Abstract

We develop a new theoretical treatment that can explain, for the first time to our knowledge, unexpectedly strong synchronized-quantum-beat-echo (SQBE) signals that we previously observed in the ground state of sodium atoms. The SQBE is obtained by successive application of two light-pulse trains tuned to an optical transition under the condition that the pulse repetition frequency or that of one of its harmonics is equal to the separation of a ground-state sublevel pair. The theory is developed for weak optical pumping for which the duration of the light-pulse trains is much longer than the spontaneous decay time of the excited state. We show that the echo is formed effectively as a result of nonuniform depopulation from the sublevel pair to the other levels because of frequency-selective optical pumping during the rapid spontaneous decay of the excited state. Details of the experiment with sodium atoms are described also.

© 1992 Optical Society of America

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  1. S. Haroche, in High-Resolution Laser Spectroscopy, K. Shimoda, ed. (Springer-Verlag, Heidelberg, 1976), p. 253.
    [CrossRef]
  2. H. Harde, H. Burggraf, J. Mlynek, W. Lange, Opt. Lett. 6, 290 (1981).
    [CrossRef] [PubMed]
  3. T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, Phys. Rev. B 28, 1983 (2869).
    [CrossRef]
  4. Y. Fukuda, K. Yamada, T. Hashi, Opt. Commun. 44, 297 (1983).
    [CrossRef]
  5. Y. Fukuda, J. Hayashi, K. Kondo, T. Hashi, Opt. Commun. 38, 357 (1981); J. Mlynek, W. Lange, H. Harde, H. Burgraff, Phys. Rev. A 24, 1099 (1981); K. Kondo, Y. Fukuda, M. Tanigawa, T. Hashi, Opt. Lett. 8, 301 (1983); K. Kondo, J. Phys. Soc. Jpn. 52, 2340 (1983).
    [CrossRef]
  6. H. Harde, H. Burggraf, Opt. Commun. 40, 441 (1982).
    [CrossRef]
  7. M. Tanigawa, Y. Fukuda, T. Kohmoto, K. Sakuno, T. Hashi, Opt. Lett. 8, 620 (1983).
    [CrossRef] [PubMed]
  8. T. Mishina, M. Tanigawa, Y. Fukuda, T. Hashi, Opt. Commun. 62, 166 (1987).
    [CrossRef]
  9. M. Rosatzin, D. Suter, J. Mlynek, Phys. Rev. A 42, 1839 (1990).
    [CrossRef] [PubMed]
  10. R. Kachru, T. W. Mossberg, S. R. Hartmann, J. Phys. B 13, 1363 (1980).
    [CrossRef]
  11. J. C. Gay, W. B. Schneider, Z. Phys. A 278, 211 (1976).
    [CrossRef]
  12. W. Happer, Rev. Mod. Phys. 44, 169 (1972).
    [CrossRef]

1990 (1)

M. Rosatzin, D. Suter, J. Mlynek, Phys. Rev. A 42, 1839 (1990).
[CrossRef] [PubMed]

1987 (1)

T. Mishina, M. Tanigawa, Y. Fukuda, T. Hashi, Opt. Commun. 62, 166 (1987).
[CrossRef]

1983 (3)

M. Tanigawa, Y. Fukuda, T. Kohmoto, K. Sakuno, T. Hashi, Opt. Lett. 8, 620 (1983).
[CrossRef] [PubMed]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, Phys. Rev. B 28, 1983 (2869).
[CrossRef]

Y. Fukuda, K. Yamada, T. Hashi, Opt. Commun. 44, 297 (1983).
[CrossRef]

1982 (1)

H. Harde, H. Burggraf, Opt. Commun. 40, 441 (1982).
[CrossRef]

1981 (2)

H. Harde, H. Burggraf, J. Mlynek, W. Lange, Opt. Lett. 6, 290 (1981).
[CrossRef] [PubMed]

Y. Fukuda, J. Hayashi, K. Kondo, T. Hashi, Opt. Commun. 38, 357 (1981); J. Mlynek, W. Lange, H. Harde, H. Burgraff, Phys. Rev. A 24, 1099 (1981); K. Kondo, Y. Fukuda, M. Tanigawa, T. Hashi, Opt. Lett. 8, 301 (1983); K. Kondo, J. Phys. Soc. Jpn. 52, 2340 (1983).
[CrossRef]

1980 (1)

R. Kachru, T. W. Mossberg, S. R. Hartmann, J. Phys. B 13, 1363 (1980).
[CrossRef]

1976 (1)

J. C. Gay, W. B. Schneider, Z. Phys. A 278, 211 (1976).
[CrossRef]

1972 (1)

W. Happer, Rev. Mod. Phys. 44, 169 (1972).
[CrossRef]

Burggraf, H.

