Waveguide transport mediated by strong coupling with atoms
aa r X i v : . [ qu a n t - ph ] A p r Waveguide transport mediated by strong coupling with atoms
Mu-Tian Cheng , , ∗ Jingping Xu , , and Girish S. Agarwal Institute for Quantum Science and Engineering andDepartment of Biological and Agricultural Engineering,Texas A & M University, College Station, Texas 77845, USA School of Electrical Engineering & Information,Anhui University of Technology, Maanshan 243002, P. R. China and MOE Key Laboratory of Advanced Micro-Structured Materials,School of Physics Science and Engineering,Tongji University, Shanghai 200092, P. R. China (Dated: April 14, 2017)
Abstract
We investigate single photon scattering properties in one-dimensional waveguide coupled to quantumemitter’s chain with dipole-dipole interaction (DDI). The photon transport is extremely sensitive to thelocation of the evanescently coupled atoms. The analytical expressions of reflection and transmission am-plitudes for the chain containing two emitters with DDI are deduced by using real-space Hamiltonian. Twocases, where the two emitters symmetrically and asymmetrically couple to the waveguide, are discussed indetail. It shows that the reflection and transmission typical spectra split into two peaks due to the DDI. TheFano minimum in the spectra can be used to estimate the strength of the DDI. Furthermore, the DDI makesspectra strongly asymmetric and create a transmission window in the region where there was zero transmis-sion. The scattering spectra for the chain consisting of multi-emitters are also given. Our key finding is thatDDI can broaden the frequency band width for high reflection when the chain consists of many emitters.
PACS numbers: 42.50.Nn, 42.50.Ct, 32.70.Jz ∗ Electronic address: [email protected] . INTRODUCTION Strong coupling between photons and atoms plays important roles in quantum information pro-cession and quantum computation. Nanocavities, which can possess ultrasmall mode volume,are often used to realize the strong coupling [1, 2]. Recently, both theoretical [3, 4] and experi-mental [5–15] works reported the strong coupling between the atoms and propagating photons inone-dimensional waveguide. Here, the strong coupling means that most of the energy from theatoms decays into the propagating modes of the waveguide. Based on the strong coupling, thephotons scattering properties in one-dimensional waveguide are extensively investigated [16–41]and reviewed in [42]. Many quantum devices, such as single photon switching [16–25], router[36–38], isolation [43], transistor [35, 44, 45], frequency comb generator [46], and single photonfrequency converter[34] have been proposed or realized. The one-dimensional waveguide can bephotonic crystal waveguide [14], metal nanowire [5, 6], superconducting microwave transmissionlines [10, 11], fiber [12], and diamond waveguide [15]. Atoms and cavities can play the role ofscatter. Atomic chain is also an important scatter. The coupling between one-dimensional waveg-uide and atomic chain can lead to many interesting phenomena, such as superradiant decays [14],and changing optical band structure [47]. It can also be used to realize Bragg mirrors [48, 49] andsingle photon isolator [50].It is well-known that if the separation between two atoms is much smaller than the resonancewavelength, the dipole-dipole interaction (DDI) can be strong [51]. It has been shown that the DDIcan change the single photon scattering properties[52–54]. But in these studies, the two atoms arelocalized in one cavity [52, 53] or the same place along the waveguide [54]. The spatial separationbetween the two atoms along