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Theory of frequency-modulation spectroscopy of coherent dark resonances
Julia Vladimirova, Boris Grishanin, Victor Zadkov, Valerio Biancalanat, Giuseppe Viadimirova, Bevilacquat, Evelina Breschit, Yordanka Danchevat, and Luigi Moit International Laser Center and Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow, Russia tDepartment of Physics, University of Siena, via Banchi di Sotto 55, 53100 Siena, Italy

ABSTRACT
Theoretical model for the frequency-modulation spectroscopy of dark resonances is discussed in detail on example of a three-level quantum system in A-configuration driven by resonant laser field(s) with and without frequency modulation using two simulation techniques--the density matrix and quantum trajectories analysis.

Keywords frequency modulation spectroscopy, Keywords:: frequency modulation spectroscopy, dark resonances

1. INTRODUCTION
The coherent population trapping (CPT) phenomenon is currently widely used in different applications such as magnetometry, metrology, and others [1--6] . It is most conspicuous for the A-transition between two closely spaced long lived levels optically coupled to a third distant short lived level by two continuous coherent radiation fields (Fig. 1). In absorption spectra, coherent superposition of closely spaced levels fields (Fig. 1) . In absorption spectra, coherent superposition of closely spaced levels leads to a very narrow dip of induced transparency or, equivalently, to a non-absorbing dark resonance when resonance fluorescence is
observed. The basics of CPT phenomenon are well understood in the frame of three-level analytical model [7] . For the case of multilevel systems, however, the simple model has to be significantly complicated and analytical results in most cases became impossible [8]. Enriched energetic structure of multilevel atoms, especially in the presence of an external magnetic field, also results in essential modification of the resonance dependencies on driving fields

parameters. Despite the conventional experimental technique for observing the dark resonances spectra with the use of two resonant laser fields described above is now widely used for many applications, still there is a need in elaborating simpler experimental techniques, which would, for instance, employ only one laser field, but with frequency modulation (FM), which also allow the spectroscopy of dark resonances in multilevel atoms. Such experiments are conducted by the group of Prof. Luigi Moi at the University of Siena in Italy [9] and they, in fact, initiated the current theoretical study of interaction between the three-level system in A-configuration with the frequency-modulated laser field. In a typical experiment on FM-spectroscopy of coherent dark resonances of Cs atoms, the atomic media is placed in a homogeneous magnetic field, whose value is in the range of few 10 juT. The coherent resonance is observed when the laser emission, which is frequency modulated, contains in its spectrum a number of frequency components that are in resonance with the atomic levels, i.e., the frequency difference between these components matches the Zeeman splitting w12 of the ground-state sublevels (Fig. 1) due to the presence of dc magnetic field. The laser spectrum is tailored via diode laser frequency modulation obtained by direct modulation of the laser junction current. The coherent structure can be then seen scanning the modulation frequency in a small range around the two-photon resonance condition, or at a fixed modulation frequency scanning the magnetic field in
the corresponding range [9--12].

Despite obvious simplicity of this method, analysis of the spectrum becomes a separate problem as a theoretical model for the FM-spectroscopy of dark resonances does not exist to our knowledge so far.
E-mail: vyulia©comsiml .phys.msu.su
ICONO 2005: Nonlinear Laser Spectroscopy, High Precision Measurements, and Laser Biomedicine and Chemistry, edited by Sergey Bagayev et al., Proc. of SPIE Vol. 6257, 625704, (2006) 0277-786X/06/$15 · doi: 10.1117/12.677920 Proc. of SPIE Vol. 6257 625704-1


mF= mF

Figure 1. A systems formed of the Zeeman sublevels in the F9 = 3 --+ Fe 2 transitions excited by the sublevels in the F9 = and a sublevels components of the respective frequencies Wi and W2,whose difference is equal to the splitting of the Zeeman sublevels respective frequencies Wi W2 , whose difference is equal to the splitting with /2mF = 2. Three-level system in A-configuration has the following parameters: WL1 , WL2 are the frequencies of the A-configuration has the following parameters: WL1 wL2 laser fields driving transitions of the system; 113 , Q23 are the respective Rabi frequencies; 5L is the frequency detuning frequencies; 5L is the frequency detuning fields driving transitions of the system; 13, Q23 rates from excited state 3) onto the low-laying Ii) and 12); from the 1) +-+ J3) transition; 731, 732 are the decay rates from excited state 13) onto the low-laying levels 1) and 12); the 1) Â+ J3) transition; pumping rates of the level Ii) via the level 12), correspondingly; F31 , F32, and 12 and 712and w are the decay and pumping rates of the level 1) via the level 12), correspondingly; F31, F32 and F12 are the dephasing rates for the transitions Ii) ++ 13) and 12) -+ 13), and Ii) -+ 12), respectively. ++ 3) and 2) ÂÂ 13), 12), respectively.
,

In this paper, we present a detailed study of the dynamics of a three-level quantum system in A-configuration driven by resonant laser field(s) with and without frequency modulation using two simulation techniques--the density matrix and quantum trajectories analysis. The paper is organized as follows. General theoretical background for both density matrix and quantum trajectories analysis used for numerical simulation of the fluorescence spectrum of the driven A-system is given in Sec. 2. In Sec. 3 we consider physical quantities that are of interest and can be measured experimentally. They include the total fluorescence intensity in equilibrium and in the transient response, resonance fluorescence spectrum, linear and nonlinear absorption coefficients, and the refractive indices. In Sec. 4 two considered above computer simulation techniques are applied for the analysis of a driven A-system in both cases, when the frequency modulation is switched off (Sec. 4.1) and on (Sec. 4.2), respectively. Calculations are made for the fluorescence intensity. Finally, the conclusions are summarized in Sec. 5.

