Transmon Qubit in a Magnetic Field: Evolution of Coherence and Transition Frequency

TL;DR

This study demonstrates that a fixed-frequency concentric transmon qubit maintains quantum coherence up to 40 mT magnetic fields, with minimal dephasing impact, enabling potential applications in quantum sensing and hybrid systems.

Contribution

It provides the first measurements of transmon qubit coherence in high magnetic fields without optimized design, and introduces an analytic formula for field-dependent qubit energies considering fabrication imperfections.

Findings

Quantum coherence persists up to 40 mT magnetic field.

Dephasing rate remains unaffected across a broad frequency range.

Analytic model for field-dependent qubit energies considering second junction effects.

Abstract

We report on spectroscopic and time-domain measurements on a fixed-frequency concentric transmon qubit in an applied in-plane magnetic field to explore its limits of magnetic field compatibility. We demonstrate quantum coherence of the qubit up to field values of $B={40}\,\mathrm{mT}$, even without an optimized chip design or material combination of the qubit. The dephasing rate $\Gamma_\varphi$ is shown to be not affected by the magnetic field in a broad range of the qubit transition frequency. For the evolution of the qubit transition frequency, we find the unintended second junction created in the shadow angle evaporation process to be non-negligible and deduce an analytic formula for the field-dependent qubit energies. We discuss the relevant field-dependent loss channels, which can not be distinguished by our measurements, inviting further theoretical and experimental investigation. Using well-known and well-studied standard components of the superconducting quantum architecture, we are able to reach a field regime relevant for quantum sensing and hybrid applications of magnetic spins and spin systems.

Tables

Table 1. Table 1: Parameters of the two Josephson junctions. B Δ subscript 𝐵 Δ B_{\Delta} is the offset field, B Φ 0 subscript 𝐵 subscript Φ 0 B_{\Phi_{0}} the periodicity, and l 𝑙 l the length of the junction.

	$E_J/h\left(\mathrm{GHz}\right)$	$B_{\Delta }\left(\mathrm{mT}\right)$	$B_{{\Phi }_0}\left(\mathrm{mT}\right)$	$l\left(\mathrm{nm}\right)$
${\mathrm{JJ}}_1$	16.15	1.8	300	209
${\mathrm{JJ}}_2$	300	-0.2	25.5	2460

Equations

$${\phi }_{\Sigma }={\phi }_1+{\phi }_2$$

$$I({\phi }_{\Sigma })$$

$$\tan {\phi }_2$$

$$I({\phi }_{\Sigma })=I_{\mathrm{c},2}\sin \left(\arctan \frac{\sin {\phi }_{\Sigma }}{r+\cos {\phi }_{\Sigma }}\right),$$

$$H=4E_{\mathrm{C}}{N}^2-E_{\mathrm{J},1}\sqrt{r^2+2r\cos \phi +1},$$

$$H$$

$$\frac{1}{T_1}={\Gamma }_1={\Gamma }_{\mathrm{hyst}}+{\Gamma }_{\mathrm{non}\text{mathchar}45hyst}+{\Gamma }_{\mathrm{const}},$$

$${\Gamma }_2^{\mathrm{Ramsey}}=\frac{1}{2}{\Gamma }_1+{\Gamma }_{\phi },$$

$${\Gamma }_{\phi }^I=\pi {\left(\frac{\partial {\omega }_{01}}{\partial I}\right)}_z^2{S}_I(\omega \ll k_{\mathrm{B}}T),$$

$$C\ddot{\varphi }=-I_{\mathrm{c},2}\sin \left(\arctan \frac{\sin \frac{2\pi \varphi }{{\Phi }_0}}{r+\cos \frac{2\pi \varphi }{{\Phi }_0}}\right),$$

$$\mathcal{L}=\frac{C}{2}{\dot{\varphi }}^2+\frac{I_{\mathrm{c},1}{\Phi }_0}{2\pi }\sqrt{r^2+2r\cos \left(\frac{2\pi \varphi }{{\Phi }_0}\right)+1}.$$

$$H=4E_{\mathrm{C}}{N}^2-E_{\mathrm{J},1}\sqrt{r^2+2r\cos \phi +1}.$$

$$H\approx 4E_{\mathrm{C}}{N}^2+E_{\mathrm{J},1}\left(\frac{r}{2r+2}{\phi }^2-\frac{r(r^2-r+1)}{24(r+1{)}^3}{\phi }^4\right).$$

