guang-sheng jin et al- generation of a supersinglet of three three-level atoms in cavity qed

8/3/2019 Guang-Sheng Jin et al- Generation of a supersinglet of three three-level atoms in cavity QED

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Generation of a supersinglet of three three-level atoms in cavity QED

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

Beijing 100083, People’s Republic of China

Received 12 July 2004; published 21 March 2005

We propose a scheme to generate a supersinglet of three three-level atoms in microwave cavity quantumelectrodynamics based on the resonant atom-cavity interaction. In the scheme, three three-level atoms in

suitable initial states are sequentially sent through three cavities originally prepared in their vacuum states.

After an appropriate atom-cavity interaction process, in the subsequent measurement on the third cavity field

the atoms are projected onto the desired supersinglet. The practical feasibility of this method is discussed.

DOI: 10.1103/PhysRevA.71.034307 PACS numbers: 03.67.Mn, 03.65.w, 42.50.p

Quantum entangled states are now recognized as a pow-erful tool. They can illustrate fundamental issues of quantummechanics 1 and are basic resources of quantum informa-tion processing 2. Recently, some attention has been paid to

so-called “ supersinglets,” S N

d , the entangled states of total

spin zero of N spin-d − 1 /2 particles 3,4. These antisym-metric states are N -lateral rotationally invariant. It has beenshown that some types of supersinglets are in connectionwith violations of Bell’s inequalities 5, some are used inproofs of Bell’s theorem without inequalities 6, and somecan be used to construct decoherence-free subspaces whichare robust to collective decoherence 4,7. In particular,

N -particle N -level supersinglets, S N

N , are N -lateral unitaryinvariant and can be used to solve several problems, such as“ N strangers,” “secret sharing,” and “liar detection” prob-lems, which have no classical solutions 3,4. These stateshave also been used in a scheme designed to probe a quan-tum gate that can realize an unknown unitary transformation

8.As Cabello remarked 3,4,9, in spite of many potential

applications of the S N

N states, preparing these states for N

3 would be a formidable physical challenge. A method

using photons for preparing S33,

S33 =

1

6012 − 021 − 102 + 120 + 201 − 210 ,

1

has been recently proposed by Gisin 10. Meanwhile, en-tanglement of massive particles instead of fast-escaping pho-tons is also very interesting and has been widely investi-gated. It has been shown that the Rydberg atoms that crosssuperconductive microwave cavities are an almost ideal sys-tem to study quantum entanglement. Based on this kind of system, many experiments have been performed and numer-ous proposals have been presented to generate various en-tangled qubits and to perform small-scale quantum informa-tion processing 11–13. Very recently, the generation of higher-dimensional entangled states has also been investi-gated in this system 14.

In this paper, we propose a microwave cavity QED ex-periment with Rydberg atoms in order to create the threethree-level atomic supersinglet expressed in Eq. 1. Initially,

cavities are only prepared in vacuum states. Based on theproper resonant atom-cavity interaction and sequential mea-surement on the cavity field, the atoms will be projected onto

the S33 state with high probability. Considering the experi-

mental imperfections, we also calculate the achievable fidel-

ity of this state.The schematic setup of our proposal is shown in Fig. 1 a.

Three identical -type three-level Rydberg atoms A1 , A2, and A3, relevant energy levels being shown in Fig. 1b, are se-quentially sent through three high-Q cavities denoted byC 1 , C 2, and C 3. Cavity C 1 sustains a single-mode field ex-actly resonant with 1 Ai

↔ 2 Aii = 1 , 2 , 3 transition of the

atoms. When the atoms are in C 1, only the levels 1 Aiand

2 Aiare appropriately affected and the state 0 Ai

of the at-

oms will not be affected during the atom-cavity interaction.Similarly, cavity C 2C 3 sustains a single-mode field exactly

resonant with 0 Ai↔ 1 A

i0 A

i↔ 2 A

i transition of the at-

oms. In ideal cases the temporal evolution of each atom in

the cavities is governed by the well-known Jaynes-Cummings model interaction Hamiltonian. We assume thatthe initial state of the three atoms and cavity C 1 is

FIG. 1. a Experimental apparatus. The atoms A1 , A2, and A3

are sequentially sent through three cavities C 1 , C 2, and C 3 initially

in their vacuum states. Detection atom A4 is used to project the

atoms onto the desired supersinglet. b Energy levels of the three-

level Rydberg atom with the corresponding frequencies.

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2 A1 1 A2

0 A3 0C 1

= 2100C 1, 2

where 0C idenotes a vacuum state in C i. First, we send atom

A1 into cavity C 1 and let the effective interaction time t 1

=arccos(36+82 /73) / g1, where gi represents the cou-

pling strength between the atoms and cavity C i. When A1

flies out of C 1, atom A2 is sent into it immediately and theeffective interaction time t 2 = / 2g1 is taken. After A2 flies

out, A3 flies through C 1. As A3 is in the state 0 A3, it does not

interact with C 1. The evolution of the atom+ C 1 system canbe written as

2100C 1→

t 1 36+ 82

732100C 1

− 37 − 82

731101C 1

→

t 2 36+ 82

73210 − 37 − 82

731200C 1

.

3

So cavity C 1 is decoupled from the atoms and is still in itsinitial vacuum state.

