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LA-UR-15-20113 Decoherence and Spin Echo in Biological Systems Alexander I. Nesterov∗ Departamento de F´ısica, CUCEI, Universidad de Guadalajara, Av. Revolución 1500, Guadalajara, CP 44420, Jalisco, México Gennady P. Berman† Theoretical Division, T-4, Los Alamos National Laboratory, and the New Mexico Consortium, Los Alamos, NM 87544, USA (Dated: January 12, 2015) 5 1 The spin echo approach is extended to include bio-complexes for which the interaction with dy- 0 namicalnoiseisstrong. Significantrestorationofthefreeinductiondecaysignalduetohomogeneous 2 (decoherence)andinhomogeneous(dephasing)broadeningisdemonstratedanalyticallyandnumer- n ically, for both an individual dimer of interacting chlorophylls and for an ensemble of dimers. This a approachisbasedonanexactandclosedsystemofordinarydifferentialequationsthatcanbeeasily J solved for a wide range of parameters that are relevant for bio-applications. 9 PACSnumbers: 03.65.Yz,87.18.Tt,82.53.Ps ] Keywords: Decoherence, noise,spinecho h p - The femtosecond spectroscopic and measur- ualtwo-levelquantumsystem(TLS)[4]. Inthis o i ing technologies, developed during last few case,inhomogeneousbroadeningisabsent. But b decades, have allowed experimental scientists still one can partly suppress the broadening of . s to examine very rapiddynamical processes,in- the FI decay resulted from a pure dynamical c cluding those in biological systems. In partic- (time-dependent) noise. In [4–7] the spin echo i s ular, it was demonstrated that, even at room approachwasappliedtoa singlequantumtwo- y h temperature, photosynthetic bio-complexes ex- level system - the superconducting qubit. In p hibitcollectivequantumcoherence(CQC)dur- thiscase,thequbitisconsideredintheso-called [ ing primary electron transfer (ET) processes diagonal representation, with the main char- that occur on the time-scale of some hundreds acteristic being the energy gap (usually tens 1 v of femtoseconds [1]. of gigahertz) between the ground and the ex- 3 The CQC is resulted from the fact that cited states. The dynamical noise, acting on 1 the primary processes of exciton transfer and thequbit,isgeneratedbythe time-fluctuations 0 charge separation are so rapid (on a time-scale of the electron charge, bias current, external 2 0 ofafewpicoseconds)thattheenvironmentdoes magnetic flux, and other sources, and these all . nothavetimetorecombinetheexcitonandde- are relatively weak. 1 0 stroy the CQC. Incontrasttoasuperconductingqubit,anin- 5 One of the methods for studying the CQC dividual effective two-level quantum system (a 1 effects in bio-systems is the spin echo spec- dimer)inabio-complexisusuallycharacterized : v troscopy, which was initially developed in the in the so-called site representation, using such i nuclear magnetic resonance [2]. This spin echo main parameters as the energy gap (redox po- X procedure allows one to reduce the effects of tential)betweentheexcitedstatesofthechloro- r a the “inhomogeneous broadening” (dephasing) phylls realizing this dimer, and the matrix ele- and to increase the time of the signal gener- mentofthedipole-dipoleortheexchangeinter- ated by the transverse magnetization (free in- actions betweenthese two excited states. Also, duction, FI). Recently, the spin echo technique the dimer usually experiences a strong interac- has also been used to analyze quantum co- tion with the protein environment, caused by herent effects in an ensemble of effective 