Robust entanglement in atomic systems via
⌳-type processes
M. Ali Can, O¨ zgu¨r C¸akır, Alexander Klyachko, and Alexander ShumovskyFaculty of Science, Bilkent University, Bilkent, Ankara 06533, Turkey 共Received 26 November 2002; published 13 August 2003兲
It is shown that the system of two three-level atoms in the⌳ configuration in a cavity can evolve into a long-lived maximum entangled state if the Stokes photons vanish from the cavity by means of either leakage or damping. The difference in the evolution picture corresponding to the general model and effective model with two-photon process in a two-level system is discussed.
DOI: 10.1103/PhysRevA.68.022305 PACS number共s兲: 03.67.Mn, 42.50.Ct
I. INTRODUCTION
During the last decade, the problem of engineered en-tanglement in atomic systems has attracted a great deal of interest共see Refs. 关1–8兴, and references therein兲. In particu-lar, the atomic entangled states were successfully realized through the use of cavity QED 关1兴 and the technique of ion traps 关3兴. At present, one of the most important problems under consideration is how to make a long-lived and easy-monitored atomic entangled state with existing experimental techniques.
An interesting scheme has been proposed recently关9兴. In this scheme, the two identical atoms are placed into a cavity tuned to resonance with one of the dipole-allowed transi-tions. Initially, both atoms are prepared in the ground state, while the cavity field consists of a single photon. It is easy to show that the atom-field interaction leads in this case to a maximum atomic entangled state such that the single excita-tion is shared between the two atoms with equal probability. It was proposed in Ref.关9兴 to consider the absence of photon leakage from a nonideal cavity as a signal that the atomic entangled state has been created. The scheme can also be generalized to the case of any even number of atoms 2n, sharing n excitations. In this case, the atomic entangled states are represented by the so-called SU共2兲 phase states 关10兴.
Another interesting proposal is to use a strong coherent drive to provide the multipartite entanglement in a system of two-level atoms in a high-Q cavity 关11兴. This approach can be used to produce the atomic entanglement as well as that of atoms and cavity modes and even of different cavity modes. In the schemes of Refs. 关9,11兴, the lifetime of the en-tanglement is defined by the specific time scale of the dipole-allowed radiative processes in atoms. Unfortunately, this life-time is usually quite short关12兴.
Generally speaking, the lifetime of atomic entanglement is specified by the interaction of atoms with environment. For example, in the model of Ref.关10兴, the environment is represented by the vacuum field that causes emission of a photon getting out of the cavity.
The interaction with environment can also be used to cre-ate a long-lived entanglement in atomic systems. For ex-ample, the initial nonentangled system may evolve to an en-tangled state connected with the atomic states that cannot be depopulated by radiation decay. In this case, the lifetime of the entangled state is specified by considerably long nonra-diative processes. A possible realization is provided by the
use of a three-level ⌳-type process instead of the two-level scheme of Refs. 关9,11兴. The process is illustrated by Fig. 1. Here, the levels 1 and 2 as well as the levels 2 and 3 are connected by the electric dipole transitions. In turn, the di-pole transition between the levels 3 and 1 is forbidden be-cause of the parity conservation 关13兴. The absorption of pumping photon by the transition 1↔2 with further jump of the electron to the level 3 can be interpreted as a kind of Raman process in atomic system with emission of Stokes photon共see Ref. 关14兴, and references therein兲. It is clear that the atom excited to the level 3 can change the state either by absorption of the Stokes photon resonant with respect to the transition 3↔2 or trough a nonradiative decay.
Now we assume that the two identical ⌳-type atoms are placed into a cavity of high quality with respect to the pump-ing photons resonant to the transition 1↔2 and also that the Stokes photons created by the transition 2→3 either leave the cavity freely or are absorbed by the cavity walls. Then, the atom-field interaction may lead to creation of the maxi-mum entangled atomic state
1
冑
2共兩3,1典⫹兩1,3典
), 共1兲FIG. 1. Scheme of the process and configuration of atomic lev-els and transitions.
whose lifetime is determined by the slow processes of non-radiative decay.
The above scheme has been proposed in Ref. 关10兴 and briefly discussed in Ref. 关15兴. The main objective of the present paper is to consider in detail the evolution towards the long-lived atomic entangled state共1兲.
