PHYSICAL REVIEW RESEARCH 3, 023003 (2021)

Transmon in a semi-infinite high-impedance transmission line: Appearance of cavity modes and Rabi oscillations

E. Wiegand ,1,* B. Rousseaux ,2 and G. Johansson 1 1Applied Laboratory, Department of Microtechnology and Nanoscience-MC2, Chalmers University of Technology, 412 96 Göteborg, Sweden 2Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France

(Received 8 December 2020; accepted 11 March 2021; published 1 April 2021)

In this paper, we investigate the dynamics of a single superconducting artificial atom capacitively coupled to a transmission line with a characteristic impedance comparable to or larger than the quantum resistance. In this regime, microwaves are reflected from the atom also at frequencies far from the atom’s transition frequency. Adding a single mirror in the transmission line then creates cavity modes between the atom and the mirror. Investigating the from the atom, we then find Rabi oscillations, where the energy oscillates between the atom and one of the cavity modes.

DOI: 10.1103/PhysRevResearch.3.023003

I. INTRODUCTION [43]. Furthermore, high-impedance resonators make it pos- sible for -matter interaction to reach strong-coupling In the past two decades, circuit regimes due to strong coupling to vacuum fluctuations [44]. (circuit QED) has become a field of growing interest for quan- In this article, we investigate the spontaneous emission tum information processing and also to realize new regimes of a [45] capacitively coupled to a 1D TL that is in [1–11]. The restriction to one-dimensional shorted at one end. This system is known as an “atom in front (1D) waveguides in circuit and waveguide QED enhances of a mirror” [46–51]. Instead of using a Markovian master directionality and reduces losses and therefore has a great advantage over higher-dimensional systems to reach strong- equation approach, we are taking the photon traveling and ultrastrong-coupling regimes [12–22]. A typical circuit fully into account, making the dynamics nontrivial [14,52– QED setup consists of a superconducting coupled to a 61]. Furthermore, we explore the above-mentioned regime of a TL impedance√ Z0 exceeding the impedance of the trans- 1D transmission line (TL) [2,6–8,23]. Superconducting = /  are artificial atoms built with a nonlinear Josephson Junction mon ZJ RQ 2EC EJ , Z0 ZJ . We find that the system (JJ) that creates an anharmonic energy spectrum [9]. There are behaves qualitatively differently compared to the well-studied  different kinds of superconducting qubits, such as flux qubits, low-impedance regime Z0 ZJ . The atom reflects strongly phase qubits, and charge qubits [8,24]. A 1D transmission at all frequencies, except its transition frequency. Together with the mirror, it thus forms a cavity, and when the transition line can be modeled by coupled LC√oscillators, each having frequency is close to a cavity mode, we find a vacuum Rabi a characteristic impedance of Z0 = L0/C0 ≈ 50 , smaller 2 splitting, resulting in Rabi oscillations in the spontaneous than the quantum resistance RQ = h¯/(2e) ≈ 1.0 k.Butre- cent studies showed that it is possible to reach impedances emission. In this regime, all dynamical timescales are inde- comparable to the quantum resistance or higher [25–31]. This pendent of the coupling capacitance and instead depend on can be realized by building circuits made of arrays of JJs the intrinsic transmon capacitance and the TL impedance. [27,28,30–34] or high-kinetic-inductance materials, called su- Other cavity-free systems that show Rabi splitting and Rabi perinductors [25,26,29,35–37]. High-impedance JJ arrays and oscillations have been found with giant atoms [62], e.g., an superinductors are, for example, used in the fluxonium qubit artificial atom coupled to surface acoustic [63]. Sys- [38–41], which has reduced charge noise sensitivity and can tems involving atoms that act like mirrors were also studied have relaxation up to milliseconds [40,42]. This also has experimentally with a single trapped cold ion in front of a an advantage for metrology since the charge noise insensitiv- dielectric mirror [64] and exploration of collective dark states ity makes it possible to measure the current very accurately of strongly coupled qubits [65].

II. CIRCUIT-QED MODEL *[email protected] Our system consists of a transmon qubit capacitively cou- Published by the American Physical Society under the terms of the pled to a semi-infinite 1D TL at a distance L from its grounded Creative Commons Attribution 4.0 International license. Further end (see Fig. 1). The transmon qubit consists of a supercon- distribution of this work must maintain attribution to the author(s) ducting anharmonic LC oscillator, where the inductive (L) and the published article’s title, journal citation, and DOI. element is formed by a JJ with characteristic energy EJ in

2643-1564/2021/3(2)/023003(9) 023003-1 Published by the American Physical Society E. WIEGAND, B. ROUSSEAUX, AND G. JOHANSSON PHYSICAL REVIEW RESEARCH 3, 023003 (2021)

