Abstract

We study the behavior of a two-level atom that is driven by a bichromatic field consisting of a strong resonant component and a weaker tunable component. In addition to the splitting of the energy levels (the multiphoton AC Stark effect), we find that the weaker component also shifts the subharmonic resonances, an effect we attribute to a dynamic Stark shift. When the weaker component is tuned to a shifted resonance, no fluorescence occurs at either the frequency of the strong component or the three-photon mixing frequency. Results are obtained with numerical techniques and explained in terms of the dressed-atom model of the system.

© 1998 Optical Society of America

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  1. Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, and T. W. Mossberg, Phys. Rev. A 41, 6574 (1990); H. S. Freedhoff and Z. Chen, Phys. Rev. A 41, 6013 (1990); S. P. Tewari and M. K. Kumari, Phys. Rev. A PLRAAN 41, 5273 (1990); G. S. Agarwal, Y. Zhu, D. J. Gauthier, and T. W. Mossberg, J. Opt. Soc. Am. B JOBPDE 8, 1163 (1991); G. S. Agarwal and Y. Zhu, Phys. Rev. A PLRAAN 46, 479 (1992); Z. Ficek and H. S. Freedhoff, Phys. Rev. A PLRAAN 48, 3092 (1993); M. Elk and P. Lambropoulos, Phys. Rev. A PLRAAN 50, 1490 (1994); G. S. Agarwal, W. Lange, and H. Walther, Phys. Rev. A PLRAAN 51, 721 (1995); H. S. Freedhoff and Z. Ficek, Phys. Rev. A PLRAAN 55, 1234 (1997); C. C. Yu, J. R. Bochinski, T. M. V. Kordich, T. W. Mossberg, and Z. Ficek, Phys. Rev. A PLRAAN 56, R4381 (1997).
    [CrossRef] [PubMed]
  2. M. F. Van Leeuwen, S. Papademetriou, and C. R. Stroud, Jr., Phys. Rev. A 53, 990 (1996); S. Papademetriou, M. F. Van Leeuwen, and C. R. Stroud, Jr., Phys. Rev. A 53, 997 (1996).
    [CrossRef] [PubMed]
  3. G. I. Toptygina and E. E. Fradkin, Zh. Eksp. Teor. Fiz. 82, 429 (1982) [Sov. Phys. JETP 55, 246 (1982)]; R. Guccione-Gush and H. P. Gush, Phys. Rev. A 10, 1474 (1974); H. Friedmann and A. D. Wilson-Gordon, Phys. Rev. A PLRAAN 36, 1333 (1987); G. S. Agarwal and N. Nayak, Phys. Rev. A PLRAAN 33, 391 (1986); J. H. Eberly and V. D. Popov, Phys. Rev. A PLRAAN 37, 2012 (1988); Y. Zhu, Q. Wu, S. Morin, and T. W. Mossberg, Phys. Rev. Lett. PRLTAO 65, 1200 (1990); S. Papademetriou, S. M. Chakmakjian, and C. R. Stroud, Jr., J. Opt. Soc. Am. B JOBPDE 9, 1182 (1992); N. B. Manson, C. Wei, and J. P. D. Martin, Phys. Rev. Lett. PRLTAO 76, 3943 (1996); B. Lounis, F. Jelezko, and M. Orrit, Phys. Rev. Lett. PRLTAO 78, 3673 (1997).
    [CrossRef] [PubMed]
  4. T. G. Rudolph, H. S. Freedhoff, and Z. Ficek, Opt. Commun. 147, 78 (1998); T. G. Rudolph, H. S. Freedhoff, and Z. Ficek, Phys. Rev. A (to be published).
    [CrossRef]
  5. T. W. Mossberg and M. Lewenstein, Phys. Rev. A 39, 163 (1989); Q. Wu, D. J. Gauthier, and T. W. Mossberg, Phys. Rev. A 49, R1519 (1994); S. F. Chien, M. R. B. Wahiddin, and Z. Ficek, Phys. Rev. A PLRAAN 57, 1295 (1998).
    [CrossRef] [PubMed]
  6. G. S. Agarwal and W. Harshawardhan, Phys. Rev. A 50, R4465 (1994).
    [CrossRef]
  7. Z. Ficek and H. S. Freedhoff, Phys. Rev. A 53, 4275 (1996).
    [CrossRef] [PubMed]
  8. E. Arimondo, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 25, p. 257; S. Y. Zhu, R. C. F. Chan, and C. P. Lee, Phys. Rev. A 52, 710 (1995); S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989); P. Zhou and S. Swain, Phys. Rev. A PLRAAN 56, 3011 (1997).
    [CrossRef] [PubMed]
  9. S. Y. Zhu and M. O. Scully, Phys. Rev. Lett. 76, 388 (1996); H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, Phys. Rev. A 55, 4454 (1997); G. S. Agarwal, Phys. Rev. A PLRAAN 55, 2457 (1997).
    [CrossRef] [PubMed]
  10. J. M. Courty and S. Reynaud, Europhys. Lett. 10, 237 (1989); C. Cabrillo, W. S. Smyth, S. Swain, and P. Zhou, Opt. Commun. 114, 344 (1995).
    [CrossRef]
  11. M. Lewenstein, T. W. Mossberg, and R. J. Glauber, Phys. Rev. Lett. 59, 775 (1987); S. Haroche and D. Kleppner, Phys. Today 42(1), 24 (1989).
    [CrossRef] [PubMed]
  12. M. Lax, Phys. Rev. 172, 350 (1968).
    [CrossRef]
  13. H. Risken, The Fokker–Planck Equation (Springer-Verlag, Berlin, 1984), Chap. 9.
  14. In the case of n=1, shown in Fig. 1(a), there is in fact a small bump at the central frequency, which indicates that even for n=1 the central component appears in the spectrum. The line is more visible for higher values of α.7
  15. C. Cohen-Tannoudji and S. Reynaud, J. Phys. B 10, 345 (1997); C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wiley, New York, 1992).
    [CrossRef]
  16. L. Ballentine, Quantum Mechanics (Prentice-Hall, Englewood Cliffs, N.J., 1990).
  17. W. M. Ruyten, J. Opt. Soc. Am. B 6, 1796 (1989); Phys. Rev. A 40, 1447 (1989); J. Opt. Soc. Am. B JOBPDE 9, 1892 (1992); Phys. Rev. A PLRAAN 46, 4077 (1992).
    [CrossRef]
  18. F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).
    [CrossRef]
  19. Inclusion of the counter-rotating terms into the interaction V2 couples doublets from different manifolds and yields additional shifts of order Ω220.

