Generation of infrared entangled light in asymmetric semiconductor quantum wells

Generation of infrared entangled light in asymmetric semiconductor quantum wells

Optics Communications 283 (2010) 5279–5284 Contents lists available at ScienceDirect Optics Communications j o u r n a l h o m e p a g e : w w w. e ...

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Optics Communications 283 (2010) 5279–5284

Contents lists available at ScienceDirect

Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Generation of infrared entangled light in asymmetric semiconductor quantum wells Xin-You Lü a,⁎, Jing Wu b, Li-Li Zheng a, Pei Huang c a b c

School of Physics, Ludong University, Yantai 264025, PR China Institute of Advanced Nanophotonics State, Key Lab of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, PR China Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, PR China

a r t i c l e

i n f o

Article history: Received 25 May 2010 Received in revised form 1 August 2010 Accepted 5 August 2010 Keywords: Continuous-variable entanglement Asymmetric semiconductor quantum wells

a b s t r a c t We proposed a scheme to achieve two-mode CV entanglement with the frequencies of entangled modes in the infrared range in an asymmetric semiconductor double-quantum-wells (DQW), where the required quantum coherence is obtained by inducing the corresponding intersubband transitions (ISBTs) with a classical field. By numerically simulating the dynamics of system, we show that the entanglement period can be prolonged via enhancing the intensity of classical field, and the generation of entanglement doesn't depend intensively on the initial condition of system in our scheme. Moreover, we also show that a bipartite entanglement amplifier can be realized in our scheme. The present research provides an efficient approach to achieve infrared entangled light in the semiconductor nanostructure, which may have significant impact on the progress of solid-state quantum information theory. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Entanglement, as the important resource of quantum information technology, has attracted a lot of interest in the past several decades [1– 12]. So far, the generation of entangled state in the discrete variables systems [13–20], such as trapped ions, quantum dot, nuclear magnetic resonance, and Josephson-junction circuits etc., have been widely researched. On the other hand, it has been pointed out that the CV entanglement has many advantages for the implementation of quantum information processes, which including unconditional quantum teleportation, quantum dense coding, universal quantum computation etc. [21– 25], and has aroused increasing attentions of physicists [26–35] in recent years. For example, in order to check if a state, generally mixed, is entangled or not, several inseparability criteria for continuous-variable system were proposed by Duan, Giedke, Cirac, and Zoller (DGCZ) [26], Simon [27], Hillery and Zubairy (HZ) [28] in the form of inequalities. Based on the corresponding inseparability criteria [26], Xiong et al. [29] proposed a scheme for realizing the two-mode CV entanglement in a correlated emission laser (CEL), in which the source of entangled light is the coherent atomic system. However, the required atomic coherence in their scheme is obtained by coupling the dipole forbidden transition with strong field, which requires a higher-order process and increases the experimental difficulty. More recently, to avoid this situation, Qamar et al. [30] proposed a new CEL-based scheme for realizing CV entanglement via improving the coherence entangled source, and shown that the required atomic coherence can be obtained via coupling corresponding

⁎ Corresponding author. E-mail address: [email protected] (X.-Y. Lü). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.08.016

dipole allowed transition in their scheme. Summing up the previous studies, we can find that most of them employ the coherent atomic system as the entangled source. However, the atoms, as gaseous medium, exist some defects in the flexibility of device fabrication, and hence it is obviously significative to study the generation of entanglement in other quantum coherence system, which is easier to be fabricated. In recent years, the semiconductor quantum wells (QWs) and coupled quantum wells (CQWs), as the solid-state media with nanoscale size, have attracted more and more attention due to its atomic-like properties, such as the discrete energy levels, and many inherent advantages compared to atomic system, such as large electric dipole moments, high nonlinear optical coefficients, and a great flexibility in the device design via choosing the materials and structure dimensions. Motivated by the above inherent advantages of the QWs and CQWs, several quantum optical coherence and interference effects have been investigated theoretically and experimentally in the QWs and CQWs system, such as coherently controlled photon-current generation [36], gain without inversion [37], coherent population trapping (CPT) [38,39], electromagnetically induced transparency (EIT) [40], and tunneling-induced transparency (TIT) [41,42], etc. All the previous research has demonstrated the nice quantum coherence effects in the QWs system, and presents a feasible platform to realize quantum entanglement in a semiconductor solid-state system, which is significant for the progress of quantum information theory. In this communication, we research the generation of two-mode CV entanglement in an asymmetric DQW system. In the present scheme, the root of realizing entanglement is the quantum coherence induced by coupling a dipole allowed IBSTs with a classical field. The major advantages of applying our scheme over other methods for realizing two-mode CV entanglement are as follows. (1): The DQW