2. THEORETICAL ANALYSIS OF TEMPORAL DYNAMICS OF A DRIVEN A-SYSTEM
Theoretical analysis of a driven A-system can be done both analytically and with the help of various computer simulation techniques. Analytical approach that can be developed, for instance, with the help of diffusion approximation of atomic system under broadband excitation, have obviously rather limited application just to the simplest case of a model three-level atom and cannot be easily extended onto the case of multilevel atoms in an external magnetic field (for example, Cs or Rb atoms) which are of practical interest. It will be discussed elsewhere in a separate publication. In this section, we will focus on the numerical simulation techniques, which can adequately be used for simulating fluorescence spectrum of both a model A-system and a multilevel atom driven by the frequencymodulated laser field(s) . With these notes in mind, there exist two key computer simulation techniques suitable for our purpose, namely, the technique based on the solving the master equations for the density matrix (we will call it the density matrix analysis) and quantum trajectories technique (we will call it the quantum trajectories
analysis) {13--16}. analysis)

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Note that the results of the density matrix and quantum trajectories analysis have to be identical within a certain error, which depends on how many trajectories we calculated and averaged over and how precisely we solved master equation and for how long time. In application to the analysis of multilevel atomic systems, the key difference between these two techniques is as follows. The density matrix technique is used primarily for analysis of atomic systems with rather limited number N of energy levels because the number N2 of master equations describing the system can became too large for real multilevel atoms (like Cs or Rb) and, therefore, such analysis can require relevant computer resources. By contrast, the quantum trajectories analysis of the multilevel atomic system with large number N of energy levels requires computational resources proportional to N and, therefore, has an advantage here. Despite this difference, each of the two techniques has its own advantages and drawbacks, so that we intentionally consider both of them to clarify which one is better and in which situation.

2.1. Density matrix analysis
We will start with the density matrix analysis, when the dynamics of a quantum system is described with the density matrix time dependence of which is defined by the following kinetic or master equation:

= -{ft,] + ¸re1ax, -[ft,] ¸reiax,

(1)

where first term in the right part of the equation describes reversible dynamics of the system with the hamiltonian H and second term--nonreversible dynamics of the system due to the stochastic interaction of the system with the reservoir, which is described with the relaxation superoperator ¸reiax. In assumption that interaction of the

system with the reservoir can be described with a diffusion type process, the relaxation superoperator can be presented as the averaged over the reservoir noise of the secondary commutator with the hamiltonian H of
the system--reservoir interaction.

Then, the relaxation term ¸relaxP in master equation (1) can be written in the general Lindblad form [17] as
¸relax ¸reiax
(2)
m

m m

where ® is the substitution symbol to be replaced with the density matrix 5, operators Cm describe interaction Om with the reservoir, operators C are conjugated to the operators Cm and both of them have dimension (1/t)'/2. It is worth to note that representation (2) preserves interpretation of j5 as the density matrix, i.e., we should representation (2) preserves interpretation of 3 should fulfill the condition Tr = 1 and the probability ('bJ ,Ò 'b) of finding the system in any state 'b) must be positive b) positive or equal to zero at any time moment and for any j3(t = 0) . Note also that Eq. (2) does not require that Cm must zero at any time moment and for any 3(t = 0).
be defined uniquely.

In Eq. (2) the first sum is the anti-commutator, which consists of the terms decreasing the total population Tr5; Tr3; the second sum contributes to (d/dt)Tr3. The number of operators Cm in Eq. (2) in general case can be (d/dt)Tr5.
rather large because each operator corresponds to a specific decay channel. For the case of spontaneous emission in a two-level system, for instance, there exists only one operator because we consider only one decay channel,

F112&, where & = i.e., spontaneous decay. This operator has the form C1 = r1/2&, where & = g) (el is the atomic transition operator from the ground state g) onto the excited state Ie) . More examples are described in Ref. 16. the excited state le). More For the case of a A-system, in order to write the operators Cm one has to take into account the following spontaneous decay from the excited state 3) incoherent pumping of relaxation processes: spontaneous decay from the excited state 3), incoherent decay and incoherent pumping of the two ground levels 1) , 2). Operator Cmin this case will consist of four terms responsible for the above listed ground levels 1), 2). Operator C fl thi5 case will consist of four terms responsible relaxation processes, namely, operators Ci and C2 describe spontaneous decay from the excited state 3) onto 1), the ground states Il), 2), operators C3 and C4 describe incoherent decay and incoherent pumping of the level I 1) to the level 2):

