Transport properties of continuous-time quantum walks on Sierpinski fractals
Zoltán Darázs, Anastasiia Anishchenko, Tamás Kiss, Alexander Blumen, Oliver Mülken
TTransport properties of continuous-time quantum walks on Sierpinski fractals
Zoltán Darázs,
1, 2, ∗ Anastasiia Anishchenko, ∗ Tamás Kiss, Alexander Blumen, and Oliver Mülken WIGNER RCP, SZFKI, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, Hungary Eötvös University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany (Dated: October 5, 2018)We model quantum transport, described by continuous-time quantum walks (CTQW), on deterministic Sier-pinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures.The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by themean survival probability when absorbing traps are present. In the case of gaskets, localization can be identifiedalready for small networks (generations). For carpets, our numerical results indicate a trend towards localiza-tion, but only for relatively large structures. The comparison of gaskets and carpets further implies that, distinctfrom the corresponding classical continuous-time random walk, the spectral dimension does not fully determinethe evolution of the CTQW.
PACS numbers: 05.60.Gg, 05.60.Cd, 05.45.Df
I. INTRODUCTION
Networks are sets of connected nodes [1, 2], and their staticand the dynamic properties are of much interest. Applicationsrange from, say, polymer science [3], over traffic and powergrid studies [4], up to social networks [5]. A special classof networks are deterministic fractals which as such can bebuilt iteratively. We remark that for them sometimes analyticresults can be obtained, see, e.g., Refs. [6–8].Now, the classical dynamics of random walks (RW) overnetworks has been extensively investigated in the last decades[9, 10]. This effort has led to a very detailed understanding ofthe influence of the network’s topology on RW. When the effi-ciency of transport is concerned, the question whether the RWis recurrent or transient boils down to determining the proba-bility of the RW to return to its origin, which is also related tothe Pólya number [11]. Moreover, the global properties of theRW can also be captured by introducing the local probabil-ity decay channels and calculating the averaged decay time ofthe excitation, known as the averaged mean first passage time(MFPT) [9, 12]. For simple undirected networks the transfermatrix of the continuous-time random walk (CTRW) is givenby the connectivity matrix of the network [13]. Many net-works show scaling behavior for the lower part of the spec-trum of the connectivity matrix, with an exponent d s whichis called spectral dimension [14]. As it turns out, d s deter-mines many of the dynamical properties of the network, e.g.,the return to the origin or the MFPT.For the quantum mechanical aspects of transport on net-works, we choose as a model the continuous-time quantumwalk (CTQW), which is related to the classical CTRW [13].In this way, the Hamiltonian is determined by the connectiv-ity of the network. Therefore, by analyzing the connectivitymatrix, we obtain results for both, CTRW and CTQW. Whilein recent years CTQW over several types of networks havebeen analyzed [13], there is no unambiguous classification ∗ These authors contributed equally to this work. according to, say, the spectral dimension. In many aspects,the quantum dynamics is much richer (i.e., more complex)than the classical CTRW counterpart, since it also involvesthe wave properties of the moving object. In several casesof tree-like networks, such as stars [15, 16] or dendrimers[17], it has been shown that the (average) quantum mechani-cal transport efficiency, defined by the return to the origin, israther low compared to structures which are translationally in-variant. Quantum walks are interesting models also from thepoint of view of quantum information processing [18]. Searchvia quantum walks on fractal graphs has been considered inRefs. [6, 19, 20].A similar mathematical model arises for condensed mattersystems, in which one considers a particle moving on an un-derlying fractal lattice (a Sierpinski gasket); here the solutionof Schrödinger’s equation has been studied within the tight-binding approximation [21, 22]. For several fractals consid-ered, the dynamics has been shown to be subject to localiza-tion effects, similar to the classical waves in fractal waveg-uides [23]. From an experimental point of view, recent yearshave seen a growing number of possible implementations ofCTQW, for example, using interference effects of light. Thoseexperiments range from photonic waveguides [24] to fiber-loops [25].In this paper we study quantum transport over fractal net-works, namely over Sierpinski gaskets (SG) and their dualstructures (DSC) as well as over Sierpinski carpets (SC) andtheir dual structures (DSC). In the case of the SG and of theirduals we find clear signatures of localization around the ini-tial starting node, indicating recurrent behavior. We seek toanswer the question whether the spectral dimension d s of thegraph determines the transport properties for CTQW. Giventhe great experimental control over, say, coupling rates and de-coherence, we believe that our results for fractal structures canalso be experimentally realized, say, through photonic waveg-uides.The paper is organized as follows, Sec. II gives an overviewover the quantities we use to determine the performance ofCTQW over networks. In Sec. III we outline the deterministicconstruction rules of the SG and SC and their dual transfor- a r X i v : . [ qu a n t - ph ] S e p mations, along with their spectral properties. These systemsare then analyzed in detail in Secs. IV-VII. We close with asummary of results in Sec. VIII. II. METHODS
