Quantum many-body theory of qubit decoherence in a finite-size spin bath
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Quantum many-body theory of qubit decoherence in a finite-size spin bath
Wen Yang and Ren-Bao Liu ∗ Department of Physics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, China
Decoherence of a center spin or qubit in a spin bath is essentially determined by the many-body bath evolu-tion. We develop a cluster-correlation expansion (CCE) theory for the spin bath dynamics relevant to the qubitdecoherence problem. A cluster correlation term is recursively defined as the evolution of a group of bath spinsdivided by the cluster correlations of all the subgroups. The so-defined correlation accounts for the authentic(non-factorizable) collective excitations within a given group. The bath propagator is the product of all possiblecluster correlation terms. For a finite-time evolution as in the qubit decoherence problem, a convergent resultcan be obtained by truncating the expansion up to a certain cluster size. The two-spin cluster truncation ofthe CCE corresponds to the pair-correlation approximation developed previously [Phys. Rev. B , 195301(2006)]. In terms of the standard linked cluster expansion, a cluster correlation term is the infinite summation ofall the connected diagrams with all and only the spins in the group flip-flopped, and thus the expansion is exactwhenever converges. When the individual contribution of each higher-order correlation term to the decoher-ence is small (while all the terms combined in product could still contribute substantially), as the usual case forrelatively large baths where the decoherence could complete well within the bath spin flip-flop time, the CCEcoincides with the cluster expansion [Phys. Rev. B , 035322 (2006)]. For small baths, however, the qubitdecoherence may not complete within the bath spin flip-flop timescale and thus individual higher-order clustercorrelations could grow significant. In such cases, only the CCE converges to the exact coherent dynamics ofmulti-spin clusters. We check the accuracy of the CCE in an exactly solvable spin-chain model. PACS numbers: 76.20. + q, 03.65.Yz, 76.60.Lz, 76.30.-v I. INTRODUCTION
The decoherence of a center spin in a spin bath hasbeen of interest in spin resonance spectroscopy for a longhistory and is also a paradigmatic model in studyingthe state collapse in quantum mechanics. Recent revisitedinterest in this problem is mostly due to the decoherenceissue in quantum computing. Most relevant are singleelectron spins in quantum dots or impurity centers wherethe dominating decoherence mechanism at low temperatures(such as below a few Kelvins) is the nuclear spins of the hostlattice.
In a small system such as a quantum dot, the center spin(hereafter referred to as qubit for clarity) and the spin bath, inthe time-scale of decoherence, is a relatively isolated subsys-tem in the whole environment. Thus the qubit decoherenceis due to the entanglement with the bath during the coherentevolution of the whole system. In this paper, we are interested in the so-called pure dephas-ing in which the qubit experiences no longitudinal relaxationbut only loses its o ff -diagonal phase coherence. The pure de-phasing is relevant to an electron spin under a moderate orstrong magnetic field ( & . H = | + i H ( + ) h + | + |−i H ( − ) h−| , (1)by which the spin bath is driven by di ff erent Hamiltonians H ( ± ) depending on the qubit states |±i . When a coherent qubitstate C + | + i + C − |−i is prepared, the initial state of the qubit-bath system is the product state ( C + | + i + C − |−i ) ⊗|Ji . At time T , the bath evolution |Ji → |J ± ( T ) i ≡ e − iH ( ± ) T |Ji predicated on the qubit state |±i establishes an entangled state C + | + i ⊗|J + ( T ) i + C − |−i ⊗ |J − ( T ) i . The qubit coherence is reducedfrom ρ + − (0) = C + C ∗− to ρ + − ( T ) = C + C ∗− hJ − ( T ) |J + ( T ) i .The decoherence is characterized by the bath state overlap hJ − ( T ) |J + ( T ) i . Thus the key is the many-body bath dynam-ics caused by the interaction within the bath. Without many-body interactions, the bath would not evolve (except for atrivial phase factor that can be eliminated by standard spin-echo ) and the qubit coherence would not decay.Recently, a variety of quantum many-body theories fornuclear spin bath dynamics have been developed includingthe density matrix cluster expansion (CE), the pair-correlation approximation, and the linked-cluster ex-pansion (LCE). In the pair-correlation approximation, eachpair-wise flip-flop of nuclear spins is identified as an elemen-tary excitation mode and is taken as independent of each other.To study the higher oder correlations, the Feynman diagramLCE is developed. The evaluation of higher-order LCE, how-ever, is rather tedious due to the increasing number of dia-grams, especially for spins higher than 1 / It servesas a simple method (without the need to count or evaluateFeynman diagrams) to include the higher-order spin interac-tion e ff ects beyond the pair-correlation approximation. TheCE calculations show that when the pair-correlation contri-bution to the decoherence is suppressed by pulse control, the higher-order correlations are not negligible. In order toobtain the decoherence exponent from the CE, however, theterms involving overlapping clusters have to be neglected,which make the accuracy problematic even when the expan-sion converges. This problem limits the CE to applications inlarge baths in which the contribution of each individual clus-ter to the decoherence is small and the overlapping correctionis unimportant.In this work, we develop a cluster-correlation expansion(CCE) method in which the bath spin evolution are factor-ized into cluster correlations. Each cluster correlation termis equivalent to the sum of all the LCE series consistingof a given set of bath spins flip-flopped. In particular, thetwo-spin cluster correlations include all diagrams with twospins flip-flopped and is equivalent to the pair-correlationapproximation. The CCE bears the accuracy of the LCE(the results are accurate whenever converge) and the simplic-ity of the CE (without the need to count or evaluate