Theory of frequency modulation spectroscopy of coherent dark resonances

ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2006, Vol. 103, No. 4, pp. 528–538. © Pleiades Publishing, Inc., 2006.Original Russian Text © Yu.V. Vladimirova, B.A. Grishanin, V.N. Zadkov, V. Biancalana, G. Bevilacqua, Y. Dancheva, L. Moi, 2006, published in Zhurnal Éksperimental’no

œ

iTeoretichesko

œ

Fiziki, 2006, Vol. 130, No. 4, pp. 609–620.

528

1. INTRODUCTION

The effect of coherent population trapping (CPT) iswidely used in various applications, including magne-tometry and metrology [1–5]. This effect is manifestedmost clearly in a three-level system for

Λ

transitionsbetween closely spaced long-lived energy levels, whichare optically coupled with a third short-lived levels bytwo continuous coherent fields (see the inset to Fig. 1).

In the absorption spectrum, coherent superposition ofclosely spaced levels leads to a very narrow dip againstthe background of the absorption line or, equivalently,to a dark resonance in the case of observation of reso-nant fluorescence.

In the three-level model, the CPT effect can be fullydescribed analytically (the corresponding referencesare given in the review by Arimondo [6]). In the case ofmultilevel systems, the model becomes much more

Theory of Frequency Modulation Spectroscopyof Coherent Dark Resonances

Yu. V. Vladimirova

a

, B. A. Grishanin

a

, V. N. Zadkov

a

, V. Biancalana

b

, G. Bevilacqua

b

, Y. Dancheva

b

, and L. Moi

b

a

International Laser Center, Moscow State University, Moscow, 119899 Russia

b

University of Siena, Siena, 53100 Italye-mail: [email protected]

Received March 21, 2006

Abstract

—The interaction of a three-level atom in the

Λ

configuration with a frequency-modulated (FM) fieldis analyzed theoretically for the first time. The two-frequency model for solving the problem of frequency mod-ulation spectroscopy of coherent dark resonances is described and analyzed for a three-level

Λ

system. Theeffectiveness of the two-frequency model is demonstrated by comparing the results obtained using this modelwith the results of solving the exact problem of interaction of a

Λ

system with an FM field, which can be solvedby the density matrix method. It is shown that the simple two-frequency model corresponds to the exact solutionand is in qualitative agreement with experimental data.

PACS numbers: 42.50.Hz, 42.50.Gy

DOI:

10.1134/S1063776106100037

ATOMS, MOLECULES, OPTICS

δ

L

ω

L

1

,

g

13

γ

31

,

Γ

13

ω

L

2

,

g

23

γ

32

,

Γ

23

γ

21

,

Γ

12

,

ω

12

ω

23

ω

13

∆|

1

⟩

|

2

⟩

|

3

⟩

12

0–1

m

F

= –2

12

3

0–1

–2

m

F

= –3

σ

+

,

ω

L

1

σ

–

,

ω

L

2

F

e

= 2

F

g

= 3

Fig. 1.

Λ

systems formed by Zeeman sublevels at transitions

F

g

= 3

F

e

= 2 induced by the

σ

+

and

σ

–

laser field componentswith corresponding frequencies, the difference between which is equal to the spacing

∆

m

F

= 2 between the Zeeman sublevels. Theinset shows a diagram of a three-level system in the

Λ

configuration and its parameters: frequencies

ω

L

1

and

ω

L

2

of laser fieldsacting on transitions in the system; Rabi frequencies

g

13

and

g

23

; laser detuning

δ

L

from transition

|

1

⟩

|

3

⟩

; rates

γ

31

and

γ

32

ofdecay from excited state

|

3

⟩

to lower-lying levels

|

1

⟩

and

|

2

⟩

; decay rate

γ

12

and rate of pumping

w

of level

|

1

⟩

via level

|

2

⟩

; anddephasing rates

Γ

13

,

Γ

23

, and

Γ

12

of transitions

|

1

⟩

|

3

⟩

,

|

2

⟩

|

3

⟩

, and

|

1

⟩

|

2

⟩

, respectively.

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THEORY OF FREQUENCY MODULATION SPECTROSCOPY 529

complicated since a more complicated energy structureof multilevel atoms (especially in the presence of exter-nal magnetic fields) leads to a substantial modificationof resonance dependences on the parameters of actingfields; for this reason, it is impossible to obtain analyticresults [7] in most cases.

