Method for generating maximally entangled states of multiple three-level atoms in cavity QED

Guang-Sheng Jin, Shu-Shen Li, Song-Lin Feng, and Hou-Zhi ZhengState Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912,

Beijing 100083, People’s Republic of China(Received 25 March 2003; published 8 March 2004)

We propose a scheme to generate maximally entangled states(MESs) of multiple three-level atoms inmicrowave cavity QED based on the resonant atom-cavity interaction. In the scheme, multiple three-levelatoms initially in their ground states are sequently sent through two suitably prepared cavities. After a processof appropriate atom-cavity interaction, a subsequent measurement on the second cavity field projects the atomsonto the MESs. The practical feasibility of this method is also discussed.

DOI: 10.1103/PhysRevA.69.034302 PACS number(s): 03.67.2a, 03.65.2w, 42.50.2p

Entanglement is recognized nowadays as a key ingredientfor fundamental tests of quantum mechanics[1] and as abasic resource of quantum information processing[2]. There-fore, numerous schemes for generating various entangledstates have been proposed during the last decade. In particu-lar, entangled states of two or more particles have been pro-duced experimentally in photonic systems[3], internal de-grees of freedom of atoms interacting with a microwavecavity [4,5], as well as the motion of trapped ions[6]. Al-though most of the research to date has been restricted toentangledtwo-level quantum systems(qubits), entanglementof higher-dimensional quantum systems has several interest-ing properties and has received a great deal of attention re-cently. To investigate these systems, the simplest place tostart is with qutrits(three-dimensional quantum systems). Ithas been shown that entangled states of qutrits lead to quan-tum predictions that differ from classical physics more radi-cally than those of qubits and are more resistant to noise[7].Also, using higher-dimensional systems can decrease the de-tection efficiency required to close the detection loophole forEPR experiments[8]. Moreover, several qutrit-based crypto-graphic protocols have been shown to be more efficient andsecure than their qubit-based counterparts[9].

There are several ways of physically realizing entangledqutrits using photons[10]. Entanglement of massive particlesinstead of fast-escaping photons has also been investigated.Very recently, Zouet al. presented a scheme to generatemaximally entangled states(MESs) of two atomic qutritswith a nonresonant cavity by cavity-assisted collisions[11].In this paper we propose an alternative cavity QED experi-ment with Rydberg atoms in order to create MESs of mul-tiple qutrits. Our method is based on the resonant atom-cavity interaction, which provides a direct mechanism toentangle the atomic and cavity states. Compared with thescheme of Ref.[11], individual addressing of atoms is notrequired and our scheme is not limited to the generation ofMESs of two atomic qutrits. Indeed, with the proper im-provement of the current experimental setups, our methodcan be straightforwardly extended to produceN-partite sNù3d three-level-atom MESs in principle. To prove the feasi-bility of this scheme, we calculate the achievable fidelity ofthe produced state, taking into account not only the dissipa-tion but also the experimental imperfections. Also, we dis-

cuss what should experimentally be done to analyze the en-tangled states.

To explain the main idea, we now consider how to pro-duce a MES of three atomic qutrits under ideal conditions.The schematic setup of our proposal is shown in Fig. 1(a).Three identicalJ-type three-level Rydberg atomsA1, A2,andA3, relevant energy levels being shown in Fig. 1(b), aresequently sent through two high-Q cavities denoted byC1andC2. Cavity C1sC2d sustains a single-mode field in exactresonance withuil↔ ugl sugl↔ ueld transition of the atoms.When the atoms are inC1sC2d, only the levelsuil andugl (uglanduel) are appropriately affected and the stateuelsuild of theatoms will not be affected during the atom-cavity interaction.The temporal evolution of each atom inC1 or C2 is governedby the Jaynes-Cummings model interaction Hamiltonian

HC1

sAkd = i"g1sa1uglAkki u − a1

†uilAkkgud,

and

FIG. 1. (a) Experimental apparatus. The atomsA1, A2, andA3

initially in their ground states are sequently sent through two suit-ably prepared cavitiesC1 and C2. A detection atomA4 is used toproject the atoms onto the MES.(b) Energy levels of the three-levelRydberg atom with the corresponding frequencies.

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HC2

sAkd = i"g2sa2uelAkkgu − a2

†uglAkkeud, sk = 1,2,3d, s1d

respectively. Hereaj† and aj s j =1,2d are the creation and

annihilation operators of the resonant field mode inCj; gjdenotes the coupling strength between each atom and cavityCj. If one of the atoms interacts with the cavity field inC1for a timeT1, the evolution of the atom+C1 system for dif-ferent initial states can be given by

uilAkunlC1

→ cossg1T1ÎnduilAk

unlC1

+ sinsg1T1ÎnduglAk

un − 1lC1,

s2d

uglAkunlC1

→ cossg1T1În + 1duglAk

unlC1

− sinsg1T1În + 1duilAk

un + 1lC1,

uelAkunlC1

→ uelAkunlC1

.

