Improving the entanglement transfer from continuous

PHYSICAL REVIEW A 75, 032336 共2007兲
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, Università di Milano, Milano, Italy
共Received 18 December 2006; revised manuscript received 12 January 2007; published 26 March 2007兲
We investigate the entanglement transfer from a bipartite continuous-variable 共CV兲 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 the case of the CV system
prepared in a Bell-like superposition and investigate the conditions for maximum entanglement transfer. Then
we analyze the general case of two-mode CV states that can be represented by a Schmidt decomposition in the
Fock number basis. This class includes both Gaussian and non-Gaussian CV states, as, for example, twin-beam
共TWB兲 and pair-coherent 共TMC, also known as two-mode-coherent兲 states, respectively. Under resonance
conditions, equal interaction times for both qubits and different initial preparations, we find that the entanglement transfer is more efficient for TMC than for TWB states. In the perspective of applications such as in
cavity QED or with superconducting qubits, we analyze in detail the effects of off-resonance interactions
共detuning兲 and different interaction times for the two qubits, and discuss conditions to preserve the entanglement transfer.
DOI: 10.1103/PhysRevA.75.032336
PACS number共s兲: 03.67.Mn, 42.50.Pq
I. INTRODUCTION
Entanglement is the main resource of quantum information processing 共QIP兲. Indeed, much attention has been devoted to generation and manipulation of entanglement either
in discrete or in continuous variable 共CV兲 systems. Crucial
and rewarding steps in the development of QIP are now the
storage of entanglement in quantum memories 关1,2兴 and the
transfer of entanglement from localized to flying registers
and vice versa. Indeed, effective protocols for the distribution of entanglement would allow one to realize quantum
cryptography over long distances 关3兴, as well as distributed
quantum computation 关4兴 and distributed network for quantum communication purposes.
Few schemes have been suggested either to entangle localized qubits, e.g., distant atoms or superconducting quantum interference devices, using squeezed radiation 关5兴 or to
transfer entanglement between qubits and radiation 关6–8兴. As
a matter of fact, efficient sources of entanglement have been
developed for CV systems, especially by quantum-optical
implementations 关9兴. Indeed, multiphoton states might be optimal when considering long-distance communication, where
they may travel through free space or optical fibers, in view
of the robustness of their entanglement against losses 关10兴.
The entanglement transfer from free propagating light to
atomic systems has been achieved experimentally in recent
years 关1,11兴. From the theoretical point of view, the resonant
entanglement transfer between a bipartite continuous variable systems and a pair of qubits has been analyzed in 关12兴
where the CV field is assumed to be a two-mode squeezed
vacuum or twin-beam 共TWB兲 state 关13兴 with the two modes
injected into spatially separate cavities. Two identical atoms,
*Electronic address: [email protected]
†
Electronic address: [email protected]
Electronic address: [email protected]
‡
1050-2947/2007/75共3兲/032336共11兲
both in the ground state, are then assumed to interact resonantly, one for each cavity, with the cavity mode field for an
interaction time shorter than the cavity lifetime. More recently, a general approach has been developed 关6兴, in which
two static qubits are isolated by the real world by their own
single mode bosonic local environment that also rules the
interaction of each qubit with an external driving field assumed to be a general broadband two-mode field. This model
may be applied to describe a cavity QED setup with two
atomic qubits trapped into remote cavities. In Ref. 关7兴 the
problems related to different interaction times for the two
qubits are pointed out, either for atomic qubits or in the case
of superconducting quantum interference devices 共SQUID兲
qubits. The possibility to transfer the entanglement of a TWB
radiation field to SQUIDS has been also investigated in 关8兴.
Very recently, in 关14兴 the entanglement transfer process
between CV and qubit bipartite systems was investigated.
Their scheme is composed by two atoms placed into two
spatially separated identical cavities where the two modes
are injected. They consider resonant interaction of two-mode
fields, such as two-photon superpositions, entangled coherent
states, and TWB, discussing conditions for maximum entanglement transfer.
The inverse problem of entanglement reciprocation from
qubits to continuous variables has been discussed in 关15兴 by
means of a model involving two atoms prepared in a maximally entangled state and then injected into two spatially
separated cavities, each one prepared in a coherent state. It
was shown that when the atoms leave the cavity their entanglement is transferred to the post-selected cavity fields.
The generated field entanglement can be then transferred
back to qubits, i.e, to another couple of atoms flying through
the cavities. In a recent paper 关16兴 the relationship between
entanglement, mixedness, and energy of two qubits and two
mode Gaussian quantum states has been analyzed, whereas a
strategy to enhance the entanglement transfer between TWB
states and multiple qubits has been suggested in 关17兴.
032336-1
©2007 The American Physical Society
CASAGRANDE, LULLI, AND PARIS
PHYSICAL REVIEW A 75, 032336 共2007兲
In this paper we investigate the dynamics of a two-mode
entangled state of radiation coupled to a pair of localized
qubits via the off-resonance Jaynes-Cummings interaction.
We focus our attention on the entanglement transfer from
radiation to atomic qubits, though our analysis may be employed also to describe the effective interaction of radiation
with superconducting qubits. In particular, compared to previous analysis, we consider in detail the effects of offresonance interactions 共detuning兲 and different interaction
times for the two qubits. As a carrier of entanglement we
consider the general case of two-mode states that can be
represented by a Schmidt decomposition in the Fock number
basis. These include Gaussian states of radiation such as twin
beams, realized by nondegenerate parametric amplifiers by
means of spontaneous down conversion in nonlinear crystals,
as well as non-Gaussian states, as, for example, pair-coherent
共TMC, also known as two-mode coherent兲 states 关18兴, that
can be obtained either by degenerate Raman processes 关19兴
or, more realistically, by conditional measurements 关20兴 and
nondegenerate parametric oscillators 关21,22兴. In fact, we find
that TMC are more effective in transferring entanglement to
qubits than TWB states and this opens perspectives on the
use of non-Gaussian states in quantum information processing.
