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Modification of atomic energy levels by two rotating rf fields
Volume 47A, number 3
PHYSICS LETTERS
25 March 1974
M O D I F I C A T I O N O F ATOMIC E N E R G Y L E V E L S BY TWO R O T A T I N G R F F I E L D S N. TSUKADA Central Research Laboratory, Mitsublshz Electric Corporation, Amagasakt, Japan Received 21 January 1974 The modification of the energy levels for spin-l/2 system by two rotating rf fields is described.
The modification of atomicg.factor due to a linearly oscillating rf field H 1 cos cot has been reported by Cohen-Tatmoudji and Haroche [1-3]. They have shown that the effective magnetic field undergoing the influence of an oscillating rf field is modified as
d
coo c%:o(col/co), =
(1)
in the case that the magnitude of a static magnetic field H 0 is very small compared with the value of co/-/, i.e., coo = -/H0 "~ co, where col = "/HI, and 7 is the gyromagnetic ratio. The atom of spin-½ has the energy or eigenvalue of ~cooJo(col/co) for IT) level and of -~rcooJo(col/co) for I'-)level, where Jo(col /6°) is the order Bessel function and I+--),I-) represent the eigenstates of the "dressed atom" [1-3]. Eq. (1) shows that the gap of the two levels of the atom is always modified as being smaller compared with the unperturbed gap coo, i.e., COo a < ¢%. In this note, we want to show the gap of the two levels can be modified as being larger compared with the unperturbed gap, i.e., coo a ~> w o, In order to obtain this effect, we have to take out two rotating rf fields with the same magnitude and different frequencies [e.g. 4]. The magnetic fields considered here are given as follows:
H=Hok
+½Hi[(coscoat+coscobOi+(sincoat+sincobt)]] ,
(2)
where we assume that the magnitude of two rotating rf fields are H 1/2, and the two rotational frequencies are coa and cob, respectively. This field can be transformed as
# = s7o +Z lcos t 7" where
(3)
"~o=Ho-(coa+cob )127 and ~ =(coa--cob)/2, (4) on the rotating coordinate system which is rotating with the angular frequency (coa+cob)/2 a~und ~0. This contains a static field, whose magnitude is H o, and a linearly oscillating field, whose magnitude and angular frequency are.//1 and G, respectively. If the condition -/H0 < ~o is satisfied, this situation becomes the same case that has been considered by Cohen-Tannoudji and Haroche [1 ]. We can expect that the atoms of spin.½ have than energy of ~ o J o ( c o l / ~ ) for {T)level and of --i ~OoJo(col /co) for IZ) level, where I~) and [Z) represent the eigenstates of the dressed atom in the rotating coordinate system, and G o = -//-I0"As the result, the energy gap of the two levels is given as ~d coo =
oSo(col/co) •
(5)
This expression has the same form as eq. (1). At the points of Jo(col/~ ) = 0, the energy gap of the two levels vanishes, i.e., the two levels are of degeneracy. When we transform eq. (5) back into the laboratory frame, the energy gap of two levels in the laboratory frame becomes as ~od = (coa+ cob)/2 + ~oJo(col/5).
(6)
This expression shows that the energy gap ff~d o of the two levels becomes smaller than coo for coo > (coa+cob)[2, on the other hand for coo < (coa+cob)/2, the energy gap becomes larger than cooFig. 1 shows the energy levels for various values of col(a), and shows the variation of the effective magnetic field/~0(= ~od/-/) for various values of the static f'teld n o = no(1 ), o, /-/and H0(2), where HO(1 ) and HO(2 ) represent a certain value of the static magnetic field which is lower than H o = 0 and is higher than H 0 = ~/-/, in the absence of the rf 265
Volume 47A, number 3
PHYSICS LETTERS
__
E
_..~
_r_/
"aoC1)
1.o
o
_-~--
___~
0
~l
%-o
--
~-~"
~.4
oJ
T
"0 " T % -~0~)
\
i ;
,((!,, 8
U'~l
I
m Ftg. 1. (a) shows the energy levels for various values of wl/~. (b) shows the variation of the effective magnetic field H~ as a function of to l/~ for various values of the magnetic field H o = He(l) , 0, ~['y and He(2). The effective magnetic field ~ 1 approach to the value ~oo = (tea+ COb)/27 as the value of tol/~ is increased. fields, respectively. At the values of 6o I for Jo(COl/~) = 0, the energy gap of the two levels becomes independent of the magnetic field H 0, and becomes (6oa+6ob)/2, m the region of the static field H 0 which satisfies the condition ~ o "¢ ~ . If 6oa = -COb, eq. (6) becomes eq. (1). I f H 0 = 0, i e., at zero magnetic field, eq. (6) becomes as - d = ~(6oa 4"6ob ) [1 -- Jo (6o 1/t~)] too
(7)
This shows that the gap o f the energy levels at zero magnetic field can be modified by choosing the adaptable values of 6oa, ~ b and 6ol. The separation of the energy levels at zero magnetic field may be considered as a fictitious field [2, 3] due to two
266
25 March 1974
