω ) ~ 4 k B T R s .
i L = q ( E ˜ c v - E ˜ v c ) 2 A Δ A .
L = m V T τ st q N c ( E ˜ c v - E ˜ v c )
R s e = L [ γ - ( E ˜ c v - E ˜ v c ) ] .
S v ( Ω ) = 2 q 2 ( E ˜ c v - E ˜ v c ) 2 L 2 [ ( E ˜ c v + E ˜ v c ) n ˜ + γ n ^ ] .
S i v ( Ω ) i ( Ω ) v ( Ω ) = - 2 q 2 ( E ˜ c v 2 - E v c 2 ) L n ^ .
d d t i L = v n - v L - R se L i L d d t Δ A ^ = - [ γ - ( E ˜ c v - E ˜ v c ) ] Δ A ^ + 1 2 A τ st Δ N ˜ + ( G ^ r + γ e f ^ r )
d d t v n = - 1 C ( 1 R + 1 R s ) v n - i L C + i C - v C R s , d d t Δ N ^ c = - ( 1 τ sp + 1 τ C R ) Δ N ˜ c - ( E ˜ c v - E ˜ v c ) 2 A ˜ Δ A ^ + F ˜ c + F ˜ th .
S F ˜ c ( Ω ) = 2 [ N ^ c τ sp + ( E ˜ c v + E ˜ v c ) n ˜ ]
S F ˜ th ( Ω ) = 4 V T q R s .
S F t h ( Ω ) = 4 V T q R s < 2 q I R S > 2 R .

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