PHYSICAL REVIEW A 83, 062302 (2011)

Preparation of n-qubit Greenberger-Horne-Zeilinger entangled states in cavity QED: An approachwith tolerance to nonidentical qubit-cavity coupling constants

Chui-Ping YangDepartment of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China and

State Key Laboratory of Precision Spectroscopy, Department of Physics, East China Normal University, Shanghai 200062, China(Received 4 February 2011; published 3 June 2011)

We propose a way for generating n-qubit Greenberger-Horne-Zeilinger (GHZ) entangled states with a three-level qubit system and (n − 1) four-level qubit systems in a cavity. This proposal does not require identicalqubit-cavity coupling constants and thus is tolerant to qubit-system parameter nonuniformity and nonexactplacement of qubits in a cavity. The proposal does not require adjustment of the qubit-system level spacingsduring the entire operation. Moreover, it is shown that entanglement can be deterministically generated usingthis method and the operation time is independent of the number of qubits. The present proposal is quite general,which can be applied to physical systems such as various types of superconducting devices coupled to a resonatoror atoms trapped in a cavity.

DOI: 10.1103/PhysRevA.83.062302 PACS number(s): 03.67.Lx, 42.50.Dv

I. INTRODUCTION

Quantum entanglement plays a crucial role in quantuminformation processing. Entanglement of 10 photons [1], eightions [2], three spins [3], two atoms in microwave cavityQED [4], or two excitons in a single quantum dot [5] hasbeen demonstrated in experiments. In addition, experimentalpreparation of three-qubit entanglement with superconductingqubits or a superconducting qubit coupled to two microscopictwo-level systems has been reported recently [6–8]. Althoughmultiparticle entanglement was experimentally created inphotons and trapped ions, it is still greatly challenging to createmultiqubit entanglement in other important physical systems.

As is well known, multiqubit GHZ (Greenberger-Horne-Zeilinger) entangled states are of great interest in the foun-dations of quantum mechanics and measurement theory, andsignificant in quantum information processing [9], quantumcommunication [10–12], error correction protocols [13], andhigh-precision spectroscopy [14]. Over the past 10 years, basedon the cavity QED technique, many different theoretical meth-ods for creating multiqubit GHZ entangled states with atomsand superconducting qubits have been presented [15–25]. Forinstance, (i) the proposals in Refs. [15–17] for implementinga GHZ entangled state are based on identical qubit-cavitycoupling constants; (ii) the approaches in Refs. [18–20]are aimed at probabilistic generation of a GHZ state; (iii)the method presented in Ref. [17] is based on the use ofan auxiliary qubit, measurement on the qubit states, andadjustment of the qubit level spacings during the entireoperation; (iv) the proposals in Refs. [21,22] are based onphoton detection outside the cavity; and (v) the proposals inRefs. [23–25] are based on sending qubits (i.e., atoms) througha cavity. These proposals are important because they openednew avenues for creating multiparticle entanglement.

In this paper, we will focus on a situation that the qubit-cavity coupling constants are nonidentical. This situationoften exists in superconducting qubits coupled to a cavityor a resonator. For solid-state devices, the device parameternonuniformity is often a problem, which results in nonidenticalqubit-cavity coupling constants though qubits are placed at

locations of a cavity where the magnetic fields or the electricfields of the cavity mode are the same. In addition, thenonidentical qubit-cavity coupling constants may also resultfrom nonexact placement of qubits in a cavity. In the following,our goal is to present a way for deterministic preparationof a n-qubit GHZ state with tolerance to the qubit-systemparameter nonuniformity and nonexact placement of qubitsin a cavity. As shown below, this proposal also has theseadvantages: (i) The operation time is independent of thenumber of qubits in the cavity; (ii) when compared with theapproach in Ref. [17], no auxiliary qubits, no measurement onthe qubit states and no adjustment of the qubit level spacingsduring the entire operation is needed (note that adjustment ofthe level spacings of the qubits during the operation is notdesired in experiments and may cause extra errors); (iii) noadjustment of the cavity mode frequency is required during theentire operation; and (iv) there is no need of photon detection.This proposal is quite general, which can be applied tovarious types of superconducting qubits and atoms trapped in acavity.

We stress that the present proposal does not require adjust-ment of the qubit level spacings during the entire operation.However, it should be mentioned that prior adjustment ofthe qubit level spacings before the operation may be needed.For instance, by prior adjustment of the qubit level spacingsbefore the operation, the level spacing inhomogeneity in eachfour-level qubit system can be made to be larger than thebandwidth of the applied pulses such that the spectral overlapof the pulses is negligible.

This paper is organized as follows. In Sec. II, we brieflyreview the basic theory of a three-level quantum system orfour-level quantum systems coupled to a single-mode cavityand/or driven by classical pulses. In Sec. III, we show how togenerate an n-qubit GHZ state with one three-level quantumsystem and (n − 1) four-level quantum systems in a cavity. InSec. IV, we compare our proposal with the previous ones. InSec. V, we give a brief discussion of the experimental issuesand possible experimental implementation with superconduct-ing qubits coupled to a resonator. A concluding summary isgiven in Sec. IV.

062302-11050-2947/2011/83(6)/062302(9) ©2011 American Physical Society

CHUI-PING YANG PHYSICAL REVIEW A 83, 062302 (2011)

II. BASIC THEORY

In this section, we will introduce three types of interactionof qubit systems with the cavity mode and/or the pulse. Theresults presented below will be employed for generation of amultiqubit GHZ state discussed in next section.