We model the quantum dynamics of an excitation over agiven fractal network by CTQW and compare this to its classi-cal counterpart, the CTRW, over the same network. A networkis determined by a set of N nodes and a set of bonds. Witheach of the nodes we associate a state | k (cid:105) corresponding to anexcitation localized at node k . For both, CTQW and CTRW,the dynamics is determined by the network’s connectivity, i.e.,by its connectivity matrix A . The off-diagonal elements of A are A kj = − if the nodes k and j are connected by a singlebond and are A kj = 0 otherwise; the diagonal elements are A kk = f k , where f k is the functionality of node k , i.e., thenumber of nodes connected to k through a single bond. Thematrix A is real and symmetric and has only real and non-negative eigenvalues. For networks without disjoint parts alleigenvalues are positive except one, E min = 0 .Now, we take for CTRW the transfer matrix T = − A and for CTQW the Hamiltonian H = A (i.e. in the fol-lowing we set (cid:126) = 1 and normalize the transfer capac-ity of each bond to unity), see also [13, 26], such that thetransition probabilities read p k,j ( t ) = (cid:104) k | exp( T t ) | j (cid:105) and π k,j ( t ) = |(cid:104) k | exp( − i H t ) | j (cid:105)| , respectively. By diagonaliz-ing A we obtain the eigenvalues E n and the eigenstates | Φ n (cid:105) (with n = 1 , . . . , N ) of A , resulting in p k,j ( t ) = N (cid:88) n =1 exp( − E n t ) (cid:104) k | Φ n (cid:105)(cid:104) Φ n | j (cid:105) (1)for CTRW and π k,j ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n =1 exp( − iE n t ) (cid:104) k | Φ n (cid:105)(cid:104) Φ n | j (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2)for CTQW. In principle all quantities of interest can be calcu-lated on the basis of the transition probabilities. In order toquantify the efficiency of the transport, we will focus on threequantities: the exact return probability and the related Pólyanumber, the average return probability, and the mean survivalprobability. A. Pólya number
The so-called Pólya number allows to assess the local trans-port properties. In classical systems, the definition of the re-currence is straightforward: it characterizes the event that thewalker returns to its initial position. For quantum walks onecan imagine different definitions depending on the envisagedmeasurement procedure [27–33].Ref. [34] suggests a possible quantum definition for thePólya number, which is directly related to the return proba- bility to the initial node ( | ψ (0) (cid:105) = | (cid:105) ): π , ( t ) = |(cid:104) | exp( − i H t ) | (cid:105)| . (3)The formal definition of the Pólya number reads P = 1 − ∞ (cid:89) i =1 [1 − π , ( t i )] , (4)where the set { t i , i = 1 , . . . ∞} is an infinite time series whichcan be chosen regularly or be determined by some randomprocess. It can be shown that its value depends on the conver-gence speed of π , ( t ) to zero: if π , ( t ) converges faster than t − then the CTQW is transient, otherwise it is recurrent [34].For a finite network of N sites the probability that we findthe walker at the origin can be written as a finite sum of cosinefunctions, π , ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n =1 (cid:104) | e − iE n t | Φ n (cid:105)(cid:104) Φ n | (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = N (cid:88) n,m =1 |(cid:104) | Φ n (cid:105)| |(cid:104) | Φ m (cid:105)| cos [( E m − E n ) t ] . (5)A finite sum of cosine functions cannot be a decaying func-tion of time and thus for any finite system the Pólya numberequals one, meaning that the walk is recurrent. On the otherhand, in an infinite network ( N → ∞ ), π , ( t ) might tendto zero in the t → ∞ limit. If the return probability has theasymptotic form π , ( t ) ∼ f ( t ) · t − δ where f ( t ) is a periodicor an almost periodic analytical function, then, with regularand Poissonian sampling, the walk is recurrent if δ ≤ , andit is transient if the envelope decays faster ( δ > ) [34]. ForCTRW on the fractals considered in the following, the decayof the probability p , ( t ) is slower than t − , which can beseen from the fact that on a fractal p , ( t ) scales as t − d s / andthe fractals considered in this paper have spectral dimension d s < [35, 36]. B. Average return probability
