Abstract

The intensity fluctuations of laser light are derived from photon number rate equations. In the limit of short times, the photon statistics for small laser devices such as typical semiconductor laser diodes show thermal characteristics, even above threshold. In the limit of long-time averages represented by the low-frequency component of the noise, the same devices exhibit squeezing. It is shown that squeezing and thermal noise can coexist in the multimode output field of laser diodes. This result implies that the squeezed light generated by regularly pumped semiconductor laser diodes is qualitatively different from single-mode squeezed light. In particular, no entanglement between photons can be generated by use of this type of collective multimode squeezing.

© 2000 Optical Society of America

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References

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  1. H. Haken, “Nonlinear theory of laser noise and coherence. I,” Z. Phys. 181, 96–124 (1964).
    [CrossRef]
  2. M. O. Scully and W. E. Lamb, “Quantum theory of the optical maser. I. General theory,” Phys. Rev. 159, 208–226 (1967).
    [CrossRef]
  3. M. Lax and W. H. Louisell, “Quantum Noise. XII. Density-operator treatment of field and population fluctuations,” Phys. Rev. 185, 568–591 (1969).
    [CrossRef]
  4. H. Risken, “Statistical properties of laser light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Chap. VIII, pp. 239–294.
  5. F. T. Arecchi and O. E. Schulz-Dubois, eds., Laser Handbook (North-Holland, Amsterdam, 1972), Vol. I.
  6. Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
    [CrossRef] [PubMed]
  7. S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
    [CrossRef] [PubMed]
  8. S. Machida and Y. Yamamoto, “Ultrabroadband amplitude squeezing in a semiconductor laser,” Phys. Rev. Lett. 60, 792–794 (1988).
    [CrossRef] [PubMed]
  9. S. Machida and Y. Yamamoto, “Observation of amplitude squeezing from semiconductor lasers by balanced direct detection with a delay line,” Opt. Lett. 14, 1045–1047 (1989).
    [CrossRef] [PubMed]
  10. S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2763 (1992).
    [CrossRef] [PubMed]
  11. M. J. Freeman, D. C. Kilper, D. G. Steel, D. Craig, and D. R. Scifres, “Room-temperature amplitude-squeezed light from an injection-locked quantum-well laser with a time-varying drive current,” Opt. Lett. 20, 183–185 (1995).
    [CrossRef] [PubMed]
  12. D. C. Kilper, D. G. Steel, D. Craig, and D. R. Scifres, “Polarization-dependent noise in photon-number squeezed light generated by quantum-well lasers,” Opt. Lett. 21, 1283–1285 (1996).
    [CrossRef] [PubMed]
  13. T. A. B. Kennedy and D. F. Walls, “Amplitude noise reduction in atomic and semiconductor lasers,” Phys. Rev. A 40, 6366–6373 (1989).
    [CrossRef] [PubMed]
  14. For a compact explanation of this procedure, see, e.g., D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994), pp. 229 ff.
  15. F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
    [CrossRef] [PubMed]
  16. A. Eschmann and C. W. Gardiner, “Master-equation theory of multimode semiconductor lasers,” Phys. Rev. A 54, 760–775 (1996).
    [CrossRef] [PubMed]
  17. For a practical introduction of the relevant device properties, see, e.g., K. J. Ebeling, Integrated Quantum Electronics (Springer-Verlag, Berlin, 1992). A theoretical derivation from band structure properties can be found in H. F. Hofmann and O. Hess, “Quantum Maxwell–Bloch equations for spatially inhomogeneous semiconductor lasers,” Phys. Rev. A 59, 2342–2358 (1998).
    [CrossRef]
  18. N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A (to be published).
  19. Y. M. Golubev, I. V. Sokolov, and M. I. Kolobov, “Possibility of suppressing quantum light fluctuations when excess photon fluctuations occur inside a cavity,” JETP 84, 864–874 (1997).
    [CrossRef]
  20. L. D. Landau and E. M. Lifschitz, Statistische Physik (Akademie Verlag, Berlin, 1987), Sec. 75.

