Entanglement and the SU(2) phase states in atomic systems

Entanglement and the SU

„2… phase states in atomic systems

M. Ali Can, Alexander A. Klyachko, and Alexander S. Shumovsky

Faculty of Science, Bilkent University, Bilkent, Ankara 06533, Turkey

共Received 6 March 2002; revised manuscript received 24 May 2002; published 19 August 2002兲

We show that a system of 2n identical two-level atoms interacting with n cavity photons manifests entangle-ment and that the set of entangled states coincides with the so-called SU共2兲 phase states. In particular, violation of classical realism in terms of the Greenberger-Horne-Zeilinger and Clauser-Horne-Shimoni-Holt conditions is proved. We discuss a property of entanglement expressed in terms of local measurements. We also show that generation of entangled states in the atom-photon systems under consideration strongly depends on the choice of initial conditions and that the parasitic influence of cavity detuning can be compensated through the use of Kerr medium.

DOI: 10.1103/PhysRevA.66.022111 PACS number共s兲: 03.65.Ud, 42.50.Ct, 03.67.⫺a

I. INTRODUCTION

It has been recognized that entanglement phenomenon touches on the conceptual problems of reality and locality in quantum physics as well as the more technological aspects of quantum communications, cryptography, and computing. In particular, the methods of quantum key distribution in com-munication channels secured from eavesdropping are based on the use of entangled states 关1–5兴 共for recent review, see Refs.关6,7兴兲. In turn, the realization of quantum computer 关8兴 is dependent on the ability to form entangled states of ini-tially uncorrelated single-particle states 关9兴.

In recent years, many successful experiments have been performed to verify the violation of Bell’s inequalities and Greenberger-Horne-Zeilinger 共GHZ兲 equality 关10,11兴 and to develop the methods of engineered entanglement for quan-tum cryptography and quanquan-tum key distribution. In particu-lar, the recent advances in the field of cavity QED and tech-niques of atom manipulation, trapping, and cooling enable a number of experiments that investigate the entanglement in the atomic systems 共see Refs. 关11–17兴 and references therein兲.

It has been shown recently关18兴 that a pure entangled state of two atoms can be obtained in an optical resonator through the exchange by a single photon. The main idea in Ref.关18兴 is that a single excitation of the system is either carried by a photon or shared between the atoms. If a photon can leak out from the resonator, the absence of photon counts in the pro-cess of continuous monitoring of the cavity decay can be associated with the presence of the pure entangled atomic state. The importance of this scheme is caused by the fact that its realization seems to be easily available with present experimental technique.

The main objective of this paper is to show that the en-tangled states in the ‘‘atoms-plus-photons’’ systems of the type discussed in Ref. 关18兴 can be represented by the so-called SU共2兲 phase states corresponding to the SU共2兲 algebra of the odd ‘‘spin’’

j⫽1 2

冋

冉

2n

n

冊

⫺1

册

, 共1兲

where 2n is the even number of atoms and n⫽1,2, . . . is the

number of cavity photons. In particular, the system consid-ered in Ref. 关18兴 corresponds to the phase states of ‘‘spin’’ j⫽1/2. The SU共2兲 phase states were introduced in Ref. 关19兴 for an arbitrary spin and then generalized in Refs.关20,21兴 to the case of the SU共2兲 subalgebra in the Weyl-Heisenberg algebra of photon operators共for recent review, see Ref. 关22兴兲. From the mathematical point of view, this is the system of

N⫽2 j⫹1 qubits defined in the Hilbert space

HN⫽共C2_兲丢N

with the componentwise action of SU(2)N. In particular, we show that these states violate the classical realism and dis-cuss their realization.

On the other hand, we will discuss a condition of en-tanglement that has been proposed recently关23兴. Let us note in this connection that, in the usual treatment of entangle-ment, the entangled states of a two-component 共in general, multicomponent兲 system are considered as the nonseparable states with respect to the subsystems共e.g., see Ref. 关24兴兲. For example, if the individual components of a two-component system are described by the states兩␰i

典

and兩␹i

典

, respectively,

the state 兩␺ent

典

⫽

兺

i bi兩␰i

典

丢兩␹i

典

,

具

␰i兩␰k

典

⫽

具

␹i兩␹k

典

⫽␦ik,

兺

i 兩bi兩 2_⫽1,

is entangled if b_i⫽” 0 for at least two distinct values of the subscript i. From the mathematical point of view, the en-tanglement is caused by the combination of the superposition principle in quantum mechanics with the tensor product structure of the space of state of the two-component or mul-ticomponent system 关25兴.

Very often, the existence of entanglement is verified in terms of violation of Bell’s inequalities and their generaliza-tions 关26–31兴. Another way is based on the use of GHZ

theorem 关10兴. A possibility to introduce more general in-equalities is also discussed关32兴.

It should be noted that the use of Bell’s inequalities and their numerous generalizations demonstrate nothing but the nonexistence of hidden variables. Moreover, it is possible to say that the unique, general, and mathematically correct defi-nition of entanglement still does not exist 共e.g., see Ref.

关32兴兲.

An interesting approach has been proposed recently关32兴. Considering the state shared between Alice and Bob as a quantum communication channel, the authors of Ref. 关32兴 concluded that the information in the case of entanglement is carried mostly by the correlations between the ends of the channel. These correlations manifest themselves by means of the local measurements on the sides of the channel 关23兴.

