on the dynamical evolution of a three-level atom with atomic motion in a lossless cavity

Optics Communications 232 (2004) 273–287

www.elsevier.com/locate/optcom

On the dynamical evolution of a three-level atomwith atomic motion in a lossless cavity

Amitabh Joshi *, Min Xiao

Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA

Received 8 June 2003; received in revised form 8 December 2003; accepted 25 December 2003

Abstract

The dynamical evolution of a moving three-level atom in K-configuration interacting with two quantized modes (or

one quantized mode and a classical field) in coherent states inside an ideal cavity is studied using density matrix

equations. The spatial field mode structures of the cavity are taken into account in this study. We also discuss analytic

results under certain parametric conditions pertaining to nonlinear transient effects similar to self-induced transparency,

adiabatic following, etc. The possibility of realizing an optical switching in such system is also described.

2004 Elsevier B.V. All rights reserved.

PACS: 42.50.Ct; 32.80.Qk; 42.50.Pq

Keywords: Cavity QED

1. Introduction

The Jaynes–Cummings model (JCM) [1] de-

scribing interaction between a two-level atom and

a single quantized mode of electromagnetic field in

an ideal cavity has been a center of attraction in

quantum optics during last few decades as many of

its predictions could be experimentally realized.Discussions related to several interesting general-

izations of this model are now available in the

* Corresponding author. Tel.: +1-479-5756402; fax: +1-479-

5754580.

E-mail address: [email protected] (A. Joshi).

0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2003.12.073

literature [2] and the model is still promising in

many applications such as motion of ions in trap,

and in the design of hardwares (such as logic

switches) for realizing quantum computation, etc.

A very significant and noteworthy generalization

of JCM is to include the effect of atomic motion so

that the spatial mode structure of the cavity field

could be incorporated into this model [3]. In thestandard JCM, the interaction between a constant

electric field and a stationary two-level atom is

considered. However, when the spatial structure of

the cavity field mode is taken into account, the

nonlinear transient effects, similar to self-induced

transparency (SIT) and adiabatic following (AF),

could be observed [3]. This kind of model provides

ed.

mail to: [email protected]

274 A. Joshi, M. Xiao / Optics Communications 232 (2004) 273–287

a most elementary system for studying the inter-

action of a single two-level atom with an electro-

magnetic pulse. These nonlinear transient effects

(SIT, AF, etc.) are found to be very sensitive to the

type of atomic transition the atom is undergoing as

well as to the number of modes sustained in thecavity. Some interesting studies for single-mode,

two-photon JCM and two-mode, two-photon

JCM with cavity spatial mode structures taken

into account are reported recently [4]. It would be

rather interesting to explore these effects when the

two-level atom is replaced by a three-level atom in

such modified JCM for studying cavity quantum

electrodynamic (QED) effects. This would lead toanother natural generalization of the modified

JCM discussed in [3] and hence we will follow [3]

closely in our this study.

In recent past a good amount of attention has

been paid in the field of quantum optics to study

three-level atomic systems, under various config-

urations (e.g., ladder (N), lambda (K), and vee (V )type models), interacting with either two quantizedelectromagnetic field modes, or two classical fields

in free space, or one quantized mode and one

classical field in cavity. Extensive studies of a

three-level atom (with different configurations)

under rotating-wave approximation interacting

with quantized fields inside an ideal cavity were

carried out in detail by Yoo and Eberly [2]. Later

on several more studies on dynamical evolutionand field statistics were reported on the similar

type of models [5]. These models could be experi-

mentally tested by utilizing three-level atoms in

various configurations in the micromaser systems.

Note that the micromaser has been experimentally

realized with both one-photon transition as well as

two-photon transition in two-level and effective

two-level atoms in the recent past [6]. The two-mode laser action from three-level atoms was

theoretically investigated to gain insight into the

photon statistics and intrinsic linewidth of such

system, which possesses complicated dynamic

Stark effect [7]. Reports on various aspects of the

interactions between three-level atoms and two

classical optical fields are available in literature

which include extensive studies on the atomicpopulation dynamics, spectral and statistical

properties of the field, and quantum jumps, etc.,

just to mention a few [8]. Also, three-level atomic

systems hold key positions in many phenomena

related to coherence and quantum interference

leading to lasing without population inversion,

materials with high refractive indices, and storage

of light in atomic vapors, etc. [9]. Recently, phe-nomena related to electromagnetically induced

transparency were investigated in three-level at-

oms, which are essential in controlling the changes

in absorption, dispersion and nonlinearity of such

atomic systems so that effects like optical bista-

bility and all-optical switching could be easily

obtained and controlled as desired [10]. In a recent

study the interaction of three-level atoms with onequantized field and a classical field has provided a

controlling mechanism for micromaser and laser

emission processes from such atoms [11]. The

system of three-level atoms in a K-type configu-

ration interacting with a single photon in cavity

along with another manipulating laser field can

give rise to entanglement between different atoms

in the cavity and hence could serve as a promisingall-optical device to realize the CNOT (control-

NOT) gate, a key element for quantum computa-

tion [12].

The studies of multi-level atoms in comparison

to their two-level counterpart are important be-

cause the multi-level atoms can give rise to a

broader range of physical effects due to the in-

duced coherence by the additional classical drivingfields or the cavity quantized fields. The motiva-

tion for further work in this direction also origi-

nates from the recent developments in the

experiments of cavity QED in both optical and

microwave regimes, e.g., transfer of entanglement

among atoms, quantum networking [13,14],

quantum state memories, quantum information

processing, and implementation of quantum logicgates using K-type three-level atoms [15], etc.

In this work, we examine the transient effects,

such as SIT and AF [16], arising from the spatial

mode structures of a microwave cavity due to a

moving three-level atom interacting with two dif-

ferent modes of quantized field or one quantized

field mode and a classical field. We will compare

our results with those of a moving two-level atomundergoing one-photon transition in a single-

mode cavity field with cavity-mode structure in-

A. Joshi, M. Xiao / Optics Communications 232 (2004) 273–287 275

cluded [3]. We will neglect the cavity damping in

our discussion of the model, which is a reasonable

assumption in the microwave regime where cavi-

ties of very high quality factor (Q 4 1010) have

been achieved [17]. The typical interaction times

for the Rydberg atoms (used in microwave cavityexperiments) having very large dipole moments are

of the order of 105 s, which is three orders of

magnitude shorter than the lifetime of photons in a

typical high-Q microwave cavity (about 102 s).

So, the assumption of negligible cavity damping

when an atom enters such cavity is not violated.

