Multilevel coherence effects in a two-level atom driven by a trichromatic field

Multilevel coherence effects in a two-level atom driven bya trichromatic field

Z. Ficeka,b,*, J. Sekea, A.V. Soldatova,c, G. Adama, N.N. Bogolubov Jr.a,c

a Institut f€uur Theoretische Physik, Technische Universit€aat Wien, Wiedner Hauptstrasse 8-10/136, A-1040 Wien, Austriab Department of Physics, School of Physical Sciences, The University of Queensland, Brisbane QLD 4072, Australiac Department of Mechanics, V.A. Steklov Mathematical Institute, Gubkin Street 8, 117966 Moscow GSP-1, Russia

Received 16 September 2002; received in revised form 6 January 2003; accepted 7 January 2003

Abstract

We study the absorption and dispersion properties of a weak probe field monitoring a two-level atom driven by a

trichromatic field. We calculate the steady-state linear susceptibility and find that the system can produce a number of

multilevel coherence effects predicted for atoms composed of three and more energy levels. Although the atom has only

one transition channel, the multilevel effects are possible because there are multichannel transitions between dressed

states induced by the driving field. In particular, we show that the system can exhibit multiple electromagnetically

induced transparency and can also produce a strong amplification at the central frequency which is not attributed to

population inversion in both the atomic bare states and in the dressed atomic states. Moreover, we show that the

absorption and dispersion of the probe field is sensitive to the initial relative phase of the components of the driving

field. In addition, we show that the group velocity of the probe field can be controlled by changing the initial relative

phases or frequencies of the driving fields and can also be varied from subluminal to superluminal.

� 2003 Elsevier Science B.V. All rights reserved.

PACS: 32.80.)t; 32.80.Qk; 42.50.Gy; 42.50.Ar

Keywords: Two-level atom; Absorption; Dispersion; Coherence; Quantum interference; Group velocity

1. Introduction

There has recently been a great interest in the

theoretical and experimental studies of changescaused by strong driving fields in the absorption

and dispersion properties of a weak probe field

monitoring a driven atomic system. In particular,

the excitation of a multilevel atom with coherent

fields has revealed new phenomena of both con-ceptual and practical importance [1,2]. It has been

shown that interference between atomic transitions

can reduce or even suppress spontaneous emission,

trap population in coherent superposition states

that can lead to the phenomenon of electromag-

netically induced transparency (EIT) [3]. The

Optics Communications 217 (2003) 299–309

www.elsevier.com/locate/optcom

*Corresponding author. Tel.: +61-7-3365-2331; fax: +61-7-

3365-1242.

E-mail address: [email protected] (Z. Ficek).

0030-4018/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0030-4018(03)01136-2

mail to: [email protected]

combination of EIT and the reduced spontaneous

emission can lead to a dramatic slowing down of

the group velocity of light [4].

The unusual effects have been predicted in

multilevel atoms that involve three or more atomic

levels, with a pair of these levels coherently cou-pled by a strong auxiliary field or by coherent

fields driving the atomic transitions [5]. Similar

effects can be observed in multilevel systems in

which coherence is created by spontaneous emis-

sion [2]. In this case spontaneous emission in one

of the atomic transitions can drive the other

transition leading to quantum interference be-

tween the amplitudes of the atomic transitions.Recently, it has been shown that multilevel co-

herence effects such as EIT, high index of refrac-

tion accompanied by vanishing absorption, and

negative dispersion in the absence of absorption

can be produced in a simple two-level atom driven

by a coherent monochromatic laser field [6–11].

Another area of interest involved a two-level atom

driven by a bichromatic or even multichromaticfield [12–15]. Various aspects of this problem have

been studied, and it has been shown that the

multilevel coherence effects, predicted for the

monochromatic case, can also occur in the bi-

chromatic case [16]. Moreover, in the transient

regime, the system can exhibit phase-dependent

dynamics sensitive to the initial relative phase of

the driving fields [17]. Phase-dependent effects andtrapping states in the transient regime have also

been predicted for a phase modulated driving field

[18].