= (73i)h/2Pi3,C = (31)1/231; C2 = (32)h/2P23,C = (32)h/2p32. (31)1/213, = (3i)"2P3i; C2 = (32)V2P32; = (12)1/2p21 C3 = (72i)1/2pi2,C = (21)h/2p21; C4 = (12)1/2p21,c = (12)h/2p12, = (21)h/2P12,C (2i)1/2P21;

(3) (3)

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where Pkl are the transition operators, which are represented, in general case, by the matrices with the only non-zero kl-element Pkl(k,l) non-zero kl-element Pkl (k, 1) = 1. With the help of Eqs. (2), (3) the relaxation term Creiaxi5in master equation (1) takes the form: (3) the relaxation term ¸re1axj in master equation (1) takes the form:
f--ii2pii(t) + 'y21p22(t) + 'y31p33(t) f--ii2pii(t) + 721p22(t) + 731p33(t)
I
--O.5(y21 --0.5(721

+ 'i2)pi2(t) i2)pi2(t)

ti2pii(t) -- I-L12pU(t)--

'\

--O.5(y31 --0.5(731 + 732 + /i2)pi3(t) i2)pi3(t)

--O.5('y31 + 732 + Li2)pi3(t)\ --O.5('y31 + 732 + ii2)pi3(t)\ --O.5('y21 ii2)pi2(t) --O.5('y21 + + ti2)pi2(t) 'y21p22(t) y32p33(t) --O.5('y31 732 + 721)p23(t) 721p22(t) + 732p33(t) --0.5(731 + + 732 +721)p23(t) ) --0.5(731 ++732++y2i)p23(t) --O.5('y31 732 721)p23(t) --(731 + 732)p33(t), --(y31 +32)p33(t), J

(4)

whereY31 , 'Y32 are the spontaneous decay rates, Y12 and ,u12 are the decay and pumping rates of the level Ii)via where 731, 732 are the spontaneous decay rates, 712 and /12 are the decay and pumping rates of the level 1) via level 2), the level 2) , respectively. Now, master equation (1) can be integrated numerically for a given Hamiltonian H of the system and then ft of the system and then all its necessary characteristics can be modeled.

2.2. Quantum trajectories analysis
Another relevant approach to model the temporal behavior of a A-system is the quantum trajectories analysis [13--16], which uses instead of the time-dependent density matrix statistically-equivalent stochastic temporal dynamics of the wave function with the following averaging of the results by analogy with the Monte-Carlo method. Modeling evolution of the wave function on the discretization interval dt, which ensures identical results with i) modeling continuous the solution of the master equation (1) , includes two parts: 1) modeling continuous variation of the current state equation (1), and ii) modeling quantum jumps occurring randomly with certain probability. Let us assume that the system at time tt is in the state kb(t)). Then, the continuous variation of the current system at time is in the state fb(t)). Then, the variation of the current state can be described with the temporal dynamics of the wave function k11)(t)), which is governed by the k1'(')(t)), SchrÆdinger equation

(')(t + 6t)) (1)(t + st))
with the non-hermitian hamiltonian

(1 + dt) I(t))

(5)

HH>Ci!nCm. =ftOiLOm.
5Jj = >6Pm '5p = :: 8Pm

(6)

New wave-function is not normalized because the hamiltonian H is a non-hermitian one and the squared 6p norm of the function is equal to (b(1)(t + + dt)./.(t + dt)) == 1-- 6p, where 5p has the form: of the function is equal to ('b(')(t dt) b(t + dt)) 1 --

dt dt

m

m

((t)I CCm P(t)), ((t)(mOm (t)),

(7) (7)

where Sp where the time step dt must fulfill the inequality 6p << 1. Random behavior of the wave function is described with the probability Sp of quantum jumps. If the quantum wave function b(')(t 6t)) must jump does not occur with the probability 1 -- 6p, the wave function b(1)(t + St)) must be renormalized to unit Sp, and then it will be mapped with the corresponding normalized function kb(t + 6t)). When the quantum jump will be mapped with the corresponding normalized function I(t 5t)). occurs, the wave-function transfers into the state C kP(t)) related probability PmISP. Thus, occurs, the wave-function transfers into the state Cm kb(t)) with the related probability 6PmI6P. Thus, at the time moment t + 6t we do have one of the two normalized wave-functions: 5t
S

with the probability 1 -- 6p the quantum jump does not occur and kb(t + 6t)) = b(1)(t + 6t))(1 -- 5p b(1)(t 6t))(1 --

· S

with the probability 6Pm the quantum jump occurs and kb(t + St)) Om 1V(t))(SPmI5t)1"2 probability 5Pm the quantum jump occurs and II(t 6t)) = Om I(t))(6PmI6t)1/2

3.

WHICH PHYSICAL QUANTITIES ARE OF INTEREST TO CALCULATE?

Physical quantities that are of interest and can be measured experimentally include the total fluorescence intensity in equilibrium and in the transient response, resonance fluorescence spectrum, linear and nonlinear absorption coefficients, and the refractive indices.

