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Theory of frequency-mo dulation sp ectroscopy of coherent dark resonances
Julia Vladimirova, Boris Grishanin, Victor Zadkov, Valerio Biancalana , Giuseppe Bevilacqua , Evelina Breschi , Yordanka Dancheva , and Luigi Moi International Laser Center and Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow, Russia Department 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 -configuration driven by resonant laser field(s) with and without frequency modulation using two simulation techniques--the density matrix and quantum tra jectories analysis. 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 [1­6]. It is most conspicuous for the -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 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 -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 µT. 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 12 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 [9­12]. 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@comsim1.phys.msu.su


Figure 1. systems formed of the Zeeman sublevels in the Fg = 3 Fe = 2 transitions excited by the + and - components of the respective frequencies 1 and 2 , whose difference is equal to the splitting of the Zeeman sublevels with mF = 2. Three-level system in -configuration has the following parameters: L1 , L2 are the frequencies of the laser fields driving transitions of the system; 13 , 23 are the respective Rabi frequencies; L is the frequency detuning from the |1 |3 transition; 31 , 32 are the decay rates from excited state |3 onto the low-laying levels |1 and |2 ; 12 and w are the decay and pumping rates of the level |1 via the level |2 , correspondingly; 31 , 32 , and 12 are the dephasing rates for the transitions |1 |3 and |2 |3 , and |1 |2 , respectively.

In this paper, we present a detailed study of the dynamics of a three-level quantum system in -configuration driven by resonant laser field(s) with and without frequency modulation using two simulation techniques--the density matrix and quantum tra jectories analysis. The paper is organized as follows. General theoretical background for both density matrix and quantum tra jectories analysis used for numerical simulation of the fluorescence spectrum of the driven -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 -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 -SYSTEM
Theoretical analysis of a driven -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 -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 tra jectories technique (we will call it the quantum trajectories analysis ) [13­16].


Note that the results of the density matrix and quantum tra jectories analysis have to be identical within a certain error, which depends on how many tra jectories 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 N 2 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 tra jectories 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 : i^ ^ = - [H , ] + L ^
relax

, ^

(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 Lrelax . 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 system­reservoir interaction. Then, the relaxation term L L
relax

in master equation (1) can be written in the general Lindblad form [17] as 1 2 ^ ^ (Cm Cm
m

relax

=-

+

^ ^ Cm Cm ) +
m

^ Cm

^ Cm ,

(2)

^ where is the substitution symbol to be replaced with the density matrix , operators Cm describe interaction ^ ^ ^ with the reservoir, operators Cm are conjugated to the operators Cm and both of them have dimension (1/t)1/2 . It is worth to note that representation (2) preserves interpretation of as the density matrix, i.e., we should ^ fulfill the condition Tr = 1 and the probability | | of finding the system in any state | must be positive ^ ^ ^ or equal to zero at any time moment and for any (t = 0). Note also that Eq. (2) does not require that Cm must ^ be defined uniquely. In Eq. (2) the first sum is the anti-commutator, which consists of the terms decreasing the total population ^ Tr; the second sum contributes to (d/dt)Tr. The number of operators Cm in Eq. (2) in general case can be ^ ^ 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, ^ i.e., spontaneous decay. This operator has the form C1 = 1/2 - , where - = |g e| is the atomic transition ^ ^ operator from the ground state |g onto the excited state |e . More examples are described in Ref. 16. ^ For the case of a -system, in order to write the operators Cm one has to take into account the following relaxation processes: spontaneous decay from the excited state |3 , incoherent decay and incoherent pumping of ^ the two ground levels |1 , |2 . Operator Cm in this case will consist of four terms responsible for the above listed ^ ^ relaxation processes, namely, operators C1 and C2 describe spontaneous decay from the excited state |3 onto ^ ^ the ground states |1 , |2 , operators C3 and C4 describe incoherent decay and incoherent pumping of the level |1 to the level |2 : ^ C1 = (31 ) ^ C3 = (21 )
1/2 1/2 ^ P13 , C1 = (31 ) ^ P12 , C3 = (21 ) 1/2 1/2

^ P31 ; ^ P21 ;

^ C2 = (32 ) ^ C4 = (µ12 )

1/2 1/2

^ ^ P23 , C2 = (32 )1/2 P32 ; 1/2 ^ ^ P21 , C4 = (µ12 ) P12 ,

(3)