1/2 dynamical noise (characterized by the recon- spins (chlorophyll-based dimers) in photosyn- struction energy), that must be taken into ac- thetic bio-complexes. (See [3], and references count in bio-applications of the spin echo tech- therein.) nique. During the last decade, it was recognized In this Letter, we analyze analytically and that the spin echo technique can be success- numericallythespinechotechniqueforbothan fully applied not only to an ensemble of spins, individualdimer andfor anensembleofdimers but alsoto anindividualspin, orto anindivid- in bio-complexes, for the case of strong inter- 2 action with the dynamical noise. We show To study the quantum decoherence, we how both the dynamical and the inhomoge- presentthe densitymatrix as,ρ(t)=(I+n(t)· neousbroadeningcanbesuppressedbythespin σ)/2, where, n(t) = Tr(ρ(t)σ), is the Bloch echo pulses. Our conclusion is that even for vector. Instead of the Liouville-von Neumann strongdynamicalnoise,thespinechoapproach equationforthedensitymatrix,i~ρ˙ =[H,ρ],it can serve as the useful complementary spec- is convenientto employ the equationof motion troscopic technique for characterizing the bio- for the Bloch vector (we set ~=1): complexes that include both individual dimers dn and an ensemble of dimers. =Ω×n+(ξ(t)/σ)ω×n. (2) Weconsideradimercomposedoftheexcited dt states of two chlorophylls, Chl1 and Chl2. We Here ω = v(sinθcosϕ,sinθsinϕ,cosθ), and assume that each chlorophyll experiences a di- Ω=(0,0,Ω). agonal noise, provided by the protein environ- Using the differential formula for the RTP ment, which is described by the random vari- [10], able,ξ(t). Inthesiterepresentation,theHamil- tonian of the system can be written as follows: d d +2γ hξ(t)R[t;ξ(τ)]i = ξ(t) R[t;ξ(τ)] , H = E1|1ih1| + E2|2ih2| + (1/2)(V12|1ih2| + (cid:16)dt (cid:17) D dt E h.c.)+ξ(t)(λ1|1ih1|+λ2|2ih2|). Wealsoassume (3) that noise is produced by the stationary ran- where, R[t;ξ(τ)], is an arbitrary functional, we dom telegraph process (RTP) with: hξ(t)i=0, obtainfromEq. (2)the closedsystemofdiffer- hξ(t)ξ(t′)i = χ(t − t′), where, χ(t − t′) = σ2e−2γ|t−t′|, is the correlation function; σ, 2γ, ential equations: and λ1,2, are the amplitude of noise, the decay dhni rate of the correlation function, and the inter- =Ω×hni+ω×hnξi, (4) dt action constants with noise, correspondingly. dhnξi In the diagonal representation of the unper- =Ω×hnξi+ω×hni−2γhnξi, (5) dt turbedHamiltonian,weobtainthetotalHamil- tonian for the effective TLS, where, hnξi = hξ(t)ni/σ. The average, h...i, is λ0 1 1 taken over the RTP. H= I+ Ωσz + Dλ,zξ(t)σz+ Homogeneous broadening. To characterize a 2 2 2 1 dimer, we introduce the dimensionless parame- D ξ(t)(cosϕσ +sinϕσ ), (1) 2 λ,⊥ x y ter,ǫ=|tanθ|=|V12/(E1−E2)|. Whenǫ≪1, we will call the dimer “weakly coupled”. In where, σx,y,z, are the Pauli matrices, λ0 = the opposite case, ǫ & 1, the dimer is called E1 + E2 + (λ1 + λ2)ξ(t), λ = λ1 − λ2, Ω = (E1−E2)2+|V12|2, Dλ,z = λcosθ, and “strongly coupled”. We first consider a weakly coupled dimer, so Dpλ,⊥ =λsinθ. We set: cosθ =(E1−E2)/Ω. one can neglect the effects of relaxation. Intro- The dynamics of a TLS is described by two ducingthecomplexvectors: hm(t)i=hn (t)i+ rates: the longitudinal relaxation rate, Γ1 = x T1−1, and the transverse relaxation rate, Γ2 = ihny(t)i and hmξ(t)i = hnξx(t)i+ihnξy(t)i, one T−1. When the noise is weak, and the con- can show that the solution of Eqs. (4) and (5) 2 can be written as, dition, τc ≪ T1,T2, is satisfied (where