Improving the entanglement transfer from continuous variable systems to localized qubits using non Gaussian states

0.46 MB·English

by  Federico Casagrande

Improving the entanglement transfer from continuous variable systems to localized
qubits using non Gaussian states
Federico Casagrande, Alfredo Lulli, and Matteo G.A. Paris
∗ † ‡
Dipartimento di Fisica dell’Universit`a di Milano, Italia
(Dated: February 1, 2008)
We investigate the entanglement transfer from a bipartite continuous-variable (CV) system to a
pair of localized qubits assuming that each CV mode couples to one qubit via the off-resonance
Jaynes-Cummings interaction with different interaction times for the two subsystems. First, we
consider thecase of theCV system prepared in a Bell-like superposition and investigate thecondi-
tionsformaximumentanglementtransfer. Thenweanalyzethegeneralcaseoftwo-modeCVstates
that can be represented bya Schmidtdecomposition in theFock numberbasis. This class includes
both Gaussian and non Gaussian CV states, as for example twin-beam (TWB) and pair-coherent
7 (TMC, also known as two-mode-coherent) states respectively. Under resonance conditions, equal
0 interaction times for both qubits and different initial preparations, we find that the entanglement
0 transfer is more efficient for TMC than for TWB states. In the perspective of applications such
2
as in cavity QED or with superconducting qubits, we analyze in details the effects of off-resonance

n interactions(detuning)and differentinteraction timesfor thetwoqubits,anddiscuss conditionsto
a preservetheentanglement transfer.
J
2 PACSnumbers: 03.67.Mn,42.50.Pq
1

2 I. INTRODUCTION partite continuous variable systems and a pair of qubits
v hasbeenfirstlyanalyzedin[12]wherethe CVfieldisas-
2 sumed to be a two-mode squeezedvacuum ortwin-beam
7 Entanglement is the main resource of quantum infor- (TWB) state [13] with the two modes injected into spa-
1
mation processing (QIP). Indeed, much attention has tiallyseparatecavities. Twoidenticalatoms,bothinthe
2
been devoted to generation and manipulation of entan- ground state, are then assumed to interact resonantly,
1
6 glementeither in discreteor incontinuousvariable (CV) one for each cavity, with the cavity mode field for an
0 systems. Crucialandrewardingstepsinthedevelopment interaction time shorter than the cavity lifetime. More
/ of QIP are now the storage of entanglement in quan- recently, a general approach has been developed [6], in
h
tum memories [1, 2] and the transfer of entanglement which two static qubits are isolated by the real world
p
- from localized to flying registers and viceversa. Indeed, bytheir ownsinglemode bosoniclocalenvironmentthat
t effective protocols for the distribution of entanglement also rules the interaction of each qubit with an exter-
n
a would allow one to realize quantum cryptography over nal driving field assumed to be a generalbroadband two
u long distances [3], as well as distributed quantum com- mode field. This model may be applied to describe a
q putation [4] and distributed network for quantum com- cavity QED setup with two atomic qubits trapped into
v: munication purposes. remotecavities. InRef.[7]theproblemsrelatedtodiffer-
i Few schemes have been suggested either to entangle ent interaction times for the two qubits are pointed out,
X
localized qubits, e.g. distant atoms or superconduct- eitherforatomicqubitsorinthecaseofsuperconducting
r ing quantum interference devices, using squeezed radi- quantum interference devices (SQUID) qubits. The pos-
a
ation[5] or to transferentanglement between qubits and sibility to transfer the entanglementof a TWB radiation
radiation [6, 7, 8]. As a matter of fact, efficient sources field to SQUIDS has been also investigated in [8].
of entanglement have been developed for CV systems, Very recently, in [14] the entanglement transfer pro-
especially by quantum-optical implementations [9]. In- cess between CV and qubit bipartite systems was inves-
deed, multiphoton states might be optimal when con- tigated. Their scheme is composed by two atoms placed
sidering long distance communication, where they may into two spatially separated identical cavities where the
travel through free space or optical fibers, in view of the two modes are injected. They consider resonant inter-
robustness of their entanglement against losses [10]. action of two-mode fields, such as two-photon superpo-
The entanglementtransferfrom free propagatinglight sitions, entangled coherent states and TWB, discussing
to atomic systems has been achieved experimentally in conditions for maximum entanglement transfer.
the recent years [1, 11]. From the theoretical point of The inverse problem of entanglement reciprocation
view, the resonant entanglement transfer between a bi- from qubits to continuous variables has been discussed
in [15] by means of a model involving two atoms pre-
pared in a maximally entangled state and then injected
intotwospatiallyseparatedcavitieseachonepreparedin
∗Electronicaddress: [email protected] acoherentstate. Itwasshownthatwhentheatomsleave
†Electronicaddress: [email protected] the cavity their entanglement is transferred to the post
‡Electronicaddress: matteo.paris@fisica.unimi.it
2
selected cavity fields. The generated field entanglement basis:
can be then transferred back to qubits, i.e to another
∞
couple of atoms flying through the cavities. In a recent x = c (x)nn (1)
paper[16]therelationshipbetweenentanglement,mixed- | i X n | i
n=0
ness and energy of two qubits and two mode Gaussian
where nn = n n and the complex coefficients
quantumstateshasbeenanalyzed,whereasastrategyto
| i | i ⊗ | i
c (x) = nnx satisfy the normalization condition
enhance the entanglement transfer between TWB states n
and multiple qubits has been suggested in [17]. P∞n=0|cn(xh)|2 |=i1. The parameter x is a complex vari-
ablethatfully characterizesthe state ofthe field. Notice
In this paper we investigate the dynamics of a two-
that a scheme for the generation of any two-mode cor-
mode entangled state of radiation coupled to a pair of
related photon number states of the form (1) has been
localized qubits via the off-resonance Jaynes-Cummings
recently proposed [20]. The simplest example within the
interaction. We focus our attention on the entangle-
class(1)isgivenbytheBell-liketwo-modesuperposition
ment transfer from radiation to atomic qubits, though
(TSS):
our analysis may be employed also to describe the effec-
tiveinteractionofradiationwithsuperconductingqubits.
