Optics Communications 284 (2011) 2919–2922

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Optics Communications

j ourna l homepage: www.e lsev ie r.com/ locate /optcom

Geometric phase for a coupled two quantum dot system

Amitabh Joshi a,⁎, Shoukry S. Hassan b

a Department of Physics, Eastern Illinois University, Charleston, IL 61920, United Statesb Department of Mathematics, College of Science, University of Bahrain, P O Box 32038, Bahrain

⁎ Corresponding author. Tel.: +1 217 581 5950; fax:E-mail addresses: ajoshi@eiu.edu (A. Joshi), shoukry

(S.S. Hassan).

0030-4018/$ – see front matter © 2011 Elsevier B.V. Aldoi:10.1016/j.optcom.2011.01.066

a b s t r a c t

a r t i c l e i n f o

Article history:Received 4 June 2010Received in revised form 20 December 2010Accepted 21 January 2011Available online 21 February 2011

The adiabatic geometric phase is calculated in a coupled two quantum dot system, which is entangled throughFörster interaction. This phase is then utilized for implementing basic quantum logic gate operation useful inquantum information processing. Such gates based on geometric phase provide fault-tolerant quantumcomputing.

+1 217 581 8548.hassan@hotmail.com

l rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The exponential faster speed of quantum computers over theirclassical counterparts has motivated several research areas inquantum computation and quantum information processing in recentyears [1–3]. Semiconductor quantum dot (QD) system is one of thepractical physical systems for quantum information processing inwhich, the exciton constitutes an alternative for the usual two-levelsystem [4–7]. When a system having more than one quantum dot isconsidered then the coupling and the interaction between quantumdots become important. A prominent interaction called Försterinteraction responsible for the transfer of exciton from one QD toanother QD has been utilized in producing quantum teleportation,optical switching, entangled Bell states, and GHZ states [8,9]. Anotherinteraction due to static exciton–exciton dipole coupling was used togenerate entangled few exciton state via ultra fast laser sequences [4].

During a cyclic evolution of any quantum mechanical systemdescribed by a Hamiltonian, the associated wave function of thesystem acquires a geometrical phase (Berry phase) in addition to thedynamical phase. Such an adiabatic geometric phase is accumulated inan instantaneous eigenstate of an adiabatically evolving Hamiltonianwhich is periodic in the parametric space [10]. The quantumcomputing and quantum information processing tasks requireimplementation of quantum logic gate operations. Such operationsare based on the dynamic evolution of the quantum system or thepure geometric based operations. The geometric based operation isquite a promising approach for the implementation of built-in fault-tolerant quantum logic gates. The geometric phase based quantumlogic gates have intrinsic advantages over the dynamic phase basedcounterparts as they are insensitive to starting state distributions, the

path shape and the passage rate to traverse the closed path and thusrobust against dephasing and significantly higher in fidelity [10-14].The normal procedure of geometric logic gate operation is to drive thequbit to have an appropriate adiabatic cyclic operation. Such schemesare proposed for the trapped ions [12], NMR systems [13] andsemiconductor nano circuits and SQUIDS in microcavites [14].Investigations of Berry phase for coupled quantum dots have been acenter of attention by several researchers and the effect ofenvironment temperature on it is also studied [15]. In another studythe Berry phase in a bipartite system was investigated includingcoupling within subsystems. It was shown that as the couplingconstants tend to infinity all the geometric phases go to zero [16]. Theeffect of Dzyaloshinnski–Moriya interaction was also included in suchstudy and the sudden change in the Berry phase for weak fields wasreported [17]. The SWAP operation in a two-qubit anisotropic XXZmodel in the presence of an inhomogeneous magnetic field wasstudied to establish the range of anisotropic parameter within whichthe SWAP operation was feasible [18]. Implementation of nonadia-batic geometrical quantum gates within semiconductor quantum dotsexploiting excitonic degrees of freedom has been discussed. Also, theeffect of geometric phases induced by either classical or quantumelectric fields acting on single electron spins in quantum dots in thepresence of spin–orbit coupling has been studied [19]. In a feasiblequantum dot model, the geometric phase of the quantum dot systemin nonunitary evolution was calculated and the effect of environmentparameters on the phase value was investigated [20].