Feyn-man diagrams), while free from the large-bath restriction ofthe CE. The CCE coincides with the CE in the leading orderof the short-time expansion, which is applicable in large spinbaths where the decoherence completes well within the bathspin flip-flop time. For small baths, however, the qubit deco-herence may not complete within the bath spin flip-flop timeand individual higher-order cluster correlations could growsignificant. In this case only the CCE converges to the exactcoherent dynamics of multi-spin clusters. Such coherent dy-namics of small clusters of bath spins is of special interest insystems with randomized qubit-bath couplings. An interest-ing example is nitrogen-vacancy centers in diamonds which are coupled to randomly located bath spins (carbon-13and nitrogen nuclear spins) in the proximity.This paper is organized as follows. In Section II, we derive,for a generic spin bath Hamiltonian, the CCE from a recur-sive cluster factorization procedure. We show that the CCEis equivalent to an infinite resummation of the LCE series andalso compare it to the CE. In section III, we check the accuracyand convergence of the truncated CCE in an exactly solvablemodel (the one-dimensional spin-1 / / II. CLUSTER-CORRELATION EXPANSIONA. Motivation: Pair-correlation approximation and beyond
As discussed in the Introduction, the Hamiltonian for thepure dephasing problem has the form of Eq. (1). For a giveninitial bath state |Ji (which could be considered as one sam-ple chosen from a thermal ensemble), the qubit coherence ischaracterized by L ( T ) = hJ| e iH ( − ) T e − iH ( + ) T |Ji . For a thermal ensemble of baths, a further ensemble averageshould be processed. The thermal fluctuation leads to the in-homogeneous broadening, which can be eliminated by spinecho. To focus on the qubit decoherence due to the quantumdynamics of the bath, throughout this paper, we consider thedecoherence for a single bath state without the ensemble aver-age. For temperatures much higher than the bath spin flip-floprates ( ∼ − K for nuclear spins in GaAs), the thermal ensem-ble has no o ff -diagonal coherence and |Ji can be taken as a (a) (b) FIG. 1: Visualization of a cluster containing (a) two or (b) three bathspins (black arrows). The spins outside the cluster (gray arrows) aretaken as frozen in calculating the cluster contribution. noninteracting product state |Ji = ⊗ n | j n i , where j n denotesthe quantum number of the n th bath spin quantized along theexternal magnetic field.In this subsection we illustrate the central idea of the CCEmethod (as an extension of the previously developed pair-correlation approximation ) using a bath consisting of N spins J , J , · · · , and J N with only pairwise secular interac-tions H ( ± ) = ± Ω + X n (cid:18) Z n ± z n (cid:19) J zn + X m , n D m , n ± d m , n ! J zm J zn + X m , n B m , n ± b m , n ! J + m J − n , (2)where Ω ( Z n ) is the qubit (bath spin) splitting energy, z n isthe diagonal qubit-bath spin interaction constant (correspond-ing to the hyperfine interaction strength in an electron-nuclearspin system), D n , m ( B n , m ) is the diagonal (o ff -diagonal) in-trinsic bath interaction strength, and d n , m ( b n , m ) is the diago-nal (o ff -diagonal) extrinsic bath interaction depending on thequbit states (which could result from the interaction mediatedby virtual flips of the qubit spin while real flips are suppressedby the large energy mismatch ).In the pair-correlation approximation, the qubit coherenceis given by the product of all possible spin pair contributionsup to a phase factor, L ≈ Y { i , j } L { i , j } , (3)where the contribution due to the flip-flops of a spin pair { i , j } [see Fig. 1(a)] is L { i , j } = (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) e ih ( − ) { i , j } T e − ih ( + ) { i , j } T (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) , (4)with the Hamiltonian governing the pair dynamics obtainedfrom the full Hamiltonian H ( ± ) by excluding the flip-flops ofall spins other than J i and J j h ( ± ) { i , j } = ± Ω + X n (cid:18) Z n ± z n (cid:19) J zn + X m , n D m , n ± d m , n ! J zm J zn + B i , j ± b i , j ! (cid:16) J + i J − j + J − i J + j (cid:17) , (5)which is equivalent to replacing in the full Hamiltonian thespins outside the pair with their mean-field averages, h ( ± ) { i , j } ≡ H ( ± ) (cid:16) J i , J j , nD J (cid:12)(cid:12)(cid:12) J n , i , j (cid:12)(cid:12)(cid:12) J Eo(cid:17) . (6)The pair-correlation approximation is valid for situationswhere pair correlations dominate. When the collective flip-flops of more spins become important, we need to considerthe higher-order correlation correction L corr defined by L = Y { i , j } L { i , j } L corr . (7)To illustrate how the high-order correction could be evaluated,let us consider a spin bath of only three spins { , , } . Obvi-ously, the result is L corr = L L { , } L { , } L { , } . (8)This result motivates a definition of non-factorizable three-spin correlations due to collective flip-flops. For a three-spincluster { i , j , k } in a bath [see Fig. 1(b)], when all the spins out-side the cluster are frozen, the qubit coherence is L { i , j , k } = (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) e ih ( − ) { i , j , k } T e − ih ( + ) { i , j , k } T (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) , with the cluster Hamiltonian h ( ± ) { i , j , k } ≡ ± Ω + X n (cid:18) Z n ± z n (cid:19) J zn + X m , n D m , n ± d m , n ! J zm J zn + X m , n ∈{ i , j , k } B m , n ± b m , n ! J + m J − n = H ( ± ) (cid:16) J i , J j , J k , nD J (cid:12)(cid:12)(cid:12) J n < { i , j , k } (cid:12)(cid:12)(cid:12) J Eo(cid:17) , obtained from the full Hamiltonian H ( ± ) by replacing the spinsoutside the cluster with their mean-field averages. The au-thentic (or non-factorizable) three-spin correlation is singledout by excluding all the pair-correlations˜ L { i , j , k } = L { i , j , k } L { i , j } L { i , k } L { j , k } . If all such three-spin correlations are picked up, the qubit co-herence is given by (up to a global phase factor)
L ≈ Y { i , j } L { i , j } · Y { i , j , k } ˜ L { i , j , k } . Thus a systematic cluster correlation expansion is motivated.
