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Electro-optic level-crossing effect
Volume 65A, number 3
PHYSICS LETTERS
6 March 1978
ELECTRO-OPTIC LEVEL-CROSSING EFFECT V. Paul KAFTANDJ IAN and Lewis KLEIN1 Centre de St. Jerome,
Université
de Proi’ence, 13397 Marseille, Cedex 4, France
and W. HANLE Pliysikalisches Institut der Universität Giessen, 63 Giessen, Germany Received 16 December 1977
The theory of the two-photon level-crossing analog of the electric-field Hanle effect is presented.
Two years after the publication of the first experiment on the zero-magnetic-field level-crossing effect [1], the corresponding experiment in an electric field was announced by one of the present authors [21. These two phenomena are related respectively to the Zeeman and Stark effects and the physical basis of both is found in the lifting of the m-state degeneracy by an externally applied field. Recently, two of the present authors [3] proposed a two-photon analog of the zero-magnetic-field levelcrossing effect, replacing the external field with intense circularly polarized laser light. The laser pumps a coupled transition and creates a magnetization in the gas through the inverse Faraday effect. Fluorescence is observed as a function of the intensity of the circularly polarized light and zero-field level-crossing resonances occur in the same manner as those observed in the usual magnetic Hanle effect. The physical basis for the two-photon level-crossing effect is found in the dynamic Stark shift of the energy level involved in the coupled transition (for the case of circularly polarized light, the shift is also termed the Zeeman light shift [41).This shift lifts the degeneracy of one of the m-substates and, hence, can be thought of as replacing the external magnetic field in the Hanle effect. From this it can be seen that the dynamic Stark 1
shift due to a linearly polarized pumping field can also be used to lift the degeneracy of the m-sublevels and thus provide a two-photon analog of the zero-electricfield level-crossing effect. To see this in detail, consider a system with the level scheme of fig. 1 where the levels a, b, and c have total angular momentum, J = 0, 1, and 0 respectively and, hence, the b level has degenerate m-substates, b 0, b~and b. This system is subjected to strong saturating light of frequency w2, near resonance with Wab, the frequency of the a b transition, propagating along the v-axis and polarized in the i-direction. In addition, the system receives weak probe radiation of frequency w1, near resonance with Wbc co- or counter propagating along i~,and polarized at 45°to the p-axis. The relative direction of propagation is unimportant as we shall neglect Doppler effects here. a —*
I ~
b
b
b
-
e,
Permanent address: Physics Department, Howard University, Washington, DC 20059, USA.
188
Fig. 1. Three-level system and light polarization directions.
PHYSICS LETTERS
Volume 65A, number 3
As is well known, the intense light field shifts the b0 sublevel an amount ~s by the dynamic Stark effect, where
Lf
j
0~ab’Yab is the line-width Here, ~“-‘ab = 0)2 with parameter = ~ab ~ the fordipole-moment ~Wab/(~W~b the a b transition + y~). matrixand element. ~ab = PabE’212h The b~ (1) sublevels are, of course, unaffected by the strong field. The fluorescence from the b state, which in the —
-~
same direction the probe field, becomes elliptically absence of the as strong field would be polarized in the polarized. This is due to the phase difference between the 2- and i-components arising from the different frequencies which are associated with the shifted b0
c and the unshifted b.,. c transitions. The intensity of the fluorescence observed with parallel, LF(ê~/4),and crossed, LF(e3~/4),polarizers is, in the fr-direction, -~
LF(e~)
.
iT.. 1.. -2
-4
0
2
4
-
Fig. 2. Intensity of fluorescence observed through parallel, LF(é 1~/4),and crossed, LF(ê3~/4)~ polarizers when probe field is resonant with ubc.
