Realizing Su-Schrieffer-Heeger topological edge states in Rydberg-atom synthetic dimensions
S. K. Kanungo, J. D. Whalen, Y. Lu, M. Yuan, S. Dasgupta, F. B. Dunning, K. R. A. Hazzard, T. C. Killian
RRealizing Su-Schrieffer-Heeger topological edge states in Rydberg-atom syntheticdimensions
S. K. Kanungo,
1, 2
J. D. Whalen,
1, 2
Y. Lu,
1, 2
M. Yuan,
1, 2, 3, ∗ S.Dasgupta,
1, 2
F. B. Dunning, K. R. A. Hazzard,
1, 2 and T. C. Killian
1, 2 Department of Physics and Astronomy, Rice University, Houston, TX 77005-1892, USA Rice Center for Quantum Materials, Rice University, Houston, TX 77005-1892, USA School of the Gifted Young, University of Science and Technology of China, Hefei 230026, China
We demonstrate a platform for synthetic dimensions based on coupled Rydberg levels in ultracoldatoms, and we implement the single-particle Su-Schrieffer-Heeger (SSH) Hamiltonian. Rydberg lev-els are interpreted as synthetic lattice sites, with tunneling introduced through resonant millimeter-wave couplings. Tunneling amplitudes are controlled through the millimeter-wave amplitudes, andon-site potentials are controlled through detunings of the millimeter waves from resonance. Usingalternating weak and strong tunneling with weak tunneling to edge lattice sites, we attain a con-figuration with symmetry-protected topological edge states. The band structure is probed throughoptical excitation to the Rydberg levels from the ground state, which reveals topological edge statesat zero energy. We verify that edge-state energies are robust to perturbation of tunneling-rates,which preserves chiral symmetry, but can be shifted by the introduction of on-site potentials.
A synthetic dimension [1, 2] is a degree of freedom en-coded into a set of internal or external states that canmimic a real-space lattice potential. Synthetic dimen-sions are powerful tools for quantum simulation, open-ing exciting possibilities such as the realization of higherdimensional systems [1, 3, 4], non-trivial real space [5]and band structure [6, 7] topologies, and artificial gaugefields [8, 9]. Experiments have utilized various degrees offreedom [2] to create synthetic dimensions, such as mo-tional [10, 11], spin [8, 12–14], and rotational [15] levelsof atoms and molecules, and frequency modes, spatialmodes, and arrival times in photonic systems [2].Prominent demonstrations of atomic synthetic dimen-sions include observation of artificial gauge fields, spin-orbit coupling, and chiral edge states in Raman-coupledground magnetic sublevels [8, 16, 17] or single-photon-coupled electronic orbitals [18, 19] grafted onto motionin a real 1D optical lattice. A synthetic dimension canalso be formed by discrete motional states [20], such asfree-particle momentum states coupled with momentum-changing two-photon Bragg transitions [21, 22]. The lat-ter has been used to observe Anderson localization [23],artificial gauge fields [24], and topological states [25, 26].Here we harness Rydberg levels of Sr to realize a syn-thetic lattice for studying quantum matter. Millimeter-wave couplings between Rydberg states introduce tunnel-ing between synthetic lattice sites. This scheme was sug-gested in [2], and it is similar to a proposal for syntheticdimensions based on molecular rotational levels [27–29].It allows for control of the connectivity, tunneling rates,and on-site potentials, and creation of a broad range ofsynthetic dimensional systems.To demonstrate this technique, we realize theSu–Schrieffer–Heeger (SSH) model [30] with six latticesites formed with three 5 sns S ( m = 1) ( ≡ ns , sites i = 1 , ,
5, with 57 s mapped to i = 1) and three 5 snp P ( ≡ np , i = 2 , ,
6) levels [Fig. 1(b)], and study its topo-
