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Ultrastrong coupling few-photon scattering theory Tao Shi,1 Yue Chang,1 and Juan José Garc´ıa-Ripoll2 1Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany 2Instituto de F´ısica Fundamental IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain (Dated: January 18, 2017) We study the scattering of photons by a two-level system ultrastrongly coupled to a one- dimensional waveguide. Using a combination of the polaron transformation with scattering theory we can compute the one-photon scattering properties of the qubit for a broad range of coupling strengths, estimating resonance frequencies, lineshapes and linewidths. We validate numerically and analytically the accuracy of this technique up to α=0.3, close to the Toulouse point α=1/2, where inelastic scattering becomes relevant. These methods model recent experiments with superconducting circuits [P. Forn-D´ıaz et al., Nat. Phys. (2016)]. 7 1 PACSnumbers: 03.65.Nk,42.50.-p,72.10.Fl 0 2 Waveguide quantum electrodynamics (QED) studies range of the USC regime. Our starting point is the spin- n the interaction between propagating photons and quan- boson model for a waveguide of length L a J tumimpuritiesin1Denvironments. Reduceddimension- 17 afelaittyureems,paonwdertshefeywb-leecvoeml esycsatpemabslewoitfhfusltlryonregflencotninligneianr- H = ∆2σz+(cid:88)k ωka†kak+ √1L(cid:88)k gkσx(ak+a†k), (1) dividualphotons[1]ormediatingastrongphoton-photon ] interaction [2]. In order for this to occur, the impurity WedonotworkwiththisHamiltonian,butbuildatrans- h formed one H = U†HU using an optimized polaron p –atwo-levelsystemorqubit—needstobeinthestrong- p p p transformation U that eliminates most of the qubit- - couplingregime,wherethespontaneousemissionintothe p t photon entanglement in the ground state [19]. The new n waveguide, Γ, dominates all other dissipation channels. a This regime is achieved in experiments with supercon- Hamiltonian Hp can be manipulated and combined with u scattering theory [11] to predict the dynamics of few- ducting circuits [1, 2], neutral atoms [3, 4] and quan- q photon wavepackets. We show results for the super- tum dots in photonic crystals [5]. In most experiments [ spontaneous emission is slower than the atom or photon conducting Ohmic spin-boson model, where the spectral 1 function is linear up to a cutoff ω oscillationfrequencies,Γ(cid:28)∆,ω,allowingforarotating- c v 9 wave approximation (RWA) and theoretical predictions 2π (cid:88) 0 basedonone-andfew-photonwavefunctions[6,7],input- J(ω)= L |gk|2δ(ω−ωk)(cid:39)παω1e−ω/ωc. (2) 7 outputtheory[8,9]andpathintegralformalism[10,11]. k 4 Superconducting circuits are waveguide QED systems Interestingly, we recover cutoff independent predictions 0 . wherethequbit-photoncouplingcanmatchthequbitand for the resonance and linewidth of elastic single-photon 1 photon energies, Γ ∼ ∆,ω. This so called ultrastrong- scatteringintheUSCregime. Theseresultsarevalidated 0 coupling regime (USC) causes the breakdown of RWA with analytics at the Toulouse point [21, 22] at α =1/2 7 1 predictions, the excitation of qubit-photon entangled andalsowithmoderate-sizematrix-productstate(MPS) : ground states [12], extremely broadband interactions numericalsimulationsofthequbitspontaneousemission. v [13], and a phase transition into the localization regime Both methods attest the qualitative (α ∈ [0.3,0.5]) and i X [14]. The USC was first demonstrated in resonators evenquantitative(α∈[0,0.3])accuracyofourtechniques r [15, 16], where it admits an analytic description [17]. in modeling