The paper is organized as follows. In Sec. II, we discuss the model Hamiltonians that can be used to describe the process under consideration. Viz, we discuss the model of the one-photon three-level interaction and an effective model of the two-photon process in a two-level system. Then, in Sec. III, we examine the irreversible dynamics, leading to state共1兲 in a cavity with leakage of Stokes photons. We show that both models describe the exponential evolution to state
共1兲. At the same time, the effective model, corresponding to
a rough time scale, is unable to take into account the possible oscillations of population between the states 1 and 2. Let us stress that the monitoring of Stokes photons outside the cav-ity can be used to detect the atomic entangled state共1兲 in this case.
Another way of creation of state共1兲 through the use of a cavity with very low quality with respect to the Stokes pho-tons is discussed in Sec. IV. Finally, in Sec. V, we discuss the possible realization of entanglement in the system of⌳-type atoms.
II. THE MODELS OF THE⌳-TYPE PROCESS Assume that a system of N identical three-level atoms with ⌳-type transitions shown in Fig. 1 interacts with the cavity mode close to resonance with 1↔2 transition and with the Stokes radiation that can leave the cavity freely. Then, following Refs. 关13,14兴, we can choose the model Hamiltonian in the following form:
H⫽H0⫹Hint, H0⫽Pa⫹PaP⫹
兺
k Sk aSk† aSk ⫹兺
f 关21 R22共 f 兲⫹31R33共 f 兲兴, 共2兲 Hint⫽兺
f P R21共 f 兲aP⫹兺
f ,k Sk R23共 f 兲aSk⫹H.c.. 共3兲Here, aPdenotes the photon annihilation operator of the cav-ity mode with frequencyP, aSkis the annihilation operator of Stokes photon with frequencySk, and 21,31are the energies of the corresponding atomic levels with respect to the ground level 1. The operator
Ri j共 f 兲⫽兩if
典具
jf兩describes the transition from level j to level i and index f marks the number of atom. In Eq. 共3兲, P and Sk are the coupling constants, specifying the dipole transitions 2↔1 and 3↔2, respectively. Summation over k in Eq. 共3兲 implies
that the Stokes photons do not feel presence of the cavity walls. This summation involves the modes, corresponding to the natural line breadth near
S⬅23⫽21⫺31. 共4兲
Apart from the total electron occupation number, the model, Eqs. 共2兲 and 共3兲, has two integrals of motion
NP⫽aP⫹aP⫹
兺
f 兵 R22共 f 兲⫹R33共 f 兲其, NS⫽兺
k aSk⫹aSk⫺兺
f R33共 f 兲. 共5兲Consider the system of only two atoms. Assume that both atoms are prepared initially in the ground state 1, the cavity contains a single photon of frequency P, and the Stokes field is in the vacuum state. Then, because of the integrals of motion 共5兲, the evolution of the system occurs in a single-excitation domain of the Hilbert space spanned by the vec-tors 兩1
典⫽兩1,1典
丢兩1P典
丢兩0S典
, 兩2 (⫾)典
⫽ 1冑
2共兩1,2典⫾兩2,1典
)丢兩0P典
丢兩0S典
, 共6兲 兩3k (⫾)典⫽
1冑
2共兩1,3典⫾兩3,1典)丢兩0P典
丢兩1Sk典
.By construction, the four states 共6兲 labeled by the super-scripts ⫾ manifest the maximum entanglement. It is easily seen that the action of operator 共3兲 cannot transform the states
兵兩1
典
,兩2 (⫹)典
,兩3k(⫹)
典
其 共7兲 into the states兵兩2 (⫺)
典
,兩3k(⫺)
典
其 共8兲 and vice versa. Thus, the evolution of the system from the initial nonexcited state 兩1典
takes place in the subspace spanned by only three vectors 共7兲. Thus, states 共8兲 can be discarded.Instead of the one-photon three-level model described by the Hamiltonian, Eqs.共2兲 and 共3兲, an effective model of two-photon process can also be used under a certain condition
关16,17兴. Viz., if the cavity is tuned consistent with
two-photon energy conservation, i.e., E3⫺E1⫽1⫺2, we are left only with one detuning parameter
⌬⫽E1⫺E2⫺1⫽E2⫺E3⫺S.
