Optics Communications 283 (2010) 2166–2173

Contents lists available at ScienceDirect

Optics Communications

journal homepage: www.elsevier .com/ locate/optcom

Controllability of entanglement by mode structure in a cavity quantumelectrodynamics system

Amitabh Joshi *

Department of Physics, Eastern Illinois University, 600, Lincoln Avenue Charleston, IL 61920, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 September 2009Received in revised form 6 January 2010Accepted 11 January 2010

0030-4018/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.optcom.2010.01.020

* Tel.: +1 217 581 5950; fax: + 1 217 581 8548.E-mail address: ajoshi@eiu.edu

It is shown that the entanglement and the purity of corresponding density operator of two initially entan-gled qubits passing through the separate cavities can be controlled by the mode structures of the electricfields sustained in the cavities. Cavity mode structure can be used as a controlling parameter to realizethe quantum gate operation and in preparing graph states.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

In the recent past there has been upsurge in the research activ-ities related to quantum entanglement. The interest in this topic isdue to the fundamental reason that entanglement is one of thedefining features of quantum mechanical world, which is nonexis-tent in the classical world. Entanglement also plays a significantrole in the realization of many quantum information protocols,e.g., quantum computing, quantum teleportation, and quantumcryptography etc. [1,2]. Hence it is interesting to know the behav-ior of entanglement during quantum measurement, quantum toclassical transition, and in any quantum information processingscheme influenced by the environmental parameters [3–5]. In a re-cent study, the reversibility of the decoherence of a mesoscopicsuperposition of radiation states was demonstarted, where a highQ cavity containing the superposed states was coupled to anothercavity. In that work it was shown that the mesoscopic quantumcoherence initially decayed rapidly, then sharply revived leadingto a periodic exchange of the energy between the two cavities[6]. Recently, entanglement sudden death (ESD): a phenomenonin which entanglement can decay to zero abruptly in finite timehas been proposed [7]. The additivity of decay rates good for deco-herence of a single system does not hold good for bipartite coher-ence of even the simplest composite system. This observation cangreatly influence the quantum entanglement, which is an essentialsource of quantum information processing [8,9]. In another inter-esting work, studying the entanglement dynamics of two cavitiesinteracting with independent reservoirs, it was found that whenthe cavity entanglement suddenly disappears, the reservoir entan-glement suddenly and necessarily appears [10]. It was also shownthat the entanglement sudden birth can manifest before, simulta-neously, or even after the ESD.

ll rights reserved.

In this work we propose a cavity quantum electrodynamics(CQED) experiment to demonstrate a controllable ESD and to furtherexplore the possibility for entanglement elongation under differentexperimental conditions. The simplest system of CQED involvesinteraction of a two-level atom with a single quantized electromag-netic field mode inside an ideal cavity. This model has been general-ized and extended in several interesting directions in past manyyears [11]. A very important and significant generalization of thismodel is to include the atomic motion in it. By doing so, the effectsof field structure of the cavity mode sustained in the cavity can be ad-dressed [11–13]. This model is a most elementary one for studyingthe interaction of a single two-level atom with an electromagneticpulse. When the cavity mode structure is taken into account, someinteresting nonlinear transient effects, which are similar to self-in-duced transparency and adiabatic following, can be observed inthe atomic inversion. Here we examine the entanglement of atomsinfluenced by these transient effects, arising from the cavity modestructure for moving two-level atoms which are injected in twoidentical ideal cavities and undergoing a one photon transition. Wealso explore the possibility of realizing quantum logic gate operationand preparation of a graph state using mode structure and movingatoms. This study is motivated by the recent CQED experimentswhich use an atomic beam passing along the axis of a cylindrical cav-ity so that one can study the interactions of an atom with differentcavity field mode structures. As we will see in the following thatfor a few selected mode structures along with cavity frequencydetuning can control, not only the ESD very effectively, but alsothe entanglement elongation. We will neglect the cavity dampingin our discussion of the model which is a reasonable assumption inthe microwave regime where cavities of very high quality factorQ � 4� 1010Þhave been achieved [14]. The typical interaction timesfor the Rydberg atoms (used in microwave cavity experiments hav-ing very large dipole moments) and the cavity field are of the order of10�5s , which is three orders of magnitude shorter than the lifetimeof photons (� 10�2s ) in a typical high-Q microwave cavity. So, theassumption of negligible cavity damping when an atom enters such

A. Joshi / Optics Communications 283 (2010) 2166–2173 2167

cavity is reasonable. Also, the radiative lifetimes of the Rydbergatoms are large enough, specifically when the circular Rydbergstates are employed in the study (about tens of ms) so that we canalso neglect the radiative damping in our model. The stability ofinteraction times in the experiments related to the observation oftrapping state dynamics of the micromaser [14] is uncertain within2–3 percent of the effective method employed in the velocity selec-tion process of the atoms. The transverse velocity spread (with re-spect to the cavity axis) is very small so that most atoms aremoving only along the cavity axis. In that experiment an uncertaintyin interaction time of �2ls over the average interaction time of80 ls was reported and that uncertainty was not mainly due tovelocity spread. Note that the stability of interaction times and min-imum transverse velocity spread are essential in observing trappingstates of electromagnetic fields in the micromaser, hence, our pre-dictions could be tested in similar kinds of experimentalarrangements.