^ where Pkl are the transition operators, which are represented, in general case, by the matrices with the only ^ non-zero k l-element Pkl (k , l) = 1. With the help of Eqs. (2), (3) the relaxation term Lrelax in master equation (1) takes the form: ^ -µ12 11 (t) + 21 22 (t) + 31 33 (t) -0.5(21 + µ12 )12 (t) -0.5(31 + 32 + µ12 )13 (t) -0.5(21 + µ12 )12 (t) µ12 11 (t) - 21 22 (t) + 32 33 (t) -0.5(31 + 32 + 21 )23 (t) -0.5(31 + 32 + µ12 )13 (t) -0.5(31 + 32 + 21 )23 (t) -(31 + 32 )33 (t), where 31 , 32 are the spontaneous decay rates, the level |2 , respectively.
12

(4)

and µ

12

are the decay and pumping rates of the level |1 via

^ Now, master equation (1) can be integrated numerically for a given Hamiltonian H of the system and then all its necessary characteristics can be modeled.

2.2. Quantum tra jectories analysis
Another relevant approach to model the temporal behavior of a -system is the quantum tra jectories analysis [13­16], 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 the solution of the master equation (1), includes two parts: i) modeling continuous variation of the current state and ii) modeling quantum jumps occurring randomly with certain probability. Let us assume that the system at time t is in the state | (t) . Then, the continuous variation of the current state can be described with the temporal dynamics of the wave function (1) (t) , which is governed by the Schr¨ odinger equation 1^ ~ (1) (t + t) (1 + H dt) | (t) (5) i with the non-hermitian hamiltonian ^ ~ ^i H=H- 2 ^ ^ Cm Cm .
m

(6)

~ New wave-function is not normalized because the hamiltonian H is a non-hermitian one and the squared (1) norm of the function is equal to (t + dt) | (t + dt) = 1 - p, where p has the form: p =
m

p

m

= dt
m

^ ^ (t)| Cm Cm | (t) , 1.

(7)

where the time step dt must fulfill the inequality p

Random behavior of the wave function is described with the probability p of quantum jumps. If the quantum jump does not occur with the probability 1 - p, the wave function (1) (t + t) must be renormalized to unit and then it will be mapped with the corresponding normalized function | (t + t) . When the quantum jump occurs, the wave-function transfers into the state Cm | (t) with the related probability pm / p. Thus, at the time moment t + t we do have one of the two normalized wave-functions: · with the probability 1 - p the quantum jump does not occur and | (t + t) = · with the probability p
m (1)

(t + t) (1 - p)
-1/2

-1/2

,

^ the quantum jump occurs and | (t + t) = Cm | (t) ( pm / t)

.

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.


3.1. Sp ontaneous fluorescence intensity
The spontaneous fluorescence intensity is proportional to the excited-state population, Ifl n3 = 3| |3 . ^ and its total intensity is proportional to the stationary value of the excited-state population: I
total fl

(8)

n ^

3

=

3| |3 .

(9)

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 finite amount of time t interacting with the laser radiation 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 p 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 |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 p , which depends on the relaxation parameters, should be at least much shorter than the longest observation time t that can be achieved in the experiment. Within the Markov approach the inverse time distribution has a Lorentzian shape.

3.2. Resonance fluorescence sp ectrum
The spectrum of resonance fluorescence of a driven -system is typically derived from the correlation function of the frequency-positive and frequency-negative operators of the complex amplitudes 13 , 23 [18] related to ^ ± ^± the corresponding Pauli matrices 13 , 23 that are the transition operators |k l| with k = l = 1, 2. Respective ^^ 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 causing the reduction of an electromagnetic wave with intensity I = c 0 E E /2 [20]: = 1 dI (z ) ^ = m( P E ), I (z ) dz 2I (z ) (10)

^ ^ where L1 , L2 . The polarizations P13 (L1 ), P23 (L2 ) acting on laser fields with frequencies and amplitudes ^ ^ EL1 , L1 and EL2 , L2 are given by ^ P13 (L1 ) = N0 d
13

|1 3|,

^ P23 (L2 ) = N0 d

23

|2 3|,

(11)
13

where N0 is the number of atoms per unit volume. Making use of Eq. (10,11) together with d 13 E (L1 ) = 2 and d23 E (L2 ) = 2 23 we find that (L1 ) = 13 N IL1 (z )
0

m |1 3| ,

(

L2

)=

23 N IL2 (z )

0

m |2 3| .

(12)

3.4. Refractive indices
Real part of the refraction index n = n + in can be calculated by analogy with the absorption coefficient from n -1= where k is the wavevector, yielding n (L1 ) - 1 = c12 N IL1 (z )
0

1 d(z ) c ^^ = e( P E ), k dz 2I (z )

(13)

e |1 3| ,

n (L2 ) - 1 =

c23 N IL2 (z )

0

e |2 3| .

