Transmon-based simulator of nonlocal -phonon coupling: A platform for observing sharp small-polaron transitions

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Citation Stojanović, Vladimir M., Mihajlo Vanević, Eugene Demler, and Lin Tian. 2014. “Transmon-Based Simulator of Nonlocal Electron- Phonon Coupling: A Platform for Observing Sharp Small-Polaron Transitions.” Physical Review B 89 (14) (April). doi:10.1103/ physrevb.89.144508.

Published Version 10.1103/PhysRevB.89.144508

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:22856730

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP arXiv:1401.4783v1 [cond-mat.mes-hall] 20 Jan 2014 aso h are sicesd–cmae otebare- the to compared – increased is effective carrier The the of vibrations. interact- lattice mass host-crystal strongly the insulator) with narrow- an ing a (or in hole) semiconductor (electron, band carrier excess an envisions since It ever interaction (e-ph) electron-phonon of studies hi netgto fcag aresi oa semiconduc polar in tors carriers charge of investigation their rions or with evi- systems ions such as simulating trapped field, for proposals the at- the in by recently denced researchers quite among go have attention that systems tracted those polaronic paradigm of particular, physics even In low-energy conventional and the with freedom, beyond models of models, degrees re- allow bosonic to enough various matured of counter- already alizations has highly-controllable field systems The simpler, many-body parts. their quantum studying complex by of physics behavior of the of frontier field current the the research at control, is and simulation manipulation quantum for methods and Lloyd ‡ † lcrncmi:[email protected] mail: Electronic [email protected] mail: Electronic ocie yLna n ea sab-rdc of by-product a as Pekar and Landau by Conceived ae ntepoern da fFymnand Feynman of ideas pioneering the on Based 11 1,2 h oao ocp a lydapvtlrl in role pivotal a played has concept polaron the , 9 n uecnutn S)circuits (SC) superconducting and , n osee ydvlpet ntechnology in developments by bolstered and , 3 t vrrhn oli ohl sunderstand us help to is goal overarching Its . ASnmes 52.p 36.c 71.38.Ht 03.67.Ac, 85.25.Cp, numbers: momentum-dep PACS strongly with tes ideal systems an in provides formation setup polaron proposed of The phonons). of (grou Hamiltoni number properties effective single-particle relevant our the in of reflected diagonalization numerical exact wfl-eeeaesalplrngon tt tnonzer at state ground small-polaron twofold-degenerate a odgnrt snl-atce rudsaecorrespon state requir ground strength coupling (single-particle) of critical nondegenerate system orders the a this reach several in readily times that can demonstrate within one We system times. this decoherence in prepared Bloc a be small-polaron a can qubits, transmon the of to array vali fields driving an microwave of entails pro realm simulator single-particle the proposed in to the nonanalyticities any belong out not rules dep that does function vertex model corresponding this its quasimomenta, Because types. mode” coupling electron-phonon (nonlocal) momentum-dependent 5,6 rnmnbsdsmltro olcleeto-hnncou electron-phonon nonlocal of simulator Transmon-based epooea nlgsprodcigqatmsmltrfor simulator quantum superconducting analog an propose We odplrmolecules polar cold , 2 .INTRODUCTION I. eateto hsc,Uiest fBlrd,Studentski Belgrade, of University Physics, of Department 3 ldmrM Stojanović M. Vladimir colo aua cecs nvriyo aiona Merc California, of University Sciences, Natural of School ltomfrosrigsapsalplrntransitions small-polaron sharp observing for platform a 1 eateto hsc,HradUiest,Cmrde A0 MA Cambridge, University, Harvard Physics, of Department 7,8 ybr atoms Rydberg , † , 10 1 . ial Vanević, Mihajlo Dtd aur 1 2014) 21, January (Dated: 12 4 - . . hnn sl-rpig.Plrnccrir aebeen semiconduc- amorphous have from tors ranging carriers materials Polaronic in deformation found virtual by lattice carrier (self-trapping). self-induced the of phonons a “dressing” effective to an causes due that – value band ti)poos wn oalrebd fwr vrthe over work of body decades large five a to past (Ein- Owing dispersionless phonons. with interac- stein) – tightly-bound momentum-independent of hence tion – local purely iso hsmdlaewl nesodb o.Recent now. interactions by e-ph understood strongly momentum-dependent on well focussed however, are have, studies model small-polaron this of ties es uha rai semiconductors organic as of signifi- properties such a transport tems, has on and electronic bearing states of cant phonon dependence upon the integrals for hopping cou- accounts off-diagonal which or Peierls-type pling), as (SSH) known Su-Schrieffer-Heeger (also by coupling furnished is example tant dnie ncl-tmcsystems cold-atomic in identified cuprates otubes edaui elo h otcytl(ml polarons). for paradigm (small problem a framework polaron-crossover crystal model, the the molecular-crystal host within Holstein studied the ex- the usually of not of are does cell carriers particle Such a unit around a cloud ceed phonon the of size fermion a field. of boson model scalar field-theoretic a a with environ- interacting to bosonic rise a giving a to of ment, coupled instance strongly any particle almost quantum includes concept polaron alized dsaeeeg,qaiatcersde average residue, quasiparticle energy, nd-state nsseswt hr-ag -hculn,atypical a coupling, e-ph short-range with systems In 2 ete.Tesprodcigcrutbehind circuit superconducting The perties. 13,14 igt eoqaioetm( quasimomentum zero to ding ueeDemler, Eugene dfrosrigtesaptasto from transition sharp the observing for ed quasimomenta o tt iha rirr quasimomentum arbitrary an with state h fS-cree-egrad“breathing- and Su-Schrieffer-Heeger of s 21 n eso o hsnnnltct is nonanalyticity this how show we an, nso ohteeeto-adphonon and electron- the both on ends dmcoaersntr.B applying By . microwave nd 16 n rpeebsdnanostructures graphene-based and , oclsa-antrssieoxides colossal-magnetoresistive to yvrigtecrutprmtr – parameters circuit the varying by – bdfrsuyn unu dynamics quantum studying for tbed neteeto-hnninteractions. electron-phonon endent oercnl,plrncbhvo a been has behavior polaronic recently, More . antd hre hntetypical the than shorter magnitude iyo h elc-öe theorem Gerlach-Löwen the of dity r 2 15 egae Serbia Belgrade, 11158 12, Trg n-iesoa oe featuring model one-dimensional a d aiona933 USA 95343, California ed, 12 ohsai n yaia proper- dynamical and static both , 1 K n i Tian Lin and gs 18 USA 2138, and − 18 K hsmdldescribes model This . gs K 17 pling: Through . gs eie,tegener- the Besides, . ‡ 0 = 3 20 to ) 19 π abnnan- carbon , nimpor- An . 15 eeto sys- -electron oundoped to 22 Such . 2 strongly momentum-dependent couplings, with vertex formation dynamics46,47. The question as to how rapidly functions that depend both on the electron and phonon upon creation (injection) of a single electron (hole), i.e., quasimomenta, are also relevant for fundamental rea- e-ph interaction quench, the cloud of virtual phonons sons. Namely, they do not belong to the realm of valid- around it forms and results in a “dressed” polaronic quasi- ity of the Gerlach-Löwen theorem23, which asserts that particle is poorly understood at present. This dynamical (single-particle) e-ph models generically show smooth de- process, which is expected to be particularly complex in pendence of the ground-state energy on the coupling systems with strongly momentum-dependent e-ph inter- strength. While this theorem was long believed to be actions, has quite recently attracted considerable atten- of quite general validity, it applies only to momentum- tion48,49. independent (Holstein-like) couplings and those that do The remainder of this paper is organized as follows. depend on the phonon- but not on the electron quasimo- The layout of the simulator circuit is presented in Sec. II, mentum. The latter are exemplified by the “breathing- together with the derivation of the effective Hamilto- mode” (BM) coupling24, relevant in the cuprates. nian and discussion of the relevant parameter regime. In The field of SC qubits25 was revolutionized by the Sec. III, we first discuss the character of the momentum development of circuit quantum electrodynamics (cir- dependence of the simulated e-ph coupling, then some cuit QED)26–28, which allowed both fast quantum-gate technical aspects related to exact numerical diagonal- realizations29,30 and demonstrations of many quantum- ization, and finally, the results obtained for the small- optics effects. Quite recently, circuit QED has come polaron . Special emphasis is placed on the to the fore as a versatile platform for on-chip analog occurrence of a sharp transition and the ensuing nonana- quantum simulation31,32. Spurred by some recent ad- lyticities in relevant single-particle properties (ground- vances in the realm of SC quantum devices, especially the state energy, quasiparticle residue, average number of striking increase in achievable times of trans- phonons). Section IV starts with a brief description of mon qubits (from 1 µs to nearly 100 µs)33,34, theoreti- our envisioned state-preparation protocol, followed by a cal proposals have already been set forth for simulating discussion of the experimental-detection and robustness Bose-Hubbard-type models35–37, coupled-spin-38–40 and aspects of the simulator. In Sec. V we lay out the scheme spin-boson models41,42, topological states of matter43, for extracting the relevant retarded Green’s functions and and gauge fields44. Along similar lines, in this paper the spectral function using the many-body Ramsey inter- we propose an SC-circuit emulation of a one-dimensional ference protocol, and explain how our setup can be used model featuring momentum-dependent e-ph couplings of for studying the dynamics of polaron formation. We sum- SSH and BM types. This analog simulator entails SC marize and conclude in Sec. VI. transmon qubits and microwave resonators, the standard building blocks of circuit-QED systems45. The role of qubits in our system is to emulate spinless-fermion ex- II. ANALOG SIMULATOR AND ITS citations, where the pseudospin-1/2 operators represent- EFFECTIVE HAMILTONIAN ing the qubits are mapped to fermionic ones through the Jordan-Wigner transformation. At the same time, the A. Circuit layout and underlying Hamiltonian modes (microwave photons) play the role of dispersionless (Einstein) phonons. A schematic of the SC circuit behind the simulator We show that the suggested setup allows realization of is shown in Fig. 1. Each building block of the simula- strong e-ph coupling regime, characterized by a small- tor consists of a transmon qubit denoted as Qn and a polaron formation. Furthermore, it enables the ob- SC resonator denoted as Rn. The transmon qubit em- servation of a sharp transition from a nondegenerate ulates fermionic excitations, while the microwave pho- single-particle ground state at zero quasimomentum to a ton modes of the SC resonator play the role of Einstein twofold-degenerate one corresponding to a pair of equal phonons. Adjacent qubits couple to each other via a con- and opposite (nonzero) quasimomenta. To demonstrate necting circuit denoted as Bn. Contrary to the more the feasibility of our simulation scheme, we show that familiar circuit-QED setup, the resonators in our cir- – by applying appropriate pump pulses on the qubits – cuit do not couple directly to the qubits. Instead, the the relevant small-polaron states can be prepared within magnetic flux of the resonator modes couples inductively time scales several orders of magnitude shorter than the to the connecting circuit Bn, which induces a nearest- relevant qubit decoherence times. neighbor qubit-qubit coupling whose strength depends The suggested simulator is particularly suitable for a on the quantum dynamics of the resonator modes. detailed characterization of small-polaron ground states The noninteracting Hamiltonian of the n-th repeating (or even excited states), by extracting their phonon unit of the simulator, containing qubit Qn and resonator content through direct counting of photons on the res- Rn, can be written as onators. This unique tool, which is not at our disposal in s † Ez z traditional solid-state systems, also opens up the possibil- H = ~ωca an + σ , (1) n n 2 n ity to address experimentally the nonequilibrium aspects of polaron physics, i.e., the complex problem of polaron- where Ez is the energy splitting of the qubit, and ωc the 3