The traditional experimental technique for observ-ing the spectra of dark resonances using two resonancelaser fields is widely used in many applications. How-ever, a simpler experimental technique should be devel-oped for spectroscopy of dark resonances in multilevelatoms, in which, for example, only one (albeit fre-quency modulated) laser field will be used. Such exper-iments are carried out by the Moi group at Siena Uni-versity (Italy) [8]; in fact, these experiments initiatedthis study devoted to the theory of frequency modula-tion (FM) spectroscopy of dark resonances.

Modulation methods in optical spectroscopy havebeen actively developed since the beginning of 1980s.The FM methods, which were well known by that timein microwave spectroscopy and NMR spectroscopy,acquired a new impetus as applied to laser spectroscopyand are widely used for stabilizing lasers. Optical FMspectroscopy continues to play the central role in preci-sion measurements in such branches of physics asdetection of gravity waves, frequency standards, andmeasurements of weak magnetic fields (see review [9]and the literature cited therein).

In spite of considerable advances in the develop-ment of optical FM spectroscopic methods, theoreticalmodels have been worked out and studied only for two-level systems, which permit the obtaining of analyticresults [9]. However, theoretical analysis of multilevel(three-level in the simplest case) systems interactingwith the FM field(s), which is of interest for the presentstudy, has not been carried out so far to our knowledge.

Here, we report on the results of theoretical analysisin FM spectroscopy of coherent dark resonances, whichmakes it possible to construct effective models of sim-ple systems interacting with the FM field and to calcu-late their dynamics and spectral dependences. In thegeneral case, the model is applicable to

Λ

systems, inwhich either hyperfine sublevels, or Zeeman sublevelsof a hyperfine level can be chosen as the ground state.In the calculations described here, the ground state ofthe

Λ

system are Zeeman sublevels of a hyperfine level.For simplicity of presentation, we will have to confineour analysis to an individual three-level

Λ

system inter-acting with a FM field to elucidate the most importantfeatures of the interaction between the

Λ

system and anFM field using this system as example. A further gener-alization of the model to multilevel systems (e.g.,chains of

Λ

systems formed by Zeeman sublevels in anapplied magnetic field; see Fig. 1) is obvious and caneasily be carried out in numerical simulation.

The two-frequency model of interaction between a

Λ

system and a FM field can essentially be reduced to

an analysis of the interaction of the system with two

monochromatic fields effectively replacing the interac-tion with the FM field. To monitor the results, we alsocalculated the absorption spectra for the FM field bysolving the exact equation for the density matrix, whichdescribes the interaction of the system with the true FMfield.

The article is organized as follows. Section 2 con-tains the main results of the CPT theory, obtained in theframework of the simplest three-level model in the

Λ

configuration. Salient features in the behavior of a

Λ

system interacting with an FM field are considered inSection 3. The mathematical apparatus used for calcu-lating CPT resonances in the case of interaction of a

Λ

system with two fields is considered, as well as the sim-plified two-frequency model of interaction of the

Λ

sys-tem with the FM field and the results of its applicationare presented in Section 4. Section 5 is devoted to theresults of calculation of absorption spectra by solvingthe exact equation for the density matrix. The main con-clusions are contained in Section 6.

2. COHERENT POPULATION TRAPPING EFFECT IN A

Λ

SYSTEM FOR BIHARMONIC EXCITATION

BY TWO MONOCHROMATIC FIELDS

In the simplest case of a three-level system ofatomic transitions in the

Λ

configuration, the two lowerlong-lived levels

|

1

⟩

and

|

2

⟩

with frequency splitting

ω

12

are coupled with the upper excited energy level

|

3

⟩

bytwo light fields (Fig. 1). If transition

|

1

⟩

|

2

⟩

in thedipole approximation is forbidden and two monochro-matic fields

E

1

exp(–

i

ω

L

1

t

–

i

ϕ

1

) and

E

2

exp(–

i

ω

L

2

t

–

i

ϕ

2

)are in resonance with the corresponding transitions

|

1

⟩

|3⟩ and |2⟩ |3⟩, a narrow CPT resonance isformed as a result of quantum interference. This reso-nance is manifested in the absorption spectrum as asharp minimum, when one of the acting fields (e.g.,ωL1) is scanned and the Raman detuning δR = ωL1 –ωL2 – ω12 passes through a zero value corresponding toexact resonance.