The paper is organized as follows: in the next section we
introduce the Hamiltonian model we are going to analyze for
entanglement transfer, as well as the different kinds of twomode CV states that provide the source of entanglement. In
Sec. III we consider resonant entanglement transfer, which is
assessed by evaluating the entanglement of formation for the
reduced density matrix of the qubits after a given interaction
time. In Secs. IV and V we analyze in some detail the effects
of detuning and of different interaction times for the two
qubits. Section VI closes the paper with some concluding
remarks.
兩x典 = c0兩00典 + c1兩11典.
II. HAMILTONIAN MODEL
We address the entanglement transfer from a bipartite CV
field to a pair of localized qubits assuming that each CV
mode couples to one qubit via the off-resonance JaynesCummings interaction 共as it happens by injecting the two
modes in two separate cavities兲. We allow for different interaction times for the two subsystems and assume 关14兴 that the
initial state of the two modes is described by a Schmidt
decomposition in the Fock number basis
⬁
兩x典 = 兺 cn共x兲兩nn典,
Equation 共1兲 also describes relevant bipartite states, as, for
example, TWB and TMC states. In these cases we can rewrite the coefficients as cn共x兲 = c0共x兲f n共x兲, where
TWB:
c0共x兲 = 冑1 − 兩x兩2,
TMC:
c0共x兲 =
1
冑I0共2兩x兩兲
,
f n共x兲 = xn ,
共3兲
xn
,
n!
共4兲
f n共x兲 =
where I0共y兲 denotes the modified Bessel function of zero
order. For TWB states the parameter 兩x兩 is related to the
squeezing parameter, and ranges from 0 共no squeezing兲 to 1
共infinite squeezing兲. For TMC states, 兩x兩 is related to the
squared field amplitude and can take any positive values. The
bipartite states described by Eq. 共1兲 show perfect photon
number correlations. The joint photon number distribution
has indeed the simple form Pnk共x兲 = ␦nk兩cn共x兲兩2. For the TSS
states the joint photon distribution is given by P00 = 兩c0兩2 and
P11 = 1 − P00, whereas for TWB and TMC it can be written as
Pnk共x兲 = ␦nk P00共x兲兩f n共x兲兩2. As we will see in the following the
photon distribution plays a fundamental role in understanding the entanglement transfer process.
The average number of photon of the states 兩x典, i.e., 具N典
⫻共x兲 = 具x 兩 a†a + b†b 兩 x典, a and b being the field mode operators, is related to the dimensionless parameter 兩x兩 by
TWB:
TMC:
兩x兩2
,
1 − 兩x兩2
共5兲
2兩x兩I1共2兩x兩兲
,
I0共2兩x兩兲
共6兲
具N典共x兲 = 2
具N典共x兲 =
where I1共y兲 denotes the modified Bessel function of first order. In Fig. 1 we show the first four terms of the photon
distribution for TMC and TWB states, respectively, as functions of the mean photon number.
The states in Eq. 共1兲 are pure states, and therefore we can
evaluate their entanglement by the Von Neumann entropy
Svn共x兲 of the reduced density matrix of each subsystem. For
the TSS case we simply have
Svn = − P00 log2 P00 − 共1 − P00兲log2共1 − P00兲,
共7兲
and, of course, the maximum value of 1 is obtained for P00
is a Bell-like
= P11 = 21 . The corresponding state 兩⌽+典 = 兩00典+兩11典
冑2
maximally entangled state. For TWB states the Von Neumann entropy can be written as
共1兲
Svn共x兲 = − log2共1 − 兩x兩2兲 −
n=0
where 兩nn典 = 兩n典 丢 兩n典 and the complex coefficients cn共x兲
⬁
= 具nn 兩 x典 satisfy the normalization condition 兺n=0
兩cn共x兲兩2 = 1.
The parameter x is a complex variable that fully characterizes the state of the field. Notice that a scheme for the generation of any two-mode correlated photon number states of
the form 共1兲 has been recently proposed 关20兴. The simplest
example within the class 共1兲 is given by the Bell-like twomode superposition 共TSS兲
共2兲
2兩x兩2
log2兩x兩,
1 − 兩x兩2
共8兲
whereas for TMC states we use the general expression
⬁
Svn共x兲 = − log2 P00共x兲 − P00共x兲 兺 兩f i共x兲兩2 log2兩f i共x兲兩2 . 共9兲
i=1
It is clear that the Von Neumann 共VN兲 entropy diverges in
the limit 兩x兩 → 1 共for TWB兲 and in the limit 兩x兩 → ⬁ 共for
TMC兲 because the probability P00共x兲 vanishes. The VN en-
032336-2
PHYSICAL REVIEW A 75, 032336 共2007兲
IMPROVING THE ENTANGLEMENT TRANSFER FROM…
P
兩␺共0兲典 = 兩x典 丢 兩␺共0兲典A 丢 兩␺共0兲典B ,
1
nn
evolves by means of the unitary operator U共␶兲 = exp
关− បi Hi␶兴 that can be factorized as the product of two offresonance Jaynes-Cummings evolution operators UA共␶兲 and
UB共␶兲 关24兴 related to each atom-mode subsystem. For the
initial state of both atoms we considered the following general superposition of their excited 共兩2典兲 and ground 共兩1典兲
states:
P
00
P11
0.5
P
22
P
33
兩␺共0兲典A = A2兩2典A + A1兩1典A ,
(a)
0
0
5
兩␺共0兲典B = B2兩2典B + B1兩1典B ,
<N> 10
where 兩A2兩2 + 兩A1兩2 = 1 and 兩B2兩2 + 兩B1兩2 = 1. This includes the
most natural and widely investigated choice of both atoms in
the ground states, but will also allow us to investigate the
effect of different interaction times.
Due to the linearity of evolution operator U共␶兲 and its
factorized form, the whole system state 兩␺共␶兲典 at a time ␶ can
be written as
1
Pnn
P00
0.5
P
⬁
11
兩␺共␶兲典 = c0共x兲 兺 f n共x兲UA共␶兲兩␺n共0兲典A 丢 UB共␶兲兩␺n共0兲典B ,
P
22
0
(b) 0
共11兲
n=0
共12兲
P
33
5
<N> 10
FIG. 1. The first four terms Pnn, n = 0 , . . . , 3 of the joint photon
distribution of the state 兩x典 as a function of average photon number
具N典. 共a兲 TMC, 共b兲 TWB.
tropy at fixed energy 共average number of photons of the two
modes兲 is maximized by the TWB expression 共8兲. For this
reason TWB states are also referred to as maximal entangled
states of bipartite CV systems.