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3.1. Spontaneous fluorescence intensity
The spontaneous fluorescence intensity is proportional to the excited-state population,
'fi ,.-) fl3 'fi

=

(3J

J3)

.

(8)

and its total intensity is proportional to the stationary value of the excited-state population:
tota1
(fl3) @13) = ((31 3)). ((31 3)).

(9) (9)

It It is worth to note here that a typical experiment spectroscopy of coherent dark resonances is carried out in a vapor cell. Therefore, every atom is spending a a finite amount of time interacting with the laser radiation before cell. Therefore, every atom is spending finite amount of time Tt interacting with the laser before it is depolarized for instance by wall collisions. Important is that an atom has a transient response to the laser

fields, or more precisely the time r it takes for an atom to reach the coherent dark state by optical pumping. Let us assume that initially atom is in an incoherent superposition of levels Ii) and 2). Then, due to the field Let us assume that initially atom is in an incoherent superposition of levels 1) and 2). Then, due to the field action onto the coupled state, it goes to the excited state. Finally, it reaches the uncoupled state due to the decay of the excited state, so that we find the excited state being significantly depopulated. Obviously the time T, which depends on the relaxation parameters, should be at least much shorter than the longest observation time Tt that can be achieved in the experiment. Within the Markov approach the inverse time distribution has
a Lorentzian shape.

3.2. Resonance fluorescence spectrum
The spectrum of resonance fluorescence of a driven A-system is typically derived from the correlation function [18] related to of the frequency-positive and frequency-negative operators of the complex amplitudes & , the corresponding Pauli matrices &, & that are thetransition operators 1k) (l with k 1 = 1, 2. Respective corresponding Pauli matrices & that are the transition operators 1k) k theoretical calculations have been made both in rotating wave approximation (RWA) and beyond it in Refs.

&

19,20.

3.3. Absorption
Absorption of the laser light transmitted through a vapor cell is characterized by the absorption coefficient c causing the reduction of an electromagnetic wave with intensity I = c0E E*/2 [20]:

=

___ = = 2I(Z)' ___

(10)

where w WL1 ,WL2. The polarizations P13(WL1), P23(wL2) acting on laser fields with frequencies and amplitudes ELi,, WL1 and EL2 , WL2are given by ELi WL1 and EL2, WL2 are given
P13(WL1) P13(WL1)

=

N0d13

Ii)

Ii) (SI,

P23(WL2) P23(WL2) = N0d23 2) (31, 2) (31,

=

(11) (11)

where

and d23E(wL2) = 2h1123 we find that d23E(wL2) = 2h123

N0 is the number of atoms per unit volume. Making use of Eq. (10,11) together with dl3E(WL1) = 2h113

(WL1) = (wL1)

=(i3NÝ) m(J1) (°) m(I1)

(31), (wL2) = (3J), a(wL2) =

(°) m(J2) m(I2)

(3J).

(12)

3.4. Refractive indices
Real part of the refraction index n = n' + in" can be calculated by analogy with the absorption coefficient from the refraction index n = + in" can be calculated by analogy with

n'-1= d(z) = -1= d(z)
where /c is the wavevector, yielding

2I(z)' 2I(Z)'
n'(wL2) -1= (°)
fl'(WL2) -1= (hc3No) e(J2) (31). e(J2) (3).

(13)

n'(WLl) -1= fl'(WLl)
'Li, 'L2

e(J1) (3J), (hc2No) e(J1) (3J),

-1= (°)

(14)

The dispersion coefficients are nonlinear ones since the density matrix elements depend on both laser light fields, dispersion coefficients are nonlinear ones since the density matrix elements depend on both laser light fields,

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4. MODELING FLUORESCENCE FROM A DRIVEN A-SYSTEM 4.1. Case when the frequency modulation is switched off
Before studying interaction of a A-system with the frequency-modulated laser field, let us first analyze a more simple problem--how the A-system interacts with a resonant laser field without frequency modulation.

4.1.1. Density matrix analysis
Let us consider a three-level quantum system in A-configuration, which interacts with the field E(t) = E0 cos(wot+ cos(wot+ Ão).The interaction hamiltonian has the form: Ão) . The interaction hamiltonian has the form:

1. =

h(O h/U 0 (o
(

\113 23 òL,J

0 113\ 13\ --W12 123 J --W12

(15)

w13 is the one-photon frequency detuning of the probe laser where 13 , 123 are the Rabi frequencies, 6L = w0 -- Wi3 15the one-photon frequency detuning of the probe laser field from the resonance, w12 is the frequency shift between two ground levels (see Fig. 1). Substituting the relaxation operator (2) in Eq. (1) we receive the following set of differential equations for all density matrix elements:

p1(t) = --/-Li2pii(t) + y2ip22(t) + i(1i3pi3(t) -- 1i3p3i(t)) + 'y31p33(t), y3ip33(t), --i2pi1(t) 123pi3(t) p2(t) = --Fi2pi2(t) -- i(w12p12(t) + 1123pi3(t) -- 1113p32(t)), --F12pi2(t) --

