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Interaction as stochastic noise Roberto D’Agosta Nano-bio Spectroscopy Group, Departamento de Fisica de Materiales, UPV/EHU, San Sebastian, 20018 Spain, and IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain∗ (Dated: January 19, 2015) Interaction is so ubiquitous that imaging a the dynamics of their quantum averages. Often, these world free from it is a difficult fantasy exercise. averages have a random behavior since they mostly con- At the same time, in understanding any com- sistofthesuperpositionofastaticmagneticmomentand plex physical system, our ability of accounting small dynamical fluctuations. 5 for the mutual interaction of its constituents is In this Letter, I will show that this analogy is even 1 often insufficient when not the restraining fac- more stringent and can be made exact for the case in 0 tor. Many strategies have been devised to con- which particles interact via a potential that depends on 2 trol particle-particle interaction and explore the one operator of particle i multiplied by an operator of n diverse regimes, from weak to strong interac- particle j. Indeed, the dynamics of such a system is a tion. Beautiful examples of these achievements exactly equivalent to the dynamics of a system of non- J aretheexperimentsonBosecondensates[1–3], or interacting particles in the presence of known stochastic 6 the recent experiments on the dynamics of spin potentials. The dynamics of the many-body system is 1 chains[4,5]. HereIintroduceanotherpossibility, then recovered by constructing the proper wave-function ] namely replacing the particle-particle interaction or density matrix and averaging the results over the h with an external stochastic field, and once again stochastic fields. This study finds direct application in p reducing the dynamics of a many-body system to the dynamics of arbitrary spin-chains and gives immedi- - t the dynamics of single-particle systems. The the- ate access to the exact high-order correlation functions n a ory is exact, in the sense that no approximations that recently have been probed experimentally [4, 5]. u are introduced in decoupling the many-body sys- Moreover, the results presented here open up the pos- q tem in its non-interacting sub-parts. Moreover, sibility of investigating the dynamics of large systems. [ the equations of motion are linear, and no un- It is well known indeed that the numerical solution of 1 known external potential is inserted. the equation of motion of the many-body density ma- v trix, ρˆ, is limited by its intrinsic large dimensionality Theideaofreplacingthemany-bodysystemunderin- 6 which scales at least quadratically with the number of 4 vestigation with a non-interacting doppelganger is not states one needs to consider, compared with the linear 0 new. From a theoretical point of view, a starting idea 4 has been to treat the interaction as an external pertur- scaling of the wave-function. Our result shows that for 0 certain cases, one can bring the exact dynamics down bation. Interaction“dresses” theparticlesandnewfunda- . to the evaluation of N relatively small density matrices, 1 mental particles appear for the description of a physical 0 phenomenon. This is the tenet of the Landau’s theory thusregainingalinearscaling. Togiveanestimateofthe 5 problem with the interacting system, with a chain of N of the Fermi gas mapping a strongly interacting electron 1 spin 1/2, the exact many-body states belongs to a space gas into a system of weakly interacting quasi-particles : v [6, 7]. More modern approaches replace the particle- of dimensionality 2N, and therefore the density matrix i has dimension 2N×2N. It should be then apparent that X particle interaction with an external effective potential, wecaninvestigatenumericallysmallchains,routinelyup e.g., the Thomas-Fermi’s theory