1997 (1)

Y. M. Golubev, I. V. Sokolov, and M. I. Kolobov, “Possibility of suppressing quantum light fluctuations when excess photon fluctuations occur inside a cavity,” JETP 84, 864–874 (1997).
[CrossRef]

1996 (2)

1995 (2)

M. J. Freeman, D. C. Kilper, D. G. Steel, D. Craig, and D. R. Scifres, “Room-temperature amplitude-squeezed light from an injection-locked quantum-well laser with a time-varying drive current,” Opt. Lett. 20, 183–185 (1995).
[CrossRef] [PubMed]

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

1992 (1)

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2763 (1992).
[CrossRef] [PubMed]

1989 (2)

1988 (1)

S. Machida and Y. Yamamoto, “Ultrabroadband amplitude squeezing in a semiconductor laser,” Phys. Rev. Lett. 60, 792–794 (1988).
[CrossRef] [PubMed]

1987 (1)

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

1986 (1)

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

1969 (1)

M. Lax and W. H. Louisell, “Quantum Noise. XII. Density-operator treatment of field and population fluctuations,” Phys. Rev. 185, 568–591 (1969).
[CrossRef]

1967 (1)

M. O. Scully and W. E. Lamb, “Quantum theory of the optical maser. I. General theory,” Phys. Rev. 159, 208–226 (1967).
[CrossRef]

1964 (1)

H. Haken, “Nonlinear theory of laser noise and coherence. I,” Z. Phys. 181, 96–124 (1964).
[CrossRef]

Bramati, A.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Craig, D.

Eschmann, A.

A. Eschmann and C. W. Gardiner, “Master-equation theory of multimode semiconductor lasers,” Phys. Rev. A 54, 760–775 (1996).
[CrossRef] [PubMed]

Freeman, M. J.

Gardiner, C. W.

A. Eschmann and C. W. Gardiner, “Master-equation theory of multimode semiconductor lasers,” Phys. Rev. A 54, 760–775 (1996).
[CrossRef] [PubMed]

Giacobino, E.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Golubev, Y. M.

Y. M. Golubev, I. V. Sokolov, and M. I. Kolobov, “Possibility of suppressing quantum light fluctuations when excess photon fluctuations occur inside a cavity,” JETP 84, 864–874 (1997).
[CrossRef]

Grangier, P.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Haken, H.

H. Haken, “Nonlinear theory of laser noise and coherence. I,” Z. Phys. 181, 96–124 (1964).
[CrossRef]

Inoue, S.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2763 (1992).
[CrossRef] [PubMed]

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Kennedy, T. A. B.

T. A. B. Kennedy and D. F. Walls, “Amplitude noise reduction in atomic and semiconductor lasers,” Phys. Rev. A 40, 6366–6373 (1989).
[CrossRef] [PubMed]

Kilper, D. C.

Kolobov, M. I.

Y. M. Golubev, I. V. Sokolov, and M. I. Kolobov, “Possibility of suppressing quantum light fluctuations when excess photon fluctuations occur inside a cavity,” JETP 84, 864–874 (1997).
[CrossRef]

Lamb, W. E.

M. O. Scully and W. E. Lamb, “Quantum theory of the optical maser. I. General theory,” Phys. Rev. 159, 208–226 (1967).
[CrossRef]

Lax, M.

M. Lax and W. H. Louisell, “Quantum Noise. XII. Density-operator treatment of field and population fluctuations,” Phys. Rev. 185, 568–591 (1969).
[CrossRef]

Louisell, W. H.

M. Lax and W. H. Louisell, “Quantum Noise. XII. Density-operator treatment of field and population fluctuations,” Phys. Rev. 185, 568–591 (1969).
[CrossRef]

Machida, S.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2763 (1992).
[CrossRef] [PubMed]

S. Machida and Y. Yamamoto, “Observation of amplitude squeezing from semiconductor lasers by balanced direct detection with a delay line,” Opt. Lett. 14, 1045–1047 (1989).
[CrossRef] [PubMed]

S. Machida and Y. Yamamoto, “Ultrabroadband amplitude squeezing in a semiconductor laser,” Phys. Rev. Lett. 60, 792–794 (1988).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Marin, F.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Nilsson, O.

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Ohzu, H.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2763 (1992).
[CrossRef] [PubMed]

Poizat, J.-Ph.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Roch, J.-F.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Scifres, D. R.

Scully, M. O.

