Temporal and diffraction effects in entanglement creation in an optical cavity
aa r X i v : . [ qu a n t - ph ] A p r Temporal and diffraction effects in entanglement creation in an optical cavity
Sonny Natali and Z. Ficek ∗ Department of Physics, School of Physical Sciences,The University of Queensland, Brisbane, Australia 4072 (Dated: October 28, 2018)A practical scheme for entanglement creation between distant atoms located inside a single-modeoptical cavity is discussed. We show that the degree of entanglement and the time it takes for theentanglement to reach its optimum value is a sensitive function the initial conditions and the positionof the atoms inside the cavity mode. It is found that the entangled properties of the two atomscan readily be extracted from dynamics of a simple two-level system. Effectively, we engineer twocoupled qubits whose the dynamics are analogous to that of a driven single two-level system. It isfound that spatial variations of the coupling constants actually help to create transient entanglementwhich may appear on the time scale much longer than that predicted for the case of equal couplingconstants. When the atoms are initially prepared in an entangled state, they may remain entangledfor all times. We also find that the entanglement exhibits an interesting phenomenon of diffractionwhen the the atoms are located between the nodes and antinodes of the cavity mode. The diffractionpattern of the entanglement varies with time and we explain this effect in terms of the quantumproperty of complementarity, which is manifested as a tradeoff between the knowledge of energy ofthe exchanged photon versus the evolution time of the system.
PACS numbers: 32.80.Qk, 42.50.Fx, 42.50.Dv
I. INTRODUCTION
The study of practical schemes for creation of quantumentanglement between atoms (or ions) is the most activearea in quantum optics and quantum information sci-ence [1]. Different schemes have been proposed includingatom trapped inside a single mode cavity [2, 3, 4], or in-side two separate cavities [5, 6, 7, 8, 9, 10, 11, 12]. Oneof the most popular scheme involves two two-level atomslocated (trapped) within a single-mode cavity field. Ithas been demonstrated that the entanglement could inprinciple be created through a continuous observation ofthe cavity field [13] or through dispersive atom-cavityfield interactions [14, 15, 16], thereby creating a strong”action at a distance”. The approach used is straightfor-ward: Provided no photon is leaking through the cavitymirrors or no photon is exchanged between the atomsand the cavity field, a pure entangled state between thetwo atoms results. However, all of these procedures forgenerating entangled atoms have suffered from a com-mon handicap: their choice of equal coupling strengthsof the atoms to the cavity mode. The difficulty is that theentanglement depends on the the coupling constant be-tween the atoms and cavity mode which depends, in turn,on the location of the atoms in the cavity mode. In astanding wave cavity, one can achieve the equal couplingconstants by locating the atoms precisely at the antin-odes of the cavity field, or by sending slowed (cooled)atoms through a cavity antinode in the direction perpen-dicular to the cavity axis. This is a relatively easy taskat microwave frequencies and, in fact, detection of entan- ∗ Electronic address: fi[email protected] gled atoms have already been performed on a beam ofRydberg atoms traversing a superconducting microwavecavity [17]. However, at optical frequencies this task maywell be hard to achieve. In practice, the dipole couplingconstants vary with the location of the atoms in the cav-ity mode. For example, in a standing-wave structure ofthe cavity mode, the coupling constant varies with theposition of the atom as [18] g ( ~r ) ≡ g ( r, z ) = g e − r /w cos ( kz ) , (1)where z determines location of the atom along the cav-ity axis, k = 2 π/λ is the wave number of the field, r = ( x + y ) / is the distance of the atom from thecavity axis, and w is the mode waist. The couplingconstant reaches maximum value g when the atom is lo-cated on the cavity axis at an antinode of the standingwave. In practice one would like the varying couplingconstant g ( ~r ) to coincide with g . However, because ofthe small wavelength, locating the atoms precisely at theantinodes of the cavity mode and thereby eliminating thevariation of g ( ~r ) with position is very difficult in an opti-cal cavity. This may change the physics completely andthus suggests that the problem of creating entanglementbetween distant atoms in an optical cavity is a significantexperimental challenge.In this paper, we are concerned with the optical frequencyregime and investigate the mechanism involved in cre-ation of entanglement between distant atoms coupled toa single-mode cavity field. Our treatment closely followsthe approach that was used Refs [14, 15, 16], but withone essential difference. We include a possible variationof the coupling constant g ( ~r ) with the location of theatoms in a standing-wave cavity mode. We are particu-larly interested in the consequences of this variation onentanglement creation between the atoms, since this willbe very pertinent to any practical experimental arrange-ments as the distances involved are very small. First ofall, we derive the general master equation for the reduceddensity operator ρ s of two two-level