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

In this letter, we investigate the dynamics of a single superconducting artificial atom capacitively coupled to a transmission line with a characteristic impedance comparable 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 spontaneous emission from the atom, we then find Rabi oscillations, where the energy oscillates between the atom and one of the cavity modes.


INTRODUCTION
In the past two decades, circuit quantum electrodynamics (circuit QED) has become a field of growing interest for quantum information processing and also to realize new regimes in quantum optics [1][2][3][4][5][6][7][8][9]. The restriction to one dimensional (1D) waveguides in circuit and waveguide QED enhances directionality and reduces losses and therefore has a great advantage over higher dimensional systems to reach strong-and ultrastrongcoupling regimes [10][11][12][13][14][15][16][17][18][19][20]. A typical circuit QED setup consists of a superconducting qubit coupled to a 1D transmission line (TL) [2,[6][7][8]21]. Superconducting qubits are artificial atoms built with a non-linear Josephson Junction (JJ), that creates an anharmonic energy spectrum [9]. There are different kinds of superconducting qubits like flux qubits, phase qubits, and charge qubits [8,22]. A 1D transmission line can be modelled by coupled LC oscillators, each having a characteristic impedance of Z 0 = L 0 /C 0 ≈ 50 Ω, smaller than the quantum resistance R Q = /(2e) 2 ≈ 1.0 kΩ. But recent studies show that it is possible to reach impedances comparable to the quantum resistance or higher [23][24][25][26][27][28][29]. This can be realized by building circuits made of arrays of JJs [25,26,[28][29][30][31][32] or high-kinetic-inductances materials, called superinductors [24,27,[33][34][35][36]. High impedance JJ arrays and superinductors are for example used in the Fluxonium qubit [37][38][39][40], which has reduced charge noise sensitivity and can have relaxation times up to milliseconds [39,41]. This also has an advantage for metrology, since the charge noise insensitivity makes it possible to measure the current very accurately [42]. Furthermore, high-impedance resonators make it possible for light-matter interaction to reach strong coupling regimes, due to strong coupling to vacuum fluctuations [43]. In this article, we investigate the spontaneous emission of a transmon [44] capacitively coupled to a 1D TL that is shorted at one end. This system is known as an "atom in front of a mirror" [45][46][47][48][49]. Instead of using a Markovian master equation approach, we are taking the photon traveling time fully into account, making the dynamics non-trivial [12,[50][51][52][53][54][55][56][57][58][59]. Furthermore, we explore the above-mentioned regime of a TL impedance exceeding the quantum resistance Z 0 R Q . We find that the system behaves qualitatively different, compared to the well studied low impedance regime Z 0 R Q . The atom reflects strongly at all frequencies, except its transition frequency. Together with the mirror, it thus forms a cavity and when the transition frequency is close to a cavity mode, we find a vacuum Rabi-splitting, resulting in Rabi oscillations in the spontaneous emission. In this regime, all dynamical timescales are independent of the coupling capacitance and instead depend on the intrinsic transmon capacitance and the TL impedance. Another cavity-free system that shows Rabi splitting is an artificial atom coupled surface acoustic waves [60].

CIRCUIT-QED MODEL
Our system consists of a transmon qubit capacitively coupled to a semi-infinite 1D TL at a distance L from its grounded end (see Fig. 1). The transmon qubit consists of a superconducting anharmonic LC-oscillator, where the inductive (L) element is formed by a Josephson junction (JJ) with characteristic energy E J in parallel with a capacitor (C) with capacitance C J . The sinusoidal current-phase relation of the JJ makes the energy spectrum of the transmon qubit anharmonic, allowing for excitation with a single microwave photon using standard harmonic microwave sources. The transmon is capacitively coupled to a microwave TL, characterised by its inductance per unit length L 0 and capacitance per unit length C 0 . The velocity and impedance of the electromagnetic field inside the TL is given by v = 1/ √ L 0 C 0 and Z 0 = L 0 /C 0 , respectively. Using the standard circuit quantization procedure [59,61], we can derive 1. a) The circuit model of a transmon coupled through the coupling capacitance Cc to a semi-infinite 1D TL with impedance Z0. The Josephson energy, flux, and capacitance of the transmon are denoted by EJ , φJ , and CJ . The flux on the coupling capacitance Cc is denoted by φ0 with the corresponding voltage V0 =φ0. b) A sketch of the system depicting an atom in front of a mirror coupled to incoming/outgoing microwave fields to the left/right characterised by their respective voltages V in/out L/R at the transmon. The mirror couples the fields to the right V in where we denoted the capacitance to ground seen by the JJ as C Σ = C c C J /(C c + C J ) and in the second equation, we introduced the Josephson inductance L J = 2 /4e 2 E J , which describes the linearized dynamics of the Josephson junction. This approximation is obviously good in the weak excitation regime |φ J (t)| < /2e and will also be sufficient to describe the spontaneous emission, where the transmon is initially excited by a single microwave photon. , and Z0/ZJ = 1000 (red) from low to high TL impedance. For low impedance, the qubit reflects only at the coupled qubit frequency ω0, but for high impedance, it reflects everywhere but the uncoupled qubit frequency ωJ .