B. General formalism of cluster-correlation expansion
We consider a generic bath Hamiltonian for the pure de-phasing problem H ( ± ) = H ( ± ) ( J , J , · · · , J N ) , …… H ( (cid:14) ) H ( (cid:16) ) t =0 t t t N+1 =Tt =0 t t (b) (a) t t H ( (cid:14) ) H ( (cid:16) ) H ( (cid:14) ) H ( (cid:16) ) t t H ( (cid:16) ) H ( (cid:14) ) (cid:87) (c) (cid:87) H ( (cid:16) ) H ( (cid:14) ) H ( (cid:14) ) H ( (cid:16) ) (cid:87) (cid:87) (cid:87) H ( (cid:16) ) H ( (cid:14) ) (cid:87) (cid:87) H ( (cid:14) ) H ( (cid:16) ) (d) H ( (cid:14) ) H ( (cid:16) ) H ( (cid:16) ) H ( (cid:14) ) (cid:87)(cid:87) FIG. 2: (a) Visualization of an arbitrary sequence of controlling π pulses at t , t , · · · , (indicated by vertical lines) and the correspondingcontour-time-dependent Hamiltonian for electron spin decoherence.(b), (c), and (d) exemplify the cases of free-induction decay, Hahnecho, and Carr-Purcell echo, respectively. which need not contain only pairwise interactions, or con-serve the spin angular momentum along any direction. Forinstance, multi-spin interaction terms like J zi J + j J − k and non-secular terms like J + i J zj could be present.For qubit decoherence under the control of an arbitrary se-quence of π -pulses applied at t , t , · · · , as shown schemati-cally in Fig. 2(a), the bath evolution predicated on the qubitstate is given by (cid:12)(cid:12)(cid:12) J ± ( T ) (cid:11) = U ( ± ) |Ji , where U ( ± ) ≡ · · · e − iH ( ± ) ( t − t ) e − iH ( ∓ ) ( t − t ) e − iH ( ± ) t . (9)The qubit coherence at the end of the evolution is L = (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) U ( − ) (cid:17) † U ( + ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) . (10)It can be written in the contour time-ordered form as L = (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c H ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) , (11)where T c is the time-ordering operator along the contour C : 0 → T →
0. As illustrated in Fig. 2(a), the contourtime-dependent Hamiltonian H ( t ) alternatively switches be-tween H ( + ) and H ( − ) each time the qubit state is flipped by a π -pulse or when the time direction is reverted at T . The ex-amples for free-induction decay, single-pulse Hahn echo, andCarr-Purcell echo are illustrated in Figs. 2 (b), (c), and (d),respectively.Following the idea illustrated in the previous subsection,the cluster correlations are recursively defined as follows.1. The empty-cluster correlation˜ L ∅ ≡ L ∅ ≡ e − i R c hJ| H ( t ) |Ji dt is a pure phase factor obtained from Eq. (11) by replac-ing the bath Hamiltonian H ( t ) with its mean-field aver-age hJ | H ( t ) | Ji .2. The single-spin correlation˜ L { i } ≡ L { i } / ˜ L ∅ , where L { i } ≡ (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c h { i } ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) is obtained from Eq. (11) by replacing the bath Hamil-tonian H ( { J n } , t ) with h { i } ( t ) ≡ H ( J i , {hJ | J n , i | Ji} , t ) . in which all the spin operators except J i are mean-fieldaveraged.3. The two-spin (pair) correlation˜ L { i , j } ≡ L { i , j } / (cid:16) ˜ L ∅ ˜ L { i } ˜ L { j } (cid:17) , where L { i , j } ≡ (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c h { i , j } ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) is obtained from Eq. (11) by replacing the bath Hamil-tonian H ( { J n } , t ) with h { i , j } ( t ) ≡ H (cid:16) J i , J j , nD J (cid:12)(cid:12)(cid:12) J n < { i , j } (cid:12)(cid:12)(cid:12) J Eo , t (cid:17) , in which all the spin operators except J i and J j aremean-field averaged.4. So on and so forth, the cluster correlation for an arbi-trary set of bath spins C is defined as˜ L C ≡ L C Q C ′ ⊂C ˜ L C ′ , (12)where L C ≡ (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c h C ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) (13)is obtained from Eq. (11) by replacing the bath Hamil-tonian H ( { J n } , t ) with h C ( t ) ≡ H (cid:16) { J n ∈C } , nD J (cid:12)(cid:12)(cid:12) J n < C (cid:12)(cid:12)(cid:12) J Eo , t (cid:17) , (14)in which all the spin operators outside the cluster aremean-field averaged or their flip-flops are frozen.Thus, by definition, the qubit coherence is factorized intoall possible cluster correlations as L = Y C⊆{ , , ··· , N } ˜ L C . (15) Calculating the CCE to the maximum order ˜ L { , , ··· , N } amountsto solving the exact bath propagator, which is in general notpossible. In the decoherence problem, we consider a finite-time evolution and it often su ffi ces to truncate the expansionby keeping cluster correlations up to a certain size M , as the M th-order truncation of the CCE ( M -CCE for short), L ( M ) = Y |C|≤ M ˜ L C , (16)where |C| is the number of spins contained in the cluster C .As an example, for the pairwise Hamiltonian in Eq. (2), thelowest nontrivial order of truncation is L (2) = ˜ L ∅ Y { i , j } ˜ L { i , j } = ˜ L ∅ Y { i , j } (cid:16) L { i , j } / ˜ L ∅ (cid:17) , (17)which is the pair-correlation approximation. C. Relation to linked-cluster expansion
Saikin et al. have recently developed an LCE method forthe qubit decoherence in a spin-1 / The detailed de-scriptions of the LCE for a generic spin bath are given in theAppendix. In general, the bath evolution can be factorizedusing Feynman diagrams so that L = exp ( π ) , (18)where π = (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c H ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) connected , (19)is the sum of all connected Feynman diagrams obtained by us-ing Wick’s theorem on the series expansion of Eq. (11). Someof the connected diagrams up to the 4th-order have been eval-uated in Ref. [23] for free-induction decay and single-pulseHahn echo with a spin-1 / ff erent clusters of spins.As an example shown in Fig. 3(a), a third-order diagram in-volving the flip-flop of a spin pair contains diagrams for spinclusters (1 , , · · · , where the numbers stand for the in-dices of the spins flip-flopped (i.e., the J z quantum numberchanged). Thus all the connected diagrams can be classifiedaccording to the spin clusters instead of the interaction orders.For an arbitrary cluster C , we define ˜ π ( C ) as the sum of allconnected diagrams in which all and only the spins in clus-ter C have been flip-flopped. For instance, some of the dia-grams constituting ˜ π ( ∅ ) and ˜ π ( i , j ) for a spin-1 / { ˜ π } functions, the LCE is expressed as π = X C⊆{ , , ··· , N } ˜ π ( C ) . (20) m = ( ) (cid:83) (cid:135) (cid:4) + + + ••• = m n (cid:122) (cid:166) ( , ) i j (cid:83) (cid:4) = + ij ij + ji + ij + jji + ij + ii jj + •••+ •••+ iij (a)(b) n (c) FIG. 3: (a) Expansion of a third-order connected diagram into dia-grams involving the flip-flops of di ff erent clusters of spins. (b) and(c) show the diagrams contained in ˜ π ( ∅ ) and ˜ π ( i , j ), respectively. Inthe diagrams, a solid arrow denotes the propagation of a spin. Awavy (dotted) line connecting two solid arrows / spins denotes a pair-wise o ff -diagonal (diagonal) interaction, and an open-ended line rep-resents the interaction with the qubit spin or an external field. Eachfilled circle / empty circle / empty square on a solid arrow denotes a J + / J − / J z operator, which raises / lowers / keeps invariant the J z quan-tum number of that spin. In particular, the infinite summation of all the connected dia-grams for a certain cluster C and all its subsets π ( C ) ≡ X C ′ ⊆C ˜ π (cid:0) C ′ (cid:1) , (21)can be obtained from the series expansion Eq. (19) by drop-ping all the terms involving the flip-flop of spins outside thecluster C , or, equivalently, by reducing the bath Hamiltonian H ( t ) to the cluster Hamiltonian h C ( t ) in which the spins out-side the cluster are mean-field averaged. Thus we have e π ( C ) = Y C ′ ⊆C e ˜ π ( C ′ ) = (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c h C ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) = L C . (22)Comparing this to Eqs. (12)-(14), we immediately have˜ L C = e ˜ π ( C ) , (23)i.e., a cluster correlation term corresponds to the infinite par-tial summation of all the connected diagrams in which all andonly the spins in the cluster have been flip-flopped.From the LCE expression of the CCE in Eq. (23), it is obvi-ous that the short-time profile of the decoherence due to clus-ters of a certain size is determined by the lowest-order dia-gram. In particular, for free-induction decay with the secularpair-interaction Hamiltonian in Eq. (2), the leading order con-tribution from a cluster of size M is˜ π ( C ) ∼ B M T M , (24)where B is the typical magnitude of the pair flip-flop interac-tion strength B m , n ± b m , n . If each spin interacts, on average,with q spins, then the number of size- M clusters is ∼ Nq M − , with N the total number of bath spins. The sum of all theleading order M -spin connected diagrams is ∼ q − N · ( qBT ) M . Thus for qBT ≪ , i.e., for a time T much shorter than thebath spin flip-flop timescale 1 / B , a truncated CCE converges.The short time condition T ≪ B − is usually satisfied for elec-tron spin decoherence caused by nuclear spins in typical quan-tum dots. The convergence of the truncated CCE, however,could go well beyond the short time restriction. One such sce-nario is small spin baths with disorder in qubit-bath couplingsor in spin splitting energies, in which multi-spin correlationcould develop at a time well beyond the short-time limit butthe size of the contributing clusters could remain bounded dueto the localization e ff ect in a disordered system, as will be ver-ified later in this paper by numerical simulations. D. Relation to cluster expansion
Witzel et al. recently developed a density matrix CE ap-proach to solving the nuclear spin dynamics in the electronspin decoherence problem, in the spirit of the clusteror virial expansion for interacting gases in grand canonicalensembles.
