Preserving Entanglement of Flying Qubits in Optical Fibers by Dynamical Decoupling
aa r X i v : . [ qu a n t - ph ] D ec Preserving Entanglement of Flying Qubits in Optical Fibers byDynamical Decoupling
Bin Yan, Chuan-Feng Li ∗ , and Guang-Can Guo Key Laboratory of Quantum Information, University of Science and Technology of China,CAS, Hefei, 230026, People’s Republic of China (Dated: October 31, 2018)We theoretically investigate the influence of dynamical decoupling sequence in preserving entan-glement of polarized photons in polarization-maintaining birefringent fibers(PMF) under a classicGauss 1/f noise. We study the dynamic evolution of entanglement along the control sequence em-bedded fibers. Decoherence due to dispersion of polarization mode in PMF can be dramaticallydepressed, even for a wild optical width. Entanglement degree can be effectively preserved whilethe control pulse is implemented.
Entanglement plays a central role in quantum commu-nication process. Due to an interaction with the uncon-trollable degree of freedom of the environment, quantumsystem may lose its entanglement degree. In an opticalfiber-based quantum communication channel, the resid-ual optical birefringence randomly accumulating alongthe fiber set up a main obstacle to maintain the fidelityof information. Polarization-entangled photons pairs dis-tributed over optical fibers even suffer the process ofabrupt disappearance of entanglement[1], known as aphenomenon of entanglement sudden death[2, 3]. Thusit is upper most important to find effective methods forpreserving entanglement of polarized photons propagat-ing in optical fibers.Several approaches have been developed to address thisissue. Notable examples are decoherence-free sbuspace[4,5] and quantum error-correction codes[6, 7], both basedon carefully encoding the quantum information into awider, while partially redundant, Hilbert space. Alter-native approach is quantum feedback[8]. In this tech-nology information channel is designed to be closed-loopwith appropriate measurements and real-time correctionto the system. However, all these strategies have thedrawback of requiring a large amount of extra resources.On the contrary, an open-lope control method known asdynamical decoupling avoids all these hindrances.Dynamical Decoupling(DD) is a simple and effectivemethod for coherence control. In this technology un-desired effects of the environment are eliminated viastrong and rapid time-dependent pulses faster than theenvironment correction time. The physical idea behindDD scheme comes from refocusing techniques in Nu-clear Magnetic Resonance (NMR) systems[9] and thenbe extended to any other physical contexts in the lastdecades, such as nuclear-quadrupole qubits and electronspins qubits[10]. Preserving of entanglement betweentwo stationary qubits by dynamical decoupling also hasbeen wildly discussed[11–13]. Some prominent exam-ples of DD schemes are the wildly used Hahn’s spin ∗ email:cfl[email protected] echo, periodic DD (PDD), Carr-Purcell-Meiboom-Gill(CPMG)[14], concatenated DD (CDD)[15], Uhrig DD(UDD)[16], and recently proposed near-optimal decou-pling (QDD)[17] used to eliminate general decoherenceof qubits.Extension of the time-dependent DD sequences to thespare-dependent dynamic evaluation process lead to theidea of applying DD controls into the optical fiber-basedquantum communication channel[18, 19]. Experimen-tal implementation has notably revealed the potentialof this extension[20, 21]. However, this delicately de-signed experiment is limited to the condition that thenoises are systematically introduced in a non-stochasticway, while in a real fiber-based channel they are dis-tributed randomly. In this paper, we investigate theperformance of CPMG sequence in preserving entangle-ment of polarized photons under stochastic classic Gaussnoises. We consider here the entangled photons are dis-tributed through polarization-maintaining(PM) birefrin-gent fibers, in which case the polarized photons suffera pure dephasing process, T >> T
2. Such buildingblocks of coherence control for a dephasing dynamic canbe extended to general decoherence processes in ordi-nary single mode optical fibers with more complicatedDD schemes like CDD[15] or QDD[17]. In the following,we show the derivation of entanglement evolution usingthe filter-design method[22] and give the numerical sim-ulation with some special initial states.We consider the situation of photon distribution de-picted in Fig. 1. Photon A is preserved by a quantumregister, while the entangled photon B, carrying the en-coded information propagates through the fiber to therecipient. The fiber is embedded with π pules realizedby half-wave plates. In our scheme the intervals betweenwaveplates are fixed to a certain scale in the fiber and thewaveplates sequence are arranged as continues CPMG cy-cles. Note that the CPMG sequence has a self-constructfeature as illustrated in Fig.1. In other words we canview two cycles of CPMG sequences with N pules eachas one cycle of CPMG sequence with 2N pules. This fea-ture of CPMG sequence allows us to analysis the dynamicevaluation of entanglement through the fiber directly, asillustrated in the following. FIG. 1: (a)Photon distribution scheme. (b)The length be-tween pulses embedded in the fiber is fixed to a certain scale.Measuremnet at length L in the fiber corresponds to a 2 pulsesCPMG sequence, while measurement at length corresponds toa 4 pulses CPMG sequence.