x =c 00 +c 11 . (2)
0 1
Inparticular,comparedtopreviousanalysis,weconsider | i | i | i
indetailsthe effects ofoff-resonanceinteractions(detun- Eq. (1) also describes relevant bipartite states, as for
ing)anddifferentinteractiontimesforthetwoqubits. As example TWB and TMC states. In these cases we can
a carrierofentanglement we consider the generalcase of rewrite the coefficients as c (x)=c (x)f (x), where:
n 0 n
two-mode states that can be represented by a Schmidt
decomposition in the Fock number basis. These include TWB: c0(x)=p1−|x|2 fn(x)=xn (3)
Gaussianstates ofradiationlike twin-beams, realizedby 1 xn
TMC: c (x)= f (x)= , (4)
nondegenerate parametric amplifiers by means of spon- 0 I (2x) n n!
taneous downconversion in nonlinear crystals, as well as p 0 | |
nonGaussianstates,asforexamplepair-coherent(TMC, where I (y) denotes the 0-th modified Bessel function
0
alsoknownastwo-modecoherent)states[18],thatcanbe of the first kind. For TWB states the parameter x
| |
obtained either by degenerate Raman processes [19] or, is related to the squeezing parameter, and ranges from
more realistically, by conditional measurements [20] and 0 (no squeezing) to 1 (infinite squeezing). For TMC
nondegenerateparametricoscillators[21,22]. Infact,we states x is related to the squared field amplitude and
| |
find that TMC are more effective in transferring entan- can take any positive values. The bipartite states de-
glementto qubitsthanTWB statesandthis opens novel scribed by (1) show perfect photon number correlations.
perspectives on the use of non Gaussian states in quan- Thejointphotonnumberdistributionhasindeedthesim-
tum information processing. ple form P (x) = δ c (x)2. For the TSS states the
nk nk n
The paper is organized as follows: in the next Section joint photon distributi|on is |given by P = c 2 and
00 0
| |
we introduce the Hamiltonian model we are goingto an- P = 1 P , whereas for TWB and TMC it can be
11 00
alyze for entanglement transfer, as well as the different written a−s P (x) = δ P (x)f (x)2. As we will see
nk nk 00 n
| |
kind of two-mode CV states that provide the source of in the following the photon distribution plays a funda-
entanglement. In Section III we consider resonant en- mental role in understanding the entanglement transfer
tanglement transfer, which is assessed by evaluating the process.
entanglement of formation for the reduced density ma- The average number of photon of the states x , i.e.
| i
trix of the qubits after a given interaction time. In Sec- N (x) = xa a+b bx , a and b being the field mode
† †
h i h | | i
tions IV and V we analyze in some details the effects operators, is related to the dimensionless parameter x
| |
ofdetuning andofdifferent interactiontimes forthe two by
qubits. SectionVIclosesthepaperwithsomeconcluding
x2
remarks. TWB: N (x)=2 | | (5)
h i 1 x2
−| |
2xI (2x)
TMC: N (x)= | | 1 | | (6)
h i I (2x)
0
II. THE HAMILTONIAN MODEL | |
where I (y) denotes the 1-th modified Bessel function of
1
the first kind. In Fig.1 we show the first four terms of
We address the entanglement transfer from a bipar-
the photon distribution for TMC and TWB states re-
tite CV field to a pair of localized qubits assuming that
spectively as functions of the mean photon number.
eachCVmodecouplestoonequbitviatheoff-resonance
The states in Eq.(1) arepure states, andtherefore we
Jaynes-Cummings interaction (as it happens by inject-
canevaluatetheirentanglementbytheVonNeumannen-
ing the two modes in two separate cavities). We allow
tropy S (x) of the reduced density matrix of each sub-
vn
fordifferentinteractiontimesforthetwosubsystemsand
system. For the TSS case we simply have:
assume [14] that the initial state of the two modes is de-
scribed by a Schmidt decomposition in the Fock number S = P log P (1 P )log (1 P ) (7)
vn − 00 2 00− − 00 2 − 00
3
1 1 evolves by means of the unitary operator U(τ) =
Pnn P00 Pnn P00 etwxpo[−off~i-Hreisτo]natnhcaetJcaaynnebse-Cfuamctmoriinzgesdeavsoluthtieonproopdeuracttoorsf
P11
0.5 P22 P33 0.5 P11 tUeAm(.τ)FaonrdtUheB(iτn)it[i2a4l]srtealtaeteodftbooethachataotmoms -wmeocdoenssuidbesryesd-
P22 a generalsuperposition of their excited (2 ) and ground
0 0 P33 (1 ) states: | i
a) 0 5 <N> 10 b) 0 5 <N> 10 | i
ψ(0) = A 2 +A 1
FIG. 1: The first four terms Pnn, n = 0,...,3 of the joint | iA 2| iA 1| iA
photon distribution of the state |xi as a function of average ψ(0) B = B2 2 B+B1 1 B (11)
| i | i | i
photon numberhNi. (a): TMC, (b): TWB.
where A 2 + A 2 = 1 and B 2 + B 2 = 1. This
2 1 2 1
| | | | | | | |
includes the most naturalandwidely investigatedchoice
and of course the maximum value of 1 is obtained for ofbothatomsinthegroundstates,butwillalsoallowus
P00 =P11 = 21. Thecorrespondingstate|Φ+i= |00i√+2|11i to investigate the effect of different interaction times.