In this work we rigorously investigate adiabatic geometric phasesof two coupled quantum dots (QDs) considered as two spin-1/2system including Förster interaction between them. For this systemthe adiabatic Berry's phase can be used to implement conditionalphase shifts and thus realization of quantum logic gate operations.The importance of Förster interaction will be investigated in thiswork. We will show that Förster interaction can be exploited togenerate the adiabatic geometric phase which then can be used to

2920 A. Joshi, S.S. Hassan / Optics Communications 284 (2011) 2919–2922

implement quantum logic gates. The Förster–Dexter resonant energytransfer has been studied in sensitized luminescence of solids inwhich an excited sensitizer atom can transfer its excitation to aneighboring acceptor atom through a virtual photon. Some interestingwork on energy transfer using this mechanism for quantum dotsystem has been reported [21,22].

The paper is organized as follows. In section 2 we present thephysical model with theoretical description, where two coupled QDsinteract through the dipole-dipole interactions. Section 3 is devoted todescribe the adiabatic evolution of this system and calculation of Berry'sphase operation of quantum logic gates. In Section4wegive a summary.

2. The model

We consider two coupled QDs situated at some distance from eachother such that they can interact through dipole–dipole interaction.EachQDhas a ground state |0⟩ and a first excited state |1⟩. The two dotscould be of different sizes and hence their dipole-coupling strengthsmay be slightly different. One can write down the interactionHamiltonian in the computational basis {|00⟩, |01⟩, |10⟩, |11⟩}, wherefirst and second digits are referring to QDI and QDII, respectively. Thesituation can be described by the following phenomenologicalHamiltonian [22]

H tð Þ =ω0 0 0 00 ω0 + ω2 W tð Þ 00 W⁎ tð Þ ω0 + ω1 00 0 0 ω0 + ω1 + ω2 + Fz

0BB@

1CCA; ð1Þ

where ω0 denotes the ground state energy of QDs, the excitonfrequency in the first (second) QD isω1 (ω2), Fz is the strength of staticexciton–exciton dipole interaction energy, which is the diagonalinteraction and is the direct Coulomb binding energy between twoexcitons, one located on each dot [22]. W(t)=W0e

− iα(t) is thestrength of Förster interaction between two QDs whose timevariation/cyclicity is determined by the variable α(t) (which takesvalue from 0 to 2π during adiabatic cyclic evolution). The Försterinteraction is off-diagonal and therefore induces the transfer of anexciton from one QD to another. For simplicity we measure allenergies from ground state and thus set ω0=0 in the subsequentdiscussion.

3. Geometric phase and quantum logic gates

In this case we need only four computational basis states ofHamiltonian (Eq. 1) defined as |00⟩, |10⟩, |01⟩, and |11⟩. We call themas qubit states of couple QD system. Any instantaneous eigenstate ofthe Hamiltonian can be represented as

jξn tð Þ⟩ = xn j00⟩ + yn j01⟩ + zn j10⟩ + wn j11⟩; ð2Þ

where n=1,2,3,4 and basis states are eigenstates of Sz(j) (j=1,2)

operators. The instantaneous normalized eigenstates are listed asfollows [22]:

jξ1⟩ = j00⟩;jξ2⟩ = a2 j01⟩ + a1e

iα tð Þ j10⟩;jξ3⟩ = −a1 j01⟩ + a2e

iα tð Þ j10⟩;jξ4⟩ = j11⟩;

ð3Þ

inwhich a1 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 1 + 1

Q

� �r, a2 =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 1− 1

Q

� �r, andQ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + 4 W0

Δ0

� �2� �s

.

The quantity Δ0=ω1−ω2 is the difference between exciton creationenergy for dot I and that for dot II and the corresponding eigen energiesare λ1=0, λ2 = ω1− Δ0

2 1 + Qð Þ, λ3 = ω1− Δ02 1−Qð Þ, and λ4=ω1+

ω2+Fz.