1996 (1)

Z. Ficek and H. S. Freedhoff, Phys. Rev. A 53, 4275 (1996).
[CrossRef] [PubMed]

1994 (1)

G. S. Agarwal and W. Harshawardhan, Phys. Rev. A 50, R4465 (1994).
[CrossRef]

1968 (1)

M. Lax, Phys. Rev. 172, 350 (1968).
[CrossRef]

1940 (1)

F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and W. Harshawardhan, Phys. Rev. A 50, R4465 (1994).
[CrossRef]

Bloch, F.

F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).
[CrossRef]

Ficek, Z.

Z. Ficek and H. S. Freedhoff, Phys. Rev. A 53, 4275 (1996).
[CrossRef] [PubMed]

Freedhoff, H. S.

Z. Ficek and H. S. Freedhoff, Phys. Rev. A 53, 4275 (1996).
[CrossRef] [PubMed]

Harshawardhan, W.

G. S. Agarwal and W. Harshawardhan, Phys. Rev. A 50, R4465 (1994).
[CrossRef]

Lax, M.

M. Lax, Phys. Rev. 172, 350 (1968).
[CrossRef]

Siegert, A.

F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).
[CrossRef]

Phys. Rev. (2)

M. Lax, Phys. Rev. 172, 350 (1968).
[CrossRef]

F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).
[CrossRef]

Phys. Rev. A (2)

G. S. Agarwal and W. Harshawardhan, Phys. Rev. A 50, R4465 (1994).
[CrossRef]

Z. Ficek and H. S. Freedhoff, Phys. Rev. A 53, 4275 (1996).
[CrossRef] [PubMed]

Other (15)

E. Arimondo, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 25, p. 257; S. Y. Zhu, R. C. F. Chan, and C. P. Lee, Phys. Rev. A 52, 710 (1995); S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989); P. Zhou and S. Swain, Phys. Rev. A PLRAAN 56, 3011 (1997).
[CrossRef] [PubMed]