A. System-pulse resonant interaction

Consider a three-level qubit system (say, qubit system1) driven by a classical pulse. Suppose that the pulse isresonant with the transition |1〉 → |2〉 of the qubit system 1but decoupled from the transition between any two other levels[Fig. 1(a)]. The interaction Hamiltonian in the interactionpicture is given by

HI = h(�reiφ|1〉〈2| + H.c.), (1)

where �r and φ are the Rabi frequency and the initial phaseof the pulse, respectively. Based on the Hamiltonian (1), it isstraightforward to show that a pulse of duration t results in thefollowing rotation

|1〉 → cos �rt |1〉 − ie−iφ sin �rt |2〉,|2〉 → −ieiφ sin �rt |1〉 + cos �rt |2〉. (2)

Note that the resonant interaction can be done within a veryshort time by increasing the pulse Rabi frequency �r (i.e., viaincreasing the pulse intensity).

In the following, we also need the resonant interaction ofthe pulse with the |0〉 ↔ |1〉 transiton of the qubit system 1[Fig. 1(c)]. In this case, we have

|0〉 → cos �r t |0〉 − ie−iφ sin �r t |1〉,|1〉 → −ieiφ sin �rt |0〉 + cos �rt |1〉, (3)

where �r is the Rabi frequency of the pulse.

B. System-cavity resonant interaction

Consider qubit system 1 coupled to a single-mode cavityfield. Suppose that the cavity mode is resonant with the |1〉 ↔|2〉 transition while decoupled (highly detuned) from the |0〉 ↔|2〉 transition and the |0〉 ↔ |1〉 transition of the qubit system1 [Fig. 1(b)]. The interaction Hamiltonian in the interactionpicture, after the rotating-wave approximation, is described by

HI = h(ga†|1〉〈2| + H.c.), (4)

where a+ and a are the creation and annihilation operators ofthe cavity mode and g is the coupling constant between thecavity mode and the |1〉 ↔ |2〉 transition of the qubit system.Under the Hamiltonian (4), the time evolution of the states|2〉|0〉c and |1〉|1〉c of the whole system are as follows:

|2〉|0〉c → cos gt |2〉|0〉c − i sin gt |1〉|1〉c,|1〉|1〉c → −i sin gt |2〉|0〉c + cos gt |1〉|1〉c, (5)

where the states |0〉c and |1〉c are the cavity-mode vacuum stateand single-photon state, respectively.

C. System-cavity-pulse off-resonant Raman coupling

Consider (n − 1) four-level qubit systems (2,3, . . . ,n). Thecavity mode is coupled to the |1〉 ↔ |3〉 transition of each qubit

2 2 2

g

~r

r

1 1 1

0 0 0

(a ) ( b ) ( c )

j

jg

j

j

3

jc,

2

1

0

(d )

FIG. 1. (Color online) (a) and (c) System-pulse resonantinteraction for qubit system 1. In (a), the pulse is resonant with the|1〉 ↔ |2〉 transition; while in (c) the pulse is resonant with the |0〉 ↔|1〉 transition. (b) System-cavity resonant interaction for qubit system1. The pulse is resonant with the |1〉 ↔ |2〉 transition. (d) System-cavity-pulse off-resonant Raman coupling for n − 1 qubit systems(2,3, . . . ,n) in a cavity. For simplicity, we here only draw a figurefor qubit system j interacting with the cavity mode and a classicalpulse (j = 2,3, . . . ,n). δj = �c,j − �j is the detuning of the cavitymode with the pulse, and �j = ω

j

31 − ωj is the detuning betweenthe pulse frequency ωj and the |1〉 ↔ |3〉 transition frequency ω

j

31

of qubit system j. Note that gj , δj , �c,j , and �j may be different forqubit systems (2,3, . . . ,n) due to nonidentical level spacings of thequbit systems which are caused by the nonuniformity of the qubitsystem parameters. The coupling constant gj may also vary withqubits due to nonexact placement of qubits in a cavity. The Rabifrequency of the pulse applied to qubit system j is denoted by �j .

system, but decoupled (highly detuned) from the transitionbetween any other twos levels [Fig. 1(d)]. In addition, aclassical pulse is applied to each one of qubit systems(2,3, . . . ,n), which is coupled to the |2〉 ↔ |3〉 transition butdecoupled from the transition between any other two levels

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[Fig. 1(d)]. In the interaction picture, the Hamiltonian for thewhole system is

H =n∑

j=2

[hgj (e−i�c,j t a+σ−13,j + H.c.)

+h�j (e−i�j tσ−23,j + H.c.)], (6)

where the subscript j represents the j th qubit system, �j

is the Rabi frequency of the pulse applied to the j th qubitsystem, σ−

13,j = |1〉j 〈3|, σ−23,j = |2〉j 〈3|, and gj is the coupling

constant between the cavity mode and the |1〉 ↔ |3〉 transitionof the j th qubit system.

