Topological edge states in periodically­driven trapped­ion chains

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Nevado, Pedro, Fernandez-Lorenzo, Samuel and Porras, Diego (2017) Topological edge states in periodically-driven trapped-ion chains. Physical Review Letters, 119 (210401). ISSN 0031-9007

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Topological edge states in periodically-driven trapped-ion chains

Pedro Nevado,1, ∗ Samuel Fernandez-Lorenzo,1 and Diego Porras1, †

1Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom(Dated: September 15, 2017)

Topological insulating phases are primarily associated with condensed-matter systems, which typically fea-ture short-range interactions. Nevertheless, many realizations of quantum matter can exhibit long-range inter-actions, and it is still largely unknown the effect that these latter may exert upon the topological phases. In thisLetter, we investigate the Su-Schrieffer-Heeger topological insulator in the presence of long-range interactions.We show that this model can be readily realized in quantum simulators with trapped ions by means of a periodicdriving. Our results indicate that the localization of the associated edge states is enhanced by the long-rangeinteractions, and that the localized components survive within the ground state of the model. These effects couldbe easily confirmed in current state-of-the-art experimental implementations.

Introduction.- Topological phases are one of the most exoticforms of quantum matter. Among their many intriguing traits,we find that they are robust against local decoherence pro-cesses, or feature fractional particle excitations with prospec-tive applications in quantum information processing [1, 2].Some of the simplest systems showcasing non-trivial topolog-ical order are the topological insulators [3–6], gapped phasesof non-interacting fermions which present gapless edge states.Despite of several experimental realizations [7, 8], the prepa-ration and measurement of topological insulators is typicallydifficult in the solid state. Analog quantum simulators [9–15], on the other hand, offer the possibility of exploring andexploiting the topological insulating phases, because of theirinherent high degree of controllability. Furthermore, interac-tions in a quantum simulator can be tuned at will, opening upthe possibility of investigating new regimes of the underlyingmodels.

Topological edge states usually occur in the insulatingphase as long as an associated bulk invariant attains a non-trivial value, and the generic symmetries of the underlyingHamiltonian are preserved [16]. This property –known as thebulk-edge correspondence– is a generic feature of topologicalinsulators. However, if interactions are taken into account, thepresence of edge states is no longer guaranteed. For instance,it has been shown that one of the edge states present in theMott insulating phase of the Bose-Hubbard model on a 1Dsuperlattice is not stable against tunneling [17]. In this work,we extend these considerations to the case of interactionswhich are explicitly long ranged. Since topological phases arecharacteristically robust against local perturbations, but long-range interactions may not qualify as such, there is an ongo-ing effort to elucidate their effect upon the topological states[18–20]. This question is not of exclusive theoretical interest,since many experimental systems implementing topologicalphases of matter feature long-range interactions. In partic-ular, we will show that trapped-ion quantum simulators canrealize a long-range interacting version of one of the simplestinstances of a topological insulator, the Su-Schrieffer-Heeger(SSH) model [21–23]

HSSH = JN−1

∑j=1

(1+δ (−1) j)(

σ+j σ−j+1 +H.c.

). (1)

The SSH model presents topological edges states for δ > 0,which, e.g., near the left end of the chain are of the form|E.S.〉 ∼ ∑

Nj=1 e(N− j+1)/ξloc σ

+j |↓↓↓ . . .〉, where the localiza-

tion length can be related to the dimerization δ through [24]

ξloc =−2/ ln1−δ

1+δ, 0 < δ < 1. (2)

The addition of long-range inter-ion couplings on (1) turnsthis model into a highly non-trivial interacting problem. How-ever, we will show that, owing to the single-particle address-ability available in trapped-ion setups, the edge states can bestudied as one-body solutions, and that their properties survivewhen interactions are taken into account.

This Letter is structured as follows. (i) We begin showinghow to implement the interacting SSH model with trapped-ion quantum matter. (ii) We then study its one-excitation sub-space, and locate the topological phase. (iii) We perform aneffective description of the low-energy sector, and establishthe dependence of the localization length with the range ofthe interactions. Also, we provide a protocol for the detectionof the edge states. (iv) Finally, we study the correlations in theground state, and establish the survival of the boundary modesagainst interactions.