isaBell-likemaximallyentangledstate. ForTWBstates Due tothe linearityofevolutionoperatorU(τ) andits
the Von Neumann Entropy can be written as: factorized form, the whole system state ψ(τ) at a time
| i
τ can be written as
2x2
S (x)= log (1 x2) | | log x (8)
vn − 2 −| | − 1 x2 2| | ψ(τ) =
−| | | i
whereas for TMC states we use the general expression: =c (x) ∞ f (x)U (τ)ψn(0) U (τ)ψn(0)
0 X n A | iA⊗ B | iB
S (x) = log P (x) n=0
vn − 2 00 − (12)
∞
P (x) f (x)2log f (x)2 (9)
− 00 X| i | 2| i | where ψn(0) = ψ(0) n . In each
A,B A,B A,B
i=1 | i | i ⊗ | i
of two atom-field subspaces A and B we expand the
Itisclearthatthe VonNeumannentropydivergesinthe wave-function on the basis 2 k , 1 k + 1
limit x 1 (for TWB) and in the limit x (for 0 1 . The coefficients c {|(0i)| =i | i|2 k ψi}n∞k(=00) ∪
TMC)| b|e→cause the probability P00(x) vanis|he|s→. T∞he VN {a|ndi|ciB},1,k(0) =B 2B k ψAn,1(0,k) B ofAthhe|Ainhit|i|al statieAs
entropy at fixed energy (average number of photons of are: h | h || i
the two modes) is maximized by the TWB expression
c (0)=A δ c (0)=B δ
(8). For this reason TWB states are also referred to as A,1,0 1 n,0 B,1,0 1 n,0
c (0)=A δ c (0)=B δ (13)
maximal entangled states of bipartite CV systems. A,2,k 2 k,n B,2,k 2 k,n
c (0)=A δ c (0)=B δ
We consider the interaction of each radiation mode A,1,k+1 1 k+1,n B,1,k+1 1 k+1,n
with a two level atom flying through the cavity. If the
The Jaynes-Cummingsinteractioncouplesonlythe coef-
interaction time is much shorter than the lifetime of the
ficients of each variety K whereas c (0), c (0) do
cavity mode and the atomic decay rates, we can neglect A,1,0 B,1,0
not evolve. Therefore, for each variety in the subspaces
dissipation in system dynamics. On the other hand, we
A and B the evolved coefficients can be obtained by ap-
considerthe generalcase ofatoms with different interac-
plying the off-resonance Jaynes-Cummings 2 2 matrix
tion times and coupling constants, prepared in superpo- ×
U so that:
jk
sition states, and off-resonance interaction between each
atomandtherelativecavitymode. Allthesefeaturescan c (τ) = U (k,τ)c (0)+U (k,τ)c (0)
2,k 11 2,k 12 1,k+1
be quite important in practical implementations such as
c (τ) = U (k,τ)c (0)+U (k,τ)c (0)
in cavity QED systems with Rydberg atoms and high-Q 1,k+1 21 2,k 22 1,k+1
microwave cavities [23], as noticed in [7]. In the inter- where
action picture, the interaction Hamiltonian H is given
i
R τ ∆ R τ
by: U (k,τ) = cos( k ) i sin( k )
11 2 − R 2
k
H = ~∆ a a ~∆ b b+
i − A † − B † 2ig√k+1 Rkτ
+ ~gA[a†SA12,−+aSA12,+]+~gB[b†SB12,−+bSB12,+] U12(k,τ) = − Rk sin( 2 )=U21(k,τ),
(10) R τ ∆ R τ
k k
U (k,τ) = cos( )+i sin( )
whereS12 andS12 aretheloweringandraisingatomic 22 2 Rk 2
A, B,
operators±of the tw±o atoms and ∆ , ∆ denote the (14)
A B
detunings between each mode frequency and the corre-
where the generalized Rabi frequencies are R =
sponding atomic transition frequency. The initial state k
4g2(k+1)+∆2. To derive the evolved atomic den-
of the whole system p
sity operator ρ1,2 we first consider the statistical opera-
a
ψ(0) = x ψ(0) ψ(0) tor of the whole system ρ(τ) = ψ(τ) ψ(τ) and then
A B
| i | i⊗| i ⊗| i | ih| |
4
we trace out the field variables. The explicit expres- In the case of both qubits initially in the groundstate
sions of the density matrix elements in the standard ba- 1 1 the expression of λPT simply reduces to ρ
| iA| iB 4 22 −
sis 2 2 , 2 1 , 1 2 , 1 1 are reported ρ , because ρ = ρ , and it is possible to derive the
A B A B A B A B 14 22 33
{| i | i | i | i | i | i | i | i } | |
in Appendix A. following simple formula:
λPT(P ,gτ)=sin2(gτ)
4 00 ×
III. ENTANGLEMENT TRANSFER AT [(1 P )cos2(gτ) (1 P )P ] (19)
× − 00 −p − 00 00
RESONANCE
We note that only the vacuum Rabi frequency is in-
volved, a fact that greatly simplifies the analysis of
Asafirstexampleweconsiderexactresonanceforboth
atom-field interaction compared to all the other atomic
atom-fieldinteractions,equalcouplingconstantgandthe
configurations. Let us first consider the Bell-like state
same interaction time τ. For the initial atomic states we
(P = 1) and look for the gτ values maximizing the
will discuss the following three cases: both atoms in the 00 2
entanglement of the two atoms. The solution of equa-
ground state (1 1 ), both atoms in the excited state
| iA| iB tion λPT(1,gτ) = 1 is given by gτ = π(2k+1) with
(|2iA|2iB), and one atom in the excited state and the k = 04,1,22,.... The−a2bove condition is r2elevant also to
other one in the ground state (1 2 ). In all these
casesthe atomicdensitymatrixa|ftieAr|thieBinteractionρ1,2 explain the entanglement transfer for TWB and TMC
a states,asdiscussedbelow. Toevaluatethe entanglement
has the following form:
transfer also for not maximally entangled TSS states in
Eq. (2), we calculate the entanglement of formation as
ρ 0 0 ρ
11 14
 0 ρ 0 0  a function of both the dimensionless interaction time gτ
ρ1,2 = 22 (15) and the probability P . As it is apparent from Fig. 2a
a 0 0 ρ 0 00
 33  therearelargeandwelldefinedregionswhereǫ >0. In
 ρ 0 0 ρ  F
∗14 44 particular, the absolute maxima (ǫ = 1) occur exactly
F
at P = 0.5 and for gτ values in agreement with the
The presence of the qubit entanglement can be revealed 00
above series. In addition, if we consider the sections at
by the Peres-Horodecki criterion [25] based on the exis-
thesegτ values,weobtainexactcoincidencewiththeVon
tence of negative eigenvalues of the partial transpose of
Neumann Entropy function S (P ). Therefore, com-
Eq. (15). From the expressions of the eigenvalues vn 00
plete entanglement transfer from the field to the atoms
λPT =ρ λPT =ρ is possible not only for the Bell State, though only for
1 44 2 11
the Bell State we may obtain the transferral of 1 ebit.