Rn Rn+1 Rn+2 to that of θn,n, depending on the geometry of the qubit- resonator coupling. Here we choose θn,n+1 = θn,n δθ. In terms of the resonator parameters, δθ is given by≡51 2e A ~ω δθ = eff c , (5) ~ d c C 0 r 0 where Aeff is an effective coupling area, C0 the capaci- Bn Bn+1 tance of the resonator, and d0 the effective spacing in the Qn Qn Qn+2 +1 resonator. In writing Eqs. (4) and (5), we assumed that the magnitudes of fluxes from all resonators are equal t and that the signs of the fluxes from adjacent resonators φn b are opposite to each other. The flux φn in the bottom b loop includes an external ac driving and a tunable dc flux φn φb0: Qn Bn π φb = cos(ω t)+ φ . (6) n − 2 0 b0 Figure 1: (Color online) Schematic of the simulator circuit Note that the ac part of φb has the same frequency, but containing transmon qubits Qn, SC resonators Rn, and con- n necting circuits Bn with three Josephson junctions. External opposite sign of its amplitude compared to the ac part of b t t magnetic fluxes φn and φn are threading the bottom- and top φn. loops of each connecting circuit, respectively. Note that the The ac magnetic fluxes in this simulator can be realized circuit elements are not drawn to scale. by fabricating control wires that couple both to the top- and bottom loops, and applying a microwave pump to the wires. At the same time, the dc magnetic flux in † frequency of the resonator mode; an (an) is the annihi- the bottom loops can be implemented by applying a dc lation (creation) for the modes of the resonator current to a separate control wire designed near those z Rn, while the Pauli matrix σn represents the qubit Qn. loops. Such an implementation should ensure that no The connecting circuit Bn consists of three Joseph- significant dc bias couples to the top loops. It should son junctions and can be viewed as a generalized SQUID also be emphasized that the only tunable parameters in loop. Let ϕn be the gauge-invariant phase variable of the this circuit are the dc bias φb0 and the frequency of the i SC island of the n-th qubit, and ϕn (i = 1, 2, 3) the re- ac drive ω0. spective phase drops on the three junctions in the circuit By generalizing the standard expression for the effec- 50 Bn. Based on flux-quantization rules , we then have tive Josephson energy of a SQUID loop (with two junc- tions, threaded by a magnetic flux)52, the total Josephson φt 1 n energy of the connecting circuit Bn can be written as ϕn = ϕn ϕn+1 + , − 2 3 t 2 φn J i i ϕ = ϕ ϕ , (2) Hn = EJ cos ϕn , (7) n n n+1 − − − 2 i=1 t X 3 φn b i ϕn = ϕn ϕn+1 φn , where EJ are the Josephson energies of the three junc- − − 2 − 1 2 tions. We will hereafter assume that EJ = EJ EJ and 3 ≡ t b E = EJb. where φn and φn are the respective magnetic fluxes in J the top- and bottom loops of Bn; both are expressed in units of Φ /2π, where Φ hc/(2e) is the flux quantum. 0 0 ≡ B. Effective Hamiltonian in the rotating frame of The resonator modes an and an+1 couple inductively the drive to the top loop of Bn (see Fig. 1). The total magnetic flux in the top loop is given by Given the explicit time dependence originating from t the driving terms, we resort to studying this system in φn = π cos(ω0t)+ φn,res , (3) the rotating frame of the drive, i.e., adopt the interaction where the first term is the flux from an external ac drive picture defined by a reference Hamiltonian with amplitude π and frequency ω0, while the second one ~ † H0 = ω0 anan . (8) † † n φn,res = θn,n+1(an+1 + an+1) θn,n(an + an) (4) X − The system dynamics in this rotating frame are described is the flux from the resonator modes. The coupling con- by the Hamiltonian stants θn,n′ quantify the contribution of the resonator i i modes an′ to the connecting circuit Bn. The sign of H = e ~ H0t (Hs + HJ ) H e− ~ H0t , (9) I n n − 0 θn,n+1 can be designed to be either the same or opposite " n # X 4 which can be recast as the (spinless-fermion) excitation hopping term

