bility

hendition for

Physics Letters A 315 (2003) 203–207

www.elsevier.com/locate/pla

Effect of spontaneously generated coherence on optical bistain three-levelΛ-type atomic system

Amitabh Joshi∗, Wenge Yang, Min Xiao

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

Received 4 June 2003; received in revised form 4 July 2003; accepted 9 July 2003

Communicated by P.R. Holland

Abstract

Optical bistability is studied for a three-level atomic system inΛ-configuration contained in an optical ring cavity and teffects of spontaneously generated coherence in the presence of two arbitrary coherent fields are investigated. Coobserving optical multistable behavior in this system has also been specified. 2003 Elsevier B.V. All rights reserved.

PACS: 42.65.Pc; 42.65.Sf; 42.50.Gy

pro-eenoci-the

lencthe[4].eGCofctsncye-on-ue

ys-

dlisin-tal

rustan-es.elet-i-eenmssub-

lar-ag-liedfor

ta-

In recent years some new types of coherenceduced due to decays of closely lying states have bdiscussed [1–5]. One of the main phenomena assated with these coherences is the modification ofline shapes of the spectral lines. InΛ-type three-leveatomic system the spontaneously generated coher(SGC) due to interaction with the vacuum bath ofradiation field has been discussed by JavanainenThe important finding in the study of Ref. [4] is thdisappearance of the dark state due to SGC. The Sis very sensitive to the alignment of dipole momentstwo transitions with respect to each other. The effeof SGC on electromagnetically induced transpare(EIT) [6] and coherent population trapping [7] phnomena were also examined [8]. Suppression of sptaneous emission [2] and subnatural linewidths [3] d

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

0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/S0375-9601(03)01046-6

e

to SGC were predicted in the three-level atomic stems.

Optical bistability (OB) was extensively studieinitially in the two-level system comprising of alkaatomic beams inside an optical resonator where agle mode beam circulating [9,10]. The experimendemonstration of OB in this system gave a great thto many potential applications such as optical trsistors, memory elements, and all optical switchThis led to further investigation of OB in three-levatomic system inside an optical cavity both theorically [11,12] as well as experimentally [13]. Multstable (multiple hysteresis) behaviors has also bobserved in Fabry–Perot cavities filled with atohaving several degenerate or nearly degeneratelevels in the ground state and driven by linearly poized light [14]. Most of these experiments used mnetic fields and high pressure buffer gases, and reon Zeeman coherence as an efficient mechanismthe observed behavior. More specifically, optical tris

.

204 A. Joshi et al. / Physics Letters A 315 (2003) 203–207

deri-

ndof

ex-mper

el.on-pulade-er,Bdes orvelionder

e is

sillesityed

iare

twol.

le

ionling

ftedionng.herremtumtly,tures ineld3],umlsof

ably

Fig. 1. Schematics of a three-levelΛ-type atom.

bility in a three-levelΛ-configuration was calculateunder different conditions and also observed expmentally [15–18].

In this Letter, we deal with the role of SGC oOB in a three-levelΛ-type atomic system containein a ring optical cavity. The atomic system consiststwo lower sublevels of same parity and a singlecited level of different parity. The effect of quantuinterference in spontaneous emission from the uplevel to the lower two levels is included in the modSuch a model with quantum interference from sptaneous emission shows interference-assisted potion inversion, trapping and probe gain at one siband of the Autler–Townes spectrum [19]. Howevthe previous theoretical works [11,12] studying Obehavior in three-level atomic systems did not inclusuch quantum interference in the decay channelSGC in their models. We expect to see some nochanges in the usual OB brought out by the inclusof this SGC in the three-level atomic system unconsideration.

The three-level atomic system considered hershown in Fig. 1. It is a closedΛ-type configurationwith one excited state|2〉 and two closely-lying lowerstates|1〉 and |3〉. The transition between|2〉 and|1〉 (with resonant frequencyω21) is mediated bythe probe laser fieldEP (frequencyω1) while thetransition|2〉 to |3〉 (with resonant frequencyω23) isdriven by another laser fieldEC (frequencyω2) calledcoupling field in this work. The atomic dynamicof the system can be described by the Liouvequation for the density operator and the denmatrix equations [4,8] with all decay terms includunder rotating-wave approximation are

-

ρ11 = 2γ1ρ22 − iΩP (ρ12 − ρ21),

ρ22 = −2(γ1 + γ2)ρ22

+ iΩP (ρ12 − ρ21) + iΩC(ρ32 − ρ23),

ρ33 = 2γ2ρ22 − iΩC(ρ32 − ρ23),

ρ21 = −(γ1 + γ2 − i∆P )ρ21 + iΩCρ31

+ iΩP (ρ11 − ρ22),

ρ23 = −(γ1 + γ2 − i∆C)ρ23 + iΩP ρ13

− iΩC(ρ22 − ρ33),

ρ13 = −i(∆P − ∆C)ρ13 − iΩCρ12

(1)+ iΩP ρ23 + 2√

γ1γ2 ηρ22.