The multilevel effects predicted in a two-level

atom driven by monochromatic or bichromatic

fields are only a few examples of coherence effects

that can occur in multilevel atoms. In this paper,

we study the absorption and dispersion propertiesof a probe field monitoring a two-level atom dri-

ven by a trichromatic field and show that this

system can produce a number of the multilevel

coherence effects which are not present in the

monochromatic or bichromatic case. We will show

that apart from the effects predicted with the

monochromatic driving field, the system can pro-

duce a strong amplification without any popula-tion inversion and phase-dependent absorption

and dispersion of a probe field monitoring the

driven atom. In fact, we predict that absorption

can change to amplification and the group velocity

can change from subluminal to superluminal even

if the only difference between the experimental

setups is the presence of a phase shifter in one of

the frequency component of the driving field.

2. The model

We consider a two-level atom with ground state

j1i, excited state j2i and the transition frequencyxa. The atomic states are connected by the tran-

sition dipole moment ~ll that is represented by theatomic spin raising (Sþ ¼ j2ih1j) and lowering

(S� ¼ j1ih2j) operators. The zero of energy for theatom is taken to be midway between the ground

and excited states such that the ground state en-

ergy (in units of �hxa) equals to �1=2.We assume that the atom is driven by a coher-

ent field and also is coupled to the remaining

modes of the three-dimensional electromagnetic(EM) field, which are in the vacuum state. The

coupling to the vacuum field leads to spontaneous

emission with a rate C. The driving field is taken asa trichromatic field with the amplitude

~EEðtÞ ¼ 1

2~EE�1ðtÞeiðx�1tþw�1Þh

þ~EE0ðtÞeiðx0tþw0Þ

þ~EE1ðtÞeiðx1tþw1Þiþ c:c:

¼ 1

2eiðx0tþw0Þ ~EE0

hþ~EE�1ðtÞeiðd�1tþ/�1Þ

þ~EE1ðtÞeiðd1tþ/1Þiþ c:c:; ð1Þ

where~EEnðtÞ and xn (n ¼ �1; 0; 1) are the amplitudeand frequency of the nth component, respectively,

d1 ¼ x1 � x0 is the detuning between the side-

band (n ¼ 1) and the central (n ¼ 0) compo-

nents, wn is the phase of the nth component and

/1 ¼ w1 � w0 is the relative initial (t ¼ 0) phasebetween the central and sideband components.

We analyze the multilevel coherence effects in

the driven two-level atom by calculating the re-

sponse of the system to a weak tunable probe field

of frequency xp. The response is given by the lin-

ear susceptibility vðxpÞ that can be written in termsof the Fourier transform of the average value of

300 Z. Ficek et al. / Optics Communications 217 (2003) 299–309

the two-time commutator of the atomic dipole

operators as [19–21]

vðxpÞ ¼ C1

T

Z T

0

dt0Z t0

0

dt ½~SS�ðtÞ; ~SSþðt0Þ�D E

eiðxp�x0Þðt�t0Þ

!; ð2Þ

where T is the integrating time of the detector, and

~SSðtÞ ¼ iSðtÞe�ix0t ð3Þare slowly varying parts of the atomic dipole op-

erators.

The two-time correlation functions of the

atomic dipole operators, appearing in Eq. (2), can

be found from the master equation of the system,which has the form [22]

o

otqðtÞ ¼ � i

�h½ ~HH ; qðtÞ� � 1

2C ~SSþ ~SS�qðtÞ�

þ qðtÞ~SSþ ~SS� � 2~SS�qðtÞ~SSþ; ð4Þ

where

~HH ¼ �hDSz � 12i�h½XðtÞ~SSþ �H:c:�: ð5Þ

Here, Sz ¼ ðj2ih2j � j1ih1jÞ=2 is the population

inversion operator, D ¼ xa � x0 is the detuning of

the central component of the field from the atomic

resonance, and

XðtÞ ¼ X0 þ X�1 e�iðd�1tþ/�1Þ þ X1 e

�iðd1tþ/1Þ ð6Þ

is the Rabi frequency of the driving field, with

Xn ¼~ll �~EEn=�h.From the master equation (4) and the quantum

regression theorem [23], we find that the two-time

commutator satisfies optical Bloch equation,which can be written in a matrix form as