atoms coupled to asingle mode cavity field. Our approach holds for atoms atrest or slowly moving through a single-mode cavity. Forsufficiently cold or slow atoms, radiative equilibrium isreached with an essentially fixed coupling constant, g ( ~r ),at every point inside the cavity, permitting studies of en-tangled properties of the system without performing theaverage over random locations and atomic dipole orienta-tions. We assume the atoms are far enough apart that thedirect dipole-dipole coupling or other direct interactionsbetween the atoms can be neglected. This also allowsa selective preparation of the atoms such that a givenset of initial conditions for the atomic states is achieved.We solve the master equation for two atoms coupled to acavity mode in the limit of a large detuning of the cavitymode from the atomic resonance frequency. This enablesto eliminate the cavity mode and obtain a new masterequation in which we will recognize some terms equiva-lent to the dipole-dipole interaction between the atomsand to the Stark shift of the atomic resonance frequen-cies. The analogy of this system with that of a singletwo-level atom driven by a detuned coherent field is ex-ploited and discussed. This analogy provides a simpledescription of the process of entanglement creation andleads to a useful pictorial representation of the systemin terms of the Bloch vector model. To quantify thedegree of entanglement, we use the concurrence that isthe widely accepted measure of two-atom entanglement.Simple analytical expressions are obtained for the concur-rence that are valid for arbitrary initial conditions andarbitrary positions of the atoms inside the cavity mode.We obtain the interesting result that spatial variations ofthe coupling constants actually help to create transiententanglement which may appear on the time scale muchlonger than that predicted for the case of equal couplingconstants. We explain this effect in terms of the degree oflocalization of the energy induced in the field by the in-teracting atoms. Moreover, we find that for an imperfectlocation of the atoms inside the cavity mode the entangle-ment exhibits an interesting time-dependent diffractionphenomenon. II. MASTER EQUATION
The system we consider consists of two identical two-levelatoms (qubits) with upper levels | e i i , ( i = 1 , | g i i , and separated by energy ~ ω . The atoms arecoupled to a standing-wave cavity mode with the positiondependent coupling constants g ( ~r i ), and damped at therate γ by spontaneous emission to modes other than theprivileged cavity mode. The cavity mode is damped withthe rate κ and its frequency ω c is significantly detunedfrom the atomic transition frequency ω , so there is nodirect exchange of photons between the atoms and the cavity mode. The behavior of the system is described bythe density operator ρ , which in the interaction picturesatisfies the master equation ∂ρ∂t = − i ~ [ H, ρ ] + 12 γ L a ρ + 12 κ L c ρ, (2)where H = ~ X j =1 (cid:2) g ( ~r j ) aS + j e − i ∆ t + H . c . (cid:3) (3)describes the interaction between the cavity field and theatoms, L c ρ = (cid:0) aρa † − a † aρ − ρa † a (cid:1) , (4)and L a ρ = X j =1 (cid:0) S − j ρS + j − S + j S − j ρ − ρS + j S − j (cid:1) . (5)are operators representing the damping of the atoms byspontaneous emission and of the cavity field by the cav-ity decay, respectively. The operators S + j and S − j arethe raising and lowering operators of the j th atom, S zj describes its energy, a and a † are the cavity-mode an-nihilation and creation operators, ∆ = ω c − ω is thedetuning of the cavity-mode frequency from the atomictransition frequency, and ~r j is the position coordinate ofthe j th atom within the cavity mode.The atoms located at different positions may experiencedifferent coupling constants, that is g ( ~r ) = g ( ~r ). Be-cause it is precisely the effect of unequal coupling con-stants that interest us most here, we choose the referenceframe such that g ( ~r ) = g , and g ( ~r ) = g cos ( kr ) , (6)where r = z − z is the distance between the atoms.This choice of the reference frame corresponds to a sit-uation where atom 1 is kept exactly at an antinode ofthe standing wave and the atom 2 is moved through suc-cessive nodes and antinodes of the standing wave. Thischoice, of course, involves no loss of generality.We also assume that the atoms are stationary during theinteraction with the cavity mode, i.e. the distance be-tween the atoms is independent of time (the Raman-Nathapproximation). This is a good approximation for manyexperiments on cooling of trapped atoms, where the stor-age time of the trapped atoms is long, so that they areessentially motionless and lie at known and controllabledistances from one another [19].In order to study the dynamics of the system, we intro-duce density-matrix elements with respect to the cavityfield mode, denoting h n | ρ | m i by ρ nm , and find from themaster equation (2) that the populations of the two low-est energy levels and coherence between them satisfy thefollowing equations of motion˙ ρ = − i X j =1 g j (cid:0) S + j ˜ ρ − ˜ ρ S − j (cid:1) + 12 γ L a ρ + κρ , ˙˜ ρ = i ∆˜ ρ − i X j =1 g j (cid:0) S + j ρ − ρ S + j (cid:1) + 12 γ L a ˜ ρ − κ ˜ ρ , ˙˜ ρ = − i ∆˜ ρ + i X j =1 g j (cid:0) ρ S − j − S − j ρ (cid:1) + 12 γ L a ˜ ρ − κ ˜ ρ , ˙ ρ = − i X j =1 g j (cid:0) S − j ˜ ρ − ˜ ρ S + j (cid:1) + 12 γ L a ρ − κρ , (7)where g j ≡ g ( ~r j ) and ˜ ρ nm are slowly varying parts of thecoherences, ˜ ρ = ρ exp( i ∆ t ) and ˜ ρ = ρ exp( − i ∆ t ).We have considered only the two lowest energy levels asin the limit of a large detuning ∆ only the ground state( n = 0) and the one-photon state ( n = 1) of the cavitymode are populated.Now, we explicitly apply the adiabatic approximationthat for a large detuning, the coherences ˜ ρ and ˜ ρ vary slowly in time, so we can assume that ˙˜ ρ ≈ ρ ≈