Second, we send these atoms going through cavity C 2 oneafter another. Similarly, the initial state of C 2 is a vacuumstate 0C 2

. We let atoms A1 , A2, and A3 interact with the same

single-mode field in C 2 with t 1=arccos(10+42 /17) / g2,

t 2= / 4g2 and t 3= / 2g2, respectively. This evolution can

be described by

36+ 82

73210 − 37− 82

731200C 2

→

t 1 36 + 82

732100C 2

− 18+ 42

731200C 2

+ 19 − 122

730201C 2

→

t 2 18 + 42

732100C 2

− 2001C 2− 1200C 2

+ 19 − 122

730201C 2

→

t 3

18 + 42

73

210 − 201 − 120

+ 19 − 122

730210C 2

. 4

Thus cavity C 2 and the atoms are also decoupled.Third, these atoms are sequentially sent through cavity C 3

which is also prepared in a vacuum state 0C 3. We let the

three atoms interact with the single-mode field in C 3 with theeffective interaction times t 1= / 4g3, t 2= / 4g3, and t 3

= / 2g3. This evolution can be expressed as

18+ 42

73210 − 201 − 120

+ 19 − 122

730210C 3

→

t 1 9 + 22

732100C 3

− 0101C 3− 2010C 3

+ 0011C 3− 21200C 3

+ 19 − 122

730210C 3

→

t 2 9 + 22

732100C 3

− 0101C 3− 2010C 3

+ 1

20011C 3

+ 1

20210C 3

− 1200C 3

+ 1001C 3 + 19 − 122

1460210C 3

− 0011C 3

→

t 3 9 + 22

73210 − 012 − 201 + 021

− 120 + 1020C 3+ 19 − 122

730011C 3

. 5

Then, if C 3 is measured and found in the vacuum state 0C 3,

the three atoms are projected onto the state S33 given in Eq.

1 except for an overall minus. Recently, some interestingproperties of this supersinglet have been reported in Ref.15 and it has been used in the solution of the so-called Byzantine agreement problem 3,4,16.

To detect the cavity state, we send into C 3 a fourth atom A4, with two relevant levels 0 A4

and 2 A4. Atom A4, initially

in state 0 A4, interacts with the resonant mode in C 3 for a

time / 2g

3. If A

4is detected and found still in state

0

A4,

we can conclude that cavity C 3 is in the vacuum state. Wehave optimized the atom-cavity interaction times to obtain a97.22% success probability with a fidelity of 1 in ideal cases.We note that, even if C 3 is not detected, we can also obtain

S33 with a fidelity of more than 97%. Three single-mode

cavities are necessary to generate this state in our experimen-tal setup.

Based mainly on Rydberg atom microwave cavity QEDexperiments performed at ENS 11, we now discuss thepractical feasibility of this proposal. Our scheme requires anexperimental configuration with three cavities, which can beconsidered as a natural development of the present configu-rations where only one cavity is available. The circular Ry-dberg atom’s lifetime 30 ms is much longer than the pro-tocol duration and is not bound to be a limiting factor. Themain cause of decoherence in the present setup is the cavitymode relaxation. To reduce the times for storing the quantuminformation in the cavities, we let each atom enter the cavityimmediately after the preceding one has left it. The atom-cavity coupling constant is position-dependent and is about2 25 kHz at cavity center. Thus, the quantum information

storage time in each cavity is of order 0.1 ms, which issmaller than the photon storage time of the microwave cavity1 ms in recent experiments. Decoherence effects due to the

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loss of a photon should then be relatively small. The inter-action time of the atoms with the cavities can be controlled

by using a velocity selector and applying Stark field adjust-ment in the cavities. The length LC iof the superconducting

cavities is on the order of centimeters. Under the assumptionof g1 LC 1

= g2 LC 2= g3 LC 3

=100 m/s, the velocity of the injec-

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

In Fig. 2 are shown the results of a numerical simulationto estimate the achievable fidelity of the produced state. Con-sidering the dissipation in the cavities, we describe the wholesystem by an appropriate master equation 17. We also takeinto consideration the presence of fluctuations in the atom-cavity interaction times, which leads to the imperfections in

the quantum Rabi pulses 18. The fidelity is plotted for vari-

ous strengths of imperfections in the Rabi pulses. For a

strength of 3%, i.e., for the achievable precision in the cur-

rent setting, and for the cavity lifetime of 1 ms, the fidelity of

the resulting state is only about 75%. However, we can see

that a few improvements on the cavity lifetime as well as the

precision of the pulses will result in a fidelity of more than90%.

Note that our scheme requires a perfect single atom“gun”, which is able to prepare the atoms via a deterministicbut not a Poissonian process. Otherwise, a very long acqui-sition time may be needed. With the development of currentexperimental techniques in cavity QED, we expect that itcould be accomplished in the near future. Achievable detec-tion efficiency of Rydberg atomic states is 70% 12; thisonly reduces the success probability of our scheme, however,it does not decrease the fidelity. In the above discussion wedo not include imperfections, such as stray thermal and staticelectric fields, detection errors, etc., which could be stronglysuppressed by a revised experimental setup now in construc-tion.

In summary, we have presented an effective method togenerate the supersinglet of three three-level atoms with highprobability. This state has potential applications in fulfillingthose tasks which are impossible using methods in classicphysics. The present scheme demonstrates the power of cav-ity QED to generate and manipulate complex entangledstates for quantum information processing and could be re-alized in the near future based on fast developing techniques.

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

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FIG. 2. Dependence of fidelity on pulse imperfections and cav-

ity relaxation time T r .

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guang-sheng jin et al- generation of a supersinglet of three three-level atoms in cavity qed

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