The paper is organized as follows. The model under consider-ation is discussed in Section 2. The analytical solutions of the mod-el under exact resonance condition and numerical solutions innon-resonant conditions are presented in Section 3. The operationsof quantum logic gates and steps to realize a graph state are dis-cussed in Section 3.2. Finally, we summarize our investigation inSection 4 and give some concluding remarks.

2. The model

We consider two initially entangled qubits and study thedynamics of their entanglement. The qubits are composed of twotwo-level atoms coupled to two independent near-resonant idealcavities. The cavities are sustaining non-decaying single modefields in their vacuum states along with the mode structure ofthe electromagnetic field. The interaction between atoms and suchstructured field may result in the change of entanglement andcoherence of the atoms as they fly through their respective cavitiesalong the cavity axis. The situation is easily realizable in any CQEDexperiment in microwave regimes, where large dipole momentRydberg atoms serve as the ideal two-level atomic system [15].The Hamiltonian describing the situation considered in our model(under rotating wave approximation) is given by [12,16]

H ¼ HA þ HC þ HI;

HA ¼ xaraz þxbrb

z ;

HC ¼ x1ay1a1 þx2ay2a2;

HI ¼ g1f1ðzÞðay1ra� þ a1ra

þÞ þ g2f2ðzÞðay2rb� þ a2rb

þÞ:

ð1Þ

In the above Hamiltonian we have not considered interaction be-tween two atoms (described by superscripts ‘a’ and ‘b’) and thetwo cavities (described by subscripts ‘1’ and ‘2’) as such. Herexa ðxbÞ is the atomic transition frequency of atom ‘a’ (atom ‘b’)and x1 ðx2Þ is cavity resonant frequency of cavity ‘1’ (cavity ‘2’),respectively. g1 and g2 are atom field coupling coefficients andr�; rz are Pauli’s spin matrices satisfying the commutation rela-tions: ½rþ; r�� ¼ rz; ½rz; r�� ¼ 2r�. This model without involvingcavity mode structures has also been considered [16] and the timeevolution of initial entanglements as well as ESD effects were dis-cussed. In our case the atoms are moving along the cavity axis (z-direction). The atomic motion can be incorporated as fiðzÞ ¼ fiðvtÞin which v denotes the atomic velocity [12]. Hence we can defineour cavity mode TEmnp as

fiðvtÞ ¼ sinðpivpt=LiÞ; ð2Þ

where pi denotes a whole number representing number of halfwavelengths of electromagnetic field mode inside the cavities oflength Li.

For a more general treatment of the problem one can include akinetic energy term p2

2m

� �of the atom in the Hamiltonian of Eq. (1).

This kind of term accounts for the mechanical effects of light fieldon the atomic motion. In this work we assume that atoms movefast enough so that no reflection effect from the cavity occurs.We also assume that they have a constant velocity along the cavityaxis and do not possess transverse spread of velocity (or in otherwords the interaction energy of atom–field coupling is much largerthan the transverse kinetic energy spread of the atoms, i.e, a strongcoupling regime) then it is sufficient to consider only the timedependence of atom–field coupling [11].

3. Results and discussions

3.1. Controlling entanglement using mode structure

To make our analysis simple we assume identical atomsðxa ¼ xbÞ and nearly identical cavities ðx1 ¼ x2 ¼ x; L1 ¼ L2 ¼ LÞsustaining different spatial cavity mode structures ðp1 – p2Þ, andwe further assume that there is no interaction between the twoatoms and the two cavities. The eigenstates of the Hamiltonian isstraightforward to find. The cavities are prepared initially in thevacuum mode ðwF>0 ¼ j0i1 � j0i2Þ and the two atoms are entan-gled initially. This implies that there is no more than one photonin each cavity at a time and hence the measure of concurrence(to quantify quantum entanglement) is uniform for both atomsand cavity modes [16].

First we consider the following initial entangled atomic state forthe two atoms:

jwAi0 ¼ cosð/Þj1;0i þ sinð/Þj0;1i: ð3Þ

The state j1; 0iðj0; 1iÞ describes atom ‘a’ (‘b’) in the excited state andatom ‘b’(‘a’) in the ground state. The combined initial state for atomand field is given by

jwi0 ¼ jwAi0 � jwFi0 ¼ cosð/Þj1;0ij00i þ sinð/Þj0;1ij00i: ð4Þ

The state (wavefunction) of the system at any other time ’t’ can bewritten as

jwðtÞi ¼ a1ðtÞj1;0ij00i þ a2ðtÞj0;1ij00i þ a3ðtÞj0;0ij01iþ a4ðtÞj0;0ij10i: ð5Þ

Using Schrödinger equation i @wðtÞ@t ¼ HwðtÞ, we can determine the prob-

ability amplitudes ai0s ði ¼ 1; 4Þ appearing in the wavefunction jwðtÞi.These amplitudes satisfy the following coupled differential equations:

ddt

a1ðtÞ ¼ �ig1 sinðp1pvt=LÞa4ðtÞ;

ddt

a2ðtÞ ¼ �ig2 sinðp2pvt=LÞa3ðtÞ;

ddt

a3ðtÞ ¼ iDa3ðtÞ � ig2 sinðp2pvt=LÞa2ðtÞ;

ddt

a4ðtÞ ¼ iDa4ðtÞ � ig1 sinðp1pvt=LÞa1ðtÞ;