(14)

The dispersion coefficients are nonlinear ones since the density matrix elements depend on both laser light fields, IL1 , IL2 .


4. MODELING FLUORESCENCE FROM A DRIVEN -SYSTEM 4.1. Case when the frequency mo dulation is switched off
Before studying interaction of a -system with the frequency-modulated laser field, let us first analyze a more simple problem--how the -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 -configuration, ). The interaction hamiltonian has the form: 0 0 ^ -12 H= 0 2 13 23 which interacts with the field E (t) = E 0 cos(0 t+ 13 23 L ,

(15)

where 13 , 23 are the Rabi frequencies, L = 0 - 13 is the one-photon frequency detuning of the probe laser field from the resonance, 12 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: 11 12 13 21 22 23 31 32 33 (t) (t) (t) (t) (t) (t) (t) (t) (t) = = = = = = = = = -µ12 11 (t) + 21 22 (t) + i(13 13 (t) - 13 31 (t)) + 31 33 (t), -12 12 (t) - i(12 12 (t) + 23 13 (t) - 13 32 (t)), -13 13 (t) + i(13 11 (t) + 23 12 (t) + L 13 (t) - 13 33 (t)), -21 21 (t) + i(12 21 (t) + 13 23 (t) - 23 31 (t)), µ12 11 (t) - 21 22 (t) + i(23 23 (t) - 23 32 (t)) + 32 33 (t), -32 23 (t) + i(13 21 (t) + 23 22 (t) + 12 23 (t) + L 23 (t) - 23 33 (t)), -31 31 (t) + i(-13 11 (t) - 23 21 (t) - L 31 (t) + 13 33 (t)), -32 32 (t) + i(-13 12 (t) - 23 22 (t) - (12 + L )32 (t) + 23 33 (t)), i(-13 13 (t) - 23 23 (t) + 13 31 (t) + 23 32 (t)) - (31 + 32 )33 (t).

(16)

Integrating this set of differential equations we obtain a complete picture of temporal dynamics of the driven -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. refIspfl-trans. This temporal dependency is shown in Fig. 2a versus the frequency splitting 12 between two ground levels. From Fig. 2 one can see that for the case when only one laser field with the frequency L acts on both transitions |1 |2 and |2 |3 of the -system, the dark resonance is observed at 12 = L - L = 0, i.e., for the case of degenerate -system. In experiment, such situation corresponds to the case when no external magnetic field is applied to the quantum system. 4.1.2. Quantum tra jectories analysis For simplicity, we will consider only two radiation decays in the -system from excited state |3 onto the low-lying ^ ^ ^ states |1 , |2 , reducing the number of operators Cm in Eq. (2) just to two operators: C1 = (31 )1/2 P13 and ^2 = (32 )1/2 P23 . ^ C ^ ^ Inserting C1 and C2 into Eq. (2), we obtain equation for the relaxation part Lrelax : ^ 31 33 (t) 0 -0.5(31 + 32 )13 (t) 0 32 33 (t) -0.5(31 + 32 )23 (t) . (17) -0.5(31 + 32 )13 (t) -0.5(31 + 32 )23 (t) -(31 + 32 )33 (t). 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: 0 0 13 ^ -12 23 , (18) H= 0 2 13 23 -i + L


0.25 0.20 0.15
3

(a)

0.3 0.2

(b)

n

0.10 0.05 0.00 0 4 8 12 -8 -4 0 4 8

n
0.1 0.0 0 4 8 12 -8 0 4 8

3

Figure 2. Temporal dependence of the excited state population from a driven -system calculated with the help of density matrix approach (a) and using quantum tra jectories technique (b). Both plots show similar dependence of the excited state population n3 versus time since the excitation of the system and the ground state frequency splitting 12 . For simplicity, the calculations were made for the case of a symmetric -system, i.e., for the equal Rabi frequencies 13 = 23 = 1 and equal spontaneous decay rates 31 = 32 = = 1 from the excited state; relaxation parameters of the ground state were set to zero. The number of computed quantum tra jectories for figure (b) is equal to 2000.

where = (

From Schr¨ odinger equation (5) one can readily obtain the following set of differential equations for the probability amplitudes a1 (t), a2 (t), and a3 (t): a1 (t) = -i13 a3 (t) a2 (t) = -i(-12 a2 (t) + 23 a3 (t)) a3 (t) = -i(13 a3 (t) + 23 a2 (t) + (-i + L )a3 (t)). (19)

Let us assume then that initially, at t = 0, all population in the system was equally distributed among the states |1 and |2 , i.e., n1 = n2 = 0.5. Then, the set of equations (19) can be integrated using the quantum tra jectories 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 tra jectories technique is shown in Fig. 2b versus the frequency detuning 12 of the two ground states. From Fig. 2b one can clearly see that the stationary solution is reached at the time of the order of 10 31 . A more detailed comparison of the simulation results obtained by the density matrix approach and with quantum tra jectories technique is shown in Fig. 3. It clearly shows that both methods give similar results and the curves are coincide rather well.