We consider the interaction of each radiation mode with a
two-level atom flying through the cavity. If the interaction
time is much shorter than the lifetime of the cavity mode and
the atomic decay rates, we can neglect dissipation in system
dynamics. On the other hand, we consider the general case of
atoms with different interaction times and coupling constants, prepared in superposition states, and off-resonance interaction between each atom and the relative cavity mode.
All these features can be quite important in practical implementations such as in cavity QED systems with Rydberg
atoms and high-Q microwave cavities 关23兴, as noticed in 关7兴.
In the interaction picture, the interaction Hamiltonian Hi is
given by
where 兩␺n共0兲典A,B = 兩␺共0兲典A,B 丢 兩n典A,B. In each of two
atom-field subspaces A and B we expand the wave
⬁
艛 兵兩0典兩1典其. The
function on the basis 兵兩2典兩k典 , 兩1典兩k + 1典其k=0
n
coefficients
cA,1,k共0兲 = A具2兩A具k 兩 兩␺ 共0兲典A
and
cB,1,k共0兲
= B具2兩B具k 兩 兩␺n共0兲典B of the initial states are
cA,2,k共0兲 = A2␦k,n,
cB,2,k共0兲 = B2␦k,n ,
cB,1,k+1共0兲 = B1␦k+1,n .
共13兲
The Jaynes-Cummings interaction couples only the coefficients of each variety K whereas cA,1,0共0兲, cB,1,0共0兲 do not
evolve. Therefore, for each variety in the subspaces A and B
the evolved coefficients can be obtained by applying the offresonance Jaynes-Cummings 2 ⫻ 2 matrix U jk so that
c2,k共␶兲 = U11共k, ␶兲c2,k共0兲 + U12共k, ␶兲c1,k+1共0兲,
c1,k+1共␶兲 = U21共k, ␶兲c2,k共0兲 + U22共k, ␶兲c1,k+1共0兲,
where
冉 冊
冉 冊
U11共k, ␶兲 = cos
U12共k, ␶兲 = −
共10兲
12
12
and SB,±
are the lowering and raising atomic opwhere SA,±
erators of the two atoms and ⌬A, ⌬B denote the detunings
between each mode frequency and the corresponding atomic
transition frequency. The initial state of the whole system,
cB,1,0共0兲 = B1␦n,0 ,
cA,1,k+1共0兲 = A1␦k+1,n,
12
12
Hi = − ប⌬Aa†a − ប⌬Bb†b + បgA关a†SA,−
+ aSA,+
兴
12
12
+ បgB关b†SB,−
+ bSB,+
兴,
cA,1,0共0兲 = A1␦n,0,
⌬
R k␶
R k␶
− i sin
,
Rk
2
2
冉 冊
2ig冑k + 1
R k␶
sin
= U21共k, ␶兲,
Rk
2
冉 冊
U22共k, ␶兲 = cos
冉 冊
⌬
R k␶
R k␶
+ i sin
,
Rk
2
2
共14兲
where the generalized Rabi frequencies are Rk
= 冑4g2共k + 1兲 + ⌬2. To derive the evolved atomic density op-
032336-3
PHYSICAL REVIEW A 75, 032336 共2007兲
CASAGRANDE, LULLI, AND PARIS
erator ␳1,2
we first consider the statistical operator of
a
the whole system ␳共␶兲 = 兩␺共␶兲典具兩␺共␶兲兩 and then we trace
out the field variables. The explicit expressions of the
density matrix elements in the standard basis
兵兩2典A兩2典B , 兩2典A兩1典B , 兩1典A兩2典B , 兩1典A兩1典B其 are reported in the Appendix.
1
εF
0.5
III. ENTANGLEMENT TRANSFER AT RESONANCE
0
12
As an example we consider exact resonance for both
atom-field interactions, equal coupling constant g, and the
same interaction time ␶. For the initial atomic states we will
discuss the following three cases: both atoms in the ground
state 共兩1典A兩1典B兲, both atoms in the excited state 共兩2典A兩2典B兲, and
one atom in the excited state and the other one in the ground
state 共兩1典A兩2典B兲. In all these cases the atomic density matrix
after the interaction ␳1,2
a has the following form:
␳1,2
a
=
冢
␳11 0 0 ␳14
0 ␳22 0
0
0
0 ␳33 0
*
␳14
0
0
␳44
冣
gτ
(a)
PT
␭3,4
=
εF
0
12
,
(b)
5
0 0
<N>
log2
εF
共16兲
0.5
共17兲
In this way we calculate the concurrence 关28兴 C
= max兵0 , ⌳1 − ⌳2 − ⌳3 − ⌳4其, where ⌳i = 冑␭Ri are the square
roots of the eigenvalues ␭Ri selected in the decreasing order,
and then evaluate the entanglement of formation
−
10
6
1
R
␭3,4
= 共冑␳11␳44 ± 兩␳14兩兲2 .
2
gτ
␭2PT = ␳11 ,
we see that only ␭4PT can assume negative values. In the case
of TSS the expression of ␭4PT can allow us to derive in a
simple way analytical results for the conditions of maximum
entanglement transfer as a function of dimensionless interaction time g␶, as well as to better understand the results in the
case of TWB and TMC states. In order to quantify the
amount of the entanglement and, in turn, to assess the entanglement transfer we choose to adopt the entanglement of
formation ⑀F 关26兴. We rewrite the atomic density matrix in
the magic basis 关27兴 ␳aMB and we evaluate the eigenvalues of
the non-Hermitian matrix R = ␳aMB共␳aMB兲*:
⑀F = −
P00
共15兲
.
2
1 − 冑1 − C2
0 0
0.5
␳22 + ␳33 ± 冑共␳22 − ␳33兲2 + 4兩␳14兩2
R
= ␳22␳33,
␭1,2
0.5
1
The presence of the qubit entanglement can be revealed by
the Peres-Horodecki criterion 关25兴 based on the existence of
negative eigenvalues of the partial transpose of Eq. 共15兲.