$$N$$

$$\phi$$

$$H=\sqrt{8E_{\mathrm{C}}{E}_{\mathrm{J},1}\frac{r}{r+1}}{a}^{†}a-\frac{r^2-r+1}{(r+1{)}^2}\frac{E_{\mathrm{C}}}{2}{a}^{†}a(a^{†}a+1).$$

$$I_c{R}_n=\frac{\pi }{2e}\Delta (T)\tanh \frac{\Delta (T)}{2k_{B}T}$$

$$\frac{1}{T_1}={\Gamma }_1={\Gamma }_{\mathrm{hyst}}+{\Gamma }_{\mathrm{non}\text{mathchar}45hyst}+{\Gamma }_{\mathrm{const}},$$

Full text

I Introduction

Superconducting quantum circuits render a versatile platform for realizing circuit quantum electrodynamic (cQED) systems. Such systems are used in various applications as they offer a flexible and engineerable toolset to build a physical model system and employ it to study quantum mechanics in depth. They can also be used for interaction and characterization of other quantum systems and turn out to be a useful tool for investigation. Superconducting quantum bits are a promising candidate for quantum computing and quantum simulation Popkin (2016), as well as for the emerging field of quantum sensing Degen et al. (2017), which becomes more and more important with the fast growing number of quantum systems that are subject to current research. Superconducting qubits are here used to study the characteristics and dynamics of unknown systems in the quantum regime and are therefore a valuable sensing tool.

Applications like quantum sensing of magnetic excitations Degen et al. (2017), creating and harnessing Majorana Fermions Mourik et al. (2012) or quantum cavity magnonics Tabuchi et al. (2015); Pfirrmann et al. expose the qubits to magnetic fields. In particular superconducting qubits are intrinsically vulnerable to magnetic fields. So far, in the literature only influences of small magnetic fields on the order of $100\text{\,}\mathrm{\SIUnitSymbolMicro T}$ have been studied, where even a slight improvement of their coherent behavior for very small fields could be found due to the creation of quasiparticle traps by entering flux vortices Wang et al. (2014). However, the general consent is to screen magnetic fields as best as possible and a multi-layered shielding based on permalloy and superconductors is commonly used Kreikebaum et al. (2016); Flanigan et al. (2016). To our knowledge, no published effort has been spent to study the limits of magnetic field compatibility of standard Josephson junction (JJ) qubits. In fact, they have been assumed to break down at very little fields and other, more stable junctions, such as the proximitized semiconducting nanowire, have been introduced to circumvent this limitation Luthi et al. (2018).

In this article however, we study the magnetic field properties of a conventional Josephson tunneling barrier junction qubit for in-plane magnetic fields up to $40\text{\,}\mathrm{mT}$, which is well above the saturation field for magnets like permalloy, opening opportunities in hybrids of quantum circuits and magnetic materials.

This letter starts with the investigation of the magnetic field dependence of the qubit’s transition frequencies, where we find an analytic formula. In the following, we study the coherence time under the influence of a magnetic field and discuss different field-dependent loss channels. This behavior is reproducible and symmetric with respect to the applied fields up to $B=\pm$20\text{,}\mathrm{mT}. Going to stronger fields, we demonstrate measurable coherence times up to $B=$40\text{\,}\mathrm{mT}, and remanently suppressed coherence when decreasing the field again. In the last section we analyze the pure dephasing rate, which we find to be independent from the magnetic field.

II Sample and Setup

The qubit used for this experiment is a single-junction concentric transmon Braumüller et al. (2016), which was already described in Ref. Schneider et al. (2018). Its capacitance pads are made from low-loss TiN and the junction is an Al/$\mathrm{AlO_{x}}$/Al structure, fabricated by shadow angle evaporation. The sample is placed in a copper box and mounted to the base stage of a dilution refrigerator at a temperature of about $30\text{\,}\mathrm{mK}$. It reaches into a solenoid fixated at the still stage. The sample was aligned to the solenoid for an in-plane orientation of the field by eye, leaving a probability for small out-of-plane field components at the sample. Due to the structured superconductor on the chip and the resulting flux-focusing Buckel and Kleiner (2004) leading to an inhomogeneous magnetic field, we assume that an ideal in-plane configuration over the whole chip is hard to achieve. This especially holds true when looking at future applications, where a possible local magnetic environment produces stray fields.

