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Topologically protected edge states in smallRydberg systemsTo cite this article: Sebastian Weber et al 2018 Quantum Sci. Technol. 3 044001

 

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QuantumSci. Technol. 3 (2018) 044001 https://doi.org/10.1088/2058-9565/aaca47

PAPER

Topologically protected edge states in small Rydberg systems

SebastianWeber1 , Sylvain de Léséleuc2, Vincent Lienhard2, Daniel Barredo2 , Thierry Lahaye2,AntoineBrowaeys2 andHans Peter Büchler1

1 Institute for Theoretical Physics III andCenter for IntegratedQuantumScience andTechnology, University of Stuttgart, D-70550Stuttgart, Germany

2 Laboratoire Charles Fabry, Institut d’OptiqueGraduate School, CNRS,Université Paris-Saclay, F-91127 PalaiseauCedex, France

E-mail: weber@itp3.uni-stuttgart.de

Keywords: topological insulator, edge states, Rydberg atoms, dipolar interaction

AbstractWepropose a simple setupof Rydberg atoms in a honeycomb latticewhich gives rise to topologicallyprotected edge states. The proposal is basedon the combination of dipolar exchange interaction,whichcouples the internal angularmomentumand the orbital degree of freedomof aRydberg excitation, and astaticmagneticfield breaking time reversal symmetry.Wedemonstrate that for realistic experimentalparameters, signatures of topologically protected edge states are present in small systemswith as few as10 atoms.Our analysis paves theway for the experimental realizationof Rydberg systems characterizedby a topological invariant, providing a promising setup for future application inquantum information.

Introduction

Systems characterized by topological invariants give rise tomany interesting phenomena [1, 2]. Of specialsignificance are topologically protected edge states which arise infinite systems as the characteristic feature oftopological band structures [3, 4]. Topologically protected edge states possess distinguished properties likerobustness to local perturbations whichmake themhighly interesting for various applications such as theprocessing or coherent transport of quantum information [5–7]. The prime example for the occurrence oftopologically protected edge states is the integer quantumHall effect [8, 9]. In addition, topological bandstructures and topologically protected edge states are observed in systems as varied as classical phononic [10, 11]or photonic [12–14] setups, solid state systems [15–17], and cold gases [18–22]. For future application inquantum information or the realization of interestingmany-body states on the basis of topological bandstructures [23–26], highly controllable quantummechanical systems are required.

Due to recent experimental progress, Rydberg atoms have become a promising candidate for the realizationof such systems. The excellent experimental controllability of the long-ranging dipolar interaction betweenRydberg atoms [27] and the capability of preparing arbitrary arrays of up to∼50 atoms [28–30] paved theway forthe realization of exoticmatter [31–34].While for polarmolecules ideas like artificial gauge fields combinedwithdipolar exchange interactions have been proposed for topological systems [35], it recently has been pointed outthat the intrinsic spin–orbit coupling of dipolar interactions in combinationwith broken time reversalsymmetry is sufficient to achieve topological protected edge states [36]; the latter setup has the advantage that itdoes not require any spatially inhomogeneous fields. The idea has been extended to Rydberg-dressed groundstate atoms [7] as well as photonic setups [37, 38].

In this paper,we adapt this idea to systemswithRydberg atoms arranged in afinite array.Themain focus is on thesignature of edge states in these small setups.Weprovide experimentally feasible parameters for the implementationof such systemsusing recently established experimental platforms [28–30].Ourproposal relies onanopticalhoneycomb lattice occupiedbyoneRydberg atomper lattice site.We assume that the atoms are in aRydberg ñ∣nSstate except for one atomthat is excited to adipole-coupled ¢ ñ∣n P state.We show that in thepresence of static electricandmagneticfields,which are isolating the relevantRydberg state from theRydbergmanifold andwhere the laterbreaks time reversal symmetry, the band structure of such an excitation features non-zeroChernnumbers.Note thatin contrast toproposals for polarmolecules [35, 36], setups relyingon lattice shaking [18, 22, 39], laser-assisted

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tunneling [40, 41], or synthetic dimensions [20, 21, 42], weneither need time-dependent nor spatiallyinhomogeneousfields for the realizationof a topological interestingband structure. For studying edge states infinitesystems,wedevelop aband structure analog.Our analysis show that signatures of topologically protected edge statesare present in systemswith as fewas 10 atoms.Considering lattice disorder,wedemonstrate in realistic systems thatthe edge states canbeprobedbyobserving the chiralmovement of an edge excitation.The requirements for ourproposal are readilymet by recentRydberg experiments.

