Документ взят из кэша поисковой машины. Адрес оригинального документа : http://qilab.phys.msu.ru/papers/SPIE-3736-246-244(1999).pdf Дата изменения: Mon Feb 4 18:39:35 2008 Дата индексирования: Mon Oct 1 20:01:24 2012 Кодировка:

Fluorescence and absorption properties of a driven A-system
I. V. Bargatin. B. A. Grishanin. and V. N. Zadkov
Faculty of Physics and International Laser Center. M. V. Lomonosov Moscow State University
Moscow 119899. Russia

ABSTRACT
Our investigation is focused on obtaining analytical results describing the resonance fluorescence and weak probe absorption spectra of a driven A-system under the Raman resonance condition. For the saturating field limit and within the rotating wave approximation (RWA) the resonance fluorescence and weak probe absorption spectra are calculated for the case of unequal driving fields strengths and non-zero one-photon detunings. generalizing the previously known results.1 A novel contribution derived is the non-positive non-lorentzian part of the spectrum. originating from the specific quantum properties of the atomic correlation functions. This contribution is asymptotically small under given conditions and, therefore, determines behavior of the resonance fluorescence spectrum wings and the absorption spectrum in the vicinities of the driving fields frequencies. Beyond the RWA. the spectral structures due

to the off-resonant four-photon atomic excitations are calculated. They consist of a coherent and two broadened
spectral lines centered at the four-photon frequencies 2WL --w driving laser frequencies.

and 2 --

WL

respectively. where WL and

w are the

Keywords: Coherent population trapping. dark resonance. fluorescence and absorption in A-system

1. INTRODUCTION
The resonance fluorescence and weak probe absorption spectra provide a detailed information about internal dynamics

of an excited atomic system. Positions of the spectral lines in these spectra are determined by the eigenvalues of the dynamic Liouvillian. i.e. the characteristic frequencies of the systems internal dynamics. while their intensities are strongly dependent on the structure of the eigenvectors of the Liouvillian. This sensitivity of the spectroscopic properties of an atomic system has once provided one of the most precise corroborations of the quantum theory of light-atom interaction.23 In theory. we can relatively easy calculate spectroscopic properties of a two-level atom (TLA) and analytical results were published almost for any set of parameters in the literature.23 Similar calculations for a multi-level atom prove to be much more complicated. making the derivation of the analytical results almost infeasible. Threelevel atoms being an intermediate case between the TLA and multilevel atoms, allow an analytic insight, thus helping us to understand the behavior of more complicated systems. One of the most intriguing effects in the three-level case is the coherent population trapping (CPT), which is most conspicuous in the A-system (Fig. 1). when the only

transitions between the upper and each of the lower levels are allowed in the dipole approximation.4 Under the

Figure 1. Parameters of a A-system driven by two laser fields with the frequencies WL and 4. y. -y'. and 19 are the population decay rates F, F', and F12 are the dephasing rates. and w is the incoherent pumping rate to the level

I')
SPIE Vol. 3736 · 0277-786X/99/$1O.OO

246


Raman resonance condition the total population of the system is trapped in the coherent superposition of the two lower levels. which does not interact with the driving fields. The fluorescence and absorption in the system are strongly suppressed in this case. which gives rise to another name of this phenomenon. the dark resonance. In this paper, we derive analytical formulae. describing the resonance fluorescence and weak probe absorption spectra for the case of unequal driving fields strengths and non-zero one-photon detunings. which are general case of the previously published results.'56 We investigate also a novel contribution to the spectrum. that is the nonlorentzian dispersive part of the spectrum. which appears as we introduce a first order correction in the approximation of the saturating field limit . This contribution despite its asymptotically small value under the given conditions determines the behavior of the spectral intensity of the fluorescence for large frequency offsets. changing drastically

