Generation correlated four-mode states in cavity QED

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Optics Communications 277 (2007) 103–108

Generation correlated four-mode states in cavity QED

Zhi-Rong Zhong

Department of Electronic Science and Applied Physics, Fuzhou University, Fuzhou 350002, PR China

Received 13 November 2006; received in revised form 4 April 2007; accepted 27 April 2007

Abstract

A scheme for preparing correlated four-mode states with controllable weighting factors is presented. In the scheme, a sequence ofsuitably prepared four-level atoms are orderly sent through two bimodal cavities, the detection of all atoms in ground state collapsescavity fields to the desire state. The distinct advantage of our scheme is that the interaction time can be greatly shortened, which is impor-tant in view of decoherence.� 2007 Elsevier B.V. All rights reserved.

PACS: 42.50.DV

Keywords: Correlated four-mode states; Four-level atom; Resonant interaction

One of the central topics in quantum optics is quantum state engineering. So far, a number of schemes have been pre-sented for generating various quantum states. In cavity QED, schemes have been proposed for the generation of Schro-dinger cat states of a cavity field [1–3]. On the other hand, many schemes have been proposed to prepare any Fockstate superposition of an electromagnetic field. Vogel et al. [4] have firstly proposed a method for producing an arbitrarysuperposition of n + 1 photon number states from the vacuum state by injecting n approximately prepared atoms into acavity and detecting all of them in the ground state. Parkins et al. [5] have proposed a scheme for generating such super-position states via adiabatic passage. Zheng [6] has shown that such states can be generated via the interaction of a multi-level atom with a single-mode cavity field. Recently, the correlated quantum states of a multi-mode field have arousedmuch interest and many schemes have been presented for preparing such states [7–11]. Sanders et al. [8] have presenteda scheme to generate entangled coherent states with the help of a Mach–Zehnder interferometer. Davidovich et al. [9] haveproposed a method for producing quantum superpositions of coherent microwave field states located simultaneously intwo cavities by using two quantum switches.

More recently, many people are interested in the multi-dimensional (more than two) quantum systems. The proof ofBell’s theorem without the inequalities presented by Greenberger, Horne, and Zeilinger was extended to multiparticlemulti-dimensional systems [12,13]. It has been shown that quantum key distributions based on N-dimensional systemsare more secure than those based on two-dimensional systems [14,15]. It also has been demonstrated that violations of localrealism caused by two entangled N-dimensional (N P 3) systems are stronger than that by two-dimensional systems [16].

In the context of cavity QED, various schemes have been proposed for preparing maximally entangled states. Zheng [17]has proposed an alternative scheme for generating multi-dimensional entanglement between two or more multilevel atomsin a thermal cavity. Shu et al. [18] have presented a scheme for generating four-mode multiphoton entangled states.

In this paper, we propose a scheme to generate correlated four-mode states with controllable weighting factors via four-level atom resonantly interacting with two bimodal cavities, which are initially prepared in the two-mode vacuum state. In

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.04.054

E-mail address: [email protected]

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104 Z.-R. Zhong / Optics Communications 277 (2007) 103–108

the scheme, atoms which are produced in zone B by the excitation of a velocity selected atomic beam effusing from oven O,pass through the first cavity. Before the atoms enter the second cavity, they are sent through two classical fields F1 and F2 inturn. As is described in Fig. 1. The distinct advantage of our scheme is that the interaction time can be greatly shortened,which is important in view of decoherence. As the application of this scheme, we show how to generate four-mode mul-tiphoton maximally entangled states.

Suppose the atom has four states denoted by jei, jgi, jii, and jg 0i, with the energy-level xi, xg, xe and xg0 , respectively.Let us consider the resonant interaction of a four-level atom with a two-mode field, as sketched in Fig. 2. The transitionbetween jei and jgi is coupled to cavity mode 1, while the transition between jgi and jii is coupled to cavity mode 2. Thetransition between jei (jgi, jii) and jg 0i is highly detuned from both the cavity modes, respectively, thus the state jg 0i is aauxiliary level which performs the transformation jgi to jg 0i so that the interaction of the atom with the second cavity isfrozen if the atom is in the state jgi after exit the first cavity. The Hamiltonian for such a system is written as (�h = 1) [19]