To describe the dynamics of interaction of a Λ sys-tem with the field, we must use the representation of theinteraction with the Hamiltonian of unperturbed motion

where

and ∆ = ωL2 – ωL1 ≈ ω12 is the detuning of the laserfields. For analyzing the main effects of CPT, it is expe-dient to write the atom–field interaction Hamiltonian in

0 t( ) i/( )0t–[ ],exp=

0 ωL1 3| ⟩ 3⟨ | ∆ 2| ⟩ 2⟨ |–( ),=

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VLADIMIROVA et al.

the basis of a “light” and orthogonal “dark” states,respectively:

In the approximation of rotating waves [10], the inter-action Hamiltonian can be reduced to

(1)

In the basis

|

3

⟩

,

|

+

⟩

,

|

–

⟩

, the matrices corresponding tothe interaction Hamiltonian assume the form

(2)

Oscillations between state

|

+

⟩

and excited state

|

3

⟩

occur with the effective Rabi frequency

g

Λ

=

; the state itself does not oscillate in therotating wave approximation. For

δ

R

= 0, matrix (2)splits into the direct sum of a 2

×

2 matrix describing thetwo-level system

|

3

⟩

⊕ |+⟩, which is formed from theexcited and light state, and a 1 × 1 matrix, whose onlyelement describes the dark state. The eigenvalues ofHamiltonian (2) are equal to ±gΛ/2, 0 and correspond toprecession between levels |+⟩ and |3⟩ with Rabi fre-quency gΛ .On the other hand, the matrix element⟨3|HΛ|–⟩ = 0; i.e., no transitions occur from the darkstate to the excited state, which is manifested in theform of the CPT resonance.

+| ⟩ gΛ1– g13e

iϕ1–1| ⟩ g23e

iϕ2–2| ⟩+( ),=

–| ⟩ g– Λ1– g23e

iϕ1–1| ⟩ g13e

iϕ2–2| ⟩–( ).=

Λ δL δR

g132

gΛ2

-------+⎝ ⎠⎜ ⎟⎛ ⎞

+| ⟩ +⟨ |=

+ δL δR

g232

gΛ2

-------+⎝ ⎠⎜ ⎟⎛ ⎞

–| ⟩ –⟨ |

+ δR

g13g23

gΛ2

-------------- ei ϕ1 ϕ2–( )

+| ⟩ –⟨ | H.c.+[ ]

+gΛ

2--------- +| ⟩ 3⟨ | H.c.+( ).

HΛ

0 gΛ/2 0

gΛ/2 δL 0

0 0 δL⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

δR

gΛ2

---------+=

×

0 0 0

0 g132 g13g23e

i ϕ1 ϕ2–( )

0 g13g23ei ϕ1 ϕ2–( )–

g232

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.

g132 g23

2+

3. INTERACTION OF A Λ SYSTEM WITH A FREQUENCY-MODULATED FIELD

Experiments show that the interaction of a three-level Λ system with an FM field substantially compli-cates the structure of absorption spectra even in theabsence of noise. In contrast to the interaction with twofields considered above, when only one CPT resonanceis observed, additional resonances are observed in theabsorption spectrum in the case when the Λ systeminteracts with the FM field. Experiments on FM spec-troscopy of dark resonances are carried out with alkaliatoms (e.g., Cs or Rb) in weak magnetic fields. Thepresence of a magnetic field complicates the energystructure of atoms due to the Zeeman effect (see Fig. 1).For example, transition Fg = 3 Fe = 2 in the Csatom is the working transition in the experiment [8].Each of levels Fg = 3 and Fe = 2 splits in a magneticfield into 2F + 1 sublevels. The pairs of ground sublev-els with ∆mF = 2 (mF is the magnetic quantum number)and the Zeeman sublevels of an excited state form achain of Λ systems. Laser radiation is linearly polar-ized; for this reason, Λ systems are formed by the Zee-man sublevels in transitions Fg = 3 Fe = 2, whichare induced by the σ+ and σ– components of the laserfield with corresponding frequencies ωL1 and ωL2; thedifference between these frequencies is equal to thespacing between Zeeman sublevels with ∆mF = 2. Thenumber of Λ systems in a chain depends on the totalangular momentum of the ground state. In typicalexperiments on FM spectroscopy of coherent dark res-onances in Cs atoms, the atomic medium is in a uniformmagnetic field with a strength on the order of 10 µT.

In the case of harmonic modulation with modulationindex M and modulation frequency Ωmod, the field is asuperposition of harmonics of the type

(3)

The amplitudes of the corresponding spectral com-ponents are proportional to Bessel functions Jn(M). Fig-ure 2 illustrates the variation of the field spectrum upona change in modulation index M and modulation fre-quency Ωmod. If the modulation frequency Ωmod is fixed,an increase in the modulation index leads to an increasein the number of bands in the spectrum with a spacingequal to Ωmod.