pÃ3(t) = --F13p13(t) + i(Ili3pii(t) + 1123p12(t)+ 5Lp13(t) -- p3(t) = --F13pi3(t) i(Ii3pii(t) 123pi2(t) + 6LP13(t) --F21p21(t) i(wi2p2i(t) + 1i3p23(t) -- p1(t) = --F2ip2i(t) + i(w12p21(t) + 1113p23(t) -- 23p3i(t)), p2(t) = /ti2pii(t) -- 'y21p22(t) + i(123p23(t) -- 123p32(t)) + 'y32p33(t), /ti2pu(t) 1123p32(t)) y32p33(t), i(Ili3p2i(t) l23p22(t) i2p23(t) + 6Lp23(t) p3(t) = --F32p23(t) + i(1i3p2i(t) ++123p22(t) ++w12p23(t)+ 6Lp23(t) -- 23p33(t)), p1(t) = --F31p31(t) ++i(--1i3pii(t) -- 123p2i(t) -- 6LP31(t) + 1i3p33(t)), --F31p31(t) i(--1i3pii(t) -- 123p2i(t) -- 6LP31(t) li3p33(t)), p2(t) = --F32p32(t) + i(--1i3pi2(t) -- 123p22(t) -- (W12 + fL)p32(t) + 123p33(t)), --I32p32(t) 23p22(t) p3(t) = i(--1113p13(t) -- 1123p23(t) + li3p3i(t) ++1123p32(t)) -- (y31+ 732)p33(t). 1i3p3i(t) 123p32(t)) (731 + y32)p33(t). i(--Ili3pi3(t)

(16)

Integrating this set of differential equations we obtain a complete picture of temporal dynamics of the driven A-system and can calculate any its characteristic. If we are interested in the fluorescence, which is proportional to the excited state population, one needs simply to calculate the population of the excited state 3) (see Eq. reflspfl-trans. This temporal dependency is shown in Fig. 2a versus the frequency splitting w12 between two dependency is shown in Fig. 2a versus the frequency splitting w12 between two ground levels. From Fig. 2 one can see that for the case when only one laser field with the frequency WL acts on both for the case when only one laser field with the frequency wL acts on both +-* 2) and 2) -+ 3) of the A-system, the dark resonance is observed at w12 = WL -- WL = 0, i.e., transitions Ii) and 2) -+ L) of the A-system, the dark resonance observed at w12 = WL wL for the case of degenerate A-system. In experiment, such situation corresponds to the case when no external magnetic field is applied to the quantum system.

4.1.2. Quantum trajectories analysis
For simplicity, we will consider only two radiation decays in the A-system from excited stateJ3) onto the low-lying states 1), 2), reducing the number of states 1), 2), reducing the number of operators Cm in Eq. (2) just to two operators: C1 = (y31)1/2P13 and
f1 _f \1/2D C2 -- '732) '23 L'2 Inserting Inserting Oi and C2 into Eq. (2), we obtain equation for the relaxation part ¸re1axi:

(

y3ip33(t) 'y3ip33(t) 0

0
'y32p33(t)

--O.5(y31 +'y32)p13(t)\ --O.5('-y31 +y32)pi3(t)\ --O.S('y +'y32)p23(t) --O.S('y31+''32)p23(t) J

.

+732)p13(t) --O.S(y3i +732)p23(t) \--0.5(y31 +y32)p13(t) --O.5('731 +y32)p23(t)

--(yi +y)p(t). --(yi +y)p(t).

)

(17)

In accordance with the results of Sec. 2.2, the non-hermitian interaction hamiltonian is given by Eq. (6). Substituting C1 and C2 in this equation leads to the hamiltonian of the form: Ci and C2 in this equation

H=

fl/U 0
(

U 0 W2 W12

113 \
123 23
)

--iF SLJ \1i3 123 --iF ++SLJ 1123

, ,

(18) (18)

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Figure 2. Temporal dependence of the excited state population from a driven A-system calculated with the help of density matrix approach (a) and using quantum trajectories technique (b). Both plots show similar dependence of the excited state population fl3 versus time since the excitation of the system and the ground state frequency splitting W12. For simplicity, the calculations were made for the case of a symmetric A-system, i.e., for the equal Rabi frequencies 13 23 = 1 and equal spontaneous decay rates 731 = 732 = 'y = 1 from the excited state; relaxation parameters of the y ground state were set to zero. The number of computed quantum trajectories for figure (b) is equal to 2000.

where F (yi + 732)12. where F = (731 + 732)12. From SchrÆdinger equation (5) one can readily obtain the following set of differential equations for the From probability amplitudes ai(t), a2 (t), and a3(t): probability amplitudes a1 (t), a2(t), and a3(t):

a(t) = --i(i3a3(t)

a(t) =

a(t) =

--i1113a3(t) --i1l3a3(t) 23a3(t)) --i(--w12a2(t) + l23a3(t))

(19)

+ 123a2(t) + (--iF + SL)a3(t)).