Setup

Weconsider an optical lattice occupied by one Rydberg atomper lattice site, see figure 1(a). In this paper, we usea honeycomb lattice, but our proposal wouldwork similarly for other lattice geometries such as square orKagome lattices.We chose the quantization axis to be perpendicular to the lattice.We are interested in aV-levelstructure comprising the Rydberg states ñ = = ñ∣ ∣nS m0 , 1 2j1 2 , +ñ = ¢ = ñ∣ ∣n P m, 3 2j3 2 , and -ñ =∣¢ = - ñ∣n P m, 1 2j3 2 , seefigure 1(b).We apply static, homogeneous electric andmagnetic fields along the

quantization axis to lift the Zeeman degeneracy to isolate theV-level structure fromother Rydberg states. Theenergy difference m = +ñ - -ñ(∣ ) (∣ )E E can be adjusted by the fields.

We regard the statewhere all atoms are in the state ñ∣0 as the ground state. In the following,we study systemscontaining one excitation to the state +ñ∣ or -ñ∣ . The excitationpropagates through the systembymeans ofdipole–dipole interaction.Note that for our analysis, we just consider dipolar exchange interaction andneglect thestatic dipole–dipole interactionof thefinite dipolemoments of theRydberg atoms inducedby the electricfield. Inaddition,we ignore van derWaals interaction.These approximations are justified in appendixB. Bydescribing thecreationof a ñ∣ excitation at lattice site i by theoperator = ñ á ∣ ∣†b 0i i i, and introducing the spinor

y = + -( )† † †b b,i i i, , , we canwrite the operator for the dipolar interactionbetween the lattice site i and j as [36]

y y= --

+f

f+

-

-

⎛⎝⎜

⎞⎠⎟∣ ∣

( )†

RV

a t w

w t

e

eh.c ., 1ij

iji j

dd3

3

2i

2i

ij

ij

where a is the lattice constant, = -R R Rij j i is the distance vector between the lattice sites, andfij is the anglebetween the distance vector and the x-axis. The parameters t+ and t− are the amplitudes of the hopping processeswhich conserve the internal angularmomentumof the excitation. The amplitudew belongs to the hoppingprocess that flips a -ñ∣ excitation into a +ñ∣ excitation, leading to a change in internalmomentumby twowhichis compensated by a change in orbitalmomentum accounted by be the phase factor f-e 2i ij. Note that in our caseof long-range interaction, the phase factor cannot be obtained through gaugefluxes.

The totalHamiltonian of the system reads

å å y ym

m= +

-¹

⎛⎝⎜

⎞⎠⎟ ( )† /

/H V

1

2

2 0

0 2, 2

i jij

ii i

dd

where the last sum includes the energy difference between +ñ∣ and -ñ∣ excitations.

Topological band structure

For broken time reversal symmetry, i.e. m ¹ 0 or ¹+ -t t , the topological properties of thisHamiltonian arecharacterized byChern numbers [36, 43]. Note that ¹+ -t t is intrinsically fulfilled by our setup because of thedifferent Clebsch–Gordan coefficients for the creation of +ñ∣ and -ñ∣ excitations.

Figure 2(a) shows the density of states and the topological band structure of the infinite honeycomb lattice for atypical set of parameters. As the unit cell of the honeycomb lattice consists of two sites and the ñ∣ excitationhas a

Figure 1. Setup. (a)Each site of an optical honeycomb lattice is occupied by one Rydberg atom. The quantization axis z is chosenperpendicular to the lattice. Static electric andmagnetic fields are applied along z, to isolate theV-level structure (b).We consider allatoms to be in the state ñ∣0 except for one atom excited to the state +ñ∣ or -ñ∣ . The excitation propagates through the systembymeansof dipole–dipole interaction.