Also, the radiative lifetimes of the Rydberg atoms

are long, specifically when the circular Rydbergstates are employed in the study (greater than tens

of milliseconds), so that we can also neglect the

radiative damping of the atom in our model. The

stability of interaction time in an experiment is

uncertain within 2–3% because of the effective

method employed in atomic velocity selection. The

transverse velocity spread (with respect to the

cavity axis) is also very small so that the atomsbasically move along the cavity axis. Note that the

stability of interaction time and small transverse

velocity spread are essential in observing trapping

states of electromagnetic fields in the micromaser

[17]. As we will see in the following that the in-

teraction of a K-type atom with one quantized field

mode and a classical field can give a control over

the dynamical evolution of the atom and hence apossibility of obtaining a switch-like action from

this system.

g1g

2

b, n1, n2

a, n1 + 1, n2

c, n1, n2+1

ba∆bc∆

(a)

Fig. 1. (a) Model A: The three-level atom in K-configuration interactin

atom in K-configuration interacting with one quantized cavity mode

The manuscript is organized as follows. In

Section 2, we describe the two models under con-

sideration in this work. In Section 3, we study the

dynamics of the first model describing interaction

between two quantized field modes and the three-

level atom. Some analytical results along with ex-act numerical results are presented. In Section 4,

the dynamics of the second model where a quan-

tized field mode and a classical field interacting

with the three-level atom is discussed. We present

numerical results along with the possibility of ob-

taining a switch-like action in this system. Section

5 serves as a summary for our investigations.

2. Models

2.1. Density operator equations describing interac-

tion of a K-type three-level atom with two quantized

field modes

The atomic level configuration and parametersfor this model are depicted in Fig. 1(a). The upper

level is jbi (energy Eb ¼ hxb), while the two lower

levels are jai (energy Ea ¼ hxa), and jci (energy

Ec ¼ hxc), respectively. The transition between

level jbi and level jaiðjciÞ is mediated by cavity

mode of frequency x1ðx2Þ with atom–field cou-

pling parameter g1ðg2Þ. The transition between

states jai and jci is forbidden. The Hamiltonianfor the atom–field system is then given, under ro-

tating-wave approximation (RWA), by

g1

b, n1

a, n1 + 1

c, n1

Ω

ba∆bc∆

(b)

g with two quantized cavity modes. (b) Model B: The three-level

and a classical field.

ð7Þ


H ¼ hxajaihaj þ hxbjbihbj þ hxcjcihcj þ hx1ay1a1

þ hx2ay2a2 þ hg1½ay1jaihbj þ a1jbihaj

þ hg2½ay2jcihbj þ a2jbihcj; ð1Þ

where ai ðayi Þ is the annihilation (creation) opera-

tor of the ith (i ¼ 1; 2) cavity field mode. We fur-

ther assume the cavity mode shape function to be

fiðzÞ ði ¼ 1; 2Þ. In view of the successful micro-

wave-type-cavity-QED experiments as discussed

above, we restrict the atomic motion to be alongthe cylindrical axis (e.g., the z-axis) of the cavity so

that only z dependence of the cavity mode function

needs to be considered. We incorporate the atomic

motion in the following way [18]:

fiðzÞ ! fiðvtÞ ði ¼ 1; 2Þ; ð2Þ

in which v denotes the atomic velocity. Thus, forthe cavity mode Tmnpi we have

fiðvtÞ ¼ sinðpipvt=LÞ; ð3Þ

where pi (i ¼ 1; 2) is a positive integer representing

number of half wavelengths of the ith mode sus-

tained inside the cavity of length L. When theshape function of the cavity mode is included into

Eq. (1) we get

H ¼ hxajaihaj þ hxbjbihbj þ hxcjcihcj þ hx1ay1a1

þ hx2ay2a2 þ hg1f1ðzÞ½ay1jaihbj þ a1jbihaj

þ hg2f2ðzÞ½ay2jcihbj þ a2jbihcj: ð4Þ

Since the atom absorbs one photon and then emits

another in an ideal cavity, the basis vectors for the

wave function are ja; n1 þ 1; n2i, jb; n1; n2i,jc; n1; n2 þ 1i and hence it is straightforward to

write down the wave function wðtÞ for this systemas

wðtÞ ¼Xn1;n2

½an1þ1;n2ðtÞja;n1 þ 1;n2i

þ bn1;n2ðtÞjb;n1;n2i þ cn1;n2þ1ðtÞjc;n1;n2 þ 1i:ð5Þ

So, it is clear that the dynamical evolution of the

system for each value of ni (i ¼ 1; 2) is determined

by the three coupled equations for the complex

quantities an1;n2ðtÞ, bn1;n2ðtÞ and cn1;n2ðtÞ. However, it

is possible to define eight real quantities for the

density operator of the system described by Eqs.

(4) and (5) as follows:

un1;n2 2Reðan1þ1;n2bn1;n2

Þ ¼ 2ReðqabÞ;vn1;n2 2Imðan1þ1;n2b

n1;n2

Þ ¼ 2ImðqabÞ;wn1;n2 jan1þ1;n2 j

2 jbn1;n2 j2 ¼ qaa qbb;

dn1;n2 2Reðan1þ1;n2cn1;n2þ1Þ ¼ 2ReðqacÞ;

fn1;n2 2Imðan1þ1;n2cn1;n2þ1Þ ¼ 2ImðqacÞ;

pn1;n2 2Reðbn1;n2cn1;n2þ1Þ ¼ 2ReðqbcÞ;qn1;n2 2Imðbn1;n2cn1;n2þ1Þ ¼ 2ImðqbcÞ;zn1;n2 jbn1;n2 j

2 jcn1þ1;n2 j2 ¼ qbb qcc:

ð6Þ

Next, we obtain a discrete set of eight coupledequations of motion for the atom–field system.