Let us assume then that initially, at t = 0, all population in the system was equally distributed among the states equations (19) can be integrated using the quantum trajectories 2), i.e., ri Ii) and 2), i.e., ri1 = Th2 0.5. Then, the set of equations (19) can be integrated using the quantum trajectories technique, which leads to the time-dependent wave-functions and, respectively, to the temporal dependencies of the population of the system's levels. Temporal dependency of the excited state population calculated with the help of quantum trajectories technique is shown in Fig. 2b versus the frequency detuning w12 of the two ground states. From Fig. 2b one can versus the frequency detuning wi2 From Fig. 2b one can clearly see that the stationary solution is reached at the time of the order of 10731. A more detailed comparison of the simulation results obtained by the density matrix approach and with quantum trajectories technique is shown in Fig. 3. It clearly shows that both methods give similar results and the curves are coincide rather well.

2

4.2. Case when the frequency modulation is switched on
In this Section, we will analyze the spectrum of the A-system driven by a resonant frequency-modulated laser field E(t) with the carrier frequency w0, which for the case of harmonic modulation with the modulation index M and frequency modulation Il can be written as modulation 1

E(t) =

+00

E0 exp[i(wot + M sin 1t)] = E0 exp(iwot)
fl=: --00

J(M) exp(im1t). exp(iri1t).

(20)

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0.25 0.20 0.15 0.10 0.05 0.00
-10

-5

0

5

10

Figure 3. Calculated population of the excited state of the symmetric A-system ('y31 = 'y32 = 'y, F = 1) versus W12 for Y32 = 'x 123 1 123 = 1 received with the help of density matrix approach (solid line) and quantum trajectories techniques (dotted 113 (solid line) and quantum trajectories techniques (dotted line, the number of computed trajectories is equal to 2000) at t/F = 14, i.e., in the steady-state case.

In this series expansion, the Bessel functions J(M) characterize the frequency components of the frequencymodulated light, i.e., amplitudes of the respective spectrum components are proportional to the Bessel modulated light, i.e. , the amplitudes of the respective spectrum components are proportional to the Bessel

functions for the fixed modulation index M. The modulation index is equal to the ratio of the deviation frequency D to the modulation frequency : M =D/fi frequency D to the modulation frequency Il: M = D/. Fig. 4 illustrates how the variation in the modulation index M and the modulation frequency Il affect the spectrum of the A-system. One can see from this figure modulation frequency that at the fixed 1 increasing the deviation frequency leads to the increase of the modulation index and, as aa fixed increasing the deviation frequency leads modulation index and, as
result, to the increase in the number of spectrum lines, which are equidistantly distributed in the spectrum. At the fixed deviation frequency, decreasing the modulation frequency leads to the increase of M and, respectively, the frequency separation between spectral lines decreases.
Increasing the modulation index M enlarges the number enlarges the of sidebands in the fluorescence spectrum

I I

I

,

i

Increasing the modulation

frequency L frequency 1) enlarges the frequency separation between sidebands

.*I .*.1

I

I.... I,...

Figure 4. Modification of the fluorescence spectrum of the A-system excited by the frequency-modulated laser field due to the variations in the modulation frequency and the modulation index M.

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4.2.1. Density matrix analysis
The interaction hamiltonian ofthe A-system interacting with the frequency-modulated laser field E(t) =E0 exp{i(wot+ E0 M cos Ilt)] has the form: 1t)]

ft = I fr=(

I If
'·\

0 0

0
--Wi2

ei(t) i13 e(t)1i3\

\\e_(t)1i3 e(t)123 e_(t)i3

e(t) 23 e(t)123 )

6L, I

(21)

where L(t) = M sin 1t. Inserting equation for ¸relaxCOS)in the form (4) into Eq. (1), we obtain the following for ¸relax(PS) in the form (4) into Eq. (1), we obtain the following where set of differential equations:

p1(t) = p (t) p2(t) = P13 (t) = p3(t) = = p1(t) = p2(t) = p3(t) = p1(t) = p2(t) = P2 (t) p3(t) =