d

dt~vvðt; t0Þ ¼ MðtÞ~vvðt; t0Þ; ð7Þ

where ~vvðt; t0Þ is a column vector with the compo-nents

v1ðt; t0Þ ¼ h½~SS�ðtÞ; ~SSþðt0Þ�i;v2ðt; t0Þ ¼ h½~SSþðtÞ; ~SSþðt0Þ�i;v3ðt; t0Þ ¼ h½SzðtÞ; ~SSþðt0Þ�i;

ð8Þ

and MðtÞ is the 3 3 matrix

MðtÞ ¼� 1

2C þ iD

�0 XðtÞ

0 � 12C � iD

�X�ðtÞ

� 12X�ðtÞ � 1

2XðtÞ �C

0@

1A:

ð9Þ

One can see from Eq. (6) that the time modulation

of the Rabi frequency XðtÞ is quite complicated asit involves two parameters d�1 and d1. The timemodulation can be simplified to only one param-

eter when the frequencies of the field components

are equidistant and symmetrically located about

the central frequency x0, such that d1 ¼ �d�1 ¼ d.Moreover, for X�1 ¼ X1 ¼ Xs and /�1 ¼ /1 ¼ /,the trichromatic field is equivalent to an amplitude

modulated field of the Rabi frequency

XðtÞ ¼ X0 þ 2Xs e�i/ cos dt: ð10Þ

In Section 3, we present numerical solutions forthe stationary linear susceptibility vðxpÞ of a weakprobe field monitoring the system. We illustrate

the solutions for two cases, the amplitude modu-

lated field whose the Rabi frequency is given in Eq.

(10), and for a more general case of X�1 6¼ X1 and

/�1 6¼ /1. As we will see, in both cases the sus-

ceptibility reveals the multilevel coherence effects

such as EIT, large refractive index without ab-sorption, subluminal and superluminal light

propagation, amplification without population

inversion, and phase control of atomic absorption

and propagation of light.

Our calculations of the linear susceptibility

follow the Floquet method [24], in which the

atomic dynamics are described in terms of Fourier

harmonics of the dipole correlation functions. Inthis approach, we first make a harmonic decom-

position of the two-time expectation values [24]

vkðt; t0Þ ¼X1l¼�1

vðlÞk ðt; t0Þeildt; k ¼ 1; 2; 3; ð11Þ

where vðlÞk are slowly varying harmonic amplitudes.

The harmonic decomposition (11) tells us that the

atomic variables will respond at harmonics of the

modulation frequency, and knowledge of vðlÞk gives

all the information about the evolution of the two-

time correlation functions.Next, we substitute the decomposition (11) into

the two-time optical Bloch equations (7), take the

Laplace transform of the resulting equations of

Z. Ficek et al. / Optics Communications 217 (2003) 299–309 301

motion for the slowly varying amplitudes, and find

the following recurrence relation for vðlÞ3 ðzÞ:

ðzþ C þ ildÞvðlÞ3 ðzÞ þ 1

2

Xn

Xm

~XX�n~XXm

1

Pl�nðzÞ

�

þ 1

QlþmðzÞ

�vðl�nþmÞ3 ðzÞ ¼ glðzÞ; ð12Þ

where PlðzÞ ¼ zþ 12C þ iD þ ild, QlðzÞ ¼ zþ 1

2C�

iD þ ild, z is a complex (Laplace transform) pa-rameter, ~XXn ¼ Xn expð�i/nÞ, and

glðzÞ ¼ vðlÞ3 ðt0; t0Þ � 1

2

Xn

~XX�n

Pl�nðzÞvðl�nÞ1 ðt0; t0Þ ð13Þ

is an inhomogeneous term given by the initial

values (t ¼ t0) of the Floquet components of theatomic correlation functions

vðlÞ1 ðt0; t0Þ ¼ �2hSzðt0ÞiðlÞ; vðlÞ

2 ðt0; t0Þ ¼ 0;

vðlÞ3 ðt0; t0Þ ¼ hSþðt0ÞiðlÞ: ð14ÞBecause we are interested in the structure of the

linear susceptibility after a sufficiently long time,

we assume that vðlÞk ðt0; t0Þ are the stationary har-

monic amplitudes evaluated at t0 ! 1. In this

limit, we can find vðlÞk ðt0; t0Þ from a recurrence re-

lation which can be easily obtained from the re-

currence relation (12) by putting z ¼ 0 and

replacing glðzÞ by �12Cdl;0.