FIG. 2. Reflection of a transmon in an open TL for different FIG. 1. (a) Circuit model of a transmon coupled through the ratios of the TL and qubit impedance Z0/ZJ . The curves show coupling capacitance Cc to a semi-infinite 1D TL with impedance Cc the reflection for = 0.1andZ0/ZJ = 0.1 (purple), Z0/ZJ = Cc+CJ Z0. The Josephson energy, flux, and capacitance of the transmon are 1 (blue), Z /Z = 10 (green), Z /Z = 100 (yellow), and Z /Z = φ 0 J 0 J 0 J denoted by EJ , J ,andCJ . The flux on the coupling capacitance Cc is 1000 (red) from low to high TL impedance. For low impedance, the denoted by φ0, with the corresponding voltage V0 = φ˙0. (b) Sketch of qubit reflects only at its low-impedance frequency ω0,butforhigh the system depicting an atom in front of a mirror coupled to incoming impedance, it reflects everywhere but its bare frequency ωJ . and outgoing microwave fields to the left and right characterized by in/out their respective voltages VL/R , respectively, at the transmon. The in =− out − transmon is initially excited by a single microwave photon. mirror couples the fields to the right VR (t ) VR (t T ), intro- ducing the time of propagation to the mirror and back T = 2L/v. Here, we note that the equations of are linear and thus could be understood from classical dynamics. parallel with a capacitor C with capacitance C . The sinusoidal J III. REFLECTION current-phase relation of the JJ makes the energy spectrum of the transmon qubit anharmonic, allowing for excitation with a A. Open TL single microwave photon using standard harmonic microwave To characterize how the behavior of the system changes sources. The transmon is capacitively coupled to a microwave when we increase the TL impedance Z0, we first investigate TL, characterized by its inductance per unit length L0 and the reflection of microwaves from the transmon coupled to an capacitance per unit length C0. The velocity and characteristic open TL, i.e., without a mirror. Since the equations of motion impedance√ of the electromagnetic√ field inside the TL are given out are linear, we can express the reflected field VL in by v = 1/ L0C0 and Z0 = L0/C0, respectively. Using the in terms of the incoming probe field operator VL by Fourier standard circuit quantization procedure [61,66,67], we can transforming the equations of motion (1)–(4), assuming no derive the Heisenberg equations of motion for the charge p0(t ) in = incoming field from the right (VR 0). The expression for on the coupling capacitor Cc, the charge pJ (t )onCJ , and its the frequency-dependent reflection coefficient is given by conjugate flux φJ (t ), giving the phase difference 2eφJ (t )/h¯ ω2 out ω − over the JJ. Denoting the operators for the voltages of the V (ω) CcZ0 ω2 1 incoming and outgoing microwave fields to the left and right r(ω) ≡ L = J , (5) in ω − ω2 + ω ω2 − in/out VL ( ) 2i 1 ω2 CcZ0 ω2 1 of the transmon as VL/R (t ) [see Fig. 1(b)], respectively, these 0 J equations are √ where ω0 = 1/ LJ (Cc + CJ ) is the√ resonance frequency of 1 the coupled transmon and ωJ = 1/ LJCJ is the resonance ∂t φJ (t ) = [pJ (t ) + p0(t )], (1) CJ frequency of the bare (uncoupled) transmon. In Fig. 2 the re- flection around the transmon resonance frequencies is shown 2e 2e φ (t ) J for different values of Z0. We see that for low impedance ∂t pJ (t ) =−EJ sin φJ (t ) ≈− , (2) h¯ h¯ LJ Z0Ccω<1, the reflection is weak, except at ω0,whereitis unity due to resonant reflection from the transmon [68]. For 2p (t ) 2p (t ) 2 ∂ = 0 + J − in + in , ω> t p0(t ) VL (t ) VR (t ) (3) high impedance Z0Cc 1 we instead see strong reflection at Z0C Z0CJ Z0 all frequencies, except around the “new” resonance frequency ω , where we find zero reflection independent of Z .The out in Z0 J 0 V / (t ) = V / (t ) − ∂ p (t ), (4) ∼ / L R R L 2 t 0 crossover occurs at Z0 ZJCJ Cc, introducing the transmon impedance where we denoted the capacitance to the ground seen by = / + the coupling node as C CcCJ (Cc CJ ) and, in the sec- LJ 2EC = ZJ = = RQ , (6) ond equation, we introduced the Josephson inductance LJ C E 2 2 J J h¯ /4e EJ , which describes the linearized dynamics of the 2 Josephson junction. This approximation is obviously good in where EC = e /(2CJ ) is the charging energy of the transmon. the weak-excitation regime |φJ (t )| < h¯/2e and will also be In the high-impedance regime, the strong away sufficient to describe the spontaneous emission, where the from ωJ occurs due to the comparably strong capacitive

