Localization effects induced by decoherence in many-spins quantum statessuperpositions

Gonzalo A. Alvarez∗ and Dieter Suter†

Fakultat Physik, Technische Universitat Dortmund,Otto-Hahn-Strasse 4, D-44221 Dortmund, Germany.

The spurious interaction of quantum systems with their environment known as decoherence leads,as a function of time, to a decay of coherence of superposition states. Since the interactions betweensystem and environment are local, they also cause a loss of spatial coherence: correlations betweenspatially distant parts of the system are lost and the equilibrium states are localized. This effect,which has not been studied in detail so far, limits the distance over which quantum information canbe transmitted, e.g., along a spin chain. We investigate this issue in a nuclear magnetic resonancequantum simulator, where it is possible to monitor the spreading of quantum information in a three-dimensional network: states that are initially localized on individual spins (qubits) spread underthe influence of a suitable Hamiltonian apparently without limits. If we add a perturbation to thisHamiltonian, the spreading stops and the system reaches a limiting size, which becomes smalleras the strength of the perturbation increases. This limiting size appears to represent a dynamicalequilibrium. We present a phenomenological model to describe these results.

PACS numbers: 03.65.Yz, 03.67.Ac, 72.15.Rn, 76.60.-k

I. INTRODUCTION

Controlling quantum mechanical systems has receivedincreasing attention in recent years [1], mainly becausewe are starting to be able to perform computations withquantum mechanical systems. This has the potential ofsolving computational problems for which no efficient al-gorithm exists for classical computers [2–4]. However, wewill only be able to realize this potential by precise con-trol of large quantum systems. The control of small quan-tum systems has been thoroughly explored in the lastyears [1], however the study and control of large quan-tum systems hat not been tackled so far. The reason forthis is partly the difficulty of simulating large quantumsystems on classical computers, which limits the numberof qubits to about 20 when only pure state for the sys-tem are considered [5, 6]. Additionally, it is also possibleto calculate the dynamics of mixed states if the initialstate is localized by using quantum parallelism of a sin-gle pure state evolution [7, 8]. On the other side, thepresent primitive state of quantum computation only al-lows incomplete control of large quantum states. So far,the only physical system that offered this possibility isnuclear magnetic resonance (NMR) of dipolar coupledspins [9–11]. In particular, the main problems are thelack of individual adressing of qubits and decoherence.The latter degrades the quantum information of a givenstate [12]. Decoherence was shown to increase as thesize of the quantum system increases, making the largestsystems the most sensitive to perturbations [9–11, 13–15]. While reducing the perturbation strength typicallyreduces decoherence, in complex and large systems, the

∗galvarez@e3.physik.uni-dortmund.de†Dieter.Suter@tu-dortmund.de

decoherence time can be independent of the perturbationstrength over a range of coupling strengths [16, 17].

Decoherence is well known as a process that causes thedecay of quantum information. Avoiding or reducing de-coherence is thus the main ingredient for implementinglarge scale quantum computers. Several techniques havebeen proposed for this purpose, including dynamical de-coupling [18], decoherence-free subspaces [19], and quan-tum error correction [20, 21]. These proposals have beentested on small systems of nuclear spins [22], trapped ions[23] or spin model quantum memories [24, 25]. Some ofthese techniques were succefully applied to large quan-tum systems with thousands of qubits where their de-coherence time was extended by almost two orders ofmagnitude [10, 15].

Decoherence not only affects the survival time of quan-tum information, it also affects the distance over which itcan be transmitted [26–34]. For example, spin systems,and in particular spin chains, can be used to transferquantum information over large distances [35, 36]. Thiskind of systems were studied with liquid state NMR forsmall numbers of spins [37–41] and slightly larger num-bers by solid-state NMR [42–44]. However, once deco-herence is considered, in the simplest two spin quantumchannel, it was shown that when the effective system-environment interaction exceeds a given strength, thesimple SWAP operation could not be performed [33, 45].Instead of having the expected oscillatory transfer of astate going forth and back between the two spins, an over-damped dynamics due to the appearance of a localizedstate appears at a critical value or exceptional point ofthe pertubation strength [45–49]. Similarly, this could beobserved in a 3-spin chain. If one spin is suffering a per-turbation, when it exceeds a given strength the dynamicslocalizes in the remaining two spins [50]. In a more gen-eral situation, longer spin-chains, it was recently pointedout that imperfections or disorder of the spin-coupling

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that allows for state transfer can induce localization ofthe quantum information to be transmited [26–31, 34] ina process related to Anderson localization [51, 52]. Inmore complex 3D spin-network topologies, we demon-strated experimentally a similar behaviour by studyingthe localization effects induced by the finite precision ofquantum gate operations used for transfering quantumstates [32].

In this paper, we extend our previous work [32], wherewe prepared the system first as individual, uncorrelatedspins and measured the build-up of clusters of correlatedspins of increasing size. Introducing a perturbation to theHamiltonian that generates these clusters, we find thatthe size of the clusters reaches an upper bound. This up-per bound appears to be a dynamic equilibrium: if thecluster size is initially larger than this equilibrium value,it decreases under the effect of the perturbed Hamilto-nian, while the unperturbed Hamiltonian leads to an in-crease. The equilibrium size decreases with increasingstrength of the perturbation. For these experiments, weuse a solid-state NMR quantum simulator. While thissystem does not allow addressing of individual qubits,it represents an excellent test-bed for studying differentaspects of decoherence and information transfer.