E i i † z z ~ H0t J − ~ H0t † HI = ~δωa an + σ + e H e , He = t (c c + h.c.) , (17) n 2 n n − 0 n n+1 n   n n X X (10) X 2 with δω ωc ω0. The explicit form of the Josephson- with t0 2δφ tr being the effective bare hopping energy, ≡ − J ≡ 0 coupling term Hn in the rotating frame is derived in de- and the excitation-phonon coupling term He-ph whose ex- tail in Appendix A. Its time-independent part is given plicit form will be specified shortly. Note that, strictly by speaking, He also contains the diagonal (on-site energy) † z terms cncn for spinless fermions, originating from the σn J 1 H¯ = 2 t E J (π/2)φ res cos(ϕ ϕ ) , terms in Eqs. (1) and (13). Yet, in consistency with the n − r − 2 J 1 n, n − n+1   usual practice in studying coupled e-ph models, we here- (11) after disregard them as they only represent a constant where Jn(x) are Bessel functions of the first kind, and band offset for our itinerant fermionic excitations. The coupling Hamiltonian H consists of two con- t = E J (π/2)(1 + cos φ ) (12) e−ph r J 0 b0 tributions: the SSH term is determined by the Josephson energy of the bottom ~ † junction, which, for convenience, is chosen as EJb = HSSH = g δω (cncn+1 + h.c.) 2E J (π/2). Along with the first two terms in Eq. (10), n J 0 X the time-independent part H¯ J of the Josephson- † † n n an+1 + a an + a , (18) coupling term forms the effective Hamiltonian of the sys- × n+1 − n tem in the rotating frame. Namely,P the remaining (time- h  i dependent) part of this transformed Josephson-coupling and the BM term term can be neglected due to its rapidly-oscillating char- acter (for details, see Appendix A), in accordance with H = g~δω c† c BM − n n the rotating-wave approximation (RWA). n X By expanding cos(ϕn ϕn+1) to second order in the † † − an+1 + an+1 an−1 + an−1 , (19) phase variables and defining pseudospin-1/2 operators σn × − that correspond to the lowest two eigenstates of the trans- h   i mon, we find that where the dimensionless coupling strength g is defined by the relation 2 2 cos(ϕn ϕn+1) 2+4δϕ0 2 − − ≈− g~δω = δφ EJ J1(π/2)δθ . (20) σz + σz 0 2δϕ2 σ+σ− + σ−σ+ n n+1 . (13) − 0 n n+1 n n+1 − 2   The SSH term physically accounts for the dynamical dependence of the hopping integral (i.e., the excitation 2 1/2 Here δϕ (2EC1/EJ1) is the quantum displace- † 0 ≡ bandwidth) on the phonon displacements un an + an, ment of the gauge-invariant phase (EC1 and EJ1 are to first order in these displacements; it is nonlocal∝ in that the charging- and Josephson energies of an individual the hopping integral between sites n and n+1 depends on transmon, respectively); note that for a typical transmon the displacements on both sites19. The BM term, on the 2 (EJ1/EC1 100) we have δϕ0 0.15. The full expres- other hand, also describes a nonlocal e-ph interaction; ¯ J∼ ≈ sion for Hn (not shown here) can easily be obtained by it accounts for the antisymmetric coupling of the excita- combining Eqs. (11) and (13). tion density at site n with the phonon displacements on We now switch to the spinless-fermion representation the neighboring sites n 1 and n +1. Being an exam- of the pseudospin-1/2 operators via the Jordan-Wigner ple of a “density-displacement”− type coupling, it can be transformation viewed as a nonlocal generalization of the Holstein-type e-ph interaction. 1+ σz 2c† c , σ+σ− + σ−σ+ c† c + h.c. . n n n n n+1 n n+1 n n+1 It is worthwhile to mention that an example of a → → (14) real electronic system where both e-ph coupling mech- The effective Hamiltonian of the system in the rotating anisms discussed here play important roles is furnished frame by the cuprates. In those materials, both mechanisms involve the planar Cu-O bond-stretching phonon modes, Heff = Hph + He + He-ph (15) also known as breathing modes. These bond-stretching includes the free-phonon term with the effective phonon modes couple to electrons both via a direct modula- 53 frequency δω, tion of the hopping integral (SSH-type coupling) , and through electrostatic changes in the Madelung energies H = ~δω a† a , (16) that originate from displacements of orbitals (BM-type ph n n 54 n coupling) . X 5

† 6 † quasimomentum operator Ktot = k kckck + q qaqaq,

5 where ck and aq are the excitation- and phonon an- ±!=2¼ = 200 MHz P P

±!=2¼ = 300 MHz 4 nihilation operators in momentum space. For conve- ± !