As a global efficiency measure, the average return proba-bility is defined as the probability to remain or return to theinitial node j , averaged over all nodes: p ( t ) ≡ N N (cid:88) j =1 p j,j ( t ) (6)and π ( t ) ≡ N N (cid:88) j =1 π j,j ( t ) . (7)While p ( t ) only depends on the eigenvalues, π ( t ) also de-pends on the eigenstates. However, by using the Cauchy-Schwarz inequality a lower bound, independent of the eigen-states, has been introduced in [15]: π ( t ) = 1 N N (cid:88) j =1 π j,j ( t ) ≥ (cid:12)(cid:12)(cid:12) N N (cid:88) j =1 α j,j ( t ) (cid:12)(cid:12)(cid:12) ≡ (cid:12)(cid:12)(cid:12) α ( t ) (cid:12)(cid:12)(cid:12) . (8)In Eq. (8) α j,j ( t ) = (cid:104) j | exp( − i H t ) | j (cid:105) is the transition am-plitude between two nodes. In the following we will compare p ( t ) with (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) and express both quantities in terms of the(discrete) density of states (DOS) ˜ ρ ( E ) = 1 N N (cid:88) n =1 δ ( E − E n ) . (9)Here δ ( E − E n ) is the Dirac delta-function. Integrating ˜ ρ ( E ) in a very small neighborhood of an eigenvalue, say, E m , gives lim ε → + (cid:90) E m + εE m − ε ˜ ρ ( E ) d E = D ( E m ) /N ≡ ρ ( E m ) . (10)where D ( E m ) is the degeneracy of E m and we introduced ρ ( E ) . This yields p ( t ) = (cid:88) { E m } ρ ( E m ) exp( − E m t )= (cid:90) ∞−∞ ˜ ρ ( E ) exp( − Et ) d E (11)and (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:88) { E m } ρ ( E m ) exp( − iE m t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) ∞−∞ ˜ ρ ( E ) exp( − iEt ) d E (cid:12)(cid:12)(cid:12) , (12)where the sums run over the set { E m } of distinct eigenvalues.Now, if both p ( t ) and (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) decay very quickly in time,the average probability to find the excitation at any node butthe initial node increases quickly. Then we call the transportover the network efficient, because ( on average ) the excitationwill efficiently explore parts of the network away from theinitial node. In contrast, if these quantities decay very slowly,we regard the transport as being inefficient.For CTRW and not too short times, p ( t ) is dominated bythe small eigenvalues. For fractals, the DOS typically scaleswith the so-called spectral dimension d s [14], i.e., ˜ ρ ( E ) ∼ E d s / − . Then, one finds in an intermediate time range, be-fore the equilibrium value is reached, that p ( t ) ∼ t − d s / .However, for CTQW such a simple analysis does not hold dueto the coherent evolution. Instead, highly degenerate eigen-values dominate (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) , see Ref. [21, 37]. In the case thatone has a single highly degenerate eigenvalue E m , the lowerbound of the average return probability can be approximatedby [16] | ¯ α ( t ) | ≈ ˜ ρ ( E m ) + ˜ ρ ( E m ) lim ε → + (cid:34) (cid:90) E m − ε −∞ ˜ ρ ( E )cos (cid:2) ( E − E m ) t (cid:3) d E + (cid:90) ∞ E m + ε ˜ ρ ( E ) cos (cid:2) ( E − E m ) t (cid:3) d E (cid:35) . (13)If there is at least one eigenvalue for which ˜ ρ ( E m ) is O (1) ,then the average transition amplitude does not tend to zero. Then the long time average χ lb of the transition probabil-ity also allows to quantify the global performance of CTQWthrough [16] χ lb = lim T →∞ T (cid:90) T | ¯ α ( t ) | d t = (cid:88) { E m } [˜ ρ ( E m )] . (14) C. Mean survival probability
In order to corroborate our findings for the average returnprobabilities, we define another (global) transport efficiencymeasure which is based on the mean survival probability, seealso [38] for CTQWs and [39] for discrete time quantumwalks. Here, the original network is augmented by local decaychannels which act as traps for the walker. These traps are lo-calized at a set M of nodes m of the original network. For thisthe total number of nodes of the system is not changed, but thetransfer matrix T as well as the Hamiltonian H get augmentedby additional terms, such that the new (effective) matrices read T eff ≡ T − Γ and H eff ≡ H − i Γ , respectively, where thetrapping matrix is diagonal, namely Γ = Γ (cid:80) m ∈M | m (cid:105)(cid:104) m | with a trapping rate Γ which we set equal for all traps. Wenote that such an effective Hamiltonian can be obtained withinthe framework of quantum master equations of Lindblad type,where the network is only coupled to the environment at thetrap nodes, see Ref. [40]. For CTRW, such traps will still leadto a real symmetric transfer matrix, but now with only posi-tive eigenvalues [9]. For CTQW, the new Hamiltonian H eff becomes non-Hermitian. Such Hamiltonians can have com-plex eigenvalues E n = (cid:15) n − iγ n with a real part (cid:15) n and animaginary part γ n . As has been shown in [38], by averagingthe transition probabilities over all possible initial and finalnodes one obtains the mean survival probability for CTQW asa function solely of the γ n , Π( t ) ≡ N N (cid:88) j,k =1 π k,j ( t ) = 1 N N (cid:88) n =1 exp( − γ n t ) . (15)Note the slightly different definition of Π( t ) compared to theone in Ref. [38]. Here we do not exclude the trap nodes fromthe sum, thus Eq. (15) becomes exact. For CTRW a similarapproach with the new transfer matrix T eff yields [41] P ( t ) ≡ N N (cid:88) j,k =1 p k,j ( t ) = 1 N N (cid:88) n =1 exp( − λ n t ) (cid:12)(cid:12)(cid:12) N (cid:88) j =1 (cid:104) j | Ψ n (cid:105) (cid:12)(cid:12)(cid:12) , (16)where λ n and | Ψ n (cid:105) are the eigenvalues and eigenstates of T eff , respectively. Thus, P ( t ) will eventually decrease to zeroand the asymptotical behavior will be dominated by the small-est eigenvalue. Now, if Π( t ) and P ( t ) decrease quickly wealso call the transport efficient (on average) because then aninitial excitation will reach the trap rather quickly.For CTQW one can relate the γ n to the eigenstates of theoriginal Hamiltonian H within a (non-degenerate) perturba-tive treatment, γ n = Γ (cid:80) m ∈M (cid:12)(cid:12) (cid:104) m | Φ n (cid:105) (cid:12)(cid:12) [13]. Thus, theimaginary parts γ n are determined by the overlap of the eigen-states | Φ n (cid:105) of H with the locations of the traps. This impliesthat for localized eigenstates this overlap can be zero, suchthat for some n the imaginary parts vanish, γ n = 0 . Thisyields a mean survival probability which does not decay tozero but which reaches the asymptotic value Π ∞ ≡ lim t →∞ Π( t ) = N N , (17)where N is the number of eigenstates for which the γ n van-ish. For the SG it has been shown that such eigenstates exist,which in fact gives rise to localization effects [21].We have now defined the asymptotic quantity Π ∞ whichallows us to assess the transport properties of CTQW by cal-culating the probability that the walker will stay forever in thenetwork. III. THE SYSTEMS UNDER STUDY
We now discuss the systems under study and their topolog-ical properties. We consider two groups of Sierpinski fractals,namely, gaskets and carpets, along with their dual transfor-mations. These fractals are built in an iterative manner: Inorder to construct the SG, one starts from a triangle of threenodes. In the next step two additional triangles are attachedto the corner nodes by merging them, so that they form a big-ger self-similar triangle. The procedure is then iterated, seeFig. 1(a) for a gasket at generation g = 3 . A similar ideais used for creating SC, where instead of triangles the centralbuilding blocks are squares, see also Fig. 1(c). At generation g the total number of nodes of the SG is N SG = (3 g + 3) / and of the SC is N SC = · g + · g + , so that at the same(large) g the carpet has much more nodes than the gasket.The dual networks of the Sierpinski fractals are easily ob-tained by the following procedure: In the original structureone replaces each of the smallest building blocks (triangles forgaskets and squares for carpets) by a node and connects thenthe nodes which belong to building blocks sharing a node (forcarpets we only allow connections in the horizontal and in thevertical direction but not diagonally), see also Figs. 1(b) and1(d), which illustrate the procedure by also showing the un-derlying lattices of the SG and SC, respectively. The numberof nodes of the DSG of generation g is N = 3 g and of theDSC of generation g is N = 8 g .Based on real space renormalization arguments, one canshow that a structure and its dual have the same fractal d f and spectral d s dimensions. For the SG and the DSG, the cor-responding values are d f = ln(3) / ln(2) ≈ . . . . and d s = 2ln(3) / ln(5) ≈ . . . . , see Ref. [6]. For the SCand the DSC, one has d f = ln(8) / ln(3) ≈ . . . . and d s ≈ . [36].For our calculations of the average return probabilities weassume that every single node of the network can be the ori-gin of the walk with the same probability and that the averageruns over all sites j = 1 , . . . , N . For the individual return Corner traplocation Central traplocation (a) SG (b) DSG(c) SC (d) DSC
FIG. 1: (Color online) The graphs under study. The graphsare at third generation ( g = 3 ), except the DSC for which g = 2 . We denoted the g = 1 graphs with green, and theholes with a gray (striped) background. Traps are put eitherat the positions indicated by the small diamonds or at thepositions indicated by the small squares.probability π j,j ( t ) we use the outer corner node as initialnode. As for the mean survival probabilities, we will distin-guish between two situations: (1) when there are three (four)trap nodes at the outer corners of the gasket (carpet), see thered diamonds in Fig 1, and (2) when the three (four) trap nodesare placed at the corners of the largest empty inner triangle(square) of the gasket (carpet), see the red squares in Fig. 1.Since the quickest decay of Π( t ) for the linear networks stud-ied in Ref. [38] is obtained when the trapping strength Γ isof the same order of magnitude as the coupling between thenodes, we choose Γ = 1 in all calculations involving traps.Let us first consider systems without traps. Since the eigen-value distributions are crucial for determining the global effi-ciency measures, we start by considering the differences be-tween our four fractal structures. In Fig. 2 we plot for sev-eral structures the normalized cumulative eigenvalue countingfunction N ( x ) = 1 N N (cid:88) n =1 θ ( x − E n E max ) , (18)where θ ( x ) is the Heaviside-function. Now, E min = 0 is thesmallest and E max the largest eigenvalue, hence, the range of x is [0 , .Already here we can exemplify the role of highly degener-ate eigenvalues. For large N the eigenvalue counting functionfor an N × N square lattice is a quite smooth function, whichfor N → ∞ we plot as a reference in Fig. 2.Also the SC for g = 6 leads to a quite smooth form for N ( x ) . However, N ( x ) for the DSC of g = 5 displays markedsteps, but its overall shape is close to the one for the SC. Forthe SG for g = 9 and its dual, the DSG for g = 9 , N ( x ) has sharp discontinuities, which reflect the presence of manyhighly degenerate eigenvalues. Already at this point we seea clear distinction between gaskets and carpets: at similar g the carpets do not have eigenvalues of such high degeneracyas the gaskets. N x DSG SG2D DSC SC