Under a photon distribution process described above,the qubits-environment interaction Hamiltonian can bewritten as[23]:ˆ H = 12 Z dω ( | ω A >< ω | ⊗ b ( ω, L ) σ Az + | ω A >< ω | ⊗ I B )(1)In nondispersive media only the first order of ω remainsin b ( ω, L )[23]. Thus we can rewrite b as: b ( ω, L ) = ω (Ω + β ( L )) (2)Ω is a constant and β ( L ) presents for the stochastic fluc-tuation of the noises with a zero mean, and it has a two-point correlation function: S ( L − L ) = < β ( L ) , β ( L ) > (3)Where <> corresponds to the average with respect tothe noise realizations. The statistical properties of theenvironment can also be expressed as the spectral densityof noise: S ( ω ) = Z e iwt S ( L ) dL (4)In the following discussing the statistics of fluctuations isassumed to be Gaussian, in which case the noise is com-pletely defined by the first-order correlation function S(L)in equation(3). We assume the polarization entangledphoton pairs are generated via spontaneous parametricdown conversion. The quantum state of the generatedphoton pairs can then be written as[1]: | ϕ in > = Z dωf ( ω ) | ω, − ω > ⊗| P > (5) | ω, − ω > is the frequency basis vector of photon pairs, ω denotes the offset from the central frequency. | P > represents polarization modes, which can be expressed asthe superimposing of horizontal and vertical polarizationbasises | HH >, | HV >, | V H >, | V V >
Due to the stochastic dephasing noise described inHamiltonian (1), the output state of the photon pairs af-ter photon A propagating through the fiber for a lengthL without DD sequence implemented is: | ϕ out > = Z dωf ( ω ) e − R L b s ( ω,L ) dL | ω, − ω > ⊗ ( a | HH > + b | HV > )+ Z dωf ( ω ) e R L b s ( ω,L ) dL | ω, − ω > ⊗ ( c | V H > + d | V V > ) (6)where the subscript s stands for a given realization ofnoise before taking the ensemble average. The den-sity matrix that characterizes the detected polarizationqubits can be abtended by a process of tracing over thefrequency modes and taking ensemble average of the en-vironment noise: ρ = Z Dβ ( T r ω | ψ out >< ψ out | )) (7)The integration is a Gaussian functional integral with thevariable β ( L ) having a Gaussian distribution.The elements of the resulting density matrix are thengiven by: ρ ii, , = ρ ii, , (0) ,i = 1 , , , ρ ij = ρ ij (0) 1 √ σ f ( L ) exp ( − ω f ( L )1 + σ f ( L ) ) ,i, j = others, (8)where f ( L ) can be expressed as the integral of the noisespectrum S ( w ): f ( L ) = Z dω π S ( ω ) F ( ωL ) ω (9) F ( x )is a filter function depends on the shape of DD se-quence. For a free evaluation without control imple-mented F ( x ) = 2 sin x x with cos x for odd-N CPMG) embedded beforethe measurement point can be written as: F ( x ) = 8 sin ( x N ) sin ( x / cos z N (10)Note that the intervals between pulses are fixed in ourscheme. Namely, N is proportional to the length of the Length of fiber C on c u rr en c e Length of fiber C on c u rr en c e (a)(b) FIG. 2: (a)Evoluation of entanglement along the fiber fordifferent pulse intervals. (b)Evoluation of entanglement withone SE pulse embedded into the fiber. Note that in SE casehere, the point at which the pulse is embedded changes withthe measurement point.