Fig. 1.atomiccavitie

Fig. 2.couplin

H ¼ xejeihej þ xgjgihgj þ xijiihij þ x1aþ1 a1 þ x2aþ2 a2 þ g1ðaþ1 jgihej þ a1jeihgjÞ þ g2ðaþ2 jiihgj þ a2jgihijÞ ð1Þ
where aþi and ai (i = 1,2) are the creation and annihilation operators for the cavity fields, g1 (g2) is the coupling constant ofthe transition between jei (jii) and jgi with the cavity field, x1, x2 are the frequencies of two-mode cavity field. In the inter-action picture, the interaction Hamiltonian is given by
HI ¼ kðaþ1 jgihej þ aþ2 jiihgjÞ þ H :c; ð2Þ
we here assume that g1 = g2 = k. The basis states for the system are of the form
jn1; n2; n3; n4; si ¼ jn1ijn2ijn3ijn4ijsi; ð3Þ
where ni (i = 1,2,3,4) refers to the number of excitations in modes of two bimodal cavities and s refers to the state of four-level atom.
1C 2C

O 1F2F DB

The displays of the set-up for engineering correlated four-mode states. The atom is produced in zone B by the excitation of a velocity selectedbeam effusing from oven O, F1 and F2 are two classical fields tuned to the transitions jeiM jii, and jgiM jg 0i, respectively, C1, C2 are the two-mode

s, D is the detector for the state jii.

Schematic diagram of energy level of four-level atom with corresponding transition (jei, jgi, jii and jg 0i denote the states of the atom, g1, g2 are theg coefficients).

Z.-R. Zhong / Optics Communications 277 (2007) 103–108 105

We assume that the first cavity field is initially in two-mode vacuum state j0,0i. Now we send the first atom initially inthe state

jW1ai ¼ jei; ð4Þ

through the cavity. After an interaction time t1 the atom-cavity system evolves into the state

jW1i ¼1

2cos

ffiffiffi2p

kt1 þ 1� �

je; 0; 0i þffiffiffi2p

2cos

ffiffiffi2p

kt1 � 1� �

ji; 1; 1i � i2

sinffiffiffi2p

kt1jg; 1; 0i: ð5Þ

After the atom exits from the cavity, it traverses two classical fields in turn, which leads to the transition, respectively.

jei ! jii; jii ! �jei; ð6Þjgi ! jg0i: ð7Þ

These lead to

jW1i ¼1

2cos

ffiffiffi2p

kt1 þ 1� �

ji; 0; 0i �ffiffiffi2p

2cos

ffiffiffi2p

kt1 � 1� �

je; 1; 1i � i2

sinffiffiffi2p

kt1jg0; 1; 0i: ð8Þ

Then we let the atom pass through the second cavity which is also initially in two-mode vacuum state, after the same inter-action time t1, the atom-cavity system evolves into

jW1i ¼1

2cos

ffiffiffi2p

kt1 þ 1� �

ji; 0; 0; 0; 0i �ffiffiffi2p

2cos

ffiffiffi2p

kt1 � 1� �"1

2cos

ffiffiffi2p

kt1 þ 1� �

je; 0; 0; 1; 1i

þffiffiffi2p

2cos

ffiffiffi2p

kt1 � 1� �

ji; 1; 1; 1; 1i � i2

sinffiffiffi2p

kt1jg; 1; 0; 1; 1i#� i

2sin

ffiffiffi2p

kt1jg0; 1; 0; 0; 0i: ð9Þ

Now we perform a measurement on the atom. If this atom is detected in the state jii, the cavity field collapses onto thestate

jW1f i ¼ N 1

1

2cos

ffiffiffi2p

kt1 þ 1� �

j0; 0; 0; 0i � 1

2cos

ffiffiffi2p

kt1 � 1� �2

j1; 1; 1; 1i;�

ð10Þ

where N1 is a normalization factor given by

N 1 ¼1

2cos

ffiffiffi2p

kt1 þ 1� �� 2

þ 1

2cos

ffiffiffi2p

kt1 � 1� �2

� �2( )�1=2

: ð11Þ

Now, we send the second atom in the state

jW2ai ¼ jei; ð12Þ

through the first cavity, after interaction time t2, the atom-cavity field system evolves into the state

jW2ðtÞi ¼ N 1

1

2cos

ffiffiffi2p

kt1 þ 1� �"1

2cos

ffiffiffi2p

kt2 þ 1� �

je; 0; 0i þffiffiffi2p

2cos

ffiffiffi2p

kt2 � 1� �

ji; 1; 1i(

� i2

sinffiffiffi2p

kt2jg; 1; 0i#j0; 0i � 1

2cos

ffiffiffi2p

kt1 � 1� �2

"1

42 cos

ffiffiffi4p

kt2 þ 2� �

je; 1; 1i

þffiffiffi6p

4cos

ffiffiffi4p

kt2 � 1� �

ji; 2; 2i �ffiffiffi2p

4isin

ffiffiffi4p

kt2jg; 2; 1i#j1; 1i

): ð13Þ

Then we let the atom pass through two classical fields in turn, where they undergo the transition, respectively.