0. In this case, we find from (7) that in the limitof ∆ ≫ g j ≫ γ, κ ˜ ρ ≈ X j =1 g j (cid:0) S + j ρ − ρ S + j (cid:1) , ˜ ρ ≈ X j =1 g j (cid:0) ρ S − j − S − j ρ (cid:1) . (8)Knowledge of the coherences ˜ ρ and ˜ ρ allows us to de-rive the master equation for the reduced density operatorof the atoms. It is done in the following way. First wesubstitute (8) into (7), and after neglecting the popula-tion ρ , as the cavity mode will never be populated, wefind ˙ ρ = i ∆ X i,j =1 g i g j (cid:2) S + i S − j , ρ (cid:3) + 12 γ L a ρ . (9)Since ρ = Tr F ( ρ ) = ρ s is the reduced density opera-tor of the atoms, we obtain the master equation for thedensity operator of the atoms dρ s dt = i X i =1 δ i (cid:2) S + i S − i , ρ s (cid:3) + i X i = j =1 Ω ij (cid:2) S + i S − j , ρ s (cid:3) + 12 γ L a ρ s , (10) where δ i = g i ∆ , and Ω ij = Ω ji = g i g j ∆ . (11)The first two terms in the master equation (10) dependon the position coordinate of the atoms and give rise tofrequency shifts of the atomic levels and the coupling be-tween the atoms, respectively. The third term representsthe damping of the atoms through the interaction withthe environment. The parameter δ i represents the shiftin energy separation of the levels of the i th atom due tothe dispersive interaction with the cavity mode. It is ananalog of a dynamic Stark shift. One can easily see fromthe structure of the first term in the master equation that i X i =1 δ i (cid:2) S + i S − i , ρ s (cid:3) = i X i =1 δ i (cid:20) S zi + 12 , ρ s (cid:21) = i X i =1 δ i [ S zi , ρ s ] , (12)which clearly shows that this term is an analog of theenergy shift term. Thus in the interaction picture usedhere, it represents a shift of the atomic energy levels. Wenote that due to nonequivalent positions of the atoms, theshift of the energy levels is different for different atoms.The multi-atom term Ω ij represents the shift in energyseparation of the levels of atom i due to its interactionwith the atom j through the cavity mode. If the atomsare located at antinodes of the standing wave, the termΩ ij is maximal, whereas Ω ij = 0 if at least one of theatoms is located at a node of the standing wave. Fromthe structure of the second term in Eq. (10) one can rec-ognize that Ω ij is an analog of the familiar dipole-dipoleinteraction between the atoms [20, 21]. This shows thatthe interaction of the atoms with a detuned cavity fieldsproduces a structure in the master equation analogous tothe dipole-dipole interaction between the atoms.The above procedure shows that the adiabatic elimina-tion of the cavity mode creates a shift of the atomic tran-sition frequencies and an effective interaction betweentwo distant atoms. Thus, the dynamics of the systemcomposed of two identical atoms in nonequivalent po-sitions in the cavity mode is equivalent to those of twonon-identical atoms of different transition frequencies. Inother words, the procedure is an example of how one can“engineer” the dipole-dipole interaction between distantatoms. It is now easy to understand why two indepen-dent atoms coupled to a strongly detuned cavity modecan exhibit entanglement. Simply, the reduced systemis equivalent to two atoms coupled through the induceddipole-dipole interaction. Exactly, this process gives riseto the entanglement.We point out in passing that despite of the presence ofthe coherent dipole-dipole interaction term, the masterequation (10) is not fully equivalent to the master equa-tion of two collective interacting atoms [23]. This is be-cause there is no contribution from the cross-dampingterms involving dipole operators of two different atoms.In other words, the interaction with a strongly detunedcavity mode does not create the collective damping ofthe atoms. As a result, the atoms interact independentlywith the environment, so that the system does not evolveto a dark state characteristic of the completely collectivesystem [21, 22, 23]. The cross-damping terms would ap-pear in the master equation if one assumes the near res-onant interaction, ∆ ≈
0, and the ”bad-cavity” limit of κ ≫ g j ≫ ∆ [4, 24]. III. EQUIVALENT TWO-LEVEL DYNAMICS
