Implementation of a transmon qubit using superconducting granular aluminum

The high kinetic inductance offered by granular aluminum (grAl) has recently been employed for linear inductors in superconducting high-impedance qubits and kinetic inductance detectors. Due to its large critical current density compared to typical Josephson junctions, its resilience to external magnetic fields, and its low dissipation, grAl may also provide a robust source of non-linearity for strongly driven quantum circuits, topological superconductivity, and hybrid systems. Having said that, can the grAl non-linearity be sufficient to build a qubit? Here we show that a small grAl volume ($10 \times 200 \times 500 \,\mathrm{nm^3}$) shunted by a thin film aluminum capacitor results in a microwave oscillator with anharmonicity $\alpha$ two orders of magnitude larger than its spectral linewidth $\Gamma_{01}$, effectively forming a transmon qubit. With increasing drive power, we observe several multi-photon transitions starting from the ground state, from which we extract $\alpha = 2 \pi \times 4.48\,\mathrm{MHz}$. Resonance fluorescence measurements of the $|0>\rightarrow |1>$ transition yield an intrinsic qubit linewidth $\gamma = 2 \pi \times 10\,\mathrm{kHz}$, corresponding to a lifetime of $16\,\mathrm{\mu s}$. This linewidth remains below $2 \pi \times 150\,\mathrm{kHz}$ for in-plane magnetic fields up to $\sim70\,\mathrm{mT}$.

The high kinetic inductance offered by granular aluminum (grAl) has recently been employed for linear inductors in superconducting high-impedance qubits and kinetic inductance detectors.Due to its large critical current density compared to typical Josephson junctions, its resilience to external magnetic fields, and its low dissipation, grAl may also provide a robust source of non-linearity for strongly driven quantum circuits, topological superconductivity, and hybrid systems.Having said that, can the grAl non-linearity be sufficient to build a qubit?Here we show that a small grAl volume (10×200×500 nm 3 ) shunted by a thin film aluminum capacitor results in a microwave oscillator with anharmonicity α two orders of magnitude larger than its spectral linewidth Γ01, effectively forming a transmon qubit.With increasing drive power, we observe several multi-photon transitions starting from the ground state, from which we extract α = 2π×4.48MHz.Resonance fluorescence measurements of the |0⟩ → |1⟩ transition yield an intrinsic qubit linewidth γ = 2π× 10 kHz, corresponding to a lifetime of 16 µs, as confirmed by pulsed time-domain measurements.This linewidth remains below 2π×150 kHz for in-plane magnetic fields up to ∼ 70 mT.
Superconducting circuits are part of a growing group of hardware platforms which successfully demonstrated quantum information processing, from quantum error correction to quantum limited amplification [1].Some of the most promising platforms, such as spin qubits [2][3][4], topological materials [5][6][7], magnons [8] or molecular electronics [9][10][11][12], benefit from hybrid architectures where superconducting circuits can provide unique functionalities, in particular dispersive readout [13] and high impedance couplers.The success of superconducting circuits is linked to the availability of non-linear elements with high intrinsic coherence, namely Josephson junctions (JJs) fabricated by thermal oxidation from thin film aluminum (Al) [14].However, their applicability in hybrid systems [15,16] is limited by the low critical field of Al [17] and by the emergence of quantum interference effects in the JJs, even for magnetic fields aligned inplane [18].Here we show that the JJ can be replaced by a small volume of granular aluminum (grAl) [19] providing enough non-linearity to implement a superconducting transmon qubit [20], which we operate in magnetic fields up to ∼ 0.1 T.
Similar to JJ arrays, the non-linearity of the grAl kinetic inductance stems from the Josephson coupling between neighboring grains, and it is inversely proportional to the critical current density j c and the volume of the film V grAl [29].This non-linearity gives rise to a frequency shift K of the fundamental plasmon mode ω 1 for each added photon (n → n+1).Although the values K(n) depend on the transition number n, to lowest order they can be approximated by a constant self-Kerr coefficient jcV grAl , where C is a numerical factor close to unity that depends on the current distribution, e is the electron charge, and a is the grain size [29].