~ nience, we express quasimomenta in units of the in- 3

= −1/2 ikn

0 verse lattice period, so that ck = N n e cn and 2 t t −1/2 iqn aq = N n e an. The eigenvalues of the operator (a) 1 P

Ktot will hereafter be denoted as K. In particular, Kgs

0 will stand forP the quasimomentum corresponding to the ground state of the system (i.e., the band minimum of

(b) 3 the effective, dressed-excitation Bloch band).

2

¸¸ A. Momentum dependence of the resulting e-ph 1 coupling

0

0.95 0.96 0.97 0.98 0.99 Before embarking on a calculation of the ground-state of the system, it is worthwhile to analyze the momentum Á =¼ b0 dependence of the vertex function corresponding to the resulting e-ph coupling term: Figure 2: Dimensionless parameters for the simulator: (a) Ratio of the hopping integral and the phonon energy, and (b) −1/2 † † the effective coupling strength λ, both shown as a function of He−ph = N γ(k, q) ck+qck(a−q + aq) . (22) 2 = 200 300 φb0/π. For the solid and dashed lines, δω/ π and Xk,q MHz, respectively. This vertex function

C. Relevant parameter range γ(k, q)=2ig~δω sin k + sin q sin(k + q) (23) − h i A suitable set of parameters for the SC resonators consists of the k- and q-dependent SSH part γSSH(k, q)= 2 ~ is given by d0 = 25 µm, Aeff = 100 µm , C0 = 2 fF, 2ig δω [sin k sin(k + q)], and the BM part γBM(q) = ~ − ωc/2π = 24 GHz, while the effective phonon frequency 2ig δω sin q that only features q dependence. Since the can be δω/2π = 200 MHz or 300 MHz. In addition, we overall vertex function γ(k, q) depends on both k and choose EJ for the Josephson junctions in the connecting q, it does not belong to the domain of applicability of 2 ~ the Gerlach-Löwen theorem23, which rules out a nonan- circuits such that 2δφ0EJ /2π = 200 GHz. From this choice of parameter values, it follows that δθ =3.5 10−3 alytic behavior of the single-particle quantities. While and gδω/2π = 198 MHz. × k- and q-dependent couplings do not necessarilly lead 55 The hopping integral t0 can be adjusted in-situ by to nonanalyticities , such nonanalyticity indeed occurs varying the dc magnetic flux φb0 between the bottom in a model with the pure SSH coupling [vertex function 56 two junctions. This is illustrated in Fig. 2(a), where γSSH(k, q)] . In the following (see Sec. III C), we show the ratio t0/~δω is shown for φb0/π between 0.95 and that the model with combined SSH and BM couplings 0.99. For the chosen values of φb0/π we are mainly in the studied here displays a similar sharp transition between adiabatic regime (t0 > ~δω), entering the antiadiabatic a quasifree excitation and a small polaron. regime (t < ~δω) for φ /π 0.98. For the same range 0 b0 ≈ of values for φb0, the effective coupling strength B. Details of exact diagonalization ~δω λ =2g2 (21) t0 To determine the ground-state properties of our re- sulting coupled e-ph model, we employ the conventional varies from the weak-coupling (λ < 1) to the strong- 57 coupling (λ > 1) regime, as can be inferred from Lanczos diagonalization in combination with a con- Fig. 2(b). For instance, with δω/2π = 200 MHz and trolled truncation of the phonon Hilbert space. The Hilbert space of the system is spanned by states φb0/π = 0.95, we have λ = 0.34; for the same value of † given as direct products n m ph. Here, n = c 0 δω, φb0/2π =0.98 yields λ =2.1. e e n e is the state of the excitation| i ⊗| localizedi at| thei site| ni , m = (m1,...,mN ) are the phonon occupation numbers, † m N mi III. SMALL-POLARON SIGNATURES AND and ph = i=1(1/√mi!)(bi ) 0 ph. Restricting our- SHARP TRANSITION selves| toi a truncated phonon Hilbert| i space that includes states with atQ most M phonons (total number on a lat- The eigenstates of our effective single-particle Hamil- tice with N sites), we take into account all m-phonon N tonian Heff are the joint eigenstates of Heff and the total states with 0 m m, where m = m M. The ≤ i ≤ i=1 i ≤ P 6

dimension of the total Hilbert space is D = De Dph, 0 × where De = N and Dph = (M + N)!/(M!N!). To further reduce the dimension of the Hamiltonian ±!=2¼ = 200 MHz

¡2t =E matrix to be diagonalized, we exploit the discrete trans- ±!=2¼ = 300 MHz 0 -5 lational invariance of our finite system, mathematically expressed as the commutation [Heff,Ktot] = 0 of the E E = Hamiltonian Heff and the total quasimomentum operator =

-10 gs gs Ktot. This allows us to perform diagonalization of Heff E E in sectors corresponding to the eigensubspaces of Ktot. For that to accomplish, we make use of the symmetrized basis -15

N −1/2 iKn K, m = N e Tn( 1 e m ph) , (24)

| i | i ⊗ | i -20 n=1 X 0.95 0.96 0.97 0.98 0.99

where Tn denotes (discrete) translation operators. Thus, Á =¼ b0 the dimension of each K-sector of the total Hilbert space is DK = Dph. Following an established phonon Hilbert- Figure 3: Ground-state energy, expressed in units of E ≡ 58 −3 2 space truncation procedure , the maximum number of 10 δφ0 EJ = 2π~ × 100 MHz, as a function of the phonons retained (M) is increased until the convergence experimentally-tunable parameter φb0. The solid curve core- (M) 2 = 200 for the ground-state energy Egs and the phonon distri- sponds to the effective phonon frequency δω/ π MHz, 2 = 300 bution is reached. while the dashed curve corresponds to δω/ π MHz. While for the momentum-independent (completely lo- cal in real space) Holstein coupling the system size is essentially inconsequential, for nonlocal couplings of the zero quasimomentum (Kgs =0), to a twofold-degenerate kind studied here this is not the case. Therefore, the sys- one corresponding to equal and opposite (nonzero) quasi- tem size has to be chosen in such a way as to ensure that momenta Kgs and Kgs. For δω/2π = 200 MHz this − further increase of the system size does not lead to appre- critical value is (φb0)c 0.968 π, while for δω/2π = 300 ≈ ciable changes of the ground-state energy and other rel- MHz we find (φb0)c 0.972 π. The corresponding criti- ≈ evant quantities. In this particular system this happens cal values of the effective coupling strength are λc 0.83 ≈ to be satisfied for N = 10. For our system with N = 10 and 0.72, respectively. Shown in Fig. 4 are two small- sites (with periodic boundary conditions), the truncation polaron Bloch-band dispersions throughout the Brillouin of the phonon Hilbert space in accordance with the afore- zone (for δω/2π = 200 MHz and two different values of mentioned criterion requires the total of M =8 phonons. φb0), both with band minima (ground states) at nonzero quasimomenta.