τc = 1/(2γ) is the correlationtime of the noise fluc- hn (t)i=hn (0)i, hnξ(t)i=0, (6) z z z tuations), one can apply Bloch-Redfield (BR) theory [8, 9]. In BR theory, the transverse re- hm(t)i=eiΩtΦ(t)hm(0)i, (7) laxation rate, Γ2 = Γϕ + Γ1/2, where Γϕ is hmξ(t)i=− ieiΩt dΦ(t)hm(0)i. (8) the so-called “dephasing” rate. In terms of vcosθ dt the spectral density of noise, S(ω), these rates are defined as follows [4]: Γ1 = πDλ2,⊥S(Ω), Here we denote by Φ(t) the generating func- Γ = πD2 S(0). Using the spectral density tional of the RTP [10]. For the FI decay, it is ϕ λ,z given by [5–7], of RTP, S(ω) = 2γσ2/π(4γ2+ω2), we obtain the relaxation and dephasing rates provided 1 by BR theory: Γ1 = 2γv2sin2θ/(4γ2+Ω2), Φf(t)=e−γt µsinh(γµt)+cosh(γµt) , (9) (cid:16) (cid:17) Γ = (v2/2γ)cos2θ, where the renormalized ϕ interaction constant, v =λσ, is introduced. where, µ= 1−(vcosθ/γ)2. p 3 FI decay. We call noise weak if the dimensionless parameter, η = |vcosθ/γ| ≪ 1. As it follows from Eq. (9), for weak noise, the decay rate of the non-diagonal averaged density matrix element (which characterizes the decoherence) coincides with Γ , provided by BR- ϕ theory. We call noise strong if η & 1. In particular, when η > 1, the parameter, µ in Eq. (9) becomes imaginary, and the decay of the functional, Φf(t), is accompanied by oscillations with frequency, γ|µ|. Below, we compare the analytical solutions (6)–(8),whenthetransverseeffectivefield(relaxation) is neglected, with the corresponding exactsolutions obtainednumerically fromEqs. (4) and (5). In numerical simulations, we put FIG. 1: (Color online) Time dependence (in ps) ~ = 1. All energy-dimensional parameters are of m = |hm(t)i|, for the FI signal. Blue curves: measuredinps−1(1ps−1 ≈0.66meV),andtime analytical results. Dashed curves: exact solution. Parameters: Ω = 127, θ = 0.968, γ = 10, v = 20 is measured in ps. (redcurve),v=40(orangecurve). Inset: θ=1.45, v=20. In Fig. 1, the strongly coupled dimer was considered (θ ≈ 0.968, ǫ ≈ 1.45). One can see, that in spite of the noise is strong: η ≈ 1.13 where, (reddashedcurve)andη ≈2.27(orangedashed curve), the approximate analytical solutions Φf(t)=Φf(t)+e−γt 1− 1 cosh(γµ(t−2τ)) (shown by blue curves), are in a good agree- g µ2 (cid:0) (cid:1)(cid:0) mentwiththeexactnumericalsolutions. Inthe −cosh(γµt) . (11) inset, the noise amplitude is relatively strong (cid:1) (v = 20), and the matrix element, V12, of the Chl1 and Chl2 interaction, is large: ǫ ≈ 8.24 (strongly coupled dimer). At the same time, the noise is weak: η ≈ 0.24, and the BR ap- proachworks. However,onecannotneglectthe contributionfromthetransversefieldtothede- coherencerate,Γ2 =Γϕ+Γ1/2. Indeed,inthis case, Γϕ ≈0.3 and Γ1 ≈0.48. That is why the approximate solution in the inset (blue curve) deviates significantly from the exact numerical solution (red dashed curve). Ournumericalsimulationsshowthatthe analytical solution, given by Eqs. (6) – (8), is in FIG. 2: (Color online) Time dependence(in ps) of a good agreementwith the exact numericalso- m = |hm(t)i|. The FI decay: blue curve. Echo lution,uptothevalueofǫ≈1.72(θ .π/3),in signals: green curve: N =10; black curve: N =15 spite of the dimer is already strongly coupled. (τ =0.025);cyancurve: N =1;orangecurve: N = 10; red curve: N = 15 (τ = 0.05). Parameters: Echo signals. For simplicity, we assume that Ω=10, v=10, γ =10, θ=0. Inset: thesequence the π/2 and all spin echo π-pulses act practi- ofechoπ-pulsesappliedat times: t=τ,2τ,4τ,.... cally instantaneously. The spin echo pulse, ap- The duration of the echo π-pulse is: δ = 1fs, and plied at the time τ, rotates the wave function its height is: h=103π. around the x-axis, by the angle π. The corresponding analytical solution can be written as, InFig. 