the usual lorentzian dependence 1 o 1/zw2 to a more drastic T x i/zw4 in the absence of the inhomogeneous
broadening. Dependence of the weak probe absorption in the vicinities of the driving fields frequencies also takes a consistent form as the lorentzian and non-lorentzian parts compensate one another so that the total absorption vanishes. which reflects the CPT in the system. We calculate also in the first order correction to the rotating wave approximation (RWA) the spectral structure of the four-photon fluorescence. that appears at the four-photon laser frequencies 2wL --w. and 2w -- . respectively ( Fig. 2). These contributions have the same structure as the oscillations in the RF-range. corresponding to the transition between the two lower levels. but is more easily observed in the optical range offrequencies. We show that the four-photon fluorescence spectrum structures consist each of one coherent and up to five (under resonance conditions) broadened lines. Its intensity is lower than the RWA fluorescence spectrum intensity by factor ofgy/F122. where gA /g2 + g'2 is the generalized Rabi frequency and g. g' are the Rabi frequencies of the driving fields.

L

2. LIOUVILLIAN OF THE DRIVEN A-SYSTEM
Hamiltonian of the driven A-system is given by (we assume that level 1) has zero energy):

-/ = W12

2) (21 +hwi3 3) (31 hy

cos(wLt + ) 1)

(31

+?g' cos(w + a') 2) (3! +h. C.:
.

(1)

where the Rabi frequencies of the driving fields are expressed in terms of the fields amplitudes. AL
dipole transition matrix elements. d13. d23. as: g = picture. with the transformation Uo(t) = exp Hamiltonian takes the form:

A

. and

d13AL and g' = d23A . respectively. In the interaction [_(i/h)ot] where o = hwL 13) (3 --h 2) (21. and in the RWA the
(2)

Ah [_I3)(3I+8R2)(2I+I1)(3I+I2)(3I+h.c.]

where S = WL Wi3 iS the one-photon detuning and w -- WL -- Wi2 is the Raman detuning. which we will assume for simplicity to be equal to zero. The latter means that the two-photon resonance is to be broad enough to account for all the active atoms. This is possible when exciting system fields are strong enough. so that the power broadening mechanisms7 could overcome the residual Doppler broadening. Liouvillian of the A-system in the form
--2y

Adding the relaxation part of the Liouvillian to its undamped part. LA = (i/Ti) [WA. ].
0

we

can rewrite the

o o
0
-- LRWA --

712
W

Y12

0
0
--F12

0 0
0 0
--F12

g// 0 0 --g// 0 g'/'/
0 0 0 0

--w

0
0

0 0 0 0 0

--g/\/ g/\/
0

--g'/\/

g'/\/ g/2

0 0 0 0 0

0 0

g'/2
0

g'/2 --g'/2 --F 0 -- 0 g/2
0

0 --g'/2 0
5 --F

0 0 --g/2 0
0 --F
--S

--g'/\/
--g/2 0
0

(3)

0
S
--F

0

0 0

247


3

2cЭL-coL

Frequency

Figure 2. A schematic representation of the fluorescence spectrum of the driven A-system. including the four-photon spectral structures centered at the frequencies 2WL --w and 2w --L· respectively. The intensity ofthe four-photon spectral structures shown in the figure is magnified by a factor of two orders of magnitude to show these structures on the same graph with the regular resonance fluorescence spectrum structures centered at the laser frequencies WL

and w.
with the use of the following basis: {Йk} = {13)(31 1)(1I 2)(2. (I1)(2I+2)(1J)/. --i(I1)(21-- 2)(1I)/. (i') (31 + 3) ('I)/ -i(I1) (31 - 3) (12) (3f + 3) (2I)/. -i(12) 3f - 3) (2I)/}. We neglected here the population relaxation rate 'y12 and pumping rate w of the dipole forbidden transition 1) -+ 2).

('l)/

3. CALCULATION OF THE RESONANCE FLUORESCENCE AND ABSORPTION SPECTRA
The resonance fluorescence spectrum can be calculated as Fourier transformation of the corresponding atomic correlation function89: (r) = (0S(ot) &(t){S(t. + r)fr(t + r)]). (4)
where is the initial density matrix. & and &+ are the Heisenberg transition operators. oscillating at the laser frequencies. The evolution superoperator S(t. t + r) within the Markov approximation has the form:

S(tit2)=Texp
where

Itl

ft2

ё(r)dr

T is the symbol of the normal time ordering. If the Liouvillian parameters change much slower than the atomic relaxation rates. then the density matrix in Eq. (4) is simply given by the zero eigenvector of the Liouvillian (3). The correlation function then takes the form:
KRWA(r) =

[c( -- iWL)r + C3E(k -- i)r]

with the intensity coefficients given by

c3
where" .
denotes

= (O&j k))(kf&),

C3 = (Ol&. Ik))(kI&).