12Pi i (t) + 721p22(t)++i i(e_(t)i3pi3(t) -- e(t) i3P3i (t) ) + 3ip33(t), --i2pii(t) + 2i P22 (t) (e_(t) i3 Pi3 (t) -- e(t)i3p3i(t)) (t), --Fi2pi2(t) + i(--i2pi2(t)++e_(t)123pi3(t) -- e_(t)1i3p32(t)), e_(t)123pi3(t) -- e_i(t)i3p32(t)), --F12p12(t) + i(--i2pi2(t) --F13pi3(t)(t) + i(ei(t) 1i3Pii (t) + e(t) 123Pi2 (t) + 5LPi3 (t) --e(t)fi3 p33(t)), --Fi3p13 + i(e(t)1i3pii(t) + e(t)123pi2(t) + 6Lpi3(t) -- e(t)1i3p33(t)), --F21p2i(t) + i(W12p2i(t) ++ e_i(t)1i3p23(t)e(t)23p3i(t)), --F21p2i(t) i(w12p21(t) e_(t)ii3p23(t) -- e(t)123p3i(t)), i(ei(t)l23p23(t) e(t)23p32(t)) y32p33(t), i2pii(t) ILi2pii(t) -- y2ip22(t) + i(e_(t)123p23(t) -- ei(t)123p32(t)) + 'y32p33(t),
--F23p23(t) + i(1l3p2(t) e(t)123p22(t) + 8Lp23(t) + Wi2p23(t) + e(t)1123p33(t)), --F23p23(t) + i(13p21(t) + + e(t)123p22(t) + 8Lp23(t) + w12p23(t) + e(t)1123p33(t)), --F31p3i(t) + i(e_iL(t)i3pii(t) ----e_(t)1l23p2i(t) -- ÆLP3i(t) + e_(t)11i3p33(t)), --F31p3i(t) + i(_e_(t)1i3pii(t) e_(t)1l23p2i(t) -- 5Lp3i(t) + e_i(t)11i3p33(t)), --F32p32 (t) + i + i(_e_t)fi3pi2(t)-- e_(t) 123P22 (t) -- 5Lp32(t) -- Wi2P32 (t) + e_(t)23p33(t)), --F32p32(t) (_e_(t) i3Pi2 (t) -- e_i(t)123p22(t) 5LP32 (t) w12p32(t) e_(t) 23P33 (t)), + y32)p33(t). i(_e_(t)1i3pi3(t) -- e(1l23p23(t) + e(t)1i3p3i(t) + et)1l23p32(t)) -- (y3i + y32)p33(t). i(_e_i(t)1li3pi3(t) e_(t)123p23(t) e(t)1i3p3i(t) + e(t)23p32(t)) (22)

Solvingthis set of equations by analogy with Sec. 4.1.1, one can calculate the temporal dependence of the the Solving this set of equations by analogy with Sec.
populations of the A-system levels.

The temporal dynamics of forming the spectrum of the dark resonance at the fixed modulation frequency and for two values of the modulation index is shown in Fig. 5. Simple analysis of this dynamics shows that with increasing the modulation index the structure of the spectrum is enriched and the number of sideband resonances is increased, too.

Figure 5. Excited state population of the A-system versus time since the excitation of the system and W12 at the fixed values of the modulation index M = modulation frequency = 2 for two values of the modulation index M -- 1.5 (a) and M = 4 (b). Other parameters were chosen as follows: cl = 2, 13 = 23 = 0.8, '731 = 732 = 1. follows: = 2, Q13 0.8, 31 32 1.

Qualitatively, the mechanism of forming the additional dark resonances in the spectrum of a A-system under

the action of a frequency-modulated laser field is clarified in Fig. 6. Every time a narrow dark resonance is formed when the frequency splitting w12 between two ground levels of the A-system exactly matches the
frequency difference between two neighboring sidebands in the spectrum of the incident frequency-modulated

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ooo

Figure 6. Mechanism of forming the dark resonances for the case of the A-system interacting with the frequencymodulated laser field.

laser field. All pairs of the components of the incident laser field spectrum with the frequency shift between them equal to the modulation frequency 1 (for instance, the pair marked by the solid line with arrows in Fig. 6) ft contribute to the dark resonance for which w12 = 1. Also, all the pairs of the components of the incident laser
field spectrum frequency shift between which is equal to the doubled modulation frequency 211 (for example, the pair marked with the dashed lines with arrows in Fig. 6) contribute to the dark resonance for which w12 = 21 and so on. From this consideration it follows that the frequency shift between the neighboring resonances in the observed spectrum of the A-system is equal to the modulation frequency IZ.

n3

0

Figure 7. Excited state population versus W12 and the modulation index M. Other parameters were chosen as follows: modulation index M. Figure 7. Excited state population versus = 2, 13 = 23 = 0.8, 'y31 = '32 = = 1.

= = = 1.

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The resulting rather complicated spectrum of the A-system irradiated with the frequency-modulated resonant laser field is shown in Fig. 7, which plots the population of the excited state of the A-system versus the W12 frequency at the fixed modulation frequency 1 = 2 for various values of the modulation index. At M = 0, we have no modulation at all and the dark resonance is observed at w12 = 0, i.e., we have the case of degenerated = 0, A-system. Increasing further the modulation index leads to appearing of additional resonances in the spectrum is because the number of sidebands in the spectrum of the at the conditions W12 = ri1, n = w12 = n1, incident laser field increases with increasing the modulation index in accordance with Eq. (20). modulation index in accordance with Eq. (20).
Amplitudes of these sidebands in the spectrum of the incident frequency-modulated laser field are proportional

to the Bessel functions J(M) at the fixed value of the modulation index M and decreasing up to zero with increasing 12. Therefore,the number of sidebands in the spectrum of the incident laser field in the central part n. Therefore, the number of sidebands in the spectrum of the incident laser field in the central part of the spectrum is approximately equal to M. Respectively, the resulting spectrum of the A-system shows also approximately M dark resonances in the central part of the spectrum. Fig. 8 confirms this result.

2
b)
006

AA
,
24
I
·

:::
0.01

-36

-24

-12

0,

12

24

36

-72

-48

__
-24

48

72

Figure 8. Population of the excited state in the symmetric A-system versus W12 for two fixed values of the modulation Figure 8. Population of the excited state in the symmetric A-system versus 0)12 for two fixed index M = 5 (a) and M = 10 (b) at the modulation frequency = 6 au. and the Rabi frequencies l3 = 23 = 0.8. 1123 Insets show the respected squared Bessel functions J,(M).