The question we are interested in concerns the conse-quences of the spatial variation of the coupling constant g on the entanglement creation between two atoms lo-cated in a strongly detuned single-mode cavity field. Toanswer this question we consider the evolution of the di-agonal density matrix elements which correspond to theoccupation probabilities of the energy levels of the two-atom system. Using the master equation (10), we findthe following equations of motion˙ ρ = γ − γ ( ρ + ρ ) , (13)˙ ρ = − γρ , (14)˙ ρ = − γρ + γρ + i Ω ( ρ − ρ ) , (15)˙ ρ = − γρ + γρ − i Ω ( ρ − ρ ) , (16)while the off-diagonal density matrix elements that arecoupled to the diagonal elements obey the equations˙ ρ = − ( γ − iδ ) ρ + i Ω ( ρ − ρ ) , (17)˙ ρ = − ( γ + iδ ) ρ − i Ω ( ρ − ρ ) , (18)where we have used the standard direct-product basisgiven by | i = | g i| g i , | i = | g i| e i , | i = | e i| g i , | i = | e i| e i . (19)Here, δ = δ − δ is a difference between the single-atomStark shifts. This parameter is of central importance hereas it determines the relative variation of atomic transitionfrequencies with position of the atoms inside the cavitymode. In the special case of g = g , the parameter δ = 0, but this can happen only when the atoms are inequivalent positions inside the mode.It is easy to see that the set of the coupled equations (13)–(18) can be split into two independent sets of equationsof motion, which can be written in the form˙ ρ = γ − γ ( ρ + ρ ) , ˙ ρ = − γρ , ˙ ρ ++ = − γρ ++ + 2 γρ , (20)and ˙ u = − γu + δ v, ˙ v = − γv − δ u − w, ˙ w = − γw + 2Ω v, (21) where ρ ++ = ρ + ρ , u = ρ + ρ , v = i ( ρ − ρ ),and w = ρ − ρ . Note that the equations of motion(21) are the exact equivalent of the optical Bloch equa-tions of a two-level system driven by a detuned coherentfield, where the dipole-dipole interaction Ω couples tothe levels like a coherent field, and δ appears as a de-tuning of the field from the driven transition. The upperand lower energy levels | i and | i thus show a dynam-ics that is analogous to that of a driven two-level sys-tem. It should be pointed out that the analogy is notabsolute because, in contrast to the case of a single two-level atom driven by a detuned laser field, Ω and δ in Eq. (21) depend on the same parameters g and g ,i.e. on the coupling constants of the atoms to the cavityfield. Consequently, the parameters Ω and δ are notindependent.The dynamics of the effective two-level system can beeasily visualized in the Bloch vector model [25]. In thismodel, the system and the driving field are representedby vectors in a three-dimensional space, and the time evo-lution is simply visualized as a precession of the system-state vector about the driving field. In terms of the Blochvector, Eq. (21) can be written as d ~Bdt = − γ ~B + ~ Ω B × ~B, (22)where ~ Ω B = ( − , , δ ) is the pseudofield vector and ~B = ( u, v, w ) is the Bloch vector. The quantities u and v are, respectively, the real and imaginary parts of the co-herence between the levels | i and | i , and w is the popu-lation inversion. In the present problem, the coherence isinduced by the dipole-dipole interaction Ω which, as wehave already pointed out, plays a role similar to the Rabifrequency of the coherent interaction between the levels,i.e. represents a rate at which one quantum of excitationis exchanged between the atoms. It should be noted thatthere is no electric-dipole moment between the levels | i and | i , so there are no radiative transitions between thelevels of the two-level system. The damping rate γ thatappears in Eq. (21) represents spontaneous decay out ofthe system to the auxiliary level | i .Since the u and v components of the coherence are re-lated to the interaction between the atoms, their prop-erties should be reflected in the entanglement betweenthe atoms. In order to determine the amount of entan-glement between the atoms and the entanglement dy-namics, we use concurrence that is the widely acceptedmeasure of entanglement. The concurrence introducedby Wootters [26] is defined as C = max (cid:16) , p λ − p λ − p λ − p λ (cid:17) , (23)where { λ i } are the the eigenvalues of the matrix R = ρ s ˜ ρ s , (24)with ˜ ρ s given by˜ ρ s = σ y ⊗ σ y ρ ∗ s σ y ⊗ σ y , (25)and σ y is the Pauli matrix. The range of concurrenceis from 0 to 1. For unentangled atoms C = 0 whereas C = 1 for the maximally entangled atoms. In terms ofthe density matrix elements, the concurrence is given by C ( t ) = 2 max n , | ρ ( t ) | − p ρ ( t ) ρ ( t ) o , (26)which shows that the basic dynamical mechanism forentanglement creation in this system is the coherence ρ ( t ). That is, the origin of the entanglement in thesystem can be traced back to the time evolution of thecoherence ρ ( t ).Utilizing the relation ρ ( t ) = ( u ( t ) − iv ( t )) /
2, the timeevolution of the coherence ρ ( t ) can be readily foundfrom the solution of the Bloch equations (21). The gen-eral solution for u ( t ) and v ( t ), valid for arbitrary initialconditions, is given by u ( t ) = ¯ u ( t )e − γt = e − γt α [2Ω A + δ ( v α sin αt + B cos αt )] ,v ( t ) = ¯ v ( t )e − γt = e − γt α [ v α cos αt + B sin αt ] , (27)where w ≡ w (0), u ≡ u (0) and v ≡ v (0) determine theinitial population inversion and coherences in the system A = 2Ω u − δ w , B = δ u + 2Ω w , (28)and α = p + δ is the detuned Rabi frequency.It is also worthwhile to find the time evolution of thepopulation inversion w ( t ) = ¯ w ( t )e − γt = e − γt α {− δ A +2Ω ( v α sin αt + B cos αt ) } , (29)which will allow us to use the Bloch vector picture togain insight into the problem of entanglement creationin the system. For a single-quantum excitation, ρ ( t ) = ρ ( t ) = 0, and then the Bloch vector component w ( t )determines the population distribution among the levels | i and | i , and through the relation u ( t ) + v ( t ) + w ( t ) = e − γt , (30)the component, in turn, determines the entanglement C ( t ) = max n , e − γt p − ¯ w ( t ) o . (31)This implies that the concurrence can be completely de-termined by observing changes in the populations of thesystem’s energy levels. When the population is in thelevel | i or in the level | i , ¯ w ( t ) = ±