C. Results and Discussion

Á =¼ = 0:97 ±!=2¼ = 200 MHz b0

Á =¼ = 0:98

b0 We now discuss the results obtained by exact diago- -9.1 nalization of our effective model. Unlike more conven- tional situation, in which the effective coupling strength -9.5 E λ [cf. Eq. (21)] is changed by varying the dimension- E = less coupling constant g (for fixed ratio of the relevant = K 8 K -9.2 E hopping integral and the phonon energy) , here we work E with a fixed value of gδω [recall Eq. (20)]. We effectively

-9.6 change λ by varying the bare hopping integral t0 through the experimentally-tunable parameter φb0, in accordance with Eq. (12).

Our main finding is that at a critical value of φb -9.3 (i.e., of the effective coupling strength λ), there is a sharp transition (nonanalyticity) of all relevant quan- -1.0 -0.5 0.0 0.5 1.0 56 tities . This sharp transition is illustrated in Fig. 3, K=¼ where the ground-state energy (expressed in units of 10−3δφ2 E =2π~ 100 MHz) is shown as a function Figure 4: Small-polaron Bloch dispersions for two different E ≡ 0 J × of φb0. The transition originates from a (real) level cross- values of the tunable parameter φb0, shown throughout the ing and is of first order. The sharp transition physically Brillouin zone. The solid curve coresponds to the y-axis scale marked on the left, while the dash-dotted curve corresponds corresponds to a change – at a critical value of φb0 – from a non-degenerate ground state that corresponds to the to the scale on the right, as indicated by the arrows. 7

1.0 regime (quasifree excitation), its reduced values in the

±!=2¼ = 200 MHz strong-coupling regime [see Fig. 5(a)] signify the pres-

±!=2¼ = 300 MHz ence of small polarons, with these two regimes being sep- s s arated by a nonanalyticity at the critical value of φ . g g b0

0.5

Z Z It is interesting to note that, unlike for the Holstein model where Zgs 0 for strong enough coupling, here

(a) ≈ Zgs may saturate at a finite value. As can be inferred

0.0 from Fig. 5(a), for δω/2π = 300 MHz we find such satu-

4 ration at Zgs 0.15. ≈

(b)

3 Another relevant quantity is the average number of phonons in the ground state ph

2

¹ N

1 N ¯ † Nph ψgs anan ψgs . (25)

0 ≡ h | | i n=1

0.95 0.96 0.97 0.98 0.99 X

Á =¼

b0 The change of this quantity from values close to zero [see Figure 5: Characterization of the sharp transiton between a Fig. 5(b)] to a nonzero value N¯ph & 3 marks the transi- quasifree excitation and a small polaron: (a) Ground-state tion from a quasifree excitation to a small polaron. The quasiparticle residue Zgs ≡ Zk=K , and (b) average number ¯ gs aforementioned effective compensation of the SSH and of phonons Nph, versus the experimentally-tunable parameter 2 = 200 BM couplings below (φb0)c is reflected in the vanishing φb0. The solid and dashed curves corespond to δω/ π phonon-dressing of fermionic excitations, the flat parts of MHz and δω/2π = 300 MHz, respectively. the curves in Fig. 5(b).

The average phonon number N¯ph is amenable to a di- For sufficiently strong coupling – e.g., φb0 & 0.98 for rect measurement in our system, through measurements δω/2π = 200 MHz – the quasimomentum Kgs corre- of photon numbers on different resonators (see Sec. IV B). sponding to the single-particle ground state saturates at Likewise, the second moment of the effective phonon dis- around π/2 (see the dash-dotted curve in Fig. 4). It tribution can also be extracted by measuring the pho- should be stressed that – while the ground-state under- ton squeezing in the resonators10,59. More generally, goes a sharp transition – the quasimomentum Kgs itself the complex multiphononic nature of small-polaron ex- varies smoothly between Kgs = 0 and this saturation citations can be fully captured by computing the entire value as φb0 is increased beyond its critical value. phonon distribution, which is depicted in Fig. 6 for both Apart from the occurrence of a sharp transition, an- values of the effective phonon frequency δω used above. other interesting aspect of our findings is an effective While for δω/2π = 200 MHz the distribution has a broad “compensation” of SSH and BM couplings below the crit- maximum at Nph 4, the one for δω/2π = 300 MHz – corresponding to a≈ smaller effective coupling strength λ ical value of φb0. This is indicated in Fig. 3, where the ground-state energy curve essentially follows the bare- [cf. Eqs. (20) and (21)] – has a weakly-pronounced max- imum at around Nph 1. By contrast, the dotted curve dispersion curve E = 2t0(φb0) all the way up to the crit- − in Fig. 6, representing≈ a typical phonon distribution for ical value of φb0. This peculiar effect can be ascribed to the character of the resulting momentum dependence of couplings below the critical one, is very strongly peaked the e-ph vertex function in Eq. (23), being a consequence at Nph =0. of the fact that here the SSH and BM coupling strengths 19 While phonon distributions peaked at zero phonons are are the same . This phenomenon could have profound characteristic of all small-polaron models in the weak- consequences for transport properties of real electronic coupling regime (e.g., below the onset of the polaron systems with competing SSH and BM couplings. crossover in the Holstein model), the peculiarity of our The central quantity for characterizing the small- findings is that here such distribution persists all the polaron regime is the quasiparticle residue Zk way up to the critical coupling strength. Although the 2 ≡ Ψk ψk , i.e., the module squared of the overlap be- sharp transitions of the type found here do not result |h | i| † tween the bare-excitation Bloch state Ψk ck 0 from any kind of cooperative behavior (as is typical for and the (dressed) Bloch state ψ of the| coupledi ≡ e-ph| i quantum phase transitions in many-particle systems), the | ki system that corresponds to the same quasimomentum observed peculiar behavior allows us to treat N¯ph as an (K = k). In particular, having determined the ground- effective “order parameter” for the predicted sharp tran- state ψgs ψ we can compute sition. Given that N¯ph is a directly measurable quantity | i ≡ | K=Kgs i Zgs Z , a quantity characterizing the ground state in our system, this fact will facilitate experimental veri- ≡ k=Kgs of the system. While Zgs 1 indicates the weak-coupling fication of the existence of this transition. ≈ 8

1.0 Assuming that β(t)=2βp cos(ωK t), where ~ωK is the energy difference between the states G0 and ψK , in the

Á =¼ = 0:98; ±!=2¼ = 200 MHz b0 RWA the two states are Rabi-coupled| withi the| effectivei