rotating rf fields which have the same magnitude. Two rotating rf fields which have the same magnitude and different frequencies are equivalent to a linearly oscdlating rf field, axis of which rotates with the angular frequency (6oa + 600)/2. In the case for the modification of the atomic g-factor by Cohen-Tannoudji and Haroche [2, 5], the sense of free precession of the magnetic moment for given static magnetic field is unchanged as the intensity 6oI of the linearly oscillating rf field is increased. However, in our case we may observe the changes m the sense of the free precession of the magnetic moment, because of the magnitude of the effective magnetic field ~ for given H 0 (see the case o f H 0 = H0(1 ) in fig. l(b)) varies from negative to positive as the intensity of the two rotating fields is increased. We showed that the eneirgy levels are remarkably modified by two rotating rf fields which have the same magmtude and different frequencies. Especially for Jo(6ol[~) = 0, the gap of the energy levels becomes independent of the static magnetic field. Namely, the energy levels behave as levels of the zero magnetic quantum number, so this effect is useful to reduce the influence of the external magnetic field on the Zeeman sublevels and to investigate the spin echange effect between different species [6]. It should be emphasized that the result obtained in this paper might be applicable to a paxr o f levels connected by electric dipole transition, so the energy levels, connected with optical frequency, is also modified by two circularly polarized light.
References [ 1 ] C. Cohen-Tannoudji and S. Haroche, J.' de Phys. (Pans)
30 (1969) 153. [2] S. Haroche, Ann. Phys. (Paris) 6 (1971) 189. [3] S. Haroche, Ann. Phys. (Paris) 6 (1971) 327. [41 N.F. Ramsey, Phys. Rev. 100 (1955) 1191, N. Tsukada, T. Yabuzaki and T. Ogawa, J. Phys. Soc. (Japan) 35 (1973) 230. [5] C. Landre', C. Cohen-Tannoudjx, J. Dupont-Roc and S. Haroche, J. de Phys. (Paris) 31 (1970) 971 [6] C. Cohen-Tannoudjl, J. Dupont-Roc, S. Haroche and F. Lalo~, Phys. Rev. Lett. 22 (1969) 758.
PHYSICS LETTERS
25 March 1974
M O D I F I C A T I O N O F ATOMIC E N E R G Y L E V E L S BY TWO R O T A T I N G R F F I E L D S N. TSUKADA Central Research Laboratory, Mitsublshz Electric Corporation, Amagasakt, Japan Received 21 January 1974 The modification of the energy levels for spin-l/2 system by two rotating rf fields is described.
The modification of atomicg.factor due to a linearly oscillating rf field H 1 cos cot has been reported by Cohen-Tatmoudji and Haroche [1-3]. They have shown that the effective magnetic field undergoing the influence of an oscillating rf field is modified as
d
coo c%:o(col/co), =
(1)
in the case that the magnitude of a static magnetic field H 0 is very small compared with the value of co/-/, i.e., coo = -/H0 "~ co, where col = "/HI, and 7 is the gyromagnetic ratio. The atom of spin-½ has the energy or eigenvalue of ~cooJo(col/co) for IT) level and of -~rcooJo(col/co) for I'-)level, where Jo(col /6°) is the order Bessel function and I+--),I-) represent the eigenstates of the "dressed atom" [1-3]. Eq. (1) shows that the gap of the two levels of the atom is always modified as being smaller compared with the unperturbed gap coo, i.e., COo a < ¢%. In this note, we want to show the gap of the two levels can be modified as being larger compared with the unperturbed gap, i.e., coo a ~> w o, In order to obtain this effect, we have to take out two rotating rf fields with the same magnitude and different frequencies [e.g. 4]. The magnetic fields considered here are given as follows:
H=Hok
+½Hi[(coscoat+coscobOi+(sincoat+sincobt)]] ,
(2)
where we assume that the magnitude of two rotating rf fields are H 1/2, and the two rotational frequencies are coa and cob, respectively. This field can be transformed as
# = s7o +Z lcos t 7" where
(3)
"~o=Ho-(coa+cob )127 and ~ =(coa--cob)/2, (4) on the rotating coordinate system which is rotating with the angular frequency (coa+cob)/2 a~und ~0. This contains a static field, whose magnitude is H o, and a linearly oscillating field, whose magnitude and angular frequency are.//1 and G, respectively. If the condition -/H0 < ~o is satisfied, this situation becomes the same case that has been considered by Cohen-Tannoudji and Haroche [1 ]. We can expect that the atoms of spin.½ have than energy of ~ o J o ( c o l / ~ ) for {T)level and of --i ~OoJo(col /co) for IZ) level, where I~) and [Z) represent the eigenstates of the dressed atom in the rotating coordinate system, and G o = -//-I0"As the result, the energy gap of the two levels is given as ~d coo =
oSo(col/co) •
(5)
This expression has the same form as eq. (1). At the points of Jo(col/~ ) = 0, the energy gap of the two levels vanishes, i.e., the two levels are of degeneracy. When we transform eq. (5) back into the laboratory frame, the energy gap of two levels in the laboratory frame becomes as ~od = (coa+ cob)/2 + ~oJo(col/5).