Realization of the spin SSH Hamiltonian.- We consider aset of N trapped ions arranged along a 1D chain. Two opticalor hyperfine levels |↑〉 , |↓〉 encode an effective spin, such that|↑〉〈↑|− |↓〉〈↓| ≡ σ z [25]. The vibrations of the chain can beapproximated by a set of harmonic modes, H = ∑

Nn=1 ωna†

nan.We add a state-dependent force conditional on the internalstates of the ions [26–29], whose frequency is fairly off-resonant with any motional excitation,

Hf = gN

∑j,n=1

σxj

(M j,neiδnta†

n +H.c.). (3)

δn = ωn−∆ω , where ∆ω is the laser detuning with respect tothe internal level and M j,n is the phonon wave-function. Weassume that the force acts in the direction transverse to theion chain. In this case the mode n = N/2 has the minimumenergy. After tracing out the vibational bath the dynamics of

2

the spins can be described by an Ising Hamiltonian [11]

HIsing =N

∑j,l=1

J(ions)j,l σ

xj σ

xl +

Ω

2

N

∑j=1

σzj , (4)

where the extra transversal field, Ω can be realized with a mi-crowave or a Raman transition. The nature of the couplingsJ(ions)

j,l depends on the width of dispersion relation of the mo-tional modes, tC, and the detuning of the laser from the bot-tom of the band, δN/2 [30]. All through this work we assumeδN/2 > 0. In [31] we showed that depending on the relativevalues of δN/2 and tC, we can distinguish two regimes: (i)

long-range limit (δN/2 tC), in which J(ions)j,l ∼ e−| j−l|/ξint .

The spin coupling takes a Yukawa-like form, with ξint =√log(2)/2

√tC/δN/2, and (ii) short-range limit (δN/2 tC),

in which the couplings decay as ∼ | j− l|−3, and the interac-tions are effectively among nearest neighbors only.

Since σ xj = σ

+j + σ

−j , Hamiltonian (4) contains terms of

the form σ+j σ

+l ,σ−j σ

−l , which do not occur in (1). To elim-

inate these we assume a rotating wave approximation in thelimit Ω J j,l . To obtain the SSH model we consider driv-ing the chain with a time-dependent field. Periodic drivingsare known to render effective Hamiltonians in which specificterms can be adiabatically eliminated, and the interactions arenon-trivially dressed [32]. In our case this dressing must alsocontain some spatial structure to give rise to the periodicityof the couplings in the SSH model. We exploit the possibil-ity of globally imprinting inhomogeneous couplings upon thechain, by taking advantage of the optical phase of the laserfields [31],

Hdriving =ηωd

2cos(ωdt)

N

∑j=1

cos(∆kd0 j+φ)σ zj . (5)

This driving relies on a standing wave modulated in time withfrequency ωdΩ, which should be implemented by a differ-ent set of lasers than the state-dependent force in Eq. (3). η isthe (dimensionless) coupling strength, ∆k is the wave vectoralong the chain axis, and φ is a global optical shift. We assumethat the ions are equally spaced by d0, so their equilibrium po-sitions are r(0)j = d0 j. This is a good approximation in the cen-ter of a Coulomb crystal in a RF trap [33], or describes a lin-ear array of microtraps [34–36]. Now we move into a rotatingframe such that HIsing+Hdriving ≡Htotal→H ′total, with H ′total =

U(t)HtotalU†(t)− iU(t) ddt U

†(t), U(t) = exp [i∑Nj=1 ∆ j(t)σ z

j ],and

∆ j(t) =Ω

2t +

ηωd

2cos(∆kd0 j+φ)

∫ t

0cos(ωdt ′)dt ′. (6)

The condition max j,l |J(ions)j,l | Ω ensures that the anoma-

lous terms are fast rotating, whereas those that preserve thez−component of the spin are renormalized by the phasese±i(∆ j(t)−∆l(t)). These quantities can be simplified by us-ing suitable trigonometric identities along with the Jacobi-Anger expansion eizsinθ = ∑

∞n=−∞ Bn(z)einθ , where Bn(z) are

the Bessel functions of the first kind [37]. Assuming thatωd max j,l |J

(ions)j,l |, the only non-fast-rotating contribution

comes from n = 0, and H ′total ' H(ions)SSH , with

H(ions)SSH =

N

∑j,l=1

J(ions)j,l J

π/2j,l

(σ+j σ−l +σ

−j σ

+l

), (7)

where we fix ∆kd0 = π/2 to achieve the periodic couplings

Jπ/2j,l = B0

(2η sin

(π

4( j+ l)+φ

)sin

π

4( j− l)

). (8)

Since Jπ/2j, j+1 = J

π/2j+T, j+T+1 with T = 2, these couplings re-

produce the dimerization of the original SSH model in thelimit of nearest-neighbor interactions. We will refer to thespin implementation (7) as the generalized SSH model. Inanalogy with (1), the dimerization is given by the differentialratio of the couplings between sites with j even and odd, i.e.,

δ =J

π/22,3 −J

π/21,2

Jπ/2

2,3 +Jπ/2

1,2

. (9)

In Eq. (9), J(ions)j,l factors out, since J(ions)

j,l = J(ions)j−l .