The detuning between the |2〉 ↔ |3〉 transition frequencyω

j

32 of the j th qubit system and the frequency ωj of thepulse applied to the j th qubit system is �j = ω

j

32 − ωj (j =2,3, . . . ,n) [Fig. 1(d)]. In addition, the detuning between the|1〉 ↔ |3〉 transition frequency ω

j

31 of the j th qubit system andthe cavity-mode frequency ωc is �c,j = ω

j

31 − ωc [Fig. 1(d)].Note that the detuning �c,j is not the same for each qubitsystem (i.e., dependent of j ) in the case when the level spacingsare nonidentical for each qubit system. Under the condition�c,j � gj and �j � �j, the level |3〉 can be adiabaticallyeliminated [26] and the effective Hamiltonian is thus givenby [27–29]

Heff = −h

n+1∑j=2

[�2

j

�j

|2〉j 〈2| + g2j

�c,j

a+a|1〉j 〈1|.

+χj (e−iδj t a+σ−12,j + H.c.)

], (7)

where σ−12,j = |1〉j 〈2|, χj = �j gj

2 (1/�j + 1/�c,j ), and

δj = �c,j − �j = ωj

31 − ωj

32 − ωc + ωj . (8)

From Eq. (8), it can be seen that the detuning δj is adjustableby changing the pulse frequency ωj . With suitable choice ofthe pulse frequencies, we can satisfy

δ2 = δ3 = · · · = δn = δ, (9)

which will apply below.For δ � g2

j /�c,j , �2j /�j , χj , there is no energy exchange

between the qubit systems and the cavity mode. Thus, underthe condition (9), the effective Hamiltonian (7) can be writtenas [25,30,31]

Heff = −h

n∑j=2

(�2

j

�j

|2〉j 〈2| + g2j

�c,j

a†a|1〉j 〈1|)

−h

n∑j=2

[χ2

j

δ(a†a|1〉j 〈1| − aa†|2〉j 〈2|)

]

+h

n∑j �=j ′=2

χjχj ′

δ(σ †

12,j σ−12,j ′ + σ−

12,j σ†12,j ′ ), (10)

where the two terms in the second line above describethe photon-number dependent Stark shifts induced by theoff-resonant Raman coupling, and the two terms in the lastparentheses describe the “dipole” coupling between the twoqubit systems (j,j ′) mediated by the cavity mode and the

classical pulses. In the case when the level |2〉 of each qubitsystem is not populated, the Hamiltonian (10) reduces to

Heff = −h

n∑j=2

(g2

j

�c,j

+ χ2j

δ

)a+a|1〉j 〈1|. (11)

It is easy to see that the states |0〉j |0〉c, |1〉j |0〉c, and |0〉j |1〉cremain unchanged under the Hamiltonian (11). However, ifthe cavity mode is initially in the photon state |1〉c, the timeevolution of the state |1〉j of the j th qubit system under theHamiltonian (11) is given by

|1〉j |1〉c → eiλj t |1〉j |1〉c, (12)

where λj = g2j

�c,j+ χ2

j

δ, which can be further written as

λj = g2j

�c,j

+ �2j g

2j

4δ

(1

�j

+ 1

�c,j

)2

. (13)

Note that the parameters gj is not adjustable once the qubitsystems (e.g., solid-state devices) are designed and built in acavity, and the detuning �c,j = ω

j

31 − ωc is fixed when thecavity mode frequency is chosen. As can be seen in Eq. (13),the parameter λj here is adjustable by changing the pulse Rabifrequency �j .

III. GENERATION OF MULTIQUBIT GHZ STATES

Let us consider n qubit systems (1,2, . . . ,n) in a single-mode cavity. The qubit system 1 has three levels shown inFigs. 1(a), 1(b), and 1(c) while the four levels of the qubitsystems (2,3, . . . ,n) are depicted in Fig. 1(d). For qubit system1, the cavity mode is resonant with the |1〉 ↔ |2〉 transitionbut highly detuned from the transition between any two otherlevels [Fig. 1(b)]. In contrast, for qubit systems (2,3, . . . ,n),the cavity mode is off-resonant with the |1〉 ↔ |3〉 transitionbut highly detuned from the transition between any other twolevels [Fig. 1(d)]. These requirements can be achieved byan appropriate choice of qubit systems (e.g., atoms), prioradjustment of the level spacings of the qubit systems (e.g.,superconducting devices), or prior adjustment of the cavitymode frequency before the operation. Note that the cavitymode frequency for both optical cavities and microwavecavities can be changed in various experiments (e.g., see,Refs. [32–36]). And, for superconducting qubit systems, thelevel spacings can be readily adjusted by varying the externalparameters (e.g., the external magnetic flux and gate voltagefor superconducting charge-qubit systems and the current biasor flux bias in the case of superconducting phase-qubit systemsand flux-qubit systems; see, e.g., Refs. [37–39]).