Following Ref.关23兴, consider a composite system defined in the Hilbert space

H⫽丢lH(l)_{, l⭓2.}

Let G be the group of dynamical symmetry of a subsystem in the composite system. Then the Hermitian operators g asso-ciated with representation of G inH(l)define the set of local measurement on the corresponding side of the channel pro-vided by a state 兩␺

典

苸H. For example, in the case of H(l)

⫽C2_{, corresponding to the Einstein-Podolsky and Rosen} 共EPR兲 spin-1

2 system, G⫽SU(2) and the set of local mea-surements can be specified by the infinitesimal generators of the SL共2兲 group

兵g其⫽兵␴k

(l)_其

, k⫽1,2,3, which is the complexification of the SU共2兲 group.

It was shown in Ref. 关23兴 that the maximum correlation between the ends of the channel corresponds to the states such that

᭙g,

具

g

典

⫽0.

This statement can be illustrated by the atoms-plus-photons systems under consideration. Consider first the set of two identical two-level atoms. Let 兩el

典

and 兩gl

典

denote the

ex-cited and the ground atomic states of the lth _atom,

respec-tively. Then, the entangled, maximum excited atomic states in the system ‘‘2 atoms plus 1 photon’’ considered in Ref.

关18兴 are

兩␺⫾

典

⫽

1

冑

2共兩e1g2

典

⫾兩g1e2

典

). 共2兲 Then, the local measurement g can be described by the Pauli matrices ␴1 (l)_⫽兩e l

典具

gl兩⫹兩gl

典具

el兩, ␴2 (l)_⫽⫺i兩e l

典具

gl兩⫹i兩gl

典具

el兩, ␴3 (l)_⫽兩e l

典具

el兩⫺兩gl

典具

gl兩, 共3兲

i.e., by the infinitesimal generators of the algebra SL共2兲. It is now a straightforward matter to check that

᭙i,l

具

␺⫾兩␴i

(l)_兩␺

⫾

典

⫽0, 共4兲

where averaging is taken over the states 共4兲. Another ex-ample is provided by the GHZ states关10兴

兩␺_⫾(GHZ)

_典

_⫽ 1

冑

2共兩e1e2e3

典

⫾兩g1g2g3

典

), 共5兲 corresponding to the maximum atomic excitation in the 3

⫹3 system. It is easily seen that the averaging of the local

operators共3兲 over Eq. 共5兲 gives the same result as Eq. 共4兲. This property 共4兲 can be used to define the entangled states.

We will show that the SU共2兲 phase states of spin j defined by Eq. 共1兲 in a (2n⫹n)-type atom-photon system obey the nonseparability conditions, have the property共4兲, and mani-fest the violation of classical realism expressed in terms of the GHZ关10兴 and CHSH 共Clauser-Horne-Shimoni-Holt兲 关33兴 conditions.

The paper is organized as follows. In Sec. II, we consider the representation of the SU共2兲 phase states. As a particular example, we examine the system of two identical two-level atoms, interacting with a single cavity photon and show that the maximum entangled atomic states of the Ref.关18兴 belong to the class of the SU共2兲 phase states of spin j⫽1/2. Let us stress that hereafter the maximum entanglement is defined in the usual way by the maximum of reduced entropy共e.g., see Refs. 关23,25,27,32兴兲. Then, we generalize this result on the case of 2n⫹n system. As a nontrivial example, we consider in Sec. III the system of four identical two-level atoms inter-acting with the two cavity photons. In this case, the set of entangled, maximum excited atomic states is provided by the six orthogonal SU共2兲 phase states of spin j⫽5/2. For these states, we prove violation of classical realism through the use of GHZ and CHSH conditions. In Sec. IV, we discuss how the entangled atomic states can be achieved in the process of steady-state evolution. In particular, we show that the maxi-mum entanglement can be achieved if the initial state of the system contains the photons and does not contain the atomic excitations. We also show that the presence of the cavity detuning hampers the creation of pure entangled states and that the parasitic influence of detuning can be compensated through the use of the Kerr medium inside the cavity. Finally, in Sec. V, we briefly discuss the obtained results.

II. REPRESENTATION OF THE SU„2… PHASE STATES An arbitrary spin j can be described by the generators J_⫹,J_⫺,Jzof the SU共2兲 algebra such that

关J⫹,J⫺兴⫽2Jz, 关Jz,J_⫾兴⫽⫾J_⫾,

J2⫽J_z2⫹ 1

2共J⫹J⫺⫹J⫺J⫹兲⫽ j共 j⫹1兲⫻1, 共6兲 where 1 is the unit operator in the (2 j⫹1)-dimensional Hil-bert space. Since

J_⫾⫽Jx⫾iJy,

it is possible to say that the generators J_⫹,J_⫺,J_z in Eq.共6兲 correspond to the Cartesian representation of the SU共2兲 alge-bra. Following Ref. 关19兴, one can introduce the representa-tion in spherical coordinates via the polar decomposirepresenta-tion of Eq. 共6兲 of the form

J_⫹⫽Jr⑀, Jr⫽Jr⫹, ⑀⑀⫹⫽1, 共7兲

where the Hermitian operator Jr corresponds to the radial

contribution, white ⑀ gives the exponential of the azimuthal phase operator. It is a straightforward matter to show that ⑀ can be represented by the following (2 j⫹1)⫻(2 j⫹1) ma-trix: ⑀⫽

冉

0 1 0 0 ••• 0 0 0 1 0 ••• 0 ⯗ ⯗ ⯗ ⯗ ⯗ ⯗ 0 0 0 0 ••• 1 ei␺ 0 0 0 ••• 0

冊

共8兲

in the (2 j⫹1)-dimensional Hilbert space. Here␺is an arbi-trary real parameter共reference phase兲. The eigenstates of the operator共8兲 ⑀兩␾n ( j)

_典

_⫽ei␾_n( j)_兩␾ n ( j)

_典

, n⫽1, . . . ,共2 j⫹1兲, 共9兲 form the basis of the so-called phase states

兩␺n ( j)

_典

_⫽ 1

冑

2 j⫹1 k

兺

⫽0 2 j eik␾n ( j) 兩␺k

典

共10兲

dual with respect to the basis of individual states兩␺k

典

of the

Hilbert space.