ð6Þ

where D ¼ x�xa represents the atom-cavity field detuning.It is easy to obtain the solution of above equations for the

situation in which the cavity eigen frequency is on resonancewith the atomic transition frequency, i.e., D ¼ 0, and g1 ¼ g2 ¼ g;p1 ¼ p2 ¼ p. Under these conditions we can exactly solve Eq. (6).For example, the equation of motion for the amplitude a1 is givenby

@2a1

@2t�

_f ðvtÞf ðvtÞ

@a1

@tþ g2f 2ðvtÞa1 ¼ 0; ð7Þ

in which f ðvtÞ is as defined in Eq. (2). Similarly, this kind of equationcan be written for other amplitudes aiðtÞ ði ¼ 2� 4Þ also. The solu-tion aiðtÞ ði ¼ 1� 4Þ is thus given by

2168 A. Joshi / Optics Communications 283 (2010) 2166–2173

a1ðtÞ ¼ cosð/Þ cosðhðtÞÞ;a2ðtÞ ¼ sinð/Þ cosðhðtÞÞ;a3ðtÞ ¼ �i sinð/Þ sinðhðtÞÞ;a4ðtÞ ¼ �i cosð/Þ sinðhðtÞÞ:

ð8Þ

The function hðtÞ corresponds to the area of the cavity field the atompasses until the time t,

hðtÞ ¼ gZ t

0dsf ðvsÞ: ð9Þ

In the following we consider the general equations of the system gi-ven in Eq. (6) and proceed with numerical calculation of amplitudesaiðtÞ ði ¼ 1� 4Þ. With the help of these amplitudes we can get thereduced density matrix of two atoms by tracing over the photonstates. The reduced density matrix in the basis of atomic states(Eq. (5)) is thus given by

qab ¼

0 0 0 00 ja1ðtÞj2 a1ðtÞa�2ðtÞ 0

0 a�1ðtÞa2ðtÞ ja2ðtÞj2 0

0 0 0 ja3ðtÞj2 þ ja4ðtÞj2

0BBBB@

1CCCCA: ð10Þ

The concurrence CðwÞ, considered as a measure of entangle-ment, is given by [17,18]

CðwÞ ¼ jhwj~wij; ð11Þ

where the spin-flipped state of two-qubits is given asj~wi ¼ ry � ryjw�i (ry is Pauli’s spin matrix), representing standardtime reversal operation and jw�i is complex conjugate of jwi. Forthe reduced density operator qab, the concurrence [17,18] is definedas

Fig. 1. Concurrence CðqabÞ for the initial atomic state: jwAi0 ¼ cosð/Þj1; 0i þ sinð/Þj0; 1i, aand (c) are for a TEmnp modes with ðp1 ¼ 1; p2 ¼ 1Þ, ðp1 ¼ 2; p2 ¼ 2Þ; ðp1 ¼ 3; p2 ¼ 3Þ,/ ¼ p=12, respectively. Plot (d) is for TEmnp modes with / ¼ p=4 and curves A, B, C, andrespectively.

CðqabÞ ¼ Max 0;ffiffiffiffiffik1

p�

ffiffiffiffiffik2

p�

ffiffiffiffiffik3

p�

ffiffiffiffiffik4

pn o; ð12Þ

and ki are the eigenvalues of the matrix ðqabgqabÞ in non increasingorder. For two-qubits state gqab ¼ ðry � ryÞðqabÞ�ðry � ryÞ, whereðqabÞ� is complex conjugate of ðqabÞ in standard basis (5). In the fol-lowing we concentrate on the study of CðqabÞ to quantify the entan-glement. It is easy to show that for the system under considerationthe concurrence CðqabÞ is given by

CðqabÞ ¼ 2ja1jja2j: ð13Þ

For the resonant atom-cavity interaction ðD ¼ 0Þ we getCðqabÞ ¼ j sinð2/Þj cos2ðhðtÞÞ.

We now discuss the results for CðqabÞ obtained by numericalintegration of Eq. (10). We find that there are interesting situationsoccurring for the non-resonant atom-cavity interaction ðD – 0Þ.The situations arise due to the mode structure of the field sus-tained in the cavity and for a particular choice of entanglement be-tween two atoms. The results are depicted in Fig. 1 where we haveplotted CðqabÞ as a function of atom–field detuning for a fixedatomic velocity v ¼ gL=p. The curves A and B are for / ¼ p=4 and/ ¼ p=12, respectively. Fig. 1(a) is for p1 ¼ 1; p2 ¼ 1 andg1 ¼ g2 ¼ g, where we do not observe complete disentanglementof the system as D=g is increased or decreased from zero value.The change in / value causes change in initial entanglement valueor the concurrence. Also, when D=g is large, CðqabÞ maintains itsinitial value. This can be explained by the adiabatic following mod-el for the Hamiltonian (1) that includes field mode structure. WhenD=g 1, then density vector qab precesses about the torque vectorX (generalized Rabi frequency) adiabatically [12] as @qab