(6)
This expression shows that the energy gap ff~d o of the two levels becomes smaller than coo for coo > (coa+cob)[2, on the other hand for coo < (coa+cob)/2, the energy gap becomes larger than cooFig. 1 shows the energy levels for various values of col(a), and shows the variation of the effective magnetic field/~0(= ~od/-/) for various values of the static f'teld n o = no(1 ), o, /-/and H0(2), where HO(1 ) and HO(2 ) represent a certain value of the static magnetic field which is lower than H o = 0 and is higher than H 0 = ~/-/, in the absence of the rf 265
Volume 47A, number 3
PHYSICS LETTERS
__
E
_..~
_r_/
"aoC1)
1.o
o
_-~--
___~
0
~l
%-o
--
~-~"
~.4
oJ
T
"0 " T % -~0~)
\
i ;
,((!,, 8
U'~l
I
m Ftg. 1. (a) shows the energy levels for various values of wl/~. (b) shows the variation of the effective magnetic field H~ as a function of to l/~ for various values of the magnetic field H o = He(l) , 0, ~['y and He(2). The effective magnetic field ~ 1 approach to the value ~oo = (tea+ COb)/27 as the value of tol/~ is increased. fields, respectively. At the values of 6o I for Jo(COl/~) = 0, the energy gap of the two levels becomes independent of the magnetic field H 0, and becomes (6oa+6ob)/2, m the region of the static field H 0 which satisfies the condition ~ o "¢ ~ . If 6oa = -COb, eq. (6) becomes eq. (1). I f H 0 = 0, i e., at zero magnetic field, eq. (6) becomes as - d = ~(6oa 4"6ob ) [1 -- Jo (6o 1/t~)] too
(7)
This shows that the gap o f the energy levels at zero magnetic field can be modified by choosing the adaptable values of 6oa, ~ b and 6ol. The separation of the energy levels at zero magnetic field may be considered as a fictitious field [2, 3] due to two
266
25 March 1974
rotating rf fields which have the same magnitude. Two rotating rf fields which have the same magnitude and different frequencies are equivalent to a linearly oscdlating rf field, axis of which rotates with the angular frequency (6oa + 600)/2. In the case for the modification of the atomic g-factor by Cohen-Tannoudji and Haroche [2, 5], the sense of free precession of the magnetic moment for given static magnetic field is unchanged as the intensity 6oI of the linearly oscillating rf field is increased. However, in our case we may observe the changes m the sense of the free precession of the magnetic moment, because of the magnitude of the effective magnetic field ~ for given H 0 (see the case o f H 0 = H0(1 ) in fig. l(b)) varies from negative to positive as the intensity of the two rotating fields is increased. We showed that the eneirgy levels are remarkably modified by two rotating rf fields which have the same magmtude and different frequencies. Especially for Jo(6ol[~) = 0, the gap of the energy levels becomes independent of the static magnetic field. Namely, the energy levels behave as levels of the zero magnetic quantum number, so this effect is useful to reduce the influence of the external magnetic field on the Zeeman sublevels and to investigate the spin echange effect between different species [6]. It should be emphasized that the result obtained in this paper might be applicable to a paxr o f levels connected by electric dipole transition, so the energy levels, connected with optical frequency, is also modified by two circularly polarized light.
References [ 1 ] C. Cohen-Tannoudji and S. Haroche, J.' de Phys. (Pans)
30 (1969) 153. [2] S. Haroche, Ann. Phys. (Paris) 6 (1971) 189. [3] S. Haroche, Ann. Phys. (Paris) 6 (1971) 327. [41 N.F. Ramsey, Phys. Rev. 100 (1955) 1191, N. Tsukada, T. Yabuzaki and T. Ogawa, J. Phys. Soc. (Japan) 35 (1973) 230. [5] C. Landre', C. Cohen-Tannoudjx, J. Dupont-Roc and S. Haroche, J. de Phys. (Paris) 31 (1970) 971 [6] C. Cohen-Tannoudjl, J. Dupont-Roc, S. Haroche and F. Lalo~, Phys. Rev. Lett. 22 (1969) 758.