In the interaction of a Λ system with an FM field,each pair of frequency components of the spectrumcontributes to the CPT resonance, which is observedwhen the frequency difference between these compo-nent coincides with Zeeman splitting ω12 of the groundstate (see Fig. 1), which is due to the presence of theconstant magnetic field (i.e., additional CPT reso-nances appear in the spectrum at frequencies multiple

E t( ) E0 i ω0t M Ωmodt( )sin+( )[ ]exp=

= E0 iω0t( ) Jn M( ) inΩmodt( ).expn ∞–=

∞

∑exp

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 103 No. 4 2006


to Ωmod. The CPT resonances are observed in experi-ments with modulation frequency scanning in a smallrange in the vicinity of the two-photon resonance for afixed value of the external magnetic field or for a fixedmodulation frequency by scanning the magnetic field inthe corresponding range [8, 11–13]. To simplify calcu-lations, the effect of motion of atoms was disregardedin the present study.

4. TWO-FREQUENCY MODEL FOR CALCULATING FREQUENCY

MODULATION SPECTRA OF CPT RESONANCES

Let us consider the problem of interaction of a Λsystem with an FM field in the framework of the two-frequency model of the effective field. For this purpose,we will first calculate the Liouvillean of a three-levelatom for two preset monochromatic fields to introducethe corresponding field parameters for the effectivetwo-frequency model.

4.1. Calculation of the Liouvillean of a Three-Level Atom Interacting with Two Monochromatic Fields

To describe the dynamics of systems with relax-ation, we will use the superoperator technique, whichmakes it possible to represent transformations corre-sponding to superoperators (i.e., operators acting onoperators) in a physically visual form by writing super-operators in a symbolic representation using the sym-bol for the substitution of the density matrix [14].

When such a technique is used, the time evolution ofthe system with relaxation is determined by a kinetic

equation of the type

(4)

where t is the superoperator describing the total Liou-villean of an N-level atom. In the general case of a laserfield in the rotating wave approximation, this superop-erator can be represented by the sum of contributionst = r + e + δ + i , where r , e , and i aresuperoperators of radiation damping, elastic dephasing,and interaction with the laser field, respectively, whileδ is the superoperator of laser detuning, which sup-plements the chosen superoperator of unperturbed evo-lution to the superoperator describing free dynamics ofthe atom in zero laser field. The latter includes the cor-responding detunings of all acting laser fields takinginto account the fact that free precession with frequen-cies of these fields is included in the superoperator ofunperturbed dynamics.

The superoperator of radiative damping can be rep-resented in the Lindblad form

where the 2D array γkl describes the spontaneous decayrate for k > l and the pumping rate for k < l. The elasticdephasing superoperator is described by the sum

∂ρ∂t------ tρ

i--- H ρ,[ ]– rρ,+= =

r γ kl Plk Pkl12--- Pkk ,[ ]+–⎝ ⎠

⎛ ⎞ ,kl

∑=

e inkl ex

kl+( )

k l<∑=

= Γinkl Pkk Pll Pll Pkk+( )– Γex

kl Ikl ,[ ]2,–

ω0

ω0 ω0

ω0

Increase in modulation index M leads to an increasein the number of bands in the fluorescence spectrum

Increase in modulationfrequency Ω leads

to an increase in the spacingbetween the bands

Fig. 2. Modification of the FM field spectrum upon a change in modulation frequency Ω and modulation index M.

532


VLADIMIROVA et al.

where and are the dephasing rates. The super-operators of laser detuning and interaction with thelaser field have the form

respectively, where δk is the array of frequency detun-ings and Ωkl is the 2D array of Rabi frequencies of kltransitions.

The matrix representations of the N2 × N2 superop-erators i, r, e, t, and δ can be calculated usingthe formula Lmn = ( , ), where is theorthonormal basis and the parentheses describe the scalar

product of operators of the type Tr( ), which is anti-linear in the first factor and linear in the second factor.