023003-2 TRANSMON IN A SEMI-INFINITE HIGH-IMPEDANCE … PHYSICAL REVIEW RESEARCH 3, 023003 (2021) coupling to the ground at the transmon without exciting the transmon. Close to ωJ , the resonantly excited transmon coun- teracts this capacitive coupling and effectively acts like an open circuit. This is the opposite behavior compared to the low-impedance regime, where the transmon is effectively an open circuit at all frequencies, except at its resonance fre- quency ω0, where it acts like a shorted circuit, giving full reflection. By expanding Eq. (5) around its pole close to ωJ , we can extract the high-impedance coupling strength γJ = 2/Z0CJ , which, in contrast to the low-impedance expression γ0 = 2/ + 2 Z0Cc 2LJ (CJ Cc ) , does not depend on Cc and decreases = in + with increasing Z0. In this regime, the voltage V0 VL out = in + out VL VR VR at the node coupling to the TL oscillates FIG. 3. Amplitude of the electromagnetic field between the qubit = φ˙ out in Cc with the full voltage across the JJ, VJ J . Due to the large and the mirror | f (ω)|=|V (ω)/V (ω)| for = 0.1andZ0/ZJ = R L CJ Z0, the amount of current passing through the TL is small 1000. The delay time is given by T = 2πn/ωc, n = 1, and ωc varies ∝ / − ( VJ Z0) and is thus not able to change the voltage V0 VJ with respect to ωJ . The different colors show ωc/ωJ = 0.98 (red), across Cc significantly, i.e., |V0 − VJ |VJ . More details of ωc/ωJ = 0.99 (yellow), ωc/ωJ = 1 (green), ωc/ωJ = 1.01 (blue), the derivation can be found in Appendix B. and ωc/ωJ = 1.02 (purple). All curves show a splitting which can be In the low-impedance regime, we instead have large cur- understood as a vacuum Rabi splitting. Without detuning between ωJ ω ω/ω = rents flowing through the TL, keeping the voltage at the and c, the splitting is centered exactly around J 1. The inset ω ≈ ω coupling node close to zero, i.e., |V0||VJ |. These currents shows the appearance of higher cavity modes appearing at n c, where ω T = 2π for ω /ω = 1, corresponding to the green curve obviously scale with Cc, and the energy dissipation scales c c J in the main plot. with Z0, explaining the expression for γ0. In the following, we investigate how the mirror affects the scattering. which we derived by analyzing the splitting of the poles of B. Mirror the Laplace transform of the equations of motion in the high- impedance regime. More details can be found in Appendix A. The mirror couples the fields to the right of the transmon Using the resonance condition ω ≈ n ω again, we note that in =− out − J c VR (t ) VR (t T ), introducing the time of propagation the n dependence cancels on resonance, to the mirror and back T = 2L/v [see Fig 1(b)]. Like before, we can find the response to a harmonic field incoming from = √ 2 . the left by Fourier transformation of the equations of motion. T (9) TCJ Z0 Since the absolute value of the reflection for the transmon in front of a mirror is always unity, we are now interested in the As we will see in the next section, where we investigate the frequency dependence of the ratio between the trapped field spontaneous emission dynamics of the transmon, we find that (between the qubit and the mirror) and the incoming field, this coupling indeed gives rise to vacuum Rabi oscillations ω ≡ out ω / in ω which is given by f ( ) VR ( ) VL ( ), between the transmon and the cavity mode. ω< ω In the low-impedance regime Z0Cc 1, f ( ) is instead ω2 − close to unity, indicating only a little scattering from the ω2 1 ω = 0 , transmon for all frequencies far from the transmon resonance f ( ) ω2 ω ω2 (7) − − CcZ0 − iωT − 1 ω2 i ω2 1 (e 1) ω = ω . Here, f (ω ) = 0 since the field is reflected by the 0 2 J 0 0 transmon and does not reach the mirror. When the transmon which is shown in Fig. 3. In the high-impedance regime, we is located at a distance corresponding to a node of the elec- now find cavity resonances between the highly reflective atom tromagnetic field at its resonance frequency, i.e., ω0T = 2nπ, and the mirror when the frequency is close to nωc for n = it is in a dark state and is thus completely invisible to the 1, 2,... and ωc = 2π/T , as shown by the peaks in the inset incoming field at frequency ω0, giving instead f (ω0 ) = 1. In of Fig. 3. These are broadened by the coupling to the TL by the dark state, both the transmon and the field between the γ n =| ω |2/ ω c t(n c ) T , where t( ) is the transmission across the transmon and the mirror are excited. Thus, if the distance is transmon and |t(ω)|2 = 1 −|r(ω)|2. slightly longer or shorter than the node, the state is no longer We find that the effect of the mirror on the transmon completely dark, and we instead get a pronounced scatter- resonance close to ωJ is simply to reduce its broadening by ing resonance (| f (ω)|1) at frequencies slightly lower or γ m = / ω a factor of 2 to J 1 Z0CJ , away from any qubit-cavity higher than 0 (see, e.g., the red line in Fig. 3). For higher resonance ωJ ≈ n ωc. As shown in the main panel of Fig. 3, Z0, we see that this dark state resonance moves in frequency on resonance ωJ ≈ n ωc we find an avoided crossing with the towards the cavity frequency ωc = 2π/T . As shown in Ap- coupling strength pendix A, for small Cc/CJ  1 and ω0 = nωc, we can find vacuum Rabi oscillations damped towards a finite dark state 2ωJ 2ωJ population. = √ = , (8) n π ω In the following, we investigate how the high-impedance 2 nCJ Z0 J 2πn Z0 ZJ TL influences spontaneous emission of the transmon.

023003-3 E. WIEGAND, B. ROUSSEAUX, AND G. JOHANSSON PHYSICAL REVIEW RESEARCH 3, 023003 (2021)

FIG. 4. Energy of the transmon (blue), field between the mirror and transmon (pink), outgoing field to the left (orange), sum of all γ m energies (red), approximated decay J (dashed cyan), and approx- −γ mt/2 2 imated Rabi frequency T and decay e J cos (T t/2) (dashed yellow) for Cc/CJ = 0.5, Z0/ZJ = 100, T = 2π/ωc, nωc = ωJ , n = 1 as a function of periods of the Rabi oscillations. The inset shows φ2 FIG. 5. Response function (log color scale) versus normalized the energy of the transmon (blue) and the flux J (green) for the frequency ω/ωJ and linearly varied time delay ωJ T . The eigenval- first oscillation of the frequency T . This clearly shows the different ues α are computed for N = 8 cavity modes and are shown as timescales of the energy compared to the variables of the transmon. superimposed white dashed lines. The parameters are chosen to be Z0/ZJ = 100 and Cc/CJ = 0.3. IV. SPONTANEOUS EMISSION AND RABI OSCILLATIONS We consider the case of a transmon initially excited at calculating the residues of the system variables give expres- = φ > time t 0 with a finite flux J (0) 0, while the other qubit sions for the Rabi frequency and damping rate similar to = = variables are zero pJ (0) p0(0) 0 and the TL is in the the analysis of the resonances in the scattering amplitudes. vacuum state. The qubit energy More details and a comparison of the approximation to the 2 2 2 numerical results are found in Appendix A. [pJ (t ) + p0(t )] p0(t ) φJ (t ) Eq(t ) = + + , (10) 2CJ 2Cc 2LJ is the sum of the capacitive energy on the two capacitances V. EFFECTIVE QUANTUM MODEL: ATOM IN A and the inductive energy in the JJ. The current amplitude MULTIMODE CAVITY HAMILTONIAN emitted from the transmon into the TL is ∂t p0(t ), and from We now go on to demonstrate that in the high-impedance this we can write the change in the energy ER(t ) of the field regime the response function f (ω) of the field trapped between the transmon and the mirror as between the transmon and the mirror (7) reproduces the dy- Z namics of an effective Hamiltonian of a single transition atom ∂ E (t ) = 0 {[∂ p (t )]2 − [∂ p (t − T )]2}, (11) t R 4 t 0 t 0 in a multimode cavity. This Hamiltonian is of the form where the first term corresponds to the instantaneous power ∞ = ω † + 1 + ω † + 1 emitted into the TL and the second term is the instantaneous H J a a 2 n c cncn 2 power coming back from the mirror. The change in the energy n=1 of the field to the left of the transmon EL (t )isgivenbythe ∞ instantaneous left-moving power leaving the system, + n (a† + a)(c† + c ), (13) 2 n n n=1 Z0 2 ∂t EL (t ) = [∂t p0(t ) − ∂t p0(t − T )] , (12) 4 where the operators a, a† are annihilation and creation bosonic where the left-moving current amplitude is a sum of the cur- operators ([a, a†] = 1) associated with excitation in the trans- , † rent emitted by the transmon and the delayed current arriving mon qubit, while cn cn annihilate and create photons in the from the mirror. cavity modes. When weakly excited, the choice of bosonic In Fig. 4, we plot these energies for Z0/ZJ = 100 for excitations of the transmon is justified, while the orthogonality the case of resonance between the transmon and the first relations between the cavity modes is ensured by the high ω = ω , † = δ cavity resonance J c. The system energies indeed per- finesse of the latter, so that we have [cn cm] nm. Details form√ damped Rabi oscillations with the frequency T = of the diagonalization of the Hamiltonian are shown in Ap- / γ m/ = | ω | 2 TCJ Z0 and half the off-resonance damping rate J 2 pendix C. The response function f ( ) and eigenfrequencies 1/2CJ Z0, as indicated by the yellow line given by the expres- of Hamiltonian (13) are shown in Fig. 5. The eigenfrequencies −γ mt/2 2 sion e J cos (T t/2), approximating the full numerical are shown to match the peaks of | f (ω)| for all cavity modes. solution of the differential equations very well. We note that Noticeably, a dip in the response function corresponding to Laplace transforming the equations of motion (1)–(4) and the dark state is found for ω = ω0. A similar multimode