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(a)

c

1

2

(b)

Energy [meV]

E3

2. Model and equation As shown in Fig. 1(a), we consider a bipartite CEL consisting of a asymmetric DQW medium and a doubly resonant cavity. The present DQW consists of a wide well (WW) and a narrow well (NW), and all possible transitions in this system are dipole allowed because of the asymmetry which breaks the parity of the wave functions. The quantum well system interacts with a classical laser field with Rabi frequencies Ωc, and two (nondegenerate) cavity modes ν1, and ν2 with coupling constants g1 and g2, respectively. Δ = ω31 − ωc denotes the corresponding frequency detunning. In the present analysis, we use the following conditions: (i) The electron sheet density of the quantum well structure is such that electron–electron effects have very small influence in our results. Therefore, the effects of electron– electron interactions are not included in our study. (ii) We assume that all subbands have the same effective mass. Then, under the dipole and rotating wave approximations, the interaction Hamiltonian of our system can be written in the interaction picture as (ℏ = 1) [43–46] HI = Δ j3〉〈3j + ½g1 a1 j3〉〈2j + g2 a2 j2〉〈1j−Ωc j3〉〈1j + h:c:;

h i  h i h i  † † ρ˙ f = −iTre HI ;ρef = −ig1 a1 ;ρ21 −ig2 a2 ;ρ32 + h:c: ;

ð2Þ

where ρef is the full electron-field density operator and ρ21 = 〈2|ρef|1〉, ρ32 = 〈3|ρef|2〉. Then, assuming that the electrons in the SQW, which are initially in subband |1〉, are injected into the cavity at a rate γa. Via solving the

125 E2

125 E1

0

5 8

12

Growth direction z [nm]

(c) 3

1

2 c

2

ð1Þ

where a†i and ai (i = 1,2) are the creation and annihilation operators corresponding to two cavity modes. Ωc = |Ωc|e − iϕc denotes the Rabi frequencies of the classical fields, and g1, and g2 are the atom-field coupling constants. Here it should be pointed out that the SQW sample used in this investigation is very much similar to the one reported in Ref. [38]. As a rule, such DQW samples are grown by the molecular beam epitaxy (MBE) method and contains a 5-nm GaAs layer followed by a 4-nm Ga0.9Al0.1As layer and a Ga0.6Al0.4As barrier. The structure is n-doped with an electron density of ns = 7 × 1011cm − 2. The sample can be designed to have equidistant transition energies, i.e., ω21 = ω32 = ω31 /2 = 125 meV. According to the standard methods of laser theory [47], the equation of motion for the reduced density operator of field ρf can be obtained by taking a trace over the electron, which leads to

QWs

medium studied here is a solid, which is much more practical than that in gaseous medium due to its flexible design and the wide adjustable parameters. For example, the transition energies and dipole moments can be well manipulated by accurately tailoring their shapes and sizes whereas they can hardly be found in the models for cold atom medium [37]. In addition, in the proposed DQW system, all possible transitions are dipole allowed because of the asymmetry which breaks the parity of the wave functions [38], which advances the feasibility of our scheme. (2): The present scheme is robust with respect to the initial condition of the cavity field. Specifically, a twomode CV entangled state can be realized in our scheme no matter the two entangled modes are initially in the same state or different states. (3): According to the corresponding parameters in Ref. [38], the frequencies of ISBTs in the present DQW are usually in the infrared scope. So, the proposed system can be served as the source of generating infrared entangled lights (i.e. λ1 ≃ 10.5μm, λ2 ≃ 11μm in the present scheme), which may has wide application in quantum communications. The remainder of this paper is organized into four parts as follows. In Sec. 2, we describe the model under consideration and derive the equation of motion for the reduced density operator ρf of the cavity field. In Sec. 3, the generation of entanglement is discussed. Finally, we conclude with a brief summary in Sec. 4.

