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Resonant fluorescence of a three-level atom
Volume
19, number
November
OPTICS COMMUN’ICATIONS
2
RESONANT FLUORESCENCE
OF A THREE-LEVEL
1976
ATOM
B. SOBOLEWSKA Institute of Physics, Polish Academy
Received
13 July
of Sciences, AI. Lornikow 32146, 02-668
1976
The fluorescence spectrum of a three-level atom interacting The calculations are based on the quantum “fluctuation-regression contains five lines, the properties of which are discussed
The resonant interaction of an atom with a strong electromagnetic field has become recently a problem of great interest. The spectral properties of the field radiated by an atom containing a single pair of participating levels was discussed thoroughly by many authors [ 1,2]. It was found, that the fluorescence spectrum of such a system consists of three lines. Two of them are displaced from the central one by i dE, where d is the atomic dipole moment and E is the amplitude of the electric field. The heights of the side lines are equal and are three times smaller than the height of the central line. The half width of the central line is equal to i r, while the half width of the central line is equal to i r. These theoretical predictions were confirmed by experiments [4-61, which were performed on sodium where the two participating levels were 3 2P3,2 [F’ = 31 and 3 2S,12 [F’ = 21, The purpose of this letter is to discuss the more general case when three levels of an atom participate in the interaction with the field. It will be shown that the fluorescence spectrum in this case has some qualitatively new features in comparison to the model of the two-level atom. The model of the three-level atom is presented in fig. 1. The energy of the monochromatic field E(t) = E cos (w,_t) is equal to the energy difference between the atomic levels J = 0 and J = 1. The upper level is splitted by the perpendicular magnetic field into three sublevels: -1, 0 and 1. The splitting a can be tuned by changing the intensity of the applied magnetic field. The hamiltonian for the system of atom and field mJ
mJ
Warsaw, Poland
=
mJ
with strong electromagnetic theorem”. It is shown,
monochromatic field is obtained. that the fluorescence spectrum
3=0 Fig. 1. Energy-level scheme of a three-level atom. The energy difference between the ground state and the two interacting excited states is equal to w - R and w + a, respectively.
in the “rotating wave approximation” field is treated classically) is then:
“-Ii
+
- u(S+ e -iwLt
(the electric
jj
+s
_
,iwLt)
where u = i dE corresponding J=O~J=l,m=-landJ=O~J=l,m=l,
, to the transitions
=
=
The spectral distribution of the radiation can be calculated from the formula [ 1,3] 185
J(c&dt
i(w-wL)(r
,/ dt’C?+(t)S_(t’))c b
0
t’)
(1)
where T is the observation time. To this aim we shall adopt for our model the method proposed by Cohen Tannoudji for the two-level problem [I]. Two two-time correlation function can be writlen in the form: d+(t)S.
(t’)) = (S+(t)) (69~(t’)) + CSS+(t) SS~ (t’j)
The fil-st term of this sum describes the “elastic component” of the radiation. while the second one is due to quan turn fluctuations and describes the “inelastic component” which is our interest. To evaluate this two-time carrel;+theorem” [ 1.3]. According to this. the tion function it is convenient to use the quantum “fluctuation-regression differential equations for the system of two-time correlation functions are exactly the same as for the simple WCIages of the operators si. The equations for (si) or, equivalently, the Bloch equations for the elements of the atom ic density matrix p are well known and were used to study three interacting atomic levels in various configurations [7,8]. By choosing the proper combination of the elements of p it is possible to transform the Blocb equation\ into such a form d Vi/d t = ci Vi + Gi oii Vi, that their approximate solutions can be easily evaluated. The coefl‘icients ayij are in this representation functions of .Q, u and of the natural half width of rbe atomic levels 1‘. and are proportional to r. while the difference between the diagonal coefficients Ej is proportional to u. As far as we deal with a strong external field, u is much larger than r and the contributions from the terms ayiiI’; to the solutions of these equations are negligible. The time dependence of Vi can thus be written as: ~‘j(t)
=
exp [iEj(t
vj(t’)
tl)]
As we shall see, the real parts of Ej determine the positions of their maxima.