ρ +ρ (ρ ρ )2+4ρ 2
λPT = 22 33±p 22− 33 | 14| (16) In Fig. 2b we consider the entanglement of formation vs
3,4 2 gτ and mean photon number N in the TWB case. We
h i
firstnotethattheregionsofmaximumentanglementcor-
we see that only λPT can assume negative values. In
4 respondto those of TSS states and the maxima occur at
the case of TSS the expression of λP4T can allow us to gτ values close to π(2k+1), as shown in Table I. We
derive in a simple way analytical results for the condi- 2
tions of maximum entanglement transfer as function of
dimensionlessinteractiontimegτ,aswellastobetterun- gτmax hNimax ǫF,max P00 P11
derstandtheresultsinthecaseofTWBandTMCstates. 1.56 0.87 0.64 0.69 0.21
Inordertoquantifytheamountoftheentanglementand, 4.61 1.82 0.81 0.52 0.25
inturn, to assessthe entanglementtransferwe chooseto 7.85 1.07 0.68 0.65 0.23
adopttheentanglementofformationǫ [26]. Werewrite
F 11.03 1.07 0.68 0.65 0.23
the atomic density matrix in the magic basis [27] ρMB
a
andweevaluatetheeigenvaluesofthenon-hermitianma- TABLEI:MaximaofthequbitentanglementofformationǫF
trix R=ρMa B(ρMa B)∗: for the resonant interaction with TWB and for both qubits
initially in theground state (see Fig. 2b).
λR =ρ ρ λR =(√ρ ρ ρ )2 (17)
1,2 22 33 3,4 11 44±| 14|
can explain this by considering the TWB photon distri-
In this way we calculate the concurrence [28] C = bution (see Fig. 1b). We note that the terms P and
00
max 0,Λ Λ Λ Λ , where Λ = λR are the P arealwaysgreaterorequalthantheothertermsP
squa{re ro1ot−s of2t−he 3ei−genv4a}lues λR seilectepd ini the de- (n11> 1) and that for N < 2 they dominate the phnon-
i h i
creasing order, and then evaluate the entanglement of ton distribution (P +P 1). Therefore, the main
00 11
≃
formation: contribution to entanglement transfer is obtained from
the above two terms as for the TSS state. In order to
1 √1 C2 1 √1 C2 explainthe absolute maximum found inthe secondpeak
ǫ = − − log − −
F − 2 2 2 at N = 1.82, we note that in this case P and P
00 11
h i
1+√1 C2 1+√1 C2 are closer to the value 0.5 of a Bell State. In addition,
− 2− log2 2− (18) for large hNi and gτ values, there are small regions (not
5
P . The absolute maximum is in the second peak at
11
N = 1.09 because P and P are even closer to the
00 11
1 hBelilState thanintheotherpeaks,andthis alsoexplains
ε
F the largerentanglementvalue ǫF. For N >4 andlarge
h i
0.5 gτ there are regions with considerable entanglement val-
ues, due to the fact that P and P are always smaller
00 11
0 than the other terms Pnn (n > 1) that dominate the
12 1 atom-field interaction.We note that the maxima of ǫF
gτ 6 0.5 P are higher than in the TWB case.
a) 0 0 00
1.5 1
ε 3 4
F
2
0.75
1
0.5
0.5 1
(1) 0.25
(2)
0 (3) (4) 0
a) 0 10 20 30 40gτ50 b) 0 0.5 P00 1
b) FIG. 3: (a): the function λP4T(12,gτ)+ 12 as in Eq. (20) for
theBell statecase with both atoms in theexcited state. (b):
entanglementofformationǫF vsP00 forTSSstatescompared
toVonNeumannentropy(dashedline)forsomevaluesofgτ,
corresponding to the numbered minima: (1) 2.03, (2) 4.53,
(3) 11.07, (4) 26.68.