Here, Ei denotes the energy of corresponding atomic level. Then, it was shown in Ref. 关16兴 that under the condition
⌬ⰇE3⫺E1,
the dynamics of the system is governed by the effective Hamiltonian of the form
He f f⫽ PaP †a P⫹SaS †a S⫹
兺
f 31 R33共 f 兲 ⫹兺
f 关R31共 f 兲aS † aP⫹aP † asR13共 f 兲兴. 共9兲This Hamiltonian 共9兲 describes an effective level two-photon system with simultaneous absorption of pumping photon and creation of Stokes photon and vice versa. Here is an effective coupling constant.
III. DYNAMICS DESCRIBED BY THE HAMILTONIAN †EQS. „2… AND „3…‡
Under the assumption that there are only two three-level
⌳-type atoms in the cavity and that the system is initially
prepared in the state兩1
典
in Eq.共5兲, in view of the results of previous section, we should choose the time-dependent wave function as follows:兩⌿共t兲典⫽C1兩1
典
⫹C2兩2典
⫹兺
kC3k兩3k
典
, 共10兲C1共0兲⫽1, C2共0兲⫽0, ᭙ k Ck共0兲⫽0, 共11兲 using the reduced basis 共7兲. Here, we use the notations
兩2
典
⬅兩2(⫹)
典
and 兩3k
典
⬅兩3k(⫹)
典
, for simplicity. The time-dependent Schro¨dinger equation with the Hamiltonian, Eqs.共2兲 and 共3兲, then leads to the following set of equations for
the coefficients in Eq. 共11兲:
iC˙1⫽PC1⫹P
冑
2C2,iC˙2⫽21C2⫹P冑2C1⫹
兺
k SkC3k, 共12兲
iC˙3k⫽共31⫹Sk兲C3k⫹SkC2.
To find solutions of Eq. 共12兲, let us represent the last equa-tion in Eq. 共12兲 in the form
C3k共t兲⫽⫺iSk
冕
0 tC2共兲ei(31⫹Sk)(⫺t)d. 共13兲 Then, we should take the time derivative on both sides of the first equation in Eq. 共12兲 and substitute the second equation together with integral representation共13兲. We get
iC¨1⫽共P⫹21兲C˙1⫹i共21P⫺2P 2兲C 1 ⫺
兺
k Sk 2冕
0 t 共iC˙1⫺PC1兲e i(31⫹Sk)(⫺t)d.Carrying out the integration by parts, we get the following integro-differential equation with respect to only one un-known variable C1(t): iC¨1⫽共P⫹21兲C˙1⫹i
冉
21P⫺2P2⫺兺
k Sk 2冊
C 1 ⫹i兺
k Sk 2 e⫺i(31⫹Sk)t⫺兺
k Sk 2 共31⫹Sk⫺P兲 ⫻冕
0 t C1共兲ei(31⫹Sk)(⫺t)d. 共14兲In contrast to the conventional Wigner-Weisskopf theory
共e.g., see Ref. 关18兴兲, Eq. 共14兲 contains the second-order
de-rivatives. This integro-differential equation 共14兲 can be ana-lyzed through the use of Laplace transformation as in the Wigner-Weisskopf theory关18,19兴. We get
冕
0 ⬁ C1共t兲e⫺stdt⫽L共C1兲,冕
0 ⬁ C˙1共t兲e⫺stdt⫽sL⫺1,冕
0 ⬁ C¨1共t兲e⫺stdt⫽s2L⫺s⫺C˙1共0兲⫽s2L⫺s⫹iP.Then, Eq.共14兲 is reduced to the following algebraic equation with respect toL: L
冋
is2⫺s共P⫹21兲⫺i冉
21P⫺2P 2⫺兺
k Sk 2冊
册
⫽⫺共P⫹21兲⫹i兺
k Sk 2 s⫹i共31⫹Sk兲 s2⫹共 31⫹Sk兲2 ⫺冕
0 ⬁ e⫺st再
兺
k Sk 2 共 31⫹Sk⫺P兲 ⫻冕
0 t C1共兲ei(31⫹Sk)(⫺t)d冎
dt. 共15兲 The last term in the right-hand side of this expression can be represented as follows:冕
0 ⬁ e⫺st再
冕
0 t ei(31⫹Sk)(⫺t)C 1共兲d冎
dt ⫽冕
0 ⬁ C1共兲ei(31⫹Sk)d冕
⬁ e⫺[s⫹i(31⫹Sk)]tdt ⫽s⫹i共L 31⫹Sk兲 .L⫽
冋
i兺
k Sk 2 s⫹i共31⫹Sk兲 s2⫹共31⫹Sk兲2 ⫺共P⫹21兲册
⫻冋
is2⫺s共P⫹21兲⫹i冉
P21⫺2P 2⫺兺
k Sk 2冊
⫹兺
k Sk 2 共 31⫹Sk⫺P兲 s⫹i共31⫹Sk兲册