4.2. Case when the frequency mo dulation is switched on
In this Section, we will analyze the spectrum of the -system driven by a resonant frequency-modulated laser field E (t) with the carrier frequency 0 , which for the case of harmonic modulation with the modulation index M and frequency modulation can be written as
+

t/
31

t/

12/

-4

12/

+ 32 )/2.

E (t) = E0 exp[i(0 t + M sin t)] = E0 exp(i0 t)
n=-

Jn (M ) exp(int).

(20)


0.25 0.20 0.15

n
0.10 0.05 0.00 -10 -5 0 5 10

3

12/
Figure 3. Calculated population of the excited state of the symmetric -system ( 31 = 32 = , = 1) versus 12 for 13 = 23 = 1 received with the help of density matrix approach (solid line) and quantum tra jectories techniques (dotted line, the number of computed tra jectories is equal to 2000) at t/ = 14, i.e., in the steady-state case.

In this series expansion, the Bessel functions Jn (M ) characterize the frequency components of the frequencymodulated 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/. Fig. 4 illustrates how the variation in the modulation index M and the modulation frequency affect the spectrum of the -system. One can see from this figure that at the fixed increasing the deviation frequency leads to the increase of the modulation index and, as a 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.

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


4.2.1. Density matrix analysis The interaction hamiltonian of the -system interacting with the frequency-modulated laser field E (t) = E 0 exp[i(0 t+ M cos t)] has the form: 0 0 ei(t) 13 ^ H= (21) 0 -12 ei(t) 23 e-i(t) 13 e-i(t) 23 L , where (t) = M sin t. Inserting equation for L set of differential equations: 11 12 13 21 22 23 31 32 33 (t) (t) (t) (t) (t) (t) (t) (t) (t) = = = = = = = = =
relax

(S ) in the form (4) into Eq. (1), we obtain the following

-µ12 11 (t) + 21 22 (t) + i(e-i(t) 13 13 (t) - ei(t) 13 31 (t)) + 31 33 (t), -12 12 (t) + i(-12 12 (t) + e-i(t) 23 13 (t) - e-i(t) 13 32 (t)), -13 13 (t) + i(ei(t) 13 11 (t) + ei(t) 23 12 (t) + L 13 (t) - ei(t) 13 33 (t)), -21 21 (t) + i(12 21 (t) + e-i(t) 13 23 (t) - ei(t) 23 31 (t)), µ12 11 (t) - 21 22 (t) + i(e-i(t) 23 23 (t) - ei(t) 23 32 (t)) + 32 33 (t), -23 23 (t) + i(13 21 (t) + ei(t) 23 22 (t) + L 23 (t) + 12 23 (t) + ei(t) 23 33 (t)), -31 31 (t) + i(-e-i(t) 13 11 (t) - e-i(t) 23 21 (t) - L 31 (t) + e-i(t) 13 33 (t)), -32 32 (t) + i(-e-i(t) 13 12 (t) - e-i(t) 23 22 (t) - L 32 (t) - 12 32 (t) + e-i(t) 23 33 (t)), i(-e-i(t) 13 13 (t) - e-i(t) 23 23 (t) + ei(t) 13 31 (t) + ei(t) 23 32 (t)) - (31 + 32 )33 (t). (22)

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

0.2

(a)
n
3

0.15 0.10 0.05 0.00 0 4 8 12 16

(b)

n

0.1

3

0.0 0 4 8 12 16

t/

Figure 5. Excited state population of the -system versus time since the excitation of the system and 12 at the fixed modulation frequency = 2 for two values of the modulation index M = 1.5 (a) and M = 4 (b). Other parameters were chosen as follows: = 2, 13 = 23 = 0.8, 31 = 32 = 1.