As a physical example of some considerable interest, con-sider now the system of the two identical two-level atom interacting with the single cavity photon 共see Ref. 关18兴兲. If the cavity photon is absorbed by either atom, the atomic subsystem can be observed in the following states

兩␺1

典

⫽兩e1g2

典

, 兩␺2

典

⫽兩g1e2

典

, 共11兲 where兩e1g2

典

⫽兩e1

典

丢兩g2

典

and兩e

典

and兩g

典

denote the excited and ground atomic states, respectively. The subscript marks the atom. Using the atomic basis共11兲, we can construct the following representation of the SU共2兲 algebra:

J_⫹⫽兩e1g2

典具

g1e2兩, J_⫺⫽兩g1e2

典具

e1g2兩, J3⫽

1

2共兩e1g2

典具

e1g2兩⫺兩g1e2

典具

g1e2兩兲. 共12兲 This representation formally corresponds to Eq. 共6兲 at the spin j⫽1/2. Then, the corresponding exponential of the phase operator共8兲 takes the form

⑀⫽兩e1g2

典具

g1e2兩⫹ei␺兩g1e2

典具

e1g2兩. 共13兲 In turn, the phase states共9兲 and 共10兲 are

兩␾⫾

典

⫽ 1

冑

2共兩e1g2

典

⫹e i␾_⫾兩g 1e2

典

), 共14兲 ␾⫾⫽␺/2⫹共1⫿1兲␲/2.

It is easily seen that the phase states 共14兲 form the set of entangled atomic states in the two-atom system under con-sideration. Definitely, these states obey the nonseparability condition. It is also seen that Eq. 共14兲 coincides with the maximally entangled states 共2兲 of Ref. 关18兴 when the refer-ence phase ␺⫽0.

Consider now a general 2n⫹n system at n⭓1. Then, the maximum excited atomic states

兩␺i

典

⫽兩兵e其n,兵g其n

典

, 共15兲

can be used to construct a representation of the SU共2兲 alge-bra 共6兲 of spin j defined in Eq. 共1兲. Here i⫽1,2, . . . ,N and

N⫽2 j⫹1⫽

冉

2n

n

冊

is the total number of such a states. In the basis共15兲, we can construct the polar decomposition of the SU共2兲 algebra of spin共1兲 and the corresponding exponential of the phase op-erator共8兲 and the phase states 共10兲. Let us rename the states

共15兲 as follows:

兩␺k

典

→兩␺k⬘

典

, k

⬘

⬅k⫺1⫽0, . . . ,N⫺1.

Then, the SU共2兲 phase states 共10兲 take the form

兩␾k

典

⫽ 1

冑

N k

兺

⬘⫽0 N⫺1 eik⬘␾k兩␺ k⬘

典

, 共16兲 where ␾k⫽共␺⫹2k␲兲/N.

These states 共16兲 form a basis dual with respect to Eq. 共15兲 and spanning the Hilbert space of the maximum excited atomic states in the 2n⫹n system under consideration. By construction, the phase states 共16兲 are nonseparable with re-spect to contributions of individual atoms and thus entangled

关24兴. Let us stress that the choice of the phase factors in Eq. 共16兲 is irrelevant to entanglement, which holds for arbitrary

phase factors. This choice is caused by the aspiration for getting the dual with respect to basis共15兲 of entangled states. It is easily seen that the states共16兲 obey the condition 共4兲. In fact, the action of the flip operators ␴_1,2(l) in Eq.共3兲 on the states 共16兲 leads to the change of the number of either ex-cited or deexex-cited atoms:

␴1,2 (l)_兩␺

k

典

→

再

兩兵e其_n⫺1,兵g其_n⫹1

典

,l苸兵g其 兩兵e其n⫹1,兵g其n⫺1

典

,l苸兵e其

and therefore

具

␴_1,2(l)

典

⫽0 in the case of averaging over the states 共16兲. Since each state 共15兲 contains equal number of excited and deexcited atoms, the action of the parity operator in Eq. 共3兲 on the phase states 共16兲 should lead to the state

that differs from Eq.共16兲 by the multiplication of a certain n terms by the factor of⫺1. Hence

具

␴3 (l)

_典

_⫽1 N

冉

兺

i⫽1 N/2 1⫺

兺

i⫽N/2⫹1 N 1

冊

⫽0.

By construction, N is always an even number.

Thus, the SU共2兲 phase states 共16兲, corresponding to the maximum excited atomic states in the 2n⫹n system, are entangled because they are nonseparable and, at the same time, obey the condition 共4兲 for the local measurements. In the following section, we show that the states共16兲 manifest violation of classical realism as well.

Before we begin to discuss this subject, let us note that the SU共2兲 phase states of the atomic system under consideration with integer spin do not provide the entanglement. Consider as an example the system of three identical two-level atoms, interacting with a single cavity photon. There are the three excited atomic states

兩e1g2g3

典

, 兩g1e2g3

典

, 兩g1g2e3

典

共17兲 and the three dual phase states of the type as shown in Eq.