For the measurements, a time-domain as well as a spectroscopy setup is used, which are described in Appendix E together with the cryogenic setup. Data acquisition and analysis are performed via Qkit qki (2017).

To infer the qubit state, we observe the dispersive frequency shift of a $\lambda/2$ resonator coupled to the qubit by $g/2\pi=$71.5\text{,}\mathrm{MHz}. The microstrip resonator is made from low-loss TiN, with initial frequency $\omega_{\mathrm{r,0}}/2\pi=$8.573\text{\,}\mathrm{GHz} and internal quality factor around $Q_{\mathrm{i,0}}=$5100\pm 120\text{,}$$, extracted by a circle fit Probst et al. (2015). When changing the magnetic field, we see a reproducible field-dependent change of $Q_{\mathrm{i}}$ and $\omega_{\mathrm{r}}$ which is hysteretic due to the creation and annihilation of flux vortices in the material Bothner et al. (2012), see Appendix A for more data. The reducing quality factor involves a decreased signal to noise ratio (SNR) for our measurements, making it harder to find the qubit transition frequency.

III Qubit transition frequency in an in-plane magnetic field

In the following, we study the qubit transition frequency under the influence of an applied in-plane magnetic field. It is known that this field suppresses the critical current $I_{\mathrm{c}}$ of a Josephson junction periodically, where the shape follows a Fraunhofer pattern Barone and Paterno (1982). In contrast to this expectation with $\omega_{01}\propto\sqrt{I_{\mathrm{c}}}$, our measurement data show a flat top at $\Phi=0$, a much steeper slope at $\Phi=\Phi_{0}$, and an overall envelope, i.e. the first side maxima are not as high as the main maximum (see \freffig:two_jjb)).

III.1 JJ Fabrication Scheme

Due to the JJ fabrication by shadow angle evaporation, two tunnel junctions exist in series, see \freffig:two_jja). The current flows from the TiN layer through the lower Al layer, via the designed tunnel junction ($\mathrm{JJ_{1}}$) to the top layer, and then passes the oxide barrier ($\mathrm{JJ_{2}}$) to reach the second TiN electrode. $\mathrm{JJ_{2}}$ is much larger in area, has a very high critical current $I_{\mathrm{c},2}\gg I_{\mathrm{c},1}$, and is therefore commonly neglected for the qubit properties. Its large cross section however gives an increased sensitivity to the applied magnetic flux. This second junction can be avoided by a shunting bandage Dunsworth et al. (2017) or overlap junctions Wu et al. (2017). The third JJ on the left side of \freffig:two_jja) however would only shunt the $\mathrm{Al_{1}}$ layer and can therefore be neglected as long as the inductance of the lower layer is negligible.

III.2 Qubit Hamiltonian with Two JJ in Series

To calculate the current-phase-relation for two JJ in series, we start with two junctions in series, having critical currents $I_{\mathrm{c},1},\,I_{\mathrm{c},2}$ and phases $\varphi_{1},\,\varphi_{2}$. It is clear that the current through them is the same and the total phase adds up:

[TABLE]

We introduce the ratio $r=I_{\mathrm{c},2}/I_{\mathrm{c},1}$ between the junctions’ critical currents and assume $r\geq 1$ without loss of generality. From this we follow

[TABLE]

The overall current-phase-relation is thus given by

[TABLE]

resulting in the system Hamiltonian

[TABLE]

where the exact derivation can be found in Appendix B. For the approximate transmon Hamiltonian we get

[TABLE]

Here, $a^{\dagger}$ and $a$ are the harmonic oscillator creation and annihilation operators, $E_{\mathrm{C}}=\mathrm{e}^{2}/2C=$190\text{,}\mathrm{MHz}$$ is the charging energy, and $E_{\mathrm{J}}=I_{\mathrm{c}}\Phi_{0}/2\pi$ is the Josephson energy. For the limit of $r\rightarrow\infty$, where $\mathrm{JJ}_{1}$ is dominating (i.e. limiting) the circuit, this formula goes back to the unperturbed approximated transmon Hamiltonian Koch et al. (2007). We emphasize that the transmon’s anharmonicity decreases if the two junctions are comparable in $I_{\mathrm{c}}$, i.e. for $r=1$ the maximum anharmonicity is reduced by a factor of 4.