Similarly, the atom+C2 system with an evolution timeT2 is

uilAkunlC2

→ uilAkunlC2

,

uglAkunlC2

→ cossg2T2ÎnduglAk

unlC2

+ sinsg2T2ÎnduelAk

un − 1lC2, s3d

uelAkunlC2

→ cossg2T2În + 1duelAk

unlC2

− sinsg2T2În + 1duglAk

un + 1lC2.

In Eqs.(2) and(3), unlCjrepresents a Fock state with photon

numbernsn=0,1,2, . . .d in cavity Cj.Similar to the discussion in Ref.[12], we assume thatC1

is initially in a coherent superposition of Fock statesu0lC1and u3lC1

,

1Î2

su0lC1+ u3lC1

d. s4d

Some schemes had been proposed to generate this kind ofsuperposition states inside a cavityf13,14g. Very recentlyusing a single, harmonically trapped9Be+ ion, Ben-Kishetal. [15] experimentally demonstrated the generation of sucha superposition state in a harmonic trap by the Law-Eberlymethod[14]. These authors argued that the same techniquescan also be applied to prepare for the cavity field state(4) incavity QED. Then we assume that all the three-level atomsare initially in their ground stateuilAk

, respectively. First,atomA1 is sent into cavityC1 and we let the interaction timet1=p / s2Î3g1d. WhenA1 flies out ofC1, atomA2 is sent intoit with t2=p / s2Î2g1d. Atom A3 is set intoC1 after A2 fliesout with t3=p / s2g1d. The evolution of the atom+C1 systemcan be written as

uilA1uilA2

uilA3

1Î2

su0lC1+ u3lC1

d

→t1 1

Î2suilA1

u0lC1+ uglA1

u2lC1duilA2

uilA3

→t2 1

Î2suilA1

uilA2u0lC1

+ uglA1uglA2

u1lC1duilA3

→t3 1

Î2suiii l + ugggldu0lC1

, s5d

where uiii lsugggld denotes the three-atom stateuilA1

uilA2uilA3

suglA1uglA2

uglA3d. Thus all the atoms are decou-

pled from C1 and are prepared as a Greenberger-Horne-Zeilinger statef16g.

Next, we send these atoms through cavityC2, one afteranother. Similarly, the initial state ofC2 is a Fock-state su-perpositionsu0lC2

+ u3lC2d /Î2. Atoms A1, A2, or A3 interact

with the same single-mode field inC2 with t18=p / s2Î3g2d,t28=p / s2Î2g2d, andt38=p / s2g2d, respectively. This evolutioncan be described by

1Î2

suiii l + ugggld1Î2

su0lC2+ u3lC2

d

→t18

12uiii lsu0lC2

+ u3lC2d + 1

2suglA1u0lC2

+ uelA1u2lC2

duglA2uglA3

→t28

12uiii lsu0lC2

+ u3lC2d + 1

2suglA1uglA2

u0lC2

+ uelA1uelA2

u1lC2duglA3

→t38

12uiii lsu0lC2

+ u3lC2d + 1

2sugggl + ueeeldu0lC2

= 12suiii l + ugggl + ueeeldu0lC2

+ 12uiii lu3lC2

. s6d

Then if C2 is measured and found in the vacuum stateu0lC2,

the three atoms are in the MES

1Î3

suiii l + ugggl + ueeeld, s7d

for which the violation of local realism is stronger than thatfor the MES of two three-level atoms, as discussed in Ref.f17g. To detect the cavity state, we send intoC2 a fourthatom A4, with two relevant levelsuglA4

and uelA4. Atom A4,

initially in its ground stateuglA4, interacts with the resonant

mode inC2 for a time t48=p / s2Î3g2d. If A4 is detected andfound still in stateuglA4

, we can conclude that cavityC2 isin the vacuum state. The success probability of obtainingthe MES in Eq.s7d is 3/4 in ideal cases.