From the expressions of the eigenvalues
␭1PT = ␳44,
1
6
1 − 冑1 − C2
2
1 + 冑1 − C2
1 + 冑1 − C2
log2
.
2
2
共18兲
In the case of both qubits initially in the ground state,
兩1典A兩1典B, the expression of ␭4PT simply reduces to ␳22−兩␳14兩,
0
12
gτ
(c)
10
6
5
0 0
<N>
FIG. 2. Entanglement of formation ⑀F of the qubit systems as a
function of the dimensionless time g␶ and the CV state parameter
P00 共a兲 or the average number of photons 具N典 共b,c兲 for the case of
both atoms initially in the ground state. 共a兲 TSS, 共b兲 TWB, 共c兲
TMC.
because ␳22 = ␳33, and it is possible to derive the following
simple formula:
␭4PT共P00,g␶兲 = sin2共g␶兲 ⫻ 关共1 − P00兲cos2共g␶兲
− 冑共1 − P00兲P00兴.
共19兲
We note that only the vacuum Rabi frequency is involved, a
fact that greatly simplifies the analysis of atom-field interaction compared to all the other atomic configurations. Let us
first consider the Bell-like state 共 P00 = 21 兲 and look for the g␶
values maximizing the entanglement of the two atoms. The
solution of equation ␭4PT共 21 , g␶兲 = − 21 is given by g␶ = ␲2 共2k
+ 1兲 with k = 0 , 1 , 2 , . . . . The above condition is also relevant
032336-4
PHYSICAL REVIEW A 75, 032336 共2007兲
IMPROVING THE ENTANGLEMENT TRANSFER FROM…
TABLE I. Maxima of the qubit entanglement of formation ⑀F
for the resonant interaction with TWB and for both qubits initially
in the ground state 关see Fig. 2共b兲兴.
g␶max
具N典max
⑀F,max
P00
P11
1.56
4.61
7.85
11.03
0.87
1.82
1.07
1.07
0.64
0.81
0.68
0.68
0.69
0.52
0.65
0.65
0.21
0.25
0.23
0.23
1.5
1
0.5
(1)
(2)
to explain the entanglement transfer for TWB and TMC
states, as discussed below. To evaluate the entanglement
transfer also for not maximally entangled TSS states in Eq.
共2兲, we calculate the entanglement of formation as a function
of both the dimensionless interaction time g␶ and the probability P00. As it is apparent from Fig. 2共a兲, there are large
and well-defined regions where ⑀F ⬎ 0. In particular, the absolute maxima 共⑀F = 1兲 occur exactly at P00 = 0.5 and for g␶
values in agreement with the above series. In addition, if we
consider the sections at these g␶ values, we obtain exact
coincidence with the Von Neumann entropy function
Svn共P00兲. Therefore, complete entanglement transfer from the
field to the atoms is possible not only for the Bell state,
though only for the Bell state may we obtain the transferral
of 1 ebit. In Fig. 2共b兲 we consider the entanglement of formation vs g␶ and mean photon number 具N典 in the TWB case.
We note that the regions of maximum entanglement correspond to those of TSS states and the maxima occur at g␶
values close to ␲2 共2k + 1兲, as shown in Table I. We can explain this by considering the TWB photon distribution 关see
Fig. 1共b兲兴. We note that the terms P00 and P11 are always
greater or equal than the other terms Pnn 共n ⬎ 1兲 and that for
具N典 ⬍ 2 they dominate the photon distribution 共P00 + P11
⯝ 1兲. Therefore, the main contribution to entanglement transfer is obtained from the above two terms as for the TSS state.
In order to explain the absolute maximum found in the second peak at 具N典 = 1.82, we note that in this case P00 and P11
are closer to the value 0.5 of a Bell state. In addition, for
large 具N典 and g␶ values, there are small regions 共not visible
in the figure兲 where entanglement transfer is possible. This is
due to the terms Pnn 共n ⬎ 1兲 in the photon distribution. In
Fig. 2共c兲 we show the TMC case and we note that for 具N典
⬍ 4 there are four well-defined peaks where the entanglement is higher than in the TWB case. Also in this case the g␶
values of the maxima 共see Table II兲 nearly correspond to
TABLE II. Maxima of the qubit entanglement of formation ⑀F
for the resonant interaction with TMC and for both qubits initially
in the ground state 关see Fig. 2共c兲兴.
g␶max
具N典max
⑀F,max
P00
P11
1.56
4.66
7.85
11.01
0.89
1.09
0.99
0.99
0.84
0.90
0.87
0.88
0.61
0.54
0.57
0.57
0.34
0.39
0.37
0.37
0
(a) 0
εF
(3)
10
(4)
20
30
1
40 gτ 50
4
3
2
0.75
0.5
1
0.25
(b)
0
0
P
0.5
00
1
FIG. 3. 共a兲 The function ␭4PT共 2 , g␶兲 + 2 as in Eq. 共20兲 for the
Bell-state case with both atoms in the excited state. 共b兲 Entanglement of formation ⑀F vs P00 for TSS states compared to Von Neumann entropy 共dashed line兲 for some values of g␶, corresponding to
the following numbered minima: 共1兲 2.03, 共2兲 4.53, 共3兲 11.07, 共4兲
26.68.
1
1
those of TSS states. As in the previous case this can be
explained by the TMC photon distribution 关see Fig. 1共a兲兴,
where for 具N典 ⬍ 4 the dominant components of the photon
distribution are P00 and P11. The absolute maximum is in the
second peak at 具N典 = 1.09 because P00 and P11 are even
closer to the Bell state than in the other peaks, and this also
explains the larger entanglement value ⑀F. For 具N典 ⬎ 4 and
large g␶ there are regions with considerable entanglement
values, due to the fact that P00 and P11 are always smaller
than the other terms Pnn 共n ⬎ 1兲 that dominate the atom-field
interaction.We note that the maxima of ⑀F are higher than in
the TWB case.