-~
2 + 472]
LF(d~f4)= 2~c[(2~Wbc + ~s) X {(Aw~~ + 72 0)bc + ~
+ 72]
}
~
(2)
+ 72)[(~~bc + ~~)2 + 72]
}
= 0. The curves are identical in form to that of the classical Hanle electric-field level-crossing effect excited with monochromatic light.with Thebroad-band usual electric Hanle effect [21which is excited
~~bc
optical pumping has the same form also except for
,
LF(d3~/4)= 2~c~ X
6 March 1978
i
where the simplifying conditions, ~ I ~ ‘3ab’ I ~‘bc1’ and 7ab = 27bc = 2y have been assumed. In addition, level c has been considered to be the ground state with zero width. When the probe-field frequency is tuned to exact resonance with 0)bc’ eqs. (2) reduce to the familiar form,
large values the value ofexternal intensity electric is twice that offound field. The curve for asymptotic large in fig. values 2, traced of theas athe function of the dynamic Stark shift, may obtained experimentally either by varying E 2,bethe intensity of the 21 strong saturating laser field, or ~~~ab’ the detuning of the frequency of that laser from the 0)ab resonance frequency. This describes the optical analogy to the electricfield Hanle effect. In the past, level crossings in an external electric field were much more difficult to observe than those in a magnetic field since the qua~,
LF(ê~/ 4)a (1
+
~2/472)/(l
+ ~2/72)
(3)
LF(d3~/4)a (~2/472)/(l+ ~2/72) These expressions are identical to those found in refs. [3, 5] describing the magneto-optical level-crossing effect in the same approximation. These results are also found for the level-crossing effect with a static external electric field and monochromatic excitation if the Stark-effect splitting of the b substates replaces
dratic Stark-effect splitting is, in general, very small. In contrast, the electro-optic level-crossing effect has the same magnitude as the magneto-optic effect of ref. [3]. Calculations there demonstrated that for a 2 is sufficient to sweep a 1 A detuning,curve l0~W/cm level-crossing with a natural width of 108 ~l. Finally, the electro-optic effect has the additional advantage of using linearly polarized laser light instead of the circularly polarized light required in the magnetooptic effect.
~S.
Fig. 2 displays the behavior of the fluorescence as a function of the dynamic Stark shift at resonance,
Conclusion. Four different types of level-crossing effects can be distinguished. In the classical Hanle ef189
Volume 65A, number 3
PHYSICS LETTERS
fect cases, the degeneracy is lifted by an external static electric field or a static magnetic field. In the optical analogs of these classical effects, it is the dynamic Stark effect produced by high-power laser radiation which shifts the levels. Circular polarization is the analog of a magnetic field and as described in this letter, linear polarization can play the role of an electric field.
190
6 March 1978
References [1] W. Hanle, Z. Phys. 30(1924)93. 121 W. Hanle, Z.Phys. 35 (1926) 346. 131 V.P. Kaftandjian and L. Klein, Phys. Lett. 62A (1977) 317. [4] W. Happer, Progr. Quant. Electron. 1(1971)53. 151 V.P. Kaftandjian, Thesis, Univ. de Provence (1977) published.
PHYSICS LETTERS
6 March 1978
ELECTRO-OPTIC LEVEL-CROSSING EFFECT V. Paul KAFTANDJ IAN and Lewis KLEIN1 Centre de St. Jerome,
Université
de Proi’ence, 13397 Marseille, Cedex 4, France
and W. HANLE Pliysikalisches Institut der Universität Giessen, 63 Giessen, Germany Received 16 December 1977
The theory of the two-photon level-crossing analog of the electric-field Hanle effect is presented.
Two years after the publication of the first experiment on the zero-magnetic-field level-crossing effect [1], the corresponding experiment in an electric field was announced by one of the present authors [21. These two phenomena are related respectively to the Zeeman and Stark effects and the physical basis of both is found in the lifting of the m-state degeneracy by an externally applied field. Recently, two of the present authors [3] proposed a two-photon analog of the zero-magnetic-field levelcrossing effect, replacing the external field with intense circularly polarized laser light. The laser pumps a coupled transition and creates a magnetization in the gas through the inverse Faraday effect. Fluorescence is observed as a function of the intensity of the circularly polarized light and zero-field level-crossing resonances occur in the same manner as those observed in the usual magnetic Hanle effect. The physical basis for the two-photon level-crossing effect is found in the dynamic Stark shift of the energy level involved in the coupled transition (for the case of circularly polarized light, the shift is also termed the Zeeman light shift [41).This shift lifts the degeneracy of one of the m-substates and, hence, can be thought of as replacing the external magnetic field in the Hanle effect. From this it can be seen that the dynamic Stark 1
shift due to a linearly polarized pumping field can also be used to lift the degeneracy of the m-sublevels and thus provide a two-photon analog of the zero-electricfield level-crossing effect. To see this in detail, consider a system with the level scheme of fig. 1 where the levels a, b, and c have total angular momentum, J = 0, 1, and 0 respectively and, hence, the b level has degenerate m-substates, b 0, b~and b. This system is subjected to strong saturating light of frequency w2, near resonance with Wab, the frequency of the a b transition, propagating along the v-axis and polarized in the i-direction. In addition, the system receives weak probe radiation of frequency w1, near resonance with Wbc co- or counter propagating along i~,and polarized at 45°to the p-axis. The relative direction of propagation is unimportant as we shall neglect Doppler effects here. a —*
I ~
b
b
b
-
e,
Permanent address: Physics Department, Howard University, Washington, DC 20059, USA.