The operator &S+(t) can be represented tr [ ~jp] = Vi. Thus, CSi+(t)
63
(t’))
=
C
(‘j(Vj(t’)
the half widths of the spectral lines. while the imaginary parts denote
in the new base of the operators
6S
(I’))
r^;.which ale defined in such a way that
exp [iej(t ~~r’)]
i
Substituting this into eq. (1) and neglecting the transient effects by calculating all averages in the stationary (pst can be evaluated from the Bloch equations for dpjj/d t = 0). it is easy to obtain the spectral distribution the atomic radiation.
186
state 01‘
Volume
19, number
l/TJinel
2
- 2
OPTICS COMMUNICATIONS
y1 t (CZ2u2r/4M60)(w
[
t
1976
]4u6/M(6w)* 1YrJ 3 (a - %)* •t r;
[52*u4/M(60)*] __~~___
t
November
t6wyty;
(w--L
C
- oL t 6~)
[~2~41MW421y, - [r~*~*l‘W~_y)l (u-wL-6d _
___~~~~~_
(u-~L-6w)*ty: [Q2u4/M(Sw)*]
t_-m_~m
(o-cd-Fcl$
r
y7 + (u41’/M8w)(w
[2~~/M(6w)~]
[ whereM=4v4
(w - wL
+y;
- wL t 260)
1
-.I
[2u6/M(ikJ:)*]y3 ~_._~_~
t
y2 + (R’u21’/2M6u)
- (ruqMsw)
(O-C+ tv*fL*
- 26w)2
(w-q t y:
tCL4.
The fluorescence spectrum contains thus 5 lines. The main properties of these will be discussed below: 1. The central line has a lorentzian shape centered atw=wL. The spectral properties of this line strongly depend on the ratio a/u. Its height is proportional to
_ 260) + when a/v < 1 and to i I when n/v > 1. The weights of the dispersion-like parts are f3I = RVr/4MFw,
The height of these two lines is proportional h* = (4LW/M)(6d (3u2 + a2)(6u4
h, = 4&/M@*
+ a*).
When Cl/v S 1,)~ 1 tends to zero, when CL/v< 1, h 1 tends to 1. The half width of this line, yu, tends to iI? for R/v < 1 and to I for CL/v% 1. Thus, when the energies of the participating levels become far from resonance or when the intensity of the incident light decreases, the height of the central line decreases and the half width increases. 2. The two lines centered at w = wL f 6w have a very complicated structure. Each of them contains two lorentzian parts and two dispersion-like components. The weights of the lorentzian parts are equal [(u = 52*u4/4M(u2 + a2)], but their half widths [yI and y2] are different. The asymptotic behaviour is, however, the same for y1 and for y2: they tend to :r
p* = 201 .
t 7v*fl*
to
t ny
+ 9u2!C12 + 04)
’
It tends to zero when R/v < 1 and also when a/v S 1. This means that these lines disappear far from resonance and in the case of “level-crossing”. 3. The two lines centered at w = wL + 2 60 contain also lorentzian and dispersion-like parts. Their weights are respectively, (Y= 2v6/M(6w)* and p= rv4/M6w. Their half width y3 tends to i r when ~l/u < 1 and to 1 when Q/v S 1. The height of these lines is proportional to
187
Volume
19, number
November
OPTIC’S (‘OMMUNI(‘ATIONS
2
197h
The presence of the dispersion-like componentb disturbs the symmetry of each side line about its central point. It is due to the fact, that the stationary atomic density matrix contains non-zero off-diagonal elements. However, the asymmetry is rather inappreciable because p is much smaller than 01(of the order of a few percent of the line width). The “elastic component” of the radiation has al91 been calculated.