A similar analysis can be done in the case of initially
excited atoms 兩2典A兩2典B. For the TSS states we can again write
a simple equation for the eigenvalue of the partial transpose:
␭4PT共P00,g␶兲 = 共1 − P00兲sin2共冑2g␶兲cos2共冑2g␶兲 + sin2共g␶兲
⫻关P00 cos2共g␶兲 − 冑共1 − P00兲P00 cos2共冑2g␶兲兴,
共20兲
where, with respect to Eq. 共19兲, an additional frequency is
present. For the Bell state we can look for g␶ values maxi032336-5
PHYSICAL REVIEW A 75, 032336 共2007兲
CASAGRANDE, LULLI, AND PARIS
ε
1
1
1
F
ε
F
2
3
1
2
4
0.5
0.5
3
5
4
5
0
0
1
5
(a)
εF
0
0
<N> 10
5
(a)
ε
1
<N> 10
1
F
2
1
3
2
0.5
0.5
4
3
5
0
0
(b)
4
5
0
(b) 0
<N> 10
FIG. 4. The off-resonance interaction effect in the TMC case,
both atoms in the ground state and g␶ = 4.66. 共a兲 ⌬A␶ = 0 and ⌬B␶
= 0 共dashed line兲, 1, 2, 3, 4, 5. 共b兲 ⌬A␶ = ⌬B␶ = 0 共dashed line兲, 1, 2,
3, 4, 5.
mizing the entanglement transfer. In this case the problem
can be solved numerically and we found, for example, in the
range g␶ = 0 − 50, that only for g␶ = 26.68⯝ 172␲ we can solve
the equation ␭4PT共 21 , g␶兲 = − 21 with a good approximation as
shown in Fig. 3共a兲. In Fig. 3共b兲 we consider also nonmaximally entangled TSS states, showing the entanglement of
formation ⑀F vs the probability P00 for g␶ values corresponding to numbered minima in Fig. 3共a兲. We see that only for
g␶ = 26.68 a Bell state can transfer 1 ebit of entanglement,
but the entanglement transfer is complete also for all the
other P00 values. A nearly complete transfer can be obtained
also for g␶ = 11.07, but in the other cases the entanglement
transfer is only partial even for the Bell state. We note that in
关14兴 it is shown that for g␶ = 11.07 one finds maximum entanglement transfer for both atomic states 兩1典A兩1典B, 兩2典A兩2典B
but starting with a different Bell-like field state 兩⌿−典
= 兩10典−兩01典
.
冑2
In the TWB case we find large entanglement transfer for
g␶ values very close to the ones of minima 共2–4兲 for the TSS
states in Fig. 3共a兲. Some g␶ values corresponding to maxima
of ⑀F in the case of both atoms in the ground state are missing, and the best value of ⑀F in the considered range is at
g␶ = 26.65. Also for the TMC states for small 具N典 we have
large entanglement transfer corresponding to the above g␶
values, but in addition for 具N典 ⬎ 4 and large g␶ values there
are regions with considerable entanglement.
Finally, in the case of one atom in the excited state and the
other one in the ground state 共兩1典A兩2典B兲, it is not possible to
write for TSS states a simple equation such as Eq. 共20兲 because in the atomic density matrix, Eq. 共15兲, ␳22 ⫽ ␳33, unlike
5
5
<N> 10
FIG. 5. The off-resonance interaction effect in the TWB case,
both atoms in the ground state and g␶ = 4.61. 共a兲 ⌬A␶ = 0 and ⌬B␶
= 0 共dashed line兲, 1, 2, 3, 4, 5. 共b兲 ⌬A␶ = ⌬B␶ = 0 共dashed line兲, 1, 2,
3, 4, 5.
in the previous cases. However, there are again only two
frequencies involved as in the TSS case and we can do a
similar analysis as in the above case of both atoms in the
excited state. Here we only mention that for dimensionless
interaction times corresponding to common maxima for the
different atomic states, the maxima of ⑀F are rather lower
than in the cases 共兩1典A兩1典B兲 and 共兩2典A兩2典B兲, and, more general,
the transfer of entanglement is sensibly reduced as a function
of 具N典.
IV. DETUNING EFFECT
From the practical point of view it is important to evaluate
the effects of the off-resonant interaction between the atoms
and their respective cavity fields, which can be actually prepared in nondegenerate optical parametric processes. We assume equal interaction times for both atoms and we consider
the case of resonant interaction for the atom A and offresonant interaction for atom B. As an example we consider
the TMC case, both atoms in the ground state and the value
g␶ = 4.66, corresponding to the maximum entanglement
transfer. In Fig. 4共a兲 we see that up to detuning values on the
order of the inverse interaction time the entanglement is preserved by the off-resonant interaction of atom B. In Fig. 4共b兲
we show the more general case of off resonance for both
atoms, taking equal detuning values for simplicity. The effect
is greater than in the previous case but it is negligible again
up to ⌬B␶ = 1.
Figure 5 shows the analogous behavior for the TWB
states for g␶ = 4.61. We see that near the peak of entangle-
032336-6
PHYSICAL REVIEW A 75, 032336 共2007兲
IMPROVING THE ENTANGLEMENT TRANSFER FROM…
1
ε
1
ε
F
F
a
0.5
0.5
b
c
b
d
a
e
0
0
(a)
ε
5
0
0
<N> 10
(a)
1
F
c
a
b
ε
5
<N>
10
5
<N> 10
1
F
c
0.5
d
c
0.5
e
b
a
0
(b) 0
5
<N> 10
(b)
1
εF
0
0
FIG. 7. The entanglement of formation ⑀F vs 具N典 for a time g␶
of simultaneous presence of both atoms after the delayed injection
of atom A. Atom A is prepared in the ground state and atom B in
superposition states such that 兩B1兩2 = 0 共dashed line兲, 共a兲 0.25, 共b兲
0.50, 共c兲 0.75, 1 共dotted line兲. 共a兲 The TMC case with g␶ = 4.66. 共b兲
The TWB case with g␶ = 4.61.
a
b
0.5
c
ment the TMC states seem more robust to off-resonance interaction than the TWB states.
d
e
0
(c) 0
5
<N>10
V. EFFECT OF DIFFERENT INTERACTION TIMES
<N> 10
In the previous analysis we considered equal coupling
constant and interaction time for both atoms. However, experimentally we may realize conditions such that the parameter g␶ is different for the two interactions due to the limitations in the control of both atomic velocities and injection
times or in the values of the coupling constants 关7兴.
