Velocity Changing and Dephasing collisions Effect on electromagnetically induced transparency in V-type Three level Atomic System.

Transcription

1 Velocity Changing and Dehasing collisions Effect on electromagnetically induced transarency in V-tye Three level Atomic System. Anil Kumar M. and Suneel Singh University of Hyderabad, School of hysics, India * suneels@uohyd.ernet.in Abstract: We study eelectromagnetically induced transarency in an inhomogeneously broadened three level V- tye atomic system including residual Doler broadening which arises due to um-robe wave vector mismatch. We show that the velocity changing collisions (VCC) and dehasing collisions (DPC) lay a key role in creating transarency at exact resonance at relatively low couling field intensities. 1. INTRODUCTION Electromagnetically induced transarency [EIT][1] is an effect observed when a weak robe beam roagating through medium, on resonance with an atomic transition shows reduced absortion due to the resence of a strong couling beam on another linked transition. In addition to reduced absortion the modification of disersion roerties of the medium leads to subluminal or suerluminal grou velocity of the transmitted robe. Accordingly EIT has received considerable attention in over the years as it offers a variety of interesting and otentially imortant alications including light storage [-4], quantum information [5] and recision magnetometers [6,7]. Study and identification of various mechanisms that lead to broadening of EIT resonance has attracted considerable attention in recent years. Three level models of various tyes of ladder, V and Λ configurations are most commonly emloyed for studying EIT in vaors or solids. The exerimental observation of cw wave vector mismatched transarency in a V-tye Doler broadened system has been reorted by several authors earlier [8-10]. In this work we resent a general theory in an inhomogeneously broaden V-tye atomic system in which Doler broadening of both one-and two hoton transitions occurs. The V -tye three level systems involved in the resent study of EIT is shown in Fig.1. The theory is more general in the sense that it takes into account the residual Doler broadening of the two hoton coherence in different wave-vector mismatch regimes and also the collisional effects of an additional (buffer) gas in vaor cell. Both the velocity changing and dehasing asects of collisions with atoms of the buffer gas are considered. The numerical calculations are based on a simle but often alicable collision model which facilitates inclusion of the effect of both the velocity- changing collisions and dehasing collisions on level coherence. In Sec., a theoretical formalism is resented, where we solve for the steady- state density-matrix equations for this system and erform a Doler averaging over the atomic suscetibility and exressions for robe absortion coefficient are obtained. Numerical results for robe absortion rofiles are resented in Sec.. Ω Ωc Fig 1: The three level V-system. Ω c and Ω resectively are the couling and robe field Rabi frequencies alied to transitions 1 and 1.. THEORETICAL FORMULATION In order to investigate EIT in a V- tye atomic system, we use a semi-classical aroach in which the atom is a quantum object and the light is described by a classical field. We consider a tyical three level V-tye atomic system as shown Fig.1. The 1