II. THE QUANTUM SIMULATOR

We consider a system of equivalent spins I = 1/2 inthe presence of a strong magnetic field. The Hamiltonianof the system is

H = Hz + Hdd

= ωz∑i

Iiz +∑i<j

dij

[2Iiz I

jz − (IixI

jx + Iiy I

jy)], (1)

where ωz is the Larmor frequency, Iix, Iiy and Iiz are the

spin operators and dij the coupling constants. Hz repre-

sents the Zeeman interaction and Hdd the dipolar inter-action [53]. The latter is truncated to commute with thestrong Zeeman interaction (ωz � dij), assuming that theeffects of its non-communting part are negliglible. In aframe of reference rotating at the Larmor frequency [53],

the Hamiltonian of the spin system reduces to Hdd.The quantum simulations start from the high-

temperature thermal equilibrium [53]. Using the nota-

tion Iz =∑i Iiz, we can write the thermal equilibrium

state as

ρ0 ≈(

1 +~ωzkBT

Iz

)/Tr

{1}. (2)

The unity operator 1 commutes with all operators, in-cluding the Hamiltonian and the density operator. Itdoes not contribute to the observable signal and it istherefore convenient to exclude it from the density oper-ator. The resulting density operator for the initial state

of the system is then ρ0 ∝ Iz. In this state, the spins areuncorrelated and the density operators commutes with

the Hamiltonian Hdd.We performed all experiments on a home-built solid

state NMR spectrometer with a 1H resonance frequencyof ωz = 300 MHz. The spins are the protons of poly-crystalline adamantane, where the strength of the aver-age dipolar interaction, determined from the width of theresonance line, is 7.9 kHz.

III. GROWTH OF SPIN CLUSTERS

A. Cluster size

The spin clusters that we consider can be written inproduct operator form as

I lu...Iov Ipw (u, v, w = x, y, z) .

Here, the indexes l, o, p identify the spins involved in thegiven cluster. We write K for the number of terms inthis product, i.e., for the number of spins in the cluster.Experimentally, we generate these clusters using an NMRmethod developed by Pines and coworkers [54, 55]. It is

based on generating an average Hamiltonian H0 that doesnot commute with the thermal equilibrium state

H0 = −∑i<j

dij

[IixI

jx − Iiy Ijy

]= −1

4

∑i<j

dij

[Ii+I

j+ − Ii−I

j−

]. (3)

In the usual computational basis, we write the states as|M,nM 〉 where M is the total magnetic quantum num-

ber, i.e., Iz |M,nM 〉 = M |M,nM 〉, and nM distinguishesdifferent states with the same M . The Hamiltonian (3)flips simultaneously two spins with the same orientation.Starting from the thermal equilibrium state, it generatesa density operator where only elements ρij with

∆M = M(i)−M(j) = 2n, n = 0, 1, 2 . . .

are populated. Such a density operator element ρij iscalled a ∆M quantum coherence. Off-diagonal elementswith ∆M = 0 represent zero-quantum coherences anddiagonal elements correspond to populations.

Figure 1 shows a graphical representation of the den-sity matrix in the computational basis. The black diago-nal line represents the population elements of the densitymatrix. The diagonal blocks (gray) represent coherenceof ρ with ∆M = 0 and moving away from these diagonalblocks, the ∆M multiple-quantum coherence (MQC) or-der increases and they are represented by different gray

values. The solid arrows indicate the effect of H0 onthe thermal equilibrum density operator. Adding the el-ements of ρ with a given ∆M , we obtain a distribution

3

ZQCPop. 1QC2QC3QC4QC5QC6QC

0H

M=0

M=1

M=-1

M=-2

M=2

M=3

M=-3

M M

123

-1-2

-3

Figure 1: Density matrix in the computational basis. Theblack diagonal line represents the populations (Pop. in thelegend). The diagonal blocks (dark gray) represent coher-ences of ρ with ∆M = 0 (ZQC) and moving away from thesediagonal blocks, one finds the ∆M multiple-quantum coher-ences (MQC) blocks represented by different gray tones. The

solid and dashed arrows indicate the effect of H0 and theperturbation interaction Σ = Hdd respectively on the ther-mal equilibrium density operator. By adding the elements ofρ with a given ∆M , we obtain a distribution function of theMQC elements of the density matrix ρ∆M . A schematic repre-sentation of a typical distibution is given by the Gaussian-likeshape on the top-left corner.

function of the MQC elements of the density matrix ρ∆M .A schematic representation of its distibution is given bythe Gaussian-like shape on the upper-left corner of Fig.1. This distribution is initially a delta function on thediagonal of ρ and spreads as the time evolves.

To determine the average number of correlated spins,the technique relies on the fact that in a system of Kspins, the number of transitions with a given ∆M followsa binomial distribution

n (∆M,K) =(2K)!

(K + ∆M)! (K−∆M)!. (4)

For K � 1, the binomial distribution can be well approx-imated with a Gaussian

n (∆M,K) ∝ exp

(−

∆M2

K

), (5)

whose half-width at e−1 is σ =√K . Thus, we can

determine the effective size of the spin clusters in a givenstate by measuring the distribution of the MQCs of itsdensity operator ρ as a function of the coherence order∆M .