and the Hartree-Fock r a approximation [8] which have all converged now some- to N (cid:39) 10−15, after which the computer memory re- quirements will be prevailing. Our result brings down what in the Density Functional Theory [7, 9, 10]. The this requirement to store up to N 2×2 matrices. price to pay for this huge simplification is the inclusion Let us begin with considering a many-body quantum ofaunknownnon-linearpotentialinthedynamicsofthe system,whosedynamicsisdeterminedbythemany-body fictitious non-interacting system [7, 11]. Density func- Hamiltonian tional theory has been instrumental in understanding many physical, chemical, and biological phenomena at N N the nano-scale and in augmenting the theoretical predic- Hˆ =(cid:88)hî+(cid:88)λi,jxî⊗xˆj, (1) tion potential. i i<j With hindsight the results I will present in the fol- where h is a single particle Hamiltonian and xˆ is some i i lowing are not completely surprising: for example when operatoractingontheparticlei,λ theinteractioncon- i,j dealingwithmagneticsystems,acommonapproximation stant between particle i and particle j, and N the total consistsinreplacingthedynamicsofspinoperatorswith particle number. By Newton’s third law, the interaction 2 constant is symmetric, i.e, λ = λ . The many-body products of single particle operators like in Eq. (1). The i,j j,i densitymatrixofthesystemevolvesaccordingtothevon theory we are putting forward then is useful for those Neumann’s equation case in which the particle-particle interaction is written as a finite sum of products of pairs of single-particle op- (cid:104) (cid:105) (cid:126)∂ ρˆ(t)=−i Hˆ(t),ρˆ(t) . (2) erators. Finally, it should beclearthat even ifthe initial t densitymatrixρˆ(0)describesarealparticle,itstimeevo- i For simplicity, let us assume that the initial density ma- lution ρˆ(t) cannot be associated with the dynamics of a i trixisthedirectproductofsingleparticledensitymatri- real particle. This is easily seen by the fact that Eq. (4) ces does not preserve either the positivity or the unitarity of ρˆ(t),evenifweassumeρˆ(0)isadefinitepositivematrix (cid:79) i i ρˆ(t=0)= ρî(t=0), (3) of unitary trace. We will discuss later on how to reduce i=1,N themany-bodydensitymatrixtoapropersingle-particle (reduced) density matrix that can be used to investigate then we can find a set of N2 independent white complex the single-particle properties of the system. noisesω (t)forwhichthesingleparticledensitymatrix i,j The exact dynamics of the many-body density matrix ρˆ evolves according to [12] i usuallycontainsaredundantamountofinformation. For practical purposes, it is usually more convenient to trace (cid:126)dρˆ(t)=−i[h ,ρˆ(t)]dt i i i outsomeofthedegreesoffreedomandobtaintheexpec- +(cid:88)N (cid:112)(cid:126)λi,j (cid:0)[xˆ ,ρˆ]dω −i{xˆ ,ρˆ}dω∗ (cid:1) tation value of single- or two-particle operators. Within 2 i i i,j i i j,i this theory, this procedure emerges naturally from the j=1 definition of the operator ρˆ. We have for example for (4) i the time evolution of the subsystem j, and the exact total density matrix is given by (cid:89) ρˆR(t)=ρˆ (t) trρˆ(t), (7) j j i (cid:79) ρˆ(t)= ρˆ(t) (5) i(cid:54)=j i i=1,N where tr indicates the trace operation on the matrix ρˆ. i where the ··· denotes the average over all the white Notice that in general trρî(t) is a function of time, and noises. In Eqs. (2) and (4), [Aˆ,Bˆ] = AˆBˆ − BÂˆ and we cannot expect that it equals 1 at all times. On the {Aˆ,Bˆ} = AˆBˆ +BÂˆ are the standard commutator and other hand, we expect that trρˆRj (t) = trρˆ(t) = 1 at all anti-commutatorofanytwooperators, respectively. The times, sothatρˆR doesdefineaproperdensitymatrixfor j white noises ω are complex Wiener processes that sat- the subsystem j [17] i,j isfy As a first example of application we consider the case of two interacting spins. We assume there is not any