For
C2 = 0
I/TJ,,(w) Ipip. 3. l~luorcaccncc spectrum 113. (a) C2 = 0. (b) Sl = I,.
of a three-level
For !2/v < 1 it tends to 4, for n/v S 1 to XIII. We see, that by the Zeeman tuning it is possible to change the spectral properties of the radiation. When 52/v increases from zero the height of the lines 3 dccreases, while the height of the lines 2 increases. Far from resonance all lines disappear: there exists only the “elastic component” of the radiation. When R = 0 (“level-crossing”) the spectrum is similar to that of the two-level case. There arc three lines in the spectrum, the distance of the two side lines from the central line is equal to 2v. The ratios between the heights and the half widths of these lines are identical as in the model of the two-level atom. In the general case, the fluorescence spectrum has been calculated numerically for some values of n/u and r/v. The results for r/v = i are presented in fig. 2
188
- t%‘l5(w
w, ) .
atom for I./V =
this line becomes the only component of the spectt um far from resonance and in the case of a weak incident field. I wish to thank Dr. J. Mostowski for many valuable discussions and suggestions.
References [ 1 1 C. (‘ohen-Tannoudji. Proc. Second Laser Spectroscop) Conf.. Mqkvc, I‘rance. 1975 (Springer, Berlin, 1975). 121 1I.R. Mallow. Phys. Rev. 4 188 (1969) 1969. 131 l~.R.~lollo\~,Pl~y~. Rev.A 12t1975) 1919. 141 lf. Walthcr. I’roc. Second I ;iwr Spectroscopy (‘ant.. Me&c. I‘rancc, 1975 (Springer. krlin, 1975). [S) I:. Schuda, C.R. Stroud, kl. Ilcrcher. J. Phys. H: At. Mol. Phys. 7 (1974) 198. [6] I:.Y. Wu, R.E. Grove. S. Ezekiel. Phyc. Rev. Letter> 35 (1975) 1426. [7] B.J. Feldman, M.S. I?cld. Phys. Rev. 5 (1972) 899. 181 R.G. Brewer and !<.I. llahn, Phys. Rev. A 8 (1973) 464.
19, number
November
OPTICS COMMUN’ICATIONS
2
RESONANT FLUORESCENCE
OF A THREE-LEVEL
1976
ATOM
B. SOBOLEWSKA Institute of Physics, Polish Academy
Received
13 July
of Sciences, AI. Lornikow 32146, 02-668
1976
The fluorescence spectrum of a three-level atom interacting The calculations are based on the quantum “fluctuation-regression contains five lines, the properties of which are discussed
The resonant interaction of an atom with a strong electromagnetic field has become recently a problem of great interest. The spectral properties of the field radiated by an atom containing a single pair of participating levels was discussed thoroughly by many authors [ 1,2]. It was found, that the fluorescence spectrum of such a system consists of three lines. Two of them are displaced from the central one by i dE, where d is the atomic dipole moment and E is the amplitude of the electric field. The heights of the side lines are equal and are three times smaller than the height of the central line. The half width of the central line is equal to i r, while the half width of the central line is equal to i r. These theoretical predictions were confirmed by experiments [4-61, which were performed on sodium where the two participating levels were 3 2P3,2 [F’ = 31 and 3 2S,12 [F’ = 21, The purpose of this letter is to discuss the more general case when three levels of an atom participate in the interaction with the field. It will be shown that the fluorescence spectrum in this case has some qualitatively new features in comparison to the model of the two-level atom. The model of the three-level atom is presented in fig. 1. The energy of the monochromatic field E(t) = E cos (w,_t) is equal to the energy difference between the atomic levels J = 0 and J = 1. The upper level is splitted by the perpendicular magnetic field into three sublevels: -1, 0 and 1. The splitting a can be tuned by changing the intensity of the applied magnetic field. The hamiltonian for the system of atom and field mJ
mJ
Warsaw, Poland
=
mJ
with strong electromagnetic theorem”. It is shown,
monochromatic field is obtained. that the fluorescence spectrum
3=0 Fig. 1. Energy-level scheme of a three-level atom. The energy difference between the ground state and the two interacting excited states is equal to w - R and w + a, respectively.