Qualitatively, the mechanism of forming the additional dark resonances in the spectrum of 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 12 between two ground levels of the -system exactly matches the frequency difference between two neighboring sidebands in the spectrum of the incident frequency-modulated

t/

15 5 10 -5 0 -10 -20 -15 12/

-5 0 -15 -10 -20

5

10 15

12/


W

d

L

|3

2W 3W 4W nW

|2

d

R

w12
|1

Figure 6. Mechanism of forming the dark resonances for the case of the -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 (for instance, the pair marked by the solid line with arrows in Fig. 6) contribute to the dark resonance for which 12 = . Also, all the pairs of the components of the incident laser field spectrum frequency shift between which is equal to the doubled modulation frequency 2 (for example, the pair marked with the dashed lines with arrows in Fig. 6) contribute to the dark resonance for which 12 = 2 and so on. From this consideration it follows that the frequency shift between the neighboring resonances in the observed spectrum of the -system is equal to the modulation frequency .

n3

0.2 0 5 4 3 16 2 1 0 -16 0 -8 8

M

12 /

Figure 7. Excited state population versus = 2, 13 = 23 = 0.8, 31 = 32 = = 1.

12

and the modulation index M . Other parameters were chosen as follows:


The resulting rather complicated spectrum of the -system irradiated with the frequency-modulated resonant laser field is shown in Fig. 7, which plots the population of the excited state of the -system versus the 12 frequency at the fixed modulation frequency = 2 for various values of the modulation index. At M = 0, we have no modulation at all and the dark resonance is observed at 12 = 0, i.e., we have the case of degenerated -system. Increasing further the modulation index leads to appearing of additional resonances in the spectrum at the conditions 12 = n, n = ±1, ±2, ±3, . . .. This is because the number of sidebands in the spectrum of the incident laser field increases with increasing the 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 Jn (M ) at the fixed value of the modulation index M and decreasing up to zero with increasing 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 -system shows also approximately M dark resonances in the central part of the spectrum. Fig. 8 confirms this result.

Figure 8. Population of the excited state in the symmetric -system versus 12 for two fixed values of the modulation index M = 5 (a) and M = 10 (b) at the modulation frequency = 6 a.u. and the Rabi frequencies 3 = 23 = 0.8. 2 Insets show the respected squared Bessel functions Jn (M ).

One can clearly see that at M = 5 (Fig. 8a) the resonance at the frequency 12 = ±2 is practically vanishes. At M = 10 (Fig. 8b), the resonances at the frequencies 12 = ±, ±3 and ±6 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 2 proportional to the squared Bessel function Jn (M ) for the given M . Keeping in mind that the Bessel function takes zero values at n = ±2 for M = 5 and at n = ±1, ±3, ±6 for M = 10, this clarifies why the resonances at the frequencies n for these values of n are practically vanished. 4.2.2. Quantum tra jectories analysis The non-hermitian interaction hamiltonian of the -system interacting with the frequency-modulated laser field E (t) = E0 exp[i(0 t + M cos t)] has the form: 0 0 ei(t) 13 ^ ~ (23) H= 0 -12 ei(t) 23 e-i(t) 13 e-i(t) 23 L - i, where (t) = M sin t, = (
31

+ 32 )/2.


From Schr¨ odinger equation (5) one can readily obtain the following set of differential equations for the probability amplitudes a1 (t), a2 (t), and a3 (t): a1 (t) = -iei(t) 13 a3 (t) a2 (t) = -i(-12 a2 (t) + ei(t) 23 a3 (t)) a3 (t) = -i(ei(t) 13 a3 (t) + ei(t) 23 a2 (t) + (-i + L )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 is distributed in between two ground levels |1 and |2 , .i.e., n1 = n2 = 0.5. Then, solving set of equations (24) with the help of quantum tra jectories 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 tra jectories versus the frequency spacing between two ground levels 12 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 tra jectories is not an infinitive one, but equal only to 5000. Increasing the number of tra jectories in a computer experiment will lead to more precise coincidence of the results, but is timeconsuming. However, even our results show that the quantum tra jectories technique can be adequately used for simulating not only three level system in -configuration interacting with the frequency-modulated field, but also can be used for simulating more complicated multilevel systems.
0.16 0.12

(a)

n

3
0.08 0.04 0.00 0

2

4

6

8 10

-1 0

-5

0

5

10

Figure 9. a) Excited state population of the -system versus time since the excitation of the system and 12 at the fixed modulation frequency = 2 for the modulation index M = 4. Other parameters were chosen as follows: = 2, 13 = 23 = 0.8, = 1. b) Excited state population of the -system versus 12 for the parameters of (a) computed with the help of density matrix approach (solid line) and quantum tra jectories techniques (dashed line). The number of computed tra jectories is equal to 5000.

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 -configuration driven by resonant laser field(s) with and without frequency modulation using two simulation techniques--the density matrix and quantum tra jectories analysis. With these techniques, such physical quantities as the total fluorescence intensity in equilibrium and

t/

12 /

5. CONCLUSIONS


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 -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 /n (where n is an integer number) equals the Zeeman splitting of two sublevels 12 . 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 tra jectories 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. 04­02­ 17554.

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