@t ¼ X� qab.Another interesting situation occurs when we select p1 ¼ p2 ¼ 2 asthe eigenmodes of the cavities (Fig. 1(b)). We get a perfect entan-

t the cavity exit vs cavity detuning D=g for a fixed transit time tT ¼ p=g. Plots (a), (b),respectively. The curves A and B represent initial entanglement for / ¼ p=4 andD are for ðp1 ¼ 1; p2 ¼ 1Þ; ðp1 ¼ 1; p2 ¼ 2Þ, ðp1 ¼ 2; p2 ¼ 3Þ, and ðp1 ¼ 7; p2 ¼ 8Þ,

A. Joshi / Optics Communications 283 (2010) 2166–2173 2169

glement of two atoms at cavity exit when D=g ¼ 0, as evident fromthe Fig. 1(b). Physically this effect is due to the rephasing of theBloch vector or density vector in one complete cycle of the modestructures ðp1 ¼ p2 ¼ 2Þ; ðg1 ¼ g2 ¼ gÞ or during integral numberof cycles of the mode structures sustained in the cavities such thatp is even and > 2. If we change the value of D=g on either side ofD=g ¼ 0, the concurrence can change from 1 to 0 depending onthe value of D=g. In Fig. 1(c) we set p1 ¼ 3; p2 ¼ 3 andg1 ¼ g2 ¼ g and get perfect entanglement and disentangle-ment at two different values of D=g (positive as well as negative),which is very much different from Fig. 1(a,b). Fig. 1(d) shows theconcurrence when two cavities are sustaining different spatialmode structures, i.e., mixed modes ðp1 – p ¼ 2Þ and / ¼ p=4.Curves A, B, C, and D are for ðp1 ¼ 1; p2 ¼ 1Þ; ðp1 ¼ 1;p2 ¼ 2Þ; ðp1 ¼ 2; p2 ¼ 3Þ, and ðp1 ¼ 7; p2 ¼ 8Þ, respectively. Clearlyfor the larger pi values we get more flat region around D=g ¼ 0meaning initial coherence is maintained effectively. This is quiteremarkable in the sense that atoms with a fixed velocity areentering the two cavities with initial fixed entanglement, but whenthey come out of the cavities their entanglement depends on thedetuning setting for these cavities. Any desired value of entangle-ment can be achieved when these atoms are just at the exit ofcavity simply by varying or controlling the experimentally conve-nient parameter of cavity field frequency detuning. Thus a veryeffective way of controlling entanglement of atoms (i.e., the atomicqubit) can be realized experimentally.

Next, we study how the change of transit time of atoms (i.e.,when atoms having different values of velocities are sent in thecavities) can control their entanglement at the exit of cavities. Thiswe show in Fig. 2 where CðqabÞ is plotted as a function of ‘gt’ keep-

Fig. 2. Concurrence CðqabÞ for the initial atomic state: jwAi0 ¼cosð/Þj1; 0i þ sinð/Þj0; 1i, with / ¼ p=4, under variable transit times. (a) is for afixed cavity mode structure p1 ¼ p2 ¼ 2 and curves A, B, C, and D are forD=g ¼ 0; 1; 1:6; and 2.0, respectively. (b) is for a fixed detuning D=g ¼ 1:6 andcurves A, B, C, and D are for ðp1 ¼ 2; p2 ¼ 2Þ; ðp1 ¼ 2; p2 ¼ 3Þ; ðp1 ¼ 3; p2 ¼ 5Þ, andðp1 ¼ 7; p2 ¼ 8Þ, respectively.

ing p1 ¼ p2 ¼ 2; g1 ¼ g2 ¼ g; / ¼ p=4. Curves A, B, C, and D are forD=g ¼ 0; 1; 1:6; and 2.0, respectively. In curve C, we observe thephenomenon of ESD without decoherence for the non-interactingand non-communicating atoms. But in curve A the opposite of thatphenomenon is observed, i.e., the concurrence shows prolongedand perfect entanglement ðCðqabÞ ¼ 1Þ over certain values of ‘gt’.This means non-interacting and non-communicating atoms notonly exhibit the phenomenon of sudden death of entanglementbut also the sudden longevity of entanglement and this would oc-cur periodically for this choice of parameters. Thus with the cavitymode structure a variety of entanglement control can be achieved.The modification caused by mixed spatial mode structures of thetwo cavities is depicted in Fig. 2(b) when we select / ¼ p=4and D=g ¼ 1:6. Curves A, B, C, and D are for ðp1 ¼ 2; p2 ¼ 2Þ;ðp1 ¼ 2; p2 ¼ 3Þ; ðp1 ¼ 3; p2 ¼ 5Þ, and ðp1 ¼ 7; p2 ¼ 8Þ, respec-tively. For the mixed spatial modes of higher pi values the initialentanglement is retained by the atoms while passing through thecavities (curve D).

Note that we have considered a realistic model where atoms areflying past with a fixed speed through the microwave cavities sus-taining the TEmnp mode structures. Also, in the model under consid-eration atoms are non-interacting with each other and the cavitiesare lossless, so not providing any decoherence mechanism. Theatoms traveling through the loss-less cavities sustaining modestructures are losing their entanglement abruptly and getting elon-gated entanglement depending upon the value of the cavity detun-ing parameter ðDÞ and the mode parameter (p). The loss (gain) ofentanglement of two atoms over a prolonged period can be attrib-uted to the information loss (gain) of atomic dynamical evolutionto (from) cavity modes, which is coming through the trace opera-tion over the cavity modes. Since the number of modes sustainedin the cavities is just one and cavities are lossless, the lost informa-tion from these modes comes back to atoms in finite time (a mem-ory effect of cavity QED system) depending on Rabi frequencies ofoscillations, cavity detunings and cavity mode parameters [16].