By reducing the grAl volume and the critical current density, one can potentially increase K(1) to a value much larger than the transition linewidth Γ 01 , allowing to map a qubit to the first two levels |0⟩ and |1⟩, similar to a transmon qubit [20].Following this approach, we construct a circuit with a transition frequency f 1 = 7.4887 GHz by connecting an Al capacitor to a small volume of grAl, V grAl = 10×200×500 nm 3 , with critical current density j c ≈ 0.4 mA/µm 2 (cf.Fig. 1).For this structure the estimated anharmonicity α = K(1) is in the MHz range [29].Indeed, as we show below, the measured value is α = 2π×4.48MHz, which is much larger than the transition linewidth Γ 01 = 2π×50 kHz, effectively implementing a relatively low anharmonicity transmon qubit.
Figure 1 shows a typical copper waveguide sample holder, together with the circuit design of our qubit, consisting of a grAl film shunted by an Al capacitor.From finite-element simulations we extract a shunt capacitance C s ≈ 137 fF and a geometric stray inductance L s ≈ 0.45 nH (cf.App.B).The outer electrode of the capacitor surrounds the inner electrode almost completely, except for a gap of width w (cf.Fig. 1b), which is used to tune the coupling rate κ between the qubit and the waveguide sample holder (cf.App.B).
The sample is fabricated on a sapphire wafer in a single-step lithography by performing a three-angle shadow evaporation.First, a 10 nm thick grAl layer with room temperature resistivity ρ n = 1800±200 µΩ cm and corresponding critical temperature T c = 1.9 K is deposited at zero-angle, followed by two 40 nm thick Al layers evaporated at ±35 • (cf.App.J).Thanks to this procedure, only a small grAl volume, highlighted in blue in Fig. 1d, remains unshunted by the pure Al layers and participates in the electromagnetic mode with a kinetic  c) we show the real and imaginary part of the reflection coefficient Re(S11) and Im(S11), respectively, as a function of the detuning between the probe frequency f and the qubit frequency f1.When increasing the probe power Pin, the response becomes elliptic in the quadrature plane, which is the signature of resonance fluorescence of a two-level-system [34].The black lines indicate fits to the experimental data according to Eq. 1.The only fitting parameter is the Rabi-frequency ΩR; κ and γ are fixed by the fit to the low power response (cf.panel a).In d) we show Ω 2 R as a function of incident on-chip power Pin.For a two level system, given by the limit ΩR ≪ α, we expect a linear dependence, as confirmed by the black dashed line passing through the coordinate origin.
inductance L K = 2.9 nH, constituting 87 % of the total inductance.
We characterize the grAl transmon by performing a single-port measurement of the complex reflection coefficient S 11 as a function of probe frequency f , in the vicinity of the resonant frequency f 1 (cf.Fig. 2).In the limit of weak driving, Ω R ≪ α, where Ω R is the Rabi frequency, we can treat the transmon as a two-level system.If the decoherence rate is dominated by the energy relaxation rate Γ 01 , similarly to Ref. [34] the complex reflection coefficient is where, ∆ = ω 0 −ω is the frequency detuning between the drive and the qubit frequency, and i is the unit imaginary number.In contrast to a harmonic oscillator, the reflection coefficient of a qubit deviates from a circle in the quadrature plane and becomes increasingly elliptic with drive power (cf.App.C).
Figure 2 depicts the measured reflection coefficient S 11 as a function of qubit-drive detuning for incident on-chip powers P in ranging between −168 dBm and −138 dBm.From a least-square fit to Eq. 1 (solid black lines), we extract the qubit frequency f 1 = 7.4887 GHz, as well as the internal loss and external coupling rates γ = 2π×10 kHz and κ = 2π×40 kHz, respectively.The corresponding energy relaxation times due to radiation into the waveguide [35] and internal losses are T 1,κ ≈ 4 µs and T 1,γ ≈ 16 µs, respectively.As shown in Fig. 2d, the Rabi frequency shows a linear dependence with drive amplitude, as expected for a two-level system.From a linear fit passing through the coordinate origin [34], we calibrate the attenuation of the input line to 103 dB, within 3 dB from room-temperature estimates.
