Steady-state bipartite entanglement supported by a squeezed environment

Lnsi,r Physirs. H:1/. {5.

N,,.

5, 2005, pp. 751-7J}_ Oiig111i1l 1i:r1C,,11p'iglit t; 2()()5 f,_v tn1m, fJtl.

Cop)rig/11 © :!OO.i /iv MMK "N,mk<l!hlf<'.T/1.:rimlii:ri" (Russia).

MODERN TRENDS

= = = = = = = = = = = · I N

LASER PHYSICS

Steady-State Bipartite Entanglement Supported

by

a Squeezed Environment

M.A. Can

1,

0.

<;akir

2

_{,A. C. Giinhan:?,A. A. Klyachko}

_1,

_{N. ~.Pak}

_2,

_and-A.S.

_Shumovsky

1

1 _Faculty

of

_{Science, Bilkent University; Bilkent,Ankan;;"06800Tu rkey} 1 _{Department of Php·ics, M iddle~East Technical University, Ankara 065~ l Ttu*ey} ccmail: can@fen;bilkent;edu.tr: cakir<i!) f en.bilkcnL.cdu.tr; agunhan@IieWton.physics.rnctu.edu. tr:

klyachko@fen.bilkenU ... 'rlu.tr; pak@rrictu,cdu.tr. Shumii@fen.bi !kent.ed\!.!r Received November 23, 2004

Abstract,-We examine tht; entanglement of two atoms induced by an external quantum system. Both two~levcl atoms are coupled to a third.two-level system by dipole interaction. arid the third system

is

under lhe inOucnce . ofa certain environment. We examine different lypes of environments and show that. Lhe s(eady~state bipartite entanglement can be achieved. We show thata strongly fluctuating environment is more preferable because the transmission ofquantum fluctuations to the-atomic system creates and suppOrL,; Steady-state cntariglemeiJ.t with

a higher amount of concurrence than a non!] uctuating environmcm or an environment with strong dassicalth.ic~ tu::itions.

It is well known that entanglementis a key resource for quanrurn information processing and quantum com--puting [l].The practicalapplications do not d~mand an arbitrary entang1ementbut a robust one. Ttlis assumes sufficiently long lifetimes for the entangled states and the max.imum amount of entanglement.

The maximum (perfect) entangled state corresponds to a pure state of a system and

.can

be associated With the maximum amount of quantum fluctuations in this state [2]. In particular,

it was

Shown that maximum entanglement can be defined as the manifestation of

quantum fluctuations at their extreme [3]. It should be Stressed that all entangled states are equivalent to the

maxirnum

entangled states to within a certain local transformation, such as the stochastic local transforma-tions assisted by classical communications [ 4] and Lorentz tmilsforrnations:[5].

The

ma:in

idea of the approach developed in [3, 6]

consistin the defin.ition ofa fundamental set of observ-ables {Oi

J

specifying a given system Sin Hilbert space

(IHls) and

in

the calculation of the total amount of fluc-tuations of these observables.

The fundamental observables can be associated with .·the dynamic symmetry group Gin 1His[6, 7]. Namely; the fundamental observables form a basis ofthe Lie , algebra;£ such that G = exp(ie). Since the observables shoi.1ld be represented by Hermitian operators, some-· .. times it is necessary to use the complexified Lie algebra

';;/:,<'

=

9:, ®

C

instead of;£. An ex.ample_ is provided by

the dynamical symmetry group SU(2) of Hilpert space '_when the observables are n~presented

by

the spin

oper-" ators, forming an infinitesimal representation of the

SL(2, C) algebra, which is known

as

the

complexifica-ition oftheSU(2) algebra.

The quantum fluctuation of an observable 0, in a state \JI E IHlsisrepresented.by the variance

...

,

s· . ') ~'

Vi<w>

=

('l'I0]\11)- <w10,1"'r.

Then, the total amount of quantum fhJctuations in a given state takes the form

··\.

(1) where summation is over all fundamental observables of the system. This q~aotity

is

similar to the so~called skew information that has been introduced by Wigner [8] as a measure of know ledge with respect to the phys~

ical quantities whose measurement requires the use:;of macroscopic apparatuses.

.8y

definition [3], the maximum entilnglement in the system S corresponds to. the maximum of the total amount of quantum fluctpations (I):

V(\JIMd

=

max

V101(\V).

lj/ I; -~.I' .

(2} To stabilize the maximum entangled state, it is neces-sary to prepare the system Sin a state that obeys condi- ·

tion {2) together with the condition of a minimum (or at least a focal minimum) of energy [9, lO]. This cail fie achieved by setting the system S to interacrwith a cer-. tain dissipative environmentcer-.

Unfortunately, in many cases; the dissipation leads the system S to evolve toward a mixed state. Such a

mixed state can never manifest maximum entangle~

ment.

The existence of entanglement in the mixed state

can be caused eitherby the properties of the system itself (the presence of strongly entangled eigenstates) [ 1 J] or by the properties of the environment.

In this paper, we show that a system With little poteniial ability for entanglement (with · a low level of 751

752 CAN et al.

BATH

i \

..

g. .. ' g

(--(~.::~.·~~-;-~-,\

, ... , .. • ~~g • _.r ... ~,'

---y-~---·-·.