REFLECTION Open TL
To characterise how the behaviour of the system changes when we increase the TL impedance Z 0 , we first investigate the reflection of microwaves from the transmon coupled to an open TL, i.e. without a mirror. Since the equations of motion are linear, we can express the reflected field operator V out L in terms of the incoming probe field operator V in L by Fourier transforming the equations of motion (1)-(4), assuming no incoming field from the right (V in R = 0). The expression for the frequency dependent reflection coefficient is given by where ω 0 = 1/ L J (C c + C J ) is the resonance frequency of the coupled transmon and ω J = 1/ √ L J C J is the resonance frequency of the bare (uncoupled) transmon. In Fig. 2 the reflection around the transmon resonance frequencies is shown for different values of Z 0 . We see that for low impedance Z 0 C c ω < 1, the reflection is weak except at ω 0 where it is unity, due to resonant reflection from the transmon [62]. For high impedance Z 0 C c ω > 1 we instead see strong reflection at all frequencies, except around the "new" resonance frequency ω J , where we find zero reflection independent of Z 0 . The crossover occurs where E C = e 2 /(2C J ) is the charging energy of the transmon.
In the high impedance regime, the strong scattering away from ω J occurs due to the comparably strong capacitive coupling to ground at the transmon without exciting the transmon. Close to ω J , the resonantly excited transmon counteracts this capacitive coupling and effectively acts like an open circuit. This is the opposite behaviour compared to the low impedance regime, where the transmon is effectively an open circuit at all frequencies, except at its resonance frequency ω 0 , where it acts like a shorted circuit, giving full reflection. By fitting Lorentzians to Eq. (5), we can extract the high impedance coupling strength γ J = 2/Z 0 C J , which in contrast to the low-impedance expression γ 0 = does not depend on C c and decreases with increasing Z 0 . In this regime, the voltage at the node coupling to the TL oscillates with the full voltage across the JJ, In the low impedance regime, we instead have large currents flowing through the TL, keeping the voltage at the coupling node close to zero, i.e. |V 0 | |V J |. These currents obviously scale with C c and the energy dissipation scales with Z 0 . In the following, we investigate how the mirror affects the scattering.

Mirror
The mirror couples the fields to the right of the transmon V in Similarly as before, we can find the response to a harmonic field incoming from the left by Fourier transformation of the equations of motion. Since the absolute value of the reflection for the transmon in front of a mirror is always unity, we are now interested in the frequency dependence of the ratio between the trapped field (between the qubit and the mirror) and the incoming field, which is given by which is shown in Fig. 3. In the high impedance regime, we now find cavity resonances between the highly reflective atom and the mirror when the frequency is close Amplitude of the electromagnetic field between the qubit and the mirror |f to n ω c for n = 1, 2, . . . and ω c = 2π/T , as shown by the peaks in the inset of Fig. 3. These are broadened by the coupling to the TL by γ n c = |t(n ω c )| 2 /T , where t(ω) is the transmission across the transmon and |t(ω)| 2 = 1 − |r(ω)| 2 . We find that the effect on the transmon resonance close to ω J is simply to reduce its broadening with a factor of two to γ m J = 1/Z 0 C J , away from any qubit-cavity resonance ω J ≈ n ω c . As shown in the main panel of Fig. 3, on resonance ω J ≈ n ω c we find an avoided crossing with the coupling strength As we will see in the next section, where we investigate the spontaneous emission dynamics of the transmon, we find that this coupling indeed gives rise to vacuum Rabioscillations between the transmon and the cavity mode.
In the low impedance regime Z 0 C c ω < 1, f (ω) is instead close to unity, indicating only little scattering from the transmon for all frequencies far from the transmon resonance ω = ω 0 . Here, f (ω 0 ) = 0, since the field is reflected by the transmon and does not reach the mirror. When the transmon is located at a distance corresponding to a node of the electromagnetic field at its resonance frequency, i.e. ω 0 T = 2nπ, it is in a dark state and is thus completely invisible to the incoming field at frequency ω 0 , giving instead f (ω 0 ) = 1. In the dark state, both the transmon and the field between the transmon and the mirror are excited. Thus, if the distance is slightly longer/shorter than the node, the state is no longer completely dark, and we instead get a pronounced scattering resonance (|f (ω)| 1) at frequencies slightly lower/higher than ω 0 , see e.g. the purple line in Fig. 3. For higher Z 0 , we see that this dark state resonance moves in frequency towards the cavity frequency ω c = 2π/T . As shown in the supplemental material, for small C c /C J 1, and ω 0 = n ω c we can find vacuum Rabi-oscillations damped towards a finite dark state population.
In the following, we investigate how the highimpedance TL influences spontaneous emission of the transmon.