new and state-of-the-art experiments such a More recently, the USC regime has been explored using as the single-photon scattering with tuneable coupling propagatingphotonsinmicrowaveguides[18]andstudy- qubits by Forn-D´ıaz et al. [18]. This work opens the ing the resonance spectrum of the qubit in the transmis- door to studying multi-photon scattering in more com- sion line. This new generation of experiments opens a plex experiments with transmons or Λ-level schemes, or very challenging theoretical problem: the integration of thedevelopmentofaccuratemodelsforphotonmediated USC in the waveguide QED framework, moving beyond interactions in open waveguides, which would have im- the study of dissipation [14], to photon-qubit scattering portantapplicationsinthequantumsimulationsofIsing- and interactions. like Hamiltonians [23] and annealing. While this question has been addressed using numer- Model setup.— Our starting point is the Hamilto- ical methods such as Matrix Product States or MPS nian (1) that models the interaction between a two-level [13, 19], the Numerical Renormalization Group [20], in system and a photonic waveguide with periodic bound- this work we develop fully analytic predictions for the ary conditions. The Pauli matrices σx,z are defined photon-qubit interaction, which are accurate for a broad in the qubit basis |e(cid:105) and |g(cid:105) for excited and ground 2 FIG. 1. Ground state properties of the polaron Hamiltonian H computedwithMPSfordifferentcutoffs. Weplot(a)the p excitation probability of the qubit P = (cid:104)σz+1(cid:105)/2 and (b) e the total energy of photons E =(cid:80) ω (cid:104)a†a (cid:105)/∆. photon k k k k states. The qubit couples to photons with momenta k, with anihilation (creation) operators a (a†). We will k k FIG. 2. (Color online) Spontaneous emission of the two-level conduct analytic calculations with a linear dispersion systeminthepolaron-transformedmodelH . (a)Totalnum- (cid:112) p ωk = c|k| and couplings gk = παcωk/2e−ωk/2ωc that ber of excitations N = σ+σ−+(cid:80) a†a starting from state k k k reproducetheOhmicspectralfunction(2),andnumerics |↑(cid:105)|0(cid:105). (b) Excitation probability (cid:104)σ+σ−(cid:105) as a function of (cid:112) with a hard cut-off ω = ω [1−cos(k)]/2, couplings time and (c) spectrum of emitted photons at t∆=30. Lines k c g =(cid:112)παcω /2 and L equal to the number of modes. correspondtoα=0.01(solid),0.07(dashed)and0.35(dash- k k dot), simulated with Hamiltonian (3). Thick dots represent Instead of (1), we implement approximations on the the outcome from (4) for similar values of α. equivalent model H =U†HU after a polaron transfor- p p p √ mationU =exp[−σx(cid:80) f (a†−a )/ L], whichdisen- p k k k k tangles the bosonic and qubit states (smallmatrices), andasmallcut-offn ≤4, significantly i improving over earlier simulations with H [13]. H = ∆˜σzO† O +(cid:88)ω a†a +(cid:88)√Gk σx(a +a†)+E , Excitationconservingpolaron.— Inspiredbythesim- p 2 −f f k k k L k k p k k plicity of the ground state, we will now assume that the (3) low-energy dynamicsof H admitsalso asimpledescrip- p The renormalized qubit energy ∆˜ = ∆e−2(cid:80)k|fk|2/L√ap- tion as quasiparticles on a close-to-vacuum state. For pears with the operators Of = exp(2σx(cid:80)kfkak/ L) that we select the single-particle section Hamiltonian from normal ordering. The Silbey-Harris prescription f[2ec4t,iv2e5]cooupptliimngizeGskf=k ∆˜=fkg,ka/n(ωdkm+ak∆˜in)g, trhedeugcrionugndthsetaetfe- Hp(1) = ∆2˜σz+Vlocal+ (cid:88) ωkA†s,kAs,k (4) of Hp as close to |g(cid:105)|0(cid:105) as possible. √ k,s=± bleDteospditiaegtohneahliizgehHlynuonsilnogcaMltPeSrmanσszaOtz−†f[2O6f–,2i8t],isapvoassrii-- + 2√L2(cid:88)∆˜fk(A†+,kσ−+A+,kσ+). p k ational estimate of the ground state wavefunction |ψ(cid:105)= (cid:80)s,ntr[As0An11···AnNN]|s,n1...nN(cid:105), where Axi ∈ Cξ×ξ This model i√ntroduces annihilation operators A±,k>0 = are different matrices labeled by physical degrees of free- (a ± a )/ 2 for the symmetric and anti-symmetric k −k dom: the qubit states, s∈{g,e}, or the photon occupa- modes, andisrestrictedtoworkwithasingleexcitation, tion numbers n of the associated momenta k . Numeri- N = σ+σ−+(cid:80) a†a = 1, which is sufficient for single i i k k k caloptimizationswiththehardcut-offmodelω /∆=3,6 photonemissionandscattering. Notethespin-dependent c and9showlessthan2%qubitexcitationprobabilityand potential, V = −4σz∆˜ (cid:80) f f A† A /L, essen- local kp k p +,p +,k anegligibleamountofphotonsbelowtheToulousepoint tial to capture the whole dynamics α=1/2[cf. Fig.1]. Interestingly,mostqubit-photonen- We have compared both models using MPS simula- tanglementisremovedbythepolarontransformationand tions of a low-energy problem in which an excited qubit the MPS converges with a small bond dimension ξ (cid:28)20 relaxes into a vacuum of photons, |ψ(0)(cid:105) = |e(cid:105)|0(cid:105). As 3 1.0 1.0 b) 0.5 c) 0.8 α=0.25 0.4 2 0.6 0.3 R α | 0.4 0.2 α=0.1 0.2 0.1 α=0.01 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.1 0.2 0.3 0.4 0.5  ω/Δ Γ/πω reson FIG. 3. (Color Online) (a) Reflection coefficient R as a function of the spin-boson coupling strength α and the photon k frequency. In solid we plot the resonance frequency, ω , and in dashed-dot we show the half-height lines. (b) Three cuts reson of the above plot show asymmetric lineshapes for increasing α. (c) The ratio between the experimental linewidth and the resonance frequency is a lower bound for α. shown in Fig. 2a, the wavefunction at all times |ψ(t)(cid:105) |r |2andtransmissivityT =|t |2determinedbyEq.(6), k k k remains in the single excitation sector up to α ∼ 0.3. satisfyR+T =1andprovideaconcretepredictionforthe Moreover,Figs.2b-cdemonstrateanexcellentagreement lineshapeofasingle-photonscatteringexperiment,forall betweentheRWA(4)andtheSilbey-HarrisHamiltonian dispersion relations and frequency dependent couplings. (3),notonlyinthequbitdynamics,butalsointheemis- Whenthedynamicsofthespontaneousemissionisslower sion spectrum. The quantitative disagreement is largely thanthatofthequbitandphotons,i.e.,|Γ|(cid:28)∆˜,ω ,the k accounted for by (i) the additional dressing of localized Markovapproximationreflectsintoanegligiblepotential photons by the qubit [cf. Fig. 1] and (ii) inleastic three- V and Lamb-shift δ ∼ 0 and a uniform Γ(ω) ∼ Γ. local L or more-photon contributions above α=0.35 [29]. We recover the usual formula Single-photon scattering estimates.— The MPS sug- gfeeesltintghafotrwtehicsavnawluoer,klewtiuths cHup(t1)thuepwtoavαeg(cid:39)uid0.e3.toTmoagkeet aa rk (cid:39) (ωk−−∆˜i)(∆˜ωk++i(∆˜ω)kΓ2+∆˜)Γ2, (8) λ/2 resonator which is resonant with the qubit. We will √ predicting total reflection R = 1 for resonant photons find a qubit-cavity coupling g = α∆. Thus, values cav ω →∆˜,anddisplayingtheusualLorentzianprofilefrom of α = 0.3 correspond to g (cid:39) 0.55∆ inside a cavity: k cav scatteringexperimentsinthestrongcouplingregimewith a coupling so strong, that the bandwidth of photons is comparable to the qubit energy Γ/∆˜ (cid:39) 1 [cf. Fig. 2c], superconducting circuits [1, 2] or quantum dots [5]. For USC, however, the self-energy and the local potential the so called ultrastrong coupling regime. V induce significant distortions and asymmetries in Our goal is to analyse scattering in the USC regime, local the lineshapes, as expected from both earlier numerics developing formulas that can be used to model experiments [18]. We will apply scattering theory to H(1), [13] and experiments [18]. p Opentransmissionline.