Suppose that the cavity mode and each of qubit systems(1,2, . . . ,n) are initially in |0〉c and (|0〉 + |1〉)/√2, respec-tively. The initial state for each of the qubit systems here canbe easily prepared by the application of classical pulses. Thewhole procedure for preparing qubit systems (1,2, . . . ,n) in aGHZ state is shown as follows:

Step (i). Apply a classical pulse (with a frequency ω = ω21

and φ = −π/2) to qubit system 1 for a duration t1a = π/(2�r )[Fig. 1(a)], wait to have the cavity mode resonantly interactingwith the |1〉 ↔ |2〉 transition of the qubit system 1 for a timeinterval t1b = π/(2g) [Fig. 1(b)], and then apply a pulse (with

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CHUI-PING YANG PHYSICAL REVIEW A 83, 062302 (2011)

a frequency ω = ω10 and φ = −π/2) to qubit system 1 for aduration t1c = π/(2�r ) [Fig. 1(c)]. According to Eqs. (2), (3),and (5), it can be seen that after the operation of this step, thefollowing transformation is obtained:

|0〉1|0〉c|1〉1|0〉c

t1a−→ |0〉1|0〉c|2〉1|0〉c

t1b−→ |0〉1|0〉c−i|1〉1|1〉c

t1c−→ |1〉1|0〉ci|0〉1|1〉c ,

(14)

which leads the initial state∏n

j=1(|0〉j + |1〉j )|0〉c of the wholesystem to the following state:

n∏j=2

(|0〉j + |1〉j )(|1〉1|0〉c + i|0〉1|1〉c). (15)

Here and below, the normalization factor 1/2n/2 is omittedfor simplicity. Equation (15) shows that the levels |1〉 and|2〉 of qubit system 1 are not populated in the case when thecavity mode is in the single-photon state |1〉c. Hence, after theoperation of this step, the qubit system 1 is decoupled fromthe cavity mode during the operation of next step.

Step (ii). Apply a classical pulse (with a duration t2) toeach of qubit systems (2,3, . . . ,n) to induce the off-resonantRaman coupling described in Sec. II C [Fig. 1(d)]. By adjustingthe pulse frequencies, we set δ2 = δ3 = · · · = δn = δ. Sincethe level |2〉 for each of qubit systems (2,3, . . . ,n) is notpopulated during the operation of step (i) above, the effectiveHamiltonian describing this step of operation is given byEq. (11). Accordingly, when the cavity mode is in thesingle-photon state |1〉c, the time evolution of the state |1〉jfor qubit system j is then given by Eq. (12). Set λ2 = λ3 =· · · = λn = λ, which can be readily achieved by adjustingthe Rabi frequencies �2, �3, . . . ,�n of the pulses applied toqubit systems (2,3, . . . ,n). From Eq. (12), one can see that fort2 = π/λ, we have the transformation |1〉j |1〉c → −|1〉j |1〉c(j = 2,3, . . . ,n). Thus, the state (15) becomes

n∏j=2

(|0〉j + |1〉j )|1〉1|0〉c + i

n∏j=2

(|0〉j − |1〉j )|0〉1|1〉c, (16)

which shows that the n qubit systems (1,2, . . . ,n) have beenentangled to each other after the above operations. Since thequbit systems (1,2, . . . ,n) are also entangled with the cavitymode, we will need to perform the following operation todisentangle qubit systems (1,2, . . . ,n) from the cavity mode.

Step (iii). Perform a reverse operation described in step (i);namely apply a classical pulse (with a frequency ω = ω10 andan initial phase φ = π/2) to qubit system 1 for a duration t1c =π/(2�r ) [Fig. 1(c)], wait to have the cavity mode resonantlyinteracting with the |1〉 ↔ |2〉 transition of the qubit system1 for a time interval t1b = π/(2g) [Fig. 1(b)], and then applya pulse (with a frequency ω = ω21 and φ = π/2) to qubitsystem 1 for a duration t1a = π/(2�r ) [Fig. 1(a)]. Accordingto Eqs. (2), (3), and (5), it can be seen that after the operationof this step, we have:

|1〉1|0〉c|0〉1|1〉c

t1c−→ |0〉1|0〉c−|1〉1|1〉c

t1b−→ |0〉1|0〉ci|2〉1|0〉c

t1a−→ |0〉1|0〉ci|1〉1|0〉c ,

(17)

which leads the state (16) to the following state:⎡⎣ n∏j=2

(|0〉j + |1〉j )|0〉1 −n∏

j=2

(|0〉j − |1〉j )|1〉1

⎤⎦ ⊗ |0〉c. (18)

Equation (18) demonstrates that the cavity mode returns toits original vacuum state and the qubit systems (1,2, . . . ,n)have been disentangled from the cavity mode after the aboveoperations.

The left part of the product in the state (18) can be rewrittenas

|0〉1|+〉2 · · · |+〉n − |1〉1|−〉2 · · · |−〉n, (19)

where |+〉j = |0〉j + |1〉j and |−〉j = |0〉j − |1〉j (j =2,3, . . . ,n). Since |+〉j is orthogonal to |−〉j , the state (19)is a GHZ entangled state of n qubits.

To reduce the operation errors, the level-spacing inhomo-geneity in each four-level system needs to be larger than thebandwidth of the applied pulse, such that the overlappingof pulse spectra or the transition between any two irrelevantlevels is negligible. This requirement can be achieved by prioradjustment of the qubit level spacings before the operation.For superconducting qubit systems, the level spacings can bereadily adjusted by varying the external parameters [37,39].To simplify our presentation, we will not give a detaileddiscussion here. Note that how to have the irrelevant levels notaffected by the pulses or the cavity mode via prior adjustmentof the level spacings was previously discussed (e.g., seeRefs. [27,41]).

Several additional points need to be made, which are asfollows:

(i) Since the cavity mode is off-resonant with the |1〉 ↔ |3〉transition of qubit system j (j = 2,3, . . . ,n) during steps (i)and (iii), a phase shift exp(iϕj ) happens to the state |1〉 of qubitsystem j when the cavity mode is in the single photon state |1〉cfor either of steps (i) and (iii). Here, ϕj = g2

j (t1b + t1c)/�c,j .