--·--(ff

Fig. 1. Two target atoms with decny rates rare in interac-tion with a source atom or decay ratcT0-thath driven by a squeezed vacuum.

quantum fluctuations) can be prepared in a steady entangled state via inlemction with a strongly fluctuat~

ing environtnent.

Let us consider a system of three atoms S\tch that

one of the atomsis connected to the others by a

dipole-di pole interaction. This atom, occupying the central position in the arrangement, can be put on the tip of a scanning tunneling microscope {STM) and, in this way,

connected with a ''bath" providing decoherence in the system (see Fig. 1 ).

The tot.al Hamiltonian has the form

2 H =

!

_{2£.J .}

"ro,cr~

_{J .}} )"'-0 (.·

~

.+ -

a.

+ - . ) + g ~

crocr

r+

2gcr,

<>2

+

h.c. , r"' J {3)

where CO; denotes the atomic transition frequency; g is

the coupling constant of interaction of the "source" atom, labeled by the subscript 0, with the "target'' atoms;

a/2 gives the relative strength ofinteraction between the "target" atoms; and

a

denotes the atomic operators. For simplicity, we assume that the target atoms. have

sym-metric positions with respect to the source atom and. that all atoms have the same frequency ro. We also assume the Lamb-Dicke limit of short interatomic d i~tances.

One can easily see that the system defined by Hamiltonian (3)

has no

entangled eigenstates. Thus, the system on its own does not carry the potential to evolve into an entangled state. This means that entanglement in the system under consideration can be generated orily through interaction with a

propc:rdissipativeenvi-ronment. It follows from our approach [3, 6, 7, 9-11]

that the proper environment shcmld manifest a high level of quantum fhictuatiolls. To illustrate this state-ment, we compare entanglement of .target atoms pro-vided

by

the thermal environment, with

a

low

level

of

quantum fluctuations, to entanglement generated

by

the squeezed vacuum, with a high amount of quantum fluc-tmttions.

The evolution of the density matrix. p of the system is described bythe equation

p

= -

i[H,

pJ

+

$(p), (4)

where the forrriit'>f the Liouvmean term depends on the specification ofthe "bath."

brthe case ofa thermal "buth" acting on the source atom, we have

5£

=

;;£,hcrmal

~(){(ii+

l)(2cr;;paJ - crc}cr;;p - pcr;cr;,')

+

ii(2crJ pcr;i - cr~cr;p- p

CTo0"6)} (5)

j.k"' I

whereTj = tji denotes the spontaneotis decay rate of the

j

th atom,

r

12

=

r

21 is the collective emission rate of the

target atoms,. and

n

is the average number of "bath" excitations.

Inthe case of a squeezed vacuum state,

we

get

·~~

.,

' . . ..

(J) co . +. +

--,w,,

- . -

.2,ro,1

..L- == .,;z,ii,ermri1-m(OoPO'oe +O"oPO'oe ),

(6)

where 2ro~ denotes the frequency of the squeezed mode

and m specifies the amount of squeezing.

Since we are looking for robustentangleinent ofthe target atoms, we restrict our consideration to the

steady-state solutions of Eq. (4). The density matri'I. p

is defined in the eight-dimensional Hilbert space of;Jhe

three two-level atoms. As the measure of entanglern~nt of the mixed state p we calculate the concurrence [12]. The numerical calculations performed in the case of a thennal environment with the Liouvillean term of fonn (5) give the rnaximtlm value of concurrence on the

r

order of C

=

1.15 X I0-8 _{at fi = I,}

rg

_{= 1.5,}

n

_{= 0.5,}

0 0

a= 3, which is next to nothing. Thus, the thermal envi-ronment cannot generate entanglement in the system

under consideration. ·

The case of a squeezed vacuum state shows a much

more interesting dependence of concurrence on tlle parameters of the system shown in Fig. 2. In particular, the maximum value .of concurrence, C = 0.227, is achieved at u

=

5 and increases with ii. This level of

entanglement corresponds to ihat usually discussed in connection with practical applications. ·

Thus, we have shown that steady-slate entanglement Wilh a reasonable amount of concurrence can be induced in a system that has a low ability to evolve into an entangled state through interaction with a strongly

STEADY-STATE BIPARTITE ENTANGLEMENT

753

Concurrence· 0.20 O.l5. 0.10.· 0.05 0 4 {l 60.

.n.

B ₈₀

Fig.,'2. Concurrence as a function ofq and ii for

rtrcr=

l and gtr0. = 1.5. The figure indi_catcs that~he m~imum iimouni Of conci.Jrrence occurs at a Cl!:rlain value of

a

and incri,:ascs s.lowly with n .

fluctuating quantum environmerit.. This approach can

be. used for investig,ition Qf eritanglemerttjn systems e>f

different

physical

natures. For example, one can expect that the

use

of iloriidenticat

target

atoms

can

lead to an

increase ofcoricurrence-underthe. influence of the

same

environment as. compared to the use-ofidentfoal

ones.

LASER PHYSICS Vol. l S· No. 5 2005

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