According to Eqs. (2) and (8), the stationary

linear susceptibility vðxpÞ is given in terms of thezeroth-order harmonic of the v1ðzÞ component as

vðxpÞ ¼ Cvð0Þ1 ðzÞjz¼�iðxp�x0Þ; ð15Þ

where

vð0Þ1 ðzÞ ¼ 1

P0ðzÞvð0Þ1 ðt0; t0Þ

þX1n¼�1

~XXnvðnÞ3 ðzÞ

!: ð16Þ

The linear susceptibility (2) can be written as

vðxÞ ¼ v0ðxÞ � iv00ðxÞ; ð17Þwhere the real (v0) and imaginary (v00) parts of vdetermine the absorptive and dispersive properties

(spectra) of the probe field. Thus, the stationary

probe absorption and dispersion spectra are given,respectively, by the real and imaginary parts of the

zeroth-order harmonic vð0Þ1 ðzÞ as

v0ðxpÞ ¼ CRevð0Þ1 ðzÞjz¼�iðxp�x0Þ; ð18Þ

and

v00ðxpÞ ¼ �C Imvð0Þ1 ðzÞjz¼�iðxp�x0Þ: ð19Þ

The harmonics vð0Þ1 ðzÞ depends on the harmonics

vðnÞ3 ðzÞ (n ¼ �1; 0; 1) that are obtained by solvingthe recurrence relation (12) using the truncated

basis of the harmonic amplitudes. The validity ofthe truncation is ensured by requiring that vðnÞ

3 ðzÞdoes not change as the number of truncated har-

monics is increased or decreased by one. Thus,

using the results (18) and (19), the absorption and

dispersion spectra can be evaluated to any desired

accuracy and for arbitrary detunings D and d, ar-bitrary Rabi frequencies Xn, and the relative pha-

ses /n.In terms of this model, we can also calculate the

group velocity vg of the probe field given by thefollowing approximate expression [4,25]

cvg

¼ 1� 2p ImvðxpÞ � 2pxp Imovox

� �x¼xp

: ð20Þ

The ratio c=vg can be smaller than unity (super-luminal light) when the dispersion is negative in

the region of ImvðxpÞ � 0, and can be larger thanunity (subluminal light) when the dispersion is

positive in this region.

In the figures below, we plot the contribution to

the group velocity of the normalised third term

appearing in Eq. (20), that can be written as

�Im oðCv=X0Þoðx=CÞ

� �x¼xp

¼ 1

Acvg

�� 1�; ð21Þ

where A ¼ 2pxpl2=ðC2�hÞ is a parameter whichdepends on the characteristics of the dispersive

medium [4].

3. Results and discussion

Before discussing the multilevel coherence ef-

fects in a two-level atom driven by a trichromaticfield, let us briefly recall the basic features of the

absorption and dispersion predicted in the case of

a monochromatic as well as a bichromatic driving

field. The absorption spectrum for a strong

monochromatic driving field was predicted by

Mollow [19] and has been experimentally observed

by Wu et al. [20] in a beam of sodium atoms, and


recently by Tamarat et al. [26] in single molecules.

In the case of the resonance the spectrum consists

of two dispersion-like features of relatively small

amplitudes, located at x0 X0, where X0 is the

Rabi frequency of the driving field. For an off-

resonant driving field, the spectrum consists of oneabsorption and one emission components at the

Rabi sidebands, and a small dispersion-like com-

ponent at the center of the spectrum. The emission

component indicates that in one Rabi sideband

stimulated emission overweighs absorption, so

that the probe field is amplified at the expense of

the driving field. The spectral features can be well

understood in terms of the dressed-atom model[27,28], in which the spectral features are related to

the energy eigenstates of the entangled atom-

driving-field system and are viewed as arising from

the transitions between them. When the driving

field frequency x0 is equal to the atomic transition

frequency xa, the populations of the dressed states

are equal, and therefore the spectrum is composed

of small dispersion features whose the origin is insmall coherences between the dressed states [29].