⫺1 . 共16兲Then, the exact form of the time behavior of the coefficient C1(t) in Eq.共10兲 is governed by the inverse Laplace trans-formation: C1共t兲⫽ 1 2i
冕
⑀⫺i⬁ ⑀⫹i⬁ estL共s兲ds, 共17兲where⑀is a infinitesimal real positive number and s is con-sidered to be a complex parameter. As soon as the explicit time behavior of C1(t) is known, the other coefficients in Eq.共10兲 can be defined through the use of Eqs. 共12兲 and 共13兲. In particular, it follows from Eqs.共6兲, 共10兲, and 共11兲 that the probability to have the atomic entangled state 共1兲 has the form
兺
k 兩C3k兩 2⫽1⫺兩C 1共t兲兩2⫺兩C2共t兲兩2⫽1⫺兩C1共t兲兩2 ⫺兩iC˙1共t兲⫺PC1共t兲兩2 2P2 . 共18兲Thus, Eqs.共16兲 and 共17兲 completely determine the probabil-ity of having the robust entangled state共1兲. It can be shown that Eq.共16兲 describes the reversible, Poincare´-type behavior
共e.g., see Ref. 关19兴兲. The irreversible evolution can be
ob-tained under the further assumption that the atomic transition 2↔3 interacts with continuum of Stokes modes rather than with a discrete spectrum
兺
k •••→冕
⫺⬁⬁
•••共兲d, ⫽ck.
Here, measure ()d defines the density of states of Stokes photons with different frequencies.
Let us stress that, unlike the conventional Wigner-Weisskopf theory, Eqs. 共16兲 and 共17兲 describe a superposi-tion of exponential decay and harmonic oscillasuperposi-tions. The lat-ter are caused by the inlat-teraction between the 1↔2 transitions and cavity field.
Further analysis shows that the coefficients C1 and C2 have the form
C1共t兲⬇
冋
⫺ 22 共⌫⫺i⌬兲2e (⫺⌫⫹i⌬)t ⫹冉
1⫹ 2 2 共⌫⫺i⌬兲2冊
e ⫺[22/(⌫⫺i⌬)]t册
e⫺iPt, C2共t兲⬇⫺冑
2 i⌫⫹⌬ 关e ⫺⌫t⫺e⫺[22/(⌫⫺i⌬)⫹i⌬]t 兴e⫺i21t 共19兲to the second order in/(⌫⫺i⌬P). Here,
⌬P⫽P⫺12
is the detuning factor for the pumping mode and
⌫⫽共S兲Sk共k⫽S/c兲, S⫽21⫺31.
Equation 共19兲 proves to be a good approximation because
(S)Ⰷ1 and ⌫ⰇP,Sk.
It is seen that Eq.共19兲 describes the damped oscillations of the coefficient C1(t) in Eq. 共10兲. According to Eq. 共11兲, C2(t) manifests similar behavior. Thus, the probability共18兲 to get the robust entangled state tends to 1 as t→⬁ 共see Fig. 2兲. The decay time␥⫺1 is defined by the coupling constant and detuning parameter for the pumping mode and by the width of the Stokes line ⌫. The contribution of oscillations into the evolution described by Eq. 共19兲 is un-noticeable at small detuning⌬P⭐⌫ and becomes apparent at ⌬PⰇ⌫ 共see Fig. 2兲.
Similar result can also be obtained in terms of the effec-tive Hamiltonian 共9兲 关15兴. It should be stressed that the as-sumptions made in the process of derivation of Eq.共9兲 lead to an effective roughening of the time scale. In fact, the effective removal of the level 2 leads to the negligence of the Rabi oscillations between the levels 1 and 2. Therefore, the effective model共9兲 gives only rough picture of purely expo-nential evolution of probability 共18兲.
