ics

eses in the

Physics Letters A 325 (2004) 30–36

www.elsevier.com/locate/pla

Effect of spontaneously generated coherence on the dynamof multi-level atomic systems

Amitabh Joshi∗, Wenge Yang, Min Xiao

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

Received 11 December 2003; received in revised form 11 March 2004; accepted 17 March 2004

Communicated by P.R. Holland

Abstract

The effect of spontaneously generated coherence on dynamicalevolution of a multi-level atomic system consisting of threupper levels and a lower level is studied. For degenerate upper levels the population trapping in these levels increasteady state. 2004 Elsevier B.V. All rights reserved.

PACS: 42.50.Lc; 32.50.+d; 32.80.-t

Keywords: Quantum interference; Spontaneously generated coherence

tumaveaderiseticset-n-iona-duc-a

heyin-sion

cedieden

ummop-rum.elsnce.tralet-wnbydy-

uslytran-withthat

The phenomena of atomic coherence and quaninterference in many quantum optical systems hbeen discussed extensively during the past decThese phenomena are instrumental in givingto some interesting observations in quantum opsuch as lasing without inversion [1], electromagnically induced transparency [2], refractive index ehancement [3], absorption cancellation, populatinversion without emission, modification of spontneous emission process [4–6], and threshold retion in optical bistability [7] etc., just to namefew.

When two excited levels decay spontaneously, tcan strongly affect each other and give rise toterference, which modifies the spontaneous emis

* Corresponding author.E-mail address: ajoshi@uark.edu (A. Joshi).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.03.035

.

process. The effect of spontaneous-emission-induinterference on the population dynamics was studin a Λ-type three-level atomic system by Javanain[5]. On the other hand, the effect of such quantinterference on theV -type three-level atomic systewas investigated by Zhu et al. [6], in the studies of pulation dynamics and spontaneous emission spectThey predicted that the population in the upper levdoes not decay exponentially due to such interfereAlso, this kind of interference can give rise to specnarrowing and dark line in the spectrum. In this Lter we consider a four-level atomic system, as shoin Fig. 1, such that three upper levels are coupledthe same vacuum modes. We study the populationnamics in this system affected by the spontaneogenerated coherence (SGC) among the decayingsitions. We also generalize our result to the casesmore than three upper levels under the conditionall the upper levels are degenerate.

.

A. Joshi et al. / Physics Letters A 325 (2004) 30–36 31

theion)cal[9]hethesionf unm-le-d tohich

ivesion1].

ys-nce

iumim-hecalna-ontualeenrs,l-of

g.dentsan-

to

onels

es

mdes

-om.e

mione

Fig. 1. Schematics of the four-level atomic system.

In the previous studies it is well established thatdecay of an excited state (via spontaneous emisscan be modified by placing the atom in a cavity/optiwaveguide [8], in a photonic bandgap materialor using the idea of quantum Zeno effect [10]. Tquantum interference plays a significant role inmodification/suppression of the spontaneous emisprocess. The spontaneous emission is a source owanted noise in many applications like quantum coputation, quantum information processing, and teportation etc., so further research works are needecontrol such spontaneous emission processes, wmotivate the current work.

The experimental demonstration of constructand destructive interferences in spontaneous emisfrom sodium dimers was reported few years ago [1However, another experiment [12] on a similar stem failed to reproduce the same results and heconstructive and destructive interferences in soddimers could not be equivocally established. Thisplies that there are great practical difficulties in texperimental demonstration of SGC. A theoretimodel of the observed fluorescence intensity is alyzed for the experiment of Ref. [11] where correlatibetween number of observed peaks with the mupolarization of the molecular dipole moments has bmade [13]. In the experiment with sodium dimeXia et al. [11] found that the transitions with paralel and antiparallel dipole moments exist becausethe mixing of molecular states by spin–orbit couplinThe idea of preselected polarization of cavity mofor creating parallel and antiparallel dipole momehas been studied recently [14]. Generation of qutum interference between non-parallel dipoles dueanisotropic vacuum has also been proposed [15].

We consider a four-level atom in the configuratias depicted in Fig. 1. It consists of three upper lev

-

|ui〉 (i = 1,2,3) coupled by the same vacuum modto the common lower level|l〉. It is straightforward towrite down the interaction Hamiltonian of the systecomposed of the four-level atom and vacuum moin the following manner:

Hint = ih∑j

[g

(1)j bj |u1〉〈l|ei(ωu1l−ωj )t + g

(2)j bj |u2〉

× 〈l|ei(ωu2l−ωj )t + g(3)j bj |u3〉〈l|ei(ωu3l−ωj )t

]− ih

∑j

[g

(1)j b

†j |l〉〈u1|e−i(ωu1l−ωj )t + g

(2)j b

†j |l〉

× 〈u2|e−i(ωu2l−ωj )t + g(3)j b

†j |l〉

(1)× 〈u3|e−i(ωu3l−ωj )t].