A M

KK~30N 0 = 0.23ms

N 0 = 0.96ms

!"## # "##

K~3900

time

MFigure 2: (Color online) Multiple-quantum coherence (MQC)distributions for two different evolution times. The cluster-size of correlated spins is extracted from the variance σ =

√K

of the probability distribution.

B. Growth

Figure 2 shows two distribution functions of the MQCsfor two different durations Nτ0 of the evolution underH0. They clearly demonstrate the increasing width ofthe MQC distribution and thereby the increasing size ofthe spin clusters.

The elements of the density operator with a given ∆Mcan be distinguished by rotating the system around the

z−axis. A rotation φz = e−iφIz by φ changes the densityoperator as

ρ (φ, t) = φz ρ (t) φ†z =∑∆M

ρ∆M (t) ei∆Mφ, (6)

where ρ (t) = e−iH0tρ0eiH0t and ρ∆M (t) contains the ele-

ments that involve coherences of order ∆M . The experi-mental observables in NMR are Ix, Iy, and by means of a

π/2 pulse, also Iz. They are operators with ∆M = ±1 or∆M = 0, respectively. Thus we are not able to measureall the elements of the density matrix. In order to quan-tify the ρ∆M blocks, we use the pulse sequence shown inFig. 3. The system evolution is then described first by anevolution period of duration Nτ0 under the Hamiltonian

4

N

0

N

0

-

x x -x -xx -x -x '

x

Figure 3: (Color online) NMR sequence for the quantum sim-

ulations. The effective Hamiltonian H0 is generated by thesequence of π/2 pulses shown in the upper part of the figure,where ∆′ = 2∆ + τp and τp is the π/2 pulse duration [55].

(H0

)φ

= φzH0φ†z, i.e.,

ρ0

(H0)φNτ0

−−−−−−−→ ρφ (Nτ0) = φz ρ (Nτ0) φ†z

= φze−iH0Nτ0 ρ0e

iH0Nτ0 φ†z

=∑∆M

φz ρ∆M (Nτ0) φ†z

=∑∆M

ρ∆M (Nτ0) ei∆Mφ. (7)

A subsequent evolution of the same duration Nτ0 under

−H0, which causes an evolution backward in time, yieldsthe final density operator

(−H0)Nτ0−−−−−−−→ ρf (2Nτ0) = eiH0Nτ0 ρφ (Nτ0) e−iH0Nτ0

=∑∆M

[eiH0Nτ0 ρ∆M (Nτ0) e−iH0Nτ0

]ei∆Mφ. (8)

If Iz is the NMR observable, we obtain the signal

S (φ,Nτ0) = Tr{Iz ρf (2Nτ0)

}= Tr

{e−iH0Nτ0 ρ0e

iH0Nτ0 ρφ (Nτ0)}

= Tr {ρ (Nτ0) ρφ (Nτ0)}

=∑∆M

eiφ∆MTr{ρ2

∆M (Nτ0)}

(9)

=∑∆M

eiφ∆MA (∆M) , (10)

where A(∆M) are the amplitudes of the MQ spectrum.To extract these amplitudes from the experimental data,we measure the signal S (φ,Nτ0) as a function of φ andperform a Fourier transform with respect to φ. Wequantify the cluster size K by inverting the relationA (∆M) = n (∆M,K) (see Ref. [55] for details).

If the system evolves under the Hamiltonian (3), thewidth of the A (∆M) distribution increases indefinitelyas shown in Fig. 4. The main panel shows the MQC

Figure 4: (Color online) Time evolution of the multiple quan-tum coherence distribution. The main panel shows the timeevolution of the MQC spectrum A(∆M). The inset shows thetime evolution of σ, where A(σ,Nτ0)/A(0, Nτ0) = 1/e.

distributions A (∆M) for different evolution times. Wequantify the spreading of the MQC distribution by mea-suring their widths (σ) at A(σ,Nτ0)/A(0, Nτ0) = 1/e fordifferent evolution times. The inset of the figure showsthe time evolution of σ. Equivalently, its correspondingcluster size of correlated spins also grows indefinitely, asshown in Figure 5.

The Hamiltonian of Eq. (3) is prepared by means ofthe standard NMR sequence [54, 55] shown in the upperpart of Fig. 3. This sequence consists of π/2 rotationsof the spins separated by periods of free precession, ∆

and ∆′, under the dipolar Hamiltonian Hdd and it wasshown to approximate quite well the ideal Hamiltonian

H0. It is inverted experimentally by shifting the phaseof all RF pulses by ±π/2 [54]. We used pulse durationsτπ/2 = 2.8µs and delays ∆ = 2µs and ∆′ = 2∆ + τpgiving a cycle time τ0 = 57.6µs.