Basis is Hermitian and can be expressed in

terms of the transition operators, represented byN × N matrices with a single nonzero kl elementPkl(k, l) = 1. The energy levels are labeled in increasingorder of their energies E1 ≤ E2 ≤ … ≤ EN; the corre-sponding basis is constructed in the form

(5)

where j(k, l) is the numbering index (i.e., the one-to-one map of the 2D set of numbers kl: k, l = 1, N, into the1D index j = 1, N2). In particular, this index can be spec-ified in the following manner, which is universal forany N:

4.2. Calculation of Mean Values

The mean value of operator of an arbitrarydynamic variable in the interaction representation canbe written in the form

(6)

Γinkl Γex

kl

δ i δk Pkk Pkk–( ),k

∑=

ii2--- Ωkl Pkl Plk+( ) ,[ ],

k l<∑–=

em en ek

A+B

ek

Pkl

e j k l,( )

Pkk, k l=

Pkl Plk+

2--------------------, k l<

iPkl Plk–

2--------------------, k– l,>

⎩⎪⎪⎪⎨⎪⎪⎪⎧

=

j

k, k l=

2k 1–( )N k 1+( )2– k 2l, k+ + l<

2l 1–( )N l 1+( )2– l 2k 1, k+ + + l.>⎩⎪⎨⎪⎧

=

A

A t( )⟨ ⟩ TrSt t( )ρ001–

t( ) A0 t( ),=

where

are the total superoperator of perturbed evolution andthe corresponding Liouvillean. The equilibrium valuesof atomic variables in this case are given by

(7)

where is the stationary density matrix and ⟨0| is thecorresponding vector in the matrix representation ofsuperoperators. The vector representation ⟨0| of the sta-tionary density matrix ⟨ | can be calculated using theequation

(8)

It is well known that basic properties of the CPTeffect are determined by the absorption of the appliedfield, which is proportional to the mean value of the

operator for the population of the excited level ( (t) ≡n3 in the case of a three-level Λ system). The timedependence of this quantity in the adiabatic approxima-tion is determined by the nonstationarity of quasi-equi-librium state ⟨0|, which is a function of time in the caseof nonstationarity of Liouvillean t , for example,because of fluctuations of the applied field.

4.3. Calculation of the Liouvillean of a Three-Level Atom in the Caseof a Frequency-Modulated Field

The problem of interaction of a Λ system with anFM field can be reduced to the problem of interactionof the system with two effective fields using the approx-imation, which will be described below. We should pro-ceed from the fact that the dependence of populationn3(ω12) = ⟨0| ⟩ of the excited state on the ground statesplitting ω12 is of prime importance.

The FM field spectrum (3) consists of equidistantcomponents at frequencies ω0 + nΩmod, the spacingbetween which is equal to modulation frequency Ωmod(Fig. 3). For this value of ω12, we will choose two fre-quency components from the entire spectrum. The firstcomponent is = n1Ωmod with number n1 (n1 is aninteger), for which one-photon resonance at the 1–3transition is most effective, i.e., the quantity

assumes the maximum value, and the second compo-nent is = n2Ωmod with number n2 (n2 is an integer),for which one-photon resonance at the 2–3 transition is

St tt( ), texp r e i δ+ + += =

A⟨ ⟩ Trρst A t( ) 0 A t( )⟨ | ⟩,= =

ρst

ρst

0⟨ |t 0.=

A

n3

ωn1

g13Jn1M( )( )2

γ 312 ωn1

ω13–( )2+------------------------------------------

ωn2



most effective, i.e., the quantity

assumes its maximum value. Since the spacing ω13 andω23 between 1 and 3 and 2 and 3 levels, respectively,changes upon a change in the ground-state splitting ω12,such a change for each value of ω12 reduces the FM spec-trum to the two most effective components with numbersn1 and n2, which depend on ω12 and give the maximumcontribution to the formation of a CPT resonance.

Liouvillean t in the matrix representation in basis (5)in the case of the two-frequency model under investiga-tion has the form

g23Jn2M( )( )2

γ 322 ωn2

ω23–( )2+------------------------------------------

(9)Lt

w– γ 12 γ 31 0 0 0g13'

2------- 0 0

w γ 12– γ 32 0 0 0 0 0g23'

2-------

0 0 γ 31– γ 32– 0 0 0g13'

2-------– 0

g23'

2-------–

0 0 0 A B ω12– 0g23'

2------- 0

g13'

2-------

0 0 0 B– ω12+ Ag23'

2-------– 0

g13'

2------- 0

0 0 0 0g23'

2------- C δL 0 0

g13'

2-------– 0

g13'

2-------

g23'

2-------– 0 δL– C 0 0

0 0 0 0g13'

2-------– 0 0 D δL B–

0g23'

2-------–

g23'

2-------

g13'

