Abstract

We develop the quantum-jump statistical tools required to analyze the probe response in three-level systems where the probe and driving lasers have arbitrary intensities and detunings. We apply these tools to investigate the appearance of two inversionless amplification sidebands in the probe spectrum as the driving laser intensity increases.

© 2003 Optical Society of America

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References

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  1. E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, Vol. XXXV, E. Wolf, ed. (Elsevier, Amsterdam, 1996).
  2. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
    [CrossRef]
  3. O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175–190 (1992).
    [CrossRef]
  4. M. O. Scully, “From lasers and masers to phaseounium and phasers,” Phys. Rep. 219, 191–201 (1992).
    [CrossRef]
  5. P. Mandel, “Lasing without inversion: a useful concept?” Contemp. Phys. 34, 235–246 (1993).
    [CrossRef]
  6. J. Mompart and R. Corbalán, “Lasing without inversion,” J. Opt. B: Quantum Semiclassical Opt. 2, R7-R24 (2000).
    [CrossRef]
  7. C. Peters and W. Lange, “Laser action below threshold inversion due to coherent population trapping,” Appl. Phys. B 62, 221–224 (1996).
    [CrossRef]
  8. J. Mompart, R. Corbalán, and R. Vilaseca, “Giant pulse lasing in three-level systems,” Phys. Rev. A 59, 3038–3043 (1999).
    [CrossRef]
  9. G. S. Agarwal, “Origin of gain in systems without inversion in bare or dressed states,” Phys. Rev. A 44, R28-R30 (1991).
    [CrossRef] [PubMed]
  10. J. Mompart and R. Corbalán, “Inversionless amplification in three-level systems: dressed-states quantum interference and quantum-jump analyses,” Opt. Commun. 156, 133–144 (1998).
    [CrossRef]
  11. P. Mandel and O. Kocharovskaya, “Inversionless amplification of a monochromatic field by a three-level medium,” Phys. Rev. A 46, 2700–2706 (1992).
    [CrossRef] [PubMed]
  12. These results hold for the usual case in which the driven transition is not inverted; see Ref. 10.
  13. J. Kitching and L. Hollberg, “Interference-induced optical gain without population inversion in cold, trapped atoms,” Phys. Rev. A 59, 4685–4689 (1999).
    [CrossRef]
  14. A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
    [CrossRef] [PubMed]
  15. H. J. Carmichael, “An open-systems approach to quantum optics,” Lect. Notes Phys. 18 (1993).
  16. J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
    [CrossRef] [PubMed]
  17. R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
    [CrossRef] [PubMed]
  18. K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
    [CrossRef]
  19. E. Arimondo, “Mechanism in laser without inversion,” in Shangai International Symposium on Quantum Optics, D.-H. Wabg and Z. Wang, eds., Proc. SPIE 1726, 484–489 (1992).
    [CrossRef]
  20. C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, “Quantum-jump approach to dissipative processes: application to amplification without inversion,” J. Opt. Soc. Am. B 10, 2107–2120 (1993).
    [CrossRef]
  21. H. J. Carmichael, “Coherence and decoherence in the interaction of lights with atoms,” Phys. Rev. A 56, 5065–5099 (1997).
    [CrossRef]
  22. J. Mompart and R. Corbalán, “Quantum-jump approach to dipole dephasing: application to inversionless amplification,” Eur. Phys. J. D 5, 351–356 (1999).
    [CrossRef]
  23. J. Mompart and R. Corbalán, “Generalized Einstein B coefficients for coherently driven three-level systems,” Phys. Rev. A 63, 063810 (2001).
    [CrossRef]
  24. Note that it is straightforward to use these Rij with i≠j to describe bidirectional pumping. For instance, let us denote by Λ the rate of a bidirectional pumping process coupled to transition |a〉–|b〉; then Rab=Λ+γab and Rba=Λ with γab being the spontaneous emission rate from |a〉 to |b〉.
  25. As a general feature, dissipative processes associated with Rij with i≠j correspond to quantum jumps connecting different manifolds, while those associated with Rij with i=j yield a new coherent evolution period in the same manifold as the previous one; see Ref. 22.
  26. The mean change of the probe photon number per unit time relates to the mean change per period as 〈dNα/dt〉=〈ΔNα〉/T where T is the average time between two consecutive quantum jumps; see Eqs. (4.11) and (4.16) in Ref. 20.
  27. Note that as a result of the presence of dissipation, xij(τ), yij(τ)→0 in an exponential way, which guarantees the convergence of ∫0|cij(τ)|2dτ.