共16兲 兩␺k

典

⫽ 1

冑

3共兩e1g2g3

典

⫹e i␾_k兩g 1e2g3

典

⫹e2i␾k兩g1g2e3

典

). 共18兲

It is clear that the states 共18兲 are the phase states of spin j

⫽1. Here

␾k⫽共␺⫹2k␲兲/3, k⫽0,1,2.

It is easily seen that the phase states 共18兲 cannot be factor-ized with respect to atoms. At the same time, the average of the parity operator␴₃(l)in Eq.共3兲 over the states 共18兲 is

᭙k,l

具

␺k兩␴3 (l)_兩␺

k

典

⫽⫺

1 3, although the averages of the flip operators are

᭙k,l

具

␺k兩␴1,2 (l)_兩␺

k

典

⫽0.

Thus, the nonseparable states共18兲 do not obey the condition

共4兲. At the same time, these states do not manifest the

maxi-mum entanglement as well. Let us stress that the nonsepara-bility is not a sufficient condition of maximum entanglement

关24兴. For example, from the measurement of the state of the

first atom we can only learn that either the atoms 2 and 3 are both in the ground state with reliability or they are in the two-atom entangled state of the type discussed in Ref. 关18兴. Similar result can be obtained for the system of three atoms interacting with two cavity photons. The only maximum en-tangled state of the system of three atoms is provided by the superposition of GHZ states共5兲.

III. THE 4¿2 SYSTEM

To show that the phase states 共16兲 of a 2n⫹n system violate the classical realism, consider the system of four identical two-level atoms interacting with two cavity pho-tons. The maximum excited atomic states at n⫽2 are

兩e1e2g3g4

典

, 兩e1g2e3g4

典

, 兩e1g2g3e4

典

,

兩g1e2e3g4

典

, 兩g1e2g3e4

典

, 兩g1g2e3e4

典

. 共19兲 These orthonormal states form the six-dimensional basis of the Hilbert space in which the representation of the genera-tors 共6兲 has the form

J_⫹⫽

冑

5兩e₁e₂g₃g₄

典具

e₁g₂e₃g₄兩⫹

冑

8兩e₁g₂e₃g₄

典具

e₁g₂g₃e₄兩 ⫹3兩e1g2g3e4

典具

g1e2e3g4兩⫹

冑

8兩g1e2e3g4

典具

g1e2g3e4兩 ⫹

冑

5兩g₁e₂g₃e₄

典具

g₁g₂e₃e₄兩, J₃⫽5 2兩e1e2g3g4

典具

e1e2g3g4兩⫹ 3 2兩e1g2e3g4

典具

e1g2e3g4兩 ⫹1 2 兩e1g2g3e4

典具

e1g2g3e4兩⫺12兩g1e2e3g4

典具

g1e2e3g4兩 ⫺3 2 兩g1e2g3e4

典具

g1e2g3e4兩⫺ 5 2兩g1g2e3e4

典具

g1g2e3e4兩.

By construction, they describe the spin j⫽5/2 system. In turn, the exponential of the phase operator共8兲 takes the form

⑀⫽兩e1e2g3g4

典具

e1g2e3g4兩⫹兩e1g2e3g4

典具

e1g2g3e4兩 ⫹兩e1g2g3e4

典具

g1e2e3g4兩⫹兩g1e2e3g4

典具

g1e2g3e4兩 ⫹兩g1e2g3e4

典具

g1g2e3e4兩⫹ei␺兩g1g2e3e4

典具

e1e2g3g4兩. Then, the six phase states 共9兲 have the form 共16兲 with N

⫽6 and ␾k⫽ ␺ 6⫹ k␲ 3 , k⫽0,1, . . . ,5. 共20兲

As well as for Eq. 共16兲, these states are nonseparable and hence entangled and obey the condition 共4兲 for local vari-ables.

To show that these phase states violate the classical real-ism, let us first represent the states 共16兲 at N⫽6 in the fol-lowing way 兩␾k

典

⫽ 1

冑

3共兩␹1k

典

⫹e i␾_k兩␹ 2k

典

⫹e2i␾k兩␹3k

典

), 共21兲

where 兩␹1k

典

⫽ 1

冑

2共兩e1e2g3g4

典

⫹e 5i␾_k兩g 1g2e3e4

典

), 兩␹2k

典

⫽ 1

冑

2共兩g1e2e3g4

典

⫹e 3i␾_k兩e 1g2g3e4

典

), 兩␹3k

典

⫽ 1

冑

2共兩g1e2g3e4

典

⫹e i␾_k兩e 1g2e3g4

典

). 共22兲

It is easily seen that each set of six states 兩␹pk

典

with p

⫽1,2,3 and k⫽0, . . . ,5 consists of the nonseparable and

hence entangled states. Consider, for example, the states

兩␹1k

典

in Eq. 共22兲. Because of the definition of the phase angle ␾k at N⫽6, they consist of the three sets of the

pair-wise orthogonal states

兵兩␹10

典

,兩␹13

典

其, 兵兩␹11

典

,兩␹14

典

其, 兵兩␹12

典

,兩␹15

典

其. It is also seen that the second and third sets here are obtained from the first set by the successive rotations of the reference frame.