2-------– 0 0 0 –δL B+ D

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,=

where

The reduced matrix Lt has a block structure, inwhich four diagonal blocks L11, L22, L33, and L44

g23' g23Jn2M( ), g13' g13Jn1

M( ),= =

Aγ 12– w– 2Γ12–

2--------------------------------------, B ωn1

ωn2,–= =

C2γ 32– 2γ 31– 2w–

4--------------------------------------------, D

2γ 32– 2γ 31– 2γ 12–4

-----------------------------------------------.= =

describe the dynamics of populations n1, n2, n3 andcoherences of the subsystems 1–2, 1–3, and 2–3.Blocks L13, L31 and L14, L41 and the blocks conjugate tothem describe Rabi oscillations at transitions 1–3 and2–3, respectively. The remaining off-diagonal blocksdescribe intermodal coupling between the polarizationof the ground state and the polarizations of the 1–3 and2–3 transitions.

Solving Eq. (8) for a set of values ω12, we obtain thedependence of population n3(ω12) of the excited stateon the splitting ω12 of the ground state, which also

ωn1ωn2

ω13

δL

ω23

ω12

|3⟩

|2⟩

|1⟩

Fig. 3. In the two-frequency model, the FM spectrum isreplaced by two effective monochromatic fields with fre-quencies and , for which one-photon resonances at

transitions |1⟩ |3⟩ and |2⟩ |3⟩ are most effective.

ωn1ωn2

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2006

VLADIMIROVA et al.

includes the dependence on the one-photon laser detun-ing

δ

L

.

4.4. Calculation of the Absorption Spectrum

4.4.1. Absence of frequency modulation of thefield.

To verify the method of the two effective fieldsdescribed above, we consider the simplest case of inter-action of a

Λ

system with a single resonance field in theabsence of frequency modulation. In this case, the mod-ulation index

M

in Liouvillean (9) is zero and field

E

(

t

)acts on both transitions. The results in the absence ofmodulation can easily be predicted, and their verifica-tion in the calculation of FM spectra by the method oftwo effective fields in the corresponding special case(

M

= 0) is the natural tool for controlling the correct-ness of the proposed method.

Figure 4a shows the dependences of the populationof the excited state on the laser detuning and frequencysplitting

ω

12

of the ground state, which were obtainedon the basis of the two-frequency model. The corre-sponding values of the parameters are chosen proceed-ing from experimental conditions [8]. Let us estimatethe resonance width using the formula [6]

(10)

where

Γ

is the rate of transverse relaxation between thelower levels and

γ

=

γ

31

+

γ

32

. The resonance width cal-culated by this formula and estimated from Fig. 4a is0.5

γ

. It also follows from the spectrum that, in the case

∆0 Γ Ω2

γ------,+=

when the

Λ

system interacts with only one field, a darkresonance is observed when the system is degenerate(

ω

12

= 0), i.e., when the magnetic field is zero.

4.4.2. Frequency-modulated field.

Using the two-frequency model described above, we calculated the 3Ddependence of the population of the upper level on theground state splitting

ω

12

and laser detuning (Fig. 5a).However, for better visualization, we consider the crosssection corresponding to the condition

δ

L

= 0 (i.e., thecase when the frequency of the field at transition 1–3 isin exact resonance with the transition frequency ω13.Figure 6a shows the dependence of the population ofthe upper level for δL = 0 on the ground state splittingω12 for M = 10 and for modulation frequency Ωmod = 6γ(γ = 1 MHz). It follows from the graph that the spec-trum contains additional CPT resonances at frequenciesmultiple to the modulation frequency.

Let us consider qualitatively the mechanism of for-mation of dark resonances in the spectrum of a Λ sys-tem interacting with a frequency-modulated laser field(Fig. 7). A CPT resonance is formed when the spacingω12 between the lower sublevels of the ground state isexactly equal to the spacing between the bands in theemission spectrum (i.e., when the splitting between theZeeman sublevels ω12 = ω1 – ω2 = 2µBgFB , where µB isthe Bohr magneton, gF is the Lande factor, and B themagnetic field strength, is equal to modulation fre-quency Ωmod). All pairs of components with the spacingequal to modulation frequency Ωmod (e.g., the pairmarked by solid arrows in Fig. 7) contribute to the res-onance, for which ω12 = Ωmod. Analogously, all pairs ofcomponents with a spacing equal to double the modu-lation frequency 2Ωmod (e.g., the pair marked by dashed

n3, rel. units

0.16

0.12

0.08

0.04

0–8 –4 0 4 8–8–4048

ω12/γ

δ L/γ

(a)0.2

0.1

0

8

–8

80ω12/γ

t/γ

n3, rel. units

(b)

Fig. 4. Dependence of the population n3 of the excited state of a Λ system interacting with a single field in the absence of modulation(M = 0) (a) on laser detuning δL and frequency splitting ω12/γ of the ground state, calculated using the two-frequency model and(b) on time and on ω12/γ, obtained by calculating the exact dynamics of the Λ system using the method of density matrix. The fol-lowing parameters were used in calculations: g13 = g23 = 1 rel. unit, γ31 = γ32 = γ = 1 rel. unit (1 rel. unit = γ = 1 MHz); the remainingrelaxation parameters were assumed to be equal to zero.