2D latticeSC g = 6 DSC g = 5 SG g = 9 DSG g = 9 FIG. 2: (Color online) The eigenvalue counting function N ( x ) , Eq. (18), for several systems under study, compared tothe simplest case of an infinite discrete square lattice, see textfor details. IV. DUAL SIERPINSKI GASKET
We start by considering the DSG, see also Fig. 1(b). Asthe SG, the DSG is a deterministic fractal, iteratively built upgeneration by generation. CTQW on DSG of different genera-tions have been studied by us in Ref. [6]. We will recapitulatethe major results, since we will use the DSG as a referencefor our new results presented below. In fact, DSG is special,in that its eigenvalues, and hence its DOS can be determinediteratively, in a simple way. This does not hold for the otherfractals considered here.For DSG the results for the CTRW and CTQW return prob-abilities p , ( t ) and π , ( t ) , along with the CTQW lowerbound (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) of π ( t ) (see Eqs. (1), (2), and (8), respectively)have been already presented in Ref. [6]. There it has been ver-ified that for the classical average return probability, the decayto the equipartition value is determined solely by d s [42], hav-ing namely p ( t ) ∼ t − d s / . It follows that the classical walk onDSG is recurrent and that the Pólya number equals unity. Aswe will show below for all the fractal types considered here,such a quite simple law does not hold for CTQW.Turning now to the quantum case and evaluating the lowerbound (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) of π ( t ) of the quantum average return probabil- ity π ( t ) , see Eq. (8), it has been found in [6] that its envelopedoes not show a strong dependence on the size of the DSG.Since the two eigenvalues and make up for about / of alleigenvalues, they control most of the behavior of π ( t ) . Then ρ (3) and ρ (5) are known in closed form, ρ (3) = 12 · g (cid:0) g − + 3 (cid:1) (19)and ρ (5) = 12 · g (cid:0) g − − (cid:1) . (20)In particular, also the long time average χ lb can be calculatedexactly, based on Eq. (14) χ lb = 13 g (cid:20) g (cid:18) g (cid:19) + 107 2 g − (cid:21) , (21)which for large g is much larger than the equipartition value − g . The limit g → ∞ yields lim g →∞ χ lb = 1 / ≈ . . (22)For both highly degenerate eigenvalues, TABLE I shows ρ (3) and ρ (5) , Eq. (10), for successive generations g from to ,calculated according to Eqs. (19) and (20). Also the exactvalue of χ lb , see Eq. (21), is shown. Both ρ (3) and ρ (5) tendto the exact limiting value / , see Eqs. (19) and (20), ratherfast, which, together with Eq. (13), means that the transport isquite inefficient.Now, we calculate for the DSG Π (1) ∞ and Π (2) ∞ usingEq. (17). In order to do this, we numerically determine theeigenvalues of the non-Hermitian H eff , paying particular at-tention to their imaginary parts γ . We do this using the MAT-LAB / GNU Octave eig() function, and in order to be moreprecise, we employed the LAPACK zgeev() function in ourFortran code with quadruple precision. Despite these efforts,the procedure may not be exact, however. First, we cannotexclude the existence of very small, but nonzero imaginaryparts which are smaller than − and are set to zero. Sec-ond, numerical errors may induce small imaginary contribu-tions where there should be none. Thus, the values in our tablefor Π ∞ may not be as exact as their form seems to imply.Counting all the eigenvalues with vanishing imaginary partwe then obtain N . From it we readily evaluate Π (1) ∞ and Π (2) ∞ ,see Eq. (17). In the next sections, the same procedure will beemployed for the other fractals studied. The analysis of thedata of TABLE II shows that Π (1) ∞ and Π (2) ∞ increase with in-creasing g , which means that N increases faster than N . Al-ready for g = 7 , corresponding to a network of N = 2187 nodes, the probabilities Π (1) ∞ and Π (2) ∞ that the walker surviveswithin the network are close to . and to . , respec-tively. We note that the values of Π (2) ∞ are somewhat belowthe ones for Π (1) ∞ , implying that here traps on the peripheryact somewhat less efficiently than centrally located traps. g ρ (3) ρ (5) χ lb / ≈ . / ≈ . . / ≈ . / ≈ . . / ≈ . / ≈ . . / ≈ . / ≈ . . / ≈ . / ≈ . . / ≈ . / ≈ . . / ≈ . / ≈ . . TABLE I: The ρ ( E ) for the eigenvalues E = 3 and E = 5 and the long time average χ lb for different generations of the DSG. g Π (1) ∞ = N (1)0 /N Π (2) ∞ = N (2)0 /N / ≈ . / ≈ .