One can clearly see that at M = 5 (Fig. 8a) the resonance at the frequency W12 = is practically vanishes. 10 (Fig. 8b), the resonances at the frequencies w12 = and are vanished, as well. This happens because the two-photon dark resonances are observed on the background of the one-photon ones. As it has been shown in Ref. 21, for example, the power of the modulated signal transmitted through the media is proportional to the squared Bessel function J(M) for the given M. Keeping in mind that the Bessel function for M = 5 and at n = takes zero values at ri = values at n for M = 10, this clarifies why the resonances at the frequencies nil for these values of n are practically vanished.

At M =

Quantum trajectories analysis The non-hermitian interaction hamiltonian of the A-system interacting with the frequency-modulated laser field E(t) = E0 exp[i(wot + M cos [It)] has the form:
4 ..2 .2 . 2

n = (I H=(
where L(t) where L(t) =

I

0
o 0

'\ e_(t) ii3 e_(t)123 8L --1', \e_(t)1li3 e_(t)23 8L

e(t)1i3 e(t)i3\ ei(t)r W12 12 e(t)1l23 ) 23
0

J

(23)

M sin [It,F = sin t, F

(731

+ 732)12. + 732)12.

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From SchrÆdinger equation (5) one can readily obtain the following set of differential equations for the
probability amplitudes a1 (t) , a2 (t) , and a3 (t):

aÃ(t) a(t) = = --i(--w12a2(t) + e(t)c23a3(t)) a(t) a(t) = _i(e(t)1i3a3(t) + e(t)123a2(t) + (--iF + SL)a3(t)).

(24)

Let us assume by analogy with Sec. 4.1.2 that at the initial time moment t = 0 all population in the system 2 = 0.5. is distributed in between two ground levels 1) and 2), .i.e., n1 = n2= 0.5. Then, solving set of equations (24) two ground levels 1) and 2), i.e., with the help of quantum trajectories technique we will receive time-dependent wave-functions of the system and time-dependent populations of each of the energetic levels. Temporal dependency of the population of the excited state calculated with the help of quantum trajectories versus the frequency spacing between two ground levels w12 is shown in Fig. 9. The calculations were done for the same parameters as similar calculations by density matrix approach (Sec. 4.2.1). Comparison of both these methods is shown in Fig. 9b. One can easily see from this figure that results obtained by two different methods are in good agreement. Some quantitative difference is because the number of calculated trajectories is not an infinitive one, but equal only to 5000. Increasing the number of trajectories in a computer experiment will lead to more precise coincidence of the results, but is timeconsuming. However, even our results show that the quantum trajectories technique can be adequately used for simulating not only three level system in A-configuration interacting with the frequency-modulated field, but also can be used for simulating more complicated multilevel
systems.

0.16 0.12

(b)

n

0.08

0.040.00· 0.00-

Jo Jo

()

11

-15

-10

-5

0

5

10

15

Figure 9. a) Excited state population of the A-system versus time since the excitation of the system and W12 at the 4. fixed modulation frequency 1 = 2 for the modulation index M = 4. Other parameters were chosen as follows: Q = 2, 23 = 0.8, 123= 0.8, F = 1. b) Excited state population of the A-system versus W12 for the parameters of (a) computed 113 with the help of density matrix approach (solid line) and quantum trajectories techniques (dashed line). The number of computed trajectories is equal to 5000.

5. CONCLUSIONS
In conclusions, we have presented a theoretical model for the frequency-modulation spectroscopy of dark resonances on example of a three-level quantum system in A-configuration driven by resonant laser field(s) with and without frequency modulation using two simulation techniques--the density matrix and quantum trajectories analysis. With these techniques, such physical quantities as the total fluorescence intensity in equilibrium and

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in the transient response, resonance fluorescence spectrum, linear and nonlinear absorption coefficients, and the refractive indices, which can be all measured experimentally, can be modeled within the frame of the proposed model. As an example, we have calculated the total fluorescence intensity in equilibrium and in the transient response for the real atomic A-system formed of the Zeeman sublevels of one of the two alkali hyperfine ground states, specifically in Cs atoms. The calculated spectrum using such a simplified model is in a qualitative agreement with the experimental results reported in the literature [9, 10] and it is clearly seen that at high laser modulation index additional side CPT-resonances are present. Their frequency positions matches the experimental ones and it can be seen that CPT resonance appear when 1/n (where n is an integer number) equals the Zeeman splitting Il/n (where n is an integer number) equals the Zeeman splitting
of two sublevels Wi2. w12.

Our future work will be concentrated on elaboration of more complex model, where a larger number of
experimental parameters will be considered. Moreover, we plan to extend the model by considering a richer level structure, which is well reasonable using the method of quantum trajectories analysis, whose computing time increases linearly with the number N of the considered levels, in contrast with the quadratic dependency shown by the density matrix approach.

ACKNOWLEDGMENTS
This work was partially supported by the Russian Foundation for Basic Research under the grant No. 04--02--
17554.

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