1 Fig. 1. (a) Possible arrangement of the experimental apparatus. The asymmetric semiconductor quantum wells medium inside a doubly resonant cavity; an external classical fields is applied. (b) Schematic energy-band diagram of a single period of the asymmetric DQW structure. The layer thicknesses in the DQW regions are respectively, from left to right, 5 nm (GaAs well), 3 nm (Ca0.6Al0.4As barrier), and 4 nm (Ca0.9Al0.1As well). The positions of the calculated energy subbands and the corresponding modulus squared of the electronic wave functions are also displayed. (c) Schematic of the energy level arrangement under study. A classical laser field induce the ISBTs |1〉 ↔ |3〉 with Rabi frequency Ωc. The ISBTs |3〉 ↔ |2〉 and |2〉 ↔ |1〉 couple with two nondegenerate cavity modes ν1 and ν2.

density operator ρ21 and ρ32, we can obtain the following master equation for the density operator of field ρf [29–31]     ρ˙ f = −α11 a†1 ρf a1 −ρf a1 a†1 −α12 a†1 ρf a†2 −ρf a†2 a†1     † † † † † † −α21 a2 a1 ρf −a1 ρf a2 −α22 a2 a2 ρf −a2 ρf a2 + h:c:     −κ1 a†1 a1 ρf + ρf a†1 a1 −2a1 ρf a†1 −κ2 a†2 a2 ρf + ρf a†2 a2 −2a2 ρf a†2 ;

½



ð3Þ

X.-Y. Lü et al. / Optics Communications 283 (2010) 5279–5284

(a)

2

2

α11 = −g1 jΩc j γa ½iΔγ3 + ðγ3 + iΓ12 ÞΓ31  = M;

ð4aÞ

h  i 2 2 2 α12 = −iΩc g1 g2 γa 2jΩc j Γ31 + γ3 Δ + Γ31 + Γ12 ðΓ31 −iΔÞ = M;

ð4bÞ

h i 2  α21 = −iΩc g1 g2 γa 2jΩc j Γ31 + γ3 ðΔ + iΓ31 ÞðΔ + iΓ32 Þ = M ;

ð4cÞ

〈(Δu)2+(Δv)2〉

where κ1 and κ2 denote the cavity decay rates of two cavity modes, respectively, and the coefficients αij (i, j = 1, 2) are given by

n h i o 2 2 2  α22 = −ig2 γa γ3 ðΔ−iΓ31 Þ jΩc j + ðΔ + iΓ31 ÞðΔ + iΓ32 Þ + 2jΩc j Γ31 ðΔ + iΓ32 Þ = M ;

5281

2

1.5

1 |Ωc|=25 meV

ð4dÞ

|Ωc|=35 meV

with

0.5

h

2

2

M = Δ γ1 γ3 + 2jΩc j ðγ1 + γ3 ÞΓ31 +

2 γ1 γ3 Γ31

ih

i jΩc j + Γ12 ðiΔ + Γ32 Þ :

0

0.5

1

2

1.5

t

ð5Þ

(b)

2

2.5

3 x 105

(meV−1)

x 105 3.2

It should be pointed out that, in the above calculations, the population decay rates and dephasing decay rates of SQW have been added phenomenologically. Specifically, γi (i = 1,2,3) denotes the population decay rates of subband |i〉, which are due primarily to the longitudinal optical (LO) phonon emission events at low temperature. The total decay rates Γij are given by Γ31 = γ3 /2 + γ1 /2 + γdph 31 , dph dph Γ32 = γ3 /2 + γ2 /2 + γdph is 32 , and Γ21 = γ2 /2 + γ1 /2 + γ21 , where γij the dephasing decay rate of the quantum coherence of the |i〉 ↔ |j〉 transition.