100 200 300 400 500 600 700 80000.050.10.150.20.250.30.35
Number of pulses C on c u rr en c e Concurrence ploted for length=50 units
FIG. 3: Concurrence as a function of the total pulse numberwithin 50 units length. fiber. As the particular property of CPMG describedabove, we can investigate the dynamical evaluation pro-cess using the deduced filter function: F ( ωL ) = 8 sin ( ω n ) sin ( ωL / cos ω n (11)Consider the frequency-dependent character of theevaluation process. From equation (8) it can be gen-erated that the elements of the resulting density matrixincrease with σ . Namely, decoherence is inhibited, in- stead of strengthen, by the optical frequency dispersion.This effect, though can be hardly observed due to theordinary situation with σ << ω , may more generallyinspire us that under a DD control process an additionaldegree of freedom with special structure can stabilize thespin coherence. In the optical case here, since a signif-icant decoherence will emerge within a length scale of σ f ( L ) <<
1, equation (8) can then be simplified as: ρ , , , = ρ , , , (0) e − ω f ( L ) (12)which is not sensitive to the optical frequency dispersion.We now turn to the characterization of the degreeof entanglement of the two-photons state. Woottersconcurrence[24] is particularly convenient for the two-qubit case here. Other reliable measure of entanglementwill yield the same conclusion. The concurrence can becalculated explicitly from the denity matrix ρ discribedin equation (12) C ( ρ ) = max(0 , p λ − p λ − p λ − p λ ) (13)where the quantities λ i are the eigenvalues in decreasingorder of the matrix: ρ ′ = ρ ( σ Ay ⊗ σ By ) ρ ∗ ( σ Ay ⊗ σ By ) (14)In the following we will analysis the evoluation of en-tanglement of a class of bipartite density matrices ini-tially prepared with the common X-form[25]: ρ AB = ρ ρ ρ ρ ρ ρ ρ ρ which occur in many contexts including pure Bell statesas well as Werner mixed states.For the concurrence withsuch arbitrary initial state there is no compact analyticalexpression. However, it has been readily showed[26] thatentanglement sudden death always occur under a clas-sic Gauss noise, as well as the evolution with DD controlimplemented. The DD method inhibit losing of entangle-ment by reshaping the exponential factor of the diagonalelements. We now give the numerical simulation to eval-uate the performance of the CPMG in our scheme.Without losing universality, we numerical analysis theconcurrence evolution with a initial specific X-form en-tangled state: ρ AB (0) = 13 / / Thus the initial concurrence is C (0) = 13 . In Fig.2(a) we give the numerical simulation of the concurrenceevolution under a Gauss 1/f noise (the noise spectral den-sity is S ( ω ) ∝ /ω , origination of which are discussed in FIG. 4: Concurrence evolution under a general Gauss noiseenvironment with variable α ranges from 0.5 to 1.5 Ref. [27]). We observed that the concurrence of entangle-ment improves as the intervals between pulses decrease.As a contrast we also depicted the entanglement evo-lution process with SE pulse implemented in Fig.2 (b).It can be inferred that the CPMG sequence drasticallyout-perform the single SE pulse in inhibiting the entan-glement sudden death. The performance of the SE pulsehas been experimental tested as an effective method forcoherence control[28]. We also analysis the entanglement preservation in afixed fiber