Abstract

We report detailed studies of freely evolving Zeeman coherence (Zeeman quantum beats) observed after optically pumping a gas of Na atoms and suddenly turning off the light field. This Zeeman coherence corresponds to a macroscopic magnetization that precesses in an external magnetic field; it is detected with a cw optical probe beam by using polarization-selective detection of the transmitted light. The amplitude and the phase of the signal show a pronounced variation with laser intensity, laser detuning, and the strength of the magnetic field. This dependence was measured on the 3s2S1/2 ground state of sodium, using the D1 line for optical excitation and detection. The experimental data are compared with theoretical predictions based on a Bloch-type equation of motion for an optically driven spin system in a J = 1/2 ground state.

© 1990 Optical Society of America

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  1. See, e.g., S. Haroche, “Quantum beats and time-resolved fluorescence spectroscopy,” in High Resolution Laser Spectroscopy, K. Shimoda, ed. (Springer-Verlag, 1976), pp. 253–313, and references therein.
    [CrossRef]
  2. M. P. Silverman, S. Haroche, M. Gross, “General theory of laser-induced quantum beats. I. Saturation effects of single laser excitation,” Phys. Rev. A 18, 1507–1516 (1978).
    [CrossRef]
  3. J. Mlynek, W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
    [CrossRef]
  4. R. M. Shelby, A.C. Tropper, R.T. Harley, R. M. Macfarlane, “Measurement of the hyperfine structure of Pr3+:YAG by quantum-beat free induction decay, hole burning, and optically detected nuclear quadrupole resonance,” Opt. Lett. 8, 304–306 (1983).
    [CrossRef] [PubMed]
  5. T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, “Quantum beat free induction decay in Tm2+:SrF2:Fourier transform ESR spectroscopy by optical means,” Phys. Rev. B 28, 2869–2872 (1983).
    [CrossRef]
  6. See, e.g., M.S. Feld, M.M. Burns, T.U. Kühl, P.G. Pappas, “Laser-saturation spectroscopy with optical pumpings,” Opt. Lett. 5, 79–81 (1980).
    [CrossRef]
  7. E. L. Hahn, “Nuclear Induction due to free Larmor precession,” Phys. Rev. 77, 297–298 (1950).
    [CrossRef]
  8. For a preliminary report see, e.g., S. Burschka, J. Mlynek, “Optically induced spin transients in the ground state of atomic sodium,” Opt. Commun. 66, 59–64 (1988).
    [CrossRef]
  9. See, e.g., A. Kastler, “Optical methods for studying Hertzian resonances,” Science 158, 214–221 (1967), and references therein.
    [CrossRef] [PubMed]
  10. See, e.g., T. Mishina, Y. Fukuda, T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
    [CrossRef]
  11. F. Mitschke, R. Deserno, W. Lange, J. Mlynek, “Magnetically induced optical self-pulsing in a nonlinear resonator,” Phys. Rev. A 33, 3219–3231 (1986).
    [CrossRef] [PubMed]
  12. D. Suter, M. Rosatzin, J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
    [CrossRef] [PubMed]
  13. C. Cohen-Tannoudji, J. Dupont-Roc, “Experimental study of Zeeman light shifts in weak magnetic fields,” Phys. Rev. A 5, 968–984 (1972).
    [CrossRef]
  14. H. Lundberg, S. Svanberg, “Two quantum beat phenomena observed for magnetically tuned atomic sublevels,” Opt. Commun. 27, 235–238 (1978).
    [CrossRef]
  15. See, e.g., R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987), pp. 119 ff.
  16. See, e.g., R.G. Brewer, “Coherent optical transients,” Phys. Today 30 (5), 50–59 (1977), and references therein.
    [CrossRef]

1990 (1)

D. Suter, M. Rosatzin, J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef] [PubMed]

1988 (2)

See, e.g., T. Mishina, Y. Fukuda, T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

For a preliminary report see, e.g., S. Burschka, J. Mlynek, “Optically induced spin transients in the ground state of atomic sodium,” Opt. Commun. 66, 59–64 (1988).
[CrossRef]

1986 (1)

F. Mitschke, R. Deserno, W. Lange, J. Mlynek, “Magnetically induced optical self-pulsing in a nonlinear resonator,” Phys. Rev. A 33, 3219–3231 (1986).
[CrossRef] [PubMed]

1983 (2)

R. M. Shelby, A.C. Tropper, R.T. Harley, R. M. Macfarlane, “Measurement of the hyperfine structure of Pr3+:YAG by quantum-beat free induction decay, hole burning, and optically detected nuclear quadrupole resonance,” Opt. Lett. 8, 304–306 (1983).
[CrossRef] [PubMed]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, “Quantum beat free induction decay in Tm2+:SrF2:Fourier transform ESR spectroscopy by optical means,” Phys. Rev. B 28, 2869–2872 (1983).
[CrossRef]