With \frefeq:H we calculate the transmon spectrum and find good agreement with the measured data in \freffig:two_jjb). For the individual junctions, $I_{\mathrm{c}}=I_{\mathrm{c}}^{0}\left|\mathrm{sinc}\frac{B-B_{\Delta}}{B_{\Phi_{0}}}\right|$ is assumed Barone and Paterno (1982), where $B$ is the applied magnetic field, $B_{\Delta}$ is a constant offset due to background fields, and $B_{\Phi_{0}}$ is a measure for the field periodicity of the corresponding junction, see \freftab:qparams.

The flux penetrating the JJ is given by $\Phi=BA=\allowbreak B(d+2\lambda_{\mathrm{L}})l$ with $A$ the effective junction cross section area, $d=$1\text{,}\mathrm{nm} the thickness of the oxide barrier, $\lambda_{\mathrm{L}}=$16\text{\,}\mathrm{nm} the London penetration depth of Al, and $l$ the length of the junction. From that we can calculate the junctions’ lengths as $l_{1}=$209\text{,}\mathrm{nm} and $l_{2}=$2.46\text{\,}\mathrm{\SIUnitSymbolMicro m}, agreeing well with the design parameters. The reduction of the superconducting gap additionally creates an envelope to the curve, which is discussed in Appendix C. The existence of $\mathrm{JJ}_{2}$ implies that the insulating barrier exists consistently over the large junction area and therefore demonstrates the good quality of the oxide film.

IV Qubit Coherence Times

IV.1 Measurement Sequence

To measure the coherence times of the transmon qubit in a magnetic field, we construct a measurement sequence that ramps the field to a specified value, scans the readout tone to find the shifted resonator frequency $\omega_{\mathrm{r}}(B)$, and scans the probe tone to find the qubit transition $\omega_{01}$. A Rabi sequence to find the length $t_{\pi}$ of a $\pi$-pulse is applied to the qubit and finally a sequence of $T_{1},T_{2}^{\mathrm{Ramsey}}$ and/or $T_{2}^{\mathrm{Echo}}$ measurements is executed to get the desired measurement values. The number of averages and points per trace is reduced to perform the whole sequence for one field value within 10 minutes, despite the low SNR.

The results of multiple sweeps in the range of $B=\pm$23.7\text{,}\mathrm{mT}$$ are shown in \freffig:coherence_quad where red (blue) triangles mark the points taken on an up (down) sweep of the magnetic field. The qubit transition frequency $\omega_{01}$ follows \frefeq:H and shows no hysteresis.

IV.2 Loss Mechanisms

The $T_{1}$ time of the qubit shows a pronounced maximum at low fields and is clearly different on up and down sweeps. To characterize this behavior, we separate the losses of the system as

[TABLE]

where $\Gamma_{\mathrm{hyst}}$ accounts for loss mechanisms showing a hysteretic field dependence, $\Gamma_{\mathrm{non\mathchar 45\relax hyst}}$ collects losses that depend directly on the magnetic field strength, and the losses associated with $\Gamma_{\mathrm{const}}$ do not depend on the magnetic field.

We attribute the hysteretic loss mechanisms $\Gamma_{\mathrm{hyst}}$ mainly to the dissipation introduced by the entering of flux vortices in the thin film superconductor and their movement due to the oscillating RF current, which was already observed for superconducting resonators Bothner et al. (2012). The quality factor of a resonator is a measure for its excitation lifetime and is therefore equivalent to the $T_{1}$ time for the qubit. The shapes and signs of the envelopes of $\tilde{Q}_{\mathrm{l}}$ and $T_{1}$ are generally similar (\freffig:coherence_quadc) and d)), as the two mainly consist of the same material. The observed mismatch can be attributed to their very different geometries and current distributions. From the large aspect ratio of the qubit island with $554\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter and $40\text{\,}\mathrm{nm}$ thickness we conclude that the vortices are mainly generated perpendicular to the film.

Non-hysteretic losses $\Gamma_{\mathrm{non\mathchar 45\relax hyst}}$ are mainly attributed to the dissipation through excitations of the superconductor, i.e. quasiparticles (QP). A linear relation between the QP density and $\Gamma_{\mathrm{QP}}$ has been demonstrated Wang et al. (2014) as well as a quadratic dependence of the QP density on the magnetic field Kwon et al. (2018) and a reduced QP recombination rate in magnetic fields Xi et al. (2013). The QP density is not reported to have a hysteretic dependence on the effective field and the relaxation to an equilibrium QP density is expected to happen within a few $\mathrm{\SIUnitSymbolMicro s}$. The hysteretic vortex configuration however affects the effective field in the superconductor and therefore the QP density.