In the adiabatic limit each eigenstate evolves as

jξn tð Þ⟩ = e−iγn tð Þ jξn 0ð Þ⟩; ð4Þ

where the total phase γn(t) has two components

γn tð Þ = γDn tð Þ + γG

n : ð5Þ

The geometric phase γnG, given by γn

G=∫02πdΦ⟨ξn(Φ)|∂Φξn(Φ)⟩ is

independent of the detailed variation ofΦwith time. For Berry phase,evolution of Φ (which is α(t) here) is to be done adiabatically. Theunitary transformation in a cycle is diag(eiγ1, eiγ2, eiγ3, eiγ4) written inthe basis {|ξ1⟩, |ξ2⟩, |ξ3⟩,|ξ4⟩} and γn are the total phases as given inEq. (5).

The dynamical phase is given by γnD=−λnt and for the states |ξ1⟩

and |ξ4⟩, geometrical phases are γ1G=γ4

G=0. The evolutions of otherstates |ξ2⟩ and |ξ3⟩ contain both dynamical and geometric phase andthe geometric phase is given by the solid angle subtended by the qubittrajectory on the Bloch sphere. We give some outline to estimate thisgeometric phase. Here the qubit states |00⟩ and |11⟩ factor out fromother two states |01⟩ and |10⟩. Hence we have a sub-space expandedby the vectors |10⟩ and |01⟩ for which state vectors given by |ξ2⟩ and|ξ3⟩ can be written in a slightly different form as

jξ2⟩ = sin θ = 2ð Þ j01⟩ + cos θ= 2ð Þeiα tð Þ j10⟩;

jξ3⟩ = −cos θ= 2ð Þ j01⟩ + sin θ = 2ð Þeiα tð Þ j10⟩; ð6Þ

such that cos θ= 2ð Þ = 1ffiffi2

p 1 + Δ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ20 + 4W2

0

p� �1=2

and sin θ= 2ð Þ =1ffiffi2

p 1− Δ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ20 + 4W2

0

p� �1=2

. If the qubit starts in an eigen state |10⟩ or |01⟩

of Hamiltonian H(0), then it remains throughout in one of theinstantaneous eigenstates (Eq. 6). Under adiabatic variation of theHamiltonian, the initial superpositionof state suchas 1ffiffi

2p j10 N + j01 N½ �

will undergo the evolution as [13]

1ffiffiffi2

p j10⟩ + j01⟩→ 1ffiffiffi2

p e−iγ2 jξ2⟩ + e−iγ3 jξ3⟩h i

; ð7Þ

where the total phases γ2 and γ3 (containing both dynamical andgeometrical phases) are given by Eq. (4). The dynamical phase can bemade to cancel out using a spin-echo or refocusing type of technique.Thebasic ideaof this technique is to apply the cyclic evolution twice. Thesecond cyclic evolution (preceded by fast π pulses that swaps the basisstates |10⟩ and |01⟩) is performedby retracing thefirst but in anoppositedirection so the total dynamical phase factors in two cycles cancel outbut the geometric phase factors get added up as the solid angle ispreserved in either direction (see detail in Ref.[13]) .We call this schemeas a ‘two-cycle’ scheme, which is using two cycles with an oppositedirection of evolution of parameterW(t).We do not go to the details forsuch a calculation but stress that bydoing things properly only the Berryphases appear in the final evolution and the state 1ffiffi

2p j10⟩ + j01⟩½ �

evolves to 1ffiffi2

p e2iγG j10⟩ + e−2iγG j01⟩

h iunder the two cycle scheme [13].

The geometric phase associated with the isolated states |00⟩ and |11⟩is zero as discussed above. Hence the unitary transformation in the basis{|00⟩, |10⟩, |01⟩, |11⟩} is then given by

UB = diag 1; e2iγG

; e−2iγG

;1� �

; ð8Þ

where γG is Berry phase given by the solid angle enclosed by the qubit'strajectory on the Bloch's sphere [10,13]

γG = π 1−cos θð Þ½ � = π 1− Δ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2

0 + 4W20

q264

375: ð9Þ

Fig. 1. Schematics of the closed loop for implementing π/8 gate on a Bloch sphere. The

analogous generalized Rabi frequency isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2

0 + 4W20

q. The solid angle for the cone is

Ω(C)=2π(1−cosθ). The other details of implementation are given in the text.