(6)

the 3x3-matrix multiplication; Ak 1k), and (kl are the eigenvalues. ket and bra eigenvectors of

248


the Liouvillian (3). Fourier transformation of the correlation function (5) gives us the fluorescence spectrum J():
8

F(w)=2e.
k=O

13 k

C23 k WL) + k z(w -- WL) + Ak

·

2V
-

(eAk)2 + (w --w -- (mAk))2

m3(wL _ -- mAk) --CAkeC3 _ w -- mAk)2+(eAk)2 + (WL _ W -- (Ak)2 + (WL m3( -- w -- mAk) --eAkeC3 +
(eAk)2 + (W'L -- -- mAk)2

(7)

The probe field absorption spectra can be calculated then by analogy as the absorption probability of a photon from the weak probe field and is also determined by Fourier transformation of a superposition of certain atomic correlation functions'9:

P(w) = gA(w) A(w) =f
where gpr 5 the Rabi frequency of the probe field and
C(T)

C(r)e2Tdr.

(8)

= ([+

(t)

u (t + r)]) = (+ () (t + r)) - (a (t + r)a (t)).
8

For the steady-state case we therefore come to the following absorption spectrum form:

A(w) = 2g e p

.

D13 k
WL) + Ak

D23 k

2 2k--O pr \
-

k=O

(w --

--eAkeD3
(Ak)2 +(WL _ -- mAk)2

+

m3(wL _
(eAk)2 +(WL _

)

+ Ak
--

mAk)

--

(9)

--eAkeD3 + (eAk)2+(w --w --mAk)2 (eAk)2+(4 --() -- mAk)2
rith the coefficients D3 (i = 1. 2) given by

D3 = (O(&j . Ik))(kI&)

D3 =

(o&

.

Ik))(kI&) - (OI& Jk))(kJ&).
.

-

(OJ& . Ik)XkI&)

(1)

Calculation of the resonance fluorescence spectrum due to the four-photon interactions. beyond the RWA. is much more complicated and is given in detail in Ref. 5. With an extensive use of computer algebra we were able to generalize the formula of Ref. 5 for the radiation correlation function due to the four-photon interactions (see also Eq. (4)):

K49h(T)

= [2
k=O

.

m2g

'2

42 \

&+ . 12

k'k --\ --i(L+)T //\ l2/

1

(11)

where m1 = (eLd;3)2/(e d;3)2 and m2 = (e d13)2/(eLdl3)2 characterize the efficiency of the off-resonant excitations

and L =

WL.

From Eq. (11) one can clearly see that the novel resonance fluorescence structures are centered at the four-photon frequencies 2w --WL and 2wL --wi. Comparing the pre-exponential coefficients in Eq. (11) with those ones in Eq. (5) we may conclude that the four-photon resonance fluorescence structures are similar to the spectrum of oscillations

of the 1)

2) transition in the RF-range. which is. however, very weak as the transition is forbidden in dipole approximation.

249


4. ANALYTICAL CALCULATIONS RESULTS
To proceed with the calculations according to the Eq. (11). we need first to calculate the eigenvalues and eigenvectors of the Liouvillian (3). This. however. proves to be unfeasible unless we can apply the saturating field approximation. In this approximation we can decompose the original Liouvillian into two parts. one of which. the "unperturbed' Liouvillian. includes only the Rabi frequencies and the other one ёRWAIgg'O can be treated as the perturbation. Then. representing the results as a series expansion for l/g. we received with the help of computer algebra the first order corrections. which generalize the zero-order results of Ref. 1. The details of our calculations are given below.