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QuantumSci. Technol. 3 (2018) 044001 SWeber et al

two-fold internal degree of freedom, there are four bands in total. For each band,we calculate theChernnumberC[44, 45]which dependsonμ as shown infigure 2(b). Our systemexhibits a rich phase diagramwithChernnumbers ranging fromC=−4 toC=4 as a function ofμ for typical hopping amplitudesw=4.10MHz,t+=2.25MHz, and t−=0.84MHzwhose experimental realization is discussed at the endof the paper. Forfurther calculations,weuse these hopping amplitudes togetherwithμ=−16MHz.The selectedparameters havethe advantage that pronounced band gaps existwhichmake theChernnumbersC=−1 andC=1of the twolower bands robust against perturbations.Moreover, becauseμ ismuch larger thanw, the two lower bandsmainlyoverlapwith the +ñ∣ statewhatwill turn out to beuseful. In the following,wewill focus on the two lower bands.

Edge states

In systemswith boundaries, edge states are the characteristic feature of the topological bands of the infinite systemas statedby thebulk-boundary correspondence [4].Wefirst study edge states in a semi-infinite systembeforeheading towards experimentally feasible, small systems. Figure 3(a) shows the band structure anddensity of state ofan exemplary semi-infinite honeycomb lattice. For the eigenstates belonging to the bands, the color code visualizesthe expectation value of thedistance of an excitation from the edge. The analysis shows that the gap,whichwaspresent between the two lower bands in the infinite system, is nowclosed byone chiral edgemode at thebeardededge of the considered semi-infinite systemandone chiral edgemode at the zigzag edge. The dispersion relations ofthe edgemodes are nearly linear. The inset offigure 3(a) shows an exemplary edge state.

As examples of experimentallywell realizable systems,we study small disk-shaped systemswith 10–31Rydbergatoms.The implementationof such systems is realistic considering recent experimental developments [28–30]. As incase of the semi-infinite system,we can calculate thedensity of states, seefigures 3(b), (c). Thedensity of states showsthat as before, edge states close thebandgapwhichwas existing in the infinite system, seefigures 3(b), (c). Yet, foranalyzing edge states, the density of states ismuch lessmeaningful than theband structure. It neither tells thenumberof edgemodesnor their dispersion.Therefore,wedeveloped aband structure analogwhichprovides us thisinformation.Contrary to similar approaches [10, 12, 37], our band structure analog allowsus tomake full use of therotational symmetryof the considereddisk-shaped systems. In the following,we explain thedeterminationof theband structure analog. Since thedisk-shaped systemsareC3 symmetric, the eigenstates areBlochwaveswhich canbelabeledby twoquantumnumbers: the band indexb and the total angular quasimomentum Î { }j 0, 1, 2 (the orbitalangular quasimomentum isnot a goodquantumnumber because it is coupled to the internal angularmomentumofthe excitation). ByBloch’s theorem, the coefficients of an eigenstate Y ñ∣ b j, are

á Y ñ = á ñj j∣ ∣ ( )ue e , 3i b jj

i b j,i i

,i i

Figure 2. (a)Density of states (DOS) and topological band structure of the infinite honeycomb lattice plotted over the depicted paththrough the Brillouin zone for a typical parameter set. The bands are labeledwith their Chern numberC. The color code tells theoverlapwith the +ñ∣ state. (b)Chern numbers and band gaps as a function ofμ forw=4.17 MHz, t+=2.41 MHz, and t−=0.80 MHz. The dashed linemarks the parameter set used in (a)withμ=−16 MHz forwhich the Chern numbers of the two lowerbands are highly robust against perturbations due to the pronounced band gaps.We use this parameter set for all calculations.