Below we reproduce the basic procedure ofthe CE and compare it to the CCE. Instead of the ensembleCE in Ref. [17], we consider a single sample state of the bathfor a direct comparison. Defining the qubit decoherence dueto a cluster C of bath spins as W ( C ) ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c h C ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | L C | , (25)a hierarchy of cluster terms { ˜ W ( C ) } are recursively defined as W ( i ) = ˜ W ( i ) , (26a) W ( i , j ) = ˜ W ( i , j ) + ˜ W ( i ) ˜ W ( j ) , (26b) W ( i , j , k ) = ˜ W ( i , j , k ) + ˜ W ( i ) ˜ W ( j ) ˜ W ( k ) + ˜ W ( i , j ) ˜ W ( k ) + ˜ W ( i , k ) ˜ W ( j ) + ˜ W ( i ) ˜ W ( j , k ) , (26c) · · · · · · W ( C ) = ˜ W ( C ) + X {C j } Y C j ˜ W ( C j ) , (26d)where in the last line the sum runs over all possible parti-tions of the cluster C into nonoverlapping nonempty subsets C , C , · · · . The M th-order truncated CE ( M -CE for short) is W ( M ) = X {C j } , | C j | ≤ M Y C j ˜ W ( C j ) , (27)with the sum running over all possible partitions of the bathinto nonoverlapping nonempty clusters C , C , · · · of size upto M .In the cluster expansion for interacting gases in grandcanonical ensembles with translational symmetry, the contri-bution from di ff erent clusters can be factorized and theevaluation of a truncated CE amounts to the calculation of a fi-nite number of finite-size cluster contributions, similar to theCCE in this paper. For a finite-size spin bath or for a bathwithout translational symmetry, however, such factorizationof di ff erent clusters does not exist, which makes it essentiallyimpossible to calculate the sum in Eq. (27) even for a small- M -CE. For example, for a bath of N spins, the number ofall terms containing only pair clusters is O ( N !!) and all suchterms have to be individually calculated and summed in the2-CE, which is practically impossible.A remedy is possible when all the cluster terms ˜ W ( C ) are in-dividually small (but the sum could still be substantial). Undersuch a condition, the CE can be approximated by a factorizedform by adding some overlapping terms which are higher-order small. For example, with the secular pair-interactionHamiltonian in Eq. (2), for which W ( i ) = ˜ W ( i ) =
1, the M -CEis W ( M ) ≈ Y < |C|≤ M h + ˜ W ( C ) i , (28)under the small-term condition (cid:12)(cid:12)(cid:12) ˜ W ( C ) (cid:12)(cid:12)(cid:12) ≪ , for |C| > . (29)Comparing the factorization approximation in Eq. (28) to theexact M -CE in Eq. (27), the error added is δ W ( M ) = X i < j < k ˜ W ( i , j ) ˜ W ( j , k ) + X i < j < k < l ˜ W ( i , j , k ) ˜ W ( k , l ) + · · · , (30)containing products of any set of cluster terms sharing atleast one spin, i.e., the overlapping terms. Such overlappingterms are higher-order small if each individual cluster term(for |C| >
1) is small. Furthermore, under the small-term con-dition, 1 + ˜ W ( C ) ≈ exp h ˜ W ( C ) i , and the CE for an arbitrarycluster becomes W ( C ) ≈ Y C ′ ⊆C e ˜ W ( C ′ ) , (31)which takes the same form as the CCE.Thus the CCE coincides with the CE under the small-termcondition Eq. (29), which is justified for large spin bathswhere the number of contributing clusters is large and hencethe contribution from each individual cluster remains smallwithin the timescale of decoherence. The problem with theneglected overlapping terms is relevant for small spin bathswhere the coherent dynamics of a small number of multi-spinclusters dominating the decoherence may persist well beyondthe bath spin flip-flop time and the small-term condition is nolonger satisfied. In this case the CE will not converge to theexact multi-spin cluster dynamics, as will be seen in the nu-merical check in the next section. III. NUMERICAL CHECK
Here we consider an exactly solvable spin bath model (theone-dimensional spin-1 / N -spin bath Hamiltonian conditioned on the qubit state |±i (with spin splitting constants dropped) is H ( ± ) = ± N X n = z n J zn + N − X n = B n ± b n ! (cid:16) J + n + J − n + J + n J − n + (cid:17) , (32)
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10 100 40010 -11 -8 -5 -2 M agn i t ude o f c l u s t e r c on t r i bu t i on s t o - l n | L | CCE2 CCE3CCE4 CCE5CCE6(a) FID
CCE2 CCE3CCE4 CCE5 CCE6(b) Hahn
Pulse delay time
CCE6CCE5CCE4CCE3CCE2(c) CP CCE7 CCE8
FIG. 4: (Color online) The magnitude of size- m cluster contributions(denoted as CCE m ) to the exponential decoherence factor − ln |L| in(a) free-induction decay (FID), (b) Hahn echo, and (c) Carr-Purcellecho (CP) for a long “sinusoidal” spin chain with N = − ln |L| ispositive (negative). The exact − ln |L| is also shown (empty squares)for comparison. where z n denotes the qubit-bath spin interaction coe ffi cient, B n is the intrinsic bath interaction strength, and b n is the interac-tion dependent on the qubit state. The initial bath state |Ji istaken as a product state of all bath spins, in which the orienta-tion of each bath spin is randomly chosen as up or down. Thequbit-bath interaction coe ffi cients { z n } are either taken froma sinusoidal distribution z n = z max sin( n π/ N ) (referred to asa “sinusoidal” spin chain) or randomly chosen from [0 , z max ](referred to as a “random” spin chain). Hereafter z max is takenas the unit of energy. The spin-flip interaction strengths { B n } and { b n } are randomly chosen from [10 − , × − ], corre-sponding to typical bath spin flip-flop time τ sf ∼ . The ex-act solution is obtained by the Jordan-Wigner transformation,which transforms the interacting spin-1 / A. Large spin bath
As the first example, we consider a sinusoisal chain of N = | z n − z n + | for a Exact 6-CCE 6-CE Q ub i t c ohe r en c e | L | Pulse delay time
FID Hahn CP