When x0 6¼ xa, the populations of the dressed

states are different and therefore the probe tuned

to one of the Rabi sidebands would experience

absorption and when tuned to another one the

probe would be amplified due to the population

inversion between the dressed states. Apart from

the amplification at one of the Rabi sidebands, theprobe also exhibits an amplification on one side of

a small dispersion-like structure centered at x0.

The amplification originates from the complicated

multiphoton interference between absorption and

emission processes and is not associated with any

population inversion because the transition occurs

between equally populated states both in the bare

and in the dressed atom basis [30–32].In addition to the absorption spectrum, Wilson-

Gordon and Friedmann [6] and Freedhoff and

Quang [7] have calculated the dispersive spectrum

and have shown that at the Rabi sidebands, where

the absorption vanishes, the refractive index is large.

Szymanowski and Keitel [8] (see also [9]) have ex-

tended these calculations to include the Doppler

broadening.Wahiddin et al. [10] have shown that inthe presence of a squeezed vacuum both absorption

and dispersion can vanish in a broad area of the

probe frequency. Recently, Bennik et al. [11] have

reproduced the earlier results for the absorption and

dispersion spectra of the driven two-level atom and

have related the vanishing absorption to the phe-

nomenon of electromagnetically induced transpar-

ency (EIT). They have also calculated the groupvelocity and have shown that the group velocity can

be either smaller or greater than that of the velocity

of light in vacuum.

In Fig. 1, we show the real part v0ðxpÞ(absorption) and the imaginary part v00ðxpÞ (dis-persion) for a strong resonant (Fig. 1(a)) and off-

resonant (Fig. 1(b)) monochromatic driving field.

As predicted in [6–9,11], for the resonant drivingfield the system exhibits a large dispersion with

vanishing absorption. One can see from Fig. 1 that

there are two frequencies at which EIT (vanishing

absorption) can be observed. Moreover, for the

off-resonant case, regions of absorption are char-

acterized by positive dispersion, whereas regions of

amplification are characterized by negative dis-

persion.The absorption and dispersion spectra of a two-

level atom driven by a bichromatic field have also

received considerable attention [13]. In this case,

the spectra consist of a central component and a

series of dispersive or absorptive sidebands whose

-200 -150 -100 -50 0 50 100 150 200

-2

0

2

x 10-3

-200 -150 -100 -50 0 50 100 150 200

-0.1

0

0.1

0.2

χ’(ω

p ),

χ’’(ω

p )

χ’(ω

p ),

χ’’(ω

p )

(ωp

−ωa) /Γ

(ωp − ω

a) /Γ(a)

(b)

Fig. 1. The absorption v0ðxpÞ (solid line) and dispersion v00ðxpÞ(dashed line) for a monochromatic driving field with X0 ¼ 100Cand different detunings D: (a) D ¼ 0, (b) D ¼ 30C.


the amplitudes and positions depend in general on

the Rabi frequencies of the two driving fields. For

equal Rabi frequencies of the driving fields, the

absorption spectrum exhibits dispersion-like

structures at the sideband frequencies and an ab-

sorption-like structure at the central frequencywhose the amplitude oscillates with the Rabi fre-

quency of the driving fields. The situation for

the dispersive spectrum is analogous when dis-

persion-like and absorption-like structures are

interchanged. In contrast to the case of a mono-

chromatic driving field, the positions of the side-

bands are independent of the Rabi frequency of

the driving field. The positions depend only on thedetuning d of the laser fields from the atomic res-

onance. However, the magnitudes of the spectral

features depend strongly on both, the Rabi fre-

quency and the detuning d. This is shown in Fig. 2,where we plot the absorption v0ðxpÞ as a functionof xp and d for a strong bichromatic field withX�1 ¼ X1 ¼ X ¼ 50C and D ¼ 0. The absorption

spectrum is symmetric about xp � xa ¼ 0 and thecentral component displays amplitude oscillations

with transparency windows (EIT) appearing at

d � X=n; n ¼ 1; 2; 3; . . . [33,34]. However, the

spectrum is independent of the initial phases of the

driving fields [35].