Here,g(i)j (i = 1,2,3) is the coupling constant (as

sumed to be real) between the atomic transition fr|ui〉 (i = 1,2,3) to |l〉 and thej th vacuum modebj (b†

j ) is the annihilation (creation) operator for thj th vacuum mode with frequencyωj . ωuil is the fre-quency difference between levels|ui〉 (i = 1,2,3) and|l〉. Herej specifies both polarization and momentuof the j th vacuum mode. The spontaneous emissprocess is governed by theHint mentioned above. Thstate vector of the system can be written as∣∣Ψ (t)

⟩ = A(1)(t)|u1〉|0〉 + A(2)(t)|u2〉|0〉(2)+ A(3)(t)|u3〉|0〉 +

∑j

Bj (t)b†j |0〉|l〉,

which obeys the Schrödinger equation

(3)ih∂|Ψ(t)〉

∂t= Hint

∣∣Ψ (t)⟩.

The initial condition in this situation is∣∣Ψ (0)

⟩ = A(1)(0)|u1〉|0〉 + A(2)(0)|u2〉|0〉(4)+ A(3)(0)|u3〉|0〉.

It is easy to see that the coefficientsA(i)(t) andBj (t)

satisfy the following equations of motion

d

dtA(1)(t) =

∑j

g(1)j ei(ωu1l−ωj )tBj (t),

d

dtA(2)(t) =

∑j

g(2)j ei(ωu2l−ωj )tBj (t),

32 A. Joshi et al. / Physics Letters A 325 (2004) 30–36

(5)(a

the

er-

mic

be

aryicalwest

on(7)

un-

dyngvel.-als

desm-to-

ecaylev--

-ly,andin

o theen-nta-this-uchde-ic

ed.

d

dtA(3)(t) =

∑j

g(3)j ei(ωu3l−ωj )tBj (t),

(5)

d

dtBj (t) = −g

(1)j A(1)e−i(ωu1l−ωj )t

− g(2)j A(2)e−i(ωu2l−ωj )t

− g(3)j A(3)e−i(ωu3l−ωj )t .

We can formally integrate the last one of Eqs.and then substitute it in the remaining equationsWiesskopf–Wigner approximation) to obtain

d

dtA(1)(t) ≡ −γ1

2A(1)(t) −

√γ1γ2

2ei∆12tA(2)(t)

−√

γ1γ3

2ei∆13tA(3)(t),

d

dtA(2)(t) ≡ −

√γ1γ2

2A(1)(t)e−i∆12t − γ2

2A(2)(t)

−√

γ2γ3

2ei∆23tA(3)(t),

(6)

d

dtA(3)(t) ≡ −

√γ1γ3

2A(1)(t)e−i∆13t

−√

γ2γ3

2A(2)(t)e−i∆23t

− γ3

2A(2)(t).

If we substitute A(i)(t) = e−(γi/2)t A(i)(t), thenEqs. (6) can be re-casted as

d

dtA(1)(t) = −

√γ1γ2

2e(

γ1−γ22 +i∆12)t A(2)(t)

−√

γ1γ3

2e(

γ1−γ32 +i∆13)t A(3)(t),

d

dtA(2)(t) = −

√γ1γ2

2e−(

γ1−γ22 +i∆12)t A(1)(t)

−√

γ2γ3

2e(

γ2−γ32 +i∆23)t A(3)(t),

(7)

d

dtA(3)(t) = −

√γ1γ3

2e−(

γ1−γ32 +i∆13)t A(1)(t)

−√

γ2γ3

2e−(

γ2−γ32 +i∆23)t A(2)(t),

where the spontaneous emission coefficients ofthree upper levels areγi = 2(πg(i))2D(ωi) (i =1,2,3), respectively, in whichD(ωi) (i = 1,2,3) arethe mode density functions [6]. The frequency diffences among the three upper levels, i.e.,∆12, ∆23 and∆13, are assumed to be much smaller than the ato

transition frequenciesωu1l , ωu2l , andωu3l . Also, thedipole moments of the transitions are assumed toparallel.