1, and then C ( t ) = 0,whereas C ( t ) achieves its optimum value C ( t ) = 1 when¯ w ( t ) = 0. Therefore, we can interpret the entangle-ment as a consequence of a distribution of the populationamong the energy levels. In the complete Bloch vector picture, the Bloch vec-tor makes a constant angle θ = tan − ( − /δ ) with ~ Ω B , it rotates around it, tracing out a circle on theBloch sphere. When the Bloch vector does not inter-sect the ”north pole” ~B = (0 , ,
1) or the ”south pole” ~B = (0 , , − w ( t ) = ± δ , because these features are not en-countered at all under the equal coupling constants andappear never to have been investigated before. This mo-tivates our study of the effect of a spatial location of theatoms on the atom-atom entanglement. IV. ENTANGLEMENT DYNAMICS
To illustrate the influence of the initial conditions andspatial location of the atoms on the time development ofthe entanglement, we will use the solutions (29) and (31)to calculate the concurrence C ( t ) for some distances r and time t . For a fixed r , we examine the time evolutionof C ( t ) for the case where the atoms are first prepared insome pure state that can be specified by the Bloch vectorcomponents u , v and w at time t = 0. In particular,we consider three different sets of initial pure states ofthe system corresponding to single-quantum excitations.In the first, the atom 1 is assumed to reside in its lowerlevel | g i and the atom 2 in its upper level | e i , i.e. theinitial conditions for the Bloch vector components are w = 1 , u = v = 0. In the second, the system is as-sumed prepared in a pure single-quantum superposition(entangled) state in which the two atoms oscillate in theopposite phase | Ψ i = 1 √ | g i| e i + i | e i| g i ) . (32)In this case, v = 1 , u = w = 0. In the third, thesystem is assumed prepared in a pure superposition statewith the atoms oscillating with the same phase | Ψ i = 1 √ | g i| e i + | e i| g i ) . (33)For this state, the initial conditions for the Bloch vectorcomponents are u = 1 , v = w = 0. A. Preparation with w = 1 , u = v = 0 This limit corresponds to an initial condition in whichthe atom 1 is in the lower level and the atom 2 is in theupper level at time t = 0, i.e. in terms of the atomic den-sity matrix elements ρ (0) = 1 and ρ (0) = ρ (0) = ρ (0) = 0. In this limit, the concurrence C ( t ) = 0 at t = 0, and at later times is given by C ( t ) = e − γt s − (cid:20) − α sin (cid:18) αt (cid:19)(cid:21) . (34)It is seen that the qualitative features of the transientdevelopment of the entanglement depend on whether ornot δ = 0. The concurrence sinusoidally varies withfrequency α/ δ . For theparticular case of equal coupling constants when δ =0, this result is reduced to that obtained previously inRefs. [15, 16]. The most obvious effect of having δ , i.e.unequal shifts of the atomic resonances, is that the theamplitude of the oscillating term is always less than 2.As a consequence, the entanglement may appear on thetime scale twice as long as for the case of equal couplingconstants. It is easy to see from Eq. (34) that for δ = 0,the concurrence vanishes periodically at times t = nπ/α, n = 0 , , , . . . , (35)whereas for δ = 0, the concurrence behaves quite dif-ferently such that it vanishes only at times t δ = 2 t = 2 nπ/α, n = 0 , , , . . . , (36)i.e. the entanglement exists on the the time scale twiceas long as for the case of equal coupling constants. α t C (t) FIG. 1: Transient evolution of the concurrence C ( t ) for γ = 0,atom 1 located exactly at an antinode of the standing wave,and various locations δr a of the atom 2 relative to an antin-ode of the standing wave: δr a = 0 (solid line), δr a = 0 . λ (dashed line). The system is initially in the state | i . Theimperfect location δr a = 0 . λ , leads to the detuning δ =2Ω . The features described above are easily seen in Fig. 1,where we display the time evolution of C ( t ) for atom 1located exactly at an antinode of the standing wave andtwo different locations δr a of the atom 2 relative to anantinode of the standing wave. According to Eq. (11),an imperfect or ”nonideal” location of the atom 2 insidethe cavity mode leads to a nonzero detuning δ . We assume a vanishing damping γ = 0. This captures theessential dynamics of the system but makes no account-ing of the dissipative process during the transient evo-lution. We see oscillations in the transient evolution ofthe entanglement that follow the Rabi flopping of popu-lation back and forth between the atoms. In other words,this oscillation reflects nutation of the atomic populationswhich, in turn, can be associated with the precession ofthe Bloch vector ~B about the driving field vector ~ Ω B withfrequency α .The most interesting feature of the transient entangle-ment seen in Fig. 1 is that in the case of unequal cou-pling strengths, the initially unentangled system evolvesinto an entangled state, and remains in this state on thetime scale twice as long as for the case of equal cou-pling strengths. This rather surprising result can be un-derstood in terms of spatial localization of the energyinduced in the field by the initially excited atom. Forequal coupling strengths the energy levels of the atomsare equally shifted by the amount δ = δ due to the in-teraction with the cavity mode. In this case the inducedenergy by the first atom oscillates with frequency α suchthat at the particular times t n = nπ/α ( n = 0 , , , . . . )is