Here, the atomic detunings are defined as∆P =ω21 − ω1, ∆C = ω23 − ω2, respectively, and the Rabfrequencies for the probe and coupling fieldsΩP = d12. EP /h and ΩC = d32. EC/h, respectively.The transition dipole moments associated with thetransitions, e.g.,d12 and d32, can be nonorthogonaWe define a parameter

η = d12. d32

| d12|.| d32|≡ cos(θ)

which is a measure of orthogonality of the dipomoments, i.e.,η = 0 for θ = π/2. Physically, theterm η

√γ1γ2 accounts for the spontaneous emiss

induced quantum interference effect due to coupbetween emission processes in the channels|2〉 → |1〉and |2〉 → |3〉. In a recent experiment the ability ocontrolling η has been experimentally demonstra[5] in sodium dimers by considering the superpositof singlet and triplet states due to spin-orbit coupliHowever, a conflicting result was obtained in anotexperiment of similar kind [20]. It seems there apractical difficulties to find a perfect atomic systewhere dipole moments are parallel so that quaninterference due to SGC is maximum [21]. Recensome novel methods are discussed in the litera[22–24] generating the quantum interference effectmore effective manners such as use of microwave fiin a three-level system [22], multilevel schemes [2or use of anisotropy of the electromagnetic vacu[24]. The Rabi frequencies in our model are afunctions of the angleθ , however, for the sake oconvenience in comparison with different values ofθ ,the Rabi frequencies are kept unchanged by suitadjusting the field strengths in this work.

A. Joshi et al. / Physics Letters A 315 (2003) 203–207 205

ing

ngaln-ir-

ir-ner-

cell

sa-

the

n-

ric

d

o bedi-

dy.theism.tofce

ngobeheedgthe

s-ityer-ysontheateit

nd-ri-t ins

lts

r-e-

o-

A,),-fer-eres A

ion

Fig. 2. Schematic diagram of a unidirectional ring cavity havfour mirrors (M1–M4) and an atomic vapor cell of lengthL. Themirrors M3 and M4 are perfectly reflecting mirrors (R = 1 for each).The incident and the transmitted fields are represented byEI

Pand

ETP

, respectively, and the coupling fieldEC is noncirculating in thecavity.

We can study OB in this system by consideriN such Λ-type atoms contained in a unidirectionoptical ring resonator depicted in Fig. 2. The intesity reflection and transmission coefficients of mrors M1 and M2 areR and T , respectively, suchthatR + T = 1. Further we assume that both the mrors M3 and M4 are perfect reflectors. This is oof the standard model for studying OB in the liteature [9]. The total electromagnetic field seen byN

homogeneously-broadened atoms contained in aof lengthL is

(2)E = EP exp(−iω1t) + EC exp(−iω2t) + c.c.

The probe field EP is circulating in the ring cavitybut the coupling fieldEC does not. So, the Maxwell’equation, under slowly-varying envelope approximtion is appropriate to describe the dynamics ofprobe field in the optical cavity,

(3)∂EP

∂t+ c

∂EP

∂z= 2πiω1d12P(ω1),

whereP(ω1) is the induced polarization in the trasition |1〉 → |2〉 and is given byP(ω1) = Nd12ρ12.The probe fieldEI

P enters the cavity through mirroM1, propagates in the cavity to interact with the atomsample of lengthL, then circulates in the cavity, anpartially transmits out from the mirror M2 asET

P . Theprobe field at the start of the atomic sample isEP (0)

and propagates to the end of the atomic sample tEP (L, t) in a single pass. The field boundary con

tions in this configuration are

ETP (t) = √

T EP (L, t),

(4)EP (0, t) = √T EI

P (t) + REP (L, t − t).

Time taken for light to travel from M2 to M1 ist

and we keep cavity detuning to be zero in this stuThe bistable behavior is due to the nonlinearity ofatomic medium and resonator feedback mechanIf we set R = 0 in Eq. (4) above, we do not geany bistability. We do not consider the circulationthe coupling beam in the optical cavity and henallow this beam to enter in the cavity via a polarizibeam splitter such that it copropagates with the prbeam to eliminate the first order Doppler effect for tthree-level atoms in the vapor cell [25] as describexperimentally in Ref. [13]. The size of the couplinbeam is larger in comparison to the probe beam invapor cell so that there is a good overlap.