Finally, we remark that we can easily extend our deriva-tion to account for the effect of an inhomogeneous ion-ionspacing, whose main effect would be to induce an extra site-dependence in the couplings to the standing-wave. Since thetopological properties investigated below are robust againstperturbations, we can expect our results to be valid even whensmall inhomogeneities are considered.

Study in the one-excitation subspace.- The preparation ofsingle excitations can be easily realized in trapped-ion chains,as demonstrated in implementations of the Ising and XY mod-els [14, 15]. The one-particle sector is spanned by the vectors| j〉 ≡ σ

+j |↓↓↓ · · · 〉 , j = 1, . . . ,N. We can think of the state

|↓↓↓ · · · 〉 as a vacuum of particles, and accordingly | j〉 repre-sents an excitation localized at site j. Since (7) is invariantunder arbitrary rotations in the xy plane, the Hamiltonian doesnot mix | j〉with states within subspaces of different number ofexcitations. Therefore, the dynamics of | j〉 is dictated by therestriction of the Hamiltonian to the one-excitation subspace,that is given as

H(ions)SSH =

N

∑j,l=1

h j,l (| j〉〈l|+ |l〉〈 j|) , h j,l = J(ions)j,l J

π/2j,l . (10)

For φ = 3π/4 and η > 0, h j,l possesses two (quasi-) zero-energy modes, which feature localization at the edges; weshow one of these in Fig.1(a). The edge state has apprecia-ble support only on the odd sites, which is a consequence ofthe chiral symmetry [38]. Indeed, the chiral-symmetric limitsof this Hamiltonian are attained for φ = π/4 and 3π/4 (seeSupplemental Material). We have depicted the dimerization(9) as a function of η in these limits (cf. Fig.1(b)). We notethat for φ = 3π/4, δ is positive, and accordingly the model

3

Figure 1. (color online) (a) Plot of one of the mid-gap states ofHamiltonian (10) near the left end of the chain, with δN/2/tC = 4,for φ = 3π/4, N = 100 and η ' 0.62, which renders δ ' 0.1. Inset(b) shows the behavior of the dimerization as a function of η in thechiral limits. Inset (c) corresponds to the Zak phase for different val-ues of η , signaling the topologically trivial (|ν | = 0) and non-trivial(|ν | = π) phases. Inset (d) shows the dependence of ξloc with thedimerization, for δN/2/tC = 0.1,0.5,8. The dashed line correspondsto Eq. (9).

presents edge states. This is accompanied by a non-zero valueof the associated bulk invariant, the Zak phase [39], whichcan take the value 0(±π) in the trivial (topological) phase. Asshown in Fig.1(c), the Zak phase is 0 or ±π in the chiral lim-its φ = π/4 and φ = 3π/4, signaling the emergence of edgestates in this latter case.

By fitting the edge state to an exponential, we can estimateits localization length numerically. According to Eq. (2), thisquantity is a decreasing function of the dimerization. Thisholds true for H(ions)

SSH , as shown in Fig.1(d). However, wenote that ξloc decreases with the range of the interactions aswell, i.e., there is an enhancement of the localization effect.This feature is not captured by the prediction for the originalSSH model, since ξloc exclusively depends on δ , and this lat-ter quantity is insensitive to the range of the couplings (cf. Eq.(9)). To obtain the dependence of the localization on the inter-action range we have considered the effective theory for thelow-energy sector of H(ions)

SSH , which captures the long-rangeeffects by a renormalization of the parameters of the theorycompared to those of the original SSH model.