0.8 Á =¼ = 0:98; ±!=2¼ = 300 MHz b0 Rabi frequency βp√ZK . Thus, starting from the vacuum

Á =¼ = 0:95 b0 state G0 , the state ψK will be prepared within a time | i 10| i ) ) interval of duration 0.6 ph ph π~ N N τprep = . (28) ( ( 2βp√ZK 0.4 P P The form of the last expression is consistent with the ex- pectation that the more strongly-dressed states (smaller

0.2 ZK) require longer preparation times. For the small- polaron ground states with K = Kgs these preparation times are much shorter than the± decoherence time T . 0.0 2

0 1 2 3 4 5 6 7 8 For instance, assuming that the pumping amplitude is βp/(2π~)=40 MHz and taking the values for Zgs at the N ph onset of the small-polaron regime [see Fig. 5(a)], we re- spectively find that τprep 24 ns for δω/2π = 200 MHz, Figure 6: Illustration of the multiphononic nature of small- ≈ and τprep 17 ns for δω/2π = 300 MHz. The ob- polaron excitations: Full ground-state phonon distribution tained state-preparation≈ times are three orders of magni- at φb0/π = 0.98, for δω/2π = 200 MHz (squares) and 2 = 300 tude shorter than currently achievable decoherence times δω/ π MHz (circles). The dotted curve coresponds 33,34 = 0 95 T2 20 100 µs of transmon qubits , this being a to the phonon distribution for φb0/π . , i.e., below the ∼ − critical coupling. strong indication of the feasibility of our proposed pro- tocol. The above Rabi-coupling state-preparation protocol, IV. STATE PREPARATION, DETECTION, AND which ensures energy and momentum conservation10, can ROBUSTNESS in principle be adapted to other systems by taking into account their underlying symmetries. Importantly, the A. State-preparation protocol ultimate sucess of this scheme in different systems will depend on the character of their underlying absorption The feasibility of our simulation scheme is contingent spectra, as described by the corresponding spectral func- upon the ability to prepare the desired small-polaron tions (see Sec. V C). As far as our system is concerned, Bloch states. For this purpose, we make use of the it should be stressed that the absorption spectra of elec- state-preparation protocol proposed in our previous work tronic systems coupled with optical phonons (i.e., phonon (Ref. 10), which is only briefly explained in the following. modes with a gap in their spectrum) are characterized by We assume that the initial state of the system is the generic spectral functions in which the “coherent” part (with a finite spectral weight) is energetically well sep- vacuum state G0 0 e 0 ph. This state, with no ex- 12,60 citations (i.e.,| withi ≡ all | i qubits⊗ | i in their spin-down states) arated from the incoherent background . This form and all resonators in their respective vacuum states, can of absorption spectra should allow one to avoid inadver- be prepared via thermalization in a low-temperature en- tent population of other states while preparing a desired vironment. Our target state is the dressed-excitation (in small-polaron ground state in our proposed system. the special case, small polaron) Bloch state ψK , corre- sponding to the eigenvalue K of the total quasimomen-| i B. Experimental detection tum operator Ktot. The microwave driving required for preparing this state is envisioned to be of the form Here we discuss the method for measuring the average phonon number N¯ph [cf. Eq. (25)], a quantity which can ~β(t) Ω(q)= σ+e−iqn + σ−eiqn , (26) be thought of as an order parameter for the sharp small- √ n n polaron transition (cf. Sec. IIIC). As already stressed N n X  above, in our implementation N¯ph corresponds to the av- where β(t) describes its time dependence, and the phase erage total photon number on the resonators. Owing to factors e±iqn indicate that spin-flip operations are ap- the discrete translational symmetry of the system, the plied to different qubits with a q-dependent phase differ- measurement of N¯ph can be reduced to the measurement ence. It is easy to show that the transition matrix ele- of the mean photon number of one of the resonators. This ment of the operator Ω(q) between the states G0 and can be accomplished by adding an ancilla qubit which 10 | i 10 ψK is given by couples to this particular resonator , but only during | i the measurement; this qubit is assumed to be far-detuned ψ Ω(q) G = ~ β(t) Z δ . (27) from the resonator modes. By measuring the qubit state, |h K | | 0i| | | K q,K p 9 the mean photon number on this resonator can be ex- hand is given by tracted, which multiplied with the total system size N yields the result for N¯ . R i † ph G (k,t)= θ(t) G [c (t),c ] G , (29) + −~ h 0| k k +| 0i

† C. Robustness of the simulator where ck(t) is a single-particle operator in the Heisenberg representation and [...]+ stands for an anticommutator. † As in every other quantum-computation device, deco- More explicitly, ck(t) = UH (t)ckUH (t), where UH (t) is herence effects should also be present in our simulator. the time-evolution operator corresponding to the lab- Possible excitations in this system correspond either to frame counterpart H = H(t) of our effective Hamilto- flipping of the qubit states or to displacement of the res- nian Heff in the rotating frame. The explicit forms of onator modes, which are subject to the decoherence of the H and UH are not relevant for our present purposes. In qubits and resonators. In addition to very long dephas- fact, the only relevant property of the Hamiltonian H is ing times (T2 20 100 µs) of transmon qubits achieved that its symmetries in the spinless-fermion (pseudospin) in recent years,∼ for− resonators the sector of the problem are the same as those of the Hamil- damping time of the microwave photons can reach the tonian Heff, since this part of Heff preserves its form in same order of magnitude as T2, with a quality factor the . larger than 106. The relevant energy scales in our simu- It should be emphasized that, while the natural lator (effective phonon frequency, e-ph coupling strength, Green’s functions for spinless fermions are those that and hopping energy) are all of the order of 100 MHz, far involve anticommutators [cf. Eq. (29)], in the single- exceeding the decoherence rates. Besides, as shown in particle problem at hand the commutator Green’s func- Sec. IVA, even for very strongly-dressed polaron states tion the typical duration of the state-preparation protocol is three orders of magnitude smaller than the decoherence R i † G−(k,t)= ~ θ(t) G0 [ck(t),ck]− G0 . (30) times. Finally, in the low-temperature environment of − h | | i our system thermal excitations – which here have ener- contains the same physical information. Namely, given gies of a few GHz – can be safely neglected. † that G0 is a vacuum state, we have that ck(t)ck G0 = The pump pulses of the kind described in Sec. IV A | i R | i 0, which implies that in this special case G−(k,t) = may, in principle, induce unwanted transitions (leakage) R 61 G (k,t). to higher energy levels in a transmon qubit . Namely, in − + all qubits based on weakly-anharmonic oscillators leak- Anticipating the use of a real-space experimental probe age from the two-dimensional qubit Hilbert space (com- (see Sec. V B), we further note that the last momentum- putational states) is the leading source of errors at short space Green’s function can be retrieved from the real- pulse times62. This is especially pronounced if the pulse space resolved ones, i.e., bandwidth is comparable to the anharmonicity. However, ′ R −1 ik·(n−n ) R in a typical transmon with EJ1/EC1 50 100 and a G−(k,t)= N e Gnn′ (t) , (31) ∼ − ′ negative anharmonicity of 3 5 %, even pulses with n,n X durations of only a few ns∼ are sufficiently− frequency se- 28 R ~ † ′ lective that the unwanted transitions can be neglected . where Gnn′ (t) (i/ )θ(t) G0 [cn(t),cn ]− G0 . By In our system, there is an off resonance of around 500 switching to the≡ pseudospin- − 1h/2|operators and| i taking MHz for such transitions. For a typical driving ampli- into account that the Jordan-Wigner string operators act tude βp/2π 40 MHz, the probability of leakage is be- trivially on the ground state G0 , these real-space com- low one percent,≈ which is a reasonable error rate for the mutator Green’s function can| bei rewritten as simulator. R i + − G ′ (t)= θ(t) G [σ (t), σ ′ ]− G . (32) nn −~ h 0| n n | 0i V. EXTRACTING CORRELATION FUNCTIONS USING A RAMSEY SEQUENCE: They can further be transformed to the form STUDY OF THE POLARON-FORMATION R xx yy xy yx DYNAMICS G ′ (t)= ′ + ′ i( ′ ′ ) , (33) nn Gnn Gnn − Gnn − Gnn