The discreteness of these equations is due to the

photon number distribution of the quantized

electromagnetic field in the cavity. The equations

are

dwn1 ;n2ðtÞdt

¼2g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1þ1

pvn1 ;n2ðtÞ

g2f2ðzÞffiffiffiffiffiffiffiffiffiffiffiffin2þ1

pqn1 ;n2 ðtÞ;

dzn1 ;n2ðtÞdt

¼ g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1þ1

pvn1 ;n2ðtÞ

þ2g2f2ðzÞffiffiffiffiffiffiffiffiffiffiffiffin2þ1

pqn1 ;n2 ðtÞ;

dun1 ;n2ðtÞdt

¼Dbavn1 ;n2ðtÞg2f2ðzÞffiffiffiffiffiffiffiffiffiffiffiffin2þ1

pfn1 ;n2ðtÞ;

dvn1 ;n2ðtÞdt

¼Dbaun1 ;n2ðtÞþg2f2ðzÞffiffiffiffiffiffiffiffiffiffiffiffin2þ1

pdn1 ;n2ðtÞ

þ2g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1þ1

pwn1 ;n2ðtÞ;

ddn1 ;n2ðtÞdt

¼ðDbaDbcÞfn1 ;n2ðtÞg1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1þ1

pqn1 ;n2ðtÞ

g2f2ðzÞffiffiffiffiffiffiffiffiffiffiffiffin2þ1

pvn1 ;n2ðtÞ;

dfn1 ;n2ðtÞdt

¼ðDbaDbcÞdn1 ;n2ðtÞg1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1þ1

ppn1 ;n2ðtÞ

þg2f2ðzÞffiffiffiffiffiffiffiffiffiffiffiffin2þ1

pun1 ;n2 ðtÞ;

dpn1 ;n2ðtÞdt

¼Dbcqn1 ;n2ðtÞþg1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1þ1

pfn1 ;n2ðtÞ;

dqn1 ;n2ðtÞdt

¼Dbcpn1 ;n2ðtÞþg1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1þ1

pdn1 ;n2ðtÞ

2g2f2ðzÞffiffiffiffiffiffiffiffiffiffiffiffin2þ1

pzn1 ;n2ðtÞ:


In the above equations, atom–field detunings are

defined as Dba ¼ xb xa x1 and Dbc ¼ xbxc x2. Also, these equations are constrained

by the closeness condition or the trace condi-

tion of the density operator, i.e., qaa þ qbb þqcc ¼ 1.

2.2. Density operator equations describing interac-

tion of a K-type three-level atom with one quantized

field mode and a classical field

The level configuration and parameters for

this model are shown in Fig. 1(b). In this case(when compared with the model discussed in the

Section 2.1, above) we replace one of the

quantized cavity modes which mediates transi-

tion between levels jci and jbi, by a classical

field (with associated Rabi frequency X) and

assume no spatial mode structure for this clas-

sical field. Essentially, it means that the classical

field is not circulating in the cavity, which iseasily achieved in real experiments. The Hamil-

tonian of this system under RWA is

H ¼ hxajaihaj þ hxbjbihbj þ hxcjcihcjþ hx1a

y1a1 þ hg1f1ðzÞ½ay1jaihbj þ a1jbihaj

þ hX½jcihbj þ jbihcj: ð8Þ

The basis vectors for the wave function of the

above Hamiltonian are ja; n1 þ 1i, jb; n1i, and

jc; n1i, so the wave function wðtÞ (equivalent of

Eq. (5)) has the following form:

wðtÞ ¼Xn1

½an1þ1ðtÞja; n1 þ 1i þ bn1ðtÞjb; n1i

þ cn1ðtÞjc; n1i: ð9Þ

Consequently, the dynamical evolution of the

system for each value of n1 is determined by the

three coupled equations for the complex quan-

tities an1ðtÞ, bn1ðtÞ and cn1ðtÞ. We can also define

eight real quantities analogous to those defined

in Eq. (6) for the density operator in this caseand the equations of motion for these quantities

are

dwn1ðtÞdt

¼2g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

pvn1ðtÞXqn1ðtÞ;

dzn1ðtÞdt

¼ g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

pvn1ðtÞþ 2Xqn1ðtÞ;

dun1ðtÞdt

¼Dbavn1ðtÞXfn1ðtÞ;

dvn1ðtÞdt

¼ Dbaun1ðtÞþXdn1ðtÞ

þ 2g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

pwn1ðtÞ;

ddn1ðtÞdt

¼ðDba DbcÞfn1ðtÞ

g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

pqn1ðtÞXvn1ðtÞ;

dfn1ðtÞdt

¼ ðDba DbcÞdn1ðtÞ g1f1ðzÞ

ffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

ppn1ðtÞþXun1ðtÞ;

dpn1ðtÞdt

¼Dbcqn1ðtÞþ g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

pfn1ðtÞ;

dqn1ðtÞdt

¼ Dbcpn1ðtÞþ g1f1ðzÞffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

pdn1ðtÞ

2Xzn1ðtÞ;

ð10Þ

along with the trace condition of the density ma-

trix: qaa þ qbb þ qcc ¼ 1.

The equations given in (7) and (10) are the keyequations governing the dynamical evolutions of

the two different systems under consideration in

this work.

3. Solutions of equations of motion for model A

3.1. Under on-resonance condition: Dba ¼ Dbc ¼ 0

It is rather difficult to obtain a general analytic

solution of Eq. (7) when spatial mode structure

functions are included in these equations. The case

for fiðvtÞ ¼ 1 ði ¼ 1; 2Þ, i.e., stationary atom, has

been discussed in detail by Zhu et al. [7] for the

work related with two-mode laser action. Our aim

in this work is to study the dynamical evolution ofthe atom, i.e., the population in various levels

under different mode structures (pi) and under

different strengths of the cavity modes. In this

work we will restrict our discussions only to the

coherent-state cavity field modes.

Under the condition of exact resonance of

cavity fields with atomic transition frequencies,


i.e., Dba ¼ Dbc ¼ 0, and with the assumption of

same cavity mode functions (e.g., f1ðvtÞ ¼f2ðvtÞ ¼ f ðvtÞ), the equations of motion for the

complex quantities an1þ1;n2ðtÞ, bn1;n2ðtÞ, and

cn1;n2þ1ðtÞ can be rewritten as

dan1þ1;n2ðtÞdt

¼ ig1f ðvtÞbn1;n2ðtÞ;

dbn1;n2ðtÞdt

¼ ig1f ðvtÞan1þ1;n2ðtÞ

ig2f ðvtÞcn;n2þ1ðtÞ;dcn;n2þ1ðtÞ

dt¼ ig2f ðvtÞbn1;n2ðtÞ:

ð11Þ

We can recast the second equation of (11) into a

second-order differential equation

d2bn1;n2ðtÞdt2

v_f ðvtÞf ðvtÞ

dbn1;n2ðtÞdt

þ ½f ðvtÞ2½g21ðn1 þ 1Þ

þ g22ðn2 þ 1Þbn1;n2ðtÞ ¼ 0; ð12Þ

which can be further transformed into a normal

form of [19]

d2bn1;n2dn2

þ ½g21ðn1 þ 1Þ þ g22ðn2 þ 1Þbn1;n2 ¼ 0: ð13Þ

The new variable n is defined as n ¼Rf ðvtÞdt and

the solution of Eq. (13) is straightforward. Hence,

it is possible to write down the complete solution

of Eq. (11) as

an1þ1;n2ðtÞ ¼ 1 2g21ðn1 þ 1ÞQ2

sinðQn=2Þ;

bn1;n2ðtÞ ¼ ig1

ffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

p

QsinðQnÞ;

cn1;n2þ1ðtÞ ¼g1g2

ffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

p ffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ 1

p

Q2

ð1 cosðQnÞÞ;