1980 (1)

1979 (1)

J. Mlynek, W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
[CrossRef]

1978 (2)

M. P. Silverman, S. Haroche, M. Gross, “General theory of laser-induced quantum beats. I. Saturation effects of single laser excitation,” Phys. Rev. A 18, 1507–1516 (1978).
[CrossRef]

H. Lundberg, S. Svanberg, “Two quantum beat phenomena observed for magnetically tuned atomic sublevels,” Opt. Commun. 27, 235–238 (1978).
[CrossRef]

1977 (1)

See, e.g., R.G. Brewer, “Coherent optical transients,” Phys. Today 30 (5), 50–59 (1977), and references therein.
[CrossRef]

1972 (1)

C. Cohen-Tannoudji, J. Dupont-Roc, “Experimental study of Zeeman light shifts in weak magnetic fields,” Phys. Rev. A 5, 968–984 (1972).
[CrossRef]

1967 (1)

See, e.g., A. Kastler, “Optical methods for studying Hertzian resonances,” Science 158, 214–221 (1967), and references therein.
[CrossRef] [PubMed]

1950 (1)

E. L. Hahn, “Nuclear Induction due to free Larmor precession,” Phys. Rev. 77, 297–298 (1950).
[CrossRef]

Bodenhausen, G.

See, e.g., R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987), pp. 119 ff.

Brewer, R.G.

See, e.g., R.G. Brewer, “Coherent optical transients,” Phys. Today 30 (5), 50–59 (1977), and references therein.
[CrossRef]

Burschka, S.

For a preliminary report see, e.g., S. Burschka, J. Mlynek, “Optically induced spin transients in the ground state of atomic sodium,” Opt. Commun. 66, 59–64 (1988).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, “Experimental study of Zeeman light shifts in weak magnetic fields,” Phys. Rev. A 5, 968–984 (1972).
[CrossRef]

Deserno, R.

F. Mitschke, R. Deserno, W. Lange, J. Mlynek, “Magnetically induced optical self-pulsing in a nonlinear resonator,” Phys. Rev. A 33, 3219–3231 (1986).
[CrossRef] [PubMed]

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, “Experimental study of Zeeman light shifts in weak magnetic fields,” Phys. Rev. A 5, 968–984 (1972).
[CrossRef]

Ernst, R. R.

See, e.g., R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987), pp. 119 ff.

Fukuda, Y.

See, e.g., T. Mishina, Y. Fukuda, T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, “Quantum beat free induction decay in Tm2+:SrF2:Fourier transform ESR spectroscopy by optical means,” Phys. Rev. B 28, 2869–2872 (1983).
[CrossRef]

Gross, M.

M. P. Silverman, S. Haroche, M. Gross, “General theory of laser-induced quantum beats. I. Saturation effects of single laser excitation,” Phys. Rev. A 18, 1507–1516 (1978).
[CrossRef]

Hahn, E. L.

E. L. Hahn, “Nuclear Induction due to free Larmor precession,” Phys. Rev. 77, 297–298 (1950).
[CrossRef]

Harley, R.T.

Haroche, S.

M. P. Silverman, S. Haroche, M. Gross, “General theory of laser-induced quantum beats. I. Saturation effects of single laser excitation,” Phys. Rev. A 18, 1507–1516 (1978).
[CrossRef]

See, e.g., S. Haroche, “Quantum beats and time-resolved fluorescence spectroscopy,” in High Resolution Laser Spectroscopy, K. Shimoda, ed. (Springer-Verlag, 1976), pp. 253–313, and references therein.
[CrossRef]

Hashi, T.

See, e.g., T. Mishina, Y. Fukuda, T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, “Quantum beat free induction decay in Tm2+:SrF2:Fourier transform ESR spectroscopy by optical means,” Phys. Rev. B 28, 2869–2872 (1983).
[CrossRef]

Kastler, A.

See, e.g., A. Kastler, “Optical methods for studying Hertzian resonances,” Science 158, 214–221 (1967), and references therein.
[CrossRef] [PubMed]

Kohmoto, T.

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, “Quantum beat free induction decay in Tm2+:SrF2:Fourier transform ESR spectroscopy by optical means,” Phys. Rev. B 28, 2869–2872 (1983).
[CrossRef]

Lange, W.

F. Mitschke, R. Deserno, W. Lange, J. Mlynek, “Magnetically induced optical self-pulsing in a nonlinear resonator,” Phys. Rev. A 33, 3219–3231 (1986).
[CrossRef] [PubMed]

J. Mlynek, W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
[CrossRef]

Lundberg, H.