- Weparticularizethepredic- focusingonthelowpowerregimeofatmostonephoton. tions to the Ohmic coupling of a superconducting qubit The reflection and transmission coefficients with a transmission line. In the limit of large cut-off rk = 12(sk−1), and tk = 21(sk+1), (5) ωc, we may approximate ∆˜ = ∆(e∆/ωc)α/(1−α), with varyingprefactorsdependingonthedetailsofthemodel. are constructed from the chiral phase shift |s | = 1 ex- However,independentoftherenormalizationscheme,our k perienced by photons in the A modes. These can be scattering estimates lead to the same self-energy k,+ computed using scattering formalism [11] or Lippmann- Schwinger theory Σ(ω)= 2∆˜2α (cid:20)ωln(ω)−ω−∆˜ −iπω(cid:21). (9) (ω+∆˜)2 ∆˜ (ω −∆˜)∆˜ −(ω +∆˜)Σ∗(ω ) s = k k k . (6) k (ω −∆˜)∆˜ −(ω +∆˜)Σ(ω ) This prediction does not involve the cut-off: this infor- k k k mation is implicit in the value of ∆˜, which the optimal Theself-energyΣ(ω)=δL(ω)−iΓ(ω)/2containsaLamb- transformation uses to regularize the couplings in both shift the infrarred and ultraviolet limits. (cid:90) ∞ dk f2 The formula above has three important consequences: δ (ω)=4∆˜2 P k , (7) L 2π ω−ω (i) The lineshape profiles are asymmetric for even mod- 0 k erate values of α (cid:39) 0.1 [See Figs. 3a-b]. (ii) As shown and a decay rate Γ(ω) = 4∆˜2f2 |∂ω /∂k|−1 given by inFig.3a, thescatteringresonanceω , definedasthe k0 k k=k0 reson thesolutionk ≡k (ω)ofω =ω. ThereflectivityR = frequency of maximum reflection, does not necessarily 0 0 k0 k 4 1.0 1.0 θR/π a) b) 1.00 0.5 0.8 R 0.95 P1+P3 0.6 π θ/ 0.0 θT/π T0.4 P30.90 P -0.5 1 0.2 0.85 T c) -1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.00.0 0.2 0.4 0.6 0.8 1.0 0.800.0 0.2 0.4 0.6 0.8 1.0 k k k/k Δ FIG.4. (ColorOnline)Single-photonreflectionandtransmissioncoefficientsasafunctionoftheincidentphotonfrequencyat the Toulouse point α=1/2, for ω =108∆. We plot (a) the phases of the transmission and reflection amplitudes and (b) the c reflection/transmissionprobabilities, bothexact(solid)andwiththepolaron-RWA(6)(dashed). (c)Probabilitytodetectone and three out-going photons P +P (dashed) and single-photon elastic scattering probability P =R+T (solid). 1 3 1 match the value ∆˜. This is due to a very large Lamb phase and as a result, P = |R | + |T | < 1. The 1 k k shift, ofthesameorderofmagnitudeasthespontaneous true dynamics deviates from the polaron-RWA predic- emissionrateitself. (ii)ThelinewidthΓdependsonboth tions because of multi-photon processes, which, as al- αandω,anditisnotpossibletocalibratetheinteraction ready shown in Ref. [29], are dominated by three-photon √ strengthusingtheformulaΓ/∆˜ (cid:39)πα,fromthenon-USC corrections P = |(cid:104)0|(cid:81)3 A† |φ (cid:105)/ 3!|2 with mo- 3 i=1 +,pi out regime [cf. Fig. 3c]. In other words, while it is true that mentap +p +p =k. InFigs.4a-bweshowtheelastic 1 2 3 wecandistinguishtheUSCregimebytheconditionthat transmissionandreflectioncoefficientsforboththeexact Γ/π becomparableto ωreson, acalibrationofα demands Toulousewavefunctionandthepolaron-RWAapproxima- the mathematical modelization of the line shapes. tion. Consideringthatα=1/2isaverylargeinteraction Toulouse point, α = 1/2.