2001 (1)

J. Mompart and R. Corbalán, “Generalized Einstein B coefficients for coherently driven three-level systems,” Phys. Rev. A 63, 063810 (2001).
[CrossRef]

2000 (1)

J. Mompart and R. Corbalán, “Lasing without inversion,” J. Opt. B: Quantum Semiclassical Opt. 2, R7-R24 (2000).
[CrossRef]

1999 (3)

J. Mompart, R. Corbalán, and R. Vilaseca, “Giant pulse lasing in three-level systems,” Phys. Rev. A 59, 3038–3043 (1999).
[CrossRef]

J. Kitching and L. Hollberg, “Interference-induced optical gain without population inversion in cold, trapped atoms,” Phys. Rev. A 59, 4685–4689 (1999).
[CrossRef]

J. Mompart and R. Corbalán, “Quantum-jump approach to dipole dephasing: application to inversionless amplification,” Eur. Phys. J. D 5, 351–356 (1999).
[CrossRef]

1998 (1)

J. Mompart and R. Corbalán, “Inversionless amplification in three-level systems: dressed-states quantum interference and quantum-jump analyses,” Opt. Commun. 156, 133–144 (1998).
[CrossRef]

1997 (2)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[CrossRef]

H. J. Carmichael, “Coherence and decoherence in the interaction of lights with atoms,” Phys. Rev. A 56, 5065–5099 (1997).
[CrossRef]

1996 (1)

C. Peters and W. Lange, “Laser action below threshold inversion due to coherent population trapping,” Appl. Phys. B 62, 221–224 (1996).
[CrossRef]

1995 (1)

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

1993 (3)

1992 (6)

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

E. Arimondo, “Mechanism in laser without inversion,” in Shangai International Symposium on Quantum Optics, D.-H. Wabg and Z. Wang, eds., Proc. SPIE 1726, 484–489 (1992).
[CrossRef]

O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175–190 (1992).
[CrossRef]

M. O. Scully, “From lasers and masers to phaseounium and phasers,” Phys. Rep. 219, 191–201 (1992).
[CrossRef]

P. Mandel and O. Kocharovskaya, “Inversionless amplification of a monochromatic field by a three-level medium,” Phys. Rev. A 46, 2700–2706 (1992).
[CrossRef] [PubMed]

1991 (1)

G. S. Agarwal, “Origin of gain in systems without inversion in bare or dressed states,” Phys. Rev. A 44, R28-R30 (1991).
[CrossRef] [PubMed]

Agarwal, G. S.

G. S. Agarwal, “Origin of gain in systems without inversion in bare or dressed states,” Phys. Rev. A 44, R28-R30 (1991).
[CrossRef] [PubMed]

Arimondo, E.

C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, “Quantum-jump approach to dissipative processes: application to amplification without inversion,” J. Opt. Soc. Am. B 10, 2107–2120 (1993).
[CrossRef]

E. Arimondo, “Mechanism in laser without inversion,” in Shangai International Symposium on Quantum Optics, D.-H. Wabg and Z. Wang, eds., Proc. SPIE 1726, 484–489 (1992).
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, “Coherence and decoherence in the interaction of lights with atoms,” Phys. Rev. A 56, 5065–5099 (1997).
[CrossRef]

Castin, Y.

K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
[CrossRef]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

Cohen-Tannoudji, C.

Corbalán, R.

J. Mompart and R. Corbalán, “Generalized Einstein B coefficients for coherently driven three-level systems,” Phys. Rev. A 63, 063810 (2001).
[CrossRef]

J. Mompart and R. Corbalán, “Lasing without inversion,” J. Opt. B: Quantum Semiclassical Opt. 2, R7-R24 (2000).
[CrossRef]

J. Mompart, R. Corbalán, and R. Vilaseca, “Giant pulse lasing in three-level systems,” Phys. Rev. A 59, 3038–3043 (1999).
[CrossRef]

J. Mompart and R. Corbalán, “Quantum-jump approach to dipole dephasing: application to inversionless amplification,” Eur. Phys. J. D 5, 351–356 (1999).
[CrossRef]

J. Mompart and R. Corbalán, “Inversionless amplification in three-level systems: dressed-states quantum interference and quantum-jump analyses,” Opt. Commun. 156, 133–144 (1998).
[CrossRef]

Dalibard, J.

K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
[CrossRef]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

Dum, R.

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[CrossRef]

Hollberg, L.