333 6 / ≈ . / ≈ .
531 36 / ≈ . / ≈ .
679 150 / ≈ . / ≈ .
783 540 / ≈ . / ≈ .
855 1806 / ≈ . TABLE II: The asymptotic limit Π ∞ of Π( t ) for DSG ofgenerations g = 2 , . . . , ; case (1) : the traps are placed onthe corner nodes, diamonds in Fig. 1(b); case (2) : the trapsare placed on the central nodes, squares in Fig. 1(b), see textfor details. V. SIERPINSKI GASKET
While the DSG allows for partly analytical results, we haveto resort to numerical calculations for the other structuresconsidered in this paper. We proceed our investigation withthe SG. In analogy to our study on DSG, we start with theCTRW and CTQW return probabilities as well as with thelower bound for the CTQW decay. These quantities involvethe eigenvalues and (depending on the functions considered)sometimes also the eigenstates of the Hermitian operators T and H , see Eqs. (1)-(2). Unlike the DSG case, for SG generalrecursive expressions for the eigenvalues of T and H are notknown. Therefore, we calculate both the eigenvalues and theeigenfunctions numerically. For this, we again use the MAT-LAB / GNU Octave eig() and eigs() functions. For large gen-erations, we calculate only the spectrum in order to evaluate | α ( t ) | and the D ( E m ) degeneracy of the eigenvalue E m us-ing the filtered Lanczos algorithm in C++ [43], and the MAT-LAB / GNU Octave eigs() function.Figure 3 shows the classical p , ( t ) and the quantum me-chanical π , ( t ) return probabilities to the initially excitednode j = 1 for a SG with g = 7 . The red dashed line in Fig. 3gives the CTRW return probability p , ( t ) . While the alge-braic decay of p , ( t ) ∼ t − d s / holds asymptotically only foran infinite fractal, one can still recognize this scaling behav-ior in an intermediate time domain in Fig. 3, before p , ( t ) saturates to the equipartition value /N at long times. Fig-ure 3 also shows the exact quantum return probability π , ( t ) (green solid line). After an initial decay to a local minimum,the return probability starts to oscillate around its long timeaverage. In the inset of Fig. 3 we present the quantum me-chanical lower bound (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) of the quantum average returnprobability π ( t ) ; (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) does not decay eventually, but showsstrong oscillations with a long time average χ lb (dashed blackline in Fig. 3) which is orders of magnitude larger than /N .Now, in contrast to CTRW, there is no apparent relation be-tween the spectral dimension and the return probability. − − − − − − − − π , ( t ) t − − − − | α ( t ) | t FIG. 3: (Color online) Quantum return probability π , ( t ) tothe corner node j = 1 (green solid line) along with itsclassical analogue (red dashed line) for the SG of g = 7 .Inset: CTQW lower bound (cid:12)(cid:12) α ( t ) (cid:12)(cid:12) of π ( t ) on the SG at g = 7 (blue solid line) and χ lb , the long time value (blackdashed line).The spectrum of the Hamiltonian already reveals whetherCTQW shows localization. For different generations of theSG, we calculate, based on Eq. (10), the ρ ( E ) of the highlydegenerate eigenvalue , ρ (6) and, based on the r.h.s. ofEq. (14), the long-time average χ lb . The data are presentedin TABLE III. As in the case of the DSG, also for SG the ρ (6) seem to converge with increasing g to the finite limiting value / . As before, such a relatively large nonvanishing value letsus infer that the transport is not very efficient.We now turn to CTQW on SG in the presence of traps, pro-cess which introduces non-Hermitian operators. In TABLE g ρ (6) χ lb ≈ ≈ ≈ ≈ ≈ ≈ TABLE III: The ρ ( E ) for the eigenvalue E = 6 and the longtime average χ lb for different generations of the SG. g Π (1) ∞ = N (1)0 /N Π (2) ∞ = N (2)0 /N / ≈ .
27 1 / ≈ . /
42 = 0 . / ≈ . / ≈ .
67 70 / ≈ . / ≈ .
78 261 / ≈ . / ≈ .