For drive powers P in > −138 dBm, we observe additional features in the reflection coefficient S 11 emerging at frequencies f n below the qubit frequency f 1 (cf.Fig. 3a left hand panel).Similar to the high power spectroscopy of JJ transmon qubits [36,37], these features are multi-photon transitions into higher energy eigenstates E n starting from the ground state E 0 , observed at frequencies f n = (E n −E 0 )/(nh), where n is the level number (see App. D for numerical simulations of the spectrum, and App.E for two-tone spectroscopy).From the frequency detuning between the first two transitions (red markers in Fig. 3a right hand panel), we extract a qubit anharmonicity α = 2π×4.48MHz.
Generally, for a JJ transmon the anharmonicity is given by the charging energy E c,s = e 2 /2C s associated with the shunt capacitance [20], which for our geometry is E c,s /ℏ = 2π×141 MHz.In the case of an array of N JJs, the anharmonicity is reduced by N 2 [38,39], which implies N = E c,s /ℏα ≈ 6 for the JJ array implemented by our grAl volume [29].The corresponding effective junctions are therefore separated by ∼ 80 nm, spanning approximately ten grains.This result is in agreement with recent scanning tunneling microscopy measurements performed on similar grAl films, which evidenced the collective charging of clusters of grains [40].
From the measured multi-photon transition frequencies f n , we calculate the non-linear frequency shift As shown in the right hand panel of Fig. 3a (right hand axis), we find that K(n) monotonically increases with n, likely due to the contribution of higher order terms in the expansion of the Josephson potential, currently not included in the model [29].
In the top panel of Fig. 3b we show measurements for three different drive powers P in = −121, −106 and = 5.1 µs, as well as the corresponding pure dephasing times 28 µs and 46 µs, respectively.c) Time stability of the energy relaxation time (grey triangles) and the deviation ∆ of the qubit transition frequency from its average (pink markers corresponding to the right-hand axis).The pink shaded area is the uncertainty for the qubit frequency measurement.The inverted black triangles show the intrinsic energy relaxation time T1,γ = T1T1,κ/(T1,κ−T1), where T1,κ = 3.1 µs (dashed line) is the limit due to spontaneous emission into the waveguide.Notably, the T1 and frequency stability data were taken in different cool downs.
−101 dBm.At any drive power in this range several multi-photon peaks are visible.The linewidth of each transition broadens with power, as illustrated in the bottom panel of Fig. 3b for the 3rd and the 10th multiphoton transition.The visibility of the peaks and the background response of the phase is in remarkable agreement with the master-equation simulation presented in App.D. The broadening of the n = 10 transition, compared to n = 3, can be explained by offset charge dispersion, which increases exponentially with n [20].
The time evolution of the qubit can also be measured using resonance fluorescence [41].To enhance the signalto-noise ratio, we added a dimer Josephson junction array amplifier (DJJAA) [42], operated in reflection, with a power gain G 0 ≳ 20 dB.In Fig. 4a we show the measured Rabi-oscillations between the qubit's ground and first excited state, obtained by applying a Gaussian shaped manipulation pulse of duration τ m .For the readout, we demodulate the second half of a 3.2 µs long pulse with rectangular envelope and power P r ≈ −152 dBm.Both readout and manipulation pulses are applied on resonance with the qubit transition frequency.As expected, the Rabi-frequency increases linearly with the drive amplitude.
The energy relaxation time T 1 = 1/Γ 01 , shown in the top panel of Fig. 4b, is measured by preparing the qubit in the excited state using a 723 ns π-pulse, and monitoring the dependence of the qubit population inversion as a function of the wait time τ .The results are averaged over 3×10 5 repetitions during the course of 8 min.The observed energy decay is fitted by a single exponential function with characteristic decay time T 1 = 2.8 µs.The bottom panel of Fig. 4b shows Ramsey-fringes and Hahnecho coherence measurements, performed with the same π-pulse, yielding T 2 = 4.7 µs and T echo 2 = 5.1 µs, respectively.The corresponding pure dephasing times are 28 µs and 46 µs.These measurements reveal energy relaxation as the main decoherence mechanism, justifying the limit T 2 ≈ 2T 1 used for the derivation of the reflection coefficient in Eq. 1.