Localization length of the edge states of H(ions)SSH .- The ef-

fective theory of the SSH model in k-space can be describedin terms of pairs of states |k,±〉 = |k ± kF〉, where kF ≡π/2. The low energy Hamiltonian is given by Hlow−E =(N/2π)

∫π

−πh(k)dk, with

h(k) = kvF (|k,+〉〈k,−| − |k,−〉〈k,−|)− i ∆0 (|k,+〉〈k,−| − |k,−〉〈k,+|〉) . (11)

The two parameters vF = 2J and ∆0 = 2Jδ fully characterizethe low energy-sector. From them, the dimerization is directlyobtained as ∆0/vF = δ , and since the effective theory assumesthat ∆0 vF, we can approximate the localization length (2)as

ξloc ∼vF

∆0. (12)

This prediction must hold for any lattice model whose low-energy excitations are captured by a spin Hamiltonian such ash(k± kF). In particular, this is the case for H(ions)

SSH , that can be

rewritten as ∑Nj=1 ∑

N− jd=1− j h(d)j (| j〉〈 j+d|+ | j+d〉〈 j|), where

h(d)j ≡ J(ions)d (J

(+)d +J

(−)d (−1) j), with

J(±)

d =12

(J even

d ±J oddd

), (13)

and the latter quantities defined as Jπ/2j, j+d for j even or odd,

respectively. In terms of plane waves, and assuming N → ∞,we obtain

H(ions)SSH = ∑

kε′(k) |k〉〈k|+∑

k∆′(k) |k+π〉〈k|+H.c., (14)

where we have definedε ′(k) = 4

∞

∑d=1

J(ions)d J

(+)d cos(kd),

∆′(k) = 2∞

∑d=1

J(ions)d J

(−)d eikd .

(15)

From these quantities, we can obtain the parameters of theeffective theory as (see Supplemental Material)

v′F =∂ε ′(k)

∂k

∣∣∣∣k=kF

, ∆′0 = 2Im

(∆′(k = kF)

), (16)

and compute the localization length (12). We show that thisprediction accurately holds for several values of δN/2/tC inFig.2a, along with the corresponding interaction range (cf.Fig.2b).

The localization enhancement could be actually measuredin an experiment. The idea is to unveil the existence of theedge state by studying the dynamics of a single excitation atthe boundary [14, 15]. To detect an edge state located at, e.g.,the left end of the chain, we can prepare the ‘excited state’|ψ(t = 0)〉 = |↑↓↓ . . .〉, which has a large overlap with theboundary mode, and look at its survival probability at longtimes, P≡ |〈ψ(t)|σ+

1 σ−1 |ψ(t)〉|2, t→∞. This quantity can be

estimated as (see Supplemental Material)

P(

1ξloc

)'(

c1

ξ 2loc

+c2

N

)2

. (17)

Since the overlap is appreciable only if the Hamiltonianpresents an edge state, P will take negligible values exceptin the event of localization at the left end. The initial con-dition |ψ(t = 0)〉 requires applying a π pulse to the leftmost

4

0 2 4 6 8 100

2

4

6

8

10

12

-5 -4 -3

-16-14-12-10

-8

0 2 4 6 8 100

2

4

Figure 2. (color online) (a) The edge state localization length, fromthe exact diagonalization of H(ions)

SSH (solid line) for N = 100 sites,and from expression (12) (circles) with δ = 0.1 (φ = 3π/4, η '0.62). The largest enhancement of ξloc occurs for δN/2/tC < 1. Ininset (b) we show the interaction range, ξint, of the exponentiallydecaying component of J(ions)

j,l . The inset (c) shows a log-log plot of

the survival probability P as a function 1/ξloc. For ξloc→ 1, P∼ ξ−β

locwith β ' 3.8, consistent with the prediction (17). We take δN/2/tC =

1/3, N = 1000, and values of η in the interval 0.13− 0.5, for φ =3π/4.

ion in the chain, which in turn can be prepared in the ‘groundstate’ |↓↓↓ . . .〉 by optical pumping [13]. Then we can switchon the Hamiltonian H(ions)

SSH , and wait up to t ∆−10 , where ∆0

is the lowest energy scale in the Hamiltonian. Finally, we canperform a fluorescence measurement of the state of the left-most ion. We have numerically confirmed the dependence ofP on ξloc (cf. Eq. (17)) in Fig.2c. Deviations from the powerlaw P ' ξ

−β

loc , with β = 4, are the consequence of finite sizeeffects, which play a less important role when 1/ξloc N.