J. Kitching and L. Hollberg, “Interference-induced optical gain without population inversion in cold, trapped atoms,” Phys. Rev. A 59, 4685–4689 (1999).
[CrossRef]

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

Kitching, J.

J. Kitching and L. Hollberg, “Interference-induced optical gain without population inversion in cold, trapped atoms,” Phys. Rev. A 59, 4685–4689 (1999).
[CrossRef]

Kocharovskaya, O.

P. Mandel and O. Kocharovskaya, “Inversionless amplification of a monochromatic field by a three-level medium,” Phys. Rev. A 46, 2700–2706 (1992).
[CrossRef] [PubMed]

O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175–190 (1992).
[CrossRef]

Lange, W.

C. Peters and W. Lange, “Laser action below threshold inversion due to coherent population trapping,” Appl. Phys. B 62, 221–224 (1996).
[CrossRef]

Lukin, M. D.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

Mandel, P.

P. Mandel, “Lasing without inversion: a useful concept?” Contemp. Phys. 34, 235–246 (1993).
[CrossRef]

P. Mandel and O. Kocharovskaya, “Inversionless amplification of a monochromatic field by a three-level medium,” Phys. Rev. A 46, 2700–2706 (1992).
[CrossRef] [PubMed]

Mølmer, K.

K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
[CrossRef]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

Mompart, J.

J. Mompart and R. Corbalán, “Generalized Einstein B coefficients for coherently driven three-level systems,” Phys. Rev. A 63, 063810 (2001).
[CrossRef]

J. Mompart and R. Corbalán, “Lasing without inversion,” J. Opt. B: Quantum Semiclassical Opt. 2, R7-R24 (2000).
[CrossRef]

J. Mompart, R. Corbalán, and R. Vilaseca, “Giant pulse lasing in three-level systems,” Phys. Rev. A 59, 3038–3043 (1999).
[CrossRef]

J. Mompart and R. Corbalán, “Quantum-jump approach to dipole dephasing: application to inversionless amplification,” Eur. Phys. J. D 5, 351–356 (1999).
[CrossRef]

J. Mompart and R. Corbalán, “Inversionless amplification in three-level systems: dressed-states quantum interference and quantum-jump analyses,” Opt. Commun. 156, 133–144 (1998).
[CrossRef]

Nikonov, D. E.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

Peters, C.

C. Peters and W. Lange, “Laser action below threshold inversion due to coherent population trapping,” Appl. Phys. B 62, 221–224 (1996).
[CrossRef]

Ritsch, H.

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

Robinson, H. G.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

Scully, M. O.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

M. O. Scully, “From lasers and masers to phaseounium and phasers,” Phys. Rep. 219, 191–201 (1992).
[CrossRef]

Velichansky, V. L.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

Vilaseca, R.

J. Mompart, R. Corbalán, and R. Vilaseca, “Giant pulse lasing in three-level systems,” Phys. Rev. A 59, 3038–3043 (1999).
[CrossRef]

Zambon, B.

Zibrov, A. S.

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

Zoller, P.

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

Appl. Phys. B (1)

C. Peters and W. Lange, “Laser action below threshold inversion due to coherent population trapping,” Appl. Phys. B 62, 221–224 (1996).
[CrossRef]

Contemp. Phys. (1)

P. Mandel, “Lasing without inversion: a useful concept?” Contemp. Phys. 34, 235–246 (1993).
[CrossRef]

Eur. Phys. J. D (1)

J. Mompart and R. Corbalán, “Quantum-jump approach to dipole dephasing: application to inversionless amplification,” Eur. Phys. J. D 5, 351–356 (1999).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

J. Mompart and R. Corbalán, “Lasing without inversion,” J. Opt. B: Quantum Semiclassical Opt. 2, R7-R24 (2000).
[CrossRef]

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

Opt. Commun. (1)

J. Mompart and R. Corbalán, “Inversionless amplification in three-level systems: dressed-states quantum interference and quantum-jump analyses,” Opt. Commun. 156, 133–144 (1998).
[CrossRef]

Phys. Rep. (2)

O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175–190 (1992).
[CrossRef]

M. O. Scully, “From lasers and masers to phaseounium and phasers,” Phys. Rep. 219, 191–201 (1992).
[CrossRef]

Phys. Rev. A (7)

J. Mompart, R. Corbalán, and R. Vilaseca, “Giant pulse lasing in three-level systems,” Phys. Rev. A 59, 3038–3043 (1999).
[CrossRef]

G. S. Agarwal, “Origin of gain in systems without inversion in bare or dressed states,” Phys. Rev. A 44, R28-R30 (1991).
[CrossRef] [PubMed]