We first consider the effect of different interaction times
at exact resonance and simultaneous injection of both atoms
prepared in the ground state. In Figs. 6共a兲 and 6共b兲 we show
the TMC case for g␶A = 4.66 and g␶A = 11.01, corresponding
to two maxima of entanglement as discussed in the previous
section, and we investigate the effect of different dimensionless interaction times for the atom B such that g␶B 艋 g␶A. We
see that increasing the difference g␶A − g␶B the entanglement
decreases. However for 具N典 ⬍ 4, if g␶B has a value close to
the one corresponding to a maximum, as for g␶A = 1.56 in
Fig. 6共a兲 and g␶A = 7.85 in Fig. 6共b兲, i.e., if g␶A − g␶B ⬵ ␲, the
entanglement again reaches large values. The effect is more
important for g␶A = 11.01 where the entanglement transfer is
the same as for equal interaction times. In Figs. 6共c兲 and
6共d兲, we show an analogous effect for the TWB case. We
finally consider the possibility that atom B enters the cavity
1
εF
0.5
a
b
c
0
0
(d)
e
d
5
FIG. 6. The effect of different interaction times for ⌬A = ⌬B = 0,
and both atoms injected simultaneously in the ground state. In 共a兲
the TMC case for g␶A = 4.66 and g␶B values 共a兲 4.66, 共b兲 4.4, 共c兲 4.2,
共d兲 4.0, 共e兲 3.8 and 1.56 共dashed line兲. In 共b兲 the TMC case for
g␶A = 11.01 and g␶B values 共a兲 11.01, 共b兲 10.8, 共c兲 10.6, 共d兲 10.4, 共e兲
10.2 and 7.85 共dashed line兲. In 共c兲 the TWB case for g␶A = 4.61 and
g␶B values 共a兲 4.61, 共b兲 4.4, 共c兲 4.2, 共d兲 4.0, 共e兲 3.8 and 1.56 共dashed
line兲. In 共d兲 the TWB case for g␶A = 11.03 and g␶B values 共a兲 11.03,
共b兲 10.8, 共c兲 10.6, 共d兲 10.4, 共e兲 10.2 and 7.85 共dashed line兲.
032336-7
PHYSICAL REVIEW A 75, 032336 共2007兲
CASAGRANDE, LULLI, AND PARIS
We found that CV states initially prepared in a two-state
superposition are the most efficient in transferring entanglement to qubits with Bell-like states able to transfer a full ebit
of entanglement. We have then considered multiphoton
preparation as TWB and TMC states and found that there are
large and well-defined regions of interaction parameters
where the transfer of entanglement is effective. At fixed energy 共average number of photons兲 TMC states are more effective in transferring entanglement than TWB states. We
have also found that the 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 transfer of entanglement from nonGaussian states as TMC states.
Overall, we conclude that the scheme analyzed in this
paper is a reliable and robust mechanism for the engineering
of the entanglement between two atomic qubits and that bipartite non-Gaussian states are promising resources in order
to optimize this protocol. Finally, we mention that our analysis may also be employed to assess the entanglement transfer
from radiation to superconducting qubits.
just before atom A. We assume that when atom A enters its
cavity the two-mode field can still be described by Eq. 共1兲.
Due to the interaction with its cavity field the atom B will be
in a superposition state B1兩1典B + B2兩2典B. In this case the
atomic density matrix ␳12
a has only two null elements, ␳23
*
= ␳32
, hence we evaluate the eigenvalues of the nonHermitian matrix numerically. We calculate the amount of
entanglement transferred to the atoms after the time ␶ of their
simultaneous presence into the respective cavities in the case
of exact resonance, equal coupling constant and velocity, assuming atom A prepared in the ground state. In Fig. 7共a兲 we
show the TMC case for g␶ = 4.66 and different values of 兩B1兩2
ranging from 0 共that is for 兩1典A兩2典B兲 to 1 共that is 兩1典A兩1典B兲. We
note that the behavior of ⑀F gradually changes from one limit
case to the other one for 具N典 ⬍ 4, when the photon distribution approaches that of a TSS state. For larger values of 具N典
we see the occurrence of a second peak in the case 兩1典A兩2典B.
In Fig. 7共b兲 we show the TWB case for g␶ = 4.61 and we note
that the gradual change described above occurs for nearly all
values of the mean photon number 具N典.
VI. CONCLUSIONS
ACKNOWLEDGMENTS
In this paper we have addressed the transfer of entanglement from a bipartite state of a continuous-variable system to
a pair of localized qubits. We have assumed that each CV
mode couples to one qubit via the Jaynes-Cummings interaction and have taken into account the degrading effects of
detuning and of different interaction times for the two subsystems. The transfer of entanglement has been assessed by
tracing out the field degrees of freedom after the interaction,
and then evaluating the entanglement of formation of the
reduced atomic density matrix.
再
This work has been supported by MIUR through Project
No. PRIN-2005024254-002. M.G.A.P. thanks Vladylav
Usenko for useful discussions about pair-coherent 共TMC兲
states.