Next, we examine another interesting situation by consideringthe initial atomic state of the system as a combination of othertwo Bell states such that the initial combined wavefunction is gi-ven by:

jwi0 ¼ jwAi0 � jwFi0 ¼ cosð/Þj1;1ij00i þ sinð/Þj0;0ij00i: ð14Þ

The new wavefunction of the system at any other time ’t’ is

jwðtÞi ¼ a1ðtÞj1;1ij00i þ a2ðtÞj0;0ij11i þ a3ðtÞj1;0ij01iþ a4ðtÞj0;1ij10i þ a5ðtÞj0;0ij00i: ð15Þ

It is easy to show that the reduced density matrix qab in thiscase has the following form

qab ¼

ja3ðtÞj2 0 0 0

0 ja1ðtÞj2 a1ðtÞa5ðtÞ 0

0 a�1ðtÞa5ðtÞ ja2ðtÞj2 þ ja5ðtÞj2 0

0 0 0 ja4ðtÞj2

0BBBB@

1CCCCA: ð16Þ

The coefficients aiðtÞ ði ¼ 1� 4Þ can be obtained by solving the fol-lowing coupled first order differential equations

ddt

a1ðtÞ ¼ �iDa1ðtÞ � ig2 sinðp2pvt=LÞa3ðtÞ � ig1 sinðp1pvt=LÞa4ðtÞ;

ddt

a2ðtÞ ¼ iDa2ðtÞ � ig1 sinðp1pvt=LÞa3ðtÞ � ig2 sinðp2pvt=LÞa4ðtÞ;

ddt

a3ðtÞ ¼ �ig2 sinðp2pvt=LÞa1ðtÞ � ig1 sinðp1pvt=LÞa2ðtÞ;

ddt

a4ðtÞ ¼ �ig1 sinðp1pvt=LÞa1ðtÞ � ig2 sinðp2pvt=LÞa2ðtÞ:

ð17Þ

2170 A. Joshi / Optics Communications 283 (2010) 2166–2173

The concurrence CðqabÞ for this reduced density matrix is obtainedby

CðqabÞðtÞ ¼ Maxf0;KðtÞg;KðtÞ ¼ 2ja1jja5j � 2ja3jja4j:

ð18Þ

The plot of CðqabÞ as a function of D=g is shown in Fig. 3(a,b,c)under two different values of initial atomic entanglement of Belltype states. In curve A we keep / ¼ p=4 and in curve B it is/ ¼ p=12. The plots (a), (b), and (c) of Fig. 3 are forg1 ¼ g2 ¼ g; p1 ¼ p2 ¼ 1; 2; and 3, respectively. When p1 ¼ p2 ¼ 1and initial entanglement is high ð/ ¼ p=4Þ there is no disentangle-ment of two atoms at the exit of cavities (curve A), but as the initialentanglement of atoms reduces (curve B, / ¼ p=12) there is a cer-tain interval of D=g when entanglement can fall abruptly to zerobefore it recovers-a situation termed as ESD. When we setp1 ¼ p2 ¼ 2 (Fig. 3(b)) we get disentanglement in both curves Aand B in two different intervals of D=g symmetrically located withrespect to D=g ¼ 0. The intervals for the curve B are larger in com-parison to the curve A. Further increase of pi values to p1 ¼ p2 ¼ 3(Fig. 3(c)) causes ESD intervals in curve B to increase. Howevercurve A keeps showing the same two intervals in D=g. This impliesthat ESD is crucially dependent on the initial atomic entanglementof two atoms as well as on the mode structures of the cavities. Thesmaller the initial entanglement, there are longer intervals of D=gfor which state will be disentangled. Also, number of such intervalsincreases with number of modes in the cavities. Fig. 3(d) shows theeffect of two cavities sustaining different spatial modes (mixedmodes) on CðqabÞ as a function of D=g for / ¼ p=12. Here curvesA, B, C, and D are for ðp1 ¼ 1; p2 ¼ 1Þ; ðp1 ¼ 1; p2 ¼ 2Þ; ðp1 ¼

Fig. 3. Concurrence CðqabÞ for the initial atomic state: jwAi0 ¼ cosð/Þj1; 1i þ sinð/Þj0; 0i, aand (c) are for a TEmnp modes with ðp1 ¼ 1; p2 ¼ 1Þ; ðp1 ¼ 2; p2 ¼ 2Þ; ðp1 ¼ 3; p2 ¼ 3Þ/ ¼ p=12, respectively. Plot (d) is for TEmnp modes with / ¼ p=4 and curves A, B, C, andrespectively.

2; p2 ¼ 3Þ, and ðp1 ¼ 7; p2 ¼ 8Þ, respectively. For curve B and Cwe get non-zero value of CðqabÞ at D=g ¼ 0. There are additionalsmall peaks in curve C on either side of D=g ¼ 0 (symmetrically sit-uated). For curve D, where cavity mode numbers arep1 ¼ 7; p2 ¼ 8, there is modulation in CðqabÞ value in the band-width ranging from D=g ¼ �5 to D=g ¼ 5. The ESD appears in therange D=g ¼ 7 to D=g ¼ 8 (and D=g ¼ �7 to D=g ¼ �8). This is incontrast to what we had in curves A, B, C.