C. MQC distributions and their correspondingcluster-size

The most accepted and simplest model for interpretingthe MQC distributions was proposed by Baum et al. [55].It assumes equal excitation probabilities (for a given sys-tem size) for every coherence term of the density matrixand therefore predicts to a Gaussian distribution whosevariance is related to the number of correlated spins. Thisis the usually adopted model because it provides the sim-plest qualitative description of the MQC evolution. How-ever, the actual experimentally observed distributions donot always match with the Gaussian distribution [56].This is also the case in our experiments: the data shownin Fig. 4 are better described by an exponential distri-

5

0.6 0.8 1 1.2

103

104

0.0 0.2 0.4 0.6 0.8 1.0 1.2

100

101

102

103

104

∝∝∝∝ ττττ

4.3

Clu

ste

r S

ize

K

Total evolution time Nτ0 (ms)

∝∝∝∝ e

0.387 N

ττττ 0

K = σ2

Figure 5: (Color online) Time evolution of the cluster size

of correlated spins with the Hamiltonian H0 (black squarepoints). The main panel shows it in log scale manifesting theregion where the growth is exponential and the inset showsit in log-log scale manifesting its power law growth for longtimes.

bution A(∆M). A different approach was proposed byLacelle et al. [56], who predicted an exponential MQCdistribution and found agreement with earlier experimen-tal results from adamantane. Additionally, a dynamicalscale invariance in the growth process of the MQC is re-vealed by their approach.

Due to the difficulty of treating many-spin systemsrigourously, quantum mechanical predictions for theMQC profile shapes and their interpretations are stillmissing. For example, if the MQC profile is not a Gaus-sian distribution, it is not trivial how to extract from itthe cluster-size. While, most of previous works studyingthe distribution of MQCs have assumed Gaussian distri-butions, it is more convenient to use a parameter that isindependent of the MQC distribution form. Khitrin pro-posed to use the second moment of the MQC spectrumfor this purpose [57]. This parameter was shown to berelated to the second moment of the system Hamiltonian,in our case given by Eq. (3) and K(t) is proportional tothe second moment of the MQC distribution.

An exponential MQC distribution of the form

exp(−∆M/

√K(t)

)shares the half-width σ =

√K with

a Gaussian distribution at 1/e. Assuming this distribu-tion, its second moment is given by 2K while for a Gaus-sian distribution it is K/2. While the exact factor thatconverts the second moment of the MQC distribution tothe cluster size K is not exactly determined, they are ofthe same order of magnitude and thus the error of thecluster-size determination is a scale factor of order 1. Animportant ingredient is that the MQC distribution hasthe same shape during the complete range of the experi-mentally accessible evolution time.

The growth depends on the spin-coupling networktopology as was observed experimentally [55–61]. Mainlythree types of growth were observed: (i) indefinite

growth, (ii) localized growth and (iii) localized growth ata first stage (intra-molecular-like localization) and thena further indefinite growth (inter-molecular).

Results for adamantane fall tino the first category.Some models suggest a power law for the growth [59, 62]and these preliminary results [56, 59] seem to matchwith an effective 3D spin-coupling network topology.That 3D behaviour seems to be achieved only for clus-ter sizes above 1000 spins [59]. However, more recentworks suggest an exponential growth of the cluster sizewhen many neighboring spins are contained in the spin-coupling topology [60, 61]. That is the case of adaman-tane and the experiments agree well with that prediction.

Experimentally, the data shown in Fig. 5 suggest threestages of the cluster size evolution: (i) An initial periodwhere the evolution cannot be described as exponentialor power law, (ii) a period of exponential growth, and (iii)a power law regime. During the first stage, the MQC dis-tribution changes from a Gaussian-like distribution to anexponential distribution. Thus, the determination of thecluster size could be affected by two kinds of systematicerrors: (i) at small cluster sizes, the small number of cor-related spins may make the statistical assumptions ques-tionable and (ii) the change of distribution from Gaus-sian to exponential may change the appropriate scale fac-tors. The exponential growth agrees with the behaviorexpected by theoretical predictions for this kind of sys-tem [60, 61]. And finally the power law behaviour couldbe due to the following two reasons: (i) as predicted byLacelle [59] after a certain number of spins (around 1000in adamantane) the effective spin-network topology turnsinto a 3D spin-coupling network, which leads to a powerlaw growth or (ii) because the experimental generation

of H0 contains non-idealities, they can be considered asa perturbation that generates localization effects, as wasdemostrated in Ref. [32] and discussed in the followingsections. Something similar to the latter point could beinterpreted from the predictions of Ref. [61].

IV. EFFECT OF PERTURBATION

A. Perturbed evolution

The evolution under the Hamiltonian H0 can be re-versed completely by changing the Hamiltonian from H0

to −H0. If this time reversal is perfect, the signal (9) forphase φ = 0 is independent of the evolution time Nτ0,∑

∆M A(∆M,Nτ0) = const.

Experimentally, the Hamiltonian H0 is generated asan effective Hamiltonian. Because of experimental im-perfections, it always deviates from the ideal Hamilto-nian (3). As a result, the actual dynamics deviates fromthe ideal one and, in particular, we cannot exactly in-vert the perturbed Hamiltonian and thus revert the time

evolution. Because of the deviations in H0, the quantity∑∆M A(∆M) is no longer conserved, but decays with in-

6

N

0

N

0

-

Figure 6: (Color online) NMR sequence for the quantum sim-ulations. A perturbed evolution is achieved when τΣ 6= 0.

The effective Hamiltonian H0 is generated by the sequence of

π/2 pulses shown in the upper part of Fig. 3 and Σ = Hdd isthe free evolution Hamiltonian.

creasing evolution time. This decay is not uniform, butit affects mostly those components of the density oper-ator that correspond to strongly delocalized coherence.As a result, the spreading of the information is attenu-ated and the system becomes effectively localized. It isthis latter effect that we want to study here; we isolateit from the overall decrease of the signal by normalizingthe MQ spectra such that the total signal

∑∆M A(∆M)

for φ = 0 is again constant in time.