In order to solve these equations under arbitrparametric conditions, we have to resort to numermethods. However, under special circumstancescan look for the analytical solutions. The simplesituation is with degenerate upper levels, i.e.,∆12 =∆23 = ∆31 = 0 andγ1 − γ2 = γ2 − γ3 = γ3 − γ1 orγ1 = γ2 = γ3 = γ , which is a reasonable assumptiin real experimental situation. The solution of Eqs.under this condition gives

(8)A(1)(t) = A(1)(0)

[2

3+ 1

3e−(3/2)γ t

],

(9)A(2)(t) = A(1)(0)

[−1

3+ 1

3e−(3/2)γ t

]= A(3)(t).

So, the steady-state populations of the upper levelsder the assumption ofA(1)(0) = 1 are|A(1)(∞)|2 =4/9 and|A(2)(∞)|2 = |A(3)(∞)|2 = 1/9, and the to-tal population in all the upper levels under steastate is 66.67% of the initial value. The remaini33.33% population decays down to the ground leIn the standardV -type three-level atom [6] the total population trapped in the two upper levels equto 50%. We have extended Eqs. (7) toN degenerateupper levels connected by the same vacuum moto the common lower level and, by numerical siulation, found a general rule for the amount oftal population trapped in the upper levels. ForN de-generate upper levels with equal spontaneous drates the total population trapped among all upperels is (N − 1)/N . Plots of such numerical calculations are given in Fig. 2 for several differentN val-ues. In Fig. 2 we keepγ1 = γ2 = γ3 = · · · = 0.5 and∆12 = ∆23 = ∆13 = · · · = 0 and plot the total population in the upper levels as a function of time. Clearsuppression in spontaneous decay becomes moremore significant when more upper levels participatethe spontaneous decay processes. However, due tselection rule in real atomic systems, only three degerate upper Zeeman sub-levels are allowed to sponeously decay to one of the lower levels, such thatresult for largeN (N > 3) is not practical in the degenerate multi-level atomic systems. Nevertheless, ssituation with more than 3 degenerate upper levelscay to one lower level might exist in other non-atomsystems, where this interesting result can be realiz

A. Joshi et al. / Physics Letters A 325 (2004) 30–36 33

stants

d 10,

nlyfor(7)

the

or

ic

ate.e.,

ed

ady-ach

velson

ineotnce

ientnt

reen ofan

lshease

nh is

Fig. 2. Evolution of the total population (S) as a function of timefor degenerate upper level systems. We set radiative decay conequal for all the upper levels (γi = 0.5). Curves A, B, C, D, E, andF correspond to the number of upper levels as 2, 3, 4, 5, 7, anrespectively.

Next, we concentrate on the four-level system oand do not invoke the condition of degeneracythe three upper levels. We can still solve Eqs.analytically under the conditions ofγ1 − γ2 = γ2 −γ3 = γ1 − γ3 (γ1 = γ2 = γ3 = γ ) and (i) ∆13 = 0,∆12 = −∆23 = ∆d ; (ii) ∆12 = 0, ∆23 = ∆13 = ∆d ;and (iii) ∆23 = 0, ∆12 = ∆13 = ∆d . For the condition(i) we have level 1 and level 3 degenerate andequations of motion forA(i) (i = 1,2,3) become

d

dtA(1)(t) = −1

2γ ei∆d t A(2)(t) − 1

2γ A(3)(t),

d

dtA(2)(t) = −1

2γ e−i∆d t A(1)(t) − 1

2γ e−i∆d t A(3)(t),

(10)d

dtA(3)(t) = −1

2γ A(1)(t) − 1

2γ ei∆d t A(2)(t).

By solving Eqs. (10) we obtain the solution fA(1)(t), A(2)(t), andA(3)(t) as

A(1)(t) = − 1

γei∆d t

[PD1e

D1t + QD2eD2t

] + 1

2eγ t/2,

A(3)(t) = − 1

γei∆d t

[PD1e

D1t + QD2eD2t

] − 1

2eγ t/2,

(11)A(2)(t) = PeD1t + QeD2t ,

where D1 and D2 are the roots of the quadratequation

(12)D2 +(

γ

2+ i∆d

)− γ 2

2= 0,

andP,Q = ∓(γ /2)/(D1 − D2), respectively.