A similar analysis can be done in the case of initially
excitedatoms 2 2 . Forthe TSSstateswecanagain
A B
| i | i
write a simple equation for the eigenvalue of the partial
transpose:
c)
λPT(P ,gτ)=(1 P )sin2(√2gτ)cos2(√2gτ)+
FIG. 2: Entanglement of formation ǫF of the qubit systems 4 00 − 00
as a function of the dimensionless time gτ and the CV state +sin2(gτ)[P cos2(gτ) (1 P )P cos2(√2gτ)]
parameterP00(a)ortheaveragenumberofphotonshNi(b,c) 00 −p − 00 00
(20)
for the case of both atoms initially in the ground state. a)
TSS, b) TWB, c) TMC.
where, with respect to Eq. (19), an additional frequency
is present. For the Bell State we can look for gτ values
maximizing the entanglement transfer. In this case the
visible in the figure) where entanglement transfer is pos-
problemcanbe solvednumerically andwe found, for ex-
sible. ThisisduetothetermsP (n>1)inthephoton
nn ampleintherangegτ =0 50,thatonlyforgτ =26.68
distribution. In Fig. 2c we show the TMC case and we 17π we can solve the equ−ation λPT(1,gτ) = 1 with≃a
note that for N < 4 there are four well defined peaks 2 4 2 −2
h i good approximation as shown in Fig. 3a. In Fig. 3b we
wherethe entanglementis higherthaninthe TWB case.
consideralsononmaximallyentangledTSSstates,show-
Also in this case the gτ values of the maxima (see Table
ing the entanglement of formation ǫ vs the probability
II) nearly correspond to those of TSS states. As in the F
P for gτ values corresponding to numbered minima in
00
Fig. 3a. We see that only for gτ = 26.68 a Bell State
gτmax hNimax ǫF,max P00 P11 can transfer 1 ebit of entanglement, but the entangle-
1.56 0.89 0.84 0.61 0.34 ment transfer is complete also for all the other P val-
00
4.66 1.09 0.90 0.54 0.39 ues. A nearly complete transfer can be obtained also
for gτ = 11.07 but in the other cases the entanglement
7.85 0.99 0.87 0.57 0.37
transfer is only partial even for the Bell State. We note
11.01 0.99 0.88 0.57 0.37
that in [14] it is shown that for gτ = 11.07 one finds
maximum entanglement transfer for both atomic states
TABLE II: Maxima of the qubit entanglement of formation
ǫF fortheresonantinteractionwithTMCandforbothqubits |1iA|1iB, |2iA|2iB but starting with a different Bell-like
initially in the ground state (see Fig. 2c). field state Ψ = |10i−|01i.
| −i √2
In the TWB case we find large entanglement transfer
previous case this can be explained by the TMC photon for gτ values very close to the ones of minima (2-4) for
distribution (see Fig. 1a), where for N < 4 the domi- the TSS states in Fig. 3a. Some gτ values correspond-
h i
nant components of the photon distribution are P and ing to maxima of ǫ in the case of both atoms in the
00 F
6
groundstate are missing, andthe best value ofǫF in the 1 1
ε ε
considered range is at gτ = 26.65. Also for the TMC F F
states for small N we have largeentanglementtransfer 1 1
correspondingtohthieabovegτ values,butinadditionfor 0.5 2 3 0.5 2
N > 4 and large gτ values there are regions with con- 4 3
h i 4
siderable entanglement. 5
0 0 5
Finally,inthecaseofoneatomintheexcitedstateand a) 0 5 <N>10 b) 0 5 <N>10
theotheroneinthegroundstate(1 2 )itisnotpos-
A B
| i | i
sibletowriteforTSSstatesasimpleequationasEq.(20)
because in the atomic density matrix Eq. (15) ρ =ρ FIG.5: TheoffresonanceinteractioneffectintheTWBcase,
22 33
unlike in the previous cases. However, there are 6again both atoms in the ground state and gτ =4.61. a) ∆Aτ =0,
only two frequencies involved as in the TSS case and we and∆Bτ =0(dashedline), 1, 2,3, 4,5. b)∆Aτ =∆Bτ =0
(dashed line), 1, 2, 3, 4, 5.
can do a similar analysis as for both atoms in the ex-
cited state. Here we only mentionthat for dimensionless
interaction times corresponding to common maxima for
V. THE EFFECT OF DIFFERENT
the different atomic states, the maxima of ǫ are rather
F INTERACTION TIMES
lower than in the cases (1 1 ) and (2 2 ), and
A B A B
| i | i | i | i
more in general the transfer of entanglement is sensibly
In the previous analysis we considered equal coupling
reduced as a function of N .
h i constant and interaction time for both atoms. However,
experimentally we may realize conditions such that the
parameter gτ is different for the two interactions due to
IV. THE DETUNING EFFECT
the limitations in the control of both atomic velocities
and injection times or in the values of the coupling con-
¿From the practical point of view it is important to
stants [7].
evaluate the effects of the off-resonant interaction be-
We first consider the effect of different interaction
tweenthe atoms andtheir respective cavityfields, which
times at exact resonance and simultaneous injection of
canbe actually preparedinnon degenerate opticalpara-
bothatomspreparedinthegroundstate. InFig.6a,bwe
metric processes. We assume equal interaction times for
showtheTMCcaseforgτ =4.66andgτ =11.01,cor-
both atomsand we considerfirstthe caseof resonantin- A A
responding to two maxima of entanglement as discussed
teractionfor the atom A and off resonantinteractionfor
in the previous section, and we investigate the effect of
atom B. As a first example we consider the TMC case,
different dimensionless interaction times for the atom B
both atoms in the ground state and the value gτ =4.66,
such that gτ gτ . We see that increasing the dif-
corresponding to the maximum entanglement transfer. B ≤ A
ference gτ gτ the entanglement decreases. However
In Fig. 4a we see that up to detuning values on the or- A− B
for N < 4, if gτ has a value close to the one corre-
der of the inverse interaction time the entanglement is h i B
sponding to a maximum, as for gτ = 1.56 in Fig. 6a
preserved by the off-resonant interaction of atom B. In A
and gτA = 7.85 in Fig. 6b, i.e. if gτA −gτB ∼= π, the
entanglement reaches again large values. The effect is
1 1
εF 321 εF 21 mtraonresfiemrpisortthaentsafomregτaAs=for11e.q0u1awlhinerteertahcetieonntatnimgleesm. eInnt
4 3 Fig.6c,dwe show ananalogouseffect for the TWB case.