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QuantumSci. Technol. 3 (2018) 044001 SWeber et al

whereji is the angular position of the i’th lattice site and ñ∣ub j, theC3 periodic part of the Blochwave. The threepossible values of jmake up the reduced Brillouin zone of the system.However, for analyzing edge states, wewould like to plot the dispersion relation over a larger Brillouin zone because the edge states are exponentiallylocalized at the edge of the disk-shaped systemswhich acquires a higher rotational symmetry than the bulk. Inorder to show theC3 symmetric bulk states as well as the higher symmetric edge states in one plot, we plot theband structure over the extended Brillouin zone. For this purpose, we have to uniquely assign to each of theNeigenstates a quasimomentum Î -˜ { }j N0, ..., 1 from the extended Brillouin zone so that the energy of Y ñ∣ j̃

plotted over j̃ make up our band structure analog. As the periodic part ñ∣ ˜u j of a Blochwave Y ñ∣ j̃ typically

changes slowlywith the quasimomentum j̃ , the assignment should be such that

å åá ñ = áY Y ñj

=

-

+=

--

+∣ ∣ ∣ ∣ ∣ ∣ ∣ ( )˜

˜ ˜˜

˜ ˜u u e , 4j

N

j jj

N

j j0

2

12

0

2i

12

ismaximal. Finding this assignment is aNP-hardmaximization problem. Fortunately, it can bemapped to thetraveling salesman problem forwhichmany excellent heuristics were developed [46]. For finding a solution, weuse theGoogleOptimization Tools [47]. Conveniently, themaximization of (4) also ensures that eigenstateswhich belong to consecutivemomenta in the reduced Brillouin zone are belonging to consecutivemomenta inthe extended Brillouin zone aswell. Thus, the resulting band structure analog can be interpreted as the optimally‘unwrapped’ version of the band structure over the reduced Brillouin zone.

Fromthe band structure analog,we see thatwehaveone chiral edgemode even in small systemwith as few as10 atoms, seefigures 3(b), (c).Wenowhave a single edgemodeper systembecause thedisk-shaped systems justhave one boundary. Thedispersion relations of the edgemode are still nearly linear. Thus,we can conclude that themainproperties of the edgemodes stay the same ifwego fromsemi-infinite systems to small disk-shaped systems.

In order to probe the edgemodes experimentally, we propose to excite one single atom at the edge into the+ñ∣ state. Aswe have tuned our system such that the two lower bandsmainly consist of the +ñ∣ state, theexcitation has a large overlapwith the edgemode between the two lower bands. The simulated time evolution ofone edge excitation is shown infigure 4.Whilemost of the excitation stays at the edge, the small overlap of theexcitationwith bulk states leads to aminor spreadwithin the bulk of the system. The center ofmass of theexcitation performs a chiralmovementwith a velocity thatmatches the group velocity extracted from thedispersion relations shown infigure 33. As the dispersion relations are not completely linear, the excitationbroadens in time as it can be seen in the simulated time evolution. These analyses even hold for the small disk-shaped systemwith as few as 10 atoms.

Robustness

Experimentally, due to the finite temperature and imperfections in the array of tweezers, the atoms are notperfectly positioned on a regular lattice, but show a small randomdisplacement that varies from shot to shot. Forthis reason, we study the influence of lattice disorder on the propagation of the excitations.Hereto, we addnormal-distributed shifts to the positions of the lattice sites. Figure 5(a) shows the sample-averaged propagationof edge excitations as a function of the standard deviationσ of the positions of the lattice sites. The chiral

Figure 3. (a)Density of states and topological band structure of a semi-infinite honeycomb lattice. The color code visualizes theexpectation value of the distance of an excitation from the edge of the lattice, highlighting edge states which are the characteristicfeature of the non-zero Chern numbers of the infinite system. The inset shows an exemplary edge state where the area of the red orblue dots is proportional to the probability of +ñ∣ or -ñ∣ excitations at a lattice site. The slope of the dashed line fitted to the edgemodeis the group velocity vg=12.3 a μs−1 of an excitation at the upper edge. (b)Density of states convoluted by a narrowGaussian andtopological band structure of a disk-shaped systemwith 31 atoms plotted over the extended Brillouin zone. The angular group velocityof an edge excitation isωg=2π×0.68 MHz. (c) Signatures of topologically protected edge states are present even in a small disk-shaped systemwith as few as 10 atoms, whereωg=2π×1.24 MHz.