S. Y. Zhu and M. O. Scully, Phys. Rev. Lett. 76, 388 (1996); H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, Phys. Rev. A 55, 4454 (1997); G. S. Agarwal, Phys. Rev. A PLRAAN 55, 2457 (1997).
[CrossRef] [PubMed]

J. M. Courty and S. Reynaud, Europhys. Lett. 10, 237 (1989); C. Cabrillo, W. S. Smyth, S. Swain, and P. Zhou, Opt. Commun. 114, 344 (1995).
[CrossRef]

M. Lewenstein, T. W. Mossberg, and R. J. Glauber, Phys. Rev. Lett. 59, 775 (1987); S. Haroche and D. Kleppner, Phys. Today 42(1), 24 (1989).
[CrossRef] [PubMed]

Y. Zhu, Q. Wu, A. Lezama, D. J. Gauthier, and T. W. Mossberg, Phys. Rev. A 41, 6574 (1990); H. S. Freedhoff and Z. Chen, Phys. Rev. A 41, 6013 (1990); S. P. Tewari and M. K. Kumari, Phys. Rev. A PLRAAN 41, 5273 (1990); G. S. Agarwal, Y. Zhu, D. J. Gauthier, and T. W. Mossberg, J. Opt. Soc. Am. B JOBPDE 8, 1163 (1991); G. S. Agarwal and Y. Zhu, Phys. Rev. A PLRAAN 46, 479 (1992); Z. Ficek and H. S. Freedhoff, Phys. Rev. A PLRAAN 48, 3092 (1993); M. Elk and P. Lambropoulos, Phys. Rev. A PLRAAN 50, 1490 (1994); G. S. Agarwal, W. Lange, and H. Walther, Phys. Rev. A PLRAAN 51, 721 (1995); H. S. Freedhoff and Z. Ficek, Phys. Rev. A PLRAAN 55, 1234 (1997); C. C. Yu, J. R. Bochinski, T. M. V. Kordich, T. W. Mossberg, and Z. Ficek, Phys. Rev. A PLRAAN 56, R4381 (1997).
[CrossRef] [PubMed]

M. F. Van Leeuwen, S. Papademetriou, and C. R. Stroud, Jr., Phys. Rev. A 53, 990 (1996); S. Papademetriou, M. F. Van Leeuwen, and C. R. Stroud, Jr., Phys. Rev. A 53, 997 (1996).
[CrossRef] [PubMed]

G. I. Toptygina and E. E. Fradkin, Zh. Eksp. Teor. Fiz. 82, 429 (1982) [Sov. Phys. JETP 55, 246 (1982)]; R. Guccione-Gush and H. P. Gush, Phys. Rev. A 10, 1474 (1974); H. Friedmann and A. D. Wilson-Gordon, Phys. Rev. A PLRAAN 36, 1333 (1987); G. S. Agarwal and N. Nayak, Phys. Rev. A PLRAAN 33, 391 (1986); J. H. Eberly and V. D. Popov, Phys. Rev. A PLRAAN 37, 2012 (1988); Y. Zhu, Q. Wu, S. Morin, and T. W. Mossberg, Phys. Rev. Lett. PRLTAO 65, 1200 (1990); S. Papademetriou, S. M. Chakmakjian, and C. R. Stroud, Jr., J. Opt. Soc. Am. B JOBPDE 9, 1182 (1992); N. B. Manson, C. Wei, and J. P. D. Martin, Phys. Rev. Lett. PRLTAO 76, 3943 (1996); B. Lounis, F. Jelezko, and M. Orrit, Phys. Rev. Lett. PRLTAO 78, 3673 (1997).
[CrossRef] [PubMed]

T. G. Rudolph, H. S. Freedhoff, and Z. Ficek, Opt. Commun. 147, 78 (1998); T. G. Rudolph, H. S. Freedhoff, and Z. Ficek, Phys. Rev. A (to be published).
[CrossRef]

T. W. Mossberg and M. Lewenstein, Phys. Rev. A 39, 163 (1989); Q. Wu, D. J. Gauthier, and T. W. Mossberg, Phys. Rev. A 49, R1519 (1994); S. F. Chien, M. R. B. Wahiddin, and Z. Ficek, Phys. Rev. A PLRAAN 57, 1295 (1998).
[CrossRef] [PubMed]

Inclusion of the counter-rotating terms into the interaction V2 couples doublets from different manifolds and yields additional shifts of order Ω220.