4.1. Resonance Fluorescence Spectra in the RWA
Assume for simplicity that F12 << F. which is easily satisfied in many cases. With this assumption the eigenvalues of the Liouvillian (3) in the first-order approximation read:
A0 = 0.
. _ --F. A4 = --g 4 2 2

.gA+c F A5=i
2

.g--8 F 2 2' A6z 2

y = ----, "2 2

--I'. \3 =

. 37 zgA --

--

3y I, ------

A7=z

.--gA+8 F

A8z.--g-2

4

F

(12)

The resonance fluorescence spectrum in the RWA (Eq. (5)) is determined then by the intensity coefficients (6). which are listed below:
Co
13
23 13 =Co =C1

c3 =
13 C6 = '12
13 C8 =

c3 = 12 cos2 sin4

2y

cos4

"12 4 4 13 23 =C1 =0, C2 =--cos sin . C =--cos asin . i(2f -- 117) sin2 1 + j3 (Ca)*. C3 C3 0.
.

7

73

J'12

2

.

·

1 + z(2F--11)
i(2F -- ) -- S
YA

.

C3 (Ca)*. C3 C3 0,
23 23

(13)

l2
y

y

cos4 cos4

2 cc sin Г 1 --
.

.

C6
--S
.

=

F12

sin

2

1+

i(2F -- y)
YA

C8

= F'2

7

cos cos

2

sin4 ( ( c1
\\

--S -- ____________ -- i(2F y)

2

sin4
.

( (
\\

1+

i(2F

--

y)

--S

YA

cos p = 9/YA and sin = g'/gA. It is important to note that the total fluorescence intensity is proportional to the ratio F12/'y << 1. which is typically lies in the range of 10_2 and manifests therefore the CPT in the system. By contrast with the Mollow-triplet in the resonance fluorescence from a TLA. one can easily see from Eqs. (12) that the resonance fluorescence structures from the A-system consist of five incoherent lines* . If one-photon detuning a_ is not equal to zero. the inner. i.e. shifted by gA/2. lines acquire the additional shift --5/2 and become of unequal intensity. destroying the initial symmetry of the spectrum (Fig. 3). Thus the A-system proves to be more sensitive than the TLA not only to the Raman detuning. which determines the CPT resonance. but also to the one-photon
where

iO

detuning. The latter differs from the TLA case. where the corrections appearing due to the inexactness of the
resonance are only of the second order. The five spectral lines manifest different dependencies on the fields intensity distribution determined by the parameter . When one field is much stronger than another one. the fluorescence of the strongly driven transition converts to the Mollow-type triplet. while the fluorescence from another driven transition has the Autler-Towns'2 structure. which consists of the two other lines of the complete spectrum (Fig. 4). A surprising fact is that the intensity coefficients (13) are not real anymore. Their imaginary parts determine the non-lorentzian part ofthe spectrum (Fig. 5a). which. despite ofits asymptotically small value in the saturating field limit, determines several important features of the fluorescence spectra. Namely. in the absence of elastic dephasing of the dipole allowed transitions (y = F) , the dependence of the fluorescence spectral power density for large offsets zw changes from the usual lorentzian one. F(Lw) cx 1/Aw2. to a more drastic dependency. .T(zw) x i/z\w4. as the lorentzian and non-lorentzian parts cancel one another. For the coefficients determined by Eqs. (13) the asymptote for large offsets is given by:

= 2F5Fi2sin2cos21
where F = F -- is the elastic dephasing.

0 ().

(14)

*In general, for the case of off-resonant excitation, resonance fluorescence structures consist of 7

spectral

lines.'0"

250


(13
cI?·o

.

--.

1(1 u.&n.ipad L______ __ O2I7UiO1-UON IO(LW110UOD
F

(q
p
oI,_o .cO_o

-- --- -

[flJ wuipodS
UrZLQ1cq-uo

4nqi;uoJ

p
S
'rI

ot,o
coo
.oE'0

S
'Dl

,00 coo

-coo C C
CD

-000

0t'0
CD

-cro
-01.0

cto
ot_0 coo

C

-coo

'Dl -0o0 01-

'TI .oo'o

0

01

00

00-

01-

0

01

00

Auonbj SJJ
C

Aunbi psj;

i/(1coo)

pu pj

=

3U35JOflU

T/

pzqiu

isuui
Tqi

=

JTOO0 y

uinipds ioJ oz-uou uo4oqd-uo uTun4p -g qj sioniid psn Vfl = = = j-- () y = (q) iuuj jipds suq iinb opppn unbij Iqs uui suq hIM oJpun snouaomoqui ajddoj uiupDloJq ji quiui&s snqj

j

p

/g

V5

si

i4JoT4T j

pow

si

coo (13
C

-- ._---

1ED!