H. Lundberg, S. Svanberg, “Two quantum beat phenomena observed for magnetically tuned atomic sublevels,” Opt. Commun. 27, 235–238 (1978).
[CrossRef]

Macfarlane, R. M.

Mishina, T.

See, e.g., T. Mishina, Y. Fukuda, T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, “Quantum beat free induction decay in Tm2+:SrF2:Fourier transform ESR spectroscopy by optical means,” Phys. Rev. B 28, 2869–2872 (1983).
[CrossRef]

Mitschke, F.

F. Mitschke, R. Deserno, W. Lange, J. Mlynek, “Magnetically induced optical self-pulsing in a nonlinear resonator,” Phys. Rev. A 33, 3219–3231 (1986).
[CrossRef] [PubMed]

Mlynek, J.

D. Suter, M. Rosatzin, J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef] [PubMed]

For a preliminary report see, e.g., S. Burschka, J. Mlynek, “Optically induced spin transients in the ground state of atomic sodium,” Opt. Commun. 66, 59–64 (1988).
[CrossRef]

F. Mitschke, R. Deserno, W. Lange, J. Mlynek, “Magnetically induced optical self-pulsing in a nonlinear resonator,” Phys. Rev. A 33, 3219–3231 (1986).
[CrossRef] [PubMed]

J. Mlynek, W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
[CrossRef]

Rosatzin, M.

D. Suter, M. Rosatzin, J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef] [PubMed]

Shelby, R. M.

Silverman, M. P.

M. P. Silverman, S. Haroche, M. Gross, “General theory of laser-induced quantum beats. I. Saturation effects of single laser excitation,” Phys. Rev. A 18, 1507–1516 (1978).
[CrossRef]

Suter, D.

D. Suter, M. Rosatzin, J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef] [PubMed]

Svanberg, S.

H. Lundberg, S. Svanberg, “Two quantum beat phenomena observed for magnetically tuned atomic sublevels,” Opt. Commun. 27, 235–238 (1978).
[CrossRef]

Tanigawa, M.

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, “Quantum beat free induction decay in Tm2+:SrF2:Fourier transform ESR spectroscopy by optical means,” Phys. Rev. B 28, 2869–2872 (1983).
[CrossRef]

Tropper, A.C.

Wokaun, A.

See, e.g., R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987), pp. 119 ff.

Opt. Commun. (4)

J. Mlynek, W. Lange, “A simple method of observing coherent ground state transients,” Opt. Commun. 30, 337–340 (1979).
[CrossRef]

For a preliminary report see, e.g., S. Burschka, J. Mlynek, “Optically induced spin transients in the ground state of atomic sodium,” Opt. Commun. 66, 59–64 (1988).
[CrossRef]

See, e.g., T. Mishina, Y. Fukuda, T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

H. Lundberg, S. Svanberg, “Two quantum beat phenomena observed for magnetically tuned atomic sublevels,” Opt. Commun. 27, 235–238 (1978).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

E. L. Hahn, “Nuclear Induction due to free Larmor precession,” Phys. Rev. 77, 297–298 (1950).
[CrossRef]

Phys. Rev. A (4)

M. P. Silverman, S. Haroche, M. Gross, “General theory of laser-induced quantum beats. I. Saturation effects of single laser excitation,” Phys. Rev. A 18, 1507–1516 (1978).
[CrossRef]

F. Mitschke, R. Deserno, W. Lange, J. Mlynek, “Magnetically induced optical self-pulsing in a nonlinear resonator,” Phys. Rev. A 33, 3219–3231 (1986).
[CrossRef] [PubMed]

D. Suter, M. Rosatzin, J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef] [PubMed]

C. Cohen-Tannoudji, J. Dupont-Roc, “Experimental study of Zeeman light shifts in weak magnetic fields,” Phys. Rev. A 5, 968–984 (1972).
[CrossRef]

Phys. Rev. B (1)

T. Kohmoto, Y. Fukuda, M. Tanigawa, T. Mishina, T. Hashi, “Quantum beat free induction decay in Tm2+:SrF2:Fourier transform ESR spectroscopy by optical means,” Phys. Rev. B 28, 2869–2872 (1983).
[CrossRef]

Phys. Today (1)

See, e.g., R.G. Brewer, “Coherent optical transients,” Phys. Today 30 (5), 50–59 (1977), and references therein.
[CrossRef]

Science (1)

See, e.g., A. Kastler, “Optical methods for studying Hertzian resonances,” Science 158, 214–221 (1967), and references therein.
[CrossRef] [PubMed]

Other (2)

See, e.g., S. Haroche, “Quantum beats and time-resolved fluorescence spectroscopy,” in High Resolution Laser Spectroscopy, K. Shimoda, ed. (Springer-Verlag, 1976), pp. 253–313, and references therein.
[CrossRef]

See, e.g., R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987), pp. 119 ff.