ð14Þ

where Q ½g21ðn1 þ 1Þ þ g22ðn2 þ 1Þ1=2 and the

atom is assumed initially to be in the ground state

jai. Using Eq. (14), it is easy to construct all the

elements of density operator as defined in the

Section 2.1. The population inversion between

the levels jai and jbi has been defined as wn1;n2ðtÞ ¼jan1þ1;n2ðtÞj

2 jbn1;n2ðtÞj2for convenience because

we consider the atom initially to be in the state jaiso that wn1;n2ð0Þ ¼ 1. The time evolution of wn1;n2ðtÞis thus given by

wn1;n2ðtÞ ¼ jan1þ1;n2ðtÞj2 jbn1;n2ðtÞj

2;

¼ g41ðn1 þ 1Þ2

Q4

þ g21ðn1 þ 1Þ

Q2

!cos2 ðQnÞ

2g41ðn1 þ 1Þ2

Q4

g21ðn1 þ 1Þ

Q2

!cosðQnÞ

þ 1

þ 3

g21ðn1 þ 1ÞQ2

þ g41ðn1 þ 1Þ2

Q4

!:

ð15Þ

The function Qn represents the effective area under

the pulse (as defined in quantum optics literature,

see [16]) with two superimposed amplitudes of

different field modes that the atom passes until the

time t, i.e.,

Qn ¼ ½g21ðn1 þ 1Þ þ g22ðn2 þ 1Þ1=2Z t

0

f ðvsÞds:

ð16Þ

For the sinusoidal mode function (TEmnp) we ob-

tain

Qn¼ tTpp

½g21ðn1 þ 1Þþ g22ðn2þ 1Þ1=2½1 cosðppt=tTÞ;

ð17Þwhere the transit time of the atoms through the

cavity is tT ¼ L=v. The information about themode structure effect due to atomic motion in

different cavity modes is contained in Eqs. (15)–

(17). These results are rather surprising because

the qualitative behavior of the three-level system is

very much similar to that of the two-level atom

when both mode functions have the identical

structure. When p is an odd positive integer then

the function Qn does not vanish over the wave-length of the mode(s), i.e., QnpðoddÞ ¼ ð2L=pvpðoddÞÞ½g21ðn1 þ 1Þ þ g22ðn2 þ 1Þ1=2, so we do not

observe effects like spin or photon echo for the

Fock-state fields in general. It is possible to adjust

atomic velocity v in such a way that for some

combinations of n1 and n2 values (fields in the

Fock states) we can get QnpðoddÞ ¼ 2pq (q is a po-

sitive integer). This means that after undergoing qcycles of effective Rabi oscillations the atom leaves

the cavity in the same state in which it entered.

This behavior is similar to SIT of a 2pq pulse. On


the other hand, if p is an even positive integer then

Qn vanishes over the wavelength of mode(s) and

the atom comes out in the same state in which it

has entered. This situation is very much similar to

SIT of a 0p pulse in the theory of pulse propaga-tion [16]. Notice that this kind of transparency is

independent of any choice for the initial field state

or atomic state (the atom could be in any of its

states or in coherent or incoherent superposition

of its states).

3.2. Far-off-resonance condition

When the detunings Dba and Dbc are quite large

in comparison to the corresponding Rabi fre-

quencies of the transitions then the K-type three-

level atomic system nearly resembles to an effective

two-level system undergoing a non-resonant two-

photon transition where one photon is absorbed

and another is emitted simultaneously. This can

easily be shown by assuming Dba ¼ Dbc ¼ D or, inother words, cavity is tuned in such a manner that

two-photon energy conservation is maintained,

i.e.,

hðxb xaÞ hðxb xcÞ hðx1 x2Þ ¼ 0

! hðxb xa x1Þ ¼ hðxb xc x2Þ ¼ hD:

ð18Þ

Under the assumption of large detuning, e.g.,

hjDj jhðxc xaÞj, the upper-level of the K-sys-tem can be eliminated adiabatically and the effec-

tive Hamiltonian reads

H eff1int ¼ hgf1ðzÞf2ðzÞðjaihcjay1a2 þ jcihaja1ay2Þ

þ b1ay1a1jaihaj þ b2a

y2a2jcihcj; ð19Þ

in which the effective atom–field coupling coeffi-

cient is given by g 2g1g2=D and the Stark shift

parameters are b1 ¼ h½f1ðzÞ2g21=D and b2 ¼h½f2ðzÞ2g22=D, respectively. Note that under the

condition b1 ¼ b2, the effect of dynamical Stark

shifts from the two levels are equal and opposite,

and hence do not affect the dynamical evolution of

the atom[20]. Exact numerical results, which willbe discussed in Section 3.3, do take care of these

Stark terms automatically depending on the choice

of parameters.

The H eff1int represents a two-photon Raman pro-

cess taking place in an ideal microwave cavity in

which a photon (the pump, operator a1) is absorbedand another photon (the stokes, operator a2) is si-multaneously scattered back in the cavity with field

mode structure taken into account. The model ofdegenerate two-photon sequential transition as well

as non-degenerate two-photon transition including

cavity mode structure effects were treated exten-

sively in earlier works [4]. In the H eff1int , the atom–

field coupling parameter is a sensitive function of

atomic detuning and for very large atomic detuning

we observe an adiabatic following type of situation.

This means that effective Bloch vector changes onthe time scale ofD1 and precesses rapidly about the

torque vector and adiabatically follows it [3,4].