M. O. Scully and W. E. Lamb, “Quantum theory of the optical maser. I. General theory,” Phys. Rev. 159, 208–226 (1967).
[CrossRef]

Sokolov, I. V.

Y. M. Golubev, I. V. Sokolov, and M. I. Kolobov, “Possibility of suppressing quantum light fluctuations when excess photon fluctuations occur inside a cavity,” JETP 84, 864–874 (1997).
[CrossRef]

Steel, D. G.

Walls, D. F.

T. A. B. Kennedy and D. F. Walls, “Amplitude noise reduction in atomic and semiconductor lasers,” Phys. Rev. A 40, 6366–6373 (1989).
[CrossRef] [PubMed]

Yamamoto, Y.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2763 (1992).
[CrossRef] [PubMed]

S. Machida and Y. Yamamoto, “Observation of amplitude squeezing from semiconductor lasers by balanced direct detection with a delay line,” Opt. Lett. 14, 1045–1047 (1989).
[CrossRef] [PubMed]

S. Machida and Y. Yamamoto, “Ultrabroadband amplitude squeezing in a semiconductor laser,” Phys. Rev. Lett. 60, 792–794 (1988).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Zhang, T.-C.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

JETP (1)

Y. M. Golubev, I. V. Sokolov, and M. I. Kolobov, “Possibility of suppressing quantum light fluctuations when excess photon fluctuations occur inside a cavity,” JETP 84, 864–874 (1997).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (2)

M. O. Scully and W. E. Lamb, “Quantum theory of the optical maser. I. General theory,” Phys. Rev. 159, 208–226 (1967).
[CrossRef]

M. Lax and W. H. Louisell, “Quantum Noise. XII. Density-operator treatment of field and population fluctuations,” Phys. Rev. 185, 568–591 (1969).
[CrossRef]

Phys. Rev. A (4)

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

T. A. B. Kennedy and D. F. Walls, “Amplitude noise reduction in atomic and semiconductor lasers,” Phys. Rev. A 40, 6366–6373 (1989).
[CrossRef] [PubMed]

A. Eschmann and C. W. Gardiner, “Master-equation theory of multimode semiconductor lasers,” Phys. Rev. A 54, 760–775 (1996).
[CrossRef] [PubMed]

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2763 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

S. Machida and Y. Yamamoto, “Ultrabroadband amplitude squeezing in a semiconductor laser,” Phys. Rev. Lett. 60, 792–794 (1988).
[CrossRef] [PubMed]

Z. Phys. (1)

H. Haken, “Nonlinear theory of laser noise and coherence. I,” Z. Phys. 181, 96–124 (1964).
[CrossRef]

Other (6)

For a practical introduction of the relevant device properties, see, e.g., K. J. Ebeling, Integrated Quantum Electronics (Springer-Verlag, Berlin, 1992). A theoretical derivation from band structure properties can be found in H. F. Hofmann and O. Hess, “Quantum Maxwell–Bloch equations for spatially inhomogeneous semiconductor lasers,” Phys. Rev. A 59, 2342–2358 (1998).
[CrossRef]

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A (to be published).

L. D. Landau and E. M. Lifschitz, Statistische Physik (Akademie Verlag, Berlin, 1987), Sec. 75.

For a compact explanation of this procedure, see, e.g., D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994), pp. 229 ff.

H. Risken, “Statistical properties of laser light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Chap. VIII, pp. 239–294.

F. T. Arecchi and O. E. Schulz-Dubois, eds., Laser Handbook (North-Holland, Amsterdam, 1972), Vol. I.

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Figures (8)

Fig. 1
Fig. 1

Energy-flow diagram of a single-mode laser.

Fig. 2
Fig. 2

Dominant time scales of the fluctuation dynamics by average photon number n¯ and spontaneous emission factor β for nT=3/2 and 3κτ=104. Threshold photon number nth is illustrated by the dotted line.

Fig. 3
Fig. 3

Contour plot of the photon-number fluctuations as a function of average photon number n¯ and spontaneous-emission factor β for nT=3/2 and 3κτ=104. The contours correspond to constant ratios of photon-number fluctuations δn¯2 and shot-noise level n¯. This ratio is equal to 1 in the black region and increases by a factor of 105/93.6 for every contour. The initial increase at low photon numbers n¯ is thermal (δn¯2n¯2).