To analyze this deviation from the ideal evolution, we

introduce a perturbation Σ, whose strength we can con-trol experimentally and study the behavior of the systemas a function of the perturbation strength. We choose the

dipole-dipole coupling for this perturbation, Σ = Hdd,which is a local interaction: every spin interacts mostlywith its nearest neighbors, while the coupling strengthwith more distant spins drops off as 1/d3.

We add this Hamiltonian to the ideal Hamiltonian H0

by concatenating short evolution periods under Hdd with

evolution periods under H0. We label the durations ofthe two time periods τΣ and τ0, as shown in Fig. 6. Whenthe duration τc = τ0 + τΣ of each cycle is short comparedto the inverse of the dipolar couplings dij , the resultingevolution can be described by the effective Hamiltonian

Heff = (1− p)H0 + pΣ, (11)

where the relative strength p = τΣ/τc of the perturbation

Σ = Hdd can be controlled by adjusting the duration τΣ.In the experiment, we compare the artificially perturbed

evolution of Heff with the H0 evolution with its intrinsicerrors. While the intrinsic errors reduce the signal or theoverall fidelity, they do not cause localization on the timescale of our experiments (see Fig. 5).

Considering now this perturbation, starting from ther-mal equilibrium, the state of the system at the end of Ncycles is

ρHeff (Nτc) = e−iHeffNτc ρ0eiHeffNτc . (12)

Taking into account now the complete sequence of evo-

lutions given by Fig. 6

ρ0

(Heff)φNτc

−−−−−−−→ ρHeff

φ (Nτc) = φz ρHeff (Nτc) φ†z

= φze−iHeffNτc ρ0e

iHeffNτc φ†z

=∑∆M

φz ρHeff

∆M (Nτc) φ†z

=∑∆M

ρHeff

∆M (Nτc) ei∆Mφ (13)

(−H0)Nτ0−−−−−−−→ ρHeff

f (Nτc +Nτ0) =

= eiH0Nτ0 ρHeff

φ (Nτc) e−iH0Nτ0

=∑∆M

[eiH0Nτ0 ρHeff

∆M (Nτc) e−iH0Nτ0]ei∆Mφ. (14)

Thus the NMR signal, which is measured after the last

backward evolution exp{iH0Nτ0

}, can be written as

S (φ,Nτ0 +Nτc) = Tr{Iz ρHeff

f (Nτc +Nτ0)}

= Tr{e−iH0Nτ0 ρ0e

iH0Nτ0 ρHeff

φ (Nτc)}

= Tr{ρH0 (Nτ0) ρHeff

φ (Nτc)}

=∑∆M

eiφ∆MTr{ρH0

∆M (Nτ0) ρHeff

∆M (Nτc)}

(15)

=∑∆M

eiφ∆MA(∆M). (16)

Reinterpreting these expressions, we define the effec-

tive observable A = e−iH0Nτ0 ρ0eiH0Nτ0 = ρH0 (Nτ0),

where ρH0 is the density operator of the unperturbedevolution. The NMR signal becomes then S(φ,Nτc) =

Tr{AρHeff

φ (Nτc)}

.

For the ideal evolution (p = 0), Eq. (9) is recovered,where A(∆M) correspond to the squared amplitudes of

the density operator elements ρH0

∆M (Nτ0) with coherenceorder ∆M . For the perturbed evolution, (p 6= 0), theyare reduced by the overlap of the actual density opera-tor elements ρHeff

∆M (Nτc) with the ideal ones. We extractthese amplitudes by performing a Fourier transformationwith respect to φ. Figure 7 shows, as an example, the re-sulting A(∆M) as a function of time for p = 0.108. Themain panel represents the MQC distributions for diferentevolution times. The main difference compared to Fig. 4is that the MQC spectrum does not spread indefenitely[32]. We consider this saturation of the spreading of theMQC spectrum evolution as evidence of localization dueto the perturbation (see below). The localization effectsare easily visualized by directly comparing the generationof high-order multiple quantum coherences of the unper-turbed case of Fig. 4 with the perturbed one of Fig. 7.

7

-30

-20

-100102030

1E-3

0.01

0.1

1

024681012

Evo

lutio

n tim

e N!

cM

QC

Am

plitu

de A(!M)

Coherence order !M

0.0 0.4 0.80

20

40Perturbationstrength p

0.108 0

MQ

C w

idth

!

Total evolution time N!c (ms)

Figure 7: (Color online) Time evolution of the multiple quan-tum coherence distribution under a perturbation p = 0.108.The main panel shows the time evolution of the MQC spec-trum A(∆M). The inset shows the time evolution of the stan-dard deviation σ for the unperturbed case compared with theperturbed case. This comparison shows directly the localiza-tion.

While the distribution spreads continuously in Fig. 4, itreaches a limiting value in Fig. 7. The inset of Fig. 7shows the time evolution of the width σ for p = 0.108compared with the unperturbed case.