in the “rotating wave approximation” field is treated classically) is then:
“-Ii
+
- u(S+ e -iwLt
(the electric
jj
+s
_
,iwLt)
where u = i dE corresponding J=O~J=l,m=-landJ=O~J=l,m=l,
, to the transitions
=
=
The spectral distribution of the radiation can be calculated from the formula [ 1,3] 185
J(c&dt
i(w-wL)(r
,/ dt’C?+(t)S_(t’))c b
0
t’)
(1)
where T is the observation time. To this aim we shall adopt for our model the method proposed by Cohen Tannoudji for the two-level problem [I]. Two two-time correlation function can be writlen in the form: d+(t)S.
(t’)) = (S+(t)) (69~(t’)) + CSS+(t) SS~ (t’j)
The fil-st term of this sum describes the “elastic component” of the radiation. while the second one is due to quan turn fluctuations and describes the “inelastic component” which is our interest. To evaluate this two-time carrel;+theorem” [ 1.3]. According to this. the tion function it is convenient to use the quantum “fluctuation-regression differential equations for the system of two-time correlation functions are exactly the same as for the simple WCIages of the operators si. The equations for (si) or, equivalently, the Bloch equations for the elements of the atom ic density matrix p are well known and were used to study three interacting atomic levels in various configurations [7,8]. By choosing the proper combination of the elements of p it is possible to transform the Blocb equation\ into such a form d Vi/d t = ci Vi + Gi oii Vi, that their approximate solutions can be easily evaluated. The coefl‘icients ayij are in this representation functions of .Q, u and of the natural half width of rbe atomic levels 1‘. and are proportional to r. while the difference between the diagonal coefficients Ej is proportional to u. As far as we deal with a strong external field, u is much larger than r and the contributions from the terms ayiiI’; to the solutions of these equations are negligible. The time dependence of Vi can thus be written as: ~‘j(t)
=
exp [iEj(t
vj(t’)
tl)]
As we shall see, the real parts of Ej determine the positions of their maxima.
The operator &S+(t) can be represented tr [ ~jp] = Vi. Thus, CSi+(t)
63
(t’))
=
C
(‘j(Vj(t’)
the half widths of the spectral lines. while the imaginary parts denote
in the new base of the operators
6S
(I’))
r^;.which ale defined in such a way that
exp [iej(t ~~r’)]
i
Substituting this into eq. (1) and neglecting the transient effects by calculating all averages in the stationary (pst can be evaluated from the Bloch equations for dpjj/d t = 0). it is easy to obtain the spectral distribution the atomic radiation.
186
state 01‘
Volume
19, number
l/TJinel
2
- 2
OPTICS COMMUNICATIONS
y1 t (CZ2u2r/4M60)(w
[
t
1976
]4u6/M(6w)* 1YrJ 3 (a - %)* •t r;
[52*u4/M(60)*] __~~___
t
November
t6wyty;
(w--L
C
- oL t 6~)
[~2~41MW421y, - [r~*~*l‘W~_y)l (u-wL-6d _
___~~~~~_
(u-~L-6w)*ty: [Q2u4/M(Sw)*]
t_-m_~m
(o-cd-Fcl$
r
y7 + (u41’/M8w)(w
[2~~/M(6w)~]
[ whereM=4v4
(w - wL
+y;
- wL t 260)
1
-.I
[2u6/M(ikJ:)*]y3 ~_._~_~
t
y2 + (R’u21’/2M6u)
- (ruqMsw)
(O-C+ tv*fL*
- 26w)2
(w-q t y:
tCL4.
The fluorescence spectrum contains thus 5 lines. The main properties of these will be discussed below: 1. The central line has a lorentzian shape centered atw=wL. The spectral properties of this line strongly depend on the ratio a/u. Its height is proportional to
_ 260) + when a/v < 1 and to i I when n/v > 1. The weights of the dispersion-like parts are f3I = RVr/4MFw,
The height of these two lines is proportional h* = (4LW/M)(6d (3u2 + a2)(6u4
h, = 4&/M@*
+ a*).