SPONTANEOUS EMISSION AND RABI OSCILLATIONS
We consider the case of a transmon initially excited at time t = 0 with a finite flux φ J (0) > 0, while the other qubit variables are zero p J (0) = p 0 (0) = 0 and the TL is in the vacuum state. The qubit energy is the sum of the capacitive energy on the two capacitances and the inductive energy in the JJ. The current amplitude emitted from the transmon into the TL is ∂ t p 0 (t) and from this we can write the change of the energy E R (t) of the field between the transmon and the mirror as where the first term corresponds to the instantaneous power emitted into the TL and the second term is the instantaneous power coming back from the mirror. The change of the energy of the field to the left of the transmon E L (t) is given by the instantaneous left-moving power leaving the system, where the left-moving current amplitude is a sum of the current emitted by the transmon and the delayed current arriving from the mirror. In Fig. 4, we plot these energies for Z 0 /Z J = 1000 for the case of resonance between transmon and the first cavity resonance ω J = ω c . The system energies indeed perform damped Rabi-oscillations with the frequency Ω = 2/ √ T C J Z 0 and half the off-resonance damping rate γ m J /2 = 1/2C J Z0, as indicated by the yellow line given by the expression e −γ m J t/2 cos 2 (Ωt/2), approximating the full numerical solution of the differential equations very well. We note that Laplace transforming the equations of motion (1)-(4) and calculating the residues of the system variables, gives similar expressions for the Rabi frequency and damping rate as the analysis of the resonances in the scattering amplitudes. More details and a comparison of the approximation to the numerical results can be seen in the supplemental materials.

EFFECTIVE QUANTUM MODEL: ATOM IN A MULTIMODE CAVITY HAMILTONIAN
We now go on to demonstrate that in the high impedance regime the response function f (ω) of the field trapped between the transmon and the mirror Eq. (7), reproduces the dynamics of an effective Hamiltonian of a single transition atom in a multimode cavity. This Hamiltonian is of the form: where the operators a, a † are annihilation and creation bosonic operators ([a, a † ] = 1) associated with excitation in the transmon qubit, while c n , c † n annihilate/create photons in the cavity modes. When weakly excited, the choice for bosonic excitations of the transmon is justified, while the orthogonality relations between the cavity modes is ensured by the high finesse of the latter, so that we have [c n , c † m ] = δ nm . Details of the diagonalisation of the Hamiltonian are shown in the supplemental materials. The response function |f (ω)| and eigenfrequencies of Hamiltonian (12) are shown in Fig. 5. The eigenfrequencies are shown to match the peaks of |f (ω)| for all cavity modes. Noticeably, a dip in the response function corresponding to the dark state is found for ω = ω 0 .

DISCUSSION AND OUTLOOK
We have made a first theoretical investigation of the properties of a transmon capacitively coupled to a high impedance transmission line, a system which is currently becoming experimentally accessible. By linearizing the Josephson junction, we could describe the low excitation dynamics, including spontaneous emission. We find qualitatively different behaviour, compared to the low impedance regime. In particular, the atom now forms its own cavity, and we can observe a vacuum Rabi splitting, giving rise to Rabi-oscillations in the spontaneous emission. The system is well described by a Hamiltonian for an atom weakly coupled to a multimode cavity. We hope that this analysis will inspire an experimental realization of this novel system. We can Laplace transform the equations of motion of the transmon shown in the main article and extract the exact poles numerically. With the following formulas we can calculate the inverse Laplace transform of the system variables and energy: where k = s ± 1,2 are the poles of p J (s). Similarly, we calculate φ J (t). We show the results as an addition to Fig. 4 in the main article. Here we provide more figures for different system parameters. In all panels of Fig 6, the impedance is chosen to be Z 0 /Z J = 1000. In Fig. 6 a), the ratio of the coupling capacitance and the Josephson capacitance is fairly small C c /C J = 0.02 and the coupling to the TL is weak. The cavity frequency equals the resonance frequency of the transmon ω C = ω J . The decay is weak and the Rabi-oscillations are clearly visible. The parameters in Fig. 6 b) are similar to a), but now C c /(C J + C c ) = 0.02 and most importantly the cavity frequency equals the transition frequency of the qubit for low impedance, ω C = ω 0 , which is the condition for a dark state [59]. The energy of the transmon decays until it reaches the dark state, with energy Cc+C J [59]. In c), the coupling capacitance is much larger compared to a) and b), C c /(C c + C J ) = 0.3 and the transmon fulfills the dark state condition ω C = ω 0 . Anyhow, the system does not converge into a dark state and the Rabi oscillations are very weak. The main difference here is that, since C c /(C c + C J ) is rather large, it means that ω J is not close to ω C and the Rabi oscillations and coupling to the cavity are suppressed. In this parameter regime, the behaviour of the system seems to be independent of the position of the transmon with respect to the mirror. Similar to Fig. 6 c), in Fig. 6 d), the coupling capacitance is rather large too C c /C J = 0.3, but here the cavity frequency equals the frequency of the transmon ω C = ω J , which means that the Rabi condition is fulfilled. We see clear Rabi oscillations and in addition the decay is much slower than in c).
As mentioned in the main article, we find an analytical expression for the oscillation frequency in the high impedance regime by analyzing the Laplace transform of the equations of motion. We find the Rabi frequency to be Ω = 2 √ T C J Z0 . In Fig. 7, we demonstrate the deviations of the approximation from numerically calculated values. We find that the higher the ratio Cc C J Z0 Z J , the closer the approximation resembles the numerical solution.