2 sontaneous emission rate from the uer level > to intermediate level > is S and that from level > to ground level 1> is S 1. A couling field r v v v E ( = ε ex[i(k r ω t) + c.c.) of frequency c c c c ω c wave vector k v c and Rabi frequency Ω = ( µ v εv ) / h is driving the 1> > transition c 1 c and a weak robe field, r v v v E ( = ε ex[i(k r ω t) + c.c.) of frequency ω wavevector k v and Rabi frequency Ω = ( µ v εv ) / h, is alied to the 1> > 1 transition. Here µ v and µ v are the diole moment 1 1 of 1> > and 1> > transition resectively. The interaction Hamiltonian V int under near resonant conditions and rotating wave aroximation is r int r V = - h[ω ex[i(k.r + t)] 1 c c c r r - Ω ex[i(k.r + t)] 1 +H.c.] (1) Where = ( ω ω ) and = ( ω ω ) denote 1 c 1 c detuning of the robe and control field frequencies from atomic resonance frequencies ω 1 and ω 1 resectively and i j (i, j =1- ) are the atomic raising or lowering oerators. Density matrix- Equation of motion: We use the density matrix formalism as it allows us to incororate the effect of various decay mechanisms on oulations of the atomic levels involved and the coherences established between them. It is ossible to relate atomic suscetibilities to these coherences and thus insect the absortion and disersion of the light incident on the atoms. The time evolution of the density matrix of the system in the interaction icture is ρ & = - ρ V -V ρ +( ρ ) () int int jm mk jm mk Rel where ρ = j ρ k and int int = m V n, (j,k, m, n V mn = 1,, ) can be calculated using the interaction int Hamiltonian V given by Eq.(1). Doler shift of atomic resonance due to thermal motion of atoms in the medium gives rise to inhomogeneous broadening. To incororate atomic motion, the derivative ρ& on the left hand side of Eq.() can be relaced by ρ& {( / t) + v r r } ρ, () where v r is the atomic velocity. For brevity we shall henceforth denote a function f(v x, v y, v z ) by f(v). The second term in Eq.() describes henomenological inclusion of the effect of relaxation rocesses such as sontaneous emission S (j, k= 1-), radiative decay of off-diagonal elements γ (= S / ), and collisions in the system. Collisions with buffer gas atoms can cause dehasing as well result in changes in the velocity of a vaor atom. Effect of such collisions can be incororated in the density matrix formalism by including a term [ 4] ( (v,t )) coll ρ& = γ h (1 δ ) ρ (v,t) Γ ρ (v,t) W (v v) (v, t)d v (4) + ρ Here γ h is the rate of collision induced dehasing of otical coherences and Γ is some average rate of change in velocity v. For simlicity Γ is assumed to be indeendent of v and related to the collision kernel W (v v) by Γ = W (v v ) d v (5) The collision kernel in general is assumed to be of the form W (v v) = W (v α v ), (6) Where α is a constant, (1 > α > 0). Physically the second term in Eq.(.5) is the out term reresenting collisional shift of atoms with a velocity v to other velocity subclasses at some rate Γ and the third term is the in term arising due to collisional shift of atoms from other velocity subclasses into velocity subclass v. In this work we however restrict our discussion to the case of most revalent and exerimentally relevant strong collision case in which collisions result in raid thermalization of the velocity distribution of the system. Strong collision model In this case a single collision on average, thermalizes the velocity shift distribution regardless of the initial velocity i.e., the collision kernel of Eq.(.7) is assumed to indeendent of initial velocity

3 and of the form Lt W (v v) = Γ M(v), (7) α 0 where 1 M(v) = (v π) ex[ v v / v ] th th r r (8) is the Maxwellian velocity distribution and v (k T / m ) 1 / = is the most robable thermal th B A velocity at a temerature T of an atom of mass m A. The equations describing time evolution of the slowly varying comonents of the density matrix elements ~ ρ (v, t), can now be written using Eq.(1),()-(4) and (7) in Eq. () and aroriate transformations to eliminates fast oscillating terms. ( ) ρ &% = iω ρ% i Ω ρ% { γ + Γ i[ + c c ( ) } ( ) + k k v ] ρ % + Γ M (v) ρ dv (9.a) c z 11 ρ &% = iω ρ ρ i Ω ρ { γ + Γ i[ c k v ]} ρ% (9.b) z 1 ρ &% = iω ρ ρ i Ω ρ { γ + Γ i[ ( ) c c + k v ]} ρ% (9.c) c z ρ &% = iω ( ρ % ρ % ) + iω ( ρ % ρ % ) Γ ρ % ρ % ρ % + Γ M(v) ρ dv 1 11 (9.d) 11 ρ &% = iω ( ρ % ρ % ) γ ρ % Γ ρ % + Γ M(v) ρ dv (9.e) ρ &% = i Ω ( ρ % ρ % ) γ ρ Γ ρ (9.f) 11 c Our aim is to determine the velocity averaged first (1) (1) order hoton coherence Ι = dv 1 ρ the imaginary 1 and real art of which describes robe absortion and disersion, resectively, in the three level V-tye system. Under Steady state conditions the density matrix equation is to be solved. Initially all the oulation is in the level > > 1> by the alication of strong control field. The zeroth order solutions are obtained 0 0 under the assumtion ρ + ρ = The 11 M(v) Zeroth and first order matrix elements are ρ (v, t), ρ (v, t), ρ, ρ (v, t), ρ (v, t). (10) o o 0 (1) (1) The relevant first order density matrix equations are found as ρ &% = [ i( ) + (k k )v +γ +γ +Γ +γ ] ρ % (1) (1) c c z 1 1 iω ρ% iω ρ % +Γ M(v) ρ dv (11.a) (1) (1) (1) c 1 ρ &% = [ i( + k v ) +γ +Γ +γ ] ρ % + iω ρ % (1) (1) (1) 1 z c iω ρ% (0) 11 (11.b) the study state solution obtained by setting the time derivative to zero on the left-hand side of Eq.(11) yields the velocity averaged one-hoton coherence as ( A ρ i Ω ρ ) M(v)dv ρ = Ω + Ω Γ 0 (1) (1) 11 c 1 (1) dv i dv i I 1 A A + Ω A A + Ω 1 c 1 c Where A 1, A and (1) I are given below (1) A = [i( c) + (k k c)v z 1 1 +Γ ] (1.a) A 1 = [i( + kv z) +γ 1 +Γ 1 +γ ] (1.b) I i Ω ( i Ω ρ A ρ ) 0 ( 0 ) c A A + Ω 1 c = M ( v ) A d v 1 1 Γ A A + Ω 1 c. Numerical result and discussions: (1.c) We now resent numerical results for EIT alying the theory to an atomic vaor contained in a cell in which a buffer gas is also introduced. The velocity changing collision rate Γ (and dehasing rate γ ) c can be varied by changing the ressure of the buffer gas in the vaor cell. It was found in many earlier studies [11, 1] that the nature of EIT in a three level system deends critically on the sign of the residual Doler width δk = (k kc ) which deending uon the robe and control field wavevector mismatch, is either ositive ( k > k c ) or negative (k < k c ). For very large wave vector mismatch it was shown in Ref. 1 that EIT in a V system is markedly dissimilar in these two cases. In our study in the wave vector mismatch in 87 Rb D 1 and D transitions is very small comared with that in earliear studies [11, 1]. In Fig. (i) the absortion rofile is shown for