188
Fig. 1. Three-level system and light polarization directions.
PHYSICS LETTERS
Volume 65A, number 3
As is well known, the intense light field shifts the b0 sublevel an amount ~s by the dynamic Stark effect, where
Lf
j
0~ab’Yab is the line-width Here, ~“-‘ab = 0)2 with parameter = ~ab ~ the fordipole-moment ~Wab/(~W~b the a b transition + y~). matrixand element. ~ab = PabE’212h The b~ (1) sublevels are, of course, unaffected by the strong field. The fluorescence from the b state, which in the —
-~
same direction the probe field, becomes elliptically absence of the as strong field would be polarized in the polarized. This is due to the phase difference between the 2- and i-components arising from the different frequencies which are associated with the shifted b0
c and the unshifted b.,. c transitions. The intensity of the fluorescence observed with parallel, LF(ê~/4),and crossed, LF(e3~/4),polarizers is, in the fr-direction, -~
LF(e~)
.
iT.. 1.. -2
-4
0
2
4
-
Fig. 2. Intensity of fluorescence observed through parallel, LF(é 1~/4),and crossed, LF(ê3~/4)~ polarizers when probe field is resonant with ubc.
-~
2 + 472]
LF(d~f4)= 2~c[(2~Wbc + ~s) X {(Aw~~ + 72 0)bc + ~
+ 72]
}
~
(2)
+ 72)[(~~bc + ~~)2 + 72]
}
= 0. The curves are identical in form to that of the classical Hanle electric-field level-crossing effect excited with monochromatic light.with Thebroad-band usual electric Hanle effect [21which is excited
~~bc
optical pumping has the same form also except for
,
LF(d3~/4)= 2~c~ X
6 March 1978
i
where the simplifying conditions, ~ I ~ ‘3ab’ I ~‘bc1’ and 7ab = 27bc = 2y have been assumed. In addition, level c has been considered to be the ground state with zero width. When the probe-field frequency is tuned to exact resonance with 0)bc’ eqs. (2) reduce to the familiar form,
large values the value ofexternal intensity electric is twice that offound field. The curve for asymptotic large in fig. values 2, traced of theas athe function of the dynamic Stark shift, may obtained experimentally either by varying E 2,bethe intensity of the 21 strong saturating laser field, or ~~~ab’ the detuning of the frequency of that laser from the 0)ab resonance frequency. This describes the optical analogy to the electricfield Hanle effect. In the past, level crossings in an external electric field were much more difficult to observe than those in a magnetic field since the qua~,
LF(ê~/ 4)a (1
+
~2/472)/(l
+ ~2/72)
(3)
LF(d3~/4)a (~2/472)/(l+ ~2/72) These expressions are identical to those found in refs. [3, 5] describing the magneto-optical level-crossing effect in the same approximation. These results are also found for the level-crossing effect with a static external electric field and monochromatic excitation if the Stark-effect splitting of the b substates replaces
dratic Stark-effect splitting is, in general, very small. In contrast, the electro-optic level-crossing effect has the same magnitude as the magneto-optic effect of ref. [3]. Calculations there demonstrated that for a 2 is sufficient to sweep a 1 A detuning,curve l0~W/cm level-crossing with a natural width of 108 ~l. Finally, the electro-optic effect has the additional advantage of using linearly polarized laser light instead of the circularly polarized light required in the magnetooptic effect.
~S.
Fig. 2 displays the behavior of the fluorescence as a function of the dynamic Stark shift at resonance,
Conclusion. Four different types of level-crossing effects can be distinguished. In the classical Hanle ef189
Volume 65A, number 3
PHYSICS LETTERS
fect cases, the degeneracy is lifted by an external static electric field or a static magnetic field. In the optical analogs of these classical effects, it is the dynamic Stark effect produced by high-power laser radiation which shifts the levels. Circular polarization is the analog of a magnetic field and as described in this letter, linear polarization can play the role of an electric field.
190
6 March 1978
References [1] W. Hanle, Z. Phys. 30(1924)93. 121 W. Hanle, Z.Phys. 35 (1926) 346. 131 V.P. Kaftandjian and L. Klein, Phys. Lett. 62A (1977) 317. [4] W. Happer, Progr. Quant. Electron. 1(1971)53. 151 V.P. Kaftandjian, Thesis, Univ. de Provence (1977) published.