EIT and large refractive index accompanied by

vanishing absorption have been first predicted in

three- and four-level systems [3,36]. Both these

effects have been regarded as multilevel coherence

effects resulting from a coherence between two

atomic transitions. Figs. 1 and 2 show that these

multilevel coherence effects can also be observed in

a system of a driven two-level atom that is much

simpler than the three- and four-level systems.However, there is a number of coherence effects

predicted in multilevel atoms which are not present

in the simple system of a two-level atom driven by

a monochromatic field. Examples include sup-

pression of spontaneous emission, amplification

without any population inversion, phase control of

the atomic dynamics, and the frequency and phase

control of the group velocity. In the following, wewill show that a two-level atom driven by a tri-

chromatic field can exhibit all of these multilevel

coherence effects, in particular, amplification

without any population inversion and the phase-

dependent control of absorption, dispersion and

the group velocity.

We begin by considering the stationary linear

susceptibility of the system of a two-level atomdriven by an amplitude modulated field. The ear-

lier theoretical studies showed that the fluores-

cence and absorption spectra of the system are

composed of a series of features located at fre-

quencies [37]

x ¼ x0 ðX0 mdÞ; m ¼ 0; 1; 2; . . . ; ð22Þ

Fig. 2. Three-dimensional plot of the absorption v0ðxpÞ as afunction of ðxp � xaÞ=C and the detuning d=C for a bichro-

matic driving field with X0 ¼ 0;X�1 ¼ X1 ¼ 50C and D ¼ 0.

Fig. 3. The absorption v0ðxpÞ (solid line) and dispersion v00ðxpÞ(dashed line) for an amplitude modulated driving field with

X0 ¼ 100C, X�1 ¼ X1 ¼ 50C, /�1 ¼ /1 ¼ 0 and d ¼ 25C.


i.e., the system responses at the Rabi frequency of

the central component and harmonics of the

modulation frequency of the driving field. In Fig. 3

we show the absorption, v0ðxpÞ, and dispersion,v00ðxpÞ, for a strong amplitude modulated field

with X0 ¼ 100C, X�1 ¼ X1 ¼ 50C, /�1 ¼ /1 ¼ 0and d ¼ 25C. We see that the absorption vanishesat frequencies close to the harmonic resonances of

the driving field. Moreover, at the harmonic res-

onances the dispersion is less negative and is al-

most equal to zero. Therefore, at the subharmonic

resonances the system is almost transparent to the

probe field. Consequently, multiple EIT can be

observed in this system, with many transparencywindows occurring near the subharmonic reso-

nances. The positions of the transparency windows

depend on the Rabi frequency X0.

In Fig. 4, we show v0ðxpÞ and v00ðxpÞ for anasymmetric driving field with D ¼ �d ¼ 25C,X�1 ¼ 100C, and X0 ¼ X1 ¼ 50C. In this case thelower frequency sideband of the driving field is on

resonance with the atomic transition, whereas the

central component and the higher frequency side-band are detuned from the resonance by d and 2d,respectively. Here, the probe is strongly absorbed

at frequencies smaller than the atomic resonant

frequency, but is amplified at frequencies larger

than xa. Moreover, the regions of low absorption,

between the absorption peaks, are characterized

by positive dispersion, whereas the area of low

absorption between the amplification peaks is

characterized by negative dispersion.

The absorption and amplification properties at

the central frequency xp ¼ xa can be controlled bythe detuning d such that below a harmonic reso-

nance the probe is absorbed, while above the res-

onance the probe is amplified. This is shown in

Fig. 5, where we plot v0ðxpÞ for xp close to the

atomic resonance xa, and the amplitude modu-

lated field with X0 ¼ 100C; X�1 ¼ �X1 ¼ �50Cand different d. By varying the detuning d near oneof the harmonic resonances the absorption at thecentral component of the spectrum can switch into

an amplification.