A. Relevant Green’s functions where

αβ i α β ′ θ(t) G [σ (t), σ ′ ]− G ( α, β = x, y ) Generally speaking, dynamical response functions – Gnn ≡−~ h 0| n n | 0i Fourier transforms of retarded two-time Green’s function (34) – provide a natural framework for characterizing excita- and, for notational simplicity, we suppressed the time tions in many-body systems. The relevant single-particle argument and the superscript R in the notation for these retarded two-time Green’s function in the problem at Green’s functions. 10

B. Many-body Ramsey interference protocol we propose its use on pairs of qubits in a multi-qubit sys- tem, for the purpose of extracting the desired two-time The Ramsey-interference protocol is in principle ap- correlation- and Green’s functions. plicable in any system where single-site addressability is available and yields naturally the real-space and time- resolved commutator Green’s functions of spin opera- C. Spectral function and relation to the dynamics tors63,64. In the problem at hand it involves the pseu- of polaron formation dospin degree of freedom of the transmon qubits. The general Rabi pulses can be parameterized as Provided that the single-particle retarded two-time Green’s function [cf. Eq. (29)] is extracted as explained θ x y θ in Secs. V A and V B, we can also obtain the infor- Rn(θ, φ) 12×2 cos +i(σ cos φ σ sin φ) sin , (35) ≡ 2 n − n 2 mation about the spectral properties of the system. Namely, by Fourier-transforming this Green’s function where θ = Ωτ, with Ω being the Rabi frequency and to momentum-frequency space, the spectral function is τ the pulse duration; φ is the phase of the laser field. obtained as A(k,ω)= (1/π)Im GR (k,ω). The spectral The Ramsey protocol makes use of the special case + function can quite generally− be represented in the form Rn(φ) Rn(θ = π/2, φ) of such pulses, with θ = π/2 and arbitrary≡ φ. A(k,ω)= ψ(j) c† 0 2δ ω E(j)/~ , (38) Quite generally, the Ramsey-interference protocol en- |h k | k| i| − k j tails the following steps: (1) perform local π/2-rotation X   at site n (with φ = φ1); (2) evolve the system during time (j) ′ where ψ is a complete set of total-quasimomentum t; (3) perform local π/2-rotation at site n , or global π/2- | k i rotation (with φ = φ2); (4) measure the system in the σz k eigenstates of the total Hamiltonian of the coupled e- ′ (j) basis at site n . In our system, the described protocol ph system (in our case Heff), and Ek the corresponding leads to the measurement result given by the expecta- eigenvalues. In the simplest case, the above sum over j (j=0) tion value includes the polaron-ground state ψk at quasimo- mentum k and its attendant continuum| ofi states that M ′ (φ , φ ,t)= correspond to the polaron with quasimomentum k q nn 1 2 − † † † z and an unbound phonon with quasimomentum q. More G R (φ )U (t)R ′ (φ )σ ′ R ′ (φ )U (t)R (φ ) G . 0 n 1 H n 2 n n 2 H n 1 0 generally, with increasing coupling strength there will be h | | (36)i multiple coherent polaron bands below the one-phonon continuum (threshold for inelastic scattering) which sets The procedure for extracting relevant Green’s function in at energy E +~ω 60, where ω is the relevant phonon can be simplified by exploiting the symmetries of our gs 0 0 frequency (in our case δω). All these coherent (split- system in the pseudospin sector. Since its Hamiltonian off from the continuum) polaron states contribute to the involves a sum of an XY -coupling term and σz terms n above sum along with their respective continua. [recall Eq. (13)], our system has a U(1) symmetry under xx yy The spectral function is intimately related to the dy- z-axis pseudospin rotations, implying that nn′ = nn′ xy yx G G namics of polaron formation. Let us assume that at t =0 and nn′ + nn′ =0. Another symmetry of our system is G G x a bare-excitation Bloch state with quasimomentum k is that under reflections with respect to the z axis (σn x y y z z → prepared and e-ph coupling is turned on (e-ph interac- σn, σn σn, σn σn), which implies that any − → − → x tion quench); in our system, this can be achieved using expectation value involving an odd (total) number of σn y the state-preparation protocol described in Sec. IVA be- and σn operators is equal to zero. For a system with these two symmetries, the Ramsey protocol measures64 fore switching on the qubit-resonator coupling at t = 0. Then the state of the system at t = 0 is ψ(0) = c† 0 | i k| i 1 and the state at a later time t is given by ψ(t) = xx yy (j) ′ ′ ′ i (j) (j) † | i Mnn (φ1, φ2,t)= sin(φ1 φ2)( nn + nn ) − ~ Ek t −4 − G G j e ψk ψk ck 0 . It is straightforward to ver- xy yx ify that by Fourier-transforming| ih | | i the spectral function to cos( φ φ )( ′ ′ ) . (37) − 1 − 2 Gnn − Gnn P † the time domain we obtain the amplitude ψ(t) ck 0 xx yy xy yx h | | i Thus the combinations nn′ + nn′ and nn′ nn′ needed to remain in the initial state of the system at time R G G G −G 47 to recover Gnn′ (t) [recall Eq. (33)] can be obtained by t . This quantity, and the corresponding probability choosing the angles φ , φ such that φ φ = π/2 and P (t) = ψ(t) c† 0 2 are clearly relevant for describing 1 2 1 − 2 ± |h | k| i| φ1 = φ2, respectively. the dynamics of polaron formation. In the realm of SC qubits, the Ramsey-interference Polaron formation is admittedly a very complex dy- protocol is conventionally used to determine the deco- namical process even in the case of the momentum- 46,47 herence time T2 of a single qubit, a procedure known as independent e-ph coupling . Even the very funda- the Ramsey-fringe experiment34. The use of this protocol mental question related to the time it takes to form a is also envisioned for other types of manipulation, such polaron, as well as a more detailed understanding of as interaction-free measurements65. In the present work, how phonon excitations evolve into the correlated phonon 11 cloud of the polaron quasiparticle, are not fully answered pects of polaron physics, i.e., the dynamics of small- to date. In the presence of strongly-momentum depen- polaron formation. Experimental studies of