Velocity-controlled entanglement of two atoms

Cai Xun-Ming, College of Computer and Information Engineering, Guizhou

University for Nationalities Guiyang 550025 , China

Fan Meng-Hui, College of Science, Guizhou University for Nationalities

Guiyang 550025 , China

Abstract-Entanglement evolution of two atoms is related to the field mode structure in an optical resonant cavity when the atoms are moving and the optical field is in coherent state. We know the evolvement of entanglement of two atoms is disorderly when two atoms in the cavity are fixed at initial time. However, the evolvement of entanglement of two atoms becomes well-regulated orderly under the coherent cavity field when atoms are in motion at beginning. We find that it is possible to control this periodic entanglement by properly choosing the velocity of atoms and the initial field. PACS: 42.50.Ex, 03.67.Bg, 03.67.-a Key words: Periodic entanglement, Atomic evolvement, Concurrence

I. INTRODUCTION Entanglement is one of quantum effects that have no classical interpretation, which has been identified as an essential of quantum information processing and quantum computation, lies at the heart of quantum information and computation[1-4]. In recent years, most research achievements in quantum nonlocality and quantum information are based on the entanglement of two atomic qubits, especially the system in which identical two-level atoms couple with a cavity mode. Various modifications and generalizations of the system have been studied for preparing entangled state or realizing various quantum information processes[5-13]. In most of such models, atoms are deemed immobilized, that is the effect of atomic motion is neglected. However, atomic motion is a basic property, even for cold or ultracold atoms in the cavity, so it is natural to consider the effect of atomic movement. In many experiments of cavity quantum electrodynamics, atoms transit a resonant cavity along the axis of the cavity. Atomic motion is also considered in some models, such as, Schlicher investigated Jaynes-Cummings model with moving atom[14]. In this paper, we will study the entanglement property of two moving atoms in a single mode optical cavity, in which the two atoms are in different velocity and prepared in a coherent state. We will investigate the effects of atomic motion, atomic state, as well as the initial state of radiation field on the entanglement of atoms. Our aim is how to manipulate the entanglement by changing the velocity of atomic motion and the initial field.

II. MODEL Suppose that two identical two-level atoms are in a single

mode resonant cavity with different speeds along the z axes of

the cavity. In the rotating-wave approximation, the Hamiltonian of the system can be written as follows( 1= ):

11 22

1 1 1 2 2 2

( )

( )( ) ( )( )z zH a a

gf z a a gf z a a

ω ω σ σ

σ σ σ σ

+

+ − + + − +

= + +

+ + + +

, (1) Where

zi i i i ie e g gσ = − ,i i ie gσ + = ,

i i ig eσ =

( 1,2)i = are atomic operators. We consider the resonant case of the atom-cavity system, i.e., the atomic transition frequency is the same with that of the cavity mode, say ω . a+ and a are creation operator and annihilation operator for the cavity field, respectively, ( )f z is the mode-function of the cavity and assumed to

be

0

0

( ) ( ) sin( ( ))

sin( ), ( 0) ;( 1,2)

1, .

i i i i

i

pf z f v t v t z

lp v t

z for moving atomsil

for fixed atoms

π

π

= = +

= = =

,

(2) g is the coupling coefficient of the two atoms with the cavity field, p is the half-wave number of the cavity with the cavity length l, ( 1, 2)iv i = is the velocity of atoms, 0z is the initial position of atoms, which is assumed to be zero for moving

atoms and 0 2l

zp

= for fixed atoms. We suppose the state

vector of the system ( )tψ is

11 10

01 00

| ( ) [ | |

| | ]

n nn

n n

t c een c egn

c gen c ggn

ψ >= > + >

+ > + > .

(3) By Schrödinger equation, the evolution equations of four

coefficients have the following

5470978-1-61284-459-6/11/$26.00 ©2011 IEEE

forms

.

11 11 1 01 1

2 10 1.

10 1 10 1 2 11

1 00 2.

01 1 01 1 1 11

2 00 2.

00 1 00 1 1

( 1) 1

1

( 1) 1

2

( 1) 1

2

1

n n n

n

n n n

n

n n n

n

n n

c i n c igf n c

igf n c

c i n c igf n c

igf n c

c i n c igf n c

igf n c

c i nc ig n f

ω

ω

ω

ω

+

+

+ +

+

+ +

+

+ +

=− + − +

− +

=− + − +

− +

=− + − +

− +

=− − + 10

2 011n

n

c

ig n f c − +

.