the waveguide direction are not involved. However, the separationplays important role in some important phenomena, such as quantum beats [55], generation en-tanglement [56–59], single photon switching [21] and Bragg mirrors [48, 49]. Recently, Liao etal. investigated the time evolution of emitter excitations and photon pulse in the one-dimensionalwaveguide coupled to multiple emitters with DDI [60]. In this paper, we study the single photonscattering properties by using the real-space Hamiltonian. The analytical expressions for reflectionand transmission amplitudes for the case of two quantum emitters (QEs) with DDI are given. Thesingle photon scattering properties for many QEs with DDI are also exhibited. The results showthat the DDI can significantly a ff ect the single photon scattering properties.The structure of this paper is organized as follows. In Sec. II, we present the model of single2 n r t r t r t N x x =0 x L = x L =2 x N L =( -1) yz FIG. 1: (Color online) The system considered in the manuscript. A chain of N QEs with equal separation L coupled to one-dimensional waveguide. photon transport in a waveguide coupled to atoms. In Sec. III, we recall the known results for thecase of a single atom. In Sec. IV, we present new features of the two atom coupling. In Sec. V, wediscuss the trend for many atoms with conclusions in Sec. VI. II. MODEL AND HAMILTONIAN
The system considered in this paper is shown in Fig.1. N QEs with equal separation L sidecouple to a waveguide. The QEs are modeled as two-level systems with ground state | g i andexcited state | e i . The transition frequency of the QEs is ω A .When ω A is much larger than the cuto ff frequency ω c of the waveguide, the dispersion relationof the waveguide near the resonant frequency can be taken as linear [26]. Then the Hamiltonian inthe real space is given by H = H f + H i + H d , with ~ =
1, [26, 56] H f = iv g Z dx ( a † L ( x ) ∂ a L ( x ) ∂ x − a † R ( x ) ∂ a R ( x ) ∂ x ) + N X j = ( ω A − i Γ ′ j / σ ( j ) ee , (1)being the free propagation photon in the waveguide and the QEs. v g is the group velocity of thephoton. a † R ( x )( a † L ( x )) means creation a right (left) propagation photon at x . σ ( j ) ee = | e i j h e | . Wehave supposed that all the transition frequencies of the atoms are the same and the energy of QE’sground state is zero. Γ ′ j is the energy decay rate into the non-waveguide modes. H i = N X j = J j Z dx { δ ( x − x j )[ a † R ( x ) + a † L ( x )] σ j + H . c . } , (2)denotes the interaction between the QEs and the waveguide photon. J j is the couple strengthbetween the j -th QE and the waveguide photon. σ j = | g i j h e | is the ladder operator for the j -th QE.Finally, H d = Ω i , j N X i , j = ( σ † i σ j + σ † j σ i ) (3)3escribes the DDI. Ω i j = Γ [( cos xx + sin xx − cos xx ) + cos θ ( cos xx − xx − xx )] is the DDI strengthbetween the i -th QE and the j -th QE [51]. x ≡ ω A c | ~ r i − ~ r j | . cos θ = ( ~ p · ( ~ r i − ~ r j ) | p |·| ~ r i − ~ r j | ) , where ~ r j is thelocation coordinate of the j -th QE and ~ p is the dipole. We suppose that all the QEs have the samedipoles and their directions are all in − y . Γ is the decay rate of QE in free space, which is takenabout 7.5 MHz in the following calculations.Since only one exciation exists in this system, the eigenstate of H takes the form | E k i = Z dx [ φ kR ( x ) a † R ( x ) + φ kL ( x ) a † L ( x )] | , g i + N X j = e ( j ) k | , e j i , (4)where E k = ~ ω k is the eigenvalue of H . | , g i represents all the QEs in the ground state and nophoton in the system. | , e j i denotes no photon in the system and the j -th QE in the excited