3The excitation propagates anticlockwise in the disk-shaped systems for a positive group velocity becausewe consider anticlockwise angles

to be positive.

4

QuantumSci. Technol. 3 (2018) 044001 SWeber et al

propagation is robust against the disorder up toσ/a∼0.1. For comparison, we calculated the sample-averagedChern number C̄ of the lowest band of an infinite honeycomb lattice for increasing disorder. To include thedisorder into the calculation, we enlarged the unit cell of the honeycomb lattice and added random shifts to thepositions of the atomswithin the enlarged unit cells. Figure 5(b) shows C̄ , which can take non-integer valuesbecause of the averaging over several system realizations, for various sizes of the unit cell. For infinite unit cells,we expect a sharp transition between the topological and the trivial phase due to self-averaging of disorder. Thedisappearance of the chiral propagation of the edge excitation coincideswith theChern number becoming zero,which confirms nicely that the chiral edgemodes are the characteristic feature of the non-zero Chern numbers.A similar analysis indicates that lattice vacancies are tolerable up to a vacancy probability of∼20%.Note that therobustness to vacancies has previously been studied in [36, 37].

Due to the robustness, the requirements for realizing such systems are readilymet by recent experimentswhere the lattice disorder is belowσ/a<0.1 and fully loaded systemswith up to∼50 atoms exist [28–30].Because of the robustness to lattice vacancies, we can also tolerate errors in the preparation of the Rydberg states.

Experimental realization

In the following, we give realistic experimental parameters for realizing the Rydberg level structure and hoppingamplitudes whichwere used throughout the paper.We suggest to use the principal quantumnumber n=63 for

Figure 4.Chiral propagation of edge excitations into the +ñ∣ state, probing the edge states between the two lowest bands in the semi-infinite lattice (a), the disk-shaped systemwith 31 atoms (b), and the disk-shaped systemwith 10 atoms (c). As infigure 3, the area ofthe dots is proportional to the probability offinding an excitation. The center ofmassmovements of the excitations are depicted asblack lines. Their velocities agree with the group velocities extracted from the band structures shown infigure 3.

Figure 5.Effects of lattice disorder. The shifts of the lattice sites obey a normal distributionwith standard deviationσ. (a)We study theeffect on the propagation of an edge excitation averaged over 800 system realizations. (b)Average Chern number C̄ of the lowest bandof the infinite honeycomb lattice calculated for unit cells of different sizes for increasing lattice disorder. The standard error is of thesize of the width of the plotted curves. The disappearance of the chiral propagation of the edge excitation, best visible in the vanishingof the circularmovement of the center ofmass of the excitation (black line), coincides with theChern number becoming zero.

5

QuantumSci. Technol. 3 (2018) 044001 SWeber et al

the ñ∣0 state and ¢ =n 62 for the ñ∣ states. These three states can bewell isolated from the other Rydberg statesandμ can be tuned to our needs by applying an electricfield of 600 mV cm−1 and amagnetic field of−15.8 Galong the quantization axis. For a lattice constant of a=10 μm,we obtain the applied hopping amplitudes asshown in appendix A.Note that our setupwouldworkwith somewhat different lattice spacing and quantumnumbers just as well if thefields are adapted accordingly.

Conclusion and outlook

Weproposed a Rydberg systemwhich gives rise to topologically protected edge states through dipolarinteraction and identified realistic experimental parameters for the implementation of the system.Our proposalhas the advantage of low experimental requirements. Signatures of topologically protected edge states existalready in tiny systemswith as few as 10 atoms and their demonstrated robustness guaranties that lattice disorderor errors in the preparation of Rydberg states can be tolerated. Thus, our proposal is perfectly suited for recentRydberg setupswhich acquired the capability of addressing single Rydberg atoms [48].While we focused onsingle excitations, studyingmany excitationswithin our setupmight beworthwhile aswell. As two excitationscannot be on the same lattice site, there is naturally a strong interaction between the excitations, which can thenbe considered as hard-core bosons. Because strong interaction and topological bands are seen as themainingredients for the realization of fractional Chern states, we expect our setup to be promising as a starting pointfor the implementation of bosonic fractional Chern insulators, especially as contrary tomany other setups, wedo not need time-dependent fields for realizing topological bands. This eliminates a possible source of energyentry into the systemwhich is considered to prevent the experimental realization of fractional Chern insulators.