fully absorbed by the second atom. Since at thesetimes the energy is well localized in space as being com-pletely absorbed by the localized atoms, the entangle-ment, which results from an unlocalized energy, is zero.The situation changes when g = g . According to (27),in this case the energy levels of the atoms are unequallyshifted by the interaction with the cavity mode. Dueto the frequency mismatch, the energy induced by theatom 1 is not fully absorbed by the atom 2, leading toa partial spatial delocalization of the photon at discretetimes t = nπ/α , where n = 1 , , , . . . . Consequently, atthese times a partial entanglement is observed. The en-tanglement vanishes every time the excitation returns toits initial state, i.e. when it returns to atom 2. B. Preparation with v = 1 , u = w = 0 In this limit the initial population distributes equally be-tween the levels | i and | i , i.e. in terms of the bare statesdensity matrix elements are ρ (0) = ρ (0) = ρ (0) = ρ (0) = 1 /
2. It then follows from Eqs. (26) and (27)that the concurrence C ( t ) = 1 at t = 0, and its timeevolution is C ( t ) = e − γt r − α sin αt. (37)We note immediately that the amplitude of the oscil-lating term that is equal to the population inversion isalways less than unity provided δ = 0. Thus, completetransfer of the population between the atoms cannot beachieved. In terms of the atomic excitation, none of theatoms can be completely inverted. This means that apart of the initial energy is always delocalized. Conse-quently, the system initially prepared in the entangledstate of the form (32) will remain entangled for all times.We emphasize that this feature arises from the presenceof a non-zero detuning δ . That is the reason why thecontinuous in time entanglement is observed. Thus, thissystem may produce continuous in time atom-atom en-tanglement through having different values of the cou-pling constants g and g . α t C (t) FIG. 2: Concurrence C ( t ) as a function of time for γ = 0,atom 1 located exactly at an antinode of the standing wave,and different δr a : δr a = 0 (solid line), δr a = 0 . λ (dashedline). The system is initially in the state v = 1, u = w = 0. This behavior is illustrated in Fig. 2, where C ( t ) is plot-ted for various values of δ and the initial superpositionstate (32). For t = 0 the system is maximally entan-gled due to our choice of the initial state. Immediatelyafterwards, the entanglement begins to decrease becauseof the coherent oscillation of the atomic excitation. Forthe case δ = 0, the system becomes unentangled peri-odically at the times t = nπ/
2, whereas for δ = 0, theperiodic minima (nodes) of C ( t ) are reduced in magni-tude resulting in a nonzero entanglement present for alltimes.The above analysis show that, rather surprisingly, im-perfect coupling of the atoms to the cavity mode mayactually help generate continuous in time atom-atom en-tanglement, through unequal shifting of the atomic reso-nance frequencies. C. Preparation with u = 1 , v = w = 0 If the system is initially prepared in the superpositionstate of the form (33), the concurrence C ( t ) = 1 at t = 0,and the time evolution of the concurrence found fromEqs. (29) and (31) is of the form C ( t ) = e − γt s − δ α sin (cid:18) αt (cid:19) . (38)It follows from Eq. (38) that in the absence of δ , i.e.when the atoms are in equivalent positions inside thecavity mode, the entanglement oscillation is completely suppressed. When δ = 0 the concurrence varies period-ically in time. The amplitude of the oscillating term thatis equal to the population inversion is less than unity, un-less δ = 2Ω and then the entanglement is completelyquenched at the times t = nπ/α . α t C (t) FIG. 3: Transient evolution of the concurrence C ( t ) for γ = 0,atom 1 located exactly at an antinode of the standing wave,and different δr a : δr a = 0 (solid line), δr a = 0 . λ (dashedline). The system is initially in the state u = 1, v = w = 0. Figure 3 shows the influence of the detuning δ on thetime evolution of C ( t ). It is evident that δ significantlymodifies the time evolution of C ( t ). For δ = 0 andvanishing damping, the entanglement is constant in time.Otherwise, when δ = 0, the entanglement oscillatesperiodically and achieves the optimum value, C ( t ) = 1only at the particular times t = 2 nπ/α ( n = 0 , , , . . . ).It can also vanish at the times t = nπ/α . However, thishappens only for a particular separation r for which δ = 2Ω . Thus, for positions of the atoms for which δ = 2Ω , entanglement is seen to occur over all times.This behavior of the entanglement is linked to the factthat with the pure entangled state preparation of theform (33) the Bloch vector and the dipole-dipole fieldvector are initially parallel. For δ = 0, the Bloch vector ~B is effectively locked to the field vector, i.e. ~B × ~ Ω B = 0,and does not precess as it evolves on a time scale givenby the spontaneous emission rate. In analogy to the spinlocking effect, we may call this phenomenon as entangle-ment locking. When δ = 0, the concurrence oscillates intime at frequencies α and α/