85 886 / ≈ . TABLE IV: The asymptotic limit Π ∞ of Π( t ) for SG ofgenerations g = 2 , . . . , ; case (1) : the traps are placed onthe corner nodes, diamonds in Fig. 1(a); case (2) : the trapsare placed on the central nodes, squares in Fig. 1(a), see textfor details.IV, we show Π (1) ∞ and Π (2) ∞ for two situations, namely whenthe traps are placed on the corners and when the traps areplaced in the center of the structure, see Fig. 1(a) for details.TABLE IV suggests that the situation is quite similar to theone for the DSG: the higher g the less it is probable that theexcitation will be absorbed even after a very long time, see theincrease in the Π ∞ -values. However, the amount of excitationwhich stays localized in the network is higher in case (1) thanin case (2) , since Π (1) ∞ > Π (2) ∞ . VI. DUAL SIERPINSKI CARPET
We continue our analysis by considering the SC and theDSC. We start with the DSC, the spectrum and harmonic func-tions of which have been considered recently [44, 45].Figure 4 presents the lower bound | α ( t ) | of the quantum π ( t ) for g = 5 , see Eq. (8). The inset of Figure 4 depicts theclassical return probability p , ( t ) given by Eq. (1). We notethat at intermediate times p , ( t ) shows an algebraic decaywith slope d s / . Furthermore, | α ( t ) | displays at short to in-termediate time a decay of the maxima, while at longer timesit slowly approaches χ lb , given in Fig. 4 through a dotted linearound which it oscillates. Given that for DSC χ lb is muchsmaller than the corresponding χ lb for SG and for DSG, weinfer that localization effects are smaller for DSC than for SG and for DSG. − − − − − | α ( t ) | t − − − p , ( t ) t FIG. 4: (Color online) The average return amplitude | α ( t ) | on the DSC of g = 5 (blue solid line) and the long timeaverage (black dotted line). Inset: classical return probability p , ( t ) for DSC of g = 2 , , and (dotted blue, dashedgreen, solid red line, respectively) as well as the fitted decay(solid straight black line), . · t − . / . g ρ (3) χ lb / ≈ . × − . × − / ≈ . × − . × − / ≈ . × − . × − / ≈ . × − . × − / ≈ . × − – TABLE V: The ρ (3) for the eigenvalue E = 3 and the longtime average χ lb for different generations of the DSC. g Π (1) ∞ = N (1)0 /N Π (2) ∞ = N (2)0 /N / ≈ .
234 14 / ≈ . / ≈ .
246 126 / ≈ . / ≈ .
251 1030 / ≈ . TABLE VI: The asymptotic limit Π ∞ of Π( t ) for DSC ofgenerations g = 2 , , and ; case (1) : the traps are placed onthe corner nodes, diamonds in Fig. 1(d); case (2) : the trapsare placed on the central nodes, squares in Fig. 1(d), see textfor details.However, from the above results we cannot deduce whetherthe walk is recurrent or not. Therefore, for DSC we againconsider the spectrum of T and H and calculate, based onEq. (10), ρ ( E ) for the most highly degenerate eigenvalue, seeTABLE V. Clearly, our calculations are limited by computa-tional power and for the DSC we could not obtain results for g larger than ; for g = 6 there are already N = 262144 nodesin the network. It seems as if for very large g the ρ (3) serieswill converge to a value somewhat above . × − . Thisfinite limit again seems to indicate that there is localization inthe system, so that CTQW may be recurrent in general.Now, let us consider the CTQW trapping process for theDSC. TABLE VI presents Π ∞ for two different trap place-ments on DSC of g = 2 , , and , as shown in Fig. 1(d). Ourcalculations of the mean survival probabilities Π( t ) and theirasymptotic values Π ∞ do not allow for a clear-cut statementfor the DSC. The first thing to note is that Π (1) ∞ and Π (2) ∞ arevery similar and that with increasing g their values stay ratherconstant. This again is only a weak indication that also theDSC shows localization. VII. SIERPINSKI CARPET
Let us now consider the transport properties of CTRW andof CTQW on SC. We start again by calculating the lowerbound | α ( t ) | of the quantum average return probability π ( t ) ,Eq. (8), see Fig. 5 for the DSC with g = 6 . The inset showsthe behavior of the corresponding CTRW p , ( t ) for various g , which for intermediate times scales with d s as expected. − − − − | α ( t ) | t − − − p , ( t ) t FIG. 5: (Color online) Average return amplitude on the SC at g = 6 (blue solid line) and the long time average χ lb (blackdotted line). Inset: classical return probability to the cornernode for the SC at g = 2 , , , and (pink dash dotted, bluedotted, green dashed, red solid line, respectively) and thedecay according to the spectral dimension d s (black straightsolid line).Again the finite size of the network does not allow for adefinite statement about the recurrent behavior of the CTQW.Therefore, we again calculate ρ ( E ) , see Eq. (10), for thehighly degenerate eigenvalue E = 4 ; the corresponding val-ues are displayed in TABLE VII. There we also show the longtime average χ lb calculated using the r.h.s. of Eq. (14). Simi-lar to the DSC case, there is no strong evidence that CTQW onSC show localization. Again, as for DSC, the limiting valueof ρ (4) lies above × − , such that at this point we can-not make any precise statement. We could not obtain results g ρ (4) χ lb / ≈ . × − . × − /
96 = 6 . × − . × − / ≈ . × − . × − / ≈ . × − . × − / ≈ . × − . × − TABLE VII: The ρ ( E ) for the eigenvalue E = 4 and the longtime average χ lb for different generations of the SC. g Π (1) ∞ = N (1)0 /N Π (2) ∞ = N (2)0 /N /
16 = 0 .