023003-4 TRANSMON IN A SEMI-INFINITE HIGH-IMPEDANCE … PHYSICAL REVIEW RESEARCH 3, 023003 (2021)

FIG. 6. Energy of the transmon (blue), energy of the field trapped between the transmon and the mirror (magenta), outgoing field to the left side of the transmon (orange), total energy of the system (red), and semi-numerically calculated energy of the transmon using Eq. (A2)(yellow dashed) as a function of the period of the Rabi oscillations. The parameters for the panels are the following: (a) Z0/ZJ = 1000, Cc/CJ = 0.02, T = 2π/ωc,andωJ = nωc, n = 1. Since we chose Cc/CJ = 0.02, the decay is very slow. (b) Z0/ZJ = 1000, Cc/(CJ + Cc ) = 0.02, T = 2π/ωc, and ω0 = nωc, n = 1. Here, the parameters are almost the same as in (a), but now the dark state condition ωc = ω0 is fulfilled. The system converges into a dark state with a finite-excitation probability of both the transmon and the field between the transmon and the mirror. The Z ω2 2 EDS = 1 γ = 0 0 Cc / = / + = . green line indicates the analytical value of the dark state energy E + T γ 2 , with 0 2 C +C .(c)Z0 ZJ 1000, Cc (CJ Cc ) 0 3, 0 (1 2 0 ) c J T = 2π/ωc and ω0 = nωc, n = 1. Here, although ω0 = ωC as in (b), the system does not seem to converge into a dark state, and the decay Cc Z0 rate is given by γJ = 1/CJ Z0. The difference from (b) is that the ratio between here is much bigger than in (b), which also means that ωJ CJ ZJ is not close to ωC and the Rabi oscillations are barely visible. (d) Z0/ZJ = 1000, Cc/CJ = 0.3, T = 2πωc,andωJ = nωc, n = 1. Here, as in Cc Z0 (c)  1, but the “Rabi condition” ωJ = nωC is still fulfilled. We see clear Rabi oscillations, the decay is slower compared to (c), and the CJ ZJ γ m/ = / decay rate is given by J 2 1 2CJ Z0. √ Hamiltonian with a (1/ n)-dependent coupling was studied and Alice Wallenberg Foundation (KAW) through the Wal- in Refs. [69,70]. lenberg Centre for (WACQT).

VI. DISCUSSION AND OUTLOOK APPENDIX A: RABI OSCILLATIONS OF AN ATOM COUPLED TO A HIGH-IMPEDANCE SEMI-INFINITE TL: We have theoretically investigated the properties of a trans- ADDITIONAL PARAMETER VALUES mon capacitively coupled to a high-impedance transmission line, a system which is currently becoming experimentally We can Laplace transform the equations of motion of the accessible. By linearizing the Josephson junction, we could transmon shown in the main text and extract the exact poles describe the low-excitation dynamics, including spontaneous numerically. With the following formulas we can calculate the emission. We find qualitatively different behavior compared inverse Laplace transform of the system variables and energy: to that of the low-impedance regime. In particular, the atom ± = − ± , Res1,2 pJ (s) lim pJ (s)(s s1,2 ) (A1) now forms its own cavity, and we can observe a vacuum Rabi s→s± 1,2 splitting, giving rise to Rabi oscillations in the spontaneous emission. The system is well described by a Hamiltonian for −st −st pJ (t ) = pJ (s)e ds = 2πi Res pJ (s)e , (A2) an atom weakly coupled to a multimode cavity. We hope that k s=s this analysis will inspire an experimental realization of this k = ± novel system. where k s1,2 are the poles of pJ (s). Similarly, we calcu- late φJ (t ). We show the results as an addition to Fig. 4 in the main text. Here, we provide more figures for different ACKNOWLEDGMENTS system parameters. In all panels of Fig. 6, the impedance The authors thank A. Parra-Rodriguez for important com- is chosen to be Z0/ZJ = 1000. In Fig. 6(a), the ratio of the ments on the manuscript. The authors acknowledge funding coupling capacitance and the Josephson capacitance is fairly from the Swedish Research Council (VR) through Grant No. small, Cc/CJ = 0.02, and the coupling to the TL is weak. 2016-06059. G.J. also acknowledges funding from the Knut The cavity frequency equals the resonance frequency of the

023003-5 E. WIEGAND, B. ROUSSEAUX, AND G. JOHANSSON PHYSICAL REVIEW RESEARCH 3, 023003 (2021)

APPENDIX B: ANALYSIS OF THE RESPONSE FUNCTIONS To analyze the solution of the Fourier transformation of the equations of motion in the main text in a convenient manner, we introduce the following functions:

N(ω) = R0(ω) − iRJ (ω), (B1) ω2 R (ω) = 1 − (B2) 0 ω2 0 C Z ω2 R (ω) = c 0 ω 1 − . (B3) J ω2 2 J With these definitions we are able to write the transmission and the reflection amplitudes of the qubit in the open trans- / FIG. 7. Rabi frequency as a function of Z0 ZJ calculated mission line as N analytically T (solid yellow line) and numerically (dots) for dif- / / = . / = . iRJ (ω) R0(ω) ferent values of Cc CJ , Cc CJ 0 01 (purple), Cc CJ 0 05 (red), r ω = , t ω = , r ω + = t ω . ( ) ω ( ) ω ( ) 1 ( ) Cc/CJ = 0.3 (orange), for T = 2π/ωc, ωJ = nωc, n = 1. We see that N( ) N( ) the higher Cc Z0 becomes, the closer the approximated analytical (B4) CJ ZJ frequency and the numerically calculated frequency N become. T We note that the high-impedance regime corresponds to |RJ (ω)||R0(ω)| away from resonances, i.e., |CcZ0ω/2| ω = ω transmon c J . The decay is weak, and the Rabi oscil- 1, while the opposite (|RJ (ω)||R0(ω)|)istrueinthelow- lations are clearly visible. The parameters in Fig. 6(b) are impedance regime, |CcZ0ω/2|1. similar to those of Fig. 6(a), but now Cc/(CJ + Cc ) = 0.02, and most importantly, the cavity frequency equals the tran- 1. Damping rate for the open TL sition frequency of the qubit for low impedance, ωc = ω0, which is the condition for a dark state [61]. The energy of the We analyze the scattering solution of the qubit excitation transmon decays until it reaches the dark state, with energy φJ to find the damping rate for the transmon in an open TL. Z ω2 C2 EDS = 1 γ = 0 0 c The solution in frequency space reads E + T γ 2 , where 0 2 C +C [61]. In Fig. 6(c),the 0 (1 2 0 ) c J coupling capacitance is much larger than those of Figs. 6(a) C L ω φ (ω) =− c J . (B5) and 6(b), Cc/(Cc + CJ ) = 0.3, and the transmon fulfills the J R0(ω) − RJ (ω) dark state condition ωc = ω0. Anyhow, the system does not converge into a dark state, and the Rabi oscillations are very In the high-impedance regime, we perform an expansion ω = ω + δω weak. The main difference here is that, since Cc/(Cc + CJ )is around the bare qubit frequency J and find, to first δω rather large, ωJ is not close to ωc, and the Rabi oscillations order in , and coupling to the cavity are suppressed. In this parameter i 1 regime, the behavior of the system seems to be independent of φ ω + δω ≈ , J ( J ) ω − δω/γ (B6) the position of the transmon with respect to the mirror. Similar J 1 i2 J to Fig. 6(c),inFig.6(d), the coupling capacitance is rather where γJ = 2/Z0CJ is the energy damping rate for sponta- large too, Cc/CJ = 0.3, but here, the cavity frequency equals ω = ω neous emission. the frequency of the transmon c J , which means that the In the low-impedance regime, the qubit resonance is shifted Rabi condition is fulfilled. We see clear Rabi oscillations, and to ω , and we instead expand ω = ω + δω to find in addition, the decay is much slower than in Fig. 6(c). 0 0 As mentioned in the main text, we find an analytical 1 1 φ (ω + δω) = , (B7) expression for the oscillation frequency by analyzing the J 0 2 CcZ0ω 1 − i2δω/γ0 Laplace transform of the equations of motion. The denomi- 0 Z ω2 C2 nator of all system variables in Laplace transform is given by γ = 0 0 c where 0 + is the low-impedance damping rate 2 Cc CJ s2 s2 [61]. N(s) = 2esT 1 + + s(1 − esT ) 1 + C Z . (A3) ω2 ω2 c 0 0 J 2. Damping rates and Lamb shifts with a mirror If we expand the denominator N(s) around the resonance Cc Z0 With a mirror, the solution of the Josephson flux φ can be frequency s → ωJ (i + x + iy), with x, y  1 for high , J CJ ZJ we find that the splitting of the poles, and hence the Rabi written as 2ω frequency, is given by = √ J .InFig.7, we demon- iωT n 2πnC Z ω −iC L ω(1 − e ) J 0 J φ ω = c J . strate the deviations of the approximation from numerically J ( ) iωT (B8) R0(ω) − iRJ (ω)(1 − e ) calculated values. We find that the higher the ratio Cc Z0 is, CJ ZJ the more closely the approximation resembles the numerical In the low-impedance regime, we find that the qubit resonance solution. is Lamb shifted toω ˜ 0 = ω0 + γ0 sin(ω0T )/2, and we can

023003-6 TRANSMON IN A SEMI-INFINITE HIGH-IMPEDANCE … PHYSICAL REVIEW RESEARCH 3, 023003 (2021)

φ ω ω = ωn + δω expand J ( ) around the resonance and find to the resonances we can expand ˜ c and find

iω T/2 2 1 Cc e 0 1 f ω˜ n + δω ≈ , (B12) φJ (˜ω0 + δω) ≈−i , c ωn − δω/γ n + γ ω / − δω/γ m t c 1 i2 c CJ Cc 0 sin ( 0T 2) 1 i2 0 (B9) γ n ≈| ωn |2/ where c t(˜c ) T is the energy damping rate of cavity γ m = mode n. with the Purcell-modified damping rate 0 2 2γ0 sin (ω0T/2). In the high-impedance regime, the resonance frequency APPENDIX C: HOPFIELD DIAGONALIZATION OF THE ATOM IN A MULTIMODE CAVITY HAMILTONIAN is Lamb shifted toω ˜ J = ωJ + γJ cot (ωJ T/2)/4. Expanding φ ω J ( ) around this frequency, we find To diagonalize the effective quantum optical Hamiltonian [Eq. (13) in the main text], we first introduce the new polari- 2i 1 φ (˜ω + δω) =− , (B10) tonic operators J J ω + δω/γ m J 1 i2 J α α † α α † α = x a + y a + m cn + h c , (C1) γ m = γ / = / n n n where J J 2 1 CJ Z0 is the damping rate of an atom in n front of a mirror in the high-Z regime. Here, we note that the 0 with xα, yα, mα, hα being the Hopfield coefficients associ- expression for the Lamb shift diverges when sin (ω T/2) = 0, n n J ated with each bosonic operator a, a†, c , c†. To ensure the i.e., when ω is close to a cavity resonance ωn = 2πn/T .This n n J c , † = δ is when the single pole approximation is no longer valid and bosonicity of the polaritonic operators ([ α β ] αβ), the we find the vacuum Rabi splitting. Away from the Rabi condi- coefficients should satisfy the relation 2 2 tion, we also note that the damping rate is independent of the |xα|2 −|yα|2 + mα − hα = 1. (C2) distance to the mirror; that is, we see no Purcell effect. Away n n n from the Rabi condition, we can also analyze the response function The new operators should satisfy the eigenvalue problem [ α, H] = α α, (C3) ω = 1 f ( ) R ω (B11) − J ( ) − iωT where α are the eigenfrequencies labeled with a new index 1 i R (ω) (1 e ) 0 α. Typically, if we couple the atom with N cavity modes, to extract the cavity modes. Due to the finite transmission then α runs from 1 to N + 1. Expanding the commutator in through the transmon, they are slightly shifted from the per- the previous equation, it is possible to write the eigenvalue ωn = ωn + ωn / ωn fect mirror frequencies to ˜ c c R0( c ) TRJ ( c ). Close problem in matrix form:

⎛ ⎞⎛ ⎞ ⎛ ⎞ α α ωJ 0 g1 −g1 ... gN −gN x x ⎜ −ω − ... − ⎟⎜ α ⎟ ⎜ α ⎟ ⎜ 0 J g1 g1 gN gN ⎟⎜ y ⎟ ⎜ y ⎟ ⎜ − ω ... ⎟⎜ α ⎟ ⎜ α ⎟ ⎜g1 g1 1 0 00⎟⎜m1 ⎟ ⎜m1 ⎟ ⎜ − −ω ⎟⎜ α ⎟ ⎜ α ⎟ ⎜g1 g1 0 1 00⎟⎜ h1 ⎟ = α⎜ h1 ⎟. (C4) ⎜ ...... ⎟⎜ . ⎟ ⎜ . ⎟ ⎜ . . . . . ⎟⎜ . ⎟ ⎜ . ⎟ ⎝ − ω ⎠⎝ α ⎠ ⎝ α ⎠ gN gN 00 N 0 mN mN − ... −ω α α gN gN 00 0 N hN hN This eigenvalue problem can be solved analytically for N = 1, but in general, one has to diagonalize it numerically. A comparison between the full response function and the eigenvalues is shown in Fig. 5 in the main text.

[1] D. Roy, C. M. Wilson, and O. Firstenberg, Colloquium: [5] F. Quijandría, I. Strandberg, and G. Johansson, Steady-State Strongly interacting photons in one-dimensional continuum, Generation of Wigner-Negative States in One-Dimensional Rev. Mod. Phys. 89, 021001 (2017). Resonance Fluorescence, Phys. Rev. Lett. 121, 263603 (2018). [2] X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, [6] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, Microwave photonics with superconducting quantum circuits, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Strong Phys. Rep. 718–719, 1 (2017). coupling of a single photon to a superconducting qubit using [3] A. González-Tudela, V. Paulisch, H. J. Kimble, and J. I. Cirac, circuit quantum electrodynamics, Nature (London) 431, 162 Efficient Multiphoton Generation in Waveguide Quantum Elec- (2004). trodynamics, Phys. Rev. Lett. 118, 213601 (2017). [7] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and [4] V. Paulisch, H. J. Kimble, J. I. Cirac, and A. González-Tudela, R. J. Schoelkopf, Cavity quantum electrodynamics for su- Generation of single- and two-mode multiphoton states in perconducting electrical circuits: An architecture for quantum waveguide QED, Phys.Rev.A97, 053831 (2018). computation, Phys.Rev.A69, 062320 (2004).

023003-7 E. WIEGAND, B. ROUSSEAUX, AND G. JOHANSSON PHYSICAL REVIEW RESEARCH 3, 023003 (2021)