Finally, we study how the change of transit time of atoms (i.e.,when atoms having different velocities are sent in the cavities) cancontrol entanglement of Bell type states when they come out fromthe cavities. In Fig. 4 we plot CðqabÞ as a function of ‘gt’ keepingg1 ¼ g2 ¼ g; p1 ¼ p2 ¼ 2; / ¼ p=12. Curves A, B, C, and D are forD=g ¼ 0; 1; 1:6; and 3.0, respectively. For D=g ¼ 0, we get suddendeath of entanglement and resurrection of initial entanglementperiodically as indicated by curve A. When D=g ¼ 1 (curve B), theESD phenomenon becomes more prominent and its interval be-comes larger in comparison to the D=g ¼ 0 case, but the initialentanglement is not completely restored. At D=g ¼ 1:6 (curve C)we get a very large interval of time or period for these Bell statesover which they are showing their disentanglement. This periodbecomes shorter with further increase in D=g value to 3.0 (CurveD). The physical explanation of these results is the same as men-tioned above before Eq. (14). In Fig. 4(b) we have shown the effectof mixed modes when / ¼ p=12 and D=g ¼ 1:0. Curves A, B, C, andD are for ðp1 ¼ 2; p2 ¼ 2Þ; ðp1 ¼ 2; p2 ¼ 3Þ; ðp1 ¼ 3; p2 ¼ 5Þ, andðp1 ¼ 7; p2 ¼ 8Þ, respectively. Clearly curve B shows a dramaticchange over curve A. The ESD in the middle lobe is very much dif-ferent. Interestingly, with further increase in the mode number (pi

value) the ESD phenomenon disappears (curves C and D). This kind

t the cavity exit vs cavity detuning D=g for a fixed transit time tT ¼ p=g. Plots (a), (b),, respectively. The curves A and B represent initial entanglement for / ¼ p=4 and

D are for ðp1 ¼ 1; p2 ¼ 1Þ, ðp1 ¼ 1; p2 ¼ 2Þ; ðp1 ¼ 2; p2 ¼ 3Þ, and ðp1 ¼ 7;p2 ¼ 8Þ,

Fig. 4. Concurrence CðqabÞ for the initial atomic state: jwA>0 ¼ cosð/Þj1; 1 > þsinð/Þj0; 0 >, with / ¼ p=12, under variable transit times. (a) is for a fixed cavitymode structure p1 ¼ p2 ¼ p ¼ 2 and curves A, B, C, and D, are for D=g ¼ 0; 1; 1:6;and 3.0, respectively. (b) is for a fixed detuning D=g ¼ 1 and curves A, B, C, and Dare for ðp1 ¼ 2; p2 ¼ 2Þ; ðp1 ¼ 2; p2 ¼ 3Þ; ðp1 ¼ 3; p2 ¼ 5Þ, and ðp1 ¼ 7; p2 ¼ 8Þ,respectively.

Fig. 5. The quantity Tr½ðqabÞ2�, (a) for the initial atomic state: jwA>0 ¼cosð/Þj1; 0 > þ sinð/Þj0; 1 >, with / ¼ p=4, and (b) for the initial atomic statejwAi0 ¼ cosð/Þj1; 1i þ sinð/Þj0; 0i, with / ¼ p=4, respectively, as a function ocavity detuning D=g. Here curves A, B, C, and D, are for ðp1 ¼ 2; p2 ¼ 2Þðp1 ¼ 2; p2 ¼ 3Þ; ðp1 ¼ 3; p2 ¼ 3Þ, ðp1 ¼ 5; p2 ¼ 6Þ, respectively.

A. Joshi / Optics Communications 283 (2010) 2166–2173 2171

of behavior and control of CðqabÞ is very much different from pre-vious work [16] and here we have shown how effectively one cancontrol CðqabÞ using spatial mode structures of cavities.

In Fig. 5 we show the purity of density operator by plottingTr½ðqabÞ2� as a function of D=g but keeping g1 ¼ g2 ¼ g. Plot (a) isfor initial atomic state given by Eq. (3). Here curves A, B, C,and D, are for ðp1 ¼ 2; p2 ¼ 2Þ; ðp1 ¼ 2; p2 ¼ 3Þ; ðp1 ¼ 3; p2 ¼ 3Þ;ðp1 ¼ 5; p2 ¼ 6Þ, respectively. The system goes to a pure state atseveral values of D=g depending upon the values of pis (curves Aand C). For mixed modes this is not so (curves B and D). The min-imum value of Tr½ðqabÞ2� is not 0.25 meaning system never goes tothe random ensemble. Fig. 5(b) is for the initial state given by Eq.(14). The trends in the plot are quite similar to what we observedin plot (a). That is when we have mixed modes (two cavitiessustaining different spatial modes) we do not get pure state forthe system but for similar modes we do get system in the purestate. This study implies that one can select pure states for thesystem using proper combination of pi and D=g so that such statesare useful for quantum information processing.