3.3. Numerical results

We now discuss the results obtained by

numerical integration of Eq. (7). Interesting

situations occur for the intermediate values of

atom–field detunings under different cavity mode

structures. For example, we observe SIT and AF in

the dynamical evolution of the atom in the observ-

able quantity wn1;n2ðtTÞ. The SIT, which occurs ex-actly at Dba=g ¼ 0 (on-resonance condition), has asignature that wn1;n2ðtTÞ ¼ wn1;n2ð0Þ ¼ 1 in all our

figures as we start with wn1;n2ð0Þ ¼ 1 at t ¼ 0. For

AF, which we observe at very large detunings

(jDbaj=g 1), the signature is that the value of

wn1;n2ðtTÞ remains nearly around 1. In Fig. 2 we

plot the atomic inversion wn1;n2ðtÞ (between levels

jai and jbi) as a function of atomic detuning Dba

for a fixed atomic velocity v ¼ g1L=p. For sim-

plicity we keep the atom–field coupling coefficients

of both the channels of the three-level atom to be

equal, i.e., g1 ¼ g2 ¼ g. Also, the atomic detunigs

are kept equal, e.g., Dba ¼ Dbc ¼ D, and initially

the atom is assumed to be in the ground state jai.The initial field state is considered to be a bimodal

coherent state whose photon-number distribution

is given by Pn1;n2 eðn1þn2Þ nn11nn22

n1!n2!, where the mode n1

(n2) is interacts with the channel a ! b (c ! b).Curve A shows the effect of odd-cavity mode

structure (p1 ¼ 1) on the usual two-level JCM dy-

namics with g1 ¼ 1; ðg2 ¼ 0Þ; n1 ¼ 10. This curve

-10 -5 0 5 10

0.0

0.3

0.6

0.9

1.2

D

C

B

A

wn 1,

n 2(t)

∆ / g

-10 -5 0 5 10

-0.3

0.0

0.3

0.6

0.9

1.2

D

C

B

A

wn 1,

n 2(t)

∆ / g

(a)

(b)

Fig. 2. Atomic inversion wn1 ;n2 ðtÞ at the cavity exit vs detuning

D=g (Dba ¼ Dbc ¼ D, g1 ¼ g2 ¼ g) for a fixed transit time

tT ¼ p=g. (a) Curve A is for a two-level JCM with odd cavity

field mode (p1 ¼ 1; g1 ¼ 1; g2 ¼ 0) and n1 ¼ 10; curves B, C, and

D are for the two-mode coupled three-level K-system with odd

cavity modes (p1 ¼ 1; p2 ¼ 1; g1 ¼ g2 ¼ 1) and (n1; n2) equals tobe ð10; 1Þ, ð10; 10Þ and ð1; 10Þ, respectively. (b) Same as in (a),

except that the cavity modes are even (p1 ¼ 2; p2 ¼ 2) now. In

all the cases the field is initially in a coherent state and the atom

is initially in the ground state jai.


is plotted here for reference so we can compare the

three-level dynamics with it. We can see that AF

behavior is evident in curve A even at moderate

atomic detunings (D) as explained in [3]. Next, we

study the effect of the additional channel (c ! b)on the atomic inversion observed in channel(a ! b). We keep a weak coherent field (n2 ¼ 1:0,g2 ¼ 1:0) in the channel (c ! b) but a strong co-

herent field in the channel (a ! b) (n1 ¼ 10:0,g1 ¼ 1:0) with both cavity field modes having odd

mode structures (p1 ¼ p2 ¼ 1). The atomic dy-

namics gets moderately affected (curve B) because

of this additional coupling between levels jbi and

jci. The most striking change is that now the AF isno longer evident at moderate D which is present

in curve A. When we keep equally strong cavity

fields in both channels (n1 ¼ 10:0, g1 ¼ 1:0;n2 ¼ 10:0, g2 ¼ 1:0), then the atomic dynamics

modifies not only at exact resonance but also at

moderate and large detunings (curve C). The sys-

tem no longer shows AF with moderate or even

with quite large atomic detunings. If we decreasethe field strength in the channel a ! b (n1 ¼ 1:0,g1 ¼ 1:0) but keep the stronger field strength in

channel c ! b (n2 ¼ 10:0, g2 ¼ 1:0) then the

atomic inversion wn1;n2ðtÞ confines towards higher

magnitudes (curve D). There is no dipping ob-

served in the atomic inversion at the exact reso-

nance condition, nevertheless, the inversion is

slightly lower than its initial value (curve D). Thisbehavior is clearly due to increase in Stark shift

introduced by the stronger field in the channel

c ! b. The stronger field in this channel causes

upper transition level to fall outside the response

spectrum of cavity mode hence the evolution is

suppressed. Next, we see how these results get

changed when we consider the even cavity modes.

In Fig. 2(b) we replot all the curves of Fig. 2(a)with the same conditions except that we use even

cavity modes in both channels (p1 ¼ 2, p2 ¼ 2).

Here curve A is exactly the same as in Fig. 2(a) but

for p1 ¼ 2 (p2 ¼ 0) mode and serves as reference

for comparison. Note that curve A shows SIT-like

behavior at the exact resonance condition and AF

at the moderate detunings [3]. When we couple this

two-level system to another channel c ! b with aweak cavity field (n1 ¼ 10:0, g1 ¼ 1:0; n2 ¼ 1:0,g2 ¼ 1:0) we still observe SIT at the exact reso-

nance condition but AF at the moderate detunings

is reduced (curve B, Fig. 2(b)). When both chan-

nels of the K-system are coupled with cavity fields

of equal strengths (n1 ¼ 10:0, g1 ¼ 1:0; n2 ¼ 10:0,g2 ¼ 1:0) in even modes (p1 ¼ p2 ¼ 2), the SIT

condition at the exact resonance is still preservedbut the AF is not observable even at large detu-

nings (curve C, Fig. 2(b)). The system now behaves

-10 -5 0 5 10-0.4

0.0

0.4

0.8

1.2

(b)

(a)

D

C

B

A

wn 1,

n 2(t)

∆ / g

-10 -5 0 5 10

-0.4

0.0

0.4

0.8

1.2

D

C

B

A

wn 1,

n 2(t)

∆ / g

Fig. 3. Atomic inversion wn1 ;n2 ðtÞ for the two-mode coupled

three-level K-system at the cavity exit vs detuning D=g(Dba ¼ Dbc ¼ D, g1 ¼ g2 ¼ g) for a fixed transit time tT ¼ p=gwith g1 ¼ g2 ¼ 1, and mixed cavity modes: (a) p1 ¼ 1; p2 ¼ 2

and (b) p1 ¼ 2; p2 ¼ 1. Curves A, B, C, and D are for (n1; n2)equals to be ð10; 1Þ, ð10; 10Þ, ð1; 10Þ and ð1; 1Þ, respectively. Inall the cases the field is initially in a coherent state and the atom

is initially in the ground state jai.


like an effective two-photon (non-degenerate)