APPENDIX: ATOMIC DENSITY MATRIX ELEMENTS
The elements of the 4 ⫻ 4 atomic density matrix ␳12
a in the
standard basis 兵兩2典A兩2典B , 兩2典A兩1典B , 兩1典A兩2典B , 兩1典A兩1典B其 are as
follows:
⬁
⬁
*
*
␳11共x兲 = 兩c0共x兲兩2 兩A2兩2兩B2兩2 兺 兩f j共x兲兩2兩UA11共j, ␶兲兩2兩UB11共j, ␶兲兩2 + A*1A2B*1B2 兺 f j共x兲f *j+1共x兲UA11共j, ␶兲UB11共j, ␶兲UA12
共j, ␶兲UB12
共j, ␶兲
j=0
j=0
⬁
⬁
+ 兩A2兩2兩B1兩2 兺 兩f j+1共x兲兩2兩UA11共j + 1, ␶兲兩2兩UB12共j, ␶兲兩2 + 兩A1兩2兩B2兩2 兺 兩f j+1共x兲兩2兩UA12共j, ␶兲兩2兩UB11共j + 1, ␶兲兩2
j=0
冎
j=0
⬁
⬁
j=0
j=0
*
*
共j, ␶兲UB11
共j, ␶兲 ,
+ 兩A1兩2兩B1兩2 兺 兩f j+1共x兲兩2兩UA12共j, ␶兲兩2兩UB12共j, ␶兲兩2 + A1A*2B1B*2 兺 f j+1共x兲f *j 共x兲UA12共j, ␶兲UB12共j, ␶兲UA11
再
⬁
⬁
␳22共x兲 = 兩c0共x兲兩2 兩A2兩2兩B2兩2 兺 兩f j−1共x兲兩2兩UA11共j − 1, ␶兲兩2兩UB21共j − 1, ␶兲兩2 + A*1A2B*1B2 兺 f j−1共x兲f *j 共x兲UA11共j − 1, ␶兲UB21共j
j=1
冋兺
j=1
⬁
*
*
− 1, ␶兲UA12
共j − 1, ␶兲UB22
共j − 1, ␶兲 + 兩A2兩2兩B1兩2
兩f j共x兲兩2兩UA11共j, ␶兲兩2兩UB22共j − 1, ␶兲兩2 + 兩UA11共0, ␶兲兩2
j=1
⬁
⬁
+ 兩A1兩 兩B2兩
2
2
兩f j共x兲兩 兩UA12共j − 1, ␶兲兩 兩UB21共j, ␶兲兩
兺
j=1
2
2
2
+ 兩A1兩 兩B1兩
2
032336-8
2
册
兩f j共x兲兩2兩UA12共j − 1, ␶兲兩2兩UB22共j − 1, ␶兲兩2
兺
j=1
共A1兲
PHYSICAL REVIEW A 75, 032336 共2007兲
IMPROVING THE ENTANGLEMENT TRANSFER FROM…
⬁
+
␳33共x兲 = 兩c0共x兲兩
A1A*2B1B*2
2
再
*
*
f j共x兲f *j−1共x兲UA12共j − 1, ␶兲UB22共j − 1, ␶兲UA11
共j − 1, ␶兲UB21
共j − 1, ␶兲
兺
j=1
兩A2兩 兩B2兩
冋兺
2
兩f j共x兲兩 兩UA21共j, ␶兲兩 兩UB11共j, ␶兲兩
兺
j=0
2
2
2
+
A*1A2B*1B2
*
*
f j共x兲f *j+1共x兲UA21共j, ␶兲UB11共j, ␶兲UA22
共j, ␶兲UB12
共j, ␶兲
兺
j=0
⬁
+ 兩A1兩 兩B2兩
2
2
兩f j共x兲兩 兩UA22共j − 1, ␶兲兩 兩UB11共j, ␶兲兩 + 兩UB11共0, ␶兲兩
2
2
2
j=1
+ 兩A1兩 兩B1兩
2
兩f j+1共x兲兩 兩UA22共j, ␶兲兩 兩UB12共j, ␶兲兩
兺
j=0
2
再
2
2
册
⬁
+ 兩A2兩 兩B1兩
2
2
兩f j+1共x兲兩2兩UA21共j + 1, ␶兲兩2兩UB12共j, ␶兲兩2
兺
j=0
⬁
⬁
2
共A2兲
,
⬁
⬁
2
冎
2
+
A1A*2B1B*2
*
*
f j+1共x兲f *j 共x兲UA22共j, ␶兲UB12共j, ␶兲UA21
共j, ␶兲UB11
共j, ␶兲
兺
j=0
⬁
冎
,
共A3兲
⬁
␳44共x兲 = 兩c0共x兲兩2 兩A2兩2兩B2兩2 兺 兩f j−1共x兲兩2兩UA21共j − 1, ␶兲兩2兩UB21共j − 1, ␶兲兩2 + A*1A2B*1B2 兺 f j−1共x兲f *j 共x兲UA21共j − 1, ␶兲UB21共j
j=1
冋兺
j=1
⬁
*
*
− 1, ␶兲UA22
共j − 1, ␶兲UB22
共j − 1, ␶兲 + 兩A2兩2兩B1兩2
冋兺
册
兩f j共x兲兩2兩UA21共j, ␶兲兩2兩UB22共j − 1, ␶兲兩2 + 兩UA21共0, ␶兲兩2
j=1
⬁
+ 兩A1兩 兩B2兩
2
2
兩f j共x兲兩 兩UA22共j − 1, ␶兲兩 兩UB21共j, ␶兲兩 + 兩UB21共0, ␶兲兩
2
2
2
j=1
− 1, ␶兲兩 + 1 +
2
␳12共x兲 = 兩c0共x兲兩
2
2
册
⬁
+ 兩A1兩 兩B1兩
再 冋
+
2
兩f j共x兲兩2兩UA22共j − 1, ␶兲兩2 + 兩UB22共j
j=1
*
*
f j共x兲f *j−1共x兲UA22共j − 1, ␶兲UB22共j − 1, ␶兲UA21
共j − 1, ␶兲UB21
共j − 1, ␶兲
兺
j=1
⬁
*
兩f j共x兲兩2兩UA11共j, ␶兲兩2UB11共j, ␶兲UB22
共j − 1, ␶兲 + 兩UA11共0, ␶兲兩2UB11共0, ␶兲
兺
j=1
⬁
兩A1兩 B2B*1
2
⬁
A1A*2B1B*2
兩A2兩2B*1B2
2
冋兺
兩f j共x兲兩 兩UA12共j − 1, ␶兲兩
兺
j=1
2
册
冎
共A4兲
,
册
⬁
2
冋兺
*
共j
UB11共j, ␶兲UB22
− 1, ␶兲 +
A1A*2兩B2兩2
*
共j
f j共x兲f *j−1共x兲UA12共j − 1, ␶兲UB11共j, ␶兲UA11