It is easy to find that if this unwanted phase shift for qubitsystem j is considered, the fidelity of the prepared GHZ stateis given by

F 1

4

⎡⎣1 +n∏

j=2

(1 + e−i2ϕj )

2

⎤⎦ ⎡⎣1 +n∏

j=2

(1 + ei2ϕj )

2

⎤⎦ ,

(20)

which shows that for ϕj ∼ 0, i.e., when the condition

g2j (t1b + t1c)/�c,j � 1 (21)

is met, we have F ∼ 1. The condition (21) can be reached byincreasing g and �r to shorten the operation time t1b + t1c orincreasing the ratio �c,j /g

2j .

(ii) For steps (i) and (iii), the pulse process depicted inFigs. 1(a) and 1(c) must be much faster than the processof the cavity mode resonantly interacting with the |1〉 ↔ |2〉transition of the qubit system 1 [Fig. 1(b)], i.e.,

t1a,t1c � t1b, (22)

such that the internal transition between the two levels |1〉 and|2〉 of qubit system 1 during the pulses, induced by the resonantinteraction of the cavity mode with the |1〉 ↔ |2〉 transition of

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PREPARATION OF n-QUBIT GREENBERGER-HORNE- . . . PHYSICAL REVIEW A 83, 062302 (2011)

qubit system 1, is negligible. Note that the condition (22) canbe readily achieved by increasing the pulse Rabi frequencies�r and �r such that �r, �r � g.

(iii) The level |2〉 of qubit system 1 is populated for a timeinterval t1a + t1b during step (i) or step (iii), thus the condition

t1a + t1b � γ −12r ,γ −1

2p (23)

needs to be satisfied in order to reduce decoherence causeddue to spontaneous emission and dephasing of the level |2〉 ofqubit system 1. Here, γ −1

2r and γ −12p are the energy relaxation

time and dephasing time of the level |2〉 of qubit system 1,

respectively.From the above description, one can see that this proposal

has the following advantages:(i) No identical qubit-cavity coupling constants for the (n −

1) qubit systems (2,3, . . . ,n) are required, thus, this proposalis tolerant to the qubit-system parameter nonuniformity andnonexact placement of qubits in a cavity;

(ii) No adjustment of the level spacings of the qubit systemsor adjustment of the cavity mode frequency during the entireoperation is needed;

(iii) Neither auxiliary qubit systems nor measurement onthe qubit states is needed;

(iv) No photon detection is needed;(v) The entanglement preparation is deterministic;(vi) The entire operation time is given by

τ = π/g + π/�r + π/�r + π/λ, (24)

which is independent of the number of qubits in the cavity.In addition, for the description given above, it can be seen

that:(vi) During the entire operation, the levels |2〉 and |3〉 of

qubit systems (2,3, . . . ,n) are unpopulated and thus decoher-ence due to spontaneous emission and dephasing from theselevels are greatly reduced;

(vii) The level |2〉 of qubit system 1 is only populated for avery short time t1a + t1b during step (i) or step (iii); and

(viii) The level |2〉 of qubit system 1 is not occupied duringstep (ii) [note that the operation of this step (ii) requires muchlonger time t2 than both step (i) and step (iii)].

Above we have discussed how to prepare a multiqubit GHZstate with a three-level qubit system and (n − 1) four-levelqubit systems in a cavity. The discussion given above is basedon qubit systems for which the level spacings become narroweras the levels go up [see Figs. 1(a), 1(b), and 1(c) for qubitsystem 1 and Fig. 1(d) for qubit systems (2,3, . . . ,j )]. Notethat this limitation is unnecessary. Namely, this proposal isalso applicable to (i) the qubit system 1 with such three levels,for which the level spacing between the two lowest levels |0〉and |1〉 is smaller than that between the two upper levels |1〉and |2〉, and (ii) the (n − 1) qubit systems (2,3, . . . ,n) withthese four levels, for which the level spacing between the twolevels |i〉 and |i + 1〉 is larger or smaller than that between thetwo levels |i + 1〉 and |i + 2〉 (here, i = 0,1). Furthermore,since the level |0〉 of each of the qubit systems (2,3, . . . ,n)was not involved during the entire operation, one can choosethe qubit systems (2,3, . . . ,n) for which the transition betweenthe two lowest levels |0〉 and |1〉 is forbidden or weak to avoid

or reduce decoherence caused by the spontaneous emissionfrom the level |1〉.

The three-level or four-level qubit systems here are widelyavailable in natural atoms and also in artificial atoms suchas superconducting charge-qubit systems [37], phase-qubitsystems [38,39], and flux-qubit systems [37,40]. For a detaileddiscussion, see Ref. [27].

IV. COMPARING WITH PREVIOUS PROPOSALS

In this section, we will give a comparison between ourproposal and previous ones. For simplicity, we will compareour proposal with the ones in Refs. [17,25]. To the best ofour knowledge, the proposals in Refs. [17,25] are most closelyrelated to our work.