0.5 0.5
5 We finally consider the possibility that atom B enters
4
the cavity just before atom A. We assume that when
5
0 0 atom A enters its cavity the two mode field can be still
a) 0 5 <N> 10 b) 0 5 <N> 10 described by Eq. (1). Due to the interaction with its
cavity field the atom B will be in a superposition state
FIG.4: TheoffresonanceinteractioneffectintheTMCcase, B1 1 B +B2 2 B. In this case the atomic density ma-
| i | i
both atoms in the ground state and gτ =4.66. a) ∆Aτ =0, trix ρ1a2 has only two null elements, ρ23 = ρ∗32, hence
and∆Bτ =0(dashed line),1, 2,3, 4, 5. b) ∆Aτ =∆Bτ =0 we evaluate the eigenvalues of the non-hermitian matrix
(dashed line), 1, 2, 3, 4, 5. numerically. We calculate the amount of entanglement
transferred to the atoms after the time τ of their simul-
Fig. 4b we show the more general case of off-resonance taneous presence into the respective cavities in the case
forbothatoms,takingequaldetuning valuesforsimplic- ofexactresonance,equalcouplingconstantandvelocity,
ity. The effect is greater than in the previous case but it assumingatomApreparedinthegroundstate. InFig.7a
is negligible again up to ∆ τ =1 . weshowtheTMCcaseforgτ =4.66anddifferentvalues
B
Fig.5showstheanalogousbehaviourfortheTWBstates of B 2 ranging from 0 (that is for 1 2 ) to 1 (that
1 A B
| | | i | i
forgτ =4.61. Weseethatnearthepeakofentanglement is 1 1 ). We notethatthe behaviourofǫ gradually
A B F
| i | i
the TMC states seemmorerobustto offresonanceinter- changesfromonelimitcasetotheotheronefor N <4,
h i
action than the TWB states. when the photon distribution approaches that of a TSS
7
1 1 VI. CONCLUSIONS
ε ε a
F F b
c
0.5 b a 0.5 d In this paper we have addressed the transfer of entan-
c e glement from a bipartite state of a continuous-variable
d system to a pair of localized qubits. We have assumed
e
0 0 that eachCV mode couples to one qubit via the Jaynes-
a) 0 5 <N> 10 b) 0 5 <N>10
Cummings interaction and have taken into account the
1 1
ε ε degrading effects of detuning and of different interaction
F F
a times for the two subsystems. The transfer of entangle-
b ment has been assessed by tracing out the field degrees
0.5 0.5
a
c of freedomafter the interaction,and then evaluating the
b
d c entanglementofformationofthereducedatomicdensity
0 e 0 e d matrix.
c) 0 5 <N>10 d) 0 5 <N>10
We found that CV states initially prepared in a two-
state superposition are the most efficient in transferring
FIG. 6: The effect of different interaction times for ∆A = entanglementtoqubitswithBell-likestatesabletotrans-
∆B = 0, and both atoms injected simultaneously in the fer a full ebit of entanglement. We have then considered
ground state. In a) the TMC case for gτA = 4.66 and gτB multiphoton preparation as TWB and TMC states and
values: a)4.66, b)4.4,c)4.2, d)4.0, e)3.8, and1.56 (dashed
found that there are large and well defined regions of in-
line). Inb)theTMCcaseforgτA=11.01andgτB values: a)
teractionparameters where the transfer of entanglement
11.01,b)10.8,c)10.6,d)10.4,e)10.2and7.85(dashedline).
In c) the TWB case for gτA =4.61 and gτB values: a) 4.61, iseffective. Atfixedenergy(averagenumberofphotons)
b)4.4, c)4.2, d)4.0, e)3.8, and1.56 (dashedline). Ind)the TMC states are more effective in transferring entangle-
TWBcase forgτA=11.03 andgτB values: a) 11.03, b)10.8, ment than TWB states. We have also found that the
c) 10.6, d) 10.4, e) 10.2 and 7.85 (dashed line). entanglement transfer is robust against the fluctuations
of interaction times and is not dramatically affected by
detuning. This kind of robustness is enhanced for the
state. Forlargervaluesof N weseetheoccurrenceofa transfer of entanglement from non Gaussian states as
h i
secondpeakinthecase 1 A 2 B. InFig.7bweshowthe TMC states.
| i | i
TWB case for gτ = 4.61 and we note that the gradual Overall, we conclude that the scheme analyzed in this
changedescribedaboveoccursfornearlyallvaluesofthe paper is a reliable and robust mechanism for the engi-
mean photon number N . neering of the entanglement between two atomic qubits
h i
andthat bipartite non Gaussianstates are promising re-
1 1 sources in order to optimize this protocol. Finally, we
εF εF mention that our analysis may also be employed to as-
sess the entanglement transfer from radiation to super-
0.5 c 0.5 c conducting qubits.
b
b a
a
0 0
a) 0 5 <N> 10 b) 0 5 <N> 10
FIG. 7: The entanglement of formation ǫF vs hNi for a time Acknowledgments
gτ of simultaneous presence of both atoms after the delayed
injection of atom A.Atom A is prepared in theground state
and atom B in superposition states such that |B1|2 = 0 This work has been supported by MIUR through the
(dashed line), 0.25 (a), 0.50 (b), 0.75 (c), 1 (dotted line). project PRIN-2005024254-002. MGAP thanks Vladylav
a) the TMC case with gτ = 4.66. b) the TWB case with Usenkoforusefuldiscussionsaboutpair-coherent(TMC)
gτ =4.61. states.
8
APPENDIX A: ATOMIC DENSITY MATRIX ELEMENTS
The elements of the 4 4 atomic density matrix ρ12 in the standard basis 2 2 , 2 1 , 1 2 , 1 1
× a {| iA| iB | iA| iB | iA| iB | iA| iB}
are listed below.