The amplification at the central component, seen

in Fig. 5, is an example of amplificationwithout any

population inversion. We have calculated the sta-

tionary population inversion hSzi ¼ 12ðq22 � q11Þ,

where q22 and q11 are the populations of the atomicexcited and ground states, respectively, and havefound that for the same parameters as in Fig. 5,

hSzi ¼ �1:2 10�4 < 0. Thus, there is no popula-

tion inversion between the atomic bare states. In

addition, there is no population inversion between

dressed states of the system. It is easy to understand

if one recalls that the net absorption at any fre-

Fig. 4. The absorption v0ðxpÞ (solid line) and dispersion v00ðxpÞ(dashed line) for an asymmetric trichromatic driving field with

D ¼ �d ¼ 25C, X�1 ¼ 100C, and X0 ¼ X1 ¼ 50C.

Fig. 5. Three-dimensional plot of the absorption v0ðxpÞ as afunction of ðxp � xaÞ=C and d=C for the amplitude modulatedfield with X0 ¼ 100C and X�1 ¼ �X1 ¼ �50C.


quency is proportional to the difference between the

populations of the lower and upper states in the

transitions. In the dressed atom model the transi-

tions occur between dressed states of two neigh-

boringmanifolds, say jn;Ni and jm;N þ 1i, where nand m label dressed states in each manifold. Thefrequency of the central component corresponds to

transitions between dressed states with n ¼ m, thatare equally populated [13,27]. Thus, the amplifica-

tion seen in Fig. 5 is not due to a population inver-

sion between the dressed states.

The origin of the amplification at the central

component of the absorption spectrum is in the so-

called coherent population oscillations [38], whichare induced by the driving and probe fields beating

together at the difference frequency xp � x0. The

effect is similar to that appearing in the case of a

two-level atom driven by a monochromatic field

and damped by a squeezed vacuum, where the

correlations between two frequencies symmetri-

cally located about the atomic resonance enhance

the coherent population oscillations [39]. In ourcase the two sideband components of the driving

field are correlated through the interaction with

the atom.

Of particular interest is the dependence of the

stationary absorption and dispersion on the initial

relative phase of the driving fields, the phenome-

non not observed with a monochromatic as well as

a bichromatic driving field [13]. The phase depen-dence of the stationary absorption spectrum is il-

lustrated in Fig. 6, where we plot v0ðxpÞ for thesame parameters as in Fig. 5, but fixed d ¼ 65Cand different initial relative phases of the compo-

nents of the driving field. The graph shows that the

absorption spectrum is strongly sensitive to the

relative phase between the field components, in

particular, near the central component, where ab-sorption can switch into an amplification as the

phase of one of the sideband components varies

from zero to p.The analysis of multilevel systems showed that

the fluorescence and absorption spectra depend on

the phase when the driving fields form a closed

loop [4,40–43]. An interesting question arises how

this could be implemented in a two-level atomdriven by three fields of different frequencies. In

the situation presented in Fig. 6, the central com-

ponent of the amplitude modulated field is reso-nant while the sidebands are equally detuned from

the resonance. A closed loop cannot be seen be-

tween the bare atomic states, but between dressed

states produced by the central component of the

field. This is shown in Fig. 7, where we plot the

energy level diagram of the dressed states pro-

duced by the central component of the field [27].

Fig. 6. The absorption v0ðxpÞ as a function of ðxp � xaÞ=C forX0 ¼ 100C, X�1 ¼ X1 ¼ 50C, d ¼ 65C and different initial rel-

ative phases of the driving fields: /�1 ¼ /1 ¼ 0 (solid line),

/�1 ¼ p;/1 ¼ 0 (dashed line).

Fig. 7. Energy level diagram of the atom dressed by the central

component of the amplitude modulated field. The arrows in-

dicate the two sideband components of the driving field, sym-

metrically detuned from the central component by the

frequency d.