0.05

0.04

0.03

0.02

0.01

0

80706050403020100ω12/γ

–4–2

n 3, r

el. u

nits

δL /γ

0.06

0.04

0.02

0510

1520

020

4060t/γ

n 3, r

el. u

nits

ω12/γ

(b)(a)

Fig. 5. Dependence of the population n3 of the excited state of a Λ system interacting with an FM field with M = 10 (a) on laserdetuning δL and frequency splitting ω12 of the ground state, calculated using the two-frequency model, and (b) on time and on ω12 ,obtained by calculating the exact dynamics of the Λ system using the method of density matrix. The parameters used in calculationsare the same as in Fig. 4b.

2 4 6 8 10 12 14n

0.02

0.04

0.06

0.08

0.10

Jn2(10)

–48–72 –24 0 24 48 72ω12/γ

0

0.01

0.02

0.03

0.04

0.05

0.06

–48–72 –24 0 24 48 72ω12/γ

0.01

0.02

0.03

0.04

0.05

0.06(a) (b)

n3, rel. units n3, rel. units

Fig. 6. Population of the excited state of a symmetric Λ system as a function of ω12 for modulation index M = 10, modulation fre-quency Ωmod = 6γ, and Rabi frequencies g13 = g23 = 0.8γ. The results were obtained using (a) the two-frequency model and (b) bycalculating exact dynamics of the Λ system by the method of density matrix. The inset shows the square of the Bessel function

(M) for M = 10.Jn2

arrows in Fig. 7) contribute to the resonance, for whichω12 = 2Ωmod, and so on. It follows hence that the spac-ing between the adjacent resonance observed in thespectrum is equal to modulation frequency Ωmod.

The amplitudes of spectral bands are proportional toBessel functions Jn(M) for a fixed modulation index Mand tend to zero upon an increase in n; consequently,the number of bands in the spectrum is approximately

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VLADIMIROVA et al.

equal to M, and the number of actually observed reso-nances is approximately equal to M.

It can be seen from Fig. 6a that resonance at certainfrequencies are virtually absent; for example, forM = 10, resonances at frequencies ω12 = ±Ωmod ,±3Ωmod , and ±6Ωmod are practically not observed. Thiscan be explained by the fact that, in the case of inter-action of the system with modulated fields, two-pho-ton CPT resonances are observed against the back-ground of one-photon resonances. It was shown, forexample, in [15] that the dependence of the power ofa modulated signal transmitted through the mediumfor a given value of M is proportional to the square of

the Bessel function (M). Since the square of theBessel function assumes zero values for n = ±1, ±3,and ±6 for M = 10, resonances at frequencies nΩmodfor these values of n are practically absent. Calcula-tions carried out for larger values of the modulationindex on the order of 100 resulted in analogous depen-dences: resonances are observed at the same frequen-cies nΩmod , which are multiple to the modulation fre-quency and for which the square of the Bessel func-tion assumes nonzero values.

The mechanism of emergence of additional CPTresonances upon an increase in the modulation index isillustrated in Fig. 8, which shows the dependence of thepopulation of the upper level on the ground-state split-

Jn2

ting ω12 for a fixed modulation frequency Ωmod = 2 forvarious values of modulation index 0 < M < 5. For M =0, modulation is absent and a dark resonance isobserved, as expected, for ω12 = 0 (i.e., when the systemis degenerate). A further increase in the modulationindex leads to the emergence of additional resonancesfor ω12 = nΩmod, n = ±1, ±2, ±3, … . This is due to thefact that the number of side bands in the spectrumincreases with the modulation index in accordance withformula (3).

Calculations also show that the relative amplitude ofresonances and their contrast sharply decrease upon anincrease in the decay rate γ of the excited state, while anincrease in the Rabi frequency results in an increase inthe relative amplitude of the resonances and theircontrast.