While the atomic system evolves to the maximum en-tangled state共1兲, the Stokes photon leaves the cavity. Thus, the observation of Stokes photon outside the cavity can be considered as a signal that the robust entangled state has been prepared.
FIG. 2. Time evolution of probability 共18兲 to have the robust entanglement atP⫽0.001⌫ for 共I兲 ⌬P⫽0 and 共II兲 ⌬P⫽⌫.
IV. CAVITY WITH ABSORPTION OF STOKES PHOTONS The atomic entangled state 共1兲 can also be realized when the Stokes mode is strongly damped in the cavity. For sim-plicity, we again assume no damping for the pumping mode. At the same time, the Stokes photons are supposed to be absorbed by the cavity walls. This situation corresponds to a number of experiments with single-atom Rydberg maser
关20,21兴. In this case, the effect of damping can be calculated
through the use of the so-called dressed-atom approximation
关22兴.
The model Hamiltonian, describing the process under consideration, can be chosen as follows:
H⫽H0⫹Hint, H0⫽PaP † aP⫹SaS † aS⫹
兺
f 关21 R22共 f 兲⫹31R33共 f 兲兴, 共20兲 Hint⫽兺
f 关P R21共 f 兲aP⫹SR23共 f 兲aS兴⫹H.c.. This corresponds to the single-Stokes-mode approximation in Eqs.共2兲 and 共3兲. The eigenstates of Hamiltonian 共20兲 have the form 兩0典
⫽ S ⑀ 兩1典
⫺ P冑2 ⑀ 兩3典
, 兩⫾典
⫽⫾ P ⑀ 兩1典
⫹ 1冑
2兩2典
⫾ S ⑀冑
2兩3典
, 共21兲 where 兩1典
coincides with the first state in Eq. 共6兲, 兩2典
⫽兩2 (⫹)
典
, and 兩3典
⫽ 1冑
2共兩3,1典⫹兩1,3典
)丢兩0P典
丢兩1S典
. In Eq.共21兲, ⑀⫽冑
2P2⫹S2. Under the assumption of exact resonanceP⫽21⫽31⫹S
that we use hereafter for simplicity, the corresponding eigen-values are
H兩0
典
⫽P兩0典
, H兩⫾典⫽共
P⫾⑀兲兩⫾典
. Besides this, there is one more eigenstate兩4
典⫽
1冑
2共兩3,1典⫹兩1,3典)丢兩0P典
丢兩0S典
, 共22兲 such thatH兩4
典
⫽31兩4典
.It is clear that this eigenstate corresponds to the maximum atomic entanglement 共1兲. Physically, this state is achieved when the Stokes photon is absorbed by the cavity walls.
To take into account the cavity damping of Stokes pho-tons, consider the interaction with a ‘‘phonon reservoir’’ re-sponsible for the absorption of photons by cavity walls关18兴. Then, Hamiltonian 共20兲 should be supplemented with the term Hloss⫽
兺
q q共bq †a S⫹aS †b q兲⫹兺
q ⍀q bq†bq, 共23兲 where bq,bq †are the Bose operators of ‘‘phonons’’ in the cavity walls.
The density matrix of the system can be chosen as fol-lows:
共t兲⫽
兺
j,ᐉ jᐉ共t兲兩j
典具
ᐉ兩,j ,ᐉ⫽0,⫾,4, 共24兲
where 兩j
典
are eigenstates 共21兲 and 共22兲 and jᐉ(t) is the time-dependent c number.With the total Hamiltonian Htot⫽H⫹Hloss
in hand, we can now write the Master Equation, eliminating the cavity degrees of freedom共e.g., see Ref. 关23兴兲,
˙⫽⫺i关H,兴⫹兵2a SaS †⫺a S †a S⫺aS †a S其, 共25兲 so that the contribution of Eq. 共23兲 is taken into account effectively through the Liouville term. Here 1/ is the life-time of a Stokes photon in the cavity and Q⫽31/ is the quality factor with respect to the Stokes photons. Let us choose the same initial condition as in the preceding section, so that
共0兲⫽兩1
典具
1兩, 共26兲where the initial state 兩1
典
is expressed in terms of eigen-states 共21兲 as follows: 兩1典⫽
S ⑀ 兩0典⫹
P ⑀ 共兩⫹典
⫺兩⫺典
).Equation 共25兲 can now be solved numerically at different values of parameter , specifying the absorption of Stokes photons. The results are shown in Fig. 3. It is seen that the system evolves to the robust atomic entangled state 共1兲. The stairslike structure is again caused by competition between the transitions 1↔2 and 2↔3. Although such a behavior is an inherent property of the model under consideration, the stairs become more visible with increase of 共see Fig. 3兲.