unipd

urz;uazol-uoN

uo!nqu;uo
F coo

(q

-- -- --

fn1 unJpd

UiZUJOI-UON uoqnqrquo

00'0
S
'Dl

CD

0I0'0

C
s000
CD

CD

C -1 0000
0Z 01-

0 C

unboij

sjy

0

01

OZ

00

0l-

01-

0

01

OZ

icuonboij

osjjo

YMU ioj inbun uiiip sppj siTisuui daind) pui qod spg usionjj uioij uoTpsui ppx cq duirid ppj sq qoid (ppi.j 4T uiorj Jqoui uoTTsu1J4 pTx Aq qoid T?Idu4-MoIIoT,JJnpnqs siqiqx suMo1-JIny inpnqs TpTqM SsiSuo3 Jo uTuiiwi suq jo T44 4TduIo3 innipds sJTm1nd psn = J Y= 0 = JTOOO = V5 = JOT () = O/6 (q) JT4M duind
U144

j uuosj uzsionjj pj si iow ?AISU4Ui UMOU)-pM

iunpds ui q4

J

I'q o qj usionj 0/

qj pj


---- Full Spectrum

-- -

Lorentzian Contribution

- -- Non-lorentzian Contributi
CQ4

--a--- Full Spectrum Lorentrian Contribution -- -- Non-lorontzian Contribution

--
3

a3

-10

0
Frequency Offset

10

-10

0

iO

LV"

Frequency Offset

Figure 5. The resonance fluorescence (a) and probe field absorption (b) spectra of the driven A-system for the case of exact resonance. The spectral lines are not fully resolved here. but the non-lorentzian part reaches nearly its maximum relative value. The graphs clearly show that the non-lorentzian contribution fully compensates the
lorentzian one for the cases oflarge offsets in the fluorescence spectrum and small offsets in the absorption spectrum. The parameters used are g = 51' y = IT. = ir/4. S = 0. F12 = 0.00ff.

4.2. Probe Field Absorption Spectra
Calculation of the probe field absorption spectrum (Eq. (9)can be done by analogy with the calculation of the fluorescence spectrum and one can readily find that among the coefficients (10). which determines the absorption
spectrum, only the following coefficients are non-zero ones:
D5
13 13
. -- sin2

2

Г 1 + iF+6
1--

D7 = -- sin 2

.

2

g + g

D5

23 23

--

2
--

cos2 cos2

1 + _____
1--

6

D7 =

g +
g

5

15

2

from the This means that the probe field absorption spectrum structures consist ofonly two lines shifted by laser frequencies (Fig. 5b). The inexactness of the resonance will also lead to the additional shift of the lines 672.but in the opposite direction. The non-lorentzian part here plays an important role once again. In the absorption spectra the non-lorentzian part compensates the lorentzian one for the small offsets. so that the absorption is practically vanished at the laser frequencies. which is just another manifestation of the CPT. the self-induced transparency.4

4.3. Resonance Fluorescence Spectra due to the Four-Photon Interactions
The fluorescence structures due to the four-photon interactions are determined by the following coefficients in Eq. ( 1 1):
. I-clf _ m1g cos cc sin k-Jo 2
2

C1 =
C7

cos2 cc sin4 cc (i _

=

mgcos cc 2

1 4 sin cc 1 +
.

if )\
zf+

2 ,--hf m2g 2 cos Li0 --

sin4cc.
16)

.