dωi∗,jdωk,p =2δi,kδj,pdt, (6) external magnetic field. The Hamiltonian for this simple systemisHˆ =λσz⊗σz whereσz isthe2×2thirdPauli 1 2 i where δ = 1 if i = k or vanishes otherwise [13–16]. i,k matrix. Forsimplicityandwithoutanylossofgenerality We prove Eq. (5) starting from Eqs. (4) in the Methods we can set λ=(cid:126). According to Eq. (4) we need to solve section. the two independent stochastic master equations Let me now discuss two important points. First, it may appear that the initial correlation between the par- 1 dρˆ = [σz·ρˆ (dω −idω∗ )−ρˆ ·σz(idω∗ +dω )], ticles is lost in this theory. This is not the case. In- 1 2 1 12 21 1 21 12 deed, the following discussion can be generalized to the 1 dρˆ = [σz·ρˆ (dω −idω∗ )−ρˆ ·σz(idω∗ +dω )]. case in which the initial condition is given by ρˆ(0) = 2 2 2 21 12 2 12 21 (cid:78) (cid:80) γ ρˆk(0),whereγ isasetofcoefficientssuch (8) i=1,N k k i k that (cid:80) γ = 1, to ensure that if trρˆk(0) = 1 for any k k i One can easily prove that the solution of these two i, then trρˆ(0) = 1. The total density matrix in this stochastic equations are case, owing to the linearity of the equation of motion, will then be given by ρˆ(t) = (cid:80)kγk(cid:78)i=1,Nρˆki(t), where ρˆ (t)=(cid:18) eiω2∗1ρ111 eω12ρ112 (cid:19). (9) eachρˆki evolvesaccordingtoEq.(4)withinitialcondition 1 e−ω12ρ211 e−iω2∗1ρ212 ρˆk(0). Forsimplicity,inthefollowingIwillmaintainthat i Eq. (3) is satisfied by the initial density matrices. The Here, we have introduced explicitly the elements of the second point is the form of the particle-particle inter- initial state ρii. ρˆ (t) can be obtained from this ex- 1 2 action: if any operator of particle i commutes, or anti- pression by substituting 1 → 2 in the subscript of commutes, with any operator of particle j then one can the initial state, and by swapping 1 ↔ 2 in the sub- always expand any two-particle interaction as a series of script of ω. Some straightforward algebra now leads 3 to the expression of ρ˜(t), and with the properties of equations of motion. In Fig. 1, we report the dynamics the averages we will discuss in a moment, to the fi- of a few elements of the density matrix ρˆR(t) calculated 1 nal expression for the total density matrix ρˆ(t). We using Eqs. (7) and (4), after taking the average of 106 can prove that the density matrix obtained in this way independent realizations of the white noises. As initial is identical to the one obtained by evaluating ρˆ(t) = conditionwehaveassumedthetwospinsareinthemixed exp(−iHt)ρˆ(0)exp(iHt) where ρˆ(0)=ρˆ (0)⊗ρˆ (0). To state, ρi,j =ρi,j =1/2 for any i and j. 1 2 1 2 obtain these results, we need to evaluate the stochastic Thepossibilitytomeasureintime,andlocallythetime averages of two functions: namely, both exp(ω +ω ) 12 21 evolution of the correlation between two spins has re- andexp(ω +iω∗ ). Todothat, weobservethatforany 12 12 cently captured a lot of attention. [4, 5] In the experi- real Wiener process, p, and any complex number, α, we mental set-up, we could change the spin-spin interaction have exp(αp)=exp(−α2t/2) [18]. We easily then arrive in such a way to explore the transition between a XY- at exp(ω +ω )=1 and exp(ω +iω∗ )=exp(−2it). 12 21 12 12 model to Ising model. In particular, this possibility has We can now evaluate the reduced density matrix for the been investigated in linear spin chains. Of particular in- spin 1, ρˆR(t). Again with some straightforward algebra, 1 terest is the spin-spin correlation function, since it can and assuming that trρˆ (0)=1, we obtain 1 be seen as a way to measure the spin propagation speed (cid:18) ρ11 ρ12(t)(cid:19) andcorrelationtime. Toshowhowthepresentformalism ρˆR1(t)= (cid:0)ρ121(t)(cid:1)∗ 1ρ22 (10) can be applied to this case, we consider the Hamiltonian 1 1 Hˆ = (cid:80)N h σz + 1/2(cid:80)N λ (cid:0)σx⊗σx(cid:1), describing a where ρ12(t) = ρ12(cid:0)cos2t+i(ρ11−ρ22)sin2t(cid:1). With i=1 i i i,j ij i j 1 1 2 2 chain of N interacting spins in the presence of a static the reduced density matrix we can therefore calculate magnetic field B(cid:126) = (0,0,h ). In the following, we con- i i the expectation value of any single spin observable, sider the case of a chain of 11 spins, in the presence of a e.g., (cid:104)σz(cid:105) = ρ11 − ρ22 and (cid:104)σx(cid:105) = 2Re(ρ12)cos(2t) − 1 1 2 1 1 uniform magnetic field, hi = h for any i, and where the 2Im(ρ12)(ρ22−ρ11)sin(2t). 