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic representation of the J = 1/2–J = 1/2 system coupled to the optical fields. The quantization axis is parallel to the propagation direction of the laser beam so that the atomic substates remain degenerate, even in the presence of a transverse magnetic field, B. In this representation, the magnetic field induces transitions between the substates, indicated by the curved line, (b) Schematic representation of the experimental setup: P, polarizer; BS, beam splitter; AOM, acousto-optic modulator; λ/4, retardation plate; B, magnetic field; A, polarization analyzer; PD, photodiode.

Fig. 2
Fig. 2

Representation of the magnetic field ΩL and the pseudo-field Δ ¯ P due to the light-shift effect, together with the equilibrium magnetization m and its projection into the yz plane myz. For details see text.

Fig. 3
Fig. 3

(a) Projection of the magnetization vector m into the yz plane. The arrow represents the stationary magnetization and the dashed spiral the evolution after the end of the optical excitation pulse. Amplitude A and phase ϕ of the FID are given by the polar coordinates of the thick arrow. (b) z component of the magnetization as a function of time (solid curve) and the FID envelope ±A exp(−γt) (dashed curves).

Fig. 4
Fig. 4

Equilibrium magnetization myz(0) as a function of optical pump rate for two different detunings. The optical pump rate |β+|22 runs from 0 to 2 × 107 sec−1. The arrows represent the values for χ22 = 2 × 106 sec−1 in each curve. The other parameters are ΩL/2π = 159 kHz and γ = 104 sec−1.

Fig. 5
Fig. 5

Experimental realization of the polarization selective detection: BS, beam splitter; A’s, polarization analyzers; PD’s, photodiodes; AMP, amplifier.

Fig. 6
Fig. 6

FID signals measured with two different laser powers. Note the different scales. Experimental parameters: ΩL/2π = 200 kHz, Δ/2π = 10.5 GHz. The lowest trace indicates the amplitude of the laser pulse.

Fig. 7
Fig. 7

Amplitude and phase of the FID signal as a function of laser power for three different magnetic-field strengths. The experimental data points are compared with the theoretical prediction. The optical detuning was set to Δ/2π = 10.5 GHz, and the decay rate was γ = 12 × 103 sec−1. The amplitude scale represents relative ground-state polarization, and the arrows indicate where p = |β+|22ΩL = 1 for each curve.

Fig. 8
Fig. 8

Phase and amplitude of the FID signal as a function of optical detuning. The experimental data points are compared with the theoretical prediction. The parameters used for the calculation were χ22 = 1.5 × 106 sec−1, γ = 8.5 × 103 sec−1, and ΩL/2π = 102 kHz.

Equations (13)

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m = [ ρ 12 + ρ 21 , i ( ρ 12 ρ 21 ) , ρ 22 ρ 11 ] .
P + = | β + | 2 Γ 2 ( 1 + Δ ¯ 2 ) ,
m ˙ = Ω × m γ eff m + P ,
Ω = ( Ω L , 0 , Δ ¯ P + ) , Ω L = μ B g B ћ , P = ( 0 , 0 , P + ) , γ eff = γ + P + .
m = P + γ eff ( Ω L 2 + Δ ¯ 2 P + 2 + γ eff 2 ) × ( Δ ¯ P + Ω L , γ eff Ω L , Δ ¯ 2 P + 2 + γ eff 2 ) .
m ( t ) = [ m x , A sin ( Ω L t + ϕ ) , A cos ( Ω L t + ϕ ) ] e γ t .
A = [ m y ( 0 ) 2 + m z ( 0 ) 2 ] 1 / 2 = P + γ eff [ γ eff 2 Ω L 2 + ( Δ ¯ 2 P + 2 + γ eff 2 ) 2 ] 1 / 2 Ω L 2 + Δ ¯ 2 P + 2 + γ eff 2 ,
tan ϕ = m y ( 0 ) m z ( 0 ) = γ eff Ω L Δ ¯ 2 P + 2 + γ eff 2 .
A = p 1 + p 2 1 + p 2 + Δ ¯ 2 ,
tan ϕ = 1 / p ,
Δ I = I 0 exp ( α 0 l ) sin ( 2 m z d 0 ) .
Δ I 1 = m z d 0 I 0 exp ( α 0 l ) .
A = p 1 + p 2 = P + Ω L 2 + P + 2 .

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