A. Comparison with the proposal in Ref. [25]

An n-qubit GHZ state can be prepared without realexcitation of the cavity mode, by using the method introducedin Ref. [25]. To see this, consider n three-level qubit systems(1,2, . . . ,n) in a cavity. Assume that the cavity mode is coupledto the |0〉 ↔ |2〉 transition and the pulses are coupled tothe |1〉 ↔ |2〉 transition, to establish the system-cavity-pulseoff-resonant Raman coupling [Fig. 2(a)]. In this case, we canobtain an effective Hamiltonian, i.e., the Hamiltonian (10) witha replacement of |1〉,|2〉, and

∑nj=2 by |0〉, |1〉, and

∑nj=1,

respectively. When the cavity mode is initially in a vacuum

j

jg

jj

2 2

g '

jc,

1 1

0 0

( a ) ( b )

FIG. 2. (Color online) (a) System-cavity-pulse off-resonantRaman coupling for n qubit systems (1,2, . . . ,n) in a cavity. Forsimplicity, we here only draw a figure for qubit system j interactingwith the cavity mode and a classical pulse (j = 1,2, . . . ,n). δj =�c,j − �j is the detuning of the cavity mode with the pulse,�c,j = ω

j

20 − ωc is the detuning between the cavity mode frequencyωc and the |0〉 ↔ |2〉 transition frequency ω

j

20 of qubit system j,

and �j = ωj

21 − ωj is the detuning between the pulse frequencyωj and the |1〉 ↔ |2〉 transition frequency ω

j

21 of qubit system j.

The blue arrow dashed line represents a second pulse applied to thequbit system j to cancel the Stark shift of the level |1〉 induced bythe first pulse. (b) System-cavity off-resonant interaction for eachof n three-level qubit systems (1,2, . . . ,n) in Ref. [17]. g′ is thenonresonant coupling strength between the cavity mode and the|0〉 ↔ |2〉 transition. δ = ω20 − ωc is the detuning of the cavity modefrequency with the |0〉 ↔ |2〉 transition frequency.

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CHUI-PING YANG PHYSICAL REVIEW A 83, 062302 (2011)

state |0〉c, this effective Hamiltonian reduces to

Heff = − h

n∑j=1

�2j

�j

|1〉j 〈1| + h

n∑j=1

χ2j

δ|1〉j 〈1|

+ h

n∑j �=j ′=1

χjχj ′

δ(σ+

01,j σ−01,j ′ + σ−

01,j σ+01,j ′ ), (25)

where �j = ωj

21 − ωj , �c,j = ωj

20 − ωc, and δ = δj =�c,j − �j (which can be reached via adjustment of the pulsefrequencies). The notation of χj is the same as that given inSec. III. To prepare the qubit systems (1,2, . . . ,n) in a GHZstate, one will need to apply a second pulse to each of thequbit systems (1,2, . . . ,n) to cancel the Stark shifts, i.e., thefirst term of Eq. (25). After that, the Hamiltonian (25) becomes

Heff = h

n∑j=1

χ2j

δ|1〉j 〈1|

+ h

n∑j �=j ′=1

χjχj ′

δ(σ+

01,j σ−01,j ′ + σ−

01,j σ+01,j ′ ). (26)

For the case of χ1 = χ2 = · · · = χn = χ (achievable bychanging the pulse Rabi frequencies), this Hamiltonian isthe same as the Hamiltonian (6) presented in Ref. [25] forn two-level atoms interacting dispersively with a single-modecavity. According to the discussion in Ref. [25], an n -qubitGHZ state can be prepared based on the Hamiltonian (26).

From the description here, one can see that to achieveidentical Raman transition strengths χ1, χ2, . . . , and χn, thepulses applied to different qubit systems should have differentRabi frequencies if the qubit-cavity coupling constants arenonidentical. And, to obtain identical detuning δ for each qubitsystem, adjustment of the pulse frequencies would be needed.Furthermore, to generate GHZ states, a second off-resonantpulse should be applied to each qubit system to cancel the Starkshift induced by the first pulse. Namely, for generation of ann-qubit GHZ state, a total of 2n pulses would be required. Incontrast, as shown above, our present proposal, which employsfour-level qubit systems, needs only (n − 1) pulses appliedto the qubit systems (2,3, . . . ,n), i.e., one pulse for each ofqubit systems (2,3, . . . ,n). Hence, the use of the pulses issignificantly reduced in the present proposal.

B. Comparison with the proposal in Ref. [17]

In Ref. [17], an auxiliary qubit system a was used to createa single photon in the cavity mode, which was then employedto prepare the n qubit systems (1,2, . . . ,n) in a GHZ state.As discussed there, measurement on the states of the auxiliaryqubit system a and adjustment of the level spacings of eachqubit system were both needed during the entire operation.In addition, qubit systems (1,2, . . . ,n) were required to haveidentical three-level structures and the qubit-cavity couplingconstants for qubit systems (1,2, . . . ,n) were needed to be thesame.

The cavity mode in Ref. [17] was set to be off-resonantresonant with the |0〉 ↔ |2〉 transition of each of qubit systems(1,2, . . . ,n), with a detuning δ = ω20 − ωc [Fig. 2(b)]. Thecoupling constant between the cavity mode and the |0〉 ↔ |2〉

transition for each of qubit systems (1,2, . . . ,n) was denotedas g′. As shown in Ref. [17], a multiqubit GHZ state wasgenerated by simultaneously performing a common phase shifteiπ on the state |0〉 of each of qubit systems (1,2, . . . ,n) withassistance of the cavity photon, by having eiλt = eiπ (i.e.,λt = π ). Here, λ = (g′)2/δ, which was originally defined inRef. [17]. For a detailed discussion, see Ref. [17].