In Fig. 2c, we show the fluctuations of T 1 and the qubit frequency monitored over 15 h.Although spontaneous emission into the waveguide limits the energy relaxation time to T 1,κ = 3.1 µs, we can infer the intrinsic energy relaxation T 1,γ (inverted triangles) from the measured T 1 values using T 1,γ = T 1 T 1,κ /(T 1,κ −T 1 ).The obtained average intrinsic energy relaxation time is T 1,γ = 20 +22 −6 µs, consistent with the γ −1 values extracted from spectroscopy (see Fig. 2).The pink markers indicate the detuning ∆ of the qubit frequency from its average value.The observed total change is on the order of the intrinsic linewidth γ.
By applying a magnetic field B y , aligned in-plane with the sample, we observe a continuous decrease of the qubit frequency f 1 (B y ), as plotted in Fig. 5a.The measurements are performed in two seperate cooldowns, with (filled crosses) and without (open pentagons) an outer superconducting Al shield.When employing the shield, the maximal field is limited to ∼ 70 mT, after which the shield becomes affected by the field coils, introducing distortions in the field alignment.Due to the large critical field of grAl (∼ 4−5 T [24]), the change in frequency is primarily due to the lowering of the Al gap ∆ Al , which leads to an increase of the kinetic inductance of the Al wires connecting the electrodes, accounting for ∼ 10 % of the total inductance.
Figure 5b depicts the internal quality factor Q i as a function of the in-plane magnetic field B y measured with (filled triangles) and without (open pentagons and triangles) an outer Al shield.Compared to the data depicted in Fig. 2, we attribute the lower internal quality factor in these three cooldowns to the removal of the µ-metal shield, which likely results in an increase of the (stray) B z field.In the current design the Al pads are the most field susceptible components, rendering the qubit frequency and internal quality factor particularly sensitive to outof-plane magnetic fields (cf.App.F), as illustrated by the factor of three difference in Q i values in the unshielded case for nominally identical setups.Finally, it is important to note that the absolute value of the non-linear frequency shift K is not expected to change in magnetic field, because the ratio ω 2 1 /j c in the expression for K is independent of the grAl superconducting gap.Indeed, we measure a constant K up to ∼ 100 mT (cf.App.G), confirming the grAl transmon's resilience to moderate magnetic fields.
In summary, we have shown that using small volumes of grAl can provide enough non-linearity to implement magnetic field resilient superconducting qubits.We have implemented a transmon qubit [20] with an anharmonicity α = 2π×4.48MHz, which, although smaller than the typical values for JJ based transmons, is much larger than the qubit linewidth Γ 01 = 2π×50 kHz.This enables timedomain manipulation and measurement of the qubit, from which we extracted an intrinsic T 1,γ = 20 +22 −6 µs, and a pure dephasing time exceeding tens of microseconds.We observe multi-photon transitions to the 20th order, showcasing the robustness of the grAl transmon to external drives; a valuable asset for strongly driven quantum circuits [44], in particular in the context of bosonic codes for quantum information [45][46][47].Measuring the same qubit with less shielding, in the presence of in-plane magnetic fields up to ∼ 70 mT, the intrinsic linewidth remains below γ = 2π×150 kHz, limited by the pure Al capacitor pads.
Following this proof of principle demonstration, future developments will focus on replacing all pure Al components with more field resilient materials, such as low resistivity grAl [32] or Nb compounds [21,22,48].This will allow to operate coherent superconducting qubits in magnetic fields beyond 1 T. In the current design, the Al pads play the role of phonon and quasiparticle traps [32,49], enhancing the coherence.In future designs, in which Al is substituted by other superconductors with higher gap and critical field, quasiparticle poisoning could be mitigated by adding dedicated phonon traps disconnected from the qubit [50,51].In order to avoid lowering T 1 by the Purcell effect, following the recent non-perturbative design reported in Ref. [52], a cross-Kerr interaction to an ancilla mode could be used for readout.The limitations on qubit manipulation fidelity can be mitigated with DRAG pulses [53] and by adapting the design to decrease the grAl volume and critical current density, thereby increasing the anharmonicity.Furthermore, larger values of anharmonicity might be achieved in fluxonium qubits [27,54].