Fukuda, Y.

T. Mishina, M. Tanigawa, Y. Fukuda, T. Hashi, Opt. Commun. 62, 166 (1987).
[CrossRef]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, Phys. Rev. B 28, 1983 (2869).
[CrossRef]

Y. Fukuda, K. Yamada, T. Hashi, Opt. Commun. 44, 297 (1983).
[CrossRef]

M. Tanigawa, Y. Fukuda, T. Kohmoto, K. Sakuno, T. Hashi, Opt. Lett. 8, 620 (1983).
[CrossRef] [PubMed]

Y. Fukuda, J. Hayashi, K. Kondo, T. Hashi, Opt. Commun. 38, 357 (1981); J. Mlynek, W. Lange, H. Harde, H. Burgraff, Phys. Rev. A 24, 1099 (1981); K. Kondo, Y. Fukuda, M. Tanigawa, T. Hashi, Opt. Lett. 8, 301 (1983); K. Kondo, J. Phys. Soc. Jpn. 52, 2340 (1983).
[CrossRef]

Gay, J. C.

J. C. Gay, W. B. Schneider, Z. Phys. A 278, 211 (1976).
[CrossRef]

Happer, W.

W. Happer, Rev. Mod. Phys. 44, 169 (1972).
[CrossRef]

Harde, H.

Haroche, S.

S. Haroche, in High-Resolution Laser Spectroscopy, K. Shimoda, ed. (Springer-Verlag, Heidelberg, 1976), p. 253.
[CrossRef]

Hartmann, S. R.

R. Kachru, T. W. Mossberg, S. R. Hartmann, J. Phys. B 13, 1363 (1980).
[CrossRef]

Hashi, T.

T. Mishina, M. Tanigawa, Y. Fukuda, T. Hashi, Opt. Commun. 62, 166 (1987).
[CrossRef]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, Phys. Rev. B 28, 1983 (2869).
[CrossRef]

Y. Fukuda, K. Yamada, T. Hashi, Opt. Commun. 44, 297 (1983).
[CrossRef]

M. Tanigawa, Y. Fukuda, T. Kohmoto, K. Sakuno, T. Hashi, Opt. Lett. 8, 620 (1983).
[CrossRef] [PubMed]

Y. Fukuda, J. Hayashi, K. Kondo, T. Hashi, Opt. Commun. 38, 357 (1981); J. Mlynek, W. Lange, H. Harde, H. Burgraff, Phys. Rev. A 24, 1099 (1981); K. Kondo, Y. Fukuda, M. Tanigawa, T. Hashi, Opt. Lett. 8, 301 (1983); K. Kondo, J. Phys. Soc. Jpn. 52, 2340 (1983).
[CrossRef]

Hayashi, J.

Y. Fukuda, J. Hayashi, K. Kondo, T. Hashi, Opt. Commun. 38, 357 (1981); J. Mlynek, W. Lange, H. Harde, H. Burgraff, Phys. Rev. A 24, 1099 (1981); K. Kondo, Y. Fukuda, M. Tanigawa, T. Hashi, Opt. Lett. 8, 301 (1983); K. Kondo, J. Phys. Soc. Jpn. 52, 2340 (1983).
[CrossRef]

Kachru, R.

R. Kachru, T. W. Mossberg, S. R. Hartmann, J. Phys. B 13, 1363 (1980).
[CrossRef]

Kohmoto, T.

M. Tanigawa, Y. Fukuda, T. Kohmoto, K. Sakuno, T. Hashi, Opt. Lett. 8, 620 (1983).
[CrossRef] [PubMed]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, Phys. Rev. B 28, 1983 (2869).
[CrossRef]

Kondo, K.