Analysis of the response functions
To analyze the solution of the Fourier transformation of the equations of motion in the main article in a convenient manner, we introduce the following functions Here despite ω0 = ωC as in b), the system does not seem to converge into a dark state and the decay rate is given by γJ = 1/CJ Z0. The difference to b) is that the ratio between Cc C J Z 0 Z J here is much bigger than in b) which also means that ωJ is not close to ωC and the Rabi oscillations are barely visible.. d) Z0/ZJ = 1000, Cc/CJ = 0.3 and ωJ = ωC . Here, as in c) Cc C J Z 0 Z J 1 but the "Rabi condition" ωJ = ωC is still fulfilled. We see clear Rabi oscillations and the decay is slower compared to c) and the decay rate is given by γ m J /2 = 1/2CJ Z0.
With these definitions we are able to write the transmission and the reflection amplitudes of the qubit in the open transmission line as We note that the high impedance regime corresponds to that |R J (ω)| |R 0 (ω)| away from resonances, i.e. |C c Z 0 ω/2| 1, while the opposite (|R J (ω)| |R 0 (ω)|) is true in the low impedance regime |C c Z 0 ω/2| 1. . We see that the higher Cc becomes, the closer the approximated analytical frequency Ω A and the numerically calculated frequency Ω N become.

Damping rate for the open TL
We analyse the scattering solution of the qubit excitation φ J to find the damping rate for the transmon in an open TL. The solution in frequency space reads In the high impedance regime, we perform an expansion around the bare qubit frequency ω = ω J + δω and find to first order in δω where γ J = 2/Z 0 C J is the energy damping rate for spontaneous emission.
In the low impedance regime, the qubit resonance is shifted to ω 0 and we instead expand ω = ω 0 + δω to find where γ 0 = Cc+C J is the low-impedance damping rate [59].

Damping rates and Lamb shifts with a mirror
With a mirror, the solution for the Josephson flux φ J can be written as In the low impedance regime, we find that the qubit resonance is Lamb-shifted toω 0 = ω 0 + γ 0 sin(ω 0 T )/2 and we can expand φ J (ω) around the resonance and find with the Purcell-modified damping rate γ m 0 = 2γ 0 sin 2 (ω 0 T /2). In the high impedance regime, the resonance frequency is Lamb shifted toω J = ω J + γ J cot (ω J T /2)/4. Expanding φ J (ω) around this frequency we find where γ m J = γ J /2 = 1/C J Z 0 is the damping rate of an atom in front of a mirror in the high Z 0 regime. Here, we note that the expression for the Lamb-shift diverges when sin (ω J T /2) = 0, i.e. when ω J is close to a cavity resonance ω n C = 2πn/T . This is when the single pole approximation is no longer valid and we find the vacuum Rabi splitting. Away from the Rabi condition, we also note that the damping rate is independent of the distance to the mirror, i.e. we see no Purcell effect. Away from the Rabi condition, we can also analyze the response function to extract the cavity modes. Due to the finite transmission through the transmon, they are slightly shifted from the perfect mirror frequencies toω n c = ω n c + R 0 (ω n c )/T R J (ω n c ). Close to the resonances we can expand ω =ω n c + δω and find where γ n c ≈ |t(ω n C )| 2 /T is the energy damping rate of cavity mode n.
The new operators should satisfy the eigenvalue problem: where Ω α are the eigenfrequencies labeled with a new index α. Typically, if we couple the atom with N cavity modes, then α runs from 1 to N +1. Expanding the commutator in the previous equation, it is possible to write the eigenvalue problem in a matrix form: 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 are shown in Fig. 5 in the main article.