3

T (mev−1)

2.8 2.6 2.4 2.2

3. Generation of entanglement

2

In this section, we will study the generation of two-mode CV entanglement based on the sufficient inseparability criterion proposed by Duan et al. [26]. A quantum system is said to be entangled if and only if it is nonseparable. That is, the density operator for the state ρ cannot be written as, ð2Þ

j

ð6Þ

and ρ(2) are the normalized states of with pj ≥ 0 and ∑ j pj = 1. ρ(1) j j two field modes, respectively. According to the criterion derived in paper [17], the state of the system is entangled if the total variance of two Einstein-Podolsky-Rosen (EPR) type operators u and υ of the two modes satisfy the inequality    2  2 Δ uˆ + Δ υˆ b 2;

25

31

33

35

(a) 80 |Ωc|=25 meV |Ωc|=35 meV

40 20

where

0

0

0.5

1

1.5

2

2.5

3

3.5 x 105

t (meV−1)

(b) 60 |Ωc|=25 meV

〈 N2〉

uˆ = xˆ1 + xˆ2 ; υˆ = pˆ 1 −pˆ 2 ; ð8Þ   pffiffiffi   pffiffiffi with xj = aj + a†j = 2 and pj = aj −a†j = 2i (j = 1,2) are the quadrature operators for the two modes, 1 and 2. In order to prove the generation of entanglement, we present the equations of motion for various moments that are required to evaluate the quantities involved in Eq. (7) in the Appendix A. Since the exact analytical results are rather complicated in this situation, we numerically solve the Eqs. (9a)–(9h) and simulate the dynamical evolution of this system for different values of parameter, as illustrated in Figs. 2–5. In those figures, the system parameters are chosen such that they correspond to Ref. [38]. In the following paragraphs, we will mainly discuss the generation and evolution of two-mode CV entanglement from three points. First, we will demonstrate that the two-mode CV entanglement with the frequency of each entangled mode in the infrared range can be realized in the

29

D E 2 2 Fig. 2. The evolution of ðΔ uˆ Þ + ðΔ υˆ Þ for different jΩc j [panel (a)], and the entanglement period T versus jΩc j, when two cavity modes are initially in the vacuum state. Various parameters are γ 3 = γ 2 = γ 1 = 1 meV, Γ 32 = Γ 12 = Γ 31 = 5 meV, γa = 1.5 meV, g1 = g2 = 0.02 meV, κ1 = κ2 = 10 − 6 meV, Δ = 2 meV, and ϕc = − π /2.

60

ð7Þ

27

|Ωc| (meV)

〈 N1〉

ð1Þ

ρ = ∑ pj ρj ⊗ρj

1.8

40

|Ωc|=35 meV

20 0

0

0.5

1

1.5

2

t (meV−1)

2.5

3

3.5 x 105

D E D E Fig. 3. The time evolution of mean photon numbers Nˆ 1 [panel (a)] and Nˆ 2 [panel (b)] for different values of jΩc j. The parameters are same as in Fig. 2.

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(a) 2

2

1.5

1.5

〈(Δu)2+(Δv)2〉

〈(Δu)2+(Δv)2〉

(a)

1

0.5

1

0.5 0

0.5

1

1.5

t

2

2.5

3

3.5

0

0.5

1

1.5

x 105

(meV−1)

t

2

2.5

3 x 105

(meV−1)

(b)

(b)

7000

6000 〈 N1〉

〈 N1〉

6000

5000

6000 4000

4000

3000

5000

〈 N2〉

〈 N2〉 4000

4000 2000

3000 2000

10 0

2

1000 0

0.5

1

1.5

0 0

2000

4 x 105

1

2

3

4

x 105 1000 10

0

2000

3000

1000

2

t (meV−1)

2.5

3

3.5 x 105

D E   2 2 Fig. 4. The evolution of ðΔ uˆ Þ + ðΔ υˆ Þ [panel (a)], and Nj (j = 1,2,3) [panel (b)], when the mode 1 is initially in the vacuum state and mode 2 in the coherent state |α〉 (α = 10). The system parameters are the same as in Fig. 2 except jΩc j = 35 meV.