Now the violation of classical realism can be proved through the use of the GHZ theorem 关10兴. Consider first the state兩␹10

典

in Eq.共22兲. It is easy to verify that this state obey the following conditions:

᭙i,l 丢l⫽1 4 _␴ i (l)_兩␹ 10

典

⫽兩␹10

典

共23兲 and ␴1 (1) ␴1 (2) ␴2 (3) ␴2 (4)_兩 ␹10

典

⫽⫺兩␹10

典

, ␴2 (1)_␴ 2 (2)_␴ 1 (3)_␴ 1 (4)_兩␹ 10

典

⫽⫺兩␹10

典

, ␴1 (1)_␴ 2 (2)_␴ 1 (3)_␴ 2 (4)_兩␹ 10

典

⫽兩␹10

典

, ␴1 (1)_␴ 2 (2)_␴ 2 (3)_␴ 1 (4)_兩␹ 10

典

⫽兩␹10

典

, ␴2 (1)_␴ 1 (2)_␴ 2 (3)_␴ 1 (4)_兩␹ 10

典

⫽兩␹10

典

, ␴2 (1) ␴1 (2) ␴1 (3) ␴2 (4)_兩 ␹10

典

⫽兩␹10

典

. 共24兲 It is possible to say that these equalities共23兲 and 共24兲 express a kind of EPR ‘‘action at distance’’ in the maximum excited states of the system of four atoms interacting with two pho-tons. In other words, the correlations represented by Eqs.

共23兲 and 共24兲 permit us to determine in a unique way the

state of the fourth atom via measurement of the states of other three atoms.

The operator equalities 共23兲 and 共24兲 can be used to ob-tain the relations similar to those in the GHZ theorem. Fol-lowing Ref. 关10兴, we have to assign the classical quantities m_i(l) to the local operators. Here

m₁(l),m₂(l)⫽⫾1. Then, it follows from Eq.共23兲 that

兿

l⫽1

4

m₁(l)⫽1. 共25兲 At the same time, it follows from Eq.共24兲 that

关␴1 (1) ␴1 (2) ␴2 (3) ␴2 (4) 兴关␴1 (1) ␴2 (2) ␴1 (3) ␴2 (4) 兴 ⫻关␴1 (1) ␴2 (2) ␴2 (3) ␴1 (4)_兴兩 ␹10

典

⫽⫺兩␹10

典

.

Employing the classical variables instead of the local opera-tors allows this to be cast into the form

共m1 (1)_兲3_m 1 (2)_共m 2 (2)_兲2_m 1 (3)_共m 2 (3)_兲2_m 1 (4)_共m 2 (4)_兲2_⫽⫺1.

Since (m₁(l))2⫽(m₂(l))2⫽1, we get an equivalent equality m₁(1)m₁(2)m₁(3)m₁(4)⫽⫺1,

which contradicts Eq.共25兲. Hence, the state 兩␹10

典

in Eq.共22兲 obey the GHZ theorem. Similar result can be obtained for all other states in Eq.共22兲 and hence, for the phase states 共21兲. Our consideration so far have applied to the local mea-surements touching on a single atom. We now note that the phase states共21兲 allow another kind of entanglement in the case of pairwise measurement. Consider again the state兩␹10

典

in Eq. 共22兲 and assume that the measurements a and b cor-responds to a pair of atoms:

a⫽cos␪_a兩e₁e₂

典具

e₁e₂兩⫹sin␪_a共兩e₁e₂

典具

g₁g₂兩⫹兩g₁g₂

典

⫻

具

e1e2兩兲⫺cos␪a兩g1g2

典具

g1g2兩,

b⫽cos␪b兩e3e4

典具

e3e4兩⫹sin␪b共兩e3e4

典具

g3g4兩⫹兩g3g4

典

⫻

具

e₃e₄兩兲⫺cos␪_b兩g₃g₄

典具

g₃g₄兩. 共26兲 Assume now that we make the two measurements a and a

⬘

with the angles␪₁⫽␲ and␪_a

⬘

⫽␲/2 and the two more mea-surements b and b

⬘

with the angles␪_b

⬘

⫽⫺␪b, respectively.

Then, the averaging over the state兩␹10

典

gives

具

ab

典

⫽

具

ab

⬘典

⫽cos␪b,

具

a

⬘

b

典

⫽sin␪b⫽⫺

具

a

⬘

b

⬘典

.

Employing the CHSH inequality关33兴

兩

具

ab

典

⫹

具

a

⬘

b

典

⫹

具

a

⬘

b

⬘典

⫺

具

ab

⬘典

兩⭐2 共27兲 then gives

兩cos␪b⫺sin␪b兩⭐1.

Violation of this inequality and hence, of the classical real-ism occurs at small negative␪b, when we can put

兩cos␪b⫺sin␪b兩⬃1⫹兩␪b兩⬎1.

Similar consideration can be done for all states in Eq. 共22兲 through the use of proper pairwise measurements. At the same time, the phase states 共21兲 do not manifest entangle-ment with respect to the pairwise measureentangle-ments.

The phase states共16兲 for the 6⫹3,8⫹4, . . . systems, cor-responding to the spin 共1兲 equal to 19/2,69/2, . . . , respec-tively, can be considered as above.

IV. INITIAL CONDITIONS AND ATOMIC ENTANGLEMENT

It is clear that the evolution of the 2n⫹n system strongly depends on the choice of initial conditions. To trace the proper choice leading to the atomic entanglement, let us ig-nore the relaxation processes. Then, the steady-state evolu-tion of the 2n⫹n system under consideration is governed by the Hamiltonian

H⫽⌬a⫹a⫹␻₀N⫹␥

兺

l 共Rl

⫹_a⫹a⫹_R

l兲. 共28兲

Here ⌬ is the cavity detuning, ␻0 is the atomic transition frequency,␥ is the atom-field coupling constant, and opera-tors a and a⫹describe the cavity photons,

N⫽a⫹_a⫹

兺

l 兩el

典具

el兩丢l⬘⫽”l1(l),

and the atomic operators are defined as follows: Rl⫹⫽兩el

典具

gl兩丢l⬘⫽”l1(l⬘).