It is noted that the method in Ref. [17] cannot workin the case when the qubit-cavity coupling constants arenonidentical. The reason is that a common phase shift eiπ

cannot be simultaneously performed on the state |0〉 of each ofqubit systems (1,2, . . . ,n). For instance, when the qubit-cavitycoupling constants g′

j and g′k for qubits j and k are not the

same [resulting in different λj = (g′j )2/δ and λk = (g′

k)2/δ],one cannot have both of λj t and λkt to be equal to π for agiven time t. The discussion here applies to identical atomsin a cavity, for which the detuning δ of each atom with thecavity mode is the same but the qubit-cavity couplings arenonidentical due to nonexact placement of atoms in the cavity.

Let us now see if the method in Ref. [17] can workfor a situation that the detuning of each qubit system withthe cavity mode and the qubit-cavity coupling constants areboth nonidentical. This situation applies to superconductingqubit systems in a cavity. As discussed above, to preparea multiqubit GHZ state, the condition λj = λk needs to besatisfied for arbitrary two superconducting qubit systems j

and k. Here, λj = (g′j )2/δj and λk = (g′

k)2/δk. Note that oncethe superconducting qubit systems are designed and the cavitymode frequency is chosen, the detunings δj and δk are fixed.Thus, in order to obtain λj = λk , i.e., (g′

j )2/δj = (g′k)2/δk,

it would be required to exactly place the qubit systems inthe cavity to have desired qubit-cavity coupling constants g′

j

and g′k, which is not easy to achieve in experiments due to

the fabrication error. This problem becomes more apparentespecially when the number of qubit systems in a cavity islarge. In contrast, as shown above, this difficulty is avoided inour present proposal, because the effective coupling strengthλj in Eq. (13) can be adjusted by changing the intensity of thepulses, instead of exactly placing qubit systems in a cavity tohave desired qubit-cavity coupling constants.

From the discussion here, it can be concluded that ourpresent proposal differs markedly from the previous ones[17,25]. Due to the use of a four-level structure, the presentproposal can be used to prepare a multiqubit GHZ state fornonidentical qubit-cavity coupling constants (compared withthe proposal in Ref. [17]) and requires less application of pulses(compared with the proposal in Ref. [25]).

V. DISCUSSION

In this section, we will give a brief discussion on theexperimental issues. For the method to work:

(i) The occupation probability pj of the level |3〉 for qubitsystem j (j = 2,3, . . . ,n) during step (ii) is given by [28]

pj 2

4 + (�j/�j )2+ 2

4 + (�c,j /gj )2, (27)

which needs to be negligibly small in order to reduce theoperation error.

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PREPARATION OF n-QUBIT GREENBERGER-HORNE- . . . PHYSICAL REVIEW A 83, 062302 (2011)

(ii) As discussed above, the conditions (21), (22), and (23)need to be satisfied;

(iii) The total operation time τ given in Eq. (24) shouldbe much shorter than the energy relaxation time γ −1

1r anddephasing time γ −1

1p of the level |1〉 and the lifetime of thecavity mode κ−1 = Q/2πνc, where Q is the (loaded) qualityfactor of the cavity.

The above requirements can in principle be realized,since one can (i) reduce pj by increasing the ratio of�j/�j and �c,j /gj ; (ii) shorten t1a,t1b,t1c by increasing thepulse Rabi frequency �r, �r , and the coupling constant g;(iii) design qubit systems (e.g., superconducting devices) tohave sufficiently long energy relaxation times γ −1

1r and γ −22r

and dephasing times γ −11p and γ −1

2p or choose qubit systems(e.g., atoms) with long decoherence times of the levels |1〉 and|2〉; (iii) increase κ−1 by employing a high-Q cavity so thatthe cavity dissipation is negligible during the operation.

For the sake of definitiveness, let us consider the experimen-tal possibility of generation of a six-qubit GHZ state with sixsuperconducting qubits coupled to a resonator (Fig. 3). Withthe choice of �r, �r ∼ 10g, we have t1a,t1c ∼ 0.05π/g andt1b ∼ 0.5π/g. By setting �j ∼ 10�j,�c,j ∼ 10gj , and �j ∼0.9gj , we have δ = δj ∼ 10g2

j /�c,j ∼ 10�2j /�j ∼ 10χj ,

leading to λj ∼ 0.11gj . For gj ∼ 0.2g, we can set λ2 = λ3 =· · · = λ6 = λ ∼ 0.022g, which can be achieved by adjustingthe Rabi frequencies �2, �3, . . . ,�6 of the pulses applied tothe qubit systems (2,3, . . . ,6) as discussed above. Thus, wehave τ ∼ 46.6π/g. As a rough estimate, assume g/2π ∼ 220MHz, which could be reached for a superconducting qubitsystem coupled to a one-dimensional standing-wave CPW(coplanar waveguide) transmission line resonator [42]. Forthe g chosen here, we have (i) t1a,t1c ∼ 0.1 ns and t1b ∼ 1.1ns, which shows that the condition (22) is satisfied; (ii)t1a + t1b ∼ 1.2 ns, showing that the condition (23) is metfor min{γ −1

2r ,γ −12p } ∼ 0.5 µs [38,43]; and (iii) τ ∼ 0.1 µs,

which is much shorter than min{γ −11r ,γ −1

1p } ∼ 1 µs [38,43].In addition, consider a resonator with frequency νc ∼ 3 GHz(e.g., Ref. [44]) and Q ∼ 5 × 104. We have κ−1 ∼ 2.5 µs,which is much longer than the total operation time τ . Note thatsuperconducting coplanar waveguide resonators with a qualityfactor Q > 106 have been experimentally demonstrated [45].