As we can see the energy spectrum is composed of

a ladder of dressed states with multichannel tran-

sitions between them. A closed loop is formed

when the sideband fields couple simultaneously to

the j2;N � 1i ! j2;N þ 1i transition, where N is

the number of photons in the central componentof the driving field.

Having available the dispersion spectrum v00ðxpÞ,we can also calculate the group velocity of the probe

field. In Fig. 8, we show a three-dimensional plot of

the expression (c=vg � 1) as a function of xp and dfor an amplitude modulated field with X0 ¼ 100C,X�1 ¼ �X1 ¼ �50C, and /�1 ¼ /1 ¼ 0. We see

that with increasing d the factor (c=vg � 1) changesfrom negative to positive. In Fig. 9, we plot

(c=vg � 1) as a functionof the frequency of the probefield for two different initial relative phases of the

components of the driving field. It is seen that the

group velocity of the probe field can be controlled

by adjusting the relative phase of the driving fields.

Depending on the phase, the factor (c=vg � 1) can bepositive or negative. The group velocity in the sys-tem of two-level atoms can be measured by propa-

gating a weak probe pulse in amedium consisting of

cold atoms, thereby eliminating Doppler broaden-

ing. Cold atoms have already been used by Hau et

al. [44] in an experiment in which they observed

reduced group velocity in a gas of sodium atoms

cooled to temperatures below the transition tem-

perature for Bose–Einstein condensation. An al-

ternative scheme could involve radiofrequency

experiments, where the radiative and Doppler

broadenings are negligible. Recently, the absorp-

tion and dispersion spectra have been observed inexperiments performed on single molecules [45,46]

and a nitrogen-vacancy color center in diamond

[47–49] driven by a radiofrequency bichromatic

field. The experiments could be easily extended to

the case considered in this paper of a trichromatic

driving field.

4. Summary

We have calculated the absorption–dispersion

relation of the system of a two-level atom driven by

a trichromatic field, and have shown that the system

can produce a number of multilevel coherence ef-

fects such as electromagnetically induced transpar-

ency, a high dispersion (index of refraction)accompanied by vanishing absorption, amplifica-

tion without any population inversion, positive and

negative group velocities, and phase control of ab-

sorption and dispersion. We have also examined

how the group velocity of the probe field can be

affected by the detuning and the relative phase

Fig. 8. Three-dimensional plot of (c=vg � 1) as a function ofðxp � xaÞ=C and d=C for the amplitude modulated field with

X0 ¼ 100C and X�1 ¼ �X1 ¼ �50C.

Fig. 9. (c=vg � 1) as a function of ðxp � xaÞ=C for X0 ¼ 100C,X�1 ¼ X1 ¼ 50C, d ¼ 65C and different initial relative phases ofthe driving fields: /�1 ¼ p;/1 ¼ 0 (solid line), /�1 ¼ /1 ¼ 0

(dashed line).


between the components of the driving field. The

group velocity is sensitive to both the detuning and

phase and near the harmonic resonances there can

exist positive and negative group velocities.

The coherence effects have been first predicted

in multilevel atoms composed of three or moreatomic levels, with a pair of these levels coherently

coupled by a strong auxiliary field or coupled

through the vacuum modes. The later requires that

the dipole moments of the atomic transitions are

parallel to each other that allow to produce the

coherence between the transitions via quantum

interference. However, in practice, it is difficult to

find atoms with parallel dipole moments, andtherefore the proposed two-level system allows far

more freedom both in atomic selection and oper-

ation of the system.

Acknowledgements

This work has been supported by The Univer-sity of Queensland Travel Awards for Interna-

tional Collaborative Research. J.S. would like to

acknowledge the support by the Dr. Anton Oel-

zelt-Newinsche Stiftung of the Austrian Academy

of Sciences. A.V.S. and N.N.B. acknowledge the

support by the International Scientific Exchange

Program of the Austrian Academy of Sciences,

and by the RFBR program of Support for LeadingScientific Schools, Grant No. 00-15-96149.

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Multilevel coherence effects in a two-level atom driven by a trichromatic field

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