Here 1(l) denotes the unit operator in the two-dimensional Hilbert space of the lth atom. It is seen that关N,H兴⫽0. It is also seen that the atomic operators are similar, in a certain sense, to the local operators共3兲. In fact

R_l⫾⫽␴1 (l)_⫾i␴

2 (l)

2 .

Consider first the case of two atoms and single cavity photon when l⫽1,2 and the Hamiltonian 共28兲 coincides with that of Ref.关18兴. For simplicity, we use here the same cou-pling constant␥for both atoms. Our consideration can easily be generalized on the case of coupling constant depending on the atomic position. Let us note that, in the case of only two atoms, the Hamiltonian共28兲 can be represented as follows

H→H_␾⫽⌬a⫹a⫹␻0N␾⫹␥

冑

2共R⫹a⫹a⫹R兲, 共29兲 where N␾⫽a⫹a⫹

兺

k⫽⫾1 兩␾k

典具

␾k兩 and R⫹_⫽兩␾ ⫹

典具

g1g2兩. Here兩␾_⫾

典

denote the phase states共14兲.

Using the Hamiltonian共29兲 as the generator of evolution, for the time-dependent wave function we get

兩⌿共t兲

典

⫽e⫺iH␾t_{兩⌿共0兲}

_典

⫽关C⫺共t兲兩␾⫺

典

⫹C⫹共t兲兩␾⫹

典

]丢兩0

典

ph

⫹C共t兲兩g1g2

典

丢兩1

典

ph, 共30兲

where兩•••

典

ph denotes the states of the cavity field. The

co-efficients C_⫾(t) and C(t) in Eq.共30兲 are completely deter-mined by the initial conditions and the normalization condi-tion.

It is easily seen that the state 兩␾_⫺

典

丢兩0

典

ph is the

eigen-state of the Hamiltonian共29兲. Hence, at C_⫺共0兲⫽1, C_⫹共0兲⫽C共0兲⫽0,

the atomic phase state兩␾_⫺

典

in Eq.共14兲 provides the station-ary, maximum entangled atomic state in the system under consideration关18兴. At the same time, it is not very clear how to prepare such a state.

Therefore we consider a more realistic initial state pro-vided by excitation of either atom, while the cavity field is in the vacuum state. To realize such a state, we can assume, for example, that one of the atoms共initially deexcited兲 is trapped in the cavity, while the second atom共initially excited兲 slowly passes through the cavity like in the experiments discussed in Refs. 关14,15兴. assume for definiteness that

兩⌿共0兲

典

⫽兩e1g2

典

丢兩0

典

ph. 共31兲

Then, the coefficients of the wave function 共30兲 take the form C_⫺共t兲⫽ 1

冑

2e ⫺i␻0t_, C_⫹共t兲⫽ 1

冑

2

冋

cos共⍀t兲⫹ i⌬ 2⍀sin共⍀t兲

册

e ⫺i(␻0⫹⌬/2)t_, C共t兲⫽⫺i␥ ⍀e⫺i(␻0⫹⌬/2)tsin共⍀t兲,

where⍀⫽关2␥2⫹(⌬/2)2兴1/2. At first site, the probabilities P_⫾共t兲⫽兩

具

0兩ph丢

具

␾pm兩⌿共t兲

典

兩2⫽兩C⫾共t兲兩2

to observe the states 共14兲 corresponding to the maximum atomic entanglement, are

P_⫺共t兲⫽12 , P_⫹共t兲⫽ ⌬ 2 8⍀2⫹ ␥2 ⍀2cos 2_{共⍀t兲⭐}1 2,

respectively. At the same time, the absence of photon counts, which is considered in Ref. 关18兴 as a sign of the atomic entanglement, corresponds here to the case when both prob-abilities P_⫾(t_k)⫽1/2 at a certain time t_k. In other words, the mutually orthogonal entangled states 共14兲 have the same probability to be observed at t⫽tk. This means that there is

no atomic entanglement at all but we definitely know which atom is in the excited state.

Consider one more realistic initial state when both atoms are trapped in the cavity in deexcited state, while the cavity field contains a photon:

兩⌿共0兲

典

⫽兩g1g2

典

丢兩1

典

ph. 共32兲

Then, for all times we get C_⫺(t)⫽0 and

C_⫹共t兲⫽⫺i␥

冑

2

⍀ e⫺i(␻0⫹⌬/2)tsin共⍀t兲, C共t兲⫽

冋

cos共⍀t兲⫺ i⌬

2⍀sin共⍀t兲

册

e

⫺i(␻0⫹⌬/2)t_.

Hence, under this initial condition, the entangled state兩␾_⫺

典

cannot be achieved at all, while the second entangled state

兩␾⫹

典

in Eq.共14兲 can be achieved. It is seen that, in the case

of initial state共32兲, the probability to detect the photon is

Pph共t兲⫽兩C共t兲兩2⫽cos2共⍀t兲⫹

⌬2 4⍀2sin

2_共⍀t兲.