For the choice of �j ∼ 10�j and �c,j ∼ 10gj here, wehave pj ∼ 0.04, which can be further reduced by increasingthe ratio of �j/�j , and �c,j /gj . In addition, for theparameters chosen here, when the unwanted phase shiftsmentioned above are considered, a simple calculation showsthat the fidelity F is ∼0.992. We should mention that furtherinvestigation is needed for each particular experimental setup.However, this requires a rather lengthy and complex analysis,which is beyond the scope of this theoretical work.

As shown in Sec. III, this proposal requires simultaneousapplication of pulses with different frequencies and intensitiesduring the operation of step (ii). For the setup in Fig. 3(a),simultaneous application of several pulses of different frequen-cies and intensities to qubit systems 2,3,4, and 5 is feasible inexperiments [46]. According to Ref. [46], for a qubit systemcontaining a superconducting loop, one can place an ac currentloop on the qubit superconducting loop to create an ac flux�e (i.e., a classical magnetic pulse) threading the qubit loop

0.2L

JEJE

JEVg

( b )

x

yz

c

JE

c e e

EE JJ

c e

( c ) ( d )

( a )

FIG. 3. (Color online) (a) Sketch of the setup for six super-conducting qubit systems and a (gray) standing-wave quasi-one-dimensional coplanar waveguide resonator. The two blue curvedlines represent the standing wave magnetic field, which is in thez direction. The green dot represents the qubit system 1, whichis placed at an antinode of the standing wave magnetic field toachieve the maximal qubit-cavity coupling constant g. The reddots represent the qubit systems 2,3, . . . , and n, which are placedat locations where the magnetic fields are the same. Each qubitsystem could be a superconducting charge-qubit system shown in(b), flux-biased phase-qubit system in (c), and flux-qubit system in(d). The superconducting loop of each qubit system, which is a largesquare for (b) and (d) while a large circle for (c), is located in theplane of the resonator between the two lateral ground planes (i.e., thex-y plane). Each qubit system is coupled to the cavity mode througha magnetic flux �c threading the superconducting loop, which iscreated by the magnetic field of the cavity mode. A classical magneticpulse is applied to each qubit system through an ac flux �e threadingthe qubit superconducting loop, which is created by an ac currentloop (i.e., the red dashed-line loop) placed on the qubit loop. Thefrequency and intensity of the pulse can be adjusted by changing thefrequency and intensity of the ac loop current. EJ is the Josephsonjunction energy (0.6 < α < 0.8) and Vg is the gate voltage. λ is thewavelength of the resonator mode, and L is the length of the resonator.

[Figs. 3(b), 3(c), and 3(d)]; and the frequencies and intensitiesof pulses applied to qubit systems 2, 3, 4, and 5 can be readilyand simultaneously changed by varying the frequencies andintensities of the ac loop currents of the qubit systems 2, 3, 4,and 5 at the same time.

VI. CONCLUSION

In summary, we have proposed a way for creating n-qubitGHZ entangled states with a three-level qubit system and(n − 1) four-level qubit systems in a cavity or coupled toa resonator. The main advantage of this proposal is that noidentical qubit-cavity coupling constants are required. As aresult, qubit systems (e.g., solid-state qubits), which oftenhave parameter nonuniformity, can be used; and no exactplacement of qubits in a cavity is needed. Note that forsolid-state qubits (e.g., superconducting qubits), it is difficult

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CHUI-PING YANG PHYSICAL REVIEW A 83, 062302 (2011)

to design qubits with identical device parameters; and it isexperimentally challenging to place many qubits at differentlocations where the magnetic fields or electric fields areexactly the same. These hardships are avoided in this proposal.Another interesting property of this proposal is that duringthe entire operation, there is no need of adjusting the levelspacings of the qubit systems or the cavity mode frequency.In experiments, adjustment of the qubit level spacings or thecavity mode frequency during the operation is not desired;and extra errors may be induced by adjustment of the qubitlevel spacings or the cavity mode frequency. Furthermore, asshown above, this proposal has these additional advantages:(i) The n-qubit GHZ state is prepared deterministically and theoperation time is independent of the number of qubits in the

cavity; (ii) neither auxiliary qubit systems nor measurementon the qubit states is needed; and (iii) no photon detection isneeded. This proposal is quite general, which can be appliedto various types of superconducting qubits and atoms trappedin a cavity.

ACKNOWLEDGMENTS

C.P.Y. is grateful to Shi-Biao Zheng for very useful com-ments. This work is supported in part by the National NaturalScience Foundation of China under Grant No. 11074062,the Zhejiang Natural Science Foundation under Grant No.Y6100098, funds from Hangzhou Normal University, and theOpen Fund from the SKLPS of ECNU.

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