From the width of the MQC spectrum, we calculatethe size of the spin clusters. The black squares of Fig. 8show the average number of correlated spins as a func-tion of time for an unperturbed evolution, p = 0. Thisis the log scale representation of the curve of Fig. 5.The other symbols of the figure show the evolution ofthe number of correlated spins for different values of p.Initially, the cluster size K(Nτc) starts to grow as in theunperturbed evolution, but then it saturates after a timethat decreases with increasing perturbation strength p.As we explained above, we consider this as evidence oflocalization induced by the perturbation and it is relatedto spatial localization. The size of the cluster at whichthis saturation occurs is also determined by the strengthof the perturbation: increasing perturbation strength re-duces the limiting cluster size.

B. Interpretation

The terms in Eq. (15) are related to the fidelity ofthe corresponding density operator component ρ∆M withrespect to the corresponding component resulting from

0.0 0.2 0.4 0.6 0.8 1.0 1.2100

101

102

103

104

b c

Total evolution timeNc [ms]

Clu

ster

siz

e K

0.000 0.009 0.017 0.025 0.034 0.050 0.065 0.080 0.108

Perturbation strength p

Figure 8: (Color online) Time evolution of the cluster size fordifferent perturbation stengths. The black squares representthe unperturbed time evolution and the other symbols corre-spond to the perturbed evolutions according to the legend.

the unperturbed evolution

f∆M =Tr{ρH0

∆M (Nτ0) ρHeff

∆M (Nτc)}

Tr{ρH0

∆M (Nτ0) ρH0

∆M (Nτ0)} (17)

which reaches unity for vanishing pertubation (p = 0).We therefore consider the reduction as a quantitativemeasure of the effect of the perturbation. The decoher-ence of different ∆M -blocks of the density matrix wasstudied when a given quantum state prepared by evolu-

tion under H0 evolves under a pure Hdd = Σ perturba-tion [9, 13]. However, the quantification and characteri-zation of how a perturbation disturbs the different blocksof the density matrix during the creation of spin-clusterstates is still not known. This merits further studies likein Ref. [63]. Here, we focus only on the effect on the num-ber of correlated spins and do not consider the change ofthe overall amplitude. Thus we normalize the integral ofeach spectrum and determine its width by fitting it to aGaussian or an exponential.

The spreading of the MQC spectrum is generated bythe effective Hamiltonian created with the sequence ofFig. 1. The cluster size that we determine here corre-sponds to an overlap of the actual state with the idealstate resulting from unperturbed evolution. This is simi-lar to the fidelity measure in quantum computing, wherethe agreement between the actual state of the systemwith the target state is measured. In our case, the tar-get state is a growing cluster, while the actual state growsonly for some time until it reaches a limiting size - typicalfor localization. The size of this localized state decreaseswith increasing perturbation strength.

During our experiment, the magnetization is uniformthroughout the sample, so the process does not lead toa spatial redistribution of magnetization. However, since

8

a

Figure 9: (Color online) NMR pulse sequence for preparingdifferent initial clusters sizes and subsequently evolving themin the presence of a perturbation.

we measure the number of correlated spins, we can at-tribute a length scale to the resulting state. Given asuitable initial state, the same process would generatehighly entangled multi-qubit states. In the presence of aperturbation, the number of qubits that can be entangledin this way is limited to the size of the resulting cluster.

V. EVIDENCE FOR A DYNAMICALEQUILIBRIUM CLUSTER SIZE

The experimental results presented above show thatthe cluster size reaches a stationary value. It remains tosee if this limiting size results from a slow-down in thegrowth [29] or it represents a dynamic equilibrium state.We showed that actually they achieve a dynamical equi-librium state [32]. In order to do that we repeated theprevious experiment for a series of initial conditions cor-responding to different clusters sizes. Figure 9 shows thecorresponding pulse sequence: The initial state prepara-tion, consisting of an evolution of duration N0τ0 under

the unperturbed Hamiltonian H0, generates clusters ofsize K0. During the subsequent perturbed evolution ofduration Nτc, these initial clusters evolve and Eq. (15)becomes

S (φ,Nτc) =∑∆M

eiφ∆MA(∆M)

=∑∆M

eiφ∆MTr{ρH0

∆M (Nτ0, N0τ0) ρHeff

∆M (Nτc, N0τ0)}.

(18)

Figure 10 shows the results from these measurements.The symbols represent the amplitudes A(∆M) of the dif-ferent multiple quantum coherences as a function of thecoherence order ∆M for different evolution times. Start-ing from an initial cluster size K0 = 141 for two differentperturbations strengths (p = 0.080 in panel a, p = 0.025in panel b). In this figure, the A(∆M) values are notnormalized. We find that the width of the MQC distri-butions contracts as a function of time in panel a), butexpands in panel b). Like in the case where we start fromsizeK0 = 1, this expansion does not continue indefinitely,but it saturates. Similarly, the contraction also reaches

-60 -40 -20 0 20 40 60

100

101

102

103

-60 -40 -20 0 20 40 60

100

101

102

103

p=0.080

K(7c)~517-670

K0~141

MQ

C A

mp.

A(

M)

(arb

. uni

ts)

Coherence Order M

N 0 1 2 3 4 5 6 7

p=0.025

(b)

K(4c)~100

MQ

C A

mp.