P. Mandel and O. Kocharovskaya, “Inversionless amplification of a monochromatic field by a three-level medium,” Phys. Rev. A 46, 2700–2706 (1992).
[CrossRef] [PubMed]

J. Kitching and L. Hollberg, “Interference-induced optical gain without population inversion in cold, trapped atoms,” Phys. Rev. A 59, 4685–4689 (1999).
[CrossRef]

J. Mompart and R. Corbalán, “Generalized Einstein B coefficients for coherently driven three-level systems,” Phys. Rev. A 63, 063810 (2001).
[CrossRef]

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

H. J. Carmichael, “Coherence and decoherence in the interaction of lights with atoms,” Phys. Rev. A 56, 5065–5099 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb,” Phys. Rev. Lett. 75, 1499–1502 (1995).
[CrossRef] [PubMed]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[CrossRef]

Proc. SPIE (1)

E. Arimondo, “Mechanism in laser without inversion,” in Shangai International Symposium on Quantum Optics, D.-H. Wabg and Z. Wang, eds., Proc. SPIE 1726, 484–489 (1992).
[CrossRef]

Other (7)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, Vol. XXXV, E. Wolf, ed. (Elsevier, Amsterdam, 1996).

H. J. Carmichael, “An open-systems approach to quantum optics,” Lect. Notes Phys. 18 (1993).

These results hold for the usual case in which the driven transition is not inverted; see Ref. 10.

Note that it is straightforward to use these Rij with i≠j to describe bidirectional pumping. For instance, let us denote by Λ the rate of a bidirectional pumping process coupled to transition |a〉–|b〉; then Rab=Λ+γab and Rba=Λ with γab being the spontaneous emission rate from |a〉 to |b〉.

As a general feature, dissipative processes associated with Rij with i≠j correspond to quantum jumps connecting different manifolds, while those associated with Rij with i=j yield a new coherent evolution period in the same manifold as the previous one; see Ref. 22.

The mean change of the probe photon number per unit time relates to the mean change per period as 〈dNα/dt〉=〈ΔNα〉/T where T is the average time between two consecutive quantum jumps; see Eqs. (4.11) and (4.16) in Ref. 20.

Note that as a result of the presence of dissipation, xij(τ), yij(τ)→0 in an exponential way, which guarantees the convergence of ∫0|cij(τ)|2dτ.

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Figures (10)

Fig. 1
Fig. 1

Cascade three-level system under investigation. For the sake of clarity, the three-level scheme has been split into two separate figures. At the left-hand side is shown the coherent interaction in which α and Δα (β and Δβ) are the Rabi frequency and detuning of the probe (driving) laser field. At the right-hand side, the rates Rij with i,j=a,b,c account for all possible incoherent processes.

Fig. 2
Fig. 2

Different manifolds ξ(Nβ±m, Nα±n) of the three quasi-degenerate states of the total system atom plus laser fields for the cascade scheme of Fig. 1. Nα and Nβ are, respectively, the photon number of probe and driving laser modes. The solid horizontal arrows account for the continuous evolution given by the coherent interaction, while the dashed oblique and circular arrows account for the quantum jumps produced by the dissipative processes.

Fig. 3
Fig. 3

Time evolution of the atomic populations in a stochastic quantum trajectory for the following parameter setting: Rab=4.5Rcb, Rba=4.3Rcb, Rac=0.2Rcb, Rbc=3.5Rcb, α=β=5Rcb, and Δα=Δβ=Rca=0.

Fig. 4
Fig. 4

Stochastic quantum trajectory for the same parameters as in Fig. 3. (a) Time evolution of the imaginary part of the atomic coherences in the probed and driven transitions. (b) Time evolution of the number of photons of probe and driving fields. In (b) the marker 0 means no change in the probe number of photons, while markers ±1 and ±2 indicate the occurrence of a one-photon probe gain or loss or a two-photon probe gain or loss process, respectively.

Fig. 5
Fig. 5

Probability P(i) of starting a coherent evolution period in state |i as a function of the driving-field Rabi frequency for the following parameter settings: Rab=4.5Rcb, Rba=4.3Rcb, Rac=0.2Rcb, Rbc=3.5Rcb, α=0.0001Rcb, and Δα=Δβ=Rca=0. The solid symbols correspond to the P(i) values obtained by performing a Monte Carlo simulation. The solid curves are fittings to the symbols. For high driving intensities, the probabilities P(i) converge to the value given by Eqs. (17) (horizontal dotted lines in this figure).