We are grateful to H. Rotzinger, U.     Appendix B: Finite-element method simulations The eigenfrequency and external coupling rate of the circuit to the waveguide sample holder is simulated with a commercial finite-element method simulator (HFSS -High frequency structure simulator).The inductive contribution of the granular aluminum (grAl) volume is modeled with a linear lumped-element inductor L K .Capacitive contributions arising from the grAl microstructure are not considered.In order to extract the lumpedelement shunt capacitance C s and geometric stray inductance L s of our circuit design, we sweep the inductance L K while keeping the design geometry fixed (cf.Fig. 7a).The dimensions of the circuit geometry are listed in Tab.B. The simulated eigenfrequencies are fitted to the lumped-element model yielding C s = 137 fF and L s = 450 pH (cf.Fig. 7a).
The external quality factor can be varied over three orders of magnitude by changing the gap w in the outer electrode (cf.Fig. 1b main text and Fig. 7b).As shown in Fig. 7c, we confirm that Q c does not significantly change with the lateral position x, up to x = ±8 mm.The measured sample is shifted by x = 4 mm from the center.
Table I.Geometrical parameters of the sample (cf.Fig. 1b): sample width b, sample height h, gap in outer electrode w, length of the bridge connecting the capacitor pads l b , width of capacitor pads w f , gap between capacitor pads g f .b h w l b w f g f (µm) (µm) (µm) (µm) (µm) (µm) 1000 800 300 400 100 100 Appendix C: Resonance fluorescence: reflection coefficient In the limit of weak driving Ω R ≪ α, we describe our transmon [20] as an effective two-level system [34].Under this assumption, the reflection coefficient is where κ is the single-photon coupling rate to the waveguide, ⟨σ − ⟩ is the expectation value of the lowering operator in the two dimensional qubit subspace {|0⟩ , |1⟩}, and α in is the expectation value of the annihilation operator of the incident bosonic single-mode field amplitude a in [60] .Here, we assume a classical drive a in = α in e −iωt , which has in general a complex amplitude α in .The expectation value of the lowering operator is expressed using the Pauli operators σ x and σ y : ⟨σ − ⟩ = (⟨σ x ⟩−i ⟨σ y ⟩)/2.The steady state expectation values (time t → ∞) for the Pauli operators σ x , σ y , σ z under a continuous drive with amplitude Ω R and detuning ∆ = ω q −ω, in the presence of qubit energy relaxation at rate Γ 01 and qubit dephasing at rate Γ 2 = Γ 01 /2+Γ φ , (Γ φ : pure dephasing rate) are the following [60]: Using Eq.C2 and Eq.C3, Inserting Eq.C5 into Eq.C1, the reflection coefficient writes The relation between the Rabi frequency Ω R and the drive amplitude Using Eq.C7 and in the limit of negligible pure dephasing Γ φ ≪ Γ 01 , Eq. C6 simplifies to which is Eq. 1 in the main text.The factor in front of the second term of Eq.C8 is the coupling efficiency κ/Γ 01 , with Γ 01 = κ+γ.It is a measure of the relative size of the internal loss rate γ compared to the external coupling rate κ.
Figure 8 shows the Rabi frequency extracted from spectroscopy, similar to Fig. 2, and the time-domain measurements shown in Fig. 4a.The measurements follow the dependence expected for a two level system given by Eq.C7.

Appendix D: Numerical calculation of the Kerr Hamiltonian
Since our transmon qubit has a relatively small anharmonicty α (cf.Fig. 3 main text), we can simulate our circuit as an anharmonic oscillator with self-Kerr coefficient K = 2π×4.5MHz.This model is further justified by the weak dependence of K on the level number n as measured in Fig. 3 (main text).The driven Kerr Hamiltonian expressed in the rotating frame of the coherent drive applied at frequency ω is Here, ∆ = ω 1 −ω is the detuning between the fundamental transition frequency ω 1 and the drive tone, a † and a are the bosonic single-mode field amplitude creation and annihilation operators, respectively, and Ω is the drive amplitude.Notably, the drive amplitude Ω corresponds to the Rabi-frequency Ω R , only in the limits K ≫ κ and Ω ≪ K.Both criteria are met in our experiment.