HORN FIELD PLATES689 nm 320 nm MICRO-CHANNELPLATE ASSEMBLY Sr RYDBERG ATOMe-(a) t w t s t w t s t w i =1 (57 s ) i =2 (57 p ) i =3 (58 s ) i =4 (58 p ) i =5 (59 s ) i =6 (59 p )5 s S Probe n m + n m t s t w t s t w t s i =1 i =2 i =3 i =4 i =5 i =65 s S Probe n m + n m E / t w t s / t w E / t w (d)(b) (c) (e) FIG. 1. (a) Experimental schematic. (b) Configurations withand (c) without TPS of the SSH model using six Rydberg lev-els of Sr. Double-headed grey arrows denote near-resonantmillimeter-wave couplings, which induce tunneling betweensites of the synthetic lattice, and thicker lines correspond tofaster tunneling. Dashed lines show two-photon excitation toa Rydberg level of interest. (d,e) show band structure for (b)and (c) respectively vs. the ratio of tunneling amplitudes, t s /t w . The site-numbering convention is given in (b), withodd numbers corresponding to ns states. logically protected edge states (TPS) and their robust-ness to disorder. The SSH model describes a linear con-jugated polymer, such as polyacetylene, with alternat-ing weak and strong tunneling. The configuration withweak tunneling to edge sites possesses doubly degenerateTPS with energy centered in the gap between bulk states.TPS energies are robust against perturbations respectingthe chiral symmetry of the tunneling pattern [31, 32], asobserved in many systems [25, 33–35]. a r X i v : . [ phy s i c s . a t o m - ph ] J a n The Hamiltonian realized isˆH lattice = (cid:88) i =1 ( − ht i,i +1 | i (cid:105)(cid:104) i + 1 | + h.c.) + (cid:88) i =1 hδ i | i (cid:105)(cid:104) i | , (1)where t i,i +1 are the tunneling amplitudes and δ i areon-site potentials set respectively by amplitudes anddetunings of the millimeter-wave couplings, and h isPlanck’s constant. To obtain Eq. (1), we have neglectedcounter-rotating terms in the millimeter-wave couplingsand transformed into a rotating frame. The kets | i (cid:105) cor-respond to the unperturbed Rydberg levels of Sr, up toa time-dependent phase arising from the transformation. δ i = 0 yields the SSH model, and the configuration withTPS has t i,i +1 = t w ( t s ) for i = 1 , , ,
4) and t w < t s .For the configuration without TPS, the weak and strongcouplings are exchanged.The essential elements of the present apparatus areshown in Fig. 1(a). We trap 10 Sr atoms in an opticaldipole trap at a peak density of about ∼ cm − and atemperature of T = 2 µ K. The laser cooling and trappingof Sr has been described in detail elsewhere [36, 37].Before the first excitation cycle, millimeter wavesare switched on to provide ns ( m = 1) − np and np − ns ( m = 1) coupling for three different n ’s as shown inFigure 1(b). Millimeter-wave frequencies are generatedby combining outputs of five RF synthesizers ( < t w = 100 kHz. Each coupling is calibratedusing the Autler-Townes splitting [38] in a two-level con-figuration, which is equal to the coupling Rabi frequencyΩ = 2 t . A 4 Gauss magnetic field splits the ns magneticsublevels by 11 MHz, which is large compared to tunnel-ing rates. AC stark shifts are experimentally determined,and the δ i in Eq. 1 are relative to the Stark-shifted Ry-dberg levels.To populate and probe the synthetic space, the Srground state is coupled to the Rydberg levels via two-photon excitation with an intermediate detuning of +80MHz from the 5s5p P level [39–41], applied in a5 µ s pulse. The laser polarizations selects excitation to ns ( m = 1) levels. Immediately after excitation, Ryd-berg populations are detected using selective field ion-ization (SFI) [42], for which purpose an electric field ofthe form E ( t ) = E p (1 − e − t/τ ) is applied, with E p = 49V/cm and τ = 6 . µ s. An atom in level n(cid:96) ionizes at afield given by ∼ / [16( n − α (cid:96) ) ], where α = 3 .
371 and α = 2 .