5. EXACT CALCULATION OF FREQUENCY-MODULATED CPT SPECTRA

To estimate the accuracy of the results obtained withthe help of the simple two-frequency model, let us con-sider the exact problem of interaction of a Λ systemwith the FM field using the method of the densitymatrix, whose time evolution is determined by a kineticequation of the form (4). The Hamiltonian of interac-

δl

ω12

|1⟩

|2⟩

|3⟩

Ω2Ω3Ω4ΩnΩ

–12 –8 –4 0 4 8 12ω12/γ

0

0.04

0.08

0.12

0.16

n3, rel. units

Fig. 7. Mechanism of formation of dark resonances in the case of a Λ system interacting with a frequency-modulated laser field.



tion of the Λ system with FM field (3) has the form

(11)

where ∆(t) = Msin(Ωt). Substituting the expression forr and expression (11) into Eq. (4), we obtain a set ofnine differential equations; the solution of these equa-tions gives, among other things, the dependence ρ33(t),which describes the time variation of the population ofthe excited state of the Λ system for a given value ofω12. Calculating in this way the population for therange of ω12 values, we obtain a 3D dependencedescribing the formation of CPT resonances in time forvarious values of the ground state splitting. We per-formed such calculations and compared the results withthose obtained in the simple two-frequency model (seebelow).

By analogy with Section 4.4, we will first considerthe case when the system interacts with only one field(see Fig. 4b). For M = 0, modulation is absent and adark resonance is observed, as expected, for ω12 = 0(i.e., when the system is degenerate), which completelycoincides with the result obtained using the superoper-ator approach (Fig. 4a).

Figure 5b shows the dependences of the populationof the excited state on time and ground state splitting inthe case of an FM field, which were calculated using the

H

0 0 ei∆ t( )g13

0 ω12– ei∆ t( )g23

e i∆ t( )– g13 e i∆ t( )– g23 δL⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,=

density matrix method. Figure 6b shows the cross sec-tion t = 25γ–1 corresponding to the regime when CPTresonances are formed. A comparison of the curves inFigs. 6a and 6b reveals qualitative matching of theresults obtained using different methods. It can be seen,however, that the method of density matrix does notallow us to estimate exactly the width and contrast ofresonances since the stationary state is attained fort ∞, while calculations using this method are basedon a finite time interval; for this reason, the results forabsorption are exaggerated.

6. CONCLUSIONS

We proposed and analyzed a simple two-frequencymodel for solving the problem of FM spectroscopy ofdark resonances for a three-level Λ system. It wasshown that, since the spectrum of a FM field is a set ofequidistant frequency components, the interaction ofthe Λ system with the FM field can be described byreplacing the FM spectrum by the spectrum of twoeffective monochromatic fields. The validity and preci-sion of the two-frequency model was demonstrated bycomparing the results obtained using this model withthe result of solution of the exact problem of interactionof a Λ system with an FM field (see Section 5).

Using this two-frequency model, we calculated theabsorption spectra of a Λ system interacting with a FMfield. It was shown that in the case of a Λ system inter-acting with a single resonance field in the absence ofmodulation (M = 0), a dark resonance is observed in thespectrum only for zero splitting of the ground state (i.e.,for a degenerate Λ system). In the presence of modula-tion, additional CPT resonances are formed in the spec-trum at frequencies multiple of modulation frequencyΩmod; i.e., a resonance is formed when the intermodalspacing nΩmod is equal to the Zeeman splitting ω12 ofthe ground state. The number of resonances depends onmodulation index M and increases with the value of thisindex. For different values of M, the intensities of dif-ferent resonances are different and are controlled by thearguments of the Bessel functions Jn(M), which deter-mine the amplitudes of the frequency harmonics. Whenthe position of a resonance corresponds to the mini-mum of this function, its intensity is virtually equal tozero. It is shown that the results obtained using the two-frequency model qualitatively confirm the formation ofextra CPT resonances in the experimental spectra at fre-quencies multiple to the modulation frequency.

ACKNOWLEDGMENTS

This study was partly financed by the Russian Foun-dation for Basic Research (project no. 04-02-17554).

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168

0–8

–16

5

4

3

2

1

0

0.2

0

M

ω 12/γ

n 3, r

el. u

nits

Fig. 8. Population of the excited state as a function of ω12and modulation index M. Other parameters of the systemare as follows: γ31 = γ32 = γ, Ωmod = 2γ, g13 = g23 = 0.8γ,and γ = 1 MHz.

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Translated by N. Wadhwa

Theory of frequency modulation spectroscopy of coherent dark resonances

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Transcript of Theory of frequency modulation spectroscopy of coherent dark resonances