length as the total number of pulses changes.Thus the minimum number of pulses can be estimated toachieve a given level of entanglement. Fig.3 shows thatthe entanglement rebirth while a certain number of pulsesare implemented. However,it can be seen that the growthrate of concurrence declines while the number of pulsesincreases. This makes a preservation of entanglement toa extremely high level be difficult.The performance of CPMG sequence under a generalGauss 1 /f α with spectral density S ( ω ) ∝ /ω α is alsoconsidered. Fig.4 shows the concurrence evolution underthe noise spectral density with variable α ranges from 0.5to 1.5.In conclusion, we demonstrated that dynamic decou-pling can effectively overcome the stochastic dispersionin the polarization-maintaining birefringent fibers underclassic Gauss noises. Entanglement can be successful pre-served by embedding waveplates as CPMG sequence intothe fibers. Our work will evidently enhance the scope offiber-based quantum communication. We also hope thismethod be extended with more complicated DD sequenceto general decoherent environment in ordinary sigle modefibers.This work is supported by the National Basic ResearchProgram (2011CB921200) and National Natural ScienceFoundation of China (Grants No. 60921091 and No.10874162) [1] C. Antonelli, M. Shtaif, and M. Brodsky, Phys. Rev. Lett. , 080404 (2011).[2] J. H. Eberly and T. Yu, Science , 555 (2007).[3] T. Yu and J. H. Eberly, Science , 598 (2009).[4] P. Zanardi and M. Rasetti, Phys. Rev. Lett. , 3306(1997).[5] R. Prevedel, M. S. Tame, A. Stefanov, M. Paternostro,M. S. Kim, and A. Zeilinger, Phys. Rev. Lett. , 250503(2007).[6] P. W. Shor, Phys. Rev. A , R2493 (1995).[7] N. Boulant, L. Viola, E. M. Fortunato, and D. G. Cory,Phys. Rev. Lett. , 130501 (2005).[8] D. Vitali, P. Tombesi, and G. J. Milburn, Phys. Rev.Lett. , 2442 (1997).[9] L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys. , 1037 (2005).[10] J. F. Du, X. Rong, N. Zhao, Y. Wang, J. H. Yang andR.B. Liu, Nature , 1265 (2009).[11] M. Mukhtar, W.T. Soh, T.B. Saw, and J.B. Gong, Phys.Rev. A , 052338 (2010)[12] M. Mukhtar, T.B. Saw, W.T. Soh, and J.B. Gong, Phys.Rev. A , 012331 (2010)[13] Y. Wang, X. Rong, P.B. Feng, W.J. Xu, B. Chong, J.H.Su, J.B. Gong, and J. F. Du, Phys. Rev. Lett. ,040501 (2011)[14] S. Meiboom and D. Gill, Rev. Sci. Instrum. , 668(1958). [15] K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. ,180501 (2005).[16] G. S. Uhrig, Phys. Rev. Lett. , 100504 (2007).[17] J. R. West, B. H. Fong, and D. A. Lidar, Phys. Rev. Lett. , 130501 (2010)[18] Lian-Ao Wu and Daniel A. Lidar, Phys. Rev. A ,062310(2004)[19] Asoka Biswas1 and Daniel A. Lidar, Phys. Rev. A ,062303(2006)[20] S. Damodarakurup, M. Lucamarini, G. Di Giuseppe, D.Vitali, and P. Tombesi, Phys. Rev. Lett. , 040502(2009).[21] M. Lucamarini, G. Di Giuseppe, S. Damodarakurup, D.Vitali, and P. Tombesi , Phys. Rev. A , 032320 (2011).[22] L. Cywinski, Roman M. Lutchyn, Cody P. Nave, and S.Das Sarma, Phys. Rev. B , 174509 (2008).[23] Phoenix S.Y. Poon and C.K. Law, Phys. Rev. A ,032330 (2008).[24] W. K. Wootters, Phys. Rev. Lett. , 2245 (1998).[25] T. Yu, J.H. Eberly, Quant. Inf. Comput. , 459 (2007).[26] T. Yu, J.H. Eberly, Optics Communications , 676(2010).[27] J. Schriefl, Y. Makhlin, A. Shnirman, and G. Schon, NewJ. Phys. , 1 (2006).[28] C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo,Nature Physics7