2921A. Joshi, S.S. Hassan / Optics Communications 284 (2011) 2919–2922

Clearly, the expression of geometric phase turns out to bedifferent from the expression of dynamical phase mentionedabove. It is straightforward to show that UB can act as π/8 gate

when W0 =

ffiffiffiffiffiffiffiffiffi31900

rΔ0. The closed loop for implementing such gate is

illustrated on a Bloch sphere (Fig. 1). The ‘two-cycle’ scheme [13] ofgateoperationcanbedescribedbyUB = π T

2C2Tπ T

2CT inwhichT is tipping

of Bloch vector through angle θ by introduction of Förster interaction, C is2π operation of phase α, π is pi-pulse operation, and bar denotes reverseoperation. The dynamical phase in the entire operation for all theeigenstates turns out to be a global phase factor of exp[i(ω1+ω2)t](assuming Fz to be very small) and not physical and hence the gateoperation is given by Eq. (8). It is easy to construct other quantum logicgatesbasedonpuregeometric transformation.Wepropose the two-qubitgate in the equivalence class [23] of the controlled-Z gate, i.e., it onlydiffers from the controlled-Z gate by local operations to two qubits. Therelative phase in unitary transformation UB or the gate operation is givenby Φz=4πcos(θ). The local invariants of this gate are G1=(1/2)(1+cosΦz) and G2=(1+(1/2)cosΦz) and hence those of controlled-Zgate are given by G1=0 and G2=0 demanding cosθ=1/4 and can beeasily achieved with experimental parameters.

The advantage of such quantum logic gates based on puregeometric transformation is due to the robustness of the geometricphase. In quantum evolutions, holonomies and geometrical phases arenaturally occurring, which are robust to local errors [10-13]. The formof dependence of geometric phase on detuning Δ0 and the strength ofFörster interaction bring a natural type of fault tolerance not presentin the non-geometric conditional phase gate, which is quite similar tothe one discussed for the NMR system [13]. Note that any fluctuationof the parameters such as detuning or Förster interaction will resultinto errors that reside only in the dynamical phase since the solidangle of the loop is preserved on average if the fluctuations aresufficiently random. Removal of dynamical phase can give insensitiv-ity to parameter fluctuations. So the robustness (fault-tolerance) ofgeometric-phase based quantum gates is due to the cancelation ofdynamical phase over two consecutive cycles [10-13].

Geometric phase based quantum computing is normally lesssusceptible to the environmental noise. It is decoherence caused bythe environment, that constraints implementation of quantum gates.This is true for the coupled quantum dot system also. In a recent work,Berry phase of two coupled quantum dots was calculated in the frame

work of cavity quantum electrodynamics [15]. The interaction of thephonon field with quantum dots was also taken into account in thiswork. It was shown that the Berry phase changes near a criticaltemperature determined by the resonance condition of the cavity andthus the Berry phase before and after critical temperature maintainssome fixed values. This work suggests that in our case also despite thepresence of decoherence mechanism due to the environment we canmaintain a fixed Berry phase up to a certain temperature range andcan do the quantum information processing. It is possible to getfurther physical insight into the effect of decoherence on geometricphases with the help of Ref. [20], which may be applicable to ourstudy. According to [20] the physical picture emerging out is that forthe large decoherence the geometric phase becomes vanishinglysmall during the evolution of the Bloch vector over the Bloch-spherein a closed path. However, when there is moderate decoherence, theimplementation of geometric-phase quantum gate is feasible [20] andwe can achieve such a moderate decoherence in our system.

In our scheme, the value of Förster interaction W depends on thedot sizes and the confinement potential. Also,W can be modulated byapplying an electric field. The measured dipole values for CdSe dotsrange in 0.9 to 5.2 A and the correspondingW goes as 0.02 to 0.6 meVleading to energy transfer rate of several tens of picoseconds [22]. Thistime is short enough to be useful for any quantum informationprocessing using geometric-phase based quantum gates because thedecoherence times as large as a few nanoseconds have been observedin QDs [22]. Arrays of strongly interacting individual molecules mayalso behave as an excellent system for QIP tasks using Försterinteraction. In such systems the typical Förster interaction strength is8.3 mev and the transfer times are of the order of 500 ps.