∞
ρ (x)= c (x)2 A 2 B 2 f (x)2 U (j,τ)2 U (j,τ)2+
11 | 0 | {| 2| | 2| X| j | | A11 | | B11 |
j=0
∞
+A∗1A2B1∗B2Xfj(x)fj∗+1(x)UA11(j,τ)UB11(j,τ)UA∗12(j,τ)UB∗12(j,τ)+
j=0
∞ ∞
+A 2 B 2 f (x)2 U (j+1,τ)2 U (j,τ)2+ A 2 B 2 f (x)2 U (j,τ)2 U (j+1,τ)2+
| 2| | 1| X| j+1 | | A11 | | B12 | | 1| | 2| X| j+1 | | A12 | | B11 |
j=0 j=0
∞
+A 2 B 2 f (x)2 U (j,τ)2 U (j,τ)2+
| 1| | 1| X| j+1 | | A12 | | B12 |
j=0
∞
+A A B B f (x)f (x)U (j,τ)U (j,τ)U (j,τ)U (j, tau) (A1)
1 ∗2 1 2∗X j+1 j∗ A12 B12 A∗11 B∗11 }
j=0
∞
ρ (x)= c (x)2 A 2 B 2 f (x)2 U (j 1,τ)2 U (j 1,τ)2+
22 | 0 | {| 2| | 2| X| j−1 | | A11 − | | B21 − |
j=1
∞
+A A B B f (x)f (x)U (j 1,τ)U (j 1,τ)U (j 1,τ)U (j 1,τ)+
∗1 2 1∗ 2X j−1 j∗ A11 − B21 − A∗12 − B∗22 −
j=1
∞
+A 2 B 2[ f (x)2 U (j,τ)2 U (j 1,τ)2+ U (0,τ)2]+
| 2| | 1| X| j | | A11 | | B22 − | | A11 |
j=1
∞ ∞
+A 2 B 2 f (x)2 U (j 1,τ)2 U (j,τ)2+ A 2 B 2 f (x)2 U (j 1,τ)2 U (j 1,τ)2+
| 1| | 2| X| j | | A12 − | | B21 | | 1| | 1| X| j | | A12 − | | B22 − |
j=1 j=1
∞
+A A B B f (x)f (x)U (j 1,τ)U (j 1,τ)U (j 1,τ)U (j 1,τ) (A2)
1 ∗2 1 2∗X j j∗−1 A12 − B22 − A∗11 − B∗21 − }
j=1
∞
ρ (x)= c (x)2 A 2 B 2 f (x)2 U (j,τ)2 U (j,τ)2+
33 | 0 | {| 2| | 2| X| j | | A21 | | B11 |
j=0
∞
+A A B B f (x)f (x)U (j,τ)U (j,τ)U (j,τ)U (j,τ)+
∗1 2 1∗ 2X j j∗+1 A21 B11 A∗22 B∗12
j=0
∞
+A 2 B 2[ f (x)2 U (j 1,τ)2 U (j,τ)2+ U (0,τ)2]+
| 1| | 2| X| j | | A22 − | | B11 | | B11 |
j=1
∞ ∞
+A 2 B 2 f (x)2 U (j+1,τ)2 U (j,τ)2+ A 2 B 2 f (x)2 U (j,τ)2 U (j,τ)2+
| 2| | 1| X| j+1 | | A21 | | B12 | | 1| | 1| X| j+1 | | A22 | | B12 |
j=0 j=0
∞
+A1A∗2B1B2∗Xfj+1(x)fj∗(x)UA22(j,τ)UB12(j,τ)UA∗21(j,τ)UB∗11(j,τ)} (A3)
j=0
9
∞
ρ (x)= c (x)2 A 2 B 2 f (x)2 U (j 1,τ)2 U (j 1,τ)2+
44 | 0 | {| 2| | 2| X| j−1 | | A21 − | | B21 − |
j=1
∞
+A A B B f (x)f (x)U (j 1,τ)U (j 1,τ)U (j 1,τ)U (j 1,τ)+
∗1 2 1∗ 2X j−1 j∗ A21 − B21 − A∗22 − B∗22 −
j=1
∞
+A 2 B 2[ f (x)2 U (j,τ)2 U (j 1,τ)2+ U (0,τ)2]+
| 2| | 1| X| j | | A21 | | B22 − | | A21 |
j=1
∞
+A 2 B 2[ f (x)2 U (j 1,τ)2 U (j,τ)2+ U (0,τ)2]+
| 1| | 2| X| j | | A22 − | | B21 | | B21 |
j=1
∞
+A 2 B 2[ f (x)2 U (j 1,τ)2+ U (j 1,τ)2+1]+
| 1| | 1| X| j | | A22 − | | B22 − |
j=1
∞
+A A B B f (x)f (x)U (j 1,τ)U (j 1,τ)U (j 1,τ)U (j 1,τ) (A4)
1 ∗2 1 2∗X j j∗−1 A22 − B22 − A∗21 − B∗21 − }
j=1
∞
ρ (x)= c (x)2 A 2B B [ f (x)2 U (j,τ)2U (j,τ)U (j 1,τ)+ U (0,τ)2U (0,τ)]+
12 | 0 | {| 2| 1∗ 2 X| j | | A11 | B11 B∗22 − | A11 | B11
j=1
∞
+|A1|2B2B1∗X|fj(x)|2UA12(j−1,τ)|2|UB11(j,τ)UB∗22(j−1,τ)+
j=1
∞
+A1A∗2|B2|2Xfj(x)fj∗−1(x)UA12(j−1,τ)UB11(j,τ)UA∗11(j−1,τ)UB∗21(j−1,τ)+