state | e i while all other QEs in the ground state. e ( j ) k is the probability amplitude of the state | , e j i . φ kR ( x )and φ kL ( x ) are the amplitudes of the fields going to the right and left in the waveguide. These arecontinuous except at the positions of the atoms and thus we write these in the form [26, 27] φ kR ( x ) = e ikx ( x < , t j e ik ( x − jL ) ( j − L < x < jL (5) t N e ik ( x − NL ) x > ( N − L , and φ kL ( x ) = r e − ikx ( x < , r j + e − ik ( x − jL ) ( j − L < x < jL , (6)0 , x > ( N − L , where t j and r j are the coe ffi cients to be determined. Substituting Eqs.(5) and (6) into theSchr¨odinger equation H | E k i = E k | E k i , we obtain [26] t j e − ikL − t j − + iJ j e ( j ) k v g = , (7a) r j + e ikL − r j − iJ j e ( j ) k v g = , (7b) t j − + r j + P j − i = Ω j , i e ( i ) k + P Ni = j + Ω j , i e ( i ) k J j − ( ∆ ( j ) k + i Γ ′ j / e ( j ) k J j = , (7c)where ∆ ( j ) k = ω k − ω ( j ) A . t = r N + = ffi cients will be t = t N e − ikNL and r = r ,respectively. 4 II. SINGLE EMITTERS
Before showing how DDI a ff ects on the single photon scattering properties, we review singlephoton scattered by one QE first. The single photon transmission and reflection amplitudes are,respectively, given by [16, 37] t = ∆ k + i Γ ′ / i Γ + ∆ k + i Γ ′ / , (8a) r = − i Γ i Γ + ∆ k + i Γ ′ / , (8b)where, ∆ k = ω k − ω A and Γ = J / v g . Eqs. (8a) and (8b) show that reflection probability R ≡ | r | reaches the maximum and transmission probability T ≡ | t | reaches the minimum when ∆ k = T and R with di ff erent couplingstrengths between the QE and the photon in the nanowaveguide. In the numerical model, asemiconductor quantum dot (QD) with resonant wavelength λ qd =
655 nm (transition frequency ω A / (2 π ) ≈ λ sp is about 211.8 nm [3], which is much shorter than the resonant wavelength ofthe QD due to the reduced group velocity. The spontaneous emission rate Γ pl into the propagationsurface plasmon modes, the energy losses rate Γ ′ , which consists of radiating into the free spacerate Γ rad and non-radiative emission rate into the Ag nanowire Γ non − rad , is calculated by using theformulas given in Ref.[3]. IV. TWO QUANTUM EMITTERSA. Symmetric coupling
We now show how the DDI a ff ects on the single photons scattering properties for the case ofa pair of QEs coupling to the waveguide. First, we discuss the results for the symmetric coupling( J = J = J ). From Eqs. (7a) to (7c), one can obtain the analytical solutions to the t and r , whichare given by t = e − ikL {− i ΓΩ + ie ikL ΓΩ + e ikL [( ∆ k + i Γ ′ / − Ω ] } ( − + e ikL ) Γ + i Γ ( ∆ k + i Γ ′ / + e ikL Ω ) + ( ∆ k + i Γ ′ / − Ω , (9a) r = (1 − e ikL ) Γ − i Γ [(1 + e ikL )( ∆ k + i Γ ′ / + e ikL Ω ]( − + e ikL ) Γ + i Γ ( ∆ k + i Γ ′ / + e ikL Ω ) + ( ∆ k + i Γ ′ / − Ω , (9b)5here Ω is the DDI strength between the two QEs. When Ω =
0, which means that DDI is notconsidered, one can obtain t = ( ∆ k + i Γ ′ / ( − + e ikL ) Γ + i Γ ( ∆ k + i Γ ′ / + ( ∆ k + i Γ ′ / , (10a) r = (1 − e ikL ) Γ − i Γ (1 + e ikL )( ∆ k + i Γ ′ / − + e ikL ) Γ + i Γ ( ∆ k + i Γ ′ / + ( ∆ k + i Γ ′ / , (10b)which is consistent with previous reports [55, 61].Note that we can write the denominator in Eq. (10a) as ( ∆ k + i Γ ′ + i Γ ) + Γ e ikL . The term Γ e ikL arises from the waveguide mediated interactions between two QEs even if the direct DDI Ω =