present SQW based on ISBTs. At the same time, we will also discuss the gain property of our system and show that the bipartite entanglement amplifier can be realized in our scheme. Secondly, we will discuss the influences of classical field intensities on the generation of entanglement, and point out that the entanglement period can be prolonged effectively via enhancing the intensities of classical field. Lastly, we will discuss the influence of the initial condition of the cavity mode on the generation of entanglement, and show that the two-mode CV entanglement can be realized no matter if the two entangled (nondegenerate) modes are initially in the same state or different states based on our scheme.  2  2  In Fig. 2(a), we plot the time evolution of Δ uˆ + Δυˆ for different values of jΩc j, when two cavity modes are initially in the vacuum state (same state). Firstly, it is clearly seen from Fig. 2(a) that the two-mode CV entanglement can be realized during a proper time intervalD (entanglementE period, T), which corresponds to the range 2 2 when ðΔ uˆ Þ + ðΔ υˆ Þ is smaller than 2. In addition, our number results also show the entanglement period T can be prolonged effectively when the intensity of classical field jΩc j are enhanced. The above property can owe to the increases of quantum interference effects between the ISBTs |3〉 ↔ |2〉 and |2〉 ↔ |1〉. In order to further explicitly show the effects of control field intensity on entanglement period T more distinctly, we also plot the function curves for T versus jΩc j in Fig. 2(b). See from this figure, we can find that the entanglement period approximately manifests linear increase property as the intensity of two

0

0.5

1

1.5

2

2.5

3

3.5 x 105

t (meV−1)

D E   2 2 Fig. 5. The evolution of ðΔ uˆ Þ + ðΔ υˆ Þ [panel (a)], and Nj (j = 1,2,3) [panel (b)], when the cavity mode 1 is initially in the coherent state |α〉 (α = 10) and cavity mode 2 in the squeezed vacuum state SðξÞj 0〉 (ξ = 0.2). The system parameters are the same as in Fig. 2 except for jΩc j = 35 meV.

classical driving fields, which is consistent with our above discussion. Here, it also should be pointed out that, according to the corresponding parameters in Ref. [38], the wavelength of two generated entangled modes are in the infrared range (i.e., λ1 = 10.5 μm, λ2 = 11 μm) in Fig. 2. As a result, the present system can be used to obtain the infrared entangled light. In order to get more insight, in Fig. 3, we also present the evolutions of the average photon number of each entangled mode,   Nj (j = 1,2) for the same set of parameters as in Fig. 2. The corresponding curves show that the average photon number of each entangled mode increases as the evolvement of time, which implies that the intensities of entangled lights can be amplified effectively during the entanglement period. This phenomena † can owe to the † a

a

1 2 processes of ISBTs radiations j3〉 → j2〉 and j2〉 → j1〉, which are stronger than the processes of corresponding cavity modes decay due to the presence of gain property in the proposed DQW system. Summing up the above discussions, the CEL considered here can be served as a two-mode CV entanglement amplifier. In addition, it also   can be seen that the average photon number Nj still can be augmented effectively via increasing the intensity of control field jΩc j, which proposes a clue to obtain strong entangled light in our scheme. Up to now, we have shown that the two-mode CV entanglement can be realized when the two entangled cavity modes are initially in the same state (vacuum state). Whereas the question attracts us is