Y. Fukuda, J. Hayashi, K. Kondo, T. Hashi, Opt. Commun. 38, 357 (1981); J. Mlynek, W. Lange, H. Harde, H. Burgraff, Phys. Rev. A 24, 1099 (1981); K. Kondo, Y. Fukuda, M. Tanigawa, T. Hashi, Opt. Lett. 8, 301 (1983); K. Kondo, J. Phys. Soc. Jpn. 52, 2340 (1983).
[CrossRef]

Lange, W.

Mishina, T.

T. Mishina, M. Tanigawa, Y. Fukuda, T. Hashi, Opt. Commun. 62, 166 (1987).
[CrossRef]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, Phys. Rev. B 28, 1983 (2869).
[CrossRef]

Mlynek, J.

Mossberg, T. W.

R. Kachru, T. W. Mossberg, S. R. Hartmann, J. Phys. B 13, 1363 (1980).
[CrossRef]

Rosatzin, M.

M. Rosatzin, D. Suter, J. Mlynek, Phys. Rev. A 42, 1839 (1990).
[CrossRef] [PubMed]

Sakuno, K.

Schneider, W. B.

J. C. Gay, W. B. Schneider, Z. Phys. A 278, 211 (1976).
[CrossRef]

Suter, D.

M. Rosatzin, D. Suter, J. Mlynek, Phys. Rev. A 42, 1839 (1990).
[CrossRef] [PubMed]

Tanigawa, M.

T. Mishina, M. Tanigawa, Y. Fukuda, T. Hashi, Opt. Commun. 62, 166 (1987).
[CrossRef]

M. Tanigawa, Y. Fukuda, T. Kohmoto, K. Sakuno, T. Hashi, Opt. Lett. 8, 620 (1983).
[CrossRef] [PubMed]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, Phys. Rev. B 28, 1983 (2869).
[CrossRef]

Yamada, K.

Y. Fukuda, K. Yamada, T. Hashi, Opt. Commun. 44, 297 (1983).
[CrossRef]

J. Phys. B (1)

R. Kachru, T. W. Mossberg, S. R. Hartmann, J. Phys. B 13, 1363 (1980).
[CrossRef]

Opt. Commun. (4)

Y. Fukuda, K. Yamada, T. Hashi, Opt. Commun. 44, 297 (1983).
[CrossRef]

Y. Fukuda, J. Hayashi, K. Kondo, T. Hashi, Opt. Commun. 38, 357 (1981); J. Mlynek, W. Lange, H. Harde, H. Burgraff, Phys. Rev. A 24, 1099 (1981); K. Kondo, Y. Fukuda, M. Tanigawa, T. Hashi, Opt. Lett. 8, 301 (1983); K. Kondo, J. Phys. Soc. Jpn. 52, 2340 (1983).
[CrossRef]

H. Harde, H. Burggraf, Opt. Commun. 40, 441 (1982).
[CrossRef]

T. Mishina, M. Tanigawa, Y. Fukuda, T. Hashi, Opt. Commun. 62, 166 (1987).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

M. Rosatzin, D. Suter, J. Mlynek, Phys. Rev. A 42, 1839 (1990).
[CrossRef] [PubMed]

Phys. Rev. B (1)

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, Phys. Rev. B 28, 1983 (2869).
[CrossRef]

Rev. Mod. Phys. (1)

W. Happer, Rev. Mod. Phys. 44, 169 (1972).
[CrossRef]

Z. Phys. A (1)

J. C. Gay, W. B. Schneider, Z. Phys. A 278, 211 (1976).
[CrossRef]

Other (1)

S. Haroche, in High-Resolution Laser Spectroscopy, K. Shimoda, ed. (Springer-Verlag, Heidelberg, 1976), p. 253.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Experimental setup and optical-excitation sequence; fid, free-induction decay.

Fig. 2
Fig. 2

Energy-level diagram of the ground state of the sodium atom. Each vertical line indicates the sublevel pair in which the sublevel coherence may be created. The relevant values of κ for the sublevel pairs calculated under the low-magnetic-field approximation (see Section 3) are shown.

Fig. 3
Fig. 3

SQBE signals for a pair of ground-state sublevels (F, MF) = (1,1), (2,2). The static magnetic field applied is 13.2 Oe, and the splitting frequency is 1.8 GHz. The harmonic order of the resonance is 18. The intensity of the pump light beam is ~2 mW, and the durations of the first and the second pulse trains are approximately 35 and 4.5 μs, respectively. The time intervals T between the first and the second light-pulse trains are 35, 47, and 63 μs. The SQBE signals appear at time T after the second light-pulse trains and indicate the homogeneous relaxation of the sublevel coherence.