FIG. 5: (Color online). Qubit coherence (empty squares) in free-induction decay (FID), Hahn echo, and Carr-Purcell echo (CP) for along “sinusoidal” spin chain with N = pairwise flip-flop between neighboring spins varies smoothlyfrom one end to the other of the chain, so that a correlatedcluster can grow to larger and larger size as time passes by.For a time greater than the bath spin flip-flop time τ sf , thewhole bath could become correlated. But the qubit decoher-ence would be completed within a time T much shorter than τ sf , if the bath size is relatively large, or NT τ − ≫ , (33)due to the large number of small clusters contributing to thedecoherence. Under the short-time condition, the CCE can betruncated with a rather small cut-o ff size M . The total contri-butions from clusters of various sizes to the decay of the qubitcoherence as functions of the pulse delay time τ [see Figs. 2(b)-(d)] are shown in Fig. 4. For the free-induction decay inFig. 4(a), the cluster contributions decrease rapidly with in-creasing the cluster size m at a time much shorter than τ sf , sothe 2-CCE (pair correlation approximation) already convergesto the exact result. In the single-pulse Hahn echo [Fig. 4(b)],the higher-order correlations are more noticeable than in thefree-induction decay, but the pair correlations still dominatefor τ ≪ τ sf . For the two-pulse Carr-Purcell echo [Fig. 4(c)],as the decoherence due to the pair correlations is eliminated inthe leading order of the spin-flip interactions, the larger-sizecluster correlations become important and a 6-CCE is requiredto reproduce the exact solution.In the relatively large bath, due to the large number of con-tributing clusters, the qubit decoherence completes when eachindividual cluster contribution is small, i.e., the small-termcondition in Eq. (29) is satisfied. Thus the CCE coincideswith the CE. This is verified in Fig. 5, where the exact solu-tion for qubit decoherence agrees with both the CCE and theCE truncated at the 6th order. -4 -3 -2 -1 Q ub i t c ohe r en c e | L | (a) CCE FIDHahnCP
Exact 2-CE 4-CE 6-CE -4 -3 -2 -1 Pulse delay time (b) CE
FIDHahnCP
Exact 2-CE 4-CE 6-CE
FIG. 6: (Color online) Qubit coherence for a short “sinusoidal” spinchain of N =
100 spins in free-induction decay (FID), Hahn echo,and Carr-Purcell echo (CP). The results from (a) CCE and (b) CEtruncated to the 2nd (dashed lines), the 4th (solid lines) and the6th (dotted lines) order are compared to the exact solution (emptysquares).
B. Small spin bath
As the second example, we consider a short “sinusoidal”spin chain consisting of N =
100 spins. In this case the num-ber of contributing clusters is small and the decoherence pro-ceeds much slower. Fig. 6 shows that both the CCE and theCE converge to the exact result for a time much shorter than τ sf . As the time approaches and goes beyond the bath spinflip-flop time τ sf , the deviation from the exact solution is no-ticeable in both the CCE and the CE, indicating the emergenceof correlations for clusters larger than the truncation cut-o ff size. But the CCE agrees with the exact result as long as itconverges, while the CE converges to a di ff erent result (for T = τ >
250 in free-induction decay, T = τ >
300 in Hahnecho, and T = τ >
800 in Carr-Purcell echo). The devia-tion of the converged CE from the exact result is due to theoverlapping correction [see Eq. (30)] neglected in the CE: For T & τ sf , the contribution of an individual cluster could be siz-able and the small-term condition in Eq. (29) is violated, sothe overlapping correction has to be taken into account.In the example discussed above, the CCE may not convergeat a long time T & τ sf . This is because the qubit-bath spin cou-pling assumes a smooth sinusoidal distribution and hence theenergy cost of each neighboring pair is small and slow-varyingas a function of the pair’s position along the chain. The smalland slow-varying energy cost of a pair-flip makes it possiblefor one pair-flip to a ff ect its neighboring pairs, then the nextneighbors, and so on and so forth. This way large-size cluster Exact 2-CCE 4-CCE 6-CCE Q ub i t c ohe r en c e | L | (a) Pulse delay time
Exact 2-CE 4-CE 6-CE (b)
FIG. 7: (Color online) Qubit coherence in free induction decay for ashort “random” spin chain with N =
100 spins. (a) M -CCE and (b) M -CE are compared to the exact solution (empty squares). correlations may grow rapidly after the time goes beyond thepair-flip time τ sf . In a relatively small bath of smooth qubit-bath coupling distribution, the CCE may fail to converge inthe long time limit which is of interest in elongating the qubitcoherence time by pulse control.The convergence problem of the CCE for small spin bathsmay be avoided if the qubit-bath coupling is random so thatthe bath correlation is localized and the size of correlated clus-ters is upper bounded. As the last example, we consider ashort “random” spin chain consisting of N =
100 spins. Inthis case, the pair-flip energy cost is usually much larger thanthe spin-flip strength unless two neighboring spins are acci-dentally in near-resonance. Thus large-size cluster correla-tion can hardly grow significant even for a time well beyondthe bath spin flip-flop timescale τ sf . Figure 7 shows that thepair-correlation approximation already su ffi ces for the qubitdecoherence for an arbitrarily long time. Since higher-ordercorrelations are small, the overlapping correction to the CE(which corresponds to at least three-spin cluster correlations)is unimportant, good agreement between the CE and the exactsolution is also seen. Interestingly, in the small bath of ran-dom qubit-bath coupling, the qubit decoherence is not fullydeveloped but coherent oscillations persist over a long time.These coherent qubit oscillations have indeed been observedin a system of a similar nature, namely, nitrogen-vacancy cen-ters in diamonds coupled to nuclear spins in the proximity. We would like to point out that the coherent oscillations can-not be reproduced by the LCE truncated as any finite interac-tion order.As discussed in Sec. II D, the CCE and the CE di ff er in de- Q ub i t c ohe r en c e | L | (a) Exact 2-CCE 3-CCE 4-CCE 6-CCE