(4) By the transform

( 1)11 11

10 10

01 01( 1)

00 00

( )

( )

( )

( )

i n tn n

i ntn n

i ntn n

i n tn n

c e M t

c e M t

c e M t

c e M t

ω

ω

ω

ω

− +

−

−

− −

=

=

=

=

,

we have that

.

11 2 10 1

1 01 1.

10 1 2 11

1 00 2.

01 1 1 11

2 00 2.

00 1

( ) 1 ( )

1 ( )

( ) 1 ( )

2 ( )

( ) 1 ( )

2 ( )

( )

n n

n

n n

n

n n

n

n

M t igf n M t

igf n M t

M t igf n M t

igf n M t

M t igf n M t

igf n M t

M t ig

+

+

+

+

+

+

+

= − + ⋅

− + ⋅

= − + ⋅

− + ⋅

= − + ⋅

− + ⋅

= − 1 10

2 01.

100 1 001.

010 2 001

1 ( )

1 ( )

( )

( )

n

n

f n M t

igf n M t

M t igf M

M t igf M

+ ⋅

− + ⋅

= − ⋅

= − ⋅

.

(5) The initial atomic state of the system can be assumed to be in the coherent superposition state of 1 2,g g and 1 2,e e , and the field is

in the coherent state 2-| | / 2=e

!

n

n

nn

α αα . So

the general initial state vector of the system can be written as:

n 1, 2

0

1 2

| (0)>= F [cos( /2)|g

sin( / 2) | , ] |n

i

g

e e e nϕ

ψ θ

θ

∞

=> −

> >,

(6)

then 11 n0

(0) F sin( / 2) in

nc e φθ

∞⋅

== − ,

00 n0

(0) F cos( /2)nn

c θ∞

== , 01nc (0) 0= , 10 (0) 0nc = , where

2-| | / 2nF =e

!

n

nα α

. So we can obtain 11nc , 10nc , 01nc ,

00nc by numerical calculation of equation group (5). We can educe the reduced density matrix of the two atoms from Eq.(3) . By grouping state vectors 1 1 2| ,u e e= >

2 1 2| ,u g e= > 3 1 2| ,u e g= > 4 1, 2=|gu g > , the matrix can be expressed as follows

1 2 3 4

5 6 7 812

9 10 11 12

13 14 15 16

u u u u

u u u u

u u u u

u u u u

ρ = .

Criteria of entanglement for different systems have been extensively studied[15-16]. In this paper, we can use the concurrence, defined in the reference, to discuss the characteristics of entanglement of atoms. The concurrence related to density operator of mixed states is defined as follows:

1 2 3 4max{ ,0}C λ λ λ λ= − − − , (7)

where ( 1, 2,3,4)i iλ = are the square roots of the eigenvalues in decreasing order of magnitude of the “spin-flipped” density operator ,

*

12 121 2 1 2( ) ( )y y y yR ρ σ σ ρ σ σ= ⊗ ⊗ , ( 8)

12ρ is the reduced density matrix of the two atoms. The

concurrence varies from 0c = to 1c = . For an untangled state, 0c = ; whereas 1c = expresses that the system is in the maximally entanglement state.

III. NUMERICAL RESU LTS AND THEORETICAL ANALYSIS In this section we will numerically discuss the entanglement characteristics of the system for different initial states of atoms and field. We choose the half-wave number of the ca vity

20p = and ϕ π= . At first, the velocities of atoms are

chosen as 1 / 20v gl π= and 2 / 30v gl π= . In the beginning, we consider 0θ = , which means two atoms are in the ground state 1 2,g g at beginning, then

/ 2θ π= , that is the case the two atoms are in the maximal entanglement at beginn ing. The mean photon number n of the coherent cavity field is chosen to be 0.36 for figures (a) and (d); 4 for figures (b); 36 for figures (c)