Fig. 4
Fig. 4

Low-frequency noise characteristics (in decibels) relative to the shot-noise limit for nT=3/2 and β=10-3 and pump noise factors σ of 1, 0.25, 0.0625, and 0.

Fig. 5
Fig. 5

Noise threshold nδ and squeezing threshold nsq as a function of the device size given by the inverse spontaneous-emission factor β-1. The other device parameters are constant at nT=3/2 and 3κτ=104.

Fig. 6
Fig. 6

Photon-number fluctuations (PNF’s) and low-frequency intensity noise (LFN’s) characteristics (in decibels) relative to shot noise for nT=3/2 and 3κτ=104. Top, a mesoscopic laser with β=10-4; bottom, a microscopic laser with β=10-2. For comparison with the injected current, the dashed lines mark the region between twice threshold (2κn¯=jth) and ten times threshold (2κn¯=9jth).

Fig. 7
Fig. 7

Illustration of the intensity noise. The noise averages out on long time scales, even though it is thermal on short time scales.

Fig. 8
Fig. 8

Two-time correlation of intensity δI(t)δI(t+Δt) for γn/Γn=5.

Equations (36)

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ddtN=j-1τN-2 βτ(N-NT)n+qN(t),
ddtn=2βτ(N-NT)-κn+βτN+qn(t),
qN(t)qN(t+Δt)
=σj+βτ(n+1)N+2 βτNTn+βτNδ(Δt),
qn(t)qn(t+Δt)
=2κn+2 βτNTn+βτNδ(Δt),
qN(t)qn(t+Δt)
=-2 βτNTn+βτNδ(Δt).
I(t)=2κn+qI(t),
qI(t)qI(t+Δt)=2κnδ(Δt),
qn(t)qI(t+Δt)=-2κnδ(Δt),
qN(t)qI(t+Δt)=0.
βV10-14 cm3,
NTV1018 cm-3,
τ3×10-9 s.
κ<βNTτ3.33×1012 s-1.
N¯=1+(1/2nT)1+(1/2n¯)NT,
n¯=j-jth4κ-14+j-jth4κ+142+jth4κ1/2,
nT=βNT2κT,
jth=limn¯(1-β) N¯τ=2κ 1-ββ nT+12.
nth=n¯(jth)=116+jth4κ1/2-14.
ddtδN=-ΓNδN-rωRδn+qN(t),
ddtδn=r-1ωRδN-γnδn+qn(t),
ΓN=1τ(1+2βn¯),
γn=2κ nT+(1/2)n¯+(1/2),
ωR=2κβ n¯-nTκτ1/2,
r=κτβ (n¯-nT)(n¯+1/2)21/2.
δn¯2n¯21+2βn¯3(2βn¯+1)(nT+1/2)[4β(κτ+1)n¯2+n¯+2κτ(nT+1/2)]-1.
2βnδ3(2βnδ+1)(nT+1/2)[4β(κτ+1)nδ2+nδ+2κτ(nT+1/2)]=1.
δI(t)=I(t)-I¯=2κδn+qI(t).
δI(t)δI(t+Δt)=4κ2δn(t)δn(t+Δt)+2κqI(t)δn(t+Δt)+qI(t)qI(t+Δt).
δI2¯(ω0)LSN=0dτ[4κ2δnδn(t)(t+τ)+2κqI(t)δn(t+τ)]0dτqI(t)qI(t+τ)+1.
δI2¯(ω0)LSN(nT+1/2)[4βn¯3+2n¯2+(nT+1/2)][2βn¯2+(nT+1/2)]2+σ 4β2n¯4+4β(nT+1/2)n¯3[2βn¯2+(nT+1/2)]2.
nsq=12β{(nT+1/2)+[(nT+1/2)2+(nT+1/2)]1/2}nT+1/2βjth2κ.
δI(t)δI(t+Δt)2κn¯δ(Δt)+4κ2n¯2 exp(-γnΔt)-4κ2n¯2(ΓN/γn)exp(-ΓNΔt).
ρˆ=N!(M-1)(N+M-1)! n=0N (M+N-2-n)!(N-n)!|nn|MN+M n=01+MN-n|nn|,

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