It is rather difficult to have a closed form expresions for OB for this problem due to the complexof the dynamic equations (1)–(3). Also, it is cumbsome to putP(ω1) in a simple analytic form in steadstate for the three-level atomic system in comparito its two-level counterpart. Therefore, we solvedensity matrix equations (1) numerically and integrthe Maxwell’s equation (3) in the steady-state limover the length of the sample together with the bouary conditions (4) to get the results for OB under vaous parametric conditions. It should be noticed thathe limiting condition ofΩC → 0, this system reduceto the ordinary two-level atomic system.

In the following we discuss the numerical resufor OB in the three-levelΛ-system with SGC in-cluded. In Fig. 3 we plot the input–output field chaacteristics for different SGC (defined by the paramter η) with other parametersγ2/γ1 = 1, ΩC/γ1 = 5,∆C/γ1 = 1.0, ∆P /γ1 = 2.0, andC = 100. The def-inition of the cooperativity parameterC for atoms ina ring cavity is similar to what we have for the twlevel atom case [9], e.g.,C = αL/2T whereαL issingle-pass absorption by atomic medium. CurvesB, C, and D are forη = 0 (no quantum interferenceη = 0.5, η = 0.8, andη = 0.95 (large quantum interference), respectively. Clearly, the quantum interence reduces the bistabilty threshold (the point whtransition to upper branch takes place, see curveto C), which can be easily explained by reduct

206 A. Joshi et al. / Physics Letters A 315 (2003) 203–207

itytric

itytric

er-

tyme oging

abic

the

mstio

-(or-forgthsevelthe

ofep

,y.e

theks

r aofeseism

d-of

lla-oflineri-

ing

heval

15

Fig. 3. The input–output field characteristics of the optical cavfield for different values of quantum interference. The parameconditions areγ2/γ1 = 1, ΩC/γ1 = 5, ∆C/γ1 = 1, ∆P /γ1 = 2,andC = 100. Curves A, B, C and D are forη = 0.0,0.5,0.8, and0.95, respectively.

Fig. 4. The input–output field characteristics of the optical cavfield for different values of coupling field strengths. The parameconditions areγ2/γ1 = 1, ∆C/γ1 = 1, ∆P /γ1 = 2, η = 0.95, andC = 100. Curves A, B, C and D are forΩC/γ1 = 5, 4, 3, and 2.5,respectively.

in effective saturation intensity since quantum intference suppresses the radiative decay rate from|2〉to |1〉. In curve D, we observe optical multistabili(OM) in this system. The three-level atomic systehas an advantage over the two-level one becausthe additional controllability offered by the couplinfield strength and its frequency detuning. By adjustthese parameters along with SGC, one can alter thesorption and nonlinear optical properties of the atommedium for the cavity field and, therefore, change

f

-

steady-state behaviors. Also, for the two-level atothe atomic polarization responsible for OB is a raof polynomials of first order inΩP (in the numerator)and second order inΩP (in the denominator). However, the order of these polynomials can go higherder 5 in numerator and order 6 in denominator)three-level atoms depending on the relative strenof various parameters associated with the three-latoms [11,26]. The observed OM has certainlyroots in this complicated form of polarizationP(ω1)

in terms of the probe field amplitudeΩP . We haveconfirmed it in Fig. 4, where we see the sensitivityobserved OM on the coupling field strength. We kethe parameters in Fig. 4 asγ2/γ1 = 1, ∆C/γ1 = 1.0,∆P /γ1 = 2.0,η = 0.95, andC = 100. Curves A, B, Cand D are forΩC/γ1 = 5, 4, 3, and 2.5, respectivelClearly, with reducing the coupling field strength wfind disappearance of the OM as well increase inthreshold of the OB. Note that, in the earlier wor[11] it has been shown that the OB in aΛ-type three-level atomic system can be controlled using eithecoherent control field or with the initial coherencethe lower two levels. Here we have not used thmechanism but rather used an alternative mechanof SGC to control the threshold of OB. Similar stuies have been recently carried out in the collectionthree-level atomic system in a V configuration [27].

In summary, we have demonstrated the controbility of atomic OB by using the theoretical modelthree-level atoms inΛ-configuration inside an opticaring cavity. The controlling parameters is the SGCthe decay channels whose tunability has been expmentally demonstrated. The possibilities of obtainOM is also discussed, by SGC in the system.

Acknowledgements

We acknowledge the funding supports from tNational Science Foundation and the Office of NaResearch.

References

[1] D.A. Cardimona, M.G. Raymer, C.R. Stroud Jr., J. Phys. B(1982) 55.

[2] A. Imamoglu, Phys. Rev. A 40 (1989) 2835;

A. Joshi et al. / Physics Letters A 315 (2003) 203–207 207

62

10;

2.

5,

alh-rein

ev.