5471

Fig.1. The entanglement versus gt for two atoms that the

velocities of atoms are chosen as 1 / 20v gl π= and

2 / 30v gl π= . ϕ π= . For figures (a), (b), and (c), 0θ = . For figures (d) / 2θ π= . From Fig 1. when the two atoms move in a optical resonant cavity and the field is in a coherent state, one can see that the entanglement of atoms is periodic, and is crucially related to the initial condition of the field. The maximum of entanglement will change with different mean photon numbers when the two atoms are in the ground state at beginning , as in Fig.1(a), (b) and (c). The maximum of entanglement won’t change but the case of entanglement will be different in a period with different mean photon numbers when the two atoms are in maximal entanglement at beginning, as in Fig.1(d), It is well known that the entanglement of atoms in a cavity is also crucially related to the initial condition of the field. The entanglement of atoms may be controlled by thermal reservoir or by number state field. For the initial number state field the system may also show the periodic entanglement. However, it is not easy to obtain perfect number state of photons. We expect to manipulate the entanglement by the coherent state of cavity field, it is easier to obtain experimentally. We know the evolvement of entanglement for fixed atoms is disorderly. However, the evolvement of entanglement becomes orderly if the motion of atoms is considered. The reason stem from the field structure. From the Hamiltonian of Eq.(1), the coupling coefficient g is modulated by the function of field mode. The magnitude of g will be changed periodically when the two atoms move along the z axes of the cavity with constant speeds. So the coupling coefficients between two atoms and field will change with time in different periods 1T and 2T . The general period coupling between two atoms and field is the lease common multiple of 1T and 2T . So the period of

entanglement of two atoms is the lease common multiple of 1T

and 2T . We know that 1 2T π= and 2 3T π= by the chosen condition, then the period of entanglement of two atoms should be 6π , Fig.1.(a), (b), (c), (d) indicated that the period of entanglement of two atoms is 6π . The mean photon number n of the coherent cavity field is chosen to be 0.36 for figures (a) and (d); 4 for figures (b); 36 for figures (c)

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Fig.2. The entanglement versus gt for two atoms that the

velocities of atoms are chosen as 1 / 20v gl π= and 2 / 40v gl π= . ϕ π= . For figures (a), (b), and (c), 0θ = . For figures (d), / 2θ π= . From Fig.2, we can see that the swing of entanglement is crucially related to the mean photon number of field, and the period of entanglement of two atoms is decided by the

velocity of atoms. The velocities of atoms induced the periodic coupling of atoms with field. The periods of coupling coefficients are 1 2T π= and 2 4T π= . The period of entanglement of two atoms is the lease common multiple of

1T and 2T , that is 4π . Fig.2 shows the period is 4π .

IV. CONCLUSION We have investigated the entanglement of two moving atoms in a cavity with coherent field. The entanglement of two moving atoms is orderly when they move in the cavity. The period of entanglement depends on the velocity of atoms and the function of field mode, and is the lease common multiple of the coupling periods of atoms with field. The swing of entanglement is related to the atomic initial states, the initial field and the velocities of atoms. The maximum of entanglement fluctuates with different mean photons when the two atoms are in un-entangled states at beginning. So we can change the entanglement of two atoms by changing the velocities of atoms, the atomic initial states and field. Especially, if the initial state of two atoms is prepared in an entanglement state, it is possible to obtain the well-regulated periodically maximal entanglement for moving atoms. This is very valuable for manipulating the atomic entanglement.

References [1] M. A. Nielsen, Isaac. L. Chuang, Quantum Computatino

and Quantum Information, Cambridge Univ. Press, Cambridge 1997 .

[2] B. M. Terhal, M. M. Wolf, A. C. Doherty , Phys. Today56 (2003) 46. [3] O .Mandel, et al, Nature425 (2003) 937. [4] C. Bennett , D .DiVicenzo, Nature404 (2000) 247. [5] S. J. van Enk, J. I. Cirac, P. Zoller, Phys. Rev. Lett79 (1997) 5178. [6] C. Cabrillo, et al, Phys.Rev. A59 (1999) 1025. [7] L. M. Duan, A .Kuzmich, H. J. Kimble, ibid67 (2002) 032305. [8] M .Plenio, et al, Phys. Rev. A59 (1999) 2468. [9] E. Jane, M. B. Plenio, D .Jonathan, ibid. 65 (2002) 050302(R);

[10] D. E. Browne, M. B. Plenio, S. F. Huelga, Phys. Rev. Lett91 (2003) 067901. [11] S .Osnaghi, et al, Phys. Rev. Lett87 (2001) 037902. [12] A. S. S rensen, K .M lmer, Phys. Rev. Lett90 (2003) 127903.

[13] Shang.-Bin. Li , Jing.-Bo. Xu, Phys. Rev. A72 (2005) 022332.

[14] R. R. Schlicher, Opt. Commun70 (1989) 97. [15] A .Peris, Phys. Rev. Lett77 (1996) 1413. [16] P. Harodecki, Phys. Lett. A232 (1997) 333.

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