A(

M)

(arb

. uni

ts)

Coherence Order M

N 0 1 2 3 4

K0~141

(a)

Figure 10: (Color online) Time evolution of the multiple quan-tum coherence distribution in the presence of a perturbationstarting from an initial sate with K0 ' 141 correlated spins.The initial MQC distribution of the intial state is given by theblack squares and other colored symbols gives its further timeevolution for two different perturbation strengths p = 0.080(Panel a) and p = 0.025 (Panel b). For p = 0.080 the initialstate has a number of correlated spins higher than the respec-tive localization value for the given p and consequently thecluster-size shrink as a function of time until it achieves thelocalization value. This is evidenced by the parallel straightlines of the MQC distributions. Contrary for p = 0.025, K0 islower than the localization value and consequently the MQCspreads until it achieve the localization width.

an equilibrium value. This is evident at the longest evo-lution times, where the MQC distributions are parallel inthe semi-log scale representation.

To quantify this, we determined the width of the MQCspectra and from that the cluster-size. Figure 11 showsthe evolution of the cluster-size when we start fromthe initial value K0 = 141 for different perturbationstrengths (empty symbols). The solid symbols representthe cluster-size evolution starting from K0 = 1, i.e., thecurves shown in Fig. 8. For all perturbation strengths,the time series for a given perturbation strength convergeto the same limiting value, independent of the initial con-dition. Thus, we find that the evolution leads to a limit-ing cluster size that varies with the perturbation strengthp, but does not depend on the initial condition K0. The

9

1.21.00.8

0.60.4

0.20.0

0.000.02

0.04

0.06

0.08

0.10

0.12

100

101

102

103

104

Clu

ster

-siz

e K

Evolution time Nc (ms)

Pertu

rbat

ion

stre

ngth

p

Figure 11: (Color online) Time evolution of the correlatedcluster-size starting from two different initial states as a func-tion of the perturbation strength. Filled symbols are evo-lutions from an uncorrelated initial state (K0 = 1) and theempty symbols start from an initial state with K0 = 141correlated spins (marked with black squares joined with asolid line). The plot shows that the cluster-size evolutionsachieve a dynamical equilibrium that varies with the pertur-bation strength p, but does not depend on the initial conditionK0. The equilibrium cluster sizes are represented by symbolsin the K − p plane.

limiting cluster sizes are represented by the symbols onthe K − p plane in Fig. 11.

This is again confirmed by the results shown in Fig.12, which summarizes the results for additional initialvalues and two perturbation strengths, p = 0.034 andp = 0.065. The filled symbols correspond to uncorrelatedinitial states (K0 = 1) and the empty symbols to variousinitial cluster sizes. For a given perturbation strength,the size of the spin clusters tends towards the same lim-iting value, independent of the initial condition.

VI. LOCALIZATION SIZE VS. PERTURBATIONSTRENGTH

A. Experimental evidence

According to Figs. 8, 11 and 12, the size of the dy-namical equilibrium clusters decreases with increasingstrength of the perturbation. For a quantiative analy-sis of this dependence, we determined the size of the lo-calized clusters from the data shown in Figs. 8, 11 and12 and plotted them against the perturbation strength inFig. 13. The diagonal line in the figure represents a linearfit to the experimental data (yellow/light line) that givesKloc ∼ p−1.86±0.05, i.e., the cluster size decreases almostproportionally to the square of the perturbation strength.

0.0 0.2 0.4 0.6 0.8 1.0 1.2100

101

102

103

104

p=0.034 p=0.065 K0

1 92 141 209 308

p=0.034 p=0.065 K0

428 655 1092 1624 2200

Clu

ster

siz

e K

Evolution time Nc [ms]

Figure 12: (Color online) Time evolution of the correlatedcluster size starting from different initial sates. Filled sym-bols are evolutions from an uncorrelated initial state for twodifferent perturbation strengths given in the legend. Emptysymbols start from an initial state with K0 correlated spins.

The error of the fit is indicated by the line-width. Thelimiting value for p = 1, Kloc ≈ 1, is consistent withthe expectation that the system becomes completely lo-calized if the perturbation strength is significantly largerthan the unperturbed Hamiltonian.

The figure also summarizes the evolution of the clustersize before the static (localized) size is reached: If the ini-tial size is larger than the stationary value for the givenperturbation strength, K0 > Kloc, the cluster shrinks [in-set (a) in the figure, above the diagonal]. If it is smaller,K0 < Kloc, the size increases with time [inset (b), belowthe diagonal].

B. Phenomenological model

A theoretical model that describes this behavior wouldbe highly desirable, but goes far beyond the scope of thispaper. Instead, we describe here a simple phenomeno-logical model that summarizes the observed behavior.

For this purpose, we write the amplitudes of the MQCspectra as

A(∆M, t) = Ae−∆M/√K(t). (19)

The exponential form agrees with the experimental data(see, e.g. Figs. 4 and 10). According to Fig. 5, thegrowth of the cluster size is exponential during a largepart of the experiment. It may then be described by adifferential equation of the type

10

0.01 0.1 1100

101

102

103

104

c

M

Loca

lized

clu

ster

siz

e K

loc

Perturbation strength p

0.0 0.1 0.2 0.3

-40-200

2040

b

a

Cluster sizedecreasing

bN

c

Cluster sizeincreasing

a

1E-045E-040.0030.010.07

0.0 0.1 0.2 0.3 0.4

-40-2002040

M

Nc

Figure 13: (Color online) Localized cluster size Kloc (squaresymbols) of correlated spins versus the perturbation strengthp. Three dynamical regimes for the evolution of the clus-ter size are identified depending of the number of correlatedspins compared with the perturbation dependent localizationvalue: a) a cluster size decreases, b) a cluster size increases c)stationary regime.

dK

dt= αK, (20)

where α is the growth rate, which is proportinal to thesecond moment of the NMR resonance line [60].