0. Thus it plays the role of waveguide mediated DDI. To show this, if we drop Γ e ikL , thentransmission t (2) for two QE case is the square of the transmission t (1) for the single QE case. Thusin the absence of Γ e ikL term, the field transmitted by the first QE is transmitted by the secondQE leading to the result t (2) = ( t (1) ) . However, the physics is di ff erent. The field reflected by thesecond QE a ff ects the first QE changing its transmission which then changes the fields producedby the second QE. In principle, one has whole series of such processes and this just happens tobe the physics of DDI. Hence we refer to the term Γ e ikL as waveguide mediated DDI. We havechecked that this DDI a ff ects line shapes but does not produce splitting.The transmission and reflection spectra for the case of | ~ r − ~ r | = λ qd /
20 are shown in Fig.2.Here, kL ≈ . π , which is due to the short wavelength of SP. Without considering the DDI, thereflection spectrum reaches the maximum at ∆ k = Γ ′ =
0, it is found to occur at ∆ k = ∆ rmink = − Γ (tan kL + ΩΓ sec kL ), which depends onthe strength of DDI. Clearly, the position of the Fano minimum can be used to get an estimate ofthe DDI strength and this is displayed more clearly in Fig.3. The distance between the two dips inFig.3 is related to Ω sec kL . The Fano-lineshapes in the reflection spectra are reduced strongly butstill exist. Eqs. (9a) and (9b) also give that the single photon reflection spectrum splits into twomain peaks at ∆ k = ∆ rmaxk = ± p ΓΩ sin( kL ) + Ω when Γ ′ =
0. These values give the positionswhere R = , T =
0. Note that ∆ rmaxk depends on both DDI and kL but DDI is absolutely essentialfor ∆ rmaxk ,
0. These also correspond to the positions of dips in the transmission spectrum. Thedi ff erences of the main peaks of the reflection spectrum in Fig.2 result from the energy losses.6 a) (b)( )c (d) With DDIWithout DDISingle QD R R D k T T - D k - D k - D k - FIG. 2: (Color online) R and T as a function of ∆ k . The blue solid lines are the results for single QDs. Thered dashed-dotted lines denote the results for a pair of QDs with DDI and the green dashed lines for a pairof QDs without DDI. In (a) and (b), the two QDs locates at ~ r ( x , y , z ) = (0, 17 nm, 0) and ~ r ( x , y , z ) = (32.75nm, 17 nm, 0), respectively, corresponding to the separation between the two QDs L = λ qd / Γ= Γ and Γ ′ = Γ are used in the calculations. In (c) and (d), the two QDs are placed at ~ r ( x , y , z ) = (0, 37nm, 0) and ~ r ( x , y , z ) = (32.75 nm, 37 nm, 0 ), respectively. Γ = . Γ and Γ ′ = . Γ . The single QD islocated at (0,17 nm,0) and (0,37 nm, 0) when the blue solid lines are plotted in (a,b) and (c,d), respectively.In the calculations, Ω = . Γ . To show this claim clearly, we present Fig.4, which shows that when energy loss is zero, both ofthe two peaks reach the maximum of one. However, when energy loss increases from 3.43 Γ to6.86 Γ , the di ff erence between the heights of the two peaks increases from about 0.25 to about0.31. In the numerical calculations, we take kL = ( ω A + ∆ k ) / v g ≈ π L /λ sp since ω A ≫ Γ and ∆ k [21, 26] .Many reports show that single photon scattering properties and their applications such as singlephoton switching, generation entanglement, are strongly related on the distance between two QEs.The DDI strength also depends strongly on the distance between the two QEs. Fig. 5(a) shows7 ith DDIWithout DDI R D k . . . - -
40 0
FIG. 3: (Color online) Fano-shape of the reflection spectrum, [the region of minimum in Fig.2(a)]. = = = k / - 08 800 R /// FIG. 4: (Color online) Reflection spectra for the two identical QDs with di ff erent decays. In the calculations, Γ= Γ , L = .
75 nm and
Ω = . Γ . Ω as a function of L for the two QDs with resonant wavelength λ qd =
655 nm. The Ω decreasesfrom 23 . Γ to 0 . Γ as L increasing from 32.75 nm to 240 nm. Fig. 5(b) and (c) exhibit singlephoton reflection spectra for L = .