[8] G. Wendin, processing with supercon- [27] N. A. Masluk, I. M. Pop, A. Kamal, Z. K. Minev, and M. H. ducting circuits: A review, Rep. Prog. Phys. 80, 106001 (2017). Devoret, Microwave Characterization of Josephson Junction [9] A. F. Kockum and F. Nori, Quantum bits with Josephson junc- Arrays: Implementing a Low Loss Superinductance, Phys. Rev. tions, in Fundamentals and Frontiers of the Josephson Effect, Lett. 109, 137002 (2012). edited by F. Tafuri (Springer, Cham, 2019), pp. 703–741. [28] T. Weißl, B. Küng, E. Dumur, A. K. Feofanov, I. Matei, C. [10] B. Kannan, M. J. Ruckriegel, D. L. Campbell, A. Frisk Kockum, Naud, O. Buisson, F. W. J. Hekking, and W. Guichard, Kerr J. Braumüller, D. K. Kim, M. Kjaergaard, P. Krantz, A. coefficients of plasma resonances in Josephson junction chains, Melville, B. M. Niedzielski, A. Vepsäläinen, R. Winik, J. L. Phys. Rev. B 92, 104508 (2015). Yoder, F. Nori, T. P. Orlando, S. Gustavsson, and W. D. Oliver, [29] N. Samkharadze, A. Bruno, P. Scarlino, G. Zheng, D. P. Waveguide quantum electrodynamics with superconducting ar- DiVincenzo, L. DiCarlo, and L. M. K. Vandersypen, High- tificial giant atoms, Nature (London) 583, 775 (2020). kinetic-inductance superconducting nanowire resonators for [11] A. Frisk Kockum, Quantum optics with giant atoms—The circuit QED in a magnetic field, Phys. Rev. Appl. 5, 044004 first five years, in International Symposium on Mathemat- (2016). ics, Quantum , and Cryptography,editedbyT.Takagi, [30] R. Kuzmin, R. Mencia, N. Grabon, N. Mehta, Y. H. Lin, M. Wakayama, K. Tanaka, N. Kunihiro, K. Kimoto, and Y. and V. E. Manucharyan, Quantum electrodynamics of a Ikematsu (Springer, Singapore, 2021), pp. 125–146. superconductor–insulator phase transition, Nat. Phys. 15, 930 [12] H. Zheng, D. J. Gauthier, and H. U. Baranger, Waveguide QED: (2019). Many-body bound-state effects in coherent and Fock-state scat- [31] J. Puertas Martínez, S. Léger, N. Gheeraert, R. Dassonneville, tering from a two-level system, Phys. Rev. A 82, 063816 (2010). L. Planat, F. Foroughi, Y. Krupko, O. Buisson, C. Naud, W. [13] T. Tufarelli, F. Ciccarello, and M. S. Kim, Dynamics of sponta- Hasch-Guichard, S. Florens, I. Snyman, and N. Roch, A tunable neous emission in a single-end photonic waveguide, Phys. Rev. Josephson platform to explore many-body quantum optics in A 87, 013820 (2013). circuit-QED, npj Quantum Inf. 5, 19 (2019). [14] T. Tufarelli, M. S. Kim, and F. Ciccarello, Non-Markovianity of [32] M. T. Bell, I. A. Sadovskyy, L. B. Ioffe, A. Y. Kitaev, and M. E. a quantum emitter in front of a mirror, Phys. Rev. A 90, 012113 Gershenson, Quantum Superinductor with Tunable Nonlinear- (2014). ity, Phys.Rev.Lett.109, 137003 (2012). [15] E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. [33] P. Winkel, I. Takmakov, D. Rieger, L. Planat, W. Hasch- Martin-Moreno, Scattering in the Ultrastrong Regime: Non- Guichard, L. Grünhaupt, N. Maleeva, F. Foroughi, F. Henriques, linear Optics with One Photon, Phys. Rev. Lett. 113, 263604 K. Borisov, J. Ferrero, A. V. Ustinov, W. Wernsdorfer, N. Roch, (2014). and I. M. Pop, Nondegenerate parametric amplifiers based on [16] E. Sánchez-Burillo, L. Martín-Moreno, J. J. García-Ripoll, and dispersion-engineered Josephson-junction arrays, Phys. Rev. D. Zueco, Full two-photon down-conversion of a single photon, Appl. 13, 024015 (2020). Phys. Rev. A 94, 053814 (2016). [34] Y. Krupko, V. D. Nguyen, T. Weißl, E. Dumur, J. Puertas, R. [17] E. Sánchez-Burillo, D. Zueco, L. Martín-Moreno, and J. J. Dassonneville, C. Naud, F. W. J. Hekking, D. M. Basko, O. García-Ripoll, Dynamical signatures of bound states in waveg- Buisson, N. Roch, and W. Hasch-Guichard, Kerr nonlinearity uide QED, Phys.Rev.A96, 023831 (2017). in a superconducting Josephson metamaterial, Phys. Rev. B 98, [18] M. Devoret, S. Girvin, and R. Schoelkopf, Circuit-QED: How 094516 (2018). strong can the coupling between a Josephson junction atom and [35] J. T. Peltonen, P. C. J. J. Coumou, Z. H. Peng, T. M. Klapwijk, a transmission line resonator be?, Ann. Phys. (Berlin, Ger.) 16, J. S. Tsai, and O. V. Astafiev, Hybrid rf squid qubit based on 767 (2007). high kinetic inductance, Sci. Rep. 8, 10033 (2018). [19] M. Bamba and T. Ogawa, Recipe for the hamiltonian of system- [36] L. Grünhaupt, N. Maleeva, S. T. Skacel, M. Calvo, F. Levy- environment coupling applicable to the ultrastrong-light-matter- Bertrand, A. V. Ustinov, H. Rotzinger, A. Monfardini, G. interaction regime, Phys. Rev. A 89, 023817 (2014). Catelani, and I. M. Pop, Loss Mechanisms and Quasiparticle [20] A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, Dynamics in Superconducting Microwave Resonators Made of and F. Nori, Ultrastrong coupling between light and matter, Nat. Thin-Film Granular Aluminum, Phys.Rev.Lett.121, 117001 Rev. Phys. 1, 19 (2019). (2018). [21] D. Zueco and J. García-Ripoll, Ultrastrongly dissipative quan- [37] D. Niepce, J. J. Burnett, M. G. Latorre, and J. Bylander, tum Rabi model, Phys. Rev. A 99, 013807 (2019). Geometric scaling of two-level-system loss in supercon- [22] J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, ducting resonators, Supercond. Sci. Technol. 33, 025013 Observation of Dicke superradiance for two artificial atoms in a (2020). cavity with high decay rate, Nat. Commun. 5, 5186 (2014). [38] V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, [23] D. M. Pozar, Microwave Engineering (Wiley, Hoboken, NJ, Fluxonium: Single Cooper-pair circuit free of charge offsets, 2005). Science 326, 113 (2009). [24] J. Clarke and F. K. Wilhelm, Superconducting quantum bits, [39] L. Grünhaupt, M. Spiecker, D. Gusenkova, N. Maleeva, S. T. Nature, 453, 1031 (2008). Skacel, I. Takmakov, F. Valenti, P. Winkel, H. Rotzinger, W. [25] I. V.Pechenezhskiy, R. A. Mencia, L. B. Nguyen, Y.-H. Lin, and Wernsdorfer, A. V. Ustinov, and I. M. Pop, Granular aluminium V. E. Manucharyan, The superconducting quasicharge qubit, as a superconducting material for high-impedance quantum cir- Nature 585, 368 (2020). cuits, Nat. Mater. 18, 816 (2019). [26] D. Niepce, J. Burnett, and J. Bylander, High kinetic inductance [40] L. B. Nguyen, Y.-H. Lin, A. Somoroff, R. Mencia, N. Grabon, NbN nanowire superinductors, Phys. Rev. Appl. 11, 044014 and V. E. Manucharyan, High- Fluxonium Qubit, (2019). Phys. Rev. X 9, 041041 (2019).