H. Risken, The Fokker–Planck Equation (Springer-Verlag, Berlin, 1984), Chap. 9.

In the case of n=1, shown in Fig. 1(a), there is in fact a small bump at the central frequency, which indicates that even for n=1 the central component appears in the spectrum. The line is more visible for higher values of α.7

C. Cohen-Tannoudji and S. Reynaud, J. Phys. B 10, 345 (1997); C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wiley, New York, 1992).
[CrossRef]

L. Ballentine, Quantum Mechanics (Prentice-Hall, Englewood Cliffs, N.J., 1990).

W. M. Ruyten, J. Opt. Soc. Am. B 6, 1796 (1989); Phys. Rev. A 40, 1447 (1989); J. Opt. Soc. Am. B JOBPDE 9, 1892 (1992); Phys. Rev. A PLRAAN 46, 4077 (1992).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

The steady-state fluorescence spectrum for 2Ω=40Γ, α=0.4, and different n=2Ω/δ: (a) n=1, (b) n=2, and (c) n=3. Here ω2 is tuned exactly to the subharmonic resonance n.

Fig. 2
Fig. 2

The amplitude of the fluorescence spectrum at ω=ω0 as a function of δ/Γ for 2Ω=40Γ and α=0.4.

Fig. 3
Fig. 3

The fluorescence spectrum for 2Ω=40Γ, α=0.4, and different δ: (a) δ=40.4Γ, (b) δ=21.06Γ, and (c) δ=13.93Γ.

Fig. 4
Fig. 4

Energy level diagrams of (a) the combined (uncoupled) system of singly dressed atom plus weaker field and (b) the doubly dressed system.

Fig. 5
Fig. 5

The intradoublet splittings 2λn as a function of Δn for 2Ω=40Γ, α=0.4, and n=2 (solid curve), n=3 (dashed curve).

Equations (32)

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S(ω)=Re 0dτ exp(iωτ) limtS+(t)S-(t+τ),
ddtS˜-(t)=-12 ΓS˜-(t)+2Ω[1+α exp(-iδt)]Sz(t),
ddt S˜+(t)=-12 ΓS˜+(t)+2Ω[1+α exp(iδt)]Sz(t),
ddt Sz(t)=-12 Γ-ΓSz(t)-Ω[1+α exp(iδt)]S˜-(t)-Ω[1+α exp(-iδt)]S˜+(t),
S˜±(t)=S±exp(iω0t),
ω0, ω2Ω>Ω2, δ,
|N±=12 (|1, N±|2, N-1),
|(N+n-m)+, M-n+m|amn,
|(N-m)-, M+m|bmn,
Eamn=Nω¯0-Ω+m2Ωn+Δn-nΔn,
Ebmn=Nω¯0-Ω+m2Ωn+Δn,
V2=g2(a2S-+S+a2),
V2|amn=(1/2)αΩ(|am+1n+|am-1n+|bm+1-nn
-|bm-1-nn),
V2|bmn=(1/2)αΩ(-|bm+1n-|bm-1n-|am+1+nn
+|am-1+nn),
Raa1=-Rbb1=α28(2+Δ1/Ω) Ω,forn=1,
Raan=-Rbbn=n22(n2-1) α2(2+nΔn/Ω) Ω,
forn>1,
Rn=smi=a,b V2|isnisn|V2Eamn-Eisn.
Δnmin=2n Raan
Δ1min=(1/8) α2Ω,forn=1,
Δnmin=nα22(n2-1) Ω,forn>1.
T3=r,smi,j=a,b V2|jr3jr3|V2|is3is3|V2(Eam3-Eis3)(Eam3-Ejr3).
λ1=±12 ΩΔ1Ω-α282+α21/2,
λ2=±ΩΔ2Ω-α232+α2221/2,
λ3=±32 ΩΔ3Ω-3α2162+27α36421/2.
2λ1min=αΩ,
2λ2min=α2Ω,
2λ3min=8164 α3Ω.
|N¯, m±min=12 (|amn±|bmn).
Γpi,qj=Γ  N¯, ip|S+|N¯-1, qj2,

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