2,theFIdecayisshownforthetrans- verse Bloch vector (blue curve). The N echo pulses are applied, partly restoring the FI sig- Φf(t), 0<t<τ, Φe(t)= (10) nal,andreducingthe homogeneous broadening. (cid:26) Φfg(t), t>τ, Green curve: N = 10; black curve: N = 15 4 disorder. It is well-known that this leads to the inhomogeneous broadening of the FI signal decay. In our numerical simulations we as- sumed the independent Gaussian disorder for parameters, (Ω,θ,ϕ). Our numerical simulations demonstrate that the main contribution fromthestaticdisorder,forawiderangeofpa- rameters, is due to the fluctuations of the frequency,Ω. So,belowweneglectthe staticfluc- tuations of both angles, θ and ϕ. We assume a Gaussian distribution for the random parameter Ω, denoting the dispersion by σ. Note that the results can easily be extended by including thestaticfluctuationsofangles,θ andϕ,inthe FIG. 3: (Color online) Time dependence (in ps) numerical solutions of the exact Eqs. (4) and of m = |hm(t)i|. The FI decay: blue curve. Echo (5). signals: redcurve(numberofechopulses,N =20). Parameters: Ω = 150, v = 40, γ = 10, θ = 0.165, In Fig. 4, we compare the results for the τ = 0.02ps. Inset: Ω = 127, v = 20, γ = 10, decay of the FI signal for three values of the θ=0.968, τ =0.03. dispersion, σ = 0,10,20, of static fluctuations of the frequency, Ω, and for the amplitudes of noise, v = 20,40. The parameter, η ≈ 1.13, (τ = 0.025); cyan curve: N = 1; orange curve: so the noise is strong in this case. In the in- N = 10; red curve: N = 15 (τ = 0.05). Pa- set, oscillations are observed with the period, rameters: Ω = 10, v = 10, γ = 10, θ = 0. The T =2π/γ|µ|,due to the imaginaryvalue ofthe inset corresponds the the sequence of echo π- parameter, µ. pulses applied at times: t = τ,2τ,4τ,.... The duration of each echo pulse is: δ =1fs, and its 3 height is: h = 10 π. In this case, the noise is strong: η = 1, and the dimer is weakly coupled: ǫ = 0. (There is no contribution to the dynamics from the transverse field.) In Fig. 3, the time dependence (in ps) of the transverse Bloch vector, m = |hm(t)i|, is shown. The FI decay corresponds to the blue curve. Echosignals: redcurve(numberofecho pulses, N =20). Parameters: Ω=150,v =40, γ = 10, θ = 0.165, τ = 0.02. This corresponds to a weakly coupled dimer (ǫ ≈ 0.17), and to strong noise: η ≈ 3.9. The FI decay exhibits oscillations with the period: T = 2π/γ|µ| ≈ 0.16. The inset corresponds to the choice of parameters: Ω = 127, v = 20, γ = 10, θ = 0.968,τ =0.03. Thenoiseisstronginthiscase, FIG. 4: (Color online) Simultaneous action of ho- η ≈ 1.13, and the dimer is strongly coupled, mogeneous and inhomogeneous broadening on the ǫ ≈ 1.45. As one can see from Figs. 2 and 3, FI decay. The time dependence(in ps) of the generating functional, Φ(t). Red curve: σ = 0. Blue the homogeneous broadening of the FI signal curve: σ = 10. Green curve: σ =20. Parameters: can be significantly improved by applying the Ω=127, θ=0.968, γ =10, v=20. Inset: v=40. spin echo pulses for strong noise and for both weakly and strongly coupled dimers. Simultaneous action of the homogeneous and inhomogeneous disorder. Here, in addition to The analytical solution, which includes the dynamical fluctuations (noise, ξ(t)), we