3.2. Controlling quantum logic gate operation using mode structure

Next, we will show how we can control different kinds oftwo-qubit logic gate operations using mode structures. We con-sider Eq. (9) together with Eq. (2) and obtain explicit expressionfor hðtÞ as

hðtÞ ¼ gTpp

1� cos pp tT

� �� �; ð19Þ

in which T represents transit time of the atoms T ¼ L=v through thecavity. It is easy to see that when p is an even number we gethðTÞ ¼ 0 and at the exit of the cavity two atoms return to their ini-

:f;

tial states. But for p to be an odd number we get hðTÞ ¼ 2gTpp and thus

hðTÞ is tunable and controllable via the transit time and the modenumber. It is possible to obtain cos½hðTÞ� ¼ �1 provided g ¼ pp2

2T . Ifwe select p ¼ 3; v ¼ 5� 104 cm/s then g � 106 sec�1, which arevery much experimentally realizable values. Alternatively, for a gi-ven value of g one can determine the required value of T. Underthese conditions of parametric adjustment we obtain followingtransformation for the bases of Eq. (5) (initial condition (4) with/ ¼ 0),

j1;0ij00i ! �j1; 0ij00i;j0;1ij00i ! �j0;1ij00i;j0; 0ij01i ! j0;0ij01i;j0; 0ij10i ! j0;0ij10i;j1;1ij10i ! j1;1ij10i;

ð20Þ

completing quantum phase gate (QPG) operation which is con-trolled by cavity mode structures. In other words we obtain con-trolled-Z gate operation using cavity mode structure with movingatoms. Since the radiative life time of the Rydberg atoms is largerthan the transit time through the cavity, the controlled-Z gate oper-ation with such a system is not affected by any kind of decoherencedue to spontaneous emission of atomic levels, though it could belimited by the cavity life time. It is also possible to have a tuneablequantum gate by changing the transit time or the speed of atomsflying through the cavities. In Eq. (20), since we have only one pho-ton available in the process so the state j1; 1i of the system (that isboth the atoms in their excited states is not taking part) gets disen-

2172 A. Joshi / Optics Communications 283 (2010) 2166–2173

tangled or uncoupled with other three states and doesn’t evolve intime.

A CNOT gate can be implemented from a QPG operation througha rotation of second qubit before and after the QPG operation. Bythe application of Hadamard transformation before and after theQPG operation, we can implement following CNOT operation:

j1;0ij00iðbCÞ ! j0;1ij00i;

j0;1ij00iðbCÞ ! j1; 0ij00i;

j0; 0ij01iðbCÞ ! j0;0ij01i;j0; 0ij10iðbCÞ ! j0;0ij10i;

ð21Þ

where, bC represents CNOT operation here.We next show how the cavity mode structure can be used to

construct multiple-qubit entangling gates, which then can be usedin generation of graph state entanglement and one-way quantumcomputing [19–21]. For this purpose we briefly review the litera-ture [19–21] and provide guidelines to achieve them just for thesake of completeness of this work. Usually a graph is referred to adiagram in a plane, where any vertex is represented by a pointand any edge by an arc joining two vertices. In a mathematicalway, a graph is a pair: N ¼ ðV ; EÞ of a finite set V N (N is the setof natural numbers) and a set E ½V �2, the elements of which aresubsets of V with two elements each. The elements of V are calledvertices and the elements of E edges. A graph state can be associatedwith each such graph. Quantum mechanically a graph state is apure quantum state on a Hilbert space HV ¼ ðC2Þ�V , where C2 repre-sents a 2-dimensional vector space with basis vectors fj0i; j1ig.Each vertex labels a two-level quantum system or ubit preparedin the state 1ffiffi

2p ðj0i þ j1iÞ and each edge represents a two-qubit con-

trolled-Z gate having been applied to the wo connected qubits[19,20]. The symbol: ðC2Þ�V ¼ C2 � C2 � . . . : up to V times. So, theHilbert space HV ¼ ðC2Þ�V is a tensor product of ‘V’ two dimensionalvector spaces. The significance and relevance of graph states may besummarized as follows. When two two-level systems, have inter-acted via a certain interaction, the graph connecting the two asso-ciated vertices has an edge. A symmetric n � n matrix called theadjacency matrix for a system consisting of n qubits with values ta-ken from {0,1}, completely specifies any graph state. So the graphcan be understood as a summary of the interaction history of thetwo-level systems and the stabilizer of the states is encoded inthe adjacency matrix [19]. Thus graph states are stabilizer states,and play a key role in quantum information theory and quantumcomputing, such as in one-way computing. The graph state jNican be obtained by operating a sequence of two-qubit controlled-Z gate unitary operator on the empty graph state jsi�n, i.e.,

jN >¼ Pðx;yÞ�EUðx;yÞz jsi�n; ð22Þ

where x, y are vertices of the graph state, E represents edges in thestate jNi. The notation jsi�nð¼ jsi � jsi � . . .n times) means the ten-sor product of vectors jsi up to n times. The state jsi is defined by1ffiffi2p ðj0i þ j1iÞ. Hence the two-qubit empty graph state isjW2ðtÞi ¼ 1

2

Q2j¼1ðj0ij þ j1ijÞ. The two-qubit operation Uðx;yÞz on the

vertices x and y either adds or removes the edge fx; yg. Next, weconsider three atoms simultaneously entering the three cavitieshaving identical mode structure with initial three-qubit graph statejW3ðtÞi ¼ 1