Raman-coupled two-level atom. Finally, we use

weak field in channel a ! b (n1 ¼ 1:0), but stron-ger field in channel c ! b (n2 ¼ 10:0). SIT condi-

tion is still preserved at the exact resonance (curve

D, Fig. 2(b)) but the variation of the inversion isquite small throughout its evolution and it re-

mains around the value 1. As explained earlier this

is due to the increased Stark shift introduced by

the stronger field in channel c ! b. We always

observe SIT at the exact resonance condition ir-

respective of the field strength in either channel

and this exactly matches with the analytical results

obtained in Section 3.1 above.At this stage it becomes rather interesting to

know the dynamical evolution of this K-systemunder mixed cavity modes, i.e., one channel in-

teracts with an odd cavity mode while the other

channel interacts with an even cavity mode or vice

versa. The first situation (p1 ¼ 1, p2 ¼ 2) is shown

in Fig. 3(a). For curve A, we have stronger field in

channel a ! b but a weak field in channel c ! b(g1 ¼ g2 ¼ 1, n1 ¼ 10, n2 ¼ 1). The dynamical

evolution of the atom does not show any SIT at

the exact resonance or AF at the moderate detu-

nings. The same is true when we keep equally

strong fields in both of the channels, i.e., n1 ¼ 10,n2 ¼ 10 (curve B); or weak field in channel a ! band strong field in channel c ! b, i.e., n1 ¼ 1,n2 ¼ 10 (curve C). However, with weak fields inboth channels ( n1 ¼ 1, n2 ¼ 1) (curve D), the sys-

tem shows AF at the moderate detunings. In this

situation (curve D) the behavior of the atom is

more like a two-level atom interacting with co-

herent field in an odd cavity mode (see curve A of

Fig. 2(a)).

Another situation of mixed cavity mode con-

dition (p1 ¼ 2, p2 ¼ 1) is plotted in Fig. 3(b). Thecurves A, B, C, and D of Fig. 3(b) correspond

to the same parameters of Fig. 3(a) except that

now we have p1 ¼ 2 and p2 ¼ 1. By comparing

curves A, B, and C of Fig. 3(b) with curves A,

B, and C of Fig. 3(a), respectively, we observe a

qualitative difference in atomic inversion in these

two situations but again there is no SIT at the

exact resonance condition as well as no AF atthe moderate detunings. The curve D of

Fig. 3(b) is qualitatively similar to curve D of

Fig. 3(a) but with a much larger dip at the

resonance and shows AF at the moderate detu-

nings, and thus again reminds us of the situation

of the usual two-level JCM with an odd cavity

mode structure (curve A, Fig. 2(a)). Thus, the

dynamical evolution of the K-type three-level

atomic system is very sensitive to both the fieldstrengths as well as the mode structure functions

of the two channels.

-15 -10 -5 0 5 10 15

0.0

0.4

0.8

1.2

CB

AD

wn 1(t

)

∆ / g1

-15 -10 -5 0 5 10 15

0.0

0.4

0.8

1.2

D

C

B A

wn 1(t

)

∆ / g1

(a)

(b)

Fig. 4. Atomic inversion wn1 ðtÞ at the cavity exit vs detuning

D=g1 (Dba ¼ Dbc ¼ D), for a fixed transit time tT ¼ p=g1. (a)

Curve A is for a two-level JCM with an odd cavity field mode

(p1 ¼ 1; g1 ¼ 1; X=g1 ¼ 0) and n1 ¼ 10; curves B, C, and D are

for the one quantized cavity mode and a classical field coupled

three-level K-system with an odd cavity mode (p1 ¼ 1; g1 ¼ 1)

and (n1;X=g1) equals to be ð10; 1Þ, ð10; 2:7Þ, and ð10; 5:0Þ, re-spectively. (b) Same as in (a), except that the cavity mode is

even (p1 ¼ 2; g1 ¼ 1) but curves B, C, and D are for ðn1;X=g1Þequals to be ð10; 1Þ, ð10; 2:0Þ, and ð10; 7:0Þ, respectively. In all

the cases the field is initially in a coherent state and the atom is

initially in the ground state jai.


4. Solutions of equations of motions for model B

At the exact resonant condition we can get a

closed form solution of Eq. (10) provided we

consider the interaction of stationary atom withone quantized cavity mode and a classical field as

discussed in detail in [11]. By retaining the mode

structure term in Eq. (10) the problem becomes

difficult to solve analytically.

However, if we consider the situation of large

atomic detunings, i.e., (Dba ¼ Dbc ¼ D g1ffiffiffiffiffin1

p,

X), it is possible to approximate the interaction

Hamiltonian as follows:

H eff2int ¼ hg0f1ðzÞ½jaihcjay1 þ jaihcja1

þ b01a

y1a1jaihaj þ b0

2jcihcj; ð20Þ

where g0 ¼ Xg1=D and the Stark shift parameters

are b01 ¼ ½f1ðzÞ2g21=D and b0

2 ¼ X2=D, respectively.In obtaining Eq. (20), we have used the standardquantum optical technique in removing the upper

level adiabatically. Thus, H eff2int describes Rabi os-

cillation between states jci and jai mediated by the

cavity quantized mode a1 but with an effective

field–atom coupling coefficient g0 which is a sen-

sitive function of atomic detuning D. The Stark

shifts do not influence the dynamical evolution

under the condition b01 ¼ b0

2 [20]. The H eff2int then

represents Hamiltonian for an effective one-pho-

ton process with cavity mode structure taken into

account and is similar to the one discussed in [3]. If

the atomic detuning becomes very large, the con-

dition of adiabatic following prevails and the

atomic inversion does not change from its initial

value.

The numerical results under arbitrary atomicdetuning are obtained by integrating Eq. (10). We

consider the quantized cavity mode in a coherent

state having photon number distribution

Pn1 ¼ nn11 en1=n1!. We keep atomic detunings of

both the channels to be equal just for the sake of

simplicity and assume the atom to be initially in

the ground state jai and plot the atomic dynamics

as a function of detuning D=g1 (¼ Dba=g1 ¼ Dbc=g1)in Fig. 4. Curve A (which is used as a reference) in

Fig. 4(a), represents dynamical evolution of the

two-level JCM with an odd cavity mode

(p1 ¼ 1; g1 ¼ 1; n ¼ 10). Under weak classical

driving field in channel c ! b the three-level sys-

tem evolves quite similar to a two-level JCM with

mode structure included except that the AF is not

so apparent at the moderate atomic detunings.

However, we have verified that AF does appear at

very large atomic detunings. With further increase

in the strength of the classical driving field(X=g1 ¼ 1:0, curve B) the dynamical evolution of

0 2 4 6 8 10

-0.8

-0.4

0.0

0.4

0.8

1.2

D

C

A

B

wn 1(t

)

Ω / g1

0 5 10 15

-0.8

-0.4

0.0

0.4

0.8

1.2

D

C

B

A

wn 1(t

)

Ω / g1

(a)

(b)

Fig. 5. Atomic inversion wn1 ðtÞ at the cavity exit under the

exact resonant condition ðDba ¼ Dbc ¼ 0Þ as a function of X=g1for a fixed transit time tT ¼ p=g1 for one quantized cavity mode

and a classical field coupled three-level K-system. The cavity

mode is (a) an odd mode (p1 ¼ 1), (b) an even mode (p1 ¼ 2) in

a coherent state. Curves A, B, C, and D are for the mean

photon number n1 ¼ 1, 10, 25, and 40, respectively. In all cases

the atom is initially in the ground state jai.


the three-level atom gets considerably modified in

comparison with the two-level atom (curve A),

both at the exact resonant condition as well as at

the moderate and the far-off-resonance conditions.