023003-8 TRANSMON IN A SEMI-INFINITE HIGH-IMPEDANCE … PHYSICAL REVIEW RESEARCH 3, 023003 (2021)

[41] J. Koch, V. Manucharyan, M. H. Devoret, and L. I. Glazman, [55] A. L. Grimsmo, Time-Delayed Quantum Feedback Control, Charging Effects in the Inductively Shunted Josephson Junc- Phys. Rev. Lett. 115, 060402 (2015). tion, Phys.Rev.Lett.103, 217004 (2009). [56] H. Pichler and P. Zoller, Photonic Circuits with Time De- [42] I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. lays and Quantum Feedback, Phys. Rev. Lett. 116, 093601 Glazman, and M. H. Devoret, Coherent suppression of elec- (2016). tromagnetic dissipation due to superconducting quasiparticles, [57] H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, Universal pho- Nature (London) 508, 369 (2014). tonic quantum computation via time-delayed feedback, Proc. [43] W. Guichard and F. W. J. Hekking, Phase-charge duality in Natl. Acad. Sci. USA 114, 11362 (2017). Josephson junction circuits: Role of inertia and effect of mi- [58] P.-O. Guimond, M. Pletyukhov, H. Pichler, and P. Zoller, De- crowave irradiation, Phys.Rev.B81, 064508 (2010). layed coherent quantum feedback from a scattering theory and [44] A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. a matrix product state perspective, Quantum Sci. Technol. 2, Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. 044012 (2017). Wallraff, Strong Coupling Cavity QED with Gate-Defined Dou- [59] M. P. Schneider, T. Sproll, C. Stawiarski, P. Schmitteckert, ble Quantum Dots Enabled by a High Impedance Resonator, and K. Busch, Green’s-function formalism for waveguide QED Phys. Rev. X 7, 011030 (2017). applications, Phys. Rev. A 93, 013828 (2016). [45] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, [60] A. González-Tudela and J. I. Cirac, Markovian and non- J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Markovian dynamics of quantum emitters coupled to two- Schoelkopf, Charge-insensitive qubit design derived from the dimensional structured reservoirs, Phys. Rev. A 96, 043811 Cooper pair box, Phys. Rev. A 76, 042319 (2007). (2017). [46] I. C. Hoi, A. F. Kockum, L. Tornberg, A. Pourkabirian, G. [61] E. Wiegand, B. Rousseaux, and G. Johansson, Semiclassical Johansson, P. Delsing, and C. M. Wilson, Probing the quantum analysis of dark-state transient dynamics in waveguide circuit vacuum with an artificial atom in front of a mirror, Nat. Phys. QED, Phys.Rev.A101, 033801 (2020). 11, 1045 (2015). [62] L. Guo, A. F. Kockum, F. Marquardt, and G. Johansson, Oscil- [47] P. Y. Wen, A. F. Kockum, H. Ian, J. C. Chen, F. Nori, and lating bound states for a giant atom, Phys.Rev.Res.2, 043014 I.-C. Hoi, Reflective Amplification without Population Inver- (2020). sion from a Strongly Driven Superconducting Qubit, Phys. Rev. [63] A. Ask, M. Ekström, P. Delsing, and G. Johansson, Cavity-free Lett. 120, 063603 (2018). vacuum-Rabi splitting in circuit quantum acoustodynamics, [48] P. Y. Wen, K.-T. Lin, A. F. Kockum, B. Suri, H. Ian, J. C. Phys. Rev. A 99, 013840 (2019). Chen, S. Y. Mao, C. C. Chiu, P. Delsing, F. Nori, G.-D. Lin, [64] G. Hétet, L. Slodicka,ˇ M. Hennrich, and R. Blatt, Single Atom and I.-C. Hoi, Large Collective Lamb Shift of Two Distant as a Mirror of an Optical Cavity, Phys.Rev.Lett.107, 133002 Superconducting Artificial Atoms, Phys. Rev. Lett. 123, 233602 (2011). (2019). [65] M. Mirhosseini, E. Kim, X. Zhang, A. Sipahigil, P. B. [49] Z. H. Peng, S. E. de Graaf, J. S. Tsai, and O. V. Astafiev, Dieterle, A. J. Keller, A. Asenjo-Garcia, D. E. Chang, and O. Tuneable on-demand single-photon source in the microwave Painter, Cavity quantum electrodynamics with atom-like mir- range, Nat. Commun. 7, 12588 (2016). rors, Nature (London) 569, 692 (2019). [50] P. Forn-Díaz, C. W. Warren, C. W. S. Chang, A. M. Vadiraj, [66] B. Yurke and J. S. Denker, theory, Phys. Rev. and C. M. Wilson, On-demand microwave generator of shaped A 29, 1419 (1984). single photons, Phys. Rev. Appl. 8, 054015 (2017). [67] M. H. Devoret, Quantum fluctuations in electrical circuits, in [51] Y. Lu, A. Bengtsson, J. J. Burnett, E. Wiegand, B. Suri, P. Quantum Fluctuations, Les Houches Session LXIII (Elsevier, Krantz, A. F. Roudsari, A. F. Kockum, S. Gasparinetti, G. Amsterdam, 1995), p. 351. Johansson, and P. Delsing, Characterizing decoherence rates [68] B. Peropadre, J. Lindkvist, I.-C. Hoi, C. M. Wilson, J. J. Garcia- of a superconducting qubit by direct microwave scattering, npj Ripoll, P. Delsing, and G. Johansson, Scattering of coherent Quantum Inf. 7, 35 (2021). states on a single artificial atom, New J. Phys. 15, 035009 [52] U. Dorner and P. Zoller, Laser-driven atoms in half-cavities, (2013). Phys. Rev. A 66, 023816 (2002). [69] M. Malekakhlagh, A. Petrescu, and H. E. Türeci, Cutoff- [53] L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Free Circuit Quantum Electrodynamics, Phys. Rev. Lett. 119, Johansson, Giant acoustic atom: A single quantum system with 073601 (2017). a deterministic time delay, Phys.Rev.A95, 053821 (2017). [70] M. F. Gely, A. Parra-Rodriguez, D. Bothner, Y. M. Blanter, S. J. [54] M. Bradford and J.-T. Shen, Spontaneous emission in cavity Bosman, E. Solano, and G. A. Steele, Convergence of the multi- QED with a terminated waveguide, Phys.Rev.A87, 063830 mode quantum Rabi model of circuit quantum electrodynamics, (2013). Phys. Rev. B 95, 245115 (2017).

023003-9