28

96)

sta-;m

2)

3)

m-

33

1;29

1)

ev.

. 42

3)

51

.O.

M.O. Scully, S.Y. Zhu, A. Gavrielides, Phys. Rev. Lett.(1989) 2813;S.Y. Zhu, R.C.F. Chan, C.P. Lee, Phys. Rev. A 52 (1995) 7G.C. Hegerfeldt, M.B. Plenio, Phys. Rev. A 46 (1992) 373.

[3] P. Zhou, S. Swain, Phys. Rev. Lett. 77 (1996) 3995;P. Zhou, S. Swain, Phys. Rev. Lett. 78 (1997) 832.

[4] J. Javanainen, Europhys. Lett. 17 (1992) 407.[5] H.R. Xia, C.Y. Ye, S.Y. Zhu, Phys. Rev. Lett. 77 (1996) 103[6] S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033;

S.E. Harris, J.J. Macklin, Phys. Rev. A 40 (1989) 4135.[7] E. Arimondo, in: E. Wolf (Ed.), Progress in Optics, Vol. 3

North-Holland, Amsterdam, 1996.[8] S. Menon, G.S. Agarwal, Phys. Rev. A 57 (1998) 4014.[9] See, for example, a review by L.A. Ligiato, Theory of optic

bistability, in: E. Wolf (Ed.), Progress in Optics, Vol. 21, NortHolland, Amsterdam, 1984, p. 71, and references theincluding of;H.J. Gibbs, S.L. McCall, T.N.C. Venkatesan, Phys. RLett. 36 (1976) 1135.

[10] A.T. Rosenberger, L.A. Orozco, H.J. Kimble, Phys. Rev. A(1983) 2569;L.A. Orozco, et al., Phys. Rev. A 39 (1989) 1235.

[11] W. Harshawardhan, G.S. Agarwal, Phys. Rev. A 53 (191812;S. Gong, S. Du, Z. Xu, Phys. Lett. A 226 (1997) 293.

[12] Some earlier interesting work on the three-level atomic bibility by D.F. Walls, P. Zoller, Opt. Commun. 34 (1980) 260D.F. Walls, P. Zoller, M.L. Steyn-Ross, IEEE J. QuantuElectron. QE-17 (1981) 380.

[13] H. Wang, D.J. Goorskey, M. Xiao, Phys. Rev. A 65 (200011801(R);A. Joshi, A. Brown, H. Wang, M. Xiao, Phys. Rev. A 67 (200041801(R).

[14] G. Giusfredi, P. Salieri, S. Cecchi, F.T. Arecchi, Opt. Comun. 54 (1985) 39;E. Giacobino, Opt. Commun. 56 (1985) 249;M.W. Hamilton, et al., Opt. Commun. 48 (1983) 190;F. Mitschke, R. Deserno, W. Lange, J. Mlynek, Phys. Rev. A(1986) 3219;J. Nalik, W. Lange, F. Mitschke, Appl. Phys. B 49 (1989) 19J. Mlynek, F. Mitschke, R. Deserno, W. Lange, Phys. Rev. A(1984) 1297.

[15] M. Kitano, T. Yabuzaki, T. Ogawa, Phys. Rev. Lett. 46 (198926.

[16] S. Cecchi, G. Giusfredi, E. Petriella, P. Salieri, Phys. RLett. 49 (1982) 1928.

[17] C.M. Savage, H.J. Carmichael, D.F. Walls, Opt. Commun(1982) 211.

[18] F.T. Arecchi, J. Kurmann, A. Politi, Opt. Commun. 44 (198421.

[19] P. Zhou, Phys. Rev. A 63 (2001) 023810.[20] L. Li, et al., Phys. Rev. Lett. 84 (2000) 4016.[21] P. Zhou, S. Swain, L. You, Phys. Rev. A 63 (2001) 033818.[22] M.O. Scully, et al., Phys. Rev. Lett. 62 (1989) 2813;

M. Fleischhauer, et al., Opt. Commun. 94 (1992) 599;S.Y. Zhu, et al., Phys. Rev. A 52 (1995) 4791;E. Paspalakis, et al., Phys. Rev. A 58 (1998) 4868.

[23] A.K. Patnaik, G.S. Agarwal, Phys. Rev. A 59 (1999) 3015.[24] G.S. Agarwal, Phys. Rev. Lett. 84 (2000) 5500.[25] J. Gea-Banacloche, Y. Li, S. Jin, M. Xiao, Phys. Rev. A

(1995) 576.[26] U. Rathe, M. Fleischhauer, S.Y. Zhu, T.W. Hansch, M

Scully, Phys. Rev. A 47 (1993) 4994.[27] A. Joshi, et al., Phys. Rev. A (2003), in press.