A small number of pinned flux vortices can also decrease the number of QPs, as the normal conducting cores of the vortices act as QP traps. This can be seen in an increasing $T_{1}$ time for $B-B_{\mathrm{offs}}>0$ ($<0$) on the up (down) sweeps. We attribute the average offset field $B_{\mathrm{offs}}=$8.5\text{,}\mathrm{mT} to the presence of stray fields from magnetized components around the qubit chip, which are partially aligned perpendicular to the chip. Taking into account a small misalignment between coil field and chip of about $\alpha\approx 3$\mathrm{\SIUnitSymbolDegree}, an applied field of $8.5\text{\,}\mathrm{mT}$ would compensate a perpendicular magnetic field of $B\approx 450\,$\mathrm{\SIUnitSymbolMicro T}$$, being on the order of typical stray fields. Measurements after a cycle of the sample temperature above $7\text{\,}\mathrm{K}$ showed $T_{1}$ times on the order of few $\mathrm{\SIUnitSymbolMicro s}$, being comparable to the values for zero applied field at upsweeps and demonstrating the constant background field.

Relaxation sources like Purcell loss, radiative losses, and losses to two-level-systems in the junction and on the surface of the qubit islands do not depend on the magnetic field and are represented by $\Gamma_{\mathrm{const}}$.

IV.3 Increased Magnetic Field

While the previous measurements have been hysteretic but repeatable, we now further increase the magnetic field to stronger fields and see quantum coherence of the qubit up to values of $B_{\mathrm{appl}}\approx$40\text{,}\mathrm{mT} (\freffig:highfield). Although $T_{1}=$0.49\text{\,}\mathrm{\SIUnitSymbolMicro s} is significantly reduced, we can observe Rabi oscillations and an exponential decay after a $\pi$ pulse, as demonstrated in the inset of \freffig:highfield. At these fields, the quality factor $Q_{\mathrm{l}}$ of the resonator is significantly reduced, explaining the low SNR. Together with the decreased $T_{1}$, $T_{2}$ times and the resulting broadening of the qubit linewidth, the qubit transition could not be tracked for even higher magnetic fields, as visible in \freffig:two_jjb).

The subsequent down sweep does not show a pronounced maximum as before but only a slight increase in $T_{1}$ over a broader range. We also see a fine structure in the data, showing multiple drops in $T_{1}$ which coincide with the onset of a deviation from the Fraunhofer pattern, followed by a jump in frequency. We attribute this effect to the presence of flux vortices in the qubit islands due to the previously applied high fields. Their local field influences the field seen by the junction and therefore qubit frequency and coherence.

V Pure dephasing rate

To calculate the pure dephasing rate of the qubit $\Gamma_{\varphi}$ from measured values, we take

[TABLE]

where $\Gamma_{1}=1/T_{1}$ and $\Gamma_{2}^{\mathrm{Ramsey}}=1/T_{2}^{\mathrm{Ramsey}}$ are the decay and Ramsey dephasing rates. In order to have physically connected $\Gamma_{1}$ and $\Gamma_{2}^{\mathrm{Ramsey}}$ rates, we acquire the measurement points for both rates in turn, so that temporal fluctuations of the qubit properties influence both measurements likewise Schlör et al. . The resulting data are shown in \freffig:tphi and fits to a straight line of a constant pure dephasing $\Gamma_{\varphi}=$93.9\text{,}\mathrm{kHz}$$. For the regions of a steep slope of $\omega_{01}(B)$, a higher dephasing rate would be expected due to the stronger sensitivity to flux noise. However, a clear correlation between $\Gamma_{\varphi}$ and $\omega_{01}$ can not be seen from the data. The causality between noise in the solenoid current $S_{I}$ and the resultant $\Gamma_{\varphi}^{I}$ is given by Ithier et al. (2005):

[TABLE]

where the relevant scale for $\omega$ is the time between the Ramsey pulses, being on the order of $\omega/2\pi\approx$100\text{,}\mathrm{kHz}$$.