con- contributionsfromboth,thestaticdisorderand sider an ensemble of TLSs (dimers) with fluc- the dynamical noise, and corresponds to the tuating parameters, (Ω,θ,ϕ), due to the static spin echo signal applied at the time τ, can be 5 written as, hm(t)i=Φe(t)hm(0)i, where tant for many bio-applications at ambient con- g ditions. We also demonstrated the restoration σ2t2 of the free induction decay signal by the spin −  e 2 Φf(t), 0<t<τ echo pulses when both homogeneous and in- Φeg(t)= σ2(t−2τ)2 (12) hthoemforegeeninedouusctbiornoaddeecnaiyn.gTehqeuaalplyplcicoanttiroinbuotfethtoe  e− 2 Φfg(t), t>τ. sspysinteemchsowtiethchsntirqounegisloewsp-ferceiqaulleyncuysenfuolisfeo,rsbuicoh- as 1/f noise. Our approach can be easily gen- eralized for this case, as was done in [7], by introducing the corresponding ensemble of the fluctuators. Also,manydifferentsourcesofdy- namicalnoisescanbeincludedinthepresented here approach(as was done in [11]). Thisworkwascarriedoutundertheauspices of the National Nuclear Security Administra- tion of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. A.I.N. acknowledges the support from the CONACyT, Grant No. 15349. G.P.B. acknowledges the support from the LDRD program at LANL. FIG. 5: (Color online) The decay of the FI signal in the presence of both homogeneous and inhomogeneous broadening (blue curve), and the action of the echo signal (red curve). Time dependence ∗ Electronic address: [email protected] (in ps) of m = |hm(t)i|. Parameters: Ω = 127, † Electronic address: [email protected] θ=0.968, γ =10, v=20, σ=10, τ =0.075. [1] M. Mohseni, Y. Omar, G.S. Engel, and M.B. Plenio (Eds.), Quantum Effects in Biology, In Fig. 5, both static disorder of Ω and the (Cambridge UniversityPress, 2014). dynamical noise, ξ(t), are included. The decay [2] R.J. Abraham, J. Fisher and P. Loftus, Intro- duction to NMR spectroscopy, (Wiley, Chich- oftheFIsignalisshownbythebluecurve. For ester, 1988). our chosen parameters, the FI signal decays in [3] H.DongandG.R.Fleming,J.Phys.Chem.B, approximately 200fs. The spin echo signal was 118, 8956 (2014). applied at τ = 75fs. As one can see, the echo [4] G. Ithier, E. Collin, P. Joyez, P. J. Meeson, pulse restores significantly the FI decay (red D.Vion,D.Esteve,F.Chiarello,A.Shnirman, curve). Note that for the parameters chosen Y. Makhlin, J. Schriefl, and G. Schön, Phys. in Fig. 5, both dimensionless decay factors Rev.B, 72, 134519 (2005). [5] J. Bergli, Y.M. Galperin, and B.L. Altshuler, coincide at the characteristic time of the FI New Journ. Phys. 11, 025002 (2009). decay, t = 200fs: γt = σ2t2/2 = 2. So, both ∗ ∗ ∗ [6] Y.M. Galperin, B.L. Altshuler, J. Bergli, homogeneous and inhomogeneous broadening D. Shantsev, and V. Vinokur, Phys. Rev. B, arepartlycompensatedinthis casebythe spin 76, 064531 (2007). echo signal. [7] A.I.NesterovandG.P.Berman,Phys.Rev.A, 85, 052125 (2012). We presented the analytical and numerical [8] F. Bloch. Phys. Rev.,105, 1206 (1957). [9] A.G.Redfield, IBMJ.Res.Dev.1,19(1957). results for spin echo pulses for the two-level [10] V.Klyatskin,DynamicsofStochasticSystems, systems (TLSs) – chlorophyll-based dimers in (Elsevier, 2005). bio-complexes, embedded in noisy protein en- [11] A.I. Nesterov and G.P. Berman, The role vironment. We have shown that even strong of protein fluctuation correlations in elec- dynamicalbroadeningcanbesuppressedsignif- tron transfer in photosynthetic complexes, icantly by the spin echo pulses. This is impor- arXiv:1412.0512 (2014).