j=1
∞
+A A B 2[ f (x)f (x)U (j,τ)U (j,τ)U (j,τ)U (j 1,τ)+f (x)U (0,τ)U (0,τ)U (0,τ)]
1 ∗2| 1| X j+1 j∗ A12 B12 A∗11 B∗22 − 1 A12 B12 A∗11 }
j=1
(A5)
∞
ρ (x)= c (x)2 A 2B B f (x)f (x)U (j+1,τ)U (j,τ)U (j,τ)U (j,τ)+
13 | 0 | {| 2| 1 2∗X j+1 j∗ A11 B12 A∗21 B∗11
j=0
+A 2B B [ ∞ f (x)f (x)U (j,τ)U (j,τ)U (j 1,τ)U (j,τ)+f (x)U( 0,τ)U (0,τ)U (0,τ)]+
| 1| 1 2∗ X j+1 j∗ A12 B12 A∗22 − B∗11 1 A12 B12 B∗11
j=1
∞
+A A B 2[ f (x)2U (j,τ)U (j,τ)2U (j 1,τ)+U (0,τ)U (0,τ)2]+
∗1 2| 2| X| j | A11 | B11 | A∗22 − A11 | B11 |
j=1
∞
+A∗1A2|B1|2X|fj+1(x)|2UA11(j+1,τ)|UB12(j,τ)|2UA∗22(j,τ)} (A6)
j=0
10
∞
ρ (x)= c (x)2 A 2 B 2 f (x)f (x)U (j,τ)U (j,τ)U (j 1,τ)U (j 1,τ)+
14 | 0 | {| 2| | 2| X j j∗−1 A11 B11 A∗21 − B∗21 −
j=1
∞
+A A B B [ f (x)2U (j,τ)U (j,τ)U (j 1,τ)U (j 1,τ)+U (0,τ)U (0,τ)]+
∗1 2 1∗ 2 X| j | A11 B11 A∗22 − B∗22 − A11 B11
j=1
∞
+A 2 B 2[ f (x)f (x)U (j+1,τ)U (j,τ)U (j,τ)U (j 1,τ)+
| 2| | 1| X j j∗−1 A11 B12 A∗21 B∗22 −
j=1
+f (x)U (1,τ)U (0,τ)U (0,τ)]+
1 A11 B12 A∗21
∞
+A 2 B 2[ f (x)f (x)U (j,τ)U (j+1,τ)U (j 1,τ)U (j,τ)+
| 1| | 2| X j+1 j∗ A12 B11 A∗22 − B∗21
j=1
+f (x)U (0,τ)U (1,τ)U (0,τ)]+
1 A12 B11 B∗12
∞
+A 2 B 2[ f (x)f (x)U (j,τ)U (j,τ)U (j 1,τ)U (j 1,τ)+
| 1| | 1| X j+1 j∗ A12 B12 A∗22 − B∗22 −
j=1
+f (x)U (0,τ)U (0,τ)]+
1 A12 B12
∞
+A A B B f (x)f (x)U (j,τ)U (j,τ)U (j 1,τ)U (j 1,τ) (A7)
1 ∗2 1 2∗X j+1 j∗−1 A12 B12 A∗21 − B∗21 − }
j=1
∞
ρ (x)= c (x)2A A B B [ f (x)2U (j,τ)U (j 1,τ)U (j 1,τ)U (j,τ)+U (0,τ)U (0,τ)]
23 | 0 | ∗1 2 1 2∗ X| j | A11 B22 − A∗22 − B∗11 A11 B∗11
j=0
(A8)
∞
ρ (x)= c (x)2 A 2B B f (x)f (x)U (j,τ)U (j 1,τ)U (j 1,τ)U (j 1,τ)+
24 | 0 | {| 2| 1 2∗X j j∗−1 A11 B22 − A∗21 − B∗21 −
j=1
∞
+A 2B B [ f (x)f (x)U (j,τ)U (j,τ)U (j 1,τ)U (j,τ)+
| 1| 1 2∗ X j+1 j∗ A12 B22 A∗22 − B∗21
j=1
+f (x)U( 0,τ)U (0,τ)U (0,τ)]+
1 A12 B22 B∗21
∞
+A A B 2[ f (x)2U (j,τ)U (j,τ)2U (j 1,τ)+U (0,τ)U (0,τ)2]+
∗1 2| 2| X| j | A11 | B21 | A∗22 − A11 | B21 |
j=1
∞
+A A B 2[ f (x)2U (j,τ)U (j 1,τ)2U (j 1,τ)+U (0,τ)] (A9)
∗1 2| 1| X| j | A11 | B22 − | A∗12 − A11 }
j=1
∞
ρ (x)= c (x)2 A 2B B [ f (x)2 U (j,τ)2U (j,τ)U (j 1,τ)+ U (0,τ)2U (0,τ)
34 | 0 | {| 2| 1∗ 2 X| j | | A21 | B11 B∗22 − | A21 | B11
j=1
∞
+ A 2B B [ f (x)2U (j 1,τ)U (j,τ)U (j 1,τ)U (j 1,τ)+U (0,τ)]
| 1| 1∗ 2 X| j | A22 − B11 A∗22 − B∗22 − B11
j=1
∞
+ A A B 2 f (x)f (x)U (j 1,τ)U (j,τ)U (j 1,τ)U (j 1,τ)
1 ∗2| 2| X j j∗−1 A22 − B11 A∗21 − B∗21 −
j=1
∞
+ A A B 2[ f (x)f (x)U (j,τ)U (j,τ)U (j,τ)U (j 1,τ)+f (x)U (0,τ)U (0,τ)U (0,τ)]
∗1 2| 1| X j+1 j∗ A22 B12 A∗21 B∗22 − 1 A22 B12 A∗21 }
j=1
(A10)