1 1 1 interaction if restricted to first neighbors. A quantity of ThegeneralizationtoachainofN interactingspins1/2 interestforthesesystemsisthetimeevolutionofthecor- therefore follows in the same footsteps. If the start with relationfunction,definedasC =(cid:104)σzσz(cid:105)−(cid:104)σz(cid:105)(cid:104)σz(cid:105). In the Hamiltonian Hˆ = (cid:80)N λ σz ⊗ σz, with λ = i,j i j i j i,j=1 i,j i j i,i Fig. 2 we report the time evolution of C6,j(t). We have 0 and use Eq. (4), we obtain the single spin stochastic chosenasinitialconditionthatallthespinbut6areina density matrix mixed state, while at t=0 the state of spin 6 is up, that √ √ ρî(t)=(cid:32)ρρ2i1i11ee−i(cid:80)(cid:80)NjNj==11√λλii,,jjωωj∗i,,ij ρρ2i21ie2−e(cid:80)i(cid:80)Nj=Nj1=1√λiλ,ji,ωjiω,jj∗,i (cid:33). hhisa=vρe16,11cha=onsde1n,λρaij16t,=2im=0e.0sρt126e,p1if=o|fi∆−ρ62t,j2=|==0.010,,02worahni0ldeoatwhveeerrhawagiveseed.uosWveeder (11) 10000 realizations of the stochastic noise. It is seen that Fromthisresult,derivingthesinglespinreduceddensity the correlation grows rapidly for the spin 5 and 7, while matrix is now straightforward, for the other spins, that are not directly connected with (cid:89)N (cid:18) ρ11 f (t)(cid:19) spin 6, it remains rather small. This can be compared ρˆRi (t)= trρˆk(t)ρî(t)= f (it)∗ ρi22 , (12) with the results of the experiments reported in [4, 5]. i i k=1 It is well known that the von Neumann equation is k(cid:54)=i equivalenttotheSchrödingerequationtoalmostanypur- where we have introduced the functions f (t) = i pose in standard quantum mechanics [14, 22]. It then ρ12(cid:81)N (cid:0)ρ11exp(−2iλ t)+ρ22exp(2iλ t)(cid:1). In i k=1, k(cid:54)=i k ik k ik appears natural to extend the result of this Letter to the same way, we can build the 2-spin reduced den- the many-body problem described by the time evolution sity matrix starting from its definition ρˆR (t) = i,j of the many-body wave-function, |Ψ(t)(cid:105). We can prove, (cid:81)N trρˆ (t)ρˆ(t)⊗ρˆ (t). The knowledge of ρˆR (t) al- following essentially in the footstep of the proof given k=1 k i j i,j k(cid:54)=i,j for the many-body density matrix, than it is possible lows us to calculate the correlation functions that have tofindstochasticsingle-particlewave-functions|ψ (cid:105)such beenrecentlymeasured[5,19],inaccordancetonewthe- i that,ifattheinitialtime|Ψ(0)(cid:105)=(cid:78)N |ψ (0)(cid:105),thenfor oretical results [20, 21]. The present theory provides an i=1 i easy way to reproduce and generalize those results. any subsequent time |Ψ(t)(cid:105) = (cid:78)N |ψ (t)(cid:105). The states i=1 i We can investigate how the presence of a magnetic ψ (t) evolve according to a stochastic Schrödinger equa- i field,forexampleinthexˆdirectionaffectsthedynamics. tion[14,23–26]whichhasaformsimilartoEq.