23=2

Q3j¼1ðj0ij þ j1ijÞ. The Hamiltonian HI is to be redefined

for this purpose. For the interaction time as defined above for real-izing controlled-Z gate operation, we thus have a three-qubit entan-gling gate operated on the system and a graph statejNi ¼ Uð1;2Þz Uð2;3Þz Uð1;3Þz jW3ðtÞi is produced [20]. Any graph state jNiis a local unitary equivalent representation of another graph statejN0i if they can be transformed to each other by local unitary (LU)operation. A linear cluster state is required is one-way computing,which is a special case of a graph state, and performs von-Neumann

measurements on the vertices. A universal resource for quantumcomputation can be obtained through cluster state. The measure-ments so performed introduce a probabilistic outcome, however,the overall one-way computer is deterministic [19].

The three-qubit graph state is a LU-equivalent representation ofthe three-qubit linear clusters. Two three-cubit linear clusters canbe fused into a seven-qubit graph state by a three-qubit entanglinggate. This is the LU-equivalent to a seven-qubit cluster. For a n-qu-bit linear cluster state, n�1

2 three-qubit entangling gates are re-quired. If four or five atoms simultaneously interact with cavitymodes in different cavities, one can get four- or five-qubit entan-gling gate. A five-qubit graph state can then be obtained usingfive-qubit entangling gates and from two such five-qubit entan-gling gates an eight-qubit 2-dimensional graph state can be real-ized. The five-qubit entangling gates are required to generate 2-dimensional square lattice cluster states which is useful for one-way quantum computing [20,21].

4. Conclusion

We have described a novel way of controlling the entanglementof atomic qubit in a cavity quantum electrodynamic system in thestrong coupling regime using cavity field mode structure and cav-ity field detuning. These parameters are easily accessible in anyexperiment. The entanglement of a qubit can be varied from its ini-tial value to the complete disentanglement. The system not onlyexhibits sudden death of entanglement but also the longevity ofthe entanglement and thus could be very useful in several proto-cols of quantum information processing. The presence of mixedcavity modes brings a great change in evolution of concurrencewith system parameters. Realization of operations of quantum lo-gic gates and steps to generate graph state are also discussed with-in this model. The controllability of pure state generation alongwith entanglement and basic philosophy to achieve one-way quan-tum computation using spatial cavity mode structures are the no-vel features of this work. For the systems with radiative dampingthe same kind of ”control” could also be achieved simply by spon-taneous emission during an appropriately chosen interval of timein the system.

Acknowledgements

Author is thankful to M. Xiao, J. Gea-Banacloche, S. Singh formany helpful discussions. Encouragements and critical reading ofthe manuscript by S. Daniels is gratefully acknowledged.

References

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

[2] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145.[3] P.J. Dodd, J.J. Halliwell, Phys. Rev. A 69 (2004) 052105.[4] P.J. Dodd, Phys. Rev. A 69 (2004) 052106.[5] L. Diosi, in: Irreversible Quantum Dynamics, F. Benatti, R. Floreanini (Eds.),

Springer, New York, 2003.[6] J.M. Raimond, M. Brune, S. Haroche, Phys. Rev. Lett. 79 (1997) 1964.[7] T. Yu, J.H. Eberly, Phys. Rev. Lett. 93 (2004) 140404.[8] T. Yu, J.H. Eberly, Science 323 (2009) 598.[9] T. Yu, J.H. Eberly, Phys. Rev. Lett. 97 (2006) 140403.

[10] C.E. Lopez, G. Romero, F. Lastra, E. Solano, J.C. Retamal, Phys. Rev. Lett. 101(2008) 080503.

[11] There are several articles in Cavity Quantum electrodynamics, in: P. R. Berman(Ed.), Advances in Atomic and Molecular Physics, Suppl. 2, Academic, NY, 1994.

[12] R.R. Schlicher, Opt. Commun. 70 (1989) 97;A. Joshi, S.V. Lawande, Phys. Rev. A 42 (1990) 1752.

[13] A. Joshi, M. Xiao, J. Opt. Soc. Am. B 21 (2004) 1621.[14] M. Weidinger, B.T.H. Varcoe, R. Heerlein, H. Walther, Phys. Rev. Lett. 82 (1999)

3795.[15] J.M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys. 73 (2001) 565.[16] M. Yonac, T. Yu, J.H. Eberly, J. Phys. B 39 (2006) S621.[17] S. Hill, W.K. Wootters, Phys. Rev. Lett. 78 (1997) 5022.

A. Joshi / Optics Communications 283 (2010) 2166–2173 2173

[18] W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245.[19] M. Hein, J. Eisert, H.J. Briegel, Phys. Rev. A 69 (2004) 062311;

M. Hein, W. Dur, J. Eisert, R. Raussendorf, M. Van den Nest, H. J. Briegel, eprintarXive: quant-ph/0602096.

[20] G.W. Lin, X.B. Zou, M.Y. Ye, X.M. Lin1, G.C. Guo, Phys. Rev. A 77 (2008) 032308;A. Joshi, Private communication.

[21] R. Raussendorf, H.J. Briegel, Phys. Rev. Lett. 86 (2004) 5188.