In analogy to Eq. C1 in the case of a two-level system, the reflection coefficient of an (anharmonic) oscillator is [61,62] We calculate the expectation values ⟨a⟩ as a function of detuning ∆ and drive amplitude Ω by solving the corresponding master equation numerically using Qutip [63,64].The master equation in Lindblad form in the presence of energy relaxation at rate Γ 01 is Here, ρ s is the system density matrix, H Kerr is the driven Kerr Hamiltonian given in Eq.D1 and D[a](ρ) is the Lindblad superoperator introducing single-photon dissipation For our numerical calculations, we use internal and external decay rates γ = 2π×10 kHz and κ = 2π×40 kHz, respectively, and a total energy relaxation rate Γ 01 = κ+γ = 2π×50 kHz.The dimension of the considered Hilbert space is N level = 30.The drive amplitudes Ω = 2 √ κα in are chosen to coincide with the values used in the experiment, with α in = P in /(ℏw 0 ).
Figure 9 depicts the numerically calculated reflection coefficients S 11 and the corresponding least-square fits using Eq. 1.The colors are related to the drive power similarly to Fig. 2. The linear fit to the extracted Rabi frequencies shown in Fig. 9d confirms that the two-level approximation (qubit limit) is valid in the parameter space of our experiment.
We reproduce the experimental results presented in the main text in Fig. 3 by numerically calculating the reflection coefficient based on the treatment discussed in App.D, for the same range of probe frequency and power.Figure 10a depicts the phase of the reflection coefficient arg(S 11 ) as a function of the incident power P in and probe frequency f .Similar to the experiment, multi-photon transitions at frequencies f n = (E n −E 0 )/(nh) become visible.From a linear fit to the extracted multi-photon frequencies (cf.Fig. 10a right panel), we recover the self-Kerr coefficient K = 2π×4.5MHz set in the calculation, as expected.Since the Kerr Hamiltonian in Eq.D1 does not contain beyond 4th order terms, in contrast with the experimental results, the simulated K(n) is independent of the level number n.This confirms that the Hilbertspace dimension N level = 30 was chosen sufficiently large.
For comparison to Fig. 3b shown in the main text, Fig. 10b shows the phase of the calculated reflection coefficient arg(S 11 ) as a function of the probe frequency f for three drive powers P in = −121, −106 and −101 dBm.
We would like to add that multi-photon transitions have been measured in flux qubits [66,67] and Cooper pairs boxes [68,69], with the notable difference that in these previous cases the transitions are between the ground and the first excited state [70], while in our case they are between the ground and higher excited states.
Appendix E: Two-tone spectroscopy Figure 11 shows a two-tone spectroscopy of the qubit sample in the vicinity of its fundamental transition frequency f 1 .We apply a fixed frequency drive tone at ∆ drive = 2π×200 kHz above f 1 with varying drive power P drive .Simultaneously, we measure the reflection coefficient S 11 by appling a weak probe tone at varying frequency f and constant power P in = −160 dBm.
For small drive powers, only a single response at the qubit frequency is visible in the phase of the reflection coefficient arg(S 11 ) shown in Fig. 11a.With increasing drive power, the occupation of the first excited state |1⟩ increases and a second feature becomes visible 3.9 MHz below the qubit frequency, corresponding to the single-photon transition between the first and the second excited state |1⟩ → |2⟩, and quantifying the qubit anharmonicity α.The measured anharmonicity α = 2π×3.9MHz and the qubit frequency f 1 = 7.6790 GHz are slightly different compared to the values reported in the main text, due to the fact that the measurements were taken in different cooldowns and the sample parameters changed (most likely the grAl resistivity ρ n ).In total we performed four cooldowns, summarized in Tab.II.
The expectation value of the photon number operator ⟨a † a⟩ is shown in Fig. 11c, numerically calculated for a two-level (qubit, red) and a multi-level system (qudit, blue) with constant anharmonicity (cf.App.D).In contrast to a multi-level system, for a qubit the steady state occupation number versus drive power saturates at 0.5.The red shaded area in Fig. 11c highlights the qubit limit in which the difference between the occupation numbers of an ideal qubit and our system (main text α = 2π×4.48MHz) is below 1 %.The black arrow indicates the maximal drive power used for the resonance fluorescence measurements shown in Fig. 2.