such phe- dent e-ph interactions, which even allow for the occur- nomena in traditional solid-state systems are hampered rence of sharp transitions, it should be even more diffi- by the very short dynamical time scales, in addition to cult to arrive at a full understanding of this process. Our a very limited control over these systems. Our suggested system with its unique set of experimental tools – the setup therefore holds promise to become an invaluable ability to measure the average phonon number through experimental tool for future studies in this direction. measuring the photon number on different resonators, as well as the possibility of extracting the spectral function via a Ramsey protocol – paves the way for a detailed Acknowledgments and controlled experimental study of this important phe- nomenon. Useful discussions with J. Koch and R. J. Schoelkopf are gratefully acknowledged. V.M.S. was supported by VI. SUMMARY AND CONCLUSIONS the SNSF. L.T. was supported by NSF-DMR-0956064 and NSF-CCF-0916303. M.V. was supported by the Ser- bian Ministry of Science, project No. 171027. E.D. ac- To summarize, we have proposed a superconducting knowledges support from Harvard-MIT CUA, the ARO- analog for a model with nonlocal MURI on Atomtronics, and ARO MURI Quism program. electron-phonon couplings of Su-Schrieffer-Heeger and This research was supported in part by NSF PHY11- breathing-mode types. The simulator is based on an ar- 25915 through a KITP program (V.M.S. and L.T.). ray of transmon qubits and microwave resonators. Our setup allows one to reach the strong-coupling regime, characterized by the small-polaron formation, with quite realistic values of circuit parameters. Appendix A: Derivation of the Josephson-coupling The most interesting feature of the simulated model is term in the rotating frame the occurrence of a sharp (first-order) transition at a crit- ical coupling strength. This transition results from a real In the following, we present derivation of the effective level crossing and physically corresponds to the change Josephson coupling terms in the rotating frame. We will iH0t/~ −iH0t/~ −iω0t from a nondegenerate single-particle ground state at zero make use of the fact that e ane = ane , quasimomentum (Kgs = 0) to a twofold-degenerate one and that, consequently, at nonzero quasimomenta Kgs and Kgs. Aside from − the fact that our suggested setup provides a tunable ex- i H t − i H t e ~ 0 φ e ~ 0 perimentally platform for observing the sharp transition, n,res −iω0t † iω0t −iω0t † iω0t what further motivated the present work is the circum- = δθ(an+1e + a e ane a e ) , n+1 − − n stance that e-ph coupling in real materials is insufficiently (A1) strong for observing any measurable consequence of this type of transition20. and rely on the smallness of δθ and the RWA. For One of the obvious advantages of SC Josephson- an arbitrary unitary transformation with a generator S junction based systems compared to other quantum- (S† = S) applied to an analytic operator function f(A) simulation platforms (trapped ions, polar molecules, Ry- it holds− that eS f(A) e−S = f(eS Ae−S). Therefore, dberg atoms) is that they allow realization of strictly 10 nearest-neighbor hopping , thus making it possible to i i e ~ H0t cos(φ ) e− ~ H0t = cos δθ(a e−iω0t simulate relevant polaron models in a quite realistic fash- n,res n+1 † iω0t −iω0t † iω0t ion. In trapped-ion and polar-molecule systems, for in- + an+1e ane ane ) , (A2) stance, the presence of non-nearest neighbor hopping − −  is unavoidable, originating from the presence of long- with an analogous relation for sin(φn,res). range interactions between their elementary constituents We start from an expression for the Josephson coupling (Coulomb interaction between ions and dipolar interac- obtained by inserting Eqs. (2) into Eq. (7). By immedi- tion between polar molecules); these systems show simi- ately dropping the terms that will be rapidly rotating lar limitations when it comes to mimicking the behavior upon transformation to the rotating frame, the remain- 5,7,8 of dispersionless (zero-dimensional) phonons . In ad- ing Josephson coupling reads dition to these intrinsic advantages of SC systems com- pared to other available experimental platforms, a unique φt aspect of our suggested setup is the capability of the in- HJ 2E cos n + E cos φ n ≈− J 2 Jb b0 situ altering of the hopping energy through an externally-   tunable magnetic flux. cos(ϕ ϕ )+ (δθ2) . (A3) × n − n+1 O Once experimentally realized, our suggested setup t could also be used for studying the nonequilibrium as- With the aid of Eq. (3), the term cos(φn/2) can be writ- 12 ten as We now have to analyze which terms in the last two expansions will remain after the transformation to the t φ π cos(ω0t) φn,res rotating frame. The first (time-independent) term in cos n = sin 2 − 2 2 Eq. (A5) will clearly be unaffected by this transforma-   tion, while the remaining terms rotate at frequency 2ω0 π cos(ω0t) + cos + (δθ2) , (A4) or higher and can therefore be neglected by virtue of 2 O   the RWA. It is also easy to see that the n = 1 term in where the last two equations have been derived using Eq. (A6) will also give rise to time-independent terms. 2 Namely, using Eq. (A1) we straightforwardly obtain the asymptotic relations cos(φn,res/2)=1+ (δθ ) and 2 O sin (φn,res/2) = φn,res/2+ (δθ ). At the same time, we will utilize the following well-knownO expansions in terms of Bessel functions of the first kind:66

π cos(ω0t) i i φn,res cos = cos(ω t) e ~ H0t φ e− ~ H0t = + ..., (A7) 2 0 n,res 2   ∞ π π J 2 J cos(2nω t) , (A5) 0 2 − 2n 2 0 n=1   X   where the ellipses stand for the terms that rotate at fre- π cos(ω t) sin 0 = quency 2ω0, and and can therefore be neglected. In this 2   manner, after choosing EJb = 2EJ J0(π/2), we obtain ∞ Eq. (11). It should be stressed that, being approximately π n+1 2 J2n−1 ( 1) cos((2n 1)ω0t) . (A6) given by Eq. (13), the term cos(ϕ ϕ ) is unaffected 2 − − n n+1 n=1 − X   by the transformation to the rotating frame.

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