Acknowledgments

This research has received funding from the European ResearchCouncil (ERC) under the EuropeanUnionsHorizon 2020 research and innovation programme (grant agreementNo 681208).We acknowledge supportfrom the Région Il̂e-de-France, from the Labex PALM (Xylos project), and from the ‘Fondation d’entrepriseiXcore pour la Recherche’.

AppendixA.Microscopic derivation of the hopping amplitudes

In the following, we review the derivation of the interaction operator (1) and calculate the hopping amplitudes.Given that the quantization axis is the z-axis, the dipole–dipole interaction operator for two atoms i and j in thexy-plane reads

p= + +

- +f f

+ - - +

- - + + -

⎡⎣⎢

⎤⎦⎥

∣ ∣( )

( ) ( )

RV d d d d d d

d d d d

1

4

1

2

3

2e e , A1

ijij

i j i j i j

i j i j

dd

03

0 0

2i 2iij ij

where = -R R Rij j i is the distance vector between the atoms andfij is the angle between the distance vector

and the x-axis. The operators =d ez0 and J j= p ( )d er Y ,4

3 1, 1 are the electric dipole operators, where e is

the elementary charge andY1,±1 are spherical harmonics. By neglecting static dipole–dipole interactions d di j0 0

and introducing the hopping amplitudes

k k

k k

k k

=- á+ + ñ = á+ ñ

= - á- - ñ = á- ñ

= - á+ - ñ = á+ ñá- ñ

++ - +

-- + -

+ + - +

∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ ∣ ( )

t d d d

t d d d

w d d d d

10 0

10 ,

10 0

10 ,

30 0

30 0 , A2

i j i j i j

i j i j i j

i j i j i j

2

2

with k p= a8 03, we can transform the depicted dipole–dipole interaction operator(A1) into the operator (1).

For calculating the hopping amplitudes, we take into account that the appliedfields cause a∼7%admixtureof other Rydberg states into the states of theV-level structure. Following [49, 50], we evaluate all relevant dipolematrix elements and obtain the hopping amplitudes t+=2.25 MHz, t−=0.84 MHz, andw=4.10 MHz,whichwe used throughout the paper.

6

QuantumSci. Technol. 3 (2018) 044001 SWeber et al

Appendix B. Comparison of the interaction potentials of ourmodelwith precisepotentials

As discussed in themain text, ourmodel treats the interaction between a pair of Rydberg atoms in a simplifiedform.We just consider couplings within theHilbert space ñ +ñ -ñ{∣ ∣ ∣ }0 , , , neglecting van derWaals as well asstatic dipole–dipole interactions. In the following, we show that these simplifications are valid for the proposedexperimental parameters. To this, we compare the pair interaction potentials of themodel’sHamiltonian (2)with numerically calculated precise interaction potentials. The calculations are carried out using recentlydeveloped open-source software as described in [50]. The applied basis set comprises∼2000Rydberg pair stateswith n=59–66 and l=0–4, within an energy range of 6 GHz. Figure B1 shows the resulting potential curves asa function of the interatomic distance. The good agreementwith the interaction potentials of themodel’sHamiltonian justifies the simplifications used in the paper.

ORCID iDs

SebastianWeber https://orcid.org/0000-0001-9763-9131Daniel Barredo https://orcid.org/0000-0001-5307-1927

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Figure B1. Pair interaction potentials for ñ = = ñ∣ ∣ S m0 63 , 1 2j1 2 , +ñ = = ñ∣ ∣ P m62 , 3 2j3 2 , -ñ = = - ñ∣ ∣ P m62 , 1 2j3 2 ,E=600 mV cm−1, andB=−15.8 Gperpendicular to the interatomic axis. The vertical linemarks the lattice constant a=10 μm.For all relevant distances, the potentials of our simplifiedmodel (dashed lines) agree well with the numerically calculated precisepotentials (solid lines).

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QuantumSci. Technol. 3 (2018) 044001 SWeber et al