jei ! jii; jii ! �jei; ð14Þjgi ! jg0i; ð15Þ


the cavity field evolves into the state

jW2ðtÞi¼N 1

1

2cos

ffiffiffi2p

kt1þ1� � 1

2cos

ffiffiffi2p

kt1þ1� �

ji;0;0i�ffiffiffi2p

2cos

ffiffiffi2p

kt2�1� �

je;1;1i"(

� i2

sinffiffiffi2p

kt2jg0;1;0i�j0;0i

þ1

2cos

ffiffiffi2p

kt1�1� �2 1

42cos

ffiffiffi4p

kt2þ2� �

ji;1;1i�

�ffiffiffi6p

4cos

ffiffiffi4p

kt2�1� �

je;2;2i�ffiffiffi2p

4isin

ffiffiffi4p

kt2jg0;2;1i#j1;1i

):

ð16Þ
Then we let the atom pass through the second cavity, after the same interaction time t2 the atom-cavity system evolves intothe state
jW2ðtÞi ¼ N 1

1

2cos

ffiffiffi2p

kt1 þ 1� �(1

2cos

ffiffiffi2p

kt2 þ 1� �

ji; 0; 0; 0; 0i �ffiffiffi2p

2cos

ffiffiffi2p

kt2 � 1� �( "

1

2cos

ffiffiffi2p

kt2 þ 1� �

je; 0; 0i

þffiffiffi2p

2cos

ffiffiffi2p

kt2 � 1� �

ji; 1; 1i � iffiffiffi2p sin

ffiffiffi2p

kt2jg; 1; 0i#� iffiffiffi

2p sin

ffiffiffi2p

kt2jg0; 1; 0i)

þ 1

2cos

ffiffiffi2p

kt1 � 1� �2

(1

42 cos

ffiffiffi4p

kt2 þ 2� �

j1; 1i: 1

2cos

ffiffiffi2p

kt2 þ 1� �

ji; 1; 1i þ 1

2cos

ffiffiffi2p

kt2 � 1� �

je; 0; 0i�

� iffiffiffi2p sin

ffiffiffi2p

kt2jg; 1; 0i��

ffiffiffi6p

4cos

ffiffiffi4p

kt2 � 1� �

j2; 2i"

1

42 cos

ffiffiffi4p

kt2 þ 2� �

je; 1; 1i

þffiffiffi6p

4cos

ffiffiffi4p

kt2 � 1� �

ji; 2; 2i � iffiffiffi2p

4sin

ffiffiffi4p

kt2jg; 2; 1i#�

ffiffiffi2p

4i sin

ffiffiffi4p

kt2jg0; 2; 1; 1; 1i))

: ð17Þ

If the atom is detected in the state jii, the cavity field collapses onto the state

jW2f i ¼ N 2

1

4cos

ffiffiffi2p

kt1 þ 1� �

cosffiffiffi2p

kt2 þ 1� �

j0; 0; 0; 0i þ � 1

4cos

ffiffiffi2p

kt2 � 1� �2

��cos

ffiffiffi2p

kt1 þ 1� �

þ 1

8cos

ffiffiffi4p

kt2 þ 1� �

cosffiffiffi2p

kt2 þ 1� �

cosffiffiffi2p

kt1 � 1� �2

�j1; 1; 1; 1i

þ 3

16cos

ffiffiffi4p

kt2 � 1� �2

cosffiffiffi2p

kt1 � 1� �2

j2; 2; 2; 2i�; ð18Þ

where N2 is given by

N 2 ¼1

4cos

ffiffiffi2p

kt1 þ 1� �

cosffiffiffi2p

kt2 þ 1� �� 2

þ(

1

4cos

ffiffiffi2p

kt2 � 1� �2

cosffiffiffi2p

kt1 þ 1� ��

þ 1

8cos

ffiffiffi4p

kt2 þ 1� �

cosffiffiffi2p

kt2 þ 1� �

cosffiffiffi2p

kt1 � 1� �2

�2

þ 3

16cos

ffiffiffi4p

kt2 � 1� �2

cosffiffiffi2p

kt1 � 1� �2

� �2): ð19Þ

Suppose we repeat the procedure N times, and each time we detect the atom in the same state jii. In this way, we canconvert an initial two-mode vacuum state of the cavity field into correlated four-mode states. After the detection of the(N � 1)th atom in the state jii, the cavity field is in the state