It can be concluded that the total steady-stpopulation of the three upper levels in this case (iwhen t → ∞) reaches to the value 1/2 if initiallyall the population is in state|u1〉, (i.e., A(1)(0) =1, A(2)(0) = 0, A(3)(0) = 0). In the case (ii) when∆12 = 0, ∆23 = ∆13 = ∆d , and same initial conditionas in (i), we find exactly similar solution as detailin Eqs. (11) but the roles ofA(2)(t) and A(3)(t) areinterchanged. Hence in this case also the total stestate population of the three upper levels tends to rea value of 0.5. This means that out of three upper leif two are degenerate and initially all the populatiis in one of the degenerate levels, then this systemsteady-state behaves like aV -system with degeneratupper levels. The third upper level, which is ndegenerate with other two levels, does not influethe steady state provided it is not populated initially(see case (iii) discussed below). However, the transdynamical evolution of the system is quite differefrom the usual three-levelV -system. If initially thesystem is in an incoherent superposition of its thupper states then the total steady-state populatiothe upper levels will reach to a value of lower th0.5, a different behavior from theV -system. As wewill see in the following that all the three upper levehave different transient population evolutions in tthree cases under consideration. Finally, in the c(iii), when ∆23 = 0, ∆12 = ∆13 = ∆d , we find thefollowing set of equations of motion forA(i)(t) (i =1,2,3):

d

dtA(1)(t) = −γ

2ei∆d t A(2)(t) − γ

2ei∆dt A(3)(t),

d

dtA(2)(t) = −γ

2e−i∆d t A(1)(t) − γ

2A(3)(t),

(13)d

dtA(3)(t) = −γ

2e−i∆d t A(1)(t) − γ

2A(2)(t).

The solution ofA(i)(t) (i = 1,2,3) in this case is

(14)A(1)(t) = (ReD1t + SeD2t

),

(15)

A(2)(t) = − 1

γ

(RD1e

D1t + SD2eD2t

)e−i∆d t

= A(3)(t),

in which R,S = ±D2,1/(D2 − D1). D1 andD2 areroots of the quadratic equation (12) in which∆d isreplaced by−∆d . The total steady-state populatioof the upper levels in this case goes to zero whic

34 A. Joshi et al. / Physics Letters A 325 (2004) 30–36

theelvells.ered

(a)vels

tlyilarey

hewelllso

ch

nd

of1perof

.4,the

llyof

o

yto

achis

nsith

in

ellyits

ev-h

uslyvelsa

different from the first two cases. This means thatentire population will decay down to the ground levas we start with the entire population in the upper lewhich is non-degenerate with other two upper leve

In Fig. 3 we plot the transient evolution of thpopulation in upper levels for the cases consideabove by numerical simulation. We setγ1 = γ2 = γ3 =0.5 for this purpose and curves A, B, and C in Fig. 3represent the total population of the three upper lefor the cases of (∆13 = 0,∆12 = −∆23 = 0.2), (∆12 =0, ∆23 = ∆13 = 0.2), and (∆23 = 0, ∆12 = ∆13 =0.2), respectively. The curves A and B are exacidentical. The transient total population shows simdecay trend initially in all three cases and then thdiverge according to the different conditions. Tsteady-state values of the three curves matchwith the analytical results presented above. We aplot transient evolution of the population for eaindividual upper level for the cases of (∆12 = 0,∆23 = ∆13 = 0.2) and (∆23 = 0, ∆12 = ∆13 = 0.2)in Fig. 3(b) and (c), respectively. Curves A, B, aC in these figures are for|A(1)(t)|2, |A(2)(t)|2, and|A(3)(t)|2, respectively. In Fig. 3(b) the populationthe level |u1〉 decays down from its initial valueto the lower level as well as transfers to other uplevels due to SGC but eventually maintain a level0.25. So is the situation with level|u2〉 which startsinitially with 0 value and gets populated up to 0oscillates and eventually goes to a level of 0.25 insteady state. The population for level|u3〉 also startsfrom the value 0, shows oscillatory behavior but finadecays down completely. In Fig. 3(c) the populationthe level|u1〉 decays down from its initial value 1 tzero. The population of levels|u2〉 and |u3〉 increasefrom their initial zero values due to SGC followed boscillatory trend and then eventually decay downground level. Hence, the transient population of eindividual upper level shows interesting trend whichvery sensitive to the atomic parameters [6].