=
C7

cos4 cc sin2 cc (i + fg 6)

hf =

m2g 4 cos

-Г-

cc sin cc

.2

1

1--

+6 _____

if

the rest of them being equal to zero. The fluorescence spectrum due to the four-photon interactions consists of two spectral structures centered at the four-photon frequencies 2WL -- w and 2w -- wL respectively. Each of these structures consists of a coherent and two broadened lines, which have the same shape as the broadened lines in the absorption spectra (Fig. 6). The

252


80

a)

80 -

-

- ---- Non-Lorentzian Part

Full Spectrum

b)

-- Full Spectrum - - - -. Non-Lorentzian

Part

60

60-

3

3
40

20-

C

-30

11·11· I
-20 -10 0

C

020
30 -30 -20 -10 0
10

10

20

30

Frequency Offset (w -w )/F 4ph

Frequency Offset (w -w )/i-' 4ph

Figure 6. The fluorescence structures due to the four-photon interactions for the case of non-zero one-photon detuning S. The broadened spectral lines acquire additional shift their intensities are also different. These
graphs, except for the coherent line, are absolutely identical to those of the absorption spectra in the case of inexact resonance. The parameters used are g = 201'. -y = F. a = 7r/4. F12 = 0.0011'; 5 = 51' (a). S = --51' (b).
0,35
0,30Numerical calculation FirstOrcier Approximation - - - . Zero-Order Approximation

.

-

'0.25
' 020LD

Q)

0.15

C)

0 C) 0
C)

0,10
0,05

c

10

15

Frequency Offset (0-oL)/F

Figure 7. Comparison between numerical and approximate analytical calculations. Taking into account the first
order corrections. we obtain a very good agreement with the results of the numerical calculations. even if the condition gA >> F is not very well satisfied. The parameters used are YA = 51', -y = F. Г = ir/4. S = 0. F12 = 0.0011'.

four-photon fluorescence does not saturate. its intensity grows with the increasing strength of the driving fields as long as the condition g <

253


5. CONCLUSIONS
In this paper we obtained a novel contribution to both the resonance fluorescence and weak probe field absorption spectra of the driven A-system. This contribution is due to the first order corrections in the saturating field limit approximation. With an extensive use of computer algebra we received analytical formulae. which are in good agreement with the numerical calculations (Fig. 7). Analyzing our analytical results we revealed some new properties of the spectra. In our first-order approximation model only the inner lines acquire an additional shift in the case of inexact resonance. making them more sensitive to the Doppler broadening. The non-lorentzian part. though small under given approximations. strongly modifies the behavior of the fluorescence spectra for large offsets and absorption spectra for the small ones.

The four-photon interaction processes cause appearance of the four-photon resonance fluorescence structures centered at the four-photon frequencies 2w -- and 2w -- L. Their spectrum. calculated to the first-order

corrections to the RWA. consists of a coherent and two broadened lines and has the same structure as the oscillations in the RF-range, corresponding to the transition between the two lower levels. but is more easily observed in the optical range of frequencies.

L

ACKNOWLEDGMENTS
This work was supported by Volkswagen-Stiftung (grant No. 1/72944) and by the Russian Foundation for Basic Research (grant No. 96-03-32867). V. N. Z. also thanks the Alexander von Humboldt foundation for its support.

REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. A. S. Manka. H. M. Doss. L. M. Narducci. H. Pu. and J. L. Oppo Phys. Rev. A 43. p. 3749. 1991. W. H. Louisell. Quantum Statistical Propertzes ofRadiation. Wiley. New York. 1990. L. Mandel and E. Wolf. Optical Coherence and Quantum Optics. Cambridge. New York. 1995. E. Arimondo, "Coherent population trapping in laser spectroscopy" Progress in Optics 35. pp. 257--354. 1996. B. A. Grishanin, V. N. Zadkov. and D. Meschede J. of Exp. and Theor. Phys. (JETP) 113. p. 144, 1998. B. A. Grishanin, V. N. Zadkov. and D. Meschede Phys. Rev. A . (in print). P. L. Kelley, P. J. Harshman, 0. Blum, and T. K. Gustafson J. Opt. Soc. Am. B 11. p. 2298. 1994. B. R. Mollow Phys. Rev. 188, p. 1969, 1969. B. R. Mollow Phys. Rev. A 5. p. 2217. 1972. C. Cohen-Tannoudji and S. Reynaud J. Phys. B 10, p. 345, 1977. C. Cohen-Tannoudji and S. Reynaud J. Phys. B 10. p. 2311, 1977. S. H. Autler and C. H. Towns Phys. Rev. 100. p. 703, 1969.

254