Correlations in the many-body ground-state.- So far wehave been dealing with the single-excitation subspace. Never-theless, we expect that some localization at the edges featuresas well in the ground state of the many-body Hamiltonian (7).In a finite chain, states localized at each end hybridize to giverise to solutions that have support at the left and right bound-aries. We expect that the correlations between the ends arezero if there is no localization at the edges whereas they musthave a non-zero value otherwise, a result that has been es-tablished for the SSH model [24]. We illustrate this fact inFig. 3, where we have computed 〈σ z

1σzL〉 as a function of the

dimerization for both HSSH and H(ions)SSH . The correlations in

the original SSH model are non-zero for δ > 0 as expected.This holds qualitatively true for H(ions)

SSH as well. Indeed, in theregime of short range of the interactions the correlations arelarger than those of the original SSH model, which is con-sistent with the enhanced localization length predicted in theone-excitation subspace (cf. Fig. 2). Conversely, we observe

Figure 3. (color online) (a) Correlations 〈σ z1σ

zL〉 for N = 16, φ =

3π/4 and using η to tune the dimerization. The arrow shows the di-rection of decreasing range of the interactions, or increasing detun-ing from the bottom of the motional band. We have plotted curvesfor δN/2/tC = 0.5,1 and 10. The solid lines represent the exactresult from Hamiltonian (7), the dashed lines the results from thetruncated Hamiltonian (see Supplemental Material), and the circlescorrespond to the predictions of the HF approximation. Inset (b)shows the value of the parameter Z for different interaction rangesand δ = 0.1,0.3,0.5 from bottom to top.

a degradation of the correlations in the long-range interactionregime, i.e., for δN/2 → 0 there is a decrease in the local-ization effect. This result is a consequence of the mixing –induced by the interactions– of the single-particle edge stateswith the bulk modes. To quantify this effect we express ourgeneralized SSH model in terms of Jordan-Wigner fermionsas H(ions)

SSH = ∑Nl> j 2J(ions)

j,l Jπ/2j,l

(c†

jK j,lcl + c jK j,lc†l

), where

K j,l ≡ ∏l−1m= j(1 − 2c†

mcm). We neglect terms for which

| j− l| ≥ 3 and recast this problem as H(ions)SSH ' H0 + Hint,

where H0 =∑Nj=1(J

(1)j c†

jc j+1+J(2)j c†jc j+2+H.c.), with J(α)

j =

2J(ions)j, j+α

Jπ/2j, j+α

and

Hint =−2N

∑j=1

J(2)j (c†jc

†j+1c j+1c j+2 +H.c.). (18)

We deal with the interaction term within the Hartree-Fockapproximation [40], which renders a simplified Hamiltonianquadratic in fermion operators (see Supplemental Material)

HHF =N

∑µ=1

εµ c†µ cµ −2

N

∑µ,µ ′=1

Vµ,µ ′c†µ cµ ′ . (19)

HHF is expressed in terms of the eigenstates of H0, whichcorrespond to the solutions of the Hamiltonian in the one-excitation subspace (cf. Eq. (10)), that is, c j = ∑

Nµ=1 M j,µ cµ .

The one-body edge states are eigenstates of H0 and Vµ,µ ′ in-duces the mixing of these states with the bulk modes. Wequantify this effect with a parameter, Z, which measures the

5

overlap between the unperturbed boundary modes and the cor-responding states in the presence of interaction, and whichcan be estimated by elementary perturbation theory as Z '1−∑

Nµ 6=E.S. 4|VE.S.,µ |2/(εE.S.− εµ)

2. We show this quantityas a function of the range of interactions in the inset of Fig. 3.Accordingly, when δN/2 → 0 the fidelity drops significantly,signaling the decay of the edge modes into the continuum ofthe states in the bulk. Finally the average 〈σ z

1σzL〉 can be mea-

sured in an experiment by detecting the photo-luminescencefrom individual ions at the ends of the chain (e.g. by electron-shelving techniques [25]).

Conclusions and outlook.- In this work we have establishedthe feasibility of implementing a topological insulator withtrapped-ion quantum matter. We have shown that the edgestates get more localized because of the long-range interac-tions in ion chains, and that the localized solutions survive tothe interactions in the many-body ground state. An immedi-ate extension of this work would consist in the computation ofthe Zak phase of the many-body ground state, and establish-ing the symmetries of the model, to shed light on a prospectivebulk-edge correspondence in this system. Our ideas could beextended to systems of cold atoms [41, 42] or superconductingqubits [43], where dipolar interactions are available.

Acknowledgements.- We are greatly indebted to Alejan-dro Bermudez for very useful discussions. P.N. thanksDr. H. Takahashi for his feedback on the implementation.Our research has received funding from the People Pro-gramme (Marie Curie Actions) of the European Unions Sev-enth Framework Programme (FP7/2007-2013) under REAGrant Agreement No. PCIG14-GA-2013-630955.

∗ pedro.nevado.serrano@gmail.com† D.Porras@Sussex.ac.uk

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