In curve C (X=g1 ¼ 2:7) the dynamical evolution

of atom becomes drastically different at all valuesof D. When we let X=g1 ¼ 5 (curve D), then the

Stark shift becomes so large that the atomic in-

version does not show any evolution at the exact

resonance condition as well as at the moderate

detunings. The overall change in the inversion

even at larger detunings is also not very big. The

dynamic Stark shift due to the classical field causes

formation of Autler-Townes doublet so that theupper level (which is the common level to both the

channels) falls outside the spectrum of cavity field

mode and hence the dynamical evolution of the

atom is suppressed.

In Fig. 4(b) we plot the effect of even cavity

mode structure on the dynamical evolution of the

atom. We keep n1 ¼ 10, g1 ¼ 1 in this figure and

curve A represents evolution of the normal two-level JCM with even mode: p2 ¼ 2. Even with a

small classical driving field (X=g1 ¼ 0:5) in channel

c ! b the SIT is lost at the exact resonance con-

dition (curve B). AF still persists at moderate D.For a larger classical driving field (X ¼ 1:0) thereare some changes in the dynamical evolution but

the qualitative behaviors are the same as in curve

B. At X=g1 ¼ 2:0, we again observe tendency ofnearly SIT-like behavior at the exact resonance

condition (curve C) but AF is not so clear under

the off-resonance conditions. At very high driving

field (X=g1 ¼ 7:0) the tendency of not evolving far

from initial condition is observed at the exact

resonance as well as at the moderate detuning

conditions due to the large Stark shift (curve D).

To have a better idea about the relative roles ofthe strength of cavity quantized mode field under

different mode structures and the classical driving

field, we plot, in Fig. 5, the atomic inversion at the

exact resonance condition and at the cavity exit for

a fixed atomic velocity, as a function of classical

driving field strength for various values of quan-

tized cavity field. Fig. 5(a) and (b) are for the odd

mode (p1 ¼ 1) and the even mode (p1 ¼ 2), re-spectively. Curves A, B, C, and D are for n1 ¼ 1,

10, 25, and 40, respectively. For small cavity

quantized field strength (curve A) the inversion atthe cavity exit shows a large variation towards

negative side and after a minimum value it jumps

to a higher value, then shows small oscillations,

and eventually settles down to the value 1 (the

initial value). Similar type of evolution is observed

when we increase the cavity quantized field

strength to n1 ¼ 10, in curve B. There is a shift in

the minimum towards a larger X but the oscilla-tion settles down to a steady value of 1 as X be-

comes large. When the cavity field strength is

increased further (n1 ¼ 25, curve C) the minimum

0 10 20 30 40

0 10 20 30 40

-0.8

-0.4

0.0

0.4

0.8

1.2

C

D

B

A

n1

wn 1(t

)

-1 .0

-0.5

0.0

0.5

1.0

D

C

B

A

wn 1(t

)

n1

(a)

(b)


exact resonant condition ðDba ¼ Dbc ¼ 0Þ as a function of mean

photon number n1 for a fixed transit time tT ¼ p=g1 for the onequantized cavity mode and a classical field coupled three-level

K-system. The cavity mode is (a) an odd mode (p1 ¼ 1), (b) an

even mode (p1 ¼ 2) in a coherent state. Curves A, B, C, and D

are for the X=g1 ¼ 0, 1, 2.5, and 5, respectively. In all the cases

the atom is initially in the ground state jai.


inversion shifts up or the depth of inversion de-

creases, the oscillatory behavior increases and the

inversion finally reaches to the magnitude 1 for

much larger values of X. In curve D, we keepn1 ¼ 40 and the inversion initially goes up with

increasing X but then comes down to a minimumvalue (which is, however, higher in comparison to

curves A, B, and C) followed by oscillations and

then reaches to value 1. Thus, the value of n1 is

very important relative to X in governing the dy-

namical evolution of the atom. Smaller the n1 is,

quicker the steady-state value can be reached. We

come across further interesting situations in the

atomic inversion with even cavity mode (p2 ¼ 2) asplotted in Fig. 5(b), under various values of n1.Curve A (n1 ¼ 1) shows a double well meaning a

pair of minima in the inversion which is different

from curve A of Fig. 5(a). For n1 ¼ 10 (curve B)

the double-well minima disappear and, instead, a

single broader minimum is observed. At n1 ¼ 25

(curve C) we still see a single minimum but situ-

ated at a higher value of X. When we let n1 ¼ 40(curve D), again a pair of minima appears but the

two depths are unequal. All these curves show

oscillatory behavior after the minima and eventu-

ally approach to the magnitude 1. The study of

curves A–D suggests that one can control the

atomic inversion from its initial value 1 to a value

of )0.75 by applying an appropriate classical field

in the second channel of the three-level atom. So,in an experiment where such a control of popula-

tion inversion is required then it can easily be

achieved with the help of an additional classical

field. In other words, the classical driving field acts

as a switch which can alter the population inver-

sion between 1 and )0.75. So, this model can be

used to build an atomic-inversion NOT gate con-

trolled by the external driving field.Next, we plot the atomic inversion at the exact

resonance condition and at the cavity exit as a

function of mean cavity quantized field strength

(n1) for different values of classical driving fields in

Fig. 6(a) (p1 ¼ 1) and 6(b) (p1 ¼ 2). In these fig-

ures, curves A, B, C, and D are for classical driving

field with strengths X=g1 ¼ 0, 1, 2.5, and 5, re-

spectively. When there is no classical driving fieldpresent (curve A, X=g1 ¼ 0) we observe periodic

oscillations in the atomic inversion for the odd-

mode (p1 ¼ 1, Fig. 6(a)) but no evolution at all for

the even-mode (p1 ¼ 2, Fig. 6(b)). The width of

periodic oscillation increases with increase in n1value (p1 ¼ 1 mode). The no evolution observed

for the p1 ¼ 2 mode is due to SIT condition pre-

vailing at the exact resonance condition no matter

what the mean photon number is inside the cavity.With a small classical driving field (curve B,