95 nm (corresponding to L = λ sp / , kL = π/
2) and L = . L = λ sp / , kL = π ), respectively. It indicates that DDI can play significantroles even though the separation between the two QDs reaches L = λ sp /
4. However, when thedistance increases to L = λ sp /
2, the influence of DDI can be neglected, i.e., the numerical resultswith and without DDI are almost indistinguishable but not identical.It is need to point out that spatial separation between the two QDs along the waveguide directionplays important role in splitting the reflection spectrum. There are special cases when the DDIyields a shift in spectrum rather than splitting. This happens when the numerator and denominator8 ith DDIWithout DDI
60 120 k L / p -100 0 100 ( ) b R L (nm)1.00.50.0 R ( ) a D k -100 0 100 D k ( ) c
025 0.01.53.0180 240
FIG. 5: (Color online) (a) DDI strength as a function of the distance L between the two QDs, i.e. dots located( x , y , z ) and ( x + L , y , z ). (b) and (c) are the single photon reflection spectra for the case of L = λ sp / = . L = λ sp / = . ~ r ( x , y , z ) = (0,17 nm,0), ~ r ( x , y , z ) = (52.95 nm,17 nm,0), Ω = . Γ .In (c) ~ r ( x , y , z ) = (0, 17 nm,0), ~ r ( x , y , z ) = (105.9 nm, 17 nm, 0), Ω = . Γ . In both (b) and (c), Γ= Γ and Γ ′ = Γ . share a common zero. As an example if kL =
0, which can be realized, for example, the two QDslocated at ~ r ( x , y , z ) = (0 , ,
37 nm), ~ r ( x , y , z ) = (0 ,
37 nm, 0), respectively, then t = ( ∆ k − Ω + i Γ ′ / / (( ∆ k − Ω+ i Γ ′ / + i Γ ), and r = − i Γ / (( ∆ k − Ω+ i Γ ′ / + i Γ ). The collective behavior is stillpresent as the e ff ective line width parameter is changed from Γ to 2 Γ . There is no splitting in thereflection spectrum but the location of the peak in the reflection spectrum shifts to ω k = ω A + Ω . Asimilar result is obtained for kL = π , one needs to replace Ω by − Ω . Thus the relative phase factor kL produced by the propagation of the light from QE 1 and to QE 2 is important in the transportof light through a waveguide coupled to QEs. 9 . Asymmetric coupling We now discuss the single photon scattering with asymmetric coupling. If the distances be-tween the surface of the nanowire and the two QDs are di ff erent, then the coupling strengths J , J . From Eqs. (7a) to (7c), one can get t = e − ikL {− i √ Γ Γ Ω + ie ikL √ Γ Γ Ω + e ikL ( δ (1) k δ (2) k − Ω ) } ( − + e ikL ) Γ Γ + i ( Γ δ (2) k + Γ δ (1) k ) + ie ikL √ Γ Γ Ω + δ (1) k δ (2) k − Ω , (11a) r = (1 − e ikL ) Γ Γ − ie ikL Γ δ (1) k − i Γ δ (2) k − ie ikL √ Γ Γ Ω ( − + e ikL ) Γ Γ + i ( Γ δ (2) k + Γ δ (1) k ) + ie ikL √ Γ Γ Ω + δ (1) k δ (2) k − Ω , (11b)where Γ j = J j / v g , δ j = ∆ k + i Γ ′ j / j = , ∆ k = √ Γ Γ Ω sin( kL ) + Ω is satisfied, T = , R = Γ ′ j =
0. Furthermore, if L =
0, which canbe realized for the two QDs with the same location coordinates x , z but di ff erent y , the conditionchanges to be ∆ k = Ω . This means that the distance between the two peaks in the reflectionspectrum is not only dependent on the coupling strength via Γ but also on Ω . However, if Ω = t = δ (1) k δ (2) k ( − + e ikL ) Γ Γ + i ( Γ δ (2) k + Γ δ (1) k ) + δ (1) k δ (2) k , (12a) r = (1 − e ikL ) Γ Γ − ie ikL Γ δ (1) k − i Γ δ (2) k [( − + e ikL ) Γ Γ + i ( Γ δ (2) k + Γ δ (1) k ) + δ (1) k δ (2) k . (12b)There is no splitting in the reflection spectrum. Fig.(6) shows R and T as a function of ∆ k forasymmetric coupling. Both the cases of kL , kL = V. MULTI QUANTUM EMITTERS
If one QE couples to the waveguide, only the photon with the frequency equaling to the transi-tion frequency of the QE reflects perfectly. Based on the coupled-resonator waveguide, Chang etal. proposed using many atoms individually in the resonators to realize perfect reflection of singlephoton in a wide band of frequency [62]. Here, we exhibit that the DDI can broaden the bandwidth. Fig. 7 shows the single photon reflection spectra where 5 QDs couple to the Ag nanowire.The separations between the two neighbouring QDs are 32.75 nm (a,b), 52.95 nm (c,d) and 105.9nm (e,f), where the distance dependent Ω is considered, which can be found in Fig.5(a). In Fig.10 T (a) (b)(d)( )c With DDIWithout DDI D k R T D k D k D k - - FIG. 