In Fig. 4(a) and (b) we plot the transient evolutioof the populations of the three upper levels along wthe total population for the cases of (∆12 = 0.1,∆23 =0.1,∆13 = 0.2) and (∆12 = 0.5, ∆23 = 0.5, ∆13 =1.0), respectively. The radiative decay constantsFig. 4(a) and (b) areγi(i = 1,2,3) = 0.5 and (γ1 =0.5, γ2 = 1.5, γ3 = 1.0), respectively. In Fig. 4(a), thlevel |u1〉 (curve A) decays down non-exponentiato the lower level and/or the upper levels from

Fig. 3. (a) Evolution of the total population (S) for thefour-level atomic system of Fig. 1 withγ1 = γ2 = γ3 = 0.5.Curves A, B, and C are for (∆13 = 0,∆12 = −∆23 = 0.2),(∆12 = 0,∆13 = ∆23 = 0.2), and (∆23 = 0,∆12 = ∆13 = 0.2),respectively. (b) Evolution of populations of the three upper lels (|A(i)(t)|2) (i = 1,2,3), for the four-level atomic system wit∆12 = 0, ∆13 = ∆23 = 0.2 andγ1 = γ2 = γ3 = 0.5. Curves A, B,and C represents population in|u1〉, |u2〉, and |u3〉, respectively.(c) Same as (b) but for∆23 = 0,∆12 = ∆13 = 0.2.

initial value 1. The other two upper levels|u2〉 and|u3〉(curves B and C) get populated due to spontaneogenerated quantum coherence. However, these letoo decay down to the lower level eventually in

A. Joshi et al. / Physics Letters A 325 (2004) 30–36 35

,

theby

recanno

rherns

tingctede

hean

nsi-an-eri-iffi-u-n-

lyec-

is-iche toofo-

thanse-hisbtainolerese-eerm-uchif-

ectngvel.the

elsvele-rd iner ofmoreels

heval

Fig. 4. Evolution of populations of the three upper levels (|A(i)(t)|2)(i = 1,2,3) of the four-level atomic system with∆12 = 0.1,∆23 = 0.1,∆13 = 0.2 andγ1 = γ2 = γ3 = 0.5. Curves A, B, C, andD represent populations in|u1〉, |u2〉, |u3〉, and the total populationrespectively. (b) The same as (a) but for∆12 = 0.5, ∆23 = 0.5,∆13 = 1.0 andγ1 = 0.5, γ2 = 1.5, andγ3 = 1.0.

non-exponential manner. The total population ofupper levels decaying to the ground level is showncurve D. This kind of behavior is seen because theis no degeneracy in the upper levels, so the SGCno longer trap any population in the upper levels,matter from which upper level we start initially.

In Fig. 4(b), the level|u1〉 (curve A) decays fastebecause the radiative decay constants are largerin comparison to Fig. 4(a). There are rapid oscillatioin the populations of the levels|u2〉 and |u3〉 incomparison to Fig. 4(a), which are due to the beaphenomenon in quantum interference for the selechoices of∆12, ∆23, and ∆13. These features ar

e

quite different from the simple exponential decay. Ttotal population also decays non-exponentially withoscillatory trend.

To observe the effect of SGC among nearby trations in a single atom, the dipole moments of the trsitions should be non-orthogonal. Hence the expmental demonstration of these results becomes dcult but could be achieved in an atomic or moleclar system where close lying upper levels with idetical J decay mainly to a single lower level. Onthose states with the sameM quantum number of thclosely-lying upper levels coupled through the vauum will provide non-identical absorption and emsion profiles in a transition to the lower state to whthese upper states are weakly coupled and give risSGC. The main significant problem with this kindsystems is that the levels with identical angular mmentum generally have energy separation largerthe radiative decay width [16]. So, a very carefullection of atomic/molecular system is required for tpurpose. By using dressed-state picture one can omany closely-lying upper states having parallel dipmoments so that the SGC can be achieved. The plected cavity polarization can also be used to enginthe parallel dipole moments [14]. Also, the quantuwell system may be a good candidate to realize squantum interference by the decay of levels from dferent wells to continuum via tunnelling [17].

To summarize, we have generalized the effof SGC on a multi-level atomic system consistiof three upper levels and a common ground leDepending on the relative level separations ofupper levels, as well as the initial distribution ofpopulations, the total population of these upper levcan decay or not decay down to the lower lecompletely. This behavior is different from the threlevel system inV -configuration with SGC. If the uppelevels are degenerate, more population is trappethese levels in the steady state. Also, as the numbthe degenerate upper levels increases, more andtotal population will be trapped in these upper levdue to SGC.

Acknowledgements

We acknowledge the funding support from tNational Science Foundation and the Office of Na

36 A. Joshi et al. / Physics Letters A 325 (2004) 30–36

tivetion

;

62

;y,

52

90)

66

)

ev.

86,

e,

s.

8;152

10.;

47;00)

117

hys.

;

2.s.

0)

ure

Research. We thank the referee for his/her construccomments and suggestions as well as for appreciaof this Letter.

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