To derive an equation of motion for the perturbed evo-lution, we first consider the effect of the perturbationalone. As we have shown before [9], the interaction withthe environment causes a decay of the MQC amplitudesthat depends on the MQC order ∆M . We found overour range of measurement that the decay rate is ∝ ∆M .The decay of the MQ amplitudes during a time τ is thusδA(∆M) ∝ −A(∆M)∆Mpbτ, where p is the scaling fac-tor introduced above and b is the decay rate of singlequantum coherence under the full perturbation, p = 1.

During a short time τ , the MQC spectrum evolves thusfrom (19) to

A(∆M, t+ τ) = Ae−∆M

(1/√K1(t+τ)+pbτ

).

Here, K1 = K0(1 + ατ) is the size that the cluster sizewould reach, starting from K0 in the absence of the per-turbation. We can rewrite this as

A(∆M, t+ τ) = Ae−∆M/√K

′1(t+τ)

with

K′

1 =K0(1 + ατ)(

1 +√K0(1 + ατ)pbτ

)2

=K0(1 + ατ)

1 + 2√K0(1 + ατ)pbτ +K0(1 + ατ)(pbτ)2

, (21)

where for short τ

K′

1 ≈K0(1 + ατ)

1 + 2√K0(1 + ατ)pbτ

≈ K0(1 + ατ)(1− 2√K0(1 + ατ)pbτ). (22)

We now look for the stationary solution where K ′1 = K0 :

K′

1 = K0 = K0(1 + ατ)(1− 2√K0(1 + ατ)pbτ). (23)

If we consider only terms of O(τ) in the infinitesimal timeτ , this is equivalent to

(α− 2√K0pb) = 0. (24)

Solving for K0, we find the stationary cluster size as

Kloc =

(α

2bp

)2

. (25)

Within experimental error, this agrees with the experi-mentally observed behavior.

VII. DISCUSSION AND CONCLUSIONS

On a step forward to the understanding of the quan-tum evolution of large quantum systems, we have studiedthe spreading of information in a system of nuclear spins.We spread information on about of thousands of qubits.While decoherence is well known to limit the time forwhich quantum information can be used, we focused hereon its effects in the distance over which a quantum statecan be transferred. For that purpose, a locally storedstate was left to evolve in a 3D spin-coupling network.Using standard NMR techniques we generated a Hamil-

tonian H0 that spread the information and allows one toquantify the cluster size of correlated spins. By compar-

ing the cluster-size evolution under the influence of H0

and a perturbation Hamiltonian of variable strength, weshowed that the information becomes localized accord-ing to the perturbation strength. The localization sizedecreases with increasing strength of the perturbation.

Our experimental results demostrate that a commondynamic equilibrium size of the localized state is achievedindependent of the initial state conditions with differ-ent number of correlated spins. The equilibrium size de-pends of the perturbation strength. We developed a phe-nomenological model to describe qualitatively these ef-fect, which arised due to the competition of the spreading

evolution driven by H0 and its systematically reductiondue to decoherence induced by the perturbation. Theseresults differs from theoretical predictions on similar sys-tems that only indicate a slow down of the spreading [29].These previous work were on the framework where disor-der is the perturbation that produces Anderson localiza-tion. We are considering our perturbation as a model for

11

the disorder considered in the discussion of localization[26–31, 51, 52].

The results presented here showed a transition in thespin dynamics from an indefinite spreading to a localizeddynamics. A spin far away from the spin where the initialconditon is stored would receive excitation at some timeif the perturbation is below a critical value, however, ifthe perturbation exceeds this threshold no excitation willachive this site. This resemble the quantum dynamicalphase transition observed on 2-spin and 3-spin systemsunder SWAP operations [33, 45, 50]. There when a per-turbation from the environment exceeds a given strength,the local initial excitation is localized in fewer spins thanthe total system size. For example instead of having theexpected oscillatory behaviour of a state going forth andback on the 2-spin system, an overdamped dynamics dueto the appearance of a localized state appears at a criticalvalue or exceptional point of the perturbation strength[45–49]. For the 3-spin case, if the perturbation is actingon the last spin, the excitation only goes forth and backbetween the remaining 2-spins. Our experiments showthat according to this, a transition for the cluster-size

dynamics arises: if the cluster-size of the initial state ex-ceeds the localization value, the cluster-size shrinks untilthe equilibrium value, but when the initial cluster sizeis lower than the localization value, the size grows untilequilibrium.

These results may also be connected to our earlier find-ings that the decoherence rate of quantum states withmany correlated qubits increases with the size of the sys-tem [9], indicating that larger systems are more sensitiveto perturbations. As the system size increases, the ten-dency for the system to spread is therefore balanced bythe restriction due to the perturbation.

Acknowledgments

GAA thanks finantial support from an Alexander vonHumboldt Research Fellowship. We thank Marko Lovric,Hans Georg Krojanski and Ingo Niemeyer for helpful dis-cussions and technical support.

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