From \frefeq:H we calculate a slope of the qubit transition frequency of $\left({\partial\omega_{01}}/{\partial I}\right)/2\pi=$652\text{,}\mathrm{MHz}\text{,}{\mathrm{A}}^{-1} at $B=$21\text{\,}\mathrm{mT} and $\omega_{01}(B)/2\pi=$4.70\text{,}\mathrm{GHz}. Considering the measured power spectral density of our current source $S_{I}\approx${10}^{-15}\text{\,}{\mathrm{A}}^{2}\text{\,}{\mathrm{Hz}}^{-1}, this results in $\Gamma_{\varphi}^{I}=$53\text{,}\mathrm{kHz}$$, well below our measured $\Gamma_{\varphi}$. We conclude that for the main part of the qubit spectrum, $\Gamma_{\varphi}$ is not limited by current fluctuations or other fluctuating stray magnetic fields and the qubit coherence is governed by a magnetic field independent dephasing rate.

VI Conclusion

In this article, we demonstrated the quantum coherence of a superconducting transmon qubit in magnetic fields up to a flux density of $40\text{\,}\mathrm{mT}$, which increases their usability range as versatile sensors and is a promising finding for future developments of superconductor-magnet-hybrid devices. The influence of the second, spurious junction in circuits fabricated by shadow angle evaporation was shown, where its large area gives rise to a higher sensitivity to in-plane magnetic fields. To calculate the influence of this additional junction on the qubit transition frequency, an analytic formula for the approximated transmon Hamiltonian featuring two serial junctions was derived. Finally we studied the pure dephasing rate and found it to be mainly independent of the magnetic field.

Acknowledgements.

The authors are grateful for quantum circuits provided by D. Pappas, M. Sandberg, and M. Vissers. This work was supported by the European Research Council (ERC) under the Grant Agreement 648011, Deutsche Forschungsgemeinschaft (DFG) within projects WE4359/7-1 and INST 121384/138-1, and the Initiative and Networking Fund of the Helmholtz Association. We acknowledge financial support by the Carl-Zeiss-Foundation (A.S.) and the Helmholtz International Research School for Teratronics (T.W. and M.P.). A.V.U. acknowledges partial support from the Ministry of Education and Science of the Russian Federation in the framework of the contract No. K2-2016-063.

Supplementary Material for “Transmon Qubit in a Magnetic Field: Evolution of Coherence and Transition Frequency”

Appendix A Resonator in a Magnetic Field

The measured data on the field dependence of resonance frequency and quality factor of our readout resonator correspond well to already published data Bothner et al. (2012) for an in-plane magnetic field. In their publication, the loss rate $\Gamma_{\mathrm{i}}$ is calculated by using the classical Bean model Bean (1962, 1964) and their simulation matches our data very well for the case of a weakly inhomogeneous RF current distribution. Although \freffig:resc) suggests that the resonator is completely interspersed with flux vortices at $B\approx$100\text{,}\mathrm{mT}, a closer look in the data does not support this statement, as the phase signal becomes very weak in the region of $\left|B\right|>$100\text{\,}\mathrm{mT} and the circle fit Probst et al. (2015) does not converge. Fitting the measured data with a Lorentzian still shows a difference between up and down sweep for the loaded quality factor (data not shown).

From the circle fit data, we extract the coupling quality factor to be $Q_{\mathrm{c}}=$9.3\pm 1.3\text{,}\mathrm{MHz}$$. This quantity is defined by the geometric coupling of transmission line and resonator and no significant change over the measured range in $B$ can be seen.

Both $f_{\mathrm{r}}(B)$ and $Q_{\mathrm{i}}(B)$ are perfectly symmetric when taking into account the previously determined offset of $B_{\mathrm{offs}}=$8.5\text{,}\mathrm{mT}$$.

Appendix B Two Junction Model

For the derivation of the two junction transmon Hamiltonian, we start with Kirchhoff’s current law:

[TABLE]

where we use the current-phase-relation \frefeq:curphase derived in the main part. Without loss of generality, $r=I_{\mathrm{c},2}/I_{\mathrm{c},1}\geq 1$ was chosen there.