jWN�1f i ¼ NN�1

XN�1

K¼0

CN�1K jK;K;K;Ki: ð20Þ

Suppose the Nth atom is initially prepared in the state

jWNa i ¼ jei; ð21Þ

After it interacts with the field for an interaction time tn and is detected in the state jii, the field state reads

jWNf i ¼ N N

XN

K¼0

CNK jK;K;K;Ki; ð22Þ

where NN is a normalization factor given by

Z.-R. Zhong / Optics Communications 277 (2007) 103–108 107

NN ¼XN

J¼0

XN

K¼0

CNJ ðCN

K Þ�

" #�1=2

; ð23Þ

where

CN0 ¼

1

2cos

ffiffiffi2p

ktN þ 1� �

CN�10 ; ðC0

0 ¼ 1Þ; ð24Þ

CNK ¼

K þ 1

4Kcos

ffiffiffiffiffiffi2Kp

ktN � 1� �2

CN�1K�1 �

1

4cos


ktN þ 1� �

cosffiffiffi2p

kt0N þ 1� �

CN�1K ; ð25Þ

CNN ¼ �

N þ 1

Ncos

ffiffiffiffiffiffiffi2Np

ktN � 1� �

cosffiffiffiffiffiffi2Kp

kt0N � 1� �

CN�1N�1: ð26Þ

In order to generate the state jWNf i with the desired coefficients CN

K , we have to obtain the coefficients CN�1K for the state

jWN�1f i firstly. According to Eqs. (24)–(26), we can express the unknown coefficients CN�1

K in terms of CNK by

CN�1K ¼

4KðK þ 1Þ cosffiffiffiffiffiffi2Kp

ktN � 1 2

CNK�1 � 4K2 cos


ktN þ 1

cosffiffiffi2p

ktN þ 1

CNK

K cosffiffiffiffiffiffi2Kp

ktN þ 1

cosffiffiffi2p

ktN þ 1 � �2 þ ðK þ 1Þ cos


ktN � 1 2

h i2: ð27Þ

We take jWN�1f i as a new desired state which can be obtained by sending N � 1 atoms through the cavity. For the state

jWN�1f i we do the same calculations and obtain N � 1 coefficients CN�2

K for the state jWN�2f i. We repeat the process until

the cavity field is in two-mode vacuum state, we get a series of parameters t1, t2, . . ., tN which determine the interaction timeas to obtain the desired state jWN

f i. The probability of finding all N atoms in the jii state is given by

P ¼YNn¼1

P n ¼1

ðN nÞ2; ð28Þ

where Pn is the probability to find the nth atom in the jii state.As an example, we show how to generate four-mode multiphoton maximally entangled states. According to Eq. (18), if

the interaction time is chosen asffiffiffi2p

kt1 ¼ p5;ffiffiffi2p

kt2 ¼ p2, the state reads

jWi ¼ 1ffiffiffi3p ½j0; 0; 0; 0i þ j1; 1; 1; 1i þ j2; 2; 2; 2i�; ð29Þ

here the state jWi is the four-mode multiphoton maximally entangled states. The success probability to prepare this state is0.33.

It is necessary to give a brief discussion on the experimental realization of the proposed scheme. To generate four-modemultiphoton maximally entangled states, we may consider a four-level atom with states jei, jgi, jii, and jg0i, of which theradiative lifetimes are of the order of Tat = 30 ms. In the experiment reported in Ref. [20], the photon lifetime in the cavityis T c ¼ Q

2pv � 130 ms, thus the lifetime for the two-photon state is T 2p ¼ T c

2¼ 75 ms. By using a velocity selector and apply-

ing a Stark field adjustment, we can make the atom resonant with the cavity field for the right amount of time. Settingk = 2p · 25 kHz, we have t1 ¼ p

5ffiffi2p

k¼ 3 ls and t2 ¼ p

2ffiffi2p

k¼ 7 ls in our scheme. Considering the travelling time of the atom

[21], we obtain that the total time required to complete the process is about 100 ls, which is much shorter than Tat and T2P.Therefore, there is sufficient time to complete the process to generate four-mode multiphoton maximally entangled states.

In summary, we have proposed a scheme for generating four-mode correlated states. It is based on the injection of asequence of four-level atoms into two bimodal cavities one by one and the detection of them are in the ground state.The distinct advantage of our scheme is that the interaction time can be greatly shortened, which is important in viewof decoherence. Furthermore, we show how to generate four-mode multiphoton maximally entangled states.

Acknowledgements

This work supported by National Natural Science Foundation of China Under Grant No. 10674025, and Funds fromKey Lab of Quantum Information, University of Science and Technology of China, and FuJian Department of Educationunder Grant No. JB06043.

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Generation correlated four-mode states in cavity QED