— While the above scat- (g (cid:39) 0.71∆ in the cavity), we find a very good quali- cav tering formulas work for a broad range of couplings, tative and almost quantitative agreement between both α∈[0,0.3], it is interesting to study the source of devia- methods in the elastic sector. tionsforverystronginteractions,uptothephasetransi- Summary and discussion.— In this work we have tion into the Kondo regime. Fortunately, the spin-boson derived analytical estimates of the lineshapes and res- model with an Ohmic spectrum admits an analytical so- onance frequencies for single-photon scattering in the lution at the Toulouse point α=1/2, which already has USC regime, α ∈ [0,0.3] in the spin boson model —or been used to study scattering properties [29]. g/ω ∈ [0,55%] if we would cut the same line to shape a The basic idea is to realize that at α = 1/2, work- cavity—. These estimates are supported by strong nu- ing with a polaron displacement f(cid:48) = −g /ω we can k k k merical evidence that the static and dynamic properties cancel completely the linear coupling terms, G = 0, k of the spin-boson model can be well approximated by and map the spin and A modes to the density fluc- + (cid:113) a RWA version of the polaron transformation for this tuations a fermionic bath, A†q,+ = L2πq (cid:80)kc†k+qck. The rangeofcouplings. Ourpredictionsapplytoexperiments resulting model can be diagonalized, giving as ground with superconducting qubits in open transmission lines state |GS(cid:105) = |0(cid:105) |FS(cid:105) a product of the vacuum |0(cid:105) [18] and represent an important milestone in the inte- − f − of the anti-symmetric modes and the Fermi sea |FS(cid:105) gration of the USC regime in waveguide QED theory. f of the new fermions. To compute the scattering ma- The techniques presented in this work can be immedi- trix, we express the asymptotic state with one incom- atelyextendedtootherdispersionrelationsandcoupling ing photon |φ (cid:105) = a†U |0(cid:105) |FS(cid:105) using fermionic op- strengths,including,forinstance,USCscatteringinpho- in k p − f erators, and compute the out-going asymptotic state tonicbandgapsandcavityarrayswithboundstates[12]. |φ (cid:105) = lim e−iHT |φ (cid:105) using the Green function It is also possible to account for radiative losses, heating out T→∞ in approach in the infinite line limit L → ∞. The outgo- and dephasing, depending on the qubit nature and its ing wavefunction contains both a single photon (elastic) energy gap: in all cases the formulas generalize with the component, as well as multiphoton (inelastic) contribu- change Σ(ω) → Σ(ω)−iΓ , where Γ includes all addi- ϕ ϕ tions [29]. The single-photon reflection and transmission tional dephasing sources. We expect that the ideas put (5) derive from forward in this work will stimulate and simplify future experiments with superconducting circuits in the USC 2iw w s =1+ [2i(arccot2w+arctan )+ regime, as well as help in the development of a complete k 1+2iw 1+2w2 theory for USC scattering and effective interactions in +lnw2/(1+w2)], (10) multi-qubit setups. with w = π∆2/(4ω k). The value s is no longer a JJGR acknowledges support from MINECO/FEDER c k 5 Project FIS2015-70856-P and CAM PRICYT Project [13] B. Peropadre, D. Zueco, D. Porras, and J. J. Garc´ıa- QUITEMAD+ S2013/ICE-2801. This work was funded Ripoll, Phys. Rev. Lett. 111, 243602 (2013). by the European Union Integrated project Simulators [14] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, and Interfaces with Quantum Systems (SIQS). 1 (1987). [15] T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hu¨mmer, E. 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