The Lagrangian for this dynamics is then given by:

[TABLE]

Introducing the charging energy $E_{\mathrm{C}}=\frac{\mathrm{e}^{2}}{2C}$, the Josephson energy $E_{\mathrm{J,1}}=\frac{I_{\mathrm{c,1}}\Phi_{0}}{2\pi}$ and the number and phase operators $N$ and $\varphi=\frac{2\pi\phi}{\Phi_{0}}$ with $[N,\varphi]=\mathrm{i}$, we end up with the system Hamiltonian

[TABLE]

We now do a Taylor expansion in $\varphi$ to fourth order and neglecting constant terms we get for the approximate Hamiltonian

[TABLE]

Comparing the harmonic part to a standard quantum harmonic oscillator, we find

[TABLE]

Together with the bosonic commutation relation $[a,a^{\dagger}]=1$ and neglecting all constant terms and terms without pairs of $a$ and $a^{\dagger}$, we get

[TABLE]

Appendix C Qubit Transition Frequency for Reduced Superconducting Gap

In the main part of the paper, we assume that the overall envelope of $\omega_{01}(B)$ comes from magnetic interference in the main qubit JJ. However, the overall envelope can also be explained by a reduction of the superconducting gap. Taking the Ambegaokar-Baratoff relation Ambegaokar and Baratoff (1963)

[TABLE]

we see that the influence of the $\tanh$ term is negligible for our values of $T$ and $\Delta$, resulting in $I_{\mathrm{c}}\propto\Delta$. Together with $\Delta(B)=\Delta_{0}\sqrt{1-(B/B_{\mathrm{c}})^{2}}$, where $B_{\mathrm{c}}$ is the critical field, we get a relation for $I_{\mathrm{c},1}(B)$ and can calculate the qubit transitions. To reproduce the same transitions as in the main part, we use $B_{\mathrm{c}}=$168\text{,}\mathrm{mT}$$; see the blue line in \freffig:qubit_delta. In this limit, $\mathrm{JJ_{1}}$ is assumed to be point-like, i.e. the periodic Fraunhofer-like reduction of $I_{\mathrm{c}}$ is neglected. Within the magnetic field range accessible by our measurements, no deviation from the periodic interference vortex model can be found, and the two effects can not be distinguished with our data. In reality, both effects coexist at the same time and reduce the critical current.

Appendix D Modeling Boundaries for $\Gamma_{1}$

In the main part, the qubit losses are modeled by

[TABLE]

where $\Gamma_{\mathrm{const}}$ is assumed to be independent from $B$ and we take the quasiparticle losses $\Gamma_{\mathrm{QP}}\approx B^{2}$ as the main contribution to the non-hysteretic $\Gamma_{\mathrm{non\mathchar 45\relax hyst}}$. \freffig:qubit_gamma shows our measured data for $\Gamma_{1}$ together with a lower limitation modeled by $\Gamma=\Gamma_{\mathrm{const}}+C(B-B_{\mathrm{offs}})^{2}$ Wang et al. (2014); Kwon et al. (2018); Xi et al. (2013). We fit the envelope of our data and get $\Gamma_{\mathrm{const}}=$53.4\text{,}\mathrm{kHz}, $C=$0.785\text{\,}\mathrm{kHz}\text{\,}{\mathrm{mT}}^{-2} and $B_{\mathrm{offs}}=$2.25\text{,}\mathrm{mT}$$. The remaining hysteretic losses are assumed to come from the entering and movement of flux vortices.

Although the parabola shown in \freffig:qubit_gamma is a proper envelope for the measured data, we do not want to make any claim that this is a proof for our chosen partitioning of the loss mechanisms. In fact, the different loss mechanisms are not distinguishable by our measurement technique and our partitioning only represents the most obvious loss channels and their dependence on magnetic fields.

Appendix E Sample and Setup

For the measurements, we used two different measurement setups: The spectroscopic setup in \freffig:sampleb) is used for fast measurements with continuous wave signals, providing a reliable amplitude measurement. This setup was used for measuring the qubit frequency in \freffig:two_jj in the main part and for the additional resonator measurements in \freffig:res. As the other measurements require pulsed microwave sequences, a home-built time domain setup was used, cf. \freffig:samplea). With the IQ mixers as nonlinear components, this setup does not provide a linear amplitude relation and is therefore not suitable for the calculation of quality factors.

In the cryogenic setup, we attenuate the signal on different stages for thermalization and use microwave cables with low thermal conductance to reduce the heat input to the cold stages, giving a total attenuation of about $-70\text{\,}\mathrm{dB}$. The reflected signal is amplified by a cryogenic low-noise HEMT amplifier. The sample is shielded from high-frequency noise, infrared radiation and noise from the HEMT by highpass filters, infrared filters, and circulators.