Pulse delay time (b)
Exact 2-CE 3-CE 4-CE 6-CE
FIG. 8: (Color online) The same as Fig. 7 except that the qubit-bathcoupling constants { z n } are such that there are four neighboring spinsin near-resonance. Notice that the 4th-order and the 6th-order trun-cations are not distinguishable by eyes in both CCE and CE. scribing the multi-spin correlations, which may be importantfor a small spin bath. Such di ff erence can indeed be seen inFig. 6 for a short “sinusoidal” spin chain. It is interesting tosee the di ff erence between the two theories in studying the co-herent multi-spin dynamics in a small bath. For this purpose,we consider a short ( N = τ sf . Indeed, Fig. 8(a)shows coherent oscillations that are correctly reproduced bythe CCE at the 4th or higher-order truncation. The correc-tion by the 5th and higher-order clusters is actually negligi-ble, verifying that the dominating cause of the coherent qubitoscillation is due to up to four-spin cluster dynamics in thebath. In contrast, the CE, though already converges at the 4th-order truncation, does not reproduce the exact solution [seeFig. 8(b)], due to the neglect of the overlapping terms [seeEq. (30)]. IV. CONCLUSION
We have developed a CCE approach to solving the many-body dynamics of a generic interacting spin bath relevant tothe center spin decoherence problem. In this approach, thebath propagator is factorized exactly into the product of clus-ter correlation terms, each of which accounts for the corre-lated flip-flops of a group of bath spins. In terms of the stan-dard LCE, a cluster correlation term corresponds to the infinitesummation of all the connected diagrams with all and only thespins in the cluster flip-flopped. For a finite-time evolutionas in qubit decoherence, a convergent result can be obtainedby truncating the expansion up to a certain cluster size. TheCCE gives exact results whenever it converges. The lowestnontrivial order of the CCE corresponds to the previously de-veloped pair-correlation approximation. Compared to the CEmethod, the two theories yield similar results for large spinbaths, but for small spin baths only the CCE accurately takesinto account the multi-spin cluster correlations. As a simplemethod to sum over an infinite series of LCE diagrams, theCCE method can be readily applied to the case of interactingbosons or fermions.
Acknowledgments
This work was supported by Hong Kong RGC Project2160285.
APPENDIX A: LINKED CLUSTER EXPANSION FOR SPINBATH DYNAMICS1. Thermal ensemble LCE
The LCE for the propagator of an arbitrary spin bath in athermal ensemble has been derived in Ref. [45]. Here wesummarize the main results including the Feynman diagramrepresentation, as the basis of the extension to the LCE for anarbitrary noninteracting bath state.For a thermal ensemble of spin baths characterized by thenoninteracting density matrix ρ ≡ e − β H / Tr e − β H with H = P m ω m J zm , the evolution in the contour time-ordered form is L ens = Tr h ρ T c e − i R c H ( t ) dt i . (A1)Here the contour time-dependent Hamiltonian H ( t ) switchesalternatively between H ( + ) and H ( − ) on the contour [see Fig. 2(a)]. As an example, the pairwise bath Hamiltonian in Eq. (2)leads to H ( t ) = Ω ( t )2 + X n Z n ( t ) J zn ( t ) + X m , n D m , n ( t ) J zm ( t ) J zn ( t ) + X m , n B m , n ( t ) J + m ( t ) J − n ( t ) , (A2)where Ω ( t ), Z n ( t ), D m , n ( t ), and B m , n ( t ) depend on which con-tour time segment t is in, and the spin operators J zn ( t ) ≡ J zn , J ± n ( t ) ≡ J ± n are time-independent but are written with explicittime-dependence to keep track of the time-ordering.Eq. (A1) can be expanded into series as L ens = Tr ρ ∞ X n = ( − i ) n n ! Z c dt · · · Z c dt n T c H ( t ) · · · H ( t n ) , (A3)where each term is the sum of ensemble-averaged T c -productsof spin operators. Consider the ensemble average h F n i ≡ Tr (cid:2) ρ F n (cid:3) of an arbitrary product F n ≡ T c J α m ( t ) · · · J α n m n ( t n ),where the subscripts m q ∈ { , , · · · , N } label bath spins andthe superscripts α q ∈ { z , + , −} label the spin operators. In or-der for h F n i not to vanish, J + and J − operators must makepairs. An F n consisting of J z operators only is called a fullycontracted product, whose ensemble average is trivially eval-uated. So next we consider the case with at least one J + oper-ator, F n = T c J α m ( t ) · · · J α k − m k − ( t k − ) J + m k ( t k ) J α k + m k + · · · J α n m n ( t n ) . For the moment, we assume that the product in F n is alreadytime-ordered with t > t > · · · > t n and use the following pro-cedure to reduce h F n i to the sum of ensemble-averaged fullycontracted products. First we move the spin raising operator J + m k ( t k ) to the right, generating commutators between J + m k ( t k )and the operators on its right, h F n i = Tr ρ n X p = k + J α m ( t ) · · · h J + m k ( t k ) , J α p m p ( t p ) i · · · J α n m n ( t n ) + Tr h ρ J α m ( t ) · · · J α k − m k − ( t k − ) J α k + m k + ( t k + ) · · · J α n m n ( t n ) J + m k ( t k ) i . Then in the second term, we use the cyclic invari-ance of the trace to move J + m k ( t k ) to the left (before ρ ), and use J + m ( t ) ρ = e βω m ρ J + m ( t ) to move J + m k ( t k )across ρ back to its original position to get e βω mk h F n i .During this process, we get additional commutators e βω mk Tr h ρ P k − p = J α m ( t ) · · · h J + m k ( t k ) , J α p m p ( t p ) i · · · J α n m n ( t n ) i . Col-lecting all terms, we obtain h F n i = Tr ρ X p ( , k ) J α m ( t ) · · · h J α p m p ( t p ) i • h J + m k ( t k ) i • · · · J α n m n ( t n ) , (A4)where the contraction between a spin raising operator J + m ( t )and an arbitrary spin operator J α n ( t α ) is defined as (cid:2) J α n ( t α ) (cid:3) • (cid:2) J + m ( t ) (cid:3) • ≡ (cid:2) J + m ( t ) (cid:3) • (cid:2) J α n ( t α ) (cid:3) • ≡ δ m , n G m ( t α − t ) (cid:2) J α n , J + m (cid:3) ( t α ) , with G m ( t ) ≡ θ ( t ) (cid:2) + f ( ω m ) (cid:3) + θ ( − t ) f ( ω m ) being a Green’sfunction and f ( ω ) = / ( e βω − J α n ( t α )] • [ J + m ( t )] • ∼ (cid:2) J α n , J + m (cid:3) ( t α ) is still an operator associatedwith a contour time t α and should be used in subsequent con-tractions. With (cid:2) J z , J + (cid:3) = J + and (cid:2) J − , J + (cid:3) = − J z , the con-traction between J + m ( t ) and J zm ( t α ) [or J − m ( t α )] eliminates J + m ( t )and converts J zm ( t α ) [or J − m ( t α )] to J + m ( t α ) [or − J zm ( t α )], reduc-ing the number of spin operators by one. The resulting productin Eq. (A4) is still in a time-ordered sequence, so the time-ordering operator can be recovered so thatTr h ρ T c J α m ( t ) · · · J α k − m k − ( t k − ) J + m k ( t k ) J α k + m k + ( t k + ) · · · J α n m n ( t n ) i = Tr ρ X p ( , k ) T c J α m ( t ) · · · h J + m k ( t k ) i • h J α p m p ( t p ) i • · · · J α n m n ( t n ) . (A5)In the above equation the time ordering t > t > · · · > t n is nolonger assumed, since all the spin operators commute in the T c -product.0 t t + t t + t t + (cid:117) ( ) n Z t ( ) zn J t ( ) zn J t ( ) zm J t , ( ) m n D t (i) (ii) (iii) (iv) ( ) m J t (cid:14) ( ) n J t (cid:16) , 1 ( ) m n
B t , 2 ( ) m n
B t ( ) m J t (cid:16) ( ) n J t (cid:14) ( ) m J t (cid:14) ( ) n J t (cid:16) , ( ) m n B t
FIG. 9: (a) Diagrammatic representation of the single-spin term Z n ( t ) J zn ( t ) , the diagonal interaction D m , n ( t ) J zm ( t ) J zn ( t ) , and the o ff -diagonal interaction B m , n ( t ) J + m ( t ) J − n ( t ) . (b) Second-order fully con-tracted diagrams. The contraction procedure in Eq. (A5) can be repeatedwhenever there is still a spin raising operator left. Thus Wick’stheorem follows: Under the average over a noninteractingthermal ensemble, a T c -product of spin operators can be re-placed by the sum of all possible fully contracted productswhich contains only J z operators.According to Wick’s theorem, each term in Eq. (A3) gen-erates a series of fully contracted products. The fully con-tracted products can be visualized by Feynman diagrams withthe following definition of constituent elements and construc-tion rules.1. A spin operator J + , J − , J z is represented by a vertexas a filled circle, an empty circle, or an empty square,respectively;2. Each diagonal (o ff -diagonal) interaction term contain-ing n spin operators in the Hamiltonian is representedby a dashed (wavy) interaction line connecting n ver-tices;3. Each contraction (cid:2) J + m ( t ) (cid:3) • (cid:2) J α m ( t α ) (cid:3) • is represented by asolid arrow starting from the vertex J + m ( t ) and ending atthe vertex J α m ( t α ). At the end of the propagating arrowthe commutator (cid:2) J α m , J + m (cid:3) is to be taken;4. Each J + ( t ) vertex denoted by a filled circle is connectedwith one outgoing propagating arrow, each J z ( t ) vertexdenoted by an empty square is either free-standing orconnected to one incoming arrow [converting J z ( t ) to J + ( t )] and one outgoing arrow [from the resulting oper-ator J + ( t )], and each J − ( t ) vertex denoted by an emptycircle is connected to one incoming arrow [converting J − ( t ) to − J z ( t )] or two incoming arrows [the first ar-row converting J − ( t ) to − J z ( t ), and the second arrowconverting − J z ( t ) to − J + ( t )] and one outgoing arrow[from the resulting J + ( t )].Note that each fully contracted product (and hence each dia-gram) is an operator consisting of J z spin operators only.Taking the Hamiltonian in Eq. (A2) for example, thediagonal single-spin term Z n ( t ) J zn ( t ) , the diagonal inter-action D m , n ( t ) J zm ( t ) J zn ( t ) , and the o ff -diagonal interaction B m , n ( t ) J + m ( t ) J − n ( t ) are visualized in Fig. 9(a). The first-order , 1 ( ) m n B t , 2 ( ) m n
B t , 3 ( ) n k
D t ( ) n J t (cid:14) ( ) m J t (cid:16) ( ) m J t (cid:14) ( ) n J t (cid:16) ( ) zn J t ( ) zk J t ( ) m J t (cid:16) , 1 ( ) m n
B t ( ) n J t (cid:14) ( ) m J t (cid:14) ( ) n J t (cid:16) , 2 ( ) m n
B t ( ) n J t (cid:16) ( ) n J t (cid:14) , 3 ( ) n k
B t ( ) k J t (cid:16) ( ) k J t (cid:14) , 4 ( ) n k
B t (a) (b) ( ) ( ) ( ) ( ) ( ) ( ) z zn m m n n k
J t J t J t J t J t J t (cid:14) (cid:16) (cid:14) (cid:16) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n m m nn k k n
J t J t J t J tJ t J t J t J t (cid:14) (cid:16) (cid:14) (cid:16)(cid:14) (cid:16) (cid:14) (cid:16)
FIG. 10: (Color online) (a) A third-order connected diagram consist-ing of one diagonal and two o ff -diagonal interaction terms. (b) A4th-order connected diagram consisting of four o ff -diagonal interac-tions. The contraction processes contained in each diagram are givenbelow. expansion ( − i ) R c dt H ( t ) in Eq. (A3) gives two fully con-tracted products [the first two diagrams in Fig. 9(a)]. Thesecond-order expansion of Eq. (A3) gives four fully con-tracted products shown in Fig. 9(b). The first three diagrams(i) = ( − i ) Z m ( t ) Z n ( t ) J zm ( t ) J zn ( t ) , (ii) = × ( − i ) Z m ( t ) D n , l ( t ) J zm ( t ) J zn ( t ) J zl ( t ) , (iii) = ( − i ) D m , n ( t ) D p , q ( t ) J zm ( t ) J zn ( t ) J zp ( t ) J zq ( t ) , come from diagonal terms and involve no contractions. Here,as a convention, we have suppressed the sum over spin indicesand the contour time integrals. The last diagram(iv) = ( − i ) B m , n ( t ) B m , n ( t ) (cid:2) J + m ( t ) (cid:3) ⋄ (cid:2) J − n ( t ) (cid:3) • (cid:2) J + n ( t ) (cid:3) • (cid:2) J − m ( t ) (cid:3) ⋄ (A6)comes from two o ff -diagonal interaction terms and involvestwo contractions (cid:2) J + m ( t ) (cid:3) ⋄ (cid:2) J − m ( t ) (cid:3) ⋄ and (cid:2) J − n ( t ) (cid:3) • (cid:2) J + n ( t ) (cid:3) • , asindicated by the two solid arrows in Fig. 9 (b) (iv).Similarly, higher-order diagrams can be constructedby using the above Feynman rules. Figure 10 gives twoexamples. The 3rd-order diagram in Fig. 10(a) con-sists of two o ff -diagonal interactions B m , n ( t ) J + n ( t ) J − m ( t )and B m , n ( t ) J + m ( t ) J − n ( t ) and one diagonal interaction D n , k ( t ) J zn ( t ) J zk ( t ) . It contains three contractions, one onspin J m and two on spin J n , corresponding to the three solidarrows. The 4th-order one in Fig. 10(b) consists of fouro ff -diagonal interactions and contains five contractions, threeon spin J n and one on each of the other two spins.To illustrate the evaluation of the diagrams, we consideragain the secular pair-interaction Hamiltonian in Eq. (A2) asthe example. The rules for constructing the analytical formulafor a diagram are:1. A contour-time dependent constant is associated witheach interaction line, namely, Z n ( t ) for an open-ended1 FIG. 11: Topologically inequivalent connected diagrams up to the4th order for the Hamiltonian in Eq. (2). dashed line representing the spin splitting, D m , n ( t ) fora dashed line connecting two diagonal spin operatorsrepresenting the diagonal interaction, and B m , n ( t ) for awavy line representing the o ff -diagonal interaction;2. Each solid arrow from J + m ( t ) to J α m ( t α ) gives the Green’sfunction G m ( t α − t ), each freestanding J zn vertex gives J zn ,each J − n vertex connected to one incoming arrow gives( − J zn , and each J − n vertex connected with two incom-ing arrows and one outgoing arrow gives ( − − i ) k / k ! is associated with a diagramcontaining k interaction lines;4. The spin indices are summed over and the contour timesare integrated over.For example, the last diagram in Fig. 9(b) gives( − i ) B m , n ( t ) B m , n ( t ) G m ( t − t ) G n ( t − t ) (cid:0) − J zm (cid:1) (cid:0) − J zn (cid:1) , which can also be evaluated directly from Eq. (A6) by carry-ing out the two contractions.Summation of all the fully contracted diagrams leads to theLCE of the ensemble-averaged evolutionTr (cid:16) ρ T c e − i R c H ( t ) dt (cid:17) = Tr (cid:16) ρ e ˆ π (cid:17) , (A7)where ˆ π is the sum of all the connected diagrams, such as thefirst two diagrams in Fig. 9(a) and the last diagram in Fig. 9(b),but does not include the disconnected ones such as the firstthree diagrams in Fig. 9(b). As an example, for the Hamilto-nian in Eq. (A2), all the topologically inequivalent connecteddiagrams up to the 4th order are shown in Fig. 11. We see thatthe number of diagrams increases significantly with increas-ing perturbation order.
1, 1 (cid:14)
1, 01, 1 (cid:16) ,1 ,2 n n (cid:110) (cid:110) ,1 ,2 n n (cid:110) (cid:112) ,1 ,2 n n (cid:112) (cid:112) ,1 ,2 (1/ 2 ) zn n
S S (cid:14) (cid:16) ,1 ,2 (1/ 2 ) zn n
S S (cid:14) (cid:14) ,1 ,2 (1/ 2 ) zn n
S S (cid:16) (cid:14) ,1 ,2 (1/ 2 ) zn n
S S (cid:16) (cid:16)
FIG. 12: Mapping of spin-1 operators J ± to pseudo-spin-1 / /
2. Single-sample LCE for spin-1 / For a single noninteracting bath state |Ji (in contrast to thethermal ensemble), the LCE for spin-1 / / |Ji = ⊗ n | j n i can be taken as the ground state of a correspond-ing noninteracting Hamiltonian, H J = X n ω n J zn , (A8)where ω n < >
0) for | j n i = |↑i (or |↓i ). So the nonin-teracting single-sample average hJ | O | Ji becomes the zero-temperature limit ( β → + ∞ ) of the corresponding noninter-acting ensemble average Tr (cid:0) ρ J O (cid:1) with the density matrix ρ J ≡ e − β H J / Tr (cid:16) e − β H J (cid:17) . In particular, the single-sample ex-pectation value of the bath propagator L = (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c H ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) = lim β → + ∞ Tr h ρ J T c e − i R c H ( t ) dt i . Thus the single-sample LCE is obtained by simply setting theGreen’s function G n ( t ) = θ ( t ) δ j n , ↓ − θ ( − t ) δ j n , ↑ and replacingthe ensemble average Tr( ρ · · · ) with hJ |· · · | Ji , i.e., L = (cid:28) J (cid:12)(cid:12)(cid:12)(cid:12) T c e − i R c H ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) J (cid:29) = D J (cid:12)(cid:12)(cid:12) e ˆ π (cid:12)(cid:12)(cid:12) J E . (A9)Note that the connected diagrams in ˆ π contain only J z opera-tors, which commute with each other. Thus the single-sampleaverage can be performed for each diagram to convert it intoa c -number, so that L = exp ( hJ | ˆ π | Ji ) ≡ exp ( π ) .
3. Single-sample LCE for higher spins
For a higher-spin bath, a noninteracting single sample state |Ji in general is not the eigen state of a noninteracting Hamil-tonian as in Eq. (A8). Thus the single-sample LCE for higher-spin baths may not be derived from the ensemble LCE di-rectly. Here we provide a solution by mapping a higher spinto a composite of pseudo-spin-1 / (a)(b) FIG. 13: Diagram representation of (a) Z n J zn , (b) D m , n J zm J zn , and (c) B m , n J + m J − n . (d) Topologically inequivalent second-order connected di-agrams for a spin-1 bath. Without loss of generality, we consider a spin-1 J n . Themapping from the spin-1 states to the states of two pseudo-spin-1 / S n ,λ ( λ = ,
2) is | , − i n → |↓i n , |↓i n , , | , i n → |↑i n , |↓i n , , | , + i n → |↑i n , |↑i n , , as schematically shown in Fig. 12. The two pseudo-spin-1 / / J + n → √ S + n , (cid:16) / − S zn , (cid:17) + √ (cid:16) / + S zn , (cid:17) S + n , , J − n → √ S − n , (cid:16) / − S zn , (cid:17) + √ (cid:16) / + S zn , (cid:17) S − n , , J zn → S zn , + S zn , , in which the flip-flop of one pseudo-spin-1 / /
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