A similar result can also be obtained within the frame-work of effective model with Hamiltonian共9兲 and the damp-ing described by Eq. 共23兲. In this case, the density matrix consists of only six elements because the state 兩0
典
in Eq.共21兲 should be discarded and the states 兩⫾
典
are changed by兩⫾
典⫽
1
冑
2共兩1典⫾兩
3典
), with the eigenvalues⫾⫽P⫾
冑
2.It should be stressed that the effective model does not show the stairslike behavior of44(t).
V. SUMMARY AND DISCUSSION
In this paper, we have studied the quantum dynamics of a system of two three-level atoms in the⌳ configuration inter-acting with two modes of quantized electromagnetic field in a cavity under the assumption that the Stokes-mode photons either leave the cavity freely or are damped rapidly. It is shown that in both cases, the system evolves from the state when both atoms are in the ground state and cavity contains a pumping photon into the robust entangled state 共1兲. The lifetime of this final state is defined completely by the non-radiative processes and is therefore relatively long.
In the case of cavity transparent for the Stokes photons, the creation of Stokes photon signalizes the rise of atomic entanglement. Such a photon can be monitored outside the cavity.
Let us stress that the general models with Hamiltonians
共2兲, 共3兲, and 共20兲, which take into account all the three
atomic levels, admit a certain peculiarities in the evolution
towards the robust entangled state caused by the competition of transitions 1↔2 and 2↔3. The effective model with adiabatically eliminated highest excited level is incapable of description of these peculiarities, while predicts correct asymptotic behavior. Moreover, the general model admits also a number of intermediate maximum entangled states
关兩2
典
and兩3k典
in Eq.共6兲兴 that do not exist in the effective model. Unfortunately, the lifetime of these entangled states are defined by the dipole radiative processes and are there-fore too short.One of the most important conditions of experimental re-alization of the robust entanglement discussed in this paper is that the transitions 1↔2 and 2↔3, used for absorption of pumping photons and generation of Stokes photons, should have quite different frequencies. The considerable difference of frequencies 21 and 23 makes it possible to design a multimode cavity with high quality with respect to12, per-mitting either leakage or strong absorption of Stokes pho-tons. An important example is provided by the 3S↔4P and 4 P↔4S transitions in sodium atom and similar transitions in other alkaline atoms共see Ref. 关24兴兲. These atoms are widely used in quantum optics, in particular, in investigation of Bose-Einstein condensation 关25兴. ⌳-type structures obeying the condition12Ⰷ23can also be found in other atoms and molecules关24兴. The multimode cavities are also well known
关26兴. In particular, the cavities with necessary properties may
be assembled using distributed Bragg reflectors 共DBR兲 and double DBR structures to single out two different wave-lengths 关27兴.
The initial state of the system can be prepared in the same way as in Ref. 关21兴. The atoms can propagate through the cavity, using either the same opening or two different open-ings. The velocity of atoms should be chosen in a proper way so that the time they spend in the cavity will be
ⰇP⫺1,S⫺1. All measurements aimed at the detection of atomic entanglement can be performed outside the cavity. Thus, the discussed realization of robust entanglement seems to be feasible with the present experimental technique.
Although our results were obtained for a system of two atoms, they can be generalized with ease to the case of big atomic clusters, using the method of Ref. 关10兴. In fact, it is possible to show that a certain robust entanglement can be obtained in a system with any even number 2N of three-level
⌳-type atoms initially prepared in the ground state and
inter-acting with N pumping photons.
ACKNOWLEDGMENTS
One of the authors共A.Sh.兲 would like to thank Professor A.V. Andreev, Professor J.H. Eberly, Professor A. Vourdas, and Professor D.G. Welsch for many useful discussions.
关1兴 E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune,
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