When Cl/v S 1,)~ 1 tends to zero, when CL/v< 1, h 1 tends to 1. The half width of this line, yu, tends to iI? for R/v < 1 and to I for CL/v% 1. Thus, when the energies of the participating levels become far from resonance or when the intensity of the incident light decreases, the height of the central line decreases and the half width increases. 2. The two lines centered at w = wL f 6w have a very complicated structure. Each of them contains two lorentzian parts and two dispersion-like components. The weights of the lorentzian parts are equal [(u = 52*u4/4M(u2 + a2)], but their half widths [yI and y2] are different. The asymptotic behaviour is, however, the same for y1 and for y2: they tend to :r
p* = 201 .
t 7v*fl*
to
t ny
+ 9u2!C12 + 04)
’
It tends to zero when R/v < 1 and also when a/v S 1. This means that these lines disappear far from resonance and in the case of “level-crossing”. 3. The two lines centered at w = wL + 2 60 contain also lorentzian and dispersion-like parts. Their weights are respectively, (Y= 2v6/M(6w)* and p= rv4/M6w. Their half width y3 tends to i r when ~l/u < 1 and to 1 when Q/v S 1. The height of these lines is proportional to
187
Volume
19, number
November
OPTIC’S (‘OMMUNI(‘ATIONS
2
197h
The presence of the dispersion-like componentb disturbs the symmetry of each side line about its central point. It is due to the fact, that the stationary atomic density matrix contains non-zero off-diagonal elements. However, the asymmetry is rather inappreciable because p is much smaller than 01(of the order of a few percent of the line width). The “elastic component” of the radiation has al91 been calculated.
For
C2 = 0
I/TJ,,(w) Ipip. 3. l~luorcaccncc spectrum 113. (a) C2 = 0. (b) Sl = I,.
of a three-level
For !2/v < 1 it tends to 4, for n/v S 1 to XIII. We see, that by the Zeeman tuning it is possible to change the spectral properties of the radiation. When 52/v increases from zero the height of the lines 3 dccreases, while the height of the lines 2 increases. Far from resonance all lines disappear: there exists only the “elastic component” of the radiation. When R = 0 (“level-crossing”) the spectrum is similar to that of the two-level case. There arc three lines in the spectrum, the distance of the two side lines from the central line is equal to 2v. The ratios between the heights and the half widths of these lines are identical as in the model of the two-level atom. In the general case, the fluorescence spectrum has been calculated numerically for some values of n/u and r/v. The results for r/v = i are presented in fig. 2
188
- t%‘l5(w
w, ) .
atom for I./V =
this line becomes the only component of the spectt um far from resonance and in the case of a weak incident field. I wish to thank Dr. J. Mostowski for many valuable discussions and suggestions.
References [ 1 1 C. (‘ohen-Tannoudji. Proc. Second Laser Spectroscop) Conf.. Mqkvc, I‘rance. 1975 (Springer, Berlin, 1975). 121 1I.R. Mallow. Phys. Rev. 4 188 (1969) 1969. 131 l~.R.~lollo\~,Pl~y~. Rev.A 12t1975) 1919. 141 lf. Walthcr. I’roc. Second I ;iwr Spectroscopy (‘ant.. Me&c. I‘rancc, 1975 (Springer. krlin, 1975). [S) I:. Schuda, C.R. Stroud, kl. Ilcrcher. J. Phys. H: At. Mol. Phys. 7 (1974) 198. [6] I:.Y. Wu, R.E. Grove. S. Ezekiel. Phyc. Rev. Letter> 35 (1975) 1426. [7] B.J. Feldman, M.S. I?cld. Phys. Rev. 5 (1972) 899. 181 R.G. Brewer and !<.I. llahn, Phys. Rev. A 8 (1973) 464.