Fig. 6
Fig. 6

Time averaged (a) steady-state populations and (b) atomic coherence in the probed transition for different probe detunings. The rest of the parameters are as in Fig. 3.

Fig. 7
Fig. 7

(a) Probability that a random choice among all the coherent evolution periods of the cascade scheme of Fig. 1 gives a one-photon gain process P(a, b), a one-photon loss process P(b, a), a two-photon gain process P(a, c), and a two-photon loss process P(c, a) as a function of the probe detuning. (b) Mean change of the probe-field photon number per period. The driving-field Rabi frequency is β=0.01Rcb and the rest of parameters are as in Fig. 5.

Fig. 8
Fig. 8

As in Fig. 7 for β=Rcb.

Fig. 9
Fig. 9

As in Fig. 7 for β=3Rcb.

Fig. 10
Fig. 10

As in Fig. 7 for β=9Rcb.

Equations (47)

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

ξ(Nβ+m, Nα+n){|a, Nβ+m, Nα+n, |b, Nβ+m, Nα+1+n, |c, Nβ+1+m, Nα+1+n},
Rijperiod(j, k)Rklperiod(l, m)Rmn,
ddt |ψ(t)=-i Heff|ψ(t),
Heff
=-(Δα+Δβ)-iGa/2α/20α/2-Δβ-iGb/2β/20β/2-iGc/2,
period(c, a)Rabperiod(b, b)Rbaperiod(a, a)Rab,
period(a, b)ΔNα=+1;
ΔNβ=0one-photongain,
period(b, a)ΔNα=-1;
ΔNβ=0one-photonloss,
period(a, c)ΔNα=+1;
ΔNβ=+1two-photongain,
period(c, a)ΔNα=-1;
ΔNβ=-1two-photonloss.
ΔNα=P(a, b)+P(a, c)-P(b, a)-P(c, a),
P(i, j)=P(i)P(j/i).
P(j/i)=Gj0|cij(τ)|2dτ,
cij=cji.
ΔNα=ΔNα1p+ΔNα2p,
ΔNα1p=[P(a)Gb-P(b)Ga]0|cab(τ)|2dτ,
ΔNα2p=[P(a)Gc-P(c)Ga]0|cac(τ)|2dτ,
x˙aa=-(Ga/2)xaa-(Δα+Δβ)yaa+(α/2)yab,
y˙aa=-(Ga/2)yaa+(Δα+Δβ)xaa-(α/2)xab,
x˙bb=-(Gb/2)xbb-Δβybb+(α/2)yab+(β/2)ybc,
y˙bb=-(Gb/2)ybb+Δβxbb-(α/2)xab-(β/2)xbc,
x˙cc=-(Gc/2)xcc+(β/2)ybc,
y˙cc=-(Gc/2)ycc-(β/2)xbc,
x˙ab=-(Gb/2)xab-Δβyab+(α/2)yaa+(β/2)yac,
y˙ab=-(Gb/2)yab+Δβxab-(α/2)xaa-(β/2)xac,
x˙ac=-(Gc/2)xac+(β/2)yab,
y˙ac=-(Gc/2)yac-(β/2)xab,
x˙bc=-(Gc/2)xbc+(β/2)ybb,
y˙bc=-(Gc/2)ybc-(β/2)xbb,
P(i)=jP(j)Q(i/j),
Q(i/j)=RjiGj.
P(a)=(Rab+Rac)[Rbc2Rca+2Rba2(Rca+Rcb)+RbaRbc(3Rca+Rcb)](2Rba+Rbc)D1,
P(b)=(2Rba2+3RabRbc+Rbc2)[RacRcb+Rab(Rca+Rcb)](2Rba+Rbc)D1,
P(c)=(Rca+Rcb)[RabRbc+Rac(Rba+Rbc)]D1,
Q(a/a)=0,
Q(b/a)=Rab/(Rab+Rac),
Q(c/a)=Rac/(Rab+Rac).
Q(a/b)=Q(a/c)=Rba+RcaRba+Rca+Rcb+Rbc,
Q(b/b)=Q(b/c)=RcbRba+Rca+Rcb+Rbc,
Q(c/b)=Q(c/c)=RbcRba+Rca+Rcb+Rbc.
P(a)=(Rab+Rac)(Rba+Rca)D2,
P(b)=RacRcb+Rab(Rba+Rca+Rcb)D2,
P(c)=RabRbc+Rac(Rba+Rca+Rbc)D2,

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