This expression takes the minimum value

minP_ph⫽P_ph共t_m兲⫽ ⌬ 2 4⍀2

at t⫽tm⫽␲(2m⫹1)/2⍀, m⫽0,1, . . . . At the same time

tm, the probability to have the entangled atomic state兩␾⫹

典

takes the maximum value

P_⫹共tm兲⫽兩C⫹共tm兲兩2⫽

2␥2 2␥2⫹共⌬/2兲2.

It is seen that the pure atomic entanglement with P_⫹(tm)

⫽1 is realized at t⫽tm only in the absence of the cavity

detuning when⌬→0.

The parasitic influence of the cavity detuning can be com-pensated through the use of Kerr medium filling the cavity. In this case, the Hamiltonian共28兲 should be supplemented by the term关34兴

H_␬⫽␬共a⫹a兲2,

which leads to the following renormalization of the Rabi frequency:

⍀→⍀␬⫽

冑

2␥2⫹共⌬⫹␬兲2/4.

Then, the proper choice of the Kerr parameter ␬⫽⫺⌬ should lead to the pure entangled atomic state 兩␾_⫹

典

at a certain times.

Consider now the case of four atoms and two photons. In contrast to the previous case, neither phase state in Eq. 共21兲 is an eigenstate of the Hamiltonian共28兲. Then, the choice of the initial state either as a state with two excited atoms or as a state with one excited atom plus cavity photon does not lead to a pure atomic entanglement. As in the case of two atoms, the pure atomic entanglement can be reached under the choice of the state with the absence of the atomic exci-tations in the initial state. The influence of the cavity detun-ing can be compensated by the presence of Kerr medium as well as in the case of two atoms.

V. CONCLUSION

Let us briefly discuss the obtained results. For the system of two identical two-level atoms interacting with a single photon as proposed in Ref. 关18兴 it is shown that the maxi-mum entangled atomic states are represented by the SU共2兲 phase states of spin 1/2. Moreover, it is shown that the SU共2兲 phase states of the half-integer spin j共1兲 form a certain class of maximum entangled atomic states in the system of 2n atoms interacting with n photons. In particular, the violation of classical realism is shown.

It should be noted in this connection that the above con-sidered SU共2兲 phase states do not represent a unique way to construct the maximum entangled states in the multiatom systems and that some other symmetries, for example the SU(N) can be considered as well. Moreover, in some cases the SU共2兲 phase states cannot be used to determine the maxi-mum entangled states at all. Consider for example the case of two identical two-level atoms interacting with two photons, when the atomic subsystem can be specified by the four states

兩e1e2

典

, 兩e1g2

典

, 兩g1e2

典

, 兩g1g2

典

.

By performing a similar analysis to that described in Sec. II, it is easy to construct the corresponding set of the SU共2兲 phase states

兩␾k

典

⫽12共兩e1e2

典

⫹ei␾k兩e1g2

典

⫹e2i␾k兩g1g2

典

⫹e3i␾k兩g1e2

典

),

␾k⫽

␺

4⫹ k␲

2 , k⫽0,1,2,3,

which do not manifest the maximum entanglement. At the same time, the general criterion 共4兲 permits us to determine infinitely many maximum entangled states in this case 关23兴. An example is provided by the following set of orthonormal maximum entangled states:

兩␺1

典

⫽

1

2共兩e1e2

典

⫹兩g1g2

典

⫹i兩e1g2

典

⫹i兩g1e2

典

), 兩␺2

典

⫽

1

2共兩e1e2

典

⫺兩g1g2

典

⫺i兩e1g2

典

⫹i兩g1e2

典

), 兩␺3

典

⫽12共i兩e1e2

典

⫹i兩g1g2

典

⫹兩e1g2

典

⫹兩g1e2

典

), 兩␺4

典

⫽

1

2共⫺i兩e1e2

典

⫹i兩g1g2

典

⫹兩e1g2

典

⫺兩g1e2

典

). In fact, the Eq. 共4兲 gives a general condition 关23兴, while the SU共2兲 phase states can manifest the maximum entanglement only under a certain condition关special choice of the effective spin共1兲兴.

Nevertheless, the SU共2兲 phase states considered in Secs. II and III represent an important example of the atomic en-tangled states. First of all, they can be easily realized in the atomic systems in a cavity. In fact, these states have a simple physical meaning. In addition to Eq. 共9兲, the SU共2兲 phase states can be defined to be the eigenstates of the Hermitian ‘‘cosine’’ operator关19,22兴

where⑀ is defined by Eq.共8兲. This operator C can be con-sidered as a ‘‘Hamiltonian,’’ describing the correlations be-tween the different atoms. For example, in the case of the two atoms interacting with the single photon, the operator C takes the form

C⫽␴_⫹(1)␴_⫺(2)⫹␴_⫺(2)␴_⫹(1), 共33兲 where ␴_⫾(l)_⫽␴1 (l)_⫾i␴ 2 (l) 2 .

The operator structure of Eq.共33兲 coincides with that of the so-called model of plane rotator, which is a particular case of the Heisenberg model of ferromagnetism widely used in sta-tistical physics关35兴 and in quantum information theory 关36兴. Let us also stress that the SU共2兲 phase states similar to those considered in Secs. II and III have been discussed re-cently in the context of quantum coding关37兴.

It is also known that the SU共2兲 phase states have direct connection with the quantum description of polarization of spherical photons emitted by the multipole transitions in at-oms and molecules 关21,22,38兴. Therefore, the polarization entanglement of photons can be examined in direct analogy to the above discussed atomic entanglement 关39兴. At the same time, the consideration of spherical photons requires the use of more quantum degrees of freedom. Consider as an example the cascade decay of a two-level atom specified by the transition关40兴

兩J⫽2,m⫽0

典

→兩J

⬘

⫽0,m

⬘

⫽0

典

.