(4). Inter- To do that, we consider two interacting spins with the estingly, while the standard relation ρ(t) = |Ψ(t)(cid:105)(cid:104)Ψ(t)| totalHamiltoniangivenbyHˆ =h σx+h σx+λσz⊗σz. is satisfied, there not exists a similar relation between 1 1 2 2 1 2 We can write down the stochastic equations of motion the single-particles, i.e., in general we should expect for the two single-spin density matrices, but their ana- that ρ (t) (cid:54)= |ψ (t)(cid:105)(cid:104)ψ (t)|. Indeed, one can prove that i i i lytic solution of little use since σ and σ do not com- |ψ (t)(cid:105)(cid:104)ψ (t)|hasanequationofmotionthatdoesnotre- z x i i muteandwehavetoreverttoanumericalsolutionofthe duce to Eq. (4). Finally, we would like to comment that 4 a formalism based on the wave-function |ψ (cid:105) seems lim- teractionsinaLi7Bose-Einsteincondensate,” Phys.Rev. i ited, since it is not clear how one could possibly obtain Lett. 102, 090402 (2009). information on the single-particle properties of the real [2] F Dalfovo, S Giorgini, L P Pitaevskii, and S Stringari, “TheoryofBose-Einsteincondensationintrappedgases,” many-body problem. In fact, if we start with the defi- Rev. Mod. Phys. 71, 463 (1999). nition of ρˆR(t) from Eq. (7), and use the wave-function i [3] Jason H V Nguyen, Paul Dyke, De Luo, Boris A Mal- |Ψ(t)(cid:105) instead we get, omed, and Randall G Hulet, “Collisions of matter-wave solitons,” Nat. Phys. 10, 918–922 (2014). ρˆRj (t)=trρˆ(t)|but j =tr(|Ψ(t)(cid:105)(cid:104)Ψ(t)|)|but j [4] P.Jurcevic,B.P.Lanyon,P.Hauke,C.Hempel,P.Zoller,   R. Blatt, and C. F. Roos, “Quasiparticle engineering (cid:79)N (cid:79)N (13) andentanglementpropagationinaquantummany-body =tr |ψi(t)(cid:105) (cid:104)ψi(t)| . system,” Nature 511, 202–205 (2014). i=1 i=1 [5] Philip Richerme, Zhe-Xuan Gong, Aaron Lee, Crystal but j Senko,JacobSmith,MichaelFoss-Feig,SpyridonMicha- It is therefore not possible to swap the trace operation lakis, Alexey V. Gorshkov, and Christopher Monroe, with the average over the realizations of the stochastic “Non-local propagation of correlations in quantum sys- temswithlong-rangeinteractions,” Nature511,198–201 noise, thereforeprecludinganalternativewaytoEq.(7). (2014). Similarproblemsarisewhenoneconsidersanyn-bodyre- [6] Richard D. Mattuck, A Guide to Feynman Diagrams in duceddensitymatrixasobtainedfromthesingle-particle the Many Body Problem, 2nd ed. (Dover Publications, stochastic wave-functions. New York, 1992). In conclusion, we have shown that the dynamics [7] Gabriele F. Giuliani and Giovanni Vignale, Quantum of a many-body system, with multiplicative two-body Theory of the Electron Liquid (Cambridge University particle-particle interaction can be reduced to the inves- Press, Cambridge, 2005). [8] B. H. Bransden and C. J. Joachain, Physics of Atoms tigation of the dynamics of N single particle stochas- and Molecules (Longman Scientific and Technical, Har- tic systems. We have shown how to calculate any re- low, 1995). duced n-body properties starting from the solution of [9] P. Hohenberg and W. Kohn, “Inhomogeneous Electron these stochastic dynamical equations. We have applied Gas,” Phys. Rev. 136, B864–B871 (1964). this formalism to the case of spin chains in the presence [10] Walter Kohn and L J Sham, “Self-Consistent Equations of a finite magnetic field. IncludingExchangeandCorrelationEffects,” Phys.Rev. 140, A1133 (1965). [11] RMDreizlerandEKUGross,DensityFunctionalThe- ory (Springer-Verlag, Heidelberg, 1990). METHODS [12] Eq.(4)isnotunique.Indeed,afamilyofequivalentequa- tionscanbeeasilyobtainedbyshiftingtheweightofthe Here we present the proof of Eq. (5), based on the Itô interactionbetweenthetermsintheroundbracketinthe calculus [14, 16]. right hand side of that equation. The form chosen here keepsabalancedweightbetweenthetwoterms.Noneof Proof. The proof of Eq. (5), and that ρˆ(t) follows the physical results depend on the details of this choice. the dynamics induced by the von Neumann’s equation [13] C. W. Gardiner, Handbook of Stochastic Methods for (2) is a straightforward generalization of the results of Physics, Chemistry, and the Natural Sciences (Springer, Shao on the dynamics of a quantum system in contact 1983). with an external environment [27, 28]. Let us consider [14] C. W. Gardiner and P. Zoeller, Quantum Noise, 2nd ed. the stochastic total density matrix, ρ˜(t)=(cid:78) ρ (t). Ac- (Springer, Berlin, 2000). i i [15] Heinz-PeterBreuerandFrancescoPetruccione,TheThe- cording to Itô’s stochastic calculus, the dynamics of ρ˜is determined by, dρ˜(t) = (cid:80)N ρ ⊗ dρ ⊗ ρ + oryofOpenQuantumSystems (OxfordUniversityPress, j=1 1,j j j+1,N+1 New York, 2002). (cid:80)Nk<j=1(ρ1,k⊗dρk⊗ρk+1,j ⊗dρj ⊗ρj+1,N+1), where [16] Desmond J. Higham, “An Algorithmic Introduction to we have introduced the short-hand notation ρ = Numerical Simulation of Stochastic Differential Equa- i,j (cid:78)j−1ρ . The proof that ρ˜(t) = ρ(t) then continues by tions,” SIAM Rev. 43, 525–546 (2001). k=i k [17] We leave out the question if ρˆR(t) is also definitive posi- using Eq. (4), Eq. (6), and [xˆ ,ρˆ ]=0 if i(cid:54)=j, to arrive j (cid:104) (cid:105) i j tive. This point requires further investigation. at dρ˜(t) = −i Hˆ,ρ˜(t) dt. Due to the linearity of this [18] This can be easily proven by considering the stochastic differentialequationsolvedbyexp(αp)accordingtoIto’s equation of motion, and that by definition ρ˜(0) = ρˆ(0), calculus where p is a real white noise, then taking the we conclude that ρ˜(t)=ρˆ(t) at any time t≥0.(cid:3) average,andsolvingthedifferentialequationforexp(αp) obtained in this way. [19] C. Senko, J. Smith, P. Richerme, A. Lee, W. C. Camp- bell, and C. Monroe, “Quantum simulation. Coherent imagingspectroscopyofaquantummany-bodyspinsys- ∗ [email protected] tem.” Science 345, 430 (2014). [1] S. E. Pollack, D. Dries, M. Junker, Y. P. Chen, T. A. [20] Michael Foss-Feig, Kaden Hazzard, John Bollinger, and Corcovilos, and R. G. Hulet, “Extreme tunability of in- Ana Rey, “Nonequilibrium dynamics of arbitrary-range 5 Ising models with decoherence: An exact analytic solu- [28] Jiushu Shao, “Dissipative dynamics from a stochastic tion,” Phys. Rev. A 87, 042101 (2013). perspective,” Chem. Phys. 370, 29–33 (2010). [21] Mauritz van den Worm, Brian C Sawyer, John J Bollinger, and Michael Kastner, “Relaxation timescales and decay of correlations in a long-range interacting 0.6 quantum simulator,” New J. Phys. 15, 083007 (2013). [22] JJSakurai,ModernQuantumMechanics (AddisonWes- 0.4 ley, 1994). [23] WalterT.Strunz,LajosDiósi, andNicolasGisin,“Non- 0.2 Markovian Quantum State Diffusion and Open System R Dynamics,” Lect. Notes Phys. 538, 271–280 (2000). ρ10 [24] Robert Biele and Roberto D’Agosta, “A stochastic ap- -0.2 proach to open quantum systems.” J. Phys. Condens. Matter 24, 273201 (2012). Re(ρR) -0.4 1 11 [25] RobertoD’AgostaandMassimilianoDiVentra,“Stochas- Re(ρR) 1 12 tic time-dependent current-density functional theory: a -0.6 functionaltheoryofopenquantumsystems,” Phys.Rev. 0 20 40 60 80 100 120 t [arb. units] B 78, 165105 (2008). [26] RobertoD’AgostaandMassimilianoDiVentra,“Founda- tions of stochastic time-dependent current-density func- FIG. 1. The time dynamics of the real parts of (ρR) and 1 11 tional theory for open quantum systems: Potential pit- (ρR) calculated from Eqs. (7) and (4). As initial condition 1 21 falls and rigorous results,” Phys. Rev. B 87, 155129 we have set the two spin in the “mixed” state. We have av- (2013). eraged over 106 independent realizations of the white noises. [27] Jiushu Shao, “Decoupling quantum dissipation interac- Forthecalculationwehavechosenh =h =1andλ=0.02, 1 2 tion via stochastic fields.” J. Chem. Phys. 120, 5053–6 andatimestep,∆t=0.01. Wecanseethataftert∼100the (2004). finitenumberofrealizationsandthefinitenumericalaccuracy insolvingthestochasticdifferentialequationsareintroducing an error in both ρ11 and ρ12. 1 1 6 0.08 14 0.06 12 0.04 nits] 10 0.02 u 8 b. 0 C6j ar 6 -0.02 t [ 4 -0.04 2 -0.06 0 -0.08 1 2 3 4 5 6 7 8 9 10 11 j FIG.2. DynamicsofthecorrelationfunctionC (t)intime. 6,j We have chosen for the parameters h = 1 and λ = 0.01. All thespinbut6areinamixedstate,whilespin6startsinthe state “up”. We have chosen a time step of ∆t = 0.002 and averaged over 10000 realizations of the stochastic noise.