2. In the Bloch picture thiscorresponds to the fact that for t > ~B is no longer aligned along the ~ Ω B vector − precession ofthe Bloch vector translates into an oscillatory entangle-ment.Finally, let us examine in greater details the case δ =2Ω . We have seen that under this specific conditionthe concurrence exhibits interesting features. For exam-ple, in the case A, this is the value of δ at which theentanglement achieves the optimum value with imper-fect matching. In the case B, the entanglement is alwaysgreater than 50%, and in the case C, the population isinverted for all times, and consequently the entanglementcan be completely quenched at some discrete times. Thereason is that at this value of δ , the field vector ~ Ω B isin the uw plane, angle θ = − π/ w = 1) of the Bloch sphere. The Bloch vector processesabout a cone whose opening angle θ depends on the ini-tial conditions. In the case A, the the Bloch vector makesa constant angle θ = π/ ~ Ω B , whereas in the case C,it makes a constant angle θ = π/ − ~ Ω B . Thus, inthese two cases, the Bloch vector rotates in a quarter ofthe Bloch sphere such that it can reach one of the poles, w = ±
1. It therefore appears that in the case A, theBloch vector when processing about ~ Ω B it varies from w = 0 to w = 1, i.e. regularly reaches the north pole,but in the case B, it varies from w = 0 to w = −
1, i.e.regularly reaches the south pole. The case C is different,the Bloch vector makes an angle θ = π/ ~ Ω B , sothat it processes about ~ Ω B in such a way that it rotatesfrom w = − / √ w = 1 / √ w = ± V. ENTANGLEMENT DIFFRACTION
One of the more interesting aspects of the transient en-tanglement demonstrated in the previous section is itsdependence on the difference of the Stark shifts of theatomic transition frequencies. This difference arises froman imperfect coupling of the atoms to the cavity modethat is a consequence of nonequivalent positions of theatoms inside the cavity mode. We proceed here to presentmore detailed studies of the sensitivity of the transiententanglement to position of the atoms inside a standing-wave cavity mode. We choose the initial conditions tobe w = 1 , u = v = 0. When Eqs. (6) and (11) areused in Eq. (34), we readily find that the variation of theconcurrence with position of the atoms in the standingwave is given by C ( r ) = e − Γ τ ((cid:18) sin dd (cid:19) τ + (cid:18) sin d d (cid:19) τ sin kr ) | cos kr | , (39)where d = (1 + cos kr ) τ / τ = 2 g ∆ t, (40)and the dimensionless damping rateΓ = ∆2 g γ, (41)both measured in units of 2 g / ∆ which is always assumedto be much smaller than unity.The concurrence (39) exhibits an interesting modulationof the amplitude of the harmonic oscillation. One could naively think that a variation of the concurrence with r should reveal the cosine form of the cavity modefunction. However, Eq. (39) shows that the concurrenceis not a simple cosine function of r . It is given bythe product of two terms, one the absolute value of thecavity mode function | cos kr | and the other the time-and position-dependent diffraction structure. That is,the amplitude of the standing wave cavity mode is inthe form of position and time dependent diffraction pat-tern. Only at very early times ( τ ≪ | cos kr | , but for longer times, C ( r ) may vary slower or faster than the cosine func-tions. Within the diffraction structure itself, the magni-tude of the concurrence exhibits a succession of modesand of antinodes. As a consequence, the entanglementmay be completely quenched even for locations of theatom 2 close to an antinode of the cavity mode, and alter-natively may achieve its optimum value even for locationsof the atom 2 close to a node of the cavity mode. It iseasy to show that C ( r ) vanishes periodically whenevercos (cid:2)(cid:0) kr (cid:1) τ / (cid:3) = 1, i.e. for discrete times τ = 4 nπ kr , n = 1 , , , . . . (42)We may establish a relation between the number of ze-ros in C ( r ) and the time τ by an elementary argument.Since cos kr varies between zero and one, the shortesttime at which C ( r ) achieves at least one zero is τ = 2 π .Thus, there are no zeros in C ( r ) for locations of theatom between the successive nodes of the cavity modefunction, if τ < π . For a given τ the number of zerosis limited by the fact that cos kr ≤
1. This imposeslower and upper limit on n in Eq. (42). The upper limitis important, since it determines the width of the regionabout antinodes of the cavity function where the opti-mum entanglement occurs. It follows from Eq. (42) thatthe criterion for vanishing entanglement is satisfied for n ≤ τ π . (43)The largest n satisfying this inequality corresponds tothe largest value of cos kr , and therefore determinesa node that is the closest to the antinode of the cavitymode function, so that it determines the width of themain peak of the diffraction pattern.More interesting is a possibility of obtaining the optimumentanglement when the atom 2 is located between a nodeand a successive antinode of the cavity mode. The op-timum entanglement occurs at the locations of the mostintense maxima of the concurrence. However, the loca-tions of the maxima are not given by a simple relation.For a given τ , the concurrence (39) achieves the optimumvalue C ( r ) = 1 whenevercos (cid:20) (cid:0) kr (cid:1) τ (cid:21) = − (cid:18) sin kr kr (cid:19) . (44)This is not a simple relation, and we solve this equationgraphically as follows. Introducing the notation p ( r ) = cos (cid:20) (cid:0) kr (cid:1) τ (cid:21) ,q ( r ) = − (cid:18) sin kr kr (cid:19) , (45)we find solutions of the equation p ( r ) = q ( r ) by plot-ting separately p ( r ) and q ( r ). The functions p ( r )and q ( r ) are shown in Fig. 4. The intersection points / λ p ( r ) , q ( r ) FIG. 4: The parameters q ( r ) (solid line) and p ( r ) plottedas a function of r for different times τ : τ = π/ τ = 9 π/ τ = 27 π/ τ corre-sponds to optimum entanglement observed for the idealizedcase of g = g . of the two curves give the solutions of the equation (44).At these points the system attains the optimum entan-glement. We see from the figure that the equation (44)is satisfied only for discrete values of r . The numberof solutions, which gives us the number of the optimumthat the concurrence may achieve, depends on time τ .Rather than examine the situation at all times we willlook only at the particular times τ n = nπ/