Fig. 4
Fig. 4

Calculated refocusing efficiencies of SQBE in the ground state of sodium versus a normalized duration τ of the second light-pulse train. The efficiencies are 1–10% at optimum durations, which depend on the values of κ for the sublevel pairs.

Fig. 5
Fig. 5

(a) Simple model for ground-state SQBE’s. The trap state 4 accepts the spontaneous decay of the excited state but does not release its population. (b) Calculated echo-refocusing efficiencies for the simple model. τ is the normalized duration of the second light-pulse train. A parameter η denotes the branching ratio of the spontaneous decay into the trap state.

Fig. 6
Fig. 6

(a) Level scheme for a modified model with another ground-state sublevel. (b) Calculated refocusing efficiencies.

Fig. 7
Fig. 7

Various optical pumping paths; from the left, the direct pumping cycle, the auxiliary pumping cycle, and the terminative pumping paths.

Fig. 8
Fig. 8

Calculated refocusing efficiencies for various values of χ. The excited state decays entirely into the trap state.

Fig. 9
Fig. 9

Calculated refocusing efficiencies for various values of χ. There is no trap state, and the ground state has one sublevel in addition to the resonant sublevel pair.

Equations (24)

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d ρ d t = - ( W ρ + ρ W ) + α G Tr ( 2 W ρ )
W j k ( t ) = exp ( i ω j k t ) E ( t ) ¯ · n μ j n μ n k · 0 d u E ( t - u ) exp ( - γ u ) ,
Φ ( t ) = 0 d u E ( t - u ) exp ( - γ u ) E ( t ) ¯ ,
Φ ( ω ) = - d Ω ( 2 π ) - 1 / 2 γ - i Ω E ( Ω ) E ( Ω - ω ) ¯ ,
f ( t ) = ( 2 π ) - 1 / 2 - d ω exp ( - i ω t ) f ( ω ) .
Φ ( t ) = M Q M exp ( - i M ω 0 t ) ^ ^ ¯ ,
W j k ( t ) = M exp [ i ( ω j k - M ω 0 ) t ] Q M Λ j k ,
Λ j k = ^ ¯ · n μ j n μ n k · ^ .
L 12 exp ( - i ω 21 t ) ρ 21 ,
s ( t ) = exp ( - i ω 21 t ) ρ 21 ,
ρ j k = a b C j k a b ρ a b .
ρ 21 = C 21 12 ρ 12 + C 21 21 ρ 21 + a C 21 a a ρ a a .
c j k a b = exp ( - i ω j k T + i ω a b T ) C j k a b ,
s ( t ) = c 21 12 s [ T - ( t - T ) ] ¯ + c 21 21 s [ t - ( T - T ) ] + exp [ - i ω 21 ( t - T ) ] a c 21 a a ρ a a .
d ρ 21 d τ = - κ 21 ρ 11 - ρ 21 , d ρ 12 d τ = - κ 12 ρ 11 - ρ 12 , d ρ 11 d τ = - G - 2 G [ ρ 11 - ρ 33 + ½ ( κ 21 ρ 12 + κ 12 ρ 21 ) ] , d ρ 33 d τ = 2 G [ ρ 11 - ρ 33 + ½ ( κ 21 ρ 12 + κ 12 ρ 21 ) ] ,
ρ 11 = ρ 22 ,             ρ 1 j = ρ 2 j = ρ i 1 = ρ i 2 = ρ i j = 0             ( i , j = 3 , 4 , 8 ) .
d ρ 12 d t = - W 12 ( ρ 11 + ρ 22 ) - ( W 11 + W 22 ) ρ 12 ,
d d t ( ρ 11 + ρ 22 ) = - ( 2 - 4 α G ) ( W 21 ρ 12 + W 12 ρ 21 ) + remainder .
a = - sin Θ 1 + cos Θ 2 ,
tan Θ = ^ · μ 23 / ^ · μ 13 .
b = cos Θ 1 + sin Θ 2 ,
Q M / Q 0 = 1 ,
Λ 21 / ( Λ 11 Λ 22 ) 1 / 2 = 1.
χ = Q M / Q 0 Λ 21 / ( Λ 11 Λ 22 ) 1 / 2 ,

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