4 ositive mismatch ( k > k c ) regime of residual Doler broadening. The absortion rofile shown in Fig (ii) is for a negative mismatch (k < k c ) regime of residual Doler broadening. Absortion Imρ Absortion Imρ i ii P Fig. Probe absortion as a function of robe detuning at a fixed control field amlitude, Ω c / γ D = 0.01 and fixed dehasing collisions rate γ / γ D = 0.01 and for various velocity changing collisions rates Γ. The value of residual Doler broadening are: (i) δkv th /γ D = and (ii) δkv th /γ D = Solid lines corresond to the case when no velocity changing collisions are resent. The dotted, dashed and dot dashed absortion rofiles corresond to the VCC rates (a) Γ= 0.1γ D, (b) Γ=0.01γ D and (c) Γ=0.001γ D resectively. The aearance of a transarency di at exact resonance shows that with increase in VCC rates at fixed dehasing and couling field intensities absortion is reduced at exact resonance. The transarency is much more in the case of negative mismatch (k < k c ) than that comared with the ositive mismatch ( k > k c ) regime. Therefore the residual Doler broadening arising due to um- robe wave vector mismatch is suressed by VCC which also enhances transarency in both wave vector regimes. Acknowledgments This work is suorted by center for advanced studies (CAS) sonsored by UGC. 4. REFERENCES [1]. J. E. Field, K. H. Hahn and S. E. Harris, Phys. Rev. Lett. 67, 06 (1991); S. E. Harris, Phys. Today 50, 6 (1997). [].. C. Liu, Z. Dutton, C. H. Behroozi and L. V. Hau, Nature 409, 490 (001). []. M. Bajcsy, A. S. Zibrov and M. D. Lukin, Nature 46, 68 (00). [4]. M. D. Lukin, Rev. Mod. Phys. 75, 457 (00). [5]. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan and H. J. Kimble, Nature 4, 71 (00). [6]. D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, Phys. Rev. A 6, 0440 (000). [7]. T. W. Kornack, J. C. Allred, and M. V. Romalis, Nature 4, 596 (00). [8]. J.R. Boon, E.Zekou, D.J. Fulton, and M.H. Dum, Phys.Rev. A 57,1 (1998) [9]. J.R. Boon, E.Zekou, and M.H. Dum, Phys.Rev. A 58, 560 (1998) [10]. Silvije Vdovic, Ticijana Ban, Damir Aumiler, Goran Pichler Ot commun 7 (007) S. Sheherd, D. J. Fulton, and M. H. Dunn, Phys. Rev. A 54, 594 (1996). 1. J. R. Boon, E. Zekou, D. McGloin, and M. H. Dunn, Phys. Rev. A 59, 4675 (1999).

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