6: (Color online) The single photon reflection and transmission spectra for the two QDs placed indi ff erent locations. In (a) and (b), ~ r ( x , y , z ) = (0, 17 nm, 0), ~ r ( x , y , z ) = (20 nm, 37 nm, 0), correspondingto kL = . π . Γ = Γ and Γ ′ = Γ , Γ = . Γ , Γ ′ = . Γ and Ω = − . Γ . In (c) and(d), ~ r ( x , y , z ) = (0, 17 nm, 0), ~ r ( x , y , z ) = (0, 49.75 nm, 0), corresponding to kL = Γ = Γ and Γ ′ = Γ , Γ = . Γ , Γ ′ = . Γ and Ω = − . Γ . L reaches λ sp / kL = π/ L further increases to λ sp / kL = π ), the numerical results with and without DDI arealmost indistinguishable, as shown in Fig. 7(e).To exhibit how DDI broadens reflection spectra clearly, we present R as a function of ∆ k and kL with and without DDI in Fig. 8. When kL ≈ . π , Ω is about 23 . Γ . It can a ff ect the scatteringspectrum strongly, as we discussed above. It shows that the red region in the direction of ∆ k in Fig.8(a) is much larger than in Fig. 8(b). However, when kL increases to more than π , the influence ofDDI can be neglected, then there is no obvious di ff erence between Fig. 8(a) and (b).11 With DDIWithout DDI (a) (b)(d)( )c( )e (f) T D k R R T T D k - 015 0 1 05 D k - 015 0 1 05 D k - 015 0 1 05 D k - 015 0 1 05 D k FIG. 7: (Color online) The single photon reflection and transmission spectra for the case of 5 QDs coupledto the nanowire. The separations between two neighbouring QDs along x direction are λ qd / = λ sp / = λ sp / = Γ= Γ and Γ ′ = Γ .The DDI coupling for each pair has been calculated using Fig 5(a). As an example, in (a), the couplingbetween first and second is 23.08 Γ and between first and third is 2.60 Γ . VI. CONCLUSIONS
In summary, we have investigated single photon scattering properties in one-dimensionalwaveguide coupled to an array QEs with DDI by using real-space Hamiltonian. For the case12 L / p D(cid:3)(cid:3)/G k (a) k L / p D(cid:3)(cid:3)/G k (b) ( a ) ( b ) FIG. 8: (Color online) The single photon reflection spectra as a function of ∆ k and kL with (a) and without(b) DDI. In the calculations, Γ= Γ and Γ ′ = Γ . The DDI coupling has been calculated using Fig5(a). of the chain consisting of two QEs with symmetric coupling, the reflection spectrum splits intotwo peaks due to the DDI, however, the splitting depends on both the DDI coupling and spatialseparation between the two QEs along the photon propagation direction in waveguide. The spectraalso display the Fano interference minimum. With two QEs, there are new pathways which lead totransmission and reflection. For example a new pathway will consist of-the radiation after beingscattered by QE1 interacts with QE2; the scattered radiation from QE2 interacts back with QE1.Thus the transmitted wave has additional contribution from this path way. The new pathways resultin Fano minimum. For both symmetric and asymmetric couplings, the DDI can induce reflectionspectrum splitting. The distance between the two peaks in the reflection spectrum depends on theDDI strength strongly in both symmetric and asymmetrical couplings case. DDI can also broadenthe frequency band width of high reflection probability of single photon when many QEs coupleto the waveguide. Our results may find applications in design single photon devices and quantuminformation processing. Acknowledgments
MTC acknowledges discussions with Dr. Zeyang Liao, and support from the Anhui ProvincialNatural Science Foundation under Grant Nos.1608085MA09, 1408085QA22 and China Scholar-13hip Council. GSA thanks the Biophotonics initiative of the Texas A&M University for support. [1] H. Walther, B.T.H. Varcoe, Berthold-Georg Englert, and T. Becker, “Cavity quantum electrodynam-ics,”. Rep. Prog. Phys. , 1325C1382 (2006).[2] A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,”Rev. Mod. Phys. , 1379 (2015).[3] D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum Optics with Surface Plas-mons,” Phys. Rev. Lett. , 053002 (2006).[4] C.-L. Hung, S. M. Meenehan, D. E. Chang, O. Painter, and H. J. Kimble, “Trapped atoms in one-dimensional photonic crystals,” N. J. Phys. , 083026 (2013).[5] A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. 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