Appendix F: Magnetic field alignment
Figure 12a shows the change in qubit frequency δf 1 = f 1 (B z )−f 1 , as a function of B z for B y = 0.The field sweeps are the following: i) 0 mT → −0.2 mT (blue), ii) −0.2 mT → 0.2 mT (green), iii) 0.2 mT → 0 mT (purple), with the black arrow indicating the starting point.The qubit frequency decreases with increasing field magnitude and returns to its initial value when the field is swept in the opposite direction.For fields applied in positive z-direction, we observe several jumps and a much less smooth change in the qubit frequency with field.This observation is not strictly related to the positive z-direction, but depends on the order of the measurement sequence.Figure 12b shows the internal quality factor Q i extracted from the same measurement sequence as the qubit frequency shown in panel a.
The measurement shown in Fig. 12 emphasizes the circuit's pronounced susceptibility to out-of-plane magnetic fields.Interestingly, the maximum of the qubit frequency does not necessarily coincide with the maximal internal quality factor.The criterium for the in-plane field alignment is maximizing the qubit frequency versus B z .By performing similar sweeps of B z for different values of B y , we estimate the misalignment between the HH field and the sample's in-plane direction to be 0.7 • .
Appendix G: Qubit response to in-plane magnetic fields The in-plane magnetic field dependence of the qubit transition frequency f 1 (B) is calculated by mapping the qubit onto a linearized, lumped-element circuit model consisting of three inductive contributions in series, which are shunted by a capacitance C s = 137 fF.The inductive contributions arise from a field-independenent geometric inductance L s = 0.45 nH, and two fielddependent kinetic inductances associated with the pure Al and grAl thin films, L k,Al (B) and L k,grAL (B), respectively.The lumped-element model transition frequency is The field dependence of the kinetic contributions is derived from Mattis-Bardeen theory for superconductors in the dirty limit [71] L where R n is the normal state resistance and ∆(B, T ) is the magnetic field and temperature dependent gap parameter.The dependence of the gap parameter on magnetic fields applied in-plane is derived from a two-fluid transition.The measured anharmonicity α = 2π×3.9MHz, as well as the qubit frequency f1 = 7.6790 GHz are slightly different than the values reported in the main text, due to the fact that the measurements were taken in different cooldowns.In total we performed four cooldowns, summarized in Tab.II.Similar to the fundamental transition, the second transition also splits with increasing P drive , an effect denoted Autler-Townes [65].b) Extracted frequency splittings δf for the first two transitions.The black lines correspond to the theoretical predictions 2ΩR (Mollow triplet sidebands) and ΩR (Autler-Townes), respectively, with ΩR = √ ∆ 2 +Ω 2 and Ω = 4κP drive /hf drive .c) Expectation value for the photon number operator ⟨a † a⟩ as a function of the drive power P drive numerically calculated for a qubit (red line) and for an anharmonic multi-level oscillator (cf.App.D). model (cf.p. 392 and 393 in [43]) where ∆ 00 is the gap parameter at zero temperature and zero magnetic field, and B c is the critical magnetic flux density above which the pair correlation is zero.Due to the fact that the critical magnetic flux density of grAl [24] is two orders of magnitude higher than that our pure Al films [17], we consider L k,grAl (B) to be constant.However, for the Al kinetic inductance, by inserting Eq.G3 into Eq.G2, we find where L k,Al is the kinetic inductance in zero field.Using Eq.G1 we fit the data presented in Fig. 5a in the main text, and we obtain L k,Al = 200±5 pH and B c,Al = 150± 5 mT.The 200 pH value of L k,Al corresponds to the intrinsic kinetic inductance of the Al film plus the contribution of the two contact areas between the grAl inductor and the Al electrodes.The presence of contact junctions is expected because the Al grains in grAl are uniformly covered by an amorphous AlO x oxide.Taking into account the typically measured 15% kinetic inductance fraction in Al thin films of comparable geometry [58], we estimate the intrinsic kinetic contribution of the Al film to be 70 pH.The remaining 130 pH are associated with the contact junctions, from which we calculate a critical current I c ≈ 5 µA for each of them, with corresponding critical current density j c,JJ ≈ 0.13 mA/µm 2 , comparable to the critical current density of the grAl film j c = 0.4 mA/µm 2 (cf.Eq.G7).The inductance participation of one contact junction in the total inductance is p 1,J = 2%.Using the energy participation ratio method presented in Ref. [72], the nonlinear contribution per junction is where Φ 0 = h/2e is the magnetic flux quantum.