Fig. 6(a)) the periodic evolution of atomic inver-

sion for p1 ¼ 1 mode becomes non-periodic but the

same is restored back at higher values of X (curves

C and D). At a large value of X=g1 (¼ 5), the

evolution of inversion becomes very much limited

around the value 1 because of the Stark effect

0 1 2 3 4 5 6

-1.0

-0.5

0.0

0.5

1.0

CB

A

wn 1(t

)

g1tT

0 1 2 3 4 5 6

-1.0

-0.5

0.0

0.5

1.0C

B

A

wn 1(t

)

g1tT

(a)

(b)


exact resonant condition ðDba ¼ Dbc ¼ 0Þ as a function of

transit time g1tT for the one quantized cavity mode and a

classical field coupled three-level K-system. The cavity mode is

(a) an odd mode (p1 ¼ 1), (b) an even mode (p1 ¼ 2) in a co-

herent state with mean photon number n1 ¼ 10 . The atom is

initially in the ground state jai. Curves A, B, and C are for

X=g1 ¼ 0, 2.5, and 5, respectively.


discussed earlier. On the contrary, for p1 ¼ 2,

curves B, C, and D (Fig. 6(b)) do not show any

regular or periodic pattern in their evolutions withn1.

Further, we look at another important issue

associated with this system in its dependence onvariation of transit time tT ¼ L=v, for odd

(p1 ¼ 1) and even (p1 ¼ 2) modes with different

classical driving fields. We keep the cavity field to

be in a coherent state with a mean photon

number n1 ¼ 10 and the atom initially to be in its

ground state jai. In Fig. 7(a) and (b) we plot the

atomic inversion as a function of g1tT for p1 ¼ 1

and p1 ¼ 2, respectively at the exact resonancecondition (Dba ¼ Dbc ¼ 0). The curves A, B, and

C are for X=g1 ¼ 0, 2.5, and 5.0, respectively. The

curve A in both figures represents the two-level

atom excited by a cavity quantized mode with

spatial structure included. We observe a periodic

evolution for the atomic inversion with respect to

the transit time having a period of 2p for the odd

mode (p1 ¼ 1) but the period is either p or p=2for the even mode depending upon whether we

concentrate on upper or lower peak. Thus, the

system generates periodic oscillations in atomic

inversion with respect to transit time of atoms

irrespective of the mode structure sustained in the

cavity. For the odd mode (p1 ¼ 1), if we increase

X=g1 from 0 to 2.5 (curve B, Fig. 7(a)) the peri-

odic evolution is spoiled completely and it re-mains so even with a higher value of X=g1 (¼ 5)

in curve C of the same figure. There is a very

little change in the inversion from its initial value

of 1 at this value of X because of the Stark effect.

When we increase X=g1 to 2.5 in the even mode

case (p1 ¼ 2), there is no periodicity (curve B,

Fig. 7(b)). However, we get such periodicity back

in curve C (X=g1 ¼ 5) and the period is aboutp=2. So, there is switching action by the external

classical driving field. For this value of external

field the population inversion can switch between

two different values by controlling the transit time

of atoms. Hence, we get a NOT gate of atomic-

inversion controlled by the external field and the

transit time of atoms. Thus, the mode structure

plays a crucial role for the dynamical evolution ofa three-level atom in K-configuration inside an

ideal cavity.

In Fig. 8, we present the comparison when atom

is stationary (no mode structure effects included)

to the situation when the atom is moving and in-

teracting with different cavity modes. The inter-

action time (transit time) is kept fixed (tT ¼ p=g)and the classical field is assumed to be X=g1 ¼ 3:0.Fig. 8 depicts the atomic inversion as a function of

mean cavity photon number n1 in which curve A isfor a stationary atom (i.e., no mode structure ef-

fects) while curves B and C are for a moving atom

interacting with odd (p1 ¼ 1) and even (p1 ¼ 2)

cavity modes, respectively. The dynamical evolu-

0 10 20 30 40

0.0

0.4

0.8

1.2

Y

X

C

B

A

wn 1(t

)

n1

Fig. 8. Evolution of the atomic inversion wn1 ðtÞ as a function of

n1 for the one quantized cavity mode and a classical field cou-

pled three-level K-system for a fixed interaction/transit time

tT ¼ p=g1. The cavity field is in a coherent state with n1 ¼ 10,

the classical driving field is X=g1 ¼ 3, and the atom is initially in

the ground state jai. Curves A is for the stationary atom (no

cavity modes) while curves B and C are for moving atom with

odd (p1 ¼ 1) and even (p2 ¼ 2) cavity modes, respectively.


tion is oscillatory and the cycle width changes as n1is increased. One interesting observation is the

intersection of all three curves at a single point X ,

which implies that there exists a coincidence for

the parameters so selected in these curves that allthree different situations produce the same inver-

sion. Similarly, there is a point Y , where odd and

even modes produce the same inversion.

5. Summary and conclusions

In this work we have considered the interactionbetween a three-level atom in K-configuration and

(a) two quantized electromagnetic field modes or

(b) one quantized electromagnetic field mode and

a classical field in an ideal microwave cavity. In

both the cases we have allowed atom to move

along the cavity axis so that it can see the spatial

mode structure of the field sustained in the cavity.

Because of atomic movement in the cavity alongwith spatial field variation, we come across tran-

sient effects such as SIT and AF in the dynamical

evolution of the atom. We have studied these ef-

fects in the atomic inversion for different experi-

mental parameters such as odd or/and even cavity

modes, the average photon number of the cavity

mode, and the strength of the classical driving field

etc. We have compared these results with that of a

two-level atom whenever possible. We have given

analytical solution of the problem under exact

resonant condition for the case of two quantizedmodes interacting with the K-type atom. We have

discussed the situation of mixed cavity modes

showing how the transient effects get modified in

such conditions. We find that the atomic inversion

can easily be controlled at the cavity exit with the

help of the classical driving field. This means that

the system can act like a switch of flipping the

atomic inversion between two selected values,controlled by the classical field, which behaves like

an atomic-inversion NOT gate. Another applica-

tion of this work may be in the study of the motion

of an ion in a harmonic trap interacting with a

standing wave or a travelling wave. In certain

approximation the equations governing the mo-

tion of an ion [21] in such trap may be reduced to a

form similar to JCM with field variables replacedby the vibrational modes of the quantized center of

mass motion of the ion. The mode structure effects

are important for such problems also. Recently,

experimental demonstration of preparing entan-

gled Rubidium atoms in a superconducting mi-

crowave cavity was carried out and conditional

probabilities have been measured [22]. In such

system, one can verify these dynamical evolutionresults and thus see the mode structure effects.

These mode structure effects are useful in entan-

glement of atoms also, and the detail for the same

will be presented elsewhere.

Acknowledgements

We acknowledge the funding supports from the

National Science Foundation and the Office of

Naval Research.

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