Based mainly on Rydberg atom microwave cavity QEDexperiments performed at ENS[5], we now discuss the prac-tical feasibility of this proposal. Our scheme requires an ex-perimental configuration withtwo cavities, which can beconsidered as a natural development of the present configu-

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rations where onlyone cavity is available. The circularRydberg atoms’ lifetime(30 ms) is much longer than theprotocol duration and is not bound to be a limiting factor.The main cause of decoherence in the present setup is thecavity mode relaxation. The quantum information is stored inthe cavities during the time interval between the passage ofA1 andA3 for C1 and ofA1 andA4 for C2. Each atom mayenter the cavity immediately after the preceding one has leftit. The atom-cavity coupling constant at cavity center isabouts2pd25 kHz. Thus, the total quantum information stor-age time is of order 0.1 ms, which is smaller than the photonstorage time of the microwave cavity 1 ms. Decoherence ef-fects due to the loss of a photon should then be relativelysmall. The interaction time of the atoms with the cavities canbe controlled by using a velocity selector and applying Starkfield adjustment in the cavities. The lengthLCj

of the super-conducting cavities is on the order of centimeters. Under theassumption ofg1LC1

=g2LC2=100 m/s, the velocity of the

injection atoms should be several hundred meters per second,which is in the range of present experiments.

In Fig. 2 we show the results of a numerical simulation toestimate the achievable fidelity of the produced state. Con-sidering the dissipation in the cavities, we describe the wholesystem by a proper master equation. We also consider theimperfection in the preparation of the initial Fock superposi-tion states in the cavities. The fidelity is plotted for variousstrengths of imperfections in the quantum Rabi pulses. For astrength of 3%, i.e., for the achievable precision in the cur-rent setting, and for the cavity lifetime of 1ms, the fidelity ofthe resulting state is only 0.4766. However, we can see that afew improvements on the cavity lifetime as well as the pre-cision of the pulses will result in achieving a fidelity ofnearly 82%. Hence testing Bell’s inequality with this im-proved experimental realization seems to be within reach.

Note that our scheme requires a perfect single atom“gun,” which is able to prepare the atoms via a deterministicand not a Poissonian process. Otherwise, very long acquisi-tion times may be needed. With the development of currentexperimental techniques in cavity QED, we expect it could

be present in a near future. Achievable detection efficiency ofRydberg atomic states is 70%[18], which only reduces thesuccess probability of our scheme and does not decrease thefidelity. In above discussion we do not include imperfections,such as stray thermal and static electric fields, detection er-rors, etc., which could be strongly suppressed by a revisedexperimental setup in construction.

Let us now discuss what should experimentally be done toanalyze the produced MES. As discussed in Ref.[19], mi-crowave pulses and energy detectors behind the cavities willallow the analysis. First, we can check “longitudinal” corre-lations, which are a direct energy detection for the atoms. Inthe ideal case they should be detected in theuiii l, ugggl, orueeel state with the same probability 1/3. However, thesecorrelations, taken along, can be explained classically(statis-tical mixture of uiii l, ugggl, and ueeel states). To prove thatthe three terms in Eq.(7) coherently superpose, we nextstudy “transverse correlations” of the MES. After the desiredstate is produced, we apply ap /2 pulseR1 on the uil↔ ugltransition to atomA1, which may lead to the following evo-lution:

uilA1→R1 1

Î2suilA1

+ uglA1d,

uglA1→R1 1

Î2suilA1

− uglA1d, s8d

uelA1→R1

uelA1.

If, e.g.,A1 is found in the stateuilA1, the state of atomsA2 and

A3 should be projected onto

1Î2

suilA2uilA3

+ uglA2uglA3

d. s9d

To analyze the above state, we applyp /2 pulsesR2 andR3on the uil↔ ugl transition toA2 andA3, respectively. Owingto the rotation invariance of the quantum states9d, we shoulddetect the perfect correlation under ideal conditions, i.e.,whenA2 is detected inuilA2

suglA2d, A3 should also be detected

in the same stateuilA3suglA3

d with certain. The above correla-tion indicates the existence of the states9d and therefore thecoherent superposition ofuiii l andugggl with equal probabil-ity in the produced states7d. Similarly, if we apply ap /2pulse on theuil↔ uelsugl↔ ueld transition toA1 followed byproper pulses onA2 andA3, we may show the coherent su-perposition of uiii l and ueeel sugggl and ueeeld. Thus,we indirectly prove that the produced state is the MESin Eq. s7d.

Finally, we note that this scheme can be easily generalizedfor the preparation of the MESs ofN sN=2,3,4, . . .d three-level atoms. Provided the single-mode field inCj is initiallyin a superposition of the Fock states,

FIG. 2. Dependence of the fidelity on the pulse imperfectionsand the cavity relaxation timeTr.

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uFlCj=

1Î2

su0lCj+ uNlCj

d, s10d

the MES of atoms

s11d

is obtained by sendingN three-level atoms initially prepared

in their ground state, and with the controlled velocities.In summary, we present a scheme to generate MESs of

multiple three-level atoms with high probability. Based onmicrowave cavity QED techniques the present scheme couldbe realized in near future.

This work was supported by the National Natural ScienceFoundation of China and the special funds for Major StateBasic Research Project No. G2001CB309500 of China.

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