887 [43] are the quantum defects of the ns and np states respectively. Liberated electrons are detectedby a micro-channel plate, and the Rydberg level, or oc-cupied synthetic-lattice site, can be determined from thearrival time of the electron. With the current experimen- tal resolution, arrival times for states np and ( n + 1) s areunresolved. Approximately 10 excitation cycles are per-formed per sample at a 4 kHz repetition rate, and thetwo-photon drive is weak enough that either zero or oneatom is excited to the Rydberg manifold each cycle.To probe the lattice band structure, the two-photonexcitation laser is tuned, with detuning ∆ i pr , near theenergy of one of the unperturbed Rydberg levels ( | i pr (cid:105) ).Neglecting far off-resonant terms, the Hamiltonian forthe entire system can be written as:ˆH = h Ω i pr | g (cid:105)(cid:104) i pr | e i π ∆ i pr t + h.c. + ˆH lattice , (2)where Ω i pr denotes the effective two-photon Rabi fre-quency, which vanishes for even i pr ( np levels), and | g (cid:105) is the ground state vector in the frame rotating at thefrequency difference of the | i pr (cid:105) and | g (cid:105) levels. The Ryd-berg excitation rate before convolving with instrumentallinewidth isΓ(∆ i pr ) = π Ω i pr (cid:88) β | (cid:104) β | i pr (cid:105) | δ (∆ i pr − (cid:15) β /h ) , (3)where | β (cid:105) and and (cid:15) β are the eigenstates and eigenen-ergies of ˆH lattice . The collection of spectra, with eachspectrum arising from coupling | g (cid:105) to a different | i pr (cid:105) ,complement each other to provide a characterization ofthe band structure and decomposition of the eigenstatesbecause the spectral contribution from each eigenstate isproportional to its overlap with the unperturbed Ryd-berg level corresponding to the lattice site i pr .Figure 2(a,b) shows spectra for the configuration withTPS and δ i = 0 as a function of probe-laser detuningnear each of the unperturbed Rydberg ns levels (odd i pr ).Each spectrum is normalized by the total signal for its i pr , and t s is varied from 0.5 MHz to 1.5 MHz. Principalfeatures of the SSH model are readily discerned. Edgestates appear at detuning ∆ i pr = 0 in a gap of width ∼ t s between bulk states. The edge-state signal is largefor probe detuning near the 57 s level ( i pr = 1), smallfor the 58 s ( i pr = 3) spectrum, and barely observablefor 59 s ( i pr = 5). Because the integrated signal intensityaround the peak centered at detuning (cid:15) β /h reflects theoverlap of the lattice eigenstate | β (cid:105) with | i pr (cid:105) (Eq. 3), thisreveals that the edge states are localized on the weaklycoupled boundary sites, with little contribution from un-dressed bulk sites 58 s ( i = 3) and 59 s ( i = 5). Thewidely split bulk states, however, have strong and ap-proximately equal contributions in spectra for detuningnear the 58 s and 59 s undressed levels, and little contri-bution for detuning near 57 s .The band structure as a function of strong tunnelingrate t s [Fig. 2(c)] agrees with results from a direct diag-onalization of Eq. 1 with δ i = 0. For the configurationwith strong tunneling to the boundary sites, the edgestates vanish from the band structure [Fig. 2(d)]. Probe Detuning [MHz] S i g n a l [ a r b . ] (d) i pr = 1 i pr = 3 i pr = 5 t s / t w E / t w (c) 2 0 2 Probe Detuning [MHz] (b) ×2 ×2edge statesbulk states2 0 2
Probe Detuning [MHz] S i g n a l [ n o r m . ] (a) ×2 ×2 i pr =1 (57s) i pr =3 (58s) i pr =5 (59s) FIG. 2. (a,b) Rydberg excitation spectra when coupling to i pr = 1(57 s ), i pr = 3(58s), and i pr = 5(59s) for t s /t w =5 (a)and 15 (b) in the configuration with TPS. Probe detuning(∆ i pr ) is from the undressed Rydberg level. i pr = 3 , t s /t w . (d) Spectra at t s /t w = 5 for theconfiguration without TPS. Diagonalization also provides the decomposition ofeach SSH eigenstate | β (cid:105) upon the bare lattice sites, ex-pressed in the factors | (cid:104) β | i (cid:105) | . This can be comparedwith experimental measurements of the fraction of thetotal spectral area in either the edge or the bulk spectralfeatures when probing the overlap with a specific latticesite ( i pr ) in spectra such as Fig. 2(a,b). Spectral areais determined by fitting each of the three features in aspectrum with a sinc-squared lineshape corresponding tothe 5 µ s laser exposure time. Figure 3 (left) shows thatthe experimentally measured edge-state fraction matches (cid:80) β ∈ edge | (cid:104) β | i pr (cid:105) | , and Fig. 3 (right) does the same forthe bulk contribution and (cid:80) β ∈ bulk | (cid:104) β | i pr (cid:105) | . The widthof the calculated line denotes 10% variation in the Rabifrequencies. For a given t s /t w , the edge-state measure-ments in Fig. 3 add to one, while the bulk-state mea-surements add to two. This reflects the fact that thereare two edge states and four bulk states for this system,and half of the weight for the states in each group is inoverlap with even lattice sites, which the photoexcitationprobe does not detect.Because the spectral probe is only sensitive to ns con-tributions to the state vector (odd i ), it cannot estab-lish whether the edge states observed are localized onone boundary site or a superposition of both. To an-swer that question, we turn to SFI as a tool for site-population measurements in Rydberg-atom synthetic di-mensions. For Rydberg excitation near 58 s , correspond- t s / t w E dg e w e i g h t s (a) i pr = 1 (57s) i pr = 3 (58s) i pr = 5 (59s) t s / t w B u l k w e i g h t s (b) i pr = 1 (57s) i pr = 3 (58s) i pr = 5 (59s) FIG. 3. Synthetic-lattice-eigenstate decomposition obtainedfrom spectral-line areas [Fig. 2(a,b)]. (a) Fraction of the en-tire signal under the spectral features corresponding to theedge states for probe tuned near site i pr (Rydberg level) in-dicated in the legend. The line is the sum of the squares ofthe calculated overlaps of the SSH edge eigenstates with the i pr site found from a direct diagonalization of Eq. (1) with δ i = 0. (b) Fraction of the entire signal under the bulk statefeatures and calculated sum of squares of the overlaps of theSSH bulk eigenstates with the i pr site. C o un t s
59p 58s57p 57s59s58p(a) i = 1 i = 2, 3 i = 4, 5 i = 6Left bulk peak Right bulk peak
28 30 32 34 36 38 40Electric field [V/cm]0.000.050.10 C o un t s (b) Edge state spectra
FIG. 4. SFI signals for probe laser tuned near the 58 s Rydberg level ( i pr = 3) for t s /t w = 5 [Fig. 2(a)]. Verticallines indicate ionization fields for bare Rydberg levels. Datapoints are evenly spaced in time. (a) For excitation to theleft and right bulk-state peaks (∆ i pr ≈ ±
500 kHz), the stateexcited is localized on the bulk sites of the synthetic lattice.(b) For excitation to the edge-state peak, (∆ i pr ≈
0) thestate is localized on the 57 s ( i = 1) boundary site. The smallcontribution to the signal at i = 2 , ,
4, and 5 is predominantlyfrom the wings of the bulk-state peaks. ing to i pr = 3, and for t s /t w = 5 [Fig. 2(a)], if the detun-ing is set to resonance with the left or right bulk-statepeaks (∆ i pr ≈ ±
500 kHz), electrons are liberated at ion-ization fields for Rydberg levels corresponding entirely tobulk sites of the synthetic lattice ( i = 2 −
5) [Fig 4(a)].For laser detuning on the edge-state peak (∆ i pr ≈ s Rydberg state ( i = 1) [Fig 4(b)]. This indicateslocalization of the edge state on the boundary in generaland more specifically on the single boundary site con-nected to the ground-state by the two-photon excitation[Eq. (3)]. Integrals of SFI signals corresponding to eachlattice site and for each spectral feature provide statedecomposition that agrees with expectations as in Fig. 3.The pinning of the edge-state energy to ∆ i pr = 0 isthe defining feature of TPS in the SSH model. It reflectsan underlying chiral symmetry, which refers to the sys-tem’s bipartite structure (even and odd sites), with allHamiltonian matrix elements vanishing between sites ofthe same partition, including diagonal (on-site) matrixelements. To investigate the robustness of the pinningof the edge-state energy, we probe the band structure inthe presence of perturbations from the SSH form.Figure 5(a,b,c) shows spectra for i pr = 3 (58 s ) and i pr = 5 (59 s ) for balanced ( t − = t − = t s ) and imbal-anced [ t − = (1 ± . t s and t − = (1 ∓ . t s ] strongcoupling. Here, t s /t w = 5 and t w = 100 kHz. The bulkstates are strongly affected by imbalance. With increased t − [Fig. 5(b)], two bulk states that are more localizedon the i = 3 site show increased splitting. With increased t − [Fig. 5(c)], two bulk states that are more localizedon the i = 5 site show increased splitting. The energy ofthe edge-state signal, however [in Fig. 5(b,c)], is immuneto this perturbation, which preserves the protecting chi-ral symmetry because the tunneling matrix elements onlyconnect even and odd sites.Figure 