兺
j=1
⬁
*
− 1, ␶兲UB21
共j − 1, ␶兲 + A1A*2兩B1兩2
j=1
*
共0, ␶兲
+ f 1共x兲UA12共0, ␶兲UB12共0, ␶兲UA11
再
*
*
f j+1共x兲f *j 共x兲UA12共j, ␶兲UB12共j, ␶兲UA11
共j, ␶兲UB22
共j − 1, ␶兲
册冎
共A5兲
,
⬁
*
*
␳13共x兲 = 兩c0共x兲兩2 兩A2兩2B1B*2 兺 f j+1共x兲f *j 共x兲UA11共j + 1, ␶兲UB12共j, ␶兲UA21
共j, ␶兲UB11
共j, ␶兲
+ 兩A1兩2B1B*2
冋兺
冋兺
j=0
⬁
*
*
*
f j+1共x兲f *j 共x兲UA12共j, ␶兲UB12共j, ␶兲UA22
共j − 1, ␶兲UB11
共j, ␶兲 + f 1共x兲UA12共0, ␶兲UB12共0, ␶兲UB11
共0, ␶兲
册
j=1
⬁
+ A*1A2兩B2兩2
j=1
册
⬁
*
兩f j共x兲兩2UA11共j, ␶兲兩UB11共j, ␶兲兩2UA22
共j − 1, ␶兲 + UA11共0, ␶兲兩UB11共0, ␶兲兩2 + A*1A2兩B1兩2 兺 兩f j+1共x兲兩2UA11共j
冎
*
+ 1, ␶兲兩UB12共j, ␶兲兩2UA22
共j, ␶兲 ,
j=0
共A6兲
032336-9
PHYSICAL REVIEW A 75, 032336 共2007兲
CASAGRANDE, LULLI, AND PARIS
␳14共x兲 = 兩c0共x兲兩
+
2
再
⬁
兩A2兩 兩B2兩
2
冋兺
冋兺
冋兺
冋兺
2
*
*
f j共x兲f *j−1共x兲UA11共j, ␶兲UB11共j, ␶兲UA21
共j − 1, ␶兲UB21
共j − 1, ␶兲
兺
j=1
⬁
A*1A2B*1B2
*
*
兩f j共x兲兩2UA11共j, ␶兲UB11共j, ␶兲UA22
共j − 1, ␶兲UB22
共j − 1, ␶兲 + UA11共0, ␶兲UB11共0, ␶兲
j=1
册
⬁
+ 兩A2兩2兩B1兩2
*
*
*
f j共x兲f *j−1共x兲UA11共j + 1, ␶兲UB12共j, ␶兲UA21
共j, ␶兲UB22
共j − 1, ␶兲 + f 1共x兲UA11共1, ␶兲UB12共0, ␶兲UA21
共0, ␶兲
j=1
⬁
+ 兩A1兩2兩B2兩2
*
*
*
f j+1共x兲f *j 共x兲UA12共j, ␶兲UB11共j + 1, ␶兲UA22
共j − 1, ␶兲UB21
共j, ␶兲 + f 1共x兲UA12共0, ␶兲UB11共1, ␶兲UB12
共0, ␶兲
j=1
⬁
+ 兩A1兩2兩B1兩2
册
册
*
*
f j+1共x兲f *j 共x兲UA12共j, ␶兲UB12共j, ␶兲UA22
共j − 1, ␶兲UB22
共j − 1, ␶兲 + f 1共x兲UA12共0, ␶兲UB12共0, ␶兲
冎
j=1
⬁
册
*
*
+ A1A*2B1B*2 兺 f j+1共x兲f *j−1共x兲UA12共j, ␶兲UB12共j, ␶兲UA21
共j − 1, ␶兲UB21
共j − 1, ␶兲 ,
j=1
冋兺
共A7兲
册
⬁
␳23共x兲 =
␳24共x兲 = 兩c0共x兲兩
+
2
再
兩c0共x兲兩2A*1A2B1B*2
*
*
*
兩f j共x兲兩2UA11共j, ␶兲UB22共j − 1, ␶兲UA22
共j − 1, ␶兲UB11
共j, ␶兲 + UA11共0, ␶兲UB11
共0, ␶兲 ,
j=0
共A8兲
⬁
兩A2兩 B1B*2
2
冋兺
冋兺
*
*
f j共x兲f *j−1共x兲UA11共j, ␶兲UB22共j − 1, ␶兲UA21
共j − 1, ␶兲UB21
共j − 1, ␶兲
兺
j=1
⬁
兩A1兩 B1B*2
2
*
*
*
f j+1共x兲f *j 共x兲UA12共j, ␶兲UB22共j, ␶兲UA22
共j − 1, ␶兲UB21
共j, ␶兲 + f 1共x兲UA12共0, ␶兲UB22共0, ␶兲UB21
共0, ␶兲
册
j=1
⬁
+ A*1A2兩B2兩2
*
兩f j共x兲兩2UA11共j, ␶兲兩UB21共j, ␶兲兩2UA22
共j − 1, ␶兲 + UA11共0, ␶兲兩UB21共0, ␶兲兩2 + A*1A2兩B1兩2
j=1
*
共j − 1, ␶兲 + UA11共0, ␶兲
⫻兩UB22共j − 1, ␶兲兩2UA12
再 冋
␳34共x兲 = 兩c0共x兲兩2 兩A2兩2B*1B2
冋兺
册冎
⬁
兩f j共x兲兩2UA11共j, ␶兲
兺
j=1
共A9兲
,
⬁
*
兩f j共x兲兩2兩UA21共j, ␶兲兩2UB11共j, ␶兲UB22
共j − 1, ␶兲 + 兩UA21共0, ␶兲兩2UB11共0, ␶兲
兺
j=1
⬁
+ 兩A1兩2B*1B2
冋
册
*
*
兩f j共x兲兩2UA22共j − 1, ␶兲UB11共j, ␶兲UA22
共j − 1, ␶兲UB22
共j − 1, ␶兲 + UB11共0, ␶兲
j=1
册
册
⬁
+
A1A*2兩B2兩2
+
A*1A2兩B1兩2
*
*
f j共x兲f *j−1共x兲UA22共j − 1, ␶兲UB11共j, ␶兲UA21
共j − 1, ␶兲UB21
共j − 1, ␶兲
兺
j=1
冋兺
⬁
*
*
*
f j+1共x兲f *j 共x兲UA22共j, ␶兲UB12共j, ␶兲UA21
共j, ␶兲UB22
共j − 1, ␶兲 + f 1共x兲UA22共0, ␶兲UB12共0, ␶兲UA21
共0, ␶兲
j=1
册冎
.
共A10兲
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