2, correspond-ing to the evolution intervals the optimal entanglementis obtained for the idealized case of g = g . For τ = π/ τ = 9 π/ τ .Figure 5 shows C ( t n ) as a function of r /λ for differ-ent times τ . For a short time the entanglement is seento occur over a wide range of positions centered aboutthe antinodes of the cavity mode. The concurrence is abell-shaped function of position without any oscillation.As time progresses, oscillations appear and consequentlythe region of r where the optimum entanglement oc-curs, becomes narrower. The evolution of C ( r ) tendsto become increasingly oscillatory with r as time in-creases, and the optimum entanglement occurs in a stillmore restricted range of r . As a result, the atom-atom C ( r ) r / λ FIG. 5: Concurrence C ( r ) as a function of the separationbetween the atoms when γ = 0, the system is initially inthe lower level, w = 1 , u = v = 0, and (a) τ = π/
2, (b) τ = 9 π/
2, (c) τ = 27 π/ entanglement oscillates with position faster than the co-sine function, and the oscillations are more dramatic forlarger times. r / λ C ( r ) FIG. 6: Concurrence C ( r ) as a function of the separationbetween the atoms when γ = 0, the system is initially in thelower level, w = 1 , u = v = 0, and (a) τ = π , (b) τ = 10 π ,(c) τ = 28 π . In Fig. 6, we plot C ( t n ) as a function of r /λ for threevalues of time τ corresponding to the evolution intervalsat which the entanglement is quenched for the idealizedcase of g = g . Here we observe, that the concurrenceactually evolves with the position of the atom leading tothe appearance of what we may call an inverse diffractionpattern. Note that the concurrence vanishes for locationsof the atom precisely at the antinodes of the cavity mode,and may achieve its optimum value at locations of theatom close to the nodes of the cavity mode.0This effect can be understood as a consequence of theuncertainty relation between the evolution time and en-ergy [23, 27]. For the increasing time the uncertainty ofthe energy decreases which means that the energy be-comes more localized. The increase in the localization ofthe energy results in a degradation of the entanglement.In other words, with increasing time, one can in princi-ple obtain more information about the localization of theatoms inside the cavity mode. VI. SUMMARY
We have investigated the process involved in the entan-glement creation between two distant atoms coupled toa single-mode cavity field. Unlike previous publications,we have included a possible variation of the coupling con-stant g ( ~r ) with the location of the atoms in a standing-wave cavity mode. We have found that the entanglementcreation in a complex two-qubit system can be modeledin terms of the coherent dynamics of a simple single-qubit system driven by a coherent field. Effectively, wehave shown how to engineer two coupled qubits whosethe dynamics are analogous to that of a driven single two-level system. We have obtained analytical expressions forthe concurrence and have shown some new properties ofthe entanglement that are not met in the idealized caseof equal coupling constants appear for unequal or ”im-perfect” coupling constants. In particular, the degree ofentanglement and the time it takes for the concurrenceto reach its optimum value is a sensitive function of theposition of the atoms inside the cavity mode. Charac- terizing the system by the Bloch vector components, wehave examined the parameter ranges in which entangle-ment can take place for all times. We have demonstratedthat a spatial variation of the coupling constant affectslocalization of the energy induced in the field by interact-ing atoms that leads to a long-lived entanglement. Theconsequence of this imperfection is that under certain ini-tial conditions, an initially entangled system may remainentanglement for all times.Finally, we have shown that the variation of the concur-rence with the position of the atoms is that of the cavitymode function multiplied by a time-dependent diffrac-tion pattern. The diffraction formula shows explicitlythe trend of the modification of the entanglement withthe localization of the atoms when the observation timeincreases. For a short time the entanglement is seento occur over a wide range of positions centered aboutthe antinodes of the cavity mode. As time progresses,oscillations appear and consequently the spatial regionwhere the optimum entanglement occurs, becomesnarrower. This effect has been explained in terms ofthe quantum property of complementarity, which ismanifested as a tradeoff between the knowledge ofenergy of the exchanged photon versus the evolutiontime of the system. ACKNOWLEDGMENTS
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