X.-Y. Lü et al. / Optics Communications 283 (2010) 5279–5284

whether the generation of the entanglement is strongly depended on the initial condition of the cavity field in the present scheme. Therefore, in Figs. 4 and 5, we study the dynamics of system when two cavity modes are initially in the different states, i.e., mode 1 in the vacuum state, mode 2 in the coherent state |10〉 [Corresponding Fig. 4] and cavity mode 1 in the coherent state |10〉, cavity mode 2 in the squeezed vacuum state Sð0:2Þj0〉 [Corresponding Fig. 5]. It is easily seen from Figs. 4(a) and 5(b) that the two-mode CV entanglement still can be realized even though the two entangled modes are initially in different states. In addition,  in  Figs. 4(b) and 5(b) shows that the average photon numbers Nj (j = 1,2) still increase with the evolvement of time in this situation, which implies that the gain character of our system doesn't depend intensively on the initial state of field. Then, we can draw a conclusion from Figs. 4 and 5 that the present scheme is robust with respect to the initial conditions of the entangled modes. In other words, the two-mode CV entanglement can be realized no matter the two nondegenerate modes are initially in the same state or different states. Before ending this section, let us briefly discuss the differences of our scheme with the previous scheme [29]. First of all, the source of generating entanglement is the semiconductor solid-state medium, which is more practical than that in gaseous medium (in the previous scheme [29]) due to its flexible design and the wide adjustable parameters. In addition, in the previous scheme [29], the required quantum coherence effects is induced by coupling a dipole forbidden transition with strong field, which requires a higher-order process and increases the experimental difficulty. However, in the present semiconductor medium the corresponding transition is dipole allowed. Secondly, it should be pointed out that the dynamics of quantum wells are more complicated than the atomic model, such as the many-body effects arising from the electron–electron interactions. Therefore, the present scheme is more difficult than the previous scheme [29] from this point. However, this difficulty can be overcome in principle. In the present analysis we use the following condition: the semiconductor QWs with low doping are designed such that the electron–electron effects have very small influence in our results. Consequently, we assume that many-body effects arising from electron–electron interactions are not included in our study. This method has described quantitatively the results of several experimental papers [37,38,41] and has been used in several theoretical literature [39,42]. In fact, the effects of electron–electron interactions in the dynamics of ISBTs in SQWs have been studied in several recent publications, see, e.g., [48–50]. These works have shown that this dynamics can be significantly altered but for much larger electron doping than those of interest here. Lastly, it should be pointed out that we have use the same method to study the dynamics of SQWs medium as that in the atomic system [29]. However this method is valid for the SQWs with low doping, such as the previous studies [43– 47].

4. Conclusion In conclusion, we have proposed a scheme for realizing the twomode CV entanglement in asymmetric DQW system based on ISBTs. By numerically simulating the dynamics of system, we demonstrate that the proposed DQW system can be used to achieve the infrared entangled light. In addition, we also have studied the influences of classical field on the generation of entanglement, and show that the entanglement period can be prolonged via enhancing the intensity of classical field in our scheme. More importantly, our numerical results show that the two-mode CV entanglement can be realized no matter the two nondegenerate cavity modes are initially in the same state or different states. As a result, the present scheme provides an efficient approach for realizing infrared entangled light with different frequencies and initial states for each entangled mode in the

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semiconductor solid-state medium, which may result in a substantial impact on the progress of quantum information science. Acknowledgements The research was supported in part by the Natural Science Foundation of China (Grant No. 10704017, No. 10874050, No. 10904033, and No. 10975054), by the National Basic Research Program of China (Contract No. 2005CB724508), and by the Foundation from the Ministry of the National Education of China (Grant No. 200804870051). We would like to thank Professor Ying Wu for helpful discussion and his encouragement. Appendix A The equations of motion for various moments that are required to evaluate the quantities involved in in Eq. (7). D E d † ha1 i = −ðα11 + κ1 Þha1 i−α12 a2 ; dt

ð9aÞ



D † E d D †E  a1 = − α11 + κ1 a1 −α12 ha2 i; dt

ð9bÞ

D E d † ha2 i = −ðα22 + κ1 Þha2 i−α21 a1 ; dt

ð9cÞ



D † E d D †E  a2 = − α22 + κ1 a2 −α21 ha1 i; dt

ð9dÞ

D E

D † E dD † E  † †  a a = − α11 + α11 + 2κ1 a1 a1 −α12 a1 a2 −α12 ha1 a2 i dt 1 1

 − α11 + α11 ; ð9eÞ D E

D † E dD † E  † †  a2 a2 = − α22 + α22 + 2κ2 a2 a2 −α21 a1 a2 −α21 ha1 a2 i; dt ð9fÞ D E D E d † † ha a i = −ðα11 + α22 + κ1 + κ2 Þha1 a2 i−α12 a2 a2 −α21 a1 a1 −α21 ; dt 1 2

ð9gÞ D E D E 

D † † E d D † †E   †  †  a a = − α11 + α22 + κ1 + κ2 a1 a2 −α12 a2 a2 −α21 a1 a1 −α21 ; dt 1 2

ð9hÞ

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