125 2 /
16 = 0 . / ≈ .
240 22 / ≈ . / ≈ .
244 168 / ≈ . / ≈ .
249 1314 / ≈ . TABLE VIII: The asymptotic limit Π ∞ of Π( t ) for SC ofgenerations g = 2 , , , and ; case (1) : the traps are placedon the corner nodes, diamonds in Fig. 1(c); case (2) : thetraps are placed on the central nodes, squares in Fig. 1(c), seetext for details.for larger generations because (at present) we do not have thecomputational facilities to calculate ρ (4) for g > ; the size ofthe corresponding matrix for a DSC of g = 7 is already largerthan
300 000 ×
300 000 .Considering now absorption processes, when there are trapsplaced on three nodes of each network (see, e.g., Fig. 1(c)),we again calculate the quantum mechanical limit Π ∞ of Π( t ) ;the corresponding results are given in TABLE VIII. Here wefind that CTQW on SC are quite different from those on thegaskets, but that they are similar to CTQW on DSC: both, Π (1) ∞ and Π (2) ∞ , show a slow increase with increasing g . VIII. SUMMARY
Our analysis of CTQW over different types of Sierpinskifractals revealed interesting aspects of quantum mechanicaltransport. At first, for SG and for DSG we find strong local-ization effects, supported by the fact that the long time aver-ages χ lb approach a finite limiting value with increasing g , seeTABLES I and III. For the carpets we cannot make a definitestatement based on our numerical results for χ lb for genera-tions up to g = 6 .Turning now to the DOS and monitoring in each case theeigenvalues with the highest degeneracy, we find that ρ (3) and ρ (5) for DSG and ρ (6) for SG tend with growing g each to aconstant, quite significant value, see TABLES I and III. Thissupports our view that the walkers are localized even for verylarge g , in line with Refs. [13, 15]. For the carpets, for whichthe eigenvalue with the highest degeneracy is for DSC and for SC, we find that the corresponding values ρ (3) for the DSCand ρ (4) for the SC are rather small, which renders a clear cutdecision on localization difficult. We hence conclude that oneneeds much larger carpets than the ones we could (at present)numerically handle, in order to attain a definite conclusion.These results are confirmed by our findings for the meansurvival probabilities Π ( j ) ∞ ( j = 1 , ) for both arrangements oftraps. Here, the mean survival probability for SG and DSG in-creases with N , meaning that for larger networks it becomesless and less probable that the excitation will leave the net-work. Since in each case we consider only three trap nodes,with increasing g the number of nodes without traps increases.Hence, due to localization, an excitation starting from a nodefar away from the traps will not be able to reach them. For SCand DSC, the mean survival probabilities show only a slightincrease with g (for the network sizes considered here), thusthe probability of being trapped is almost independent of thesize of the network, see TABLES VI and VIII. In this respect,it would be also interesting to investigate the effect when thenumber of traps also increases with g .In addition and in contrast to the corresponding classicalCTRW, for CTQW there is no apparent scaling behavior withthe spectral dimension d s . This is obvious for the gaskets,see Ref. [6] for DSG and Fig. 3 for SG. However, for the car-pets one might still argue that for large generations g such ascaling could exist for the envelope of | α ( t ) | at intermediatetimes, see Figs. 4 and 5. But from our numerical results forgenerations up to g = 7 we cannot draw this conclusion.Nevertheless, the long time behavior of the CTQW withand without traps reveals clear-cut differences between gas-kets and carpets for the generations studied here. While clas-sically the difference is only manifested in a different scalingaccording to d s , quantum-mechanically we find there appearsto be a fundamental difference between gaskets and carpets -at least for the finite networks studied here. Our main resultsare summarized in Fig. 6, where we plot, as a function of thenumber of nodes N , the long time average of the lower boundof the averaged return probability, χ lb , and (for practical pur-poses) the long time value of the mean trapping probability, i.e., − Π ( j ) ∞ ( j = 1 , ), for the two situations of trap arrange-ments.Already the spectra of the gaskets and of the carpets aresignificantly different and contain - in principle - all the essen-tial information. Now, for the classical CTRW only the low-energy part of the spectrum is important for the intermediate-to-long time behavior, whereas for CTQW the whole spec-trum matters. Clearly, further investigations of these facts arein order, but go beyond the scope of the present paper. Ingeneral, a systematic study of the importance of the so-calledramification number (the number of nodes/bonds which hasto be removed in order for the fractal to fall apart), along witha careful analysis of localized eigenstates (e.g., “dark states”for the trap) deserves further studies. From the point of viewof bond percolation, we have studied localized eigenstates oftwo dimensional lattices with traps and with randomly placedbonds [46]. There, we found that the localization feature isalso mirrored in the survival probablity. While these aspectshave been already touched upon in Ref. [21] for the SG, nodetailed analysis has been carried out for the SC. ACKNOWLEDGMENTS
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