5(d,e) shows how the energies of the edge statesare affected by chiral-symmetry-breaking perturbations,in particular shifts of on-site potentials (i.e. millimeter-wave coupling frequencies), such that δ i (cid:54) = 0 for some i .Spectra are recorded with the probe laser tuned near the58 s level ( i pr = 3) for t s /t w = 10. For Fig. 5(d), the fre-quency of the i = 1 to i = 2 (57 s -57 p ) coupling is varied,which shifts δ , the on-site potential of the i = 1 (57 s )site in the synthetic lattice. This diagonal term in theHamiltonian breaks the chiral symmetry, and the edgestate signal shifts by an amount equal to the detuningfrom resonance. For Fig. 5(e), the frequency of the i = 5to i = 6 (59 s -59 p ) coupling is varied, shifting δ , andthe position of the edge-state signal remains unchanged.These results confirm that the edge state coupled to bythe probe laser is localized on the i = 1 (57 s ) boundarysite. The orthogonal edge state is localized on i = 6,with vanishing weight on odd sites, and this particularform of perturbation only affects the energy of one of theedge states. In general we expect that any perturbationproducing a Hamiltonian term that connects only evensites to even sites, or odd to odd, will break the chiralsymmetry and shift edge-state energies.We have demonstrated Rydberg-atom synthetic di-mensions as a promising new platform for the study ofquantum matter. The spectrum of photo-excitation tothe synthetic lattice space formed by the manifold of cou-pled Rydberg levels provides the band structure and de-composition of the lattice eigenstates. SFI of the excitedstates provides an additional diagnostic of lattice-site Probe detuning [MHz] (c)0.25 0.00 0.25
Probe detuning [MHz] (e)1 0 1
Probe detuning [MHz] (b) i pr = 3 (58s) i pr = 5 (59s) Probe detuning [MHz] S i g n a l [ a r b . ] (d) -200 kHz 0 kHz +200 kHz Probe detuning [MHz] S i g n a l [ n o r m . ] (a) FIG. 5. Band structure with Hamiltonian perturbations.(a) SSH model: δ i = 0 with balanced tunneling rates ( t − , t − = t s and t s /t w = 5). Lines mark positions of the bulkand edge peaks. (b) Strong tunneling rates are imbalancedto t − = 1 . t s and t − = 0 . t s for t s /t w = 5. Thisperturbation respects chiral symmetry. (c) Same as (b) but t − = 0 . t s and t − = 1 . t s . (d) Perturbation breakingchiral symmetry: tunneling rates are balanced as in the stan-dard topological SSH configuration, with t s /t w = 10, but thefrequency of the i = 1 to i = 2 (57 s -57 p ) coupling is varied bythe value given in the legend. (e) Same as (d), but the i = 5to i = 6 (59 s -59 p ) coupling frequency is varied. populations with two-site resolution. TPS were observedin a six-site SSH model, and the measured band struc-ture and eigenstate decomposition agree well with theory.Varying the detuning of the millimeter-wave fields thatcreate tunneling between sites introduces on-site poten-tials, and this has been used to break the chiral symme-try of the SSH model and shift the energies of edge statesaway from the center of the bandgap at ∆ i pr = 0.The Rydberg-atom synthetic-dimension platformopens up exciting future directions. Additionalmillimeter-wave-coupling schemes and tunneling config-urations are possible, such as two-photon transitionsand transitions with larger changes in principal quantumnumber. This will enable creation of higher-dimensionalsynthetic lattices [1, 3, 4] and investigation of systemswith non-trivial spatial [5] and band-structure [6, 7]topologies and higher-order topological states [44], for ex-ample. Expanding the size of the synthetic space throughthe application of more millimeter-wave frequency com-ponents should be straightforward. Through control ofmillimeter-wave phases, tunneling phases around pla-quettes and artificial gauge fields can be introduced[45]. This platform is also ideally suited for study oftime dependent phenomena, such as Floquet-symmetry-protected states [46], non-equilibrium states [47], andwave-packet dynamics in synthetic space. The limit im-posed by Rydberg-level decoherence needs further study,but based on previous work demonstrating coherent ma-nipulation of Rydberg-level populations (e.g. [48, 49]),decoherence should not prevent observation of the phe-nomena of interest. Tailored time variation of theelectric-field ramp [50] may improve site resolution of theSFI diagnostic. 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