Here J,J

⬘

and m,m

⬘

denote the angular momentum and pro-jection of the angular momentum of the excited and the ground atomic states, respectively. This transition gives rise to an entangled photon twins关40兴. Each photon carries spin 1, but because of the conservation of the angular momentum in the process of radiation, the sum of projections of the angular momenta of the two photons should be equal to zero. Denoting the state of a photon with given m by兩m

典

, we get the three possible states of the photon subsystem:

兩⫹1

典

丢兩⫺1

典

, 兩0

典

丢兩0

典

, 兩⫺1

典

丢兩⫹1

典

.

These three ‘‘individual’’ states can be used to construct the dual basis of the SU共2兲 phase states 关21兴

兩␾k

典

⫽ 1

冑

3共兩⫹1

典

丢兩⫺1

典

⫹e i␾_k兩0

_典

丢兩0

典

⫹e2i␾_k兩⫺1

_典

丢兩⫹1

典

, ␾k⫽ ␺⫹2k␲ 3 , k⫽0,1,2, 共34兲

similar to Eq. 共18兲. It can be easily seen that these states manifest the maximum entanglement.

Similar entangled states have been discussed in the con-text of the so-called biphoton excitations关41兴 共photon pairs in symmetric Fock states兲. They can also be used in quantum cryptography关42兴.

Let us stress that the general condition of the type as in Eq. 共4兲 is also valid in the case of states 共34兲. However, the definition of local measurement should be changed in this case. Because of the number of degrees of freedom per pho-ton is equal to 3, the Hermitian operators associated with the SU共3兲 group should be considered instead of the infinitesi-mal generators of the SL共2兲 group. For example, the set of Stokes operators of Ref.关21兴, corresponding to the represen-tation of the SU共3兲 subalgebra in the Weyl-Heisenberg alge-bra of spherical photons, can be used to define the complete set of local measurements in this case.

It is shown in Sec. IV that the realization of a pure atomic entanglement in the (2n⫹n)-type atom-plus-photon systems strongly depends on the choice of initial state. That is the entangled states can be reached in the process of steady-state evolution only if all 2n atoms are initially in the deexcited states, while the cavity contains just n photons. This condi-tion has an intuitively clear explanacondi-tion: the excitacondi-tions of different atoms have the same probability and therefore each photon in the 2n⫹n system is shared with a couple of atoms. It is also shown in Sec. IV that the presence of cavity detuning hampers the creation of a pure entangled atomic state. This negative effect can be compensated through the use of Kerr medium in the cavity.

We now note that the practical realization of a long-lived, maximum entanglement in a quantum-mechanical system strongly depends on the interaction between this system and environment. The point is that the state of a closed quantum-mechanical system changes periodically, providing the maxi-mum entanglement as an instant event only at a certain times

共see Sec. IV兲. Such a periodicity is caused by a finite number

of degrees of freedom in the system. To destruct such a pe-riodicity, it is necessary to connect the system to a ‘‘heat bath,’’ which would tune in the system to a required state. In Ref. 关18兴, it has been proposed to support the atomic en-tanglement by the cavity losses. In this case, the absence of the photon counting outside the cavity can be associated with the existence of the entangled atomic state in the cavity.

Let us stress that an advantage of the use of the SU共2兲 phase states as the maximum entangled atomic states con-sists in the simple preparation of the initial states discussed in Sec. IV.

In view of realization of atomic entanglement with the present experimental technique, it seems to be more conve-nient if the existence of entangled state in a cavity would manifest itself via a signal photon rather than the absence of photon leakage from the cavity. In this case, there should be

FIG. 1. Atomic Raman-type interaction with pump 共P兲 and Stokes共S兲 photons.

at least two modes such that one of them共the cavity mode兲 provides the correlation between the atoms, while the second can freely leave the resonator to signalize the existence of the entanglement. Such a process can be realized through the use of Raman process in atoms shown in Fig. 1 共e.g., see Ref.

关43兴兲. Here the dipole transitions are allowed between the

levels 1 and 2 and 2 and 3, while forbidden between 1 and 3 because of the parity conservation. In the simplest case, we should assume that the two identical atoms of this type are located in a cavity, which has a very high quality with re-spect to the pumping mode ␻P, while the Stokes photons

with frequency ␻Skcan leak away freely.

Assume that the atoms are initially in the ground state 1, the Stokes field is in the vacuum state, and the pump field consists of a single photon. The evolution of the system can lead to the absorption of the cavity photon by either atom with further emission of the Stokes photon that leaves the cavity. After that the atoms are in entangled state,

corre-sponding to the excitation of the atomic level 3 shared be-tween the atoms. Since the inverse process cannot be realized without assistance of the Stokes photon, such a state repre-sents a durable atomic entangled state.

It is clear that the above consideration of the atomic en-tanglement in the multiatom system can be generalized with ease in the case of Raman process in atoms. In other words, the SU共2兲 phase states similar to Eq. 共16兲 form the class of the maximum entangled atomic states in the case of Raman-type processes in the three-level atoms as well. An evident advantage of the use of the Raman process is the long-lived maximum entanglement in atomic subsystem.

ACKNOWLEDGMENTS

One of the authors 共A.S.兲 would like to thank Dr. A. Beige, Professor P. L. Knight, Professor A. Vourdas, and Pro-fessor A. Zeilinger for useful discussions.

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