We calculate the kinetic inductance of the grAl film L k,grAl = 2.7 nH by substracting L k,Al = 200 pH (fitted, cf.Eq.G4) and L s = 450 pH (FEM simulations, cf.App.B) from the total inductance L = 3.35 nH, which we obtained from the measured resonance frequency f 1 = 7.4887 GHz (cf.Fig. 2) using C s = 137 fF (FEM simulations, cf.App.B).Considering the fact that the grAl volume consists of approximately 2.5 squares, the corresponding grAl sheet kinetic inductance is L k,□ = 1.1 nH/□.This agrees with the value calculated from the room temperature sheet resistance R n,□ ≈ 1800±200 Ω/□ using the Mattis-Bardeen theory for superconductors in the local and dirty limit [71] L As discussed in the main text, the grAl non-linearity α is consistent with a number N ≈ 6 of effective JJs, yielding a critical current density where A grAl = 10×200 nm 2 is the grAl cross section area.This value for j c is in agreement with switching current measurements of similar grAl films [73].
In Fig. 13 we show the field dependence of the nonlinear frequency shift K(5) and K(6) measured during the same two cooldowns as the data presented in Fig. 5 (main text), with and without an outer Al shield, respectively.We chose to measure the non-linear coefficient for n > 1 in order to use a larger readout power and reduce the averaging time during the field sweep.As expected, since the ratio ω 2 1 /j c in the expression for K remains constant, we do not observe a change in K within the measurement accuracy.

Appendix H: Interactions with other mesoscopic systems
Unwanted interactions between superconducting qubits and other mesoscopic systems are commonly b) Frequency difference ∆f1 between the measured transition frequency f1,meas and the fit result f 1,fit (see Eq. G1) in three different field ranges.On this scale, discrete jumps in the qubit frequency become visible for the data taken in run #5.The jumps remain qualitatively unchanged over a 150 MHz span covered by the qubit frequency versus By, and in neither of the runs did we observe signatures of avoided level crossings.
observed [74][75][76][77].Although the nature of the mesoscopic system is difficult to identify, and depends on the architecture of the qubit in general, among the most prominent suspects are defects present in the non-stoichiometric oxide of the JJ barrier and other interfaces [77,78], adsorbates and organic residuals from the fabrication process [79], non-equilibrium quasiparticles [80][81][82][83], and magnetic vortices [84].The interaction can be either transversal, causing a change of the underlying eigenbase due to a hybridization between the two systems, or longitudinal, inducing a frequency shift of the qubit that depends on the state of the mesoscopic system.For a true longitudinal coupling, the state dependent frequency shift does not depend on the frequency detuning between the systems.In the transversal case, the hybridization becomes visible in form of an avoided level crossing, when the two systems are tuned on resonance [85].
We can tune the grAl transmon qubit transition frequency by more than 150 MHz by applying an external magnetic field in-plane.Figure 14a shows the qubit transition frequency as a function of the in-plane magnetic field B y measured in two consecutive cool downs (run #5 and run #6), as well as numerical fits according to Eq. G1 (black dashed lines).Each frequency value is extracted from a fit to the resonance fluorescence response of the qubit similar to Fig. 2. In both measurements shown, Table II.Summary of circuit parameters from all measurement runs: qubit frequency f1, external quality factor (zero-field) Qc,0, internal quality factor (zero-field) Qi,0, external coupling rate (zero-field) κ0, internal decay rate (zero-field) γ0, outer shielding configuration.1.7 1.9 1.8 1.5 1.9 1.9 1.5 Qi,0(×10  and in any other measurement of the same kind, we do not observe avoided level crossings.Figure 14b depicts the frequency difference ∆f 1 between the measured transition frequency f 1,meas and the fitted transition frequency f 1,fit , in three different field ranges.In run #5, the qubit frequency is observed to jump between two metastable states, while it shows no significant discontinuity in run #6