Finally, we would like to give a basic outline for realization of theabovework experimentally. The theoretical understanding of geometricphase for a pure quantum state is quite clear and has beenexperimentally demonstrated in NMR, single-photon and two-photoninterferometry [24]. When decoherence is present in the system then itleads to the preparation of mixed states. For the mixed state, geometricphase has been interpreted in terms of quantum interferometry. TheFörster interaction is controlled by the external electric field.When fieldamplitude is large, the Förster interaction remains suppressed. Bydecreasing field amplitude, this interaction can be initiated. The Försterinteraction is a sensitive function of applied electric field as the electricfield caused separation of electrons and holes to increase and thusoverlap integral of electron–hole decreases, consequently reduces theFörster interaction. Initially, one starts with state |1,0⟩ by selectivelyexciting QD-I. When the Förster interaction is turned on, the systemevolves into the eigenstate |ξ2⟩ or |ξ3⟩. This initiates tipping operation oftheBlochvector through anangle θ (definedafter Eq. (6))withα at zero.One can stop the evolution by applying an external electric field whenthe system is in one of these eigenstates (which are entangled states).Adiabatic variation of α(t) from 0 to 2π at this stage results intoacquisition of the adiabatic geometric phase. This can be achieved byvarying the phase of the external field that controls the Försterinteraction parameter [22]. For this purpose one needs to apply anadiabatic sweep which will provide circular motion in the parametricspace ofW. In order to cancel the dynamical phase onehas tomove backin the reverse path also (i.e., the two-cycle scheme to cancel dynamicalphase as described above). The essential ingredients of a quantuminterferometer, which can eventually be implemented in the experi-ment are described as follows. The quantum system of two QDsundergoes a series of unitary evolutions after which the probability offinding system in one of its eigenstates becomes an oscillatory function[25,26] of the control parameter, i.e., the Förster interaction. Suchoscillation inprobability is similar to anoptical interferencepattern [25].The shift of interference pattern is a function of the geometric phasesacquired by the double QDs system during the unitary evolutions, aswell aspurity (whichdependson thedecoherence) of the internal statesof the double QDs system. Thus geometric phase can be directly

2922 A. Joshi, S.S. Hassan / Optics Communications 284 (2011) 2919–2922

measured from the shift of the interference pattern. At the end ofinterferometric operation the tomographywill be required to constructthe components of density matrix. Such geometric phase has beenexperimentally measured in NMR and single photon interferometry[26]. On the lines of these experiments one can implement our proposalof experiment with QDs.

4. Summary

We have considered a Hamiltonian system of two quantum dotscoupled through their dipole–dipole Förster interaction and also via thestatic exciton–exciton dipole interaction energy. It is assumed theHamiltonian has a time-variation of the Förster interaction parameter.In anadiabatic limit of this variation, thegeometric Berryphasehas beencalculated. Based on the idea of two consecutive cyclic evolution, thenetdynamic phase can be canceled while the net geometric phase isdoubled in the final unitary transformation of the system. We canextend this idea for using several interacting quantum dots to constructthree-qubit and multi-qubit entangling gates and hence to generate n-qubit 2-dimensional graph (cluster) state entanglement which will beuseful to prepare a practical quantum information processing system.

Acknowledgements

Many useful discussionswithMin Xiao is gratefully acknowledged.We acknowledge the funding support from Research Corporation.

References

[1] G. Benenti, G. Casati, G. Strini, Prinicples of Quantum Computation andInformation, Vol. I, World Scientific, Singapore, 2004.

[2] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information,Cambridge University Press, Cambridge, England, 2000.

[3] D. Bouwmeester, A.K. Ekert, A. Zeilinger, The Physics of Quantum Information,Springer-Verlag, Berlin, 2000.

[4] E. Biolatti, R.C. Iotti, P. Zanardi, F. Rossi, Phys. Rev. Lett. 85 (2000) 5647.[5] A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. DiVincenzo, D. Loss, M. Sherwin, A.

Small, Phys. Rev. Lett. 83 (1999) 4.[6] Daniel Loss, David P. DiVincenzo, Phys. Rev. A 57 (1998) 120.

[7] T. Pellizzari, S.A. Gardiner, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 75 (1995) 3788.[8] J.H. Reina, L. Quiroga, N.F. Johnson, Phys. Rev. A 62 (2000) 012305;

L. Quiroga, N.F. Johnson, Phys. Rev. Lett. 83 (1999) 2270.[9] J.H. Reina, N.F. Johnson, Phys. Rev. A 63 (2000) 012303;

J. Gea-Banacloche, M. Mumba, M. Xiao, Phys. Rev. B 74 (2006) 165330;A. Joshi, B. Anderson, M. Xiao, Phys. Rev. B 75 (2007) 125304.

[10] M.V. Berry, Proc. R. Soc. Lond. Ser. 292 (1984) 45;A. Shappere, F. Wilczek, Geometric Phases in Physics, World Scientific, Singapore,1989;E. Sjqvist, Physics 1 (2008) 35.

[11] P. Zanardi, M. Rasetti, Phys. Lett. A 264 (1999) 94;S.L. Zhu, Z.D. Wang, Phys. Rev. Lett. 91 (2003) 187902;A. Joshi, M. Xiao, Phys. Lett. A 359 (2006) 390.

[12] L.M. Duan, J.I. Cirac, P. Zoller, Science 292 (2001) 1695.[13] J.A. Jones, V. Vedral, A. Ekert, G. Castagnoli, Nature 403 (2000) 869;

A. Ekert, M. Ericsson, P. Hayden, H. Inamori, J.A. Jones, D.K.L. Oi, V. Vedral, J. Mod.Opt. 47 (2000) 2501;A. Nazir, T.P. Spiller, W.J. Munro, Phys. Rev. A 65 (2002) 042303;S. Siddiqui, J. Gea-Banacloche, Phys. Rev. A 74 (2006) 052337.

[14] P. Solinas, P. Zanardi, N. Zanghi, F. Rossi, Phys. Rev. B 67 (2003) 121307 (R);P. Zhang, Z.D. Wang, J.D. Sun, C.P. Sun, Phys. Rev. A 71 (2005) 042301.

[15] K. Yang, et al., Chin. Phys. Lett. 20 (2003) 991;D.M. Tong, et al., Mod. Phys. Lett. B 18 (2004) 1433;X.Z. Yuan, K.D. Zhu, Phys. Rev. B 74 (2006) 073309.

[16] H.Y. Sun, L.C. Wang, X.X. Yi, Phys. Lett. A 370 (2007) 119.[17] Y. Zhou, G.F. Zhang, Opt. Commun. 281 (2008) 5278.[18] Y. Zhou, G.F. Zhang, F.H. Yang, S. Feng, Phys. Rev. A 75 (2007) 062304.[19] P. Solinas, P. Zanardi, N. Zanghy, F. Rossi, Phys. Rev. A 67 (2003) 052309;

P. San-Jose, B. Scharfenberger, G. Schon, A. Shnirman, G. Zarand, Phys. Rev. B 77(2008) 045305.

[20] S. Yin, D.M. Tong, Phys. Rev. A 79 (2009) 044303.[21] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, V.I. Klimov, Phys. Rev. Lett. 89 (2002)

186802.[22] Our model is based on the Hamiltonian discussed in:. B.W. Lovett, J.H. Reina, A.

Nazir, B. Kothari, G.A.D. Briggs, Phys. Lett. A 315 (2003) 136;B.W. Lovett, J.H. Reina, A. Nazir, G.A.D. Briggs, Phys. Rev. B 68 (2003) 205319;see also:. S. Oh, Z. Huang, U. Peskin, S. Kais, Phys. Rev. A 78 (2008) 0621068wherea slightly different Hamiltonian is considered.

[23] F. Brito, D.P. DiVincenzo, R.H. Koch, and M. Steffen, arXive:0709.1478v2[quant-ph] 19 Mar 2008.

[24] D. Suter, K.T. Mueller, A. Pines, Phys. Rev. Lett. 60 (1988) 1218;P.G. Kwiat, R.Y. Chiao, Phys. Rev. Lett. 66 (1991) 588;J. Brendel, W. Dultz, W. Martienssen, Phys. Rev. A 52 (1995) 2551.

[25] E. Sjöqvist, et al., Phys. Rev. Lett. 85 (2000) 2845.[26] J. Du, et al., Phys. Rev. Lett. 91 (2003) 100403;

M. Ericsson, et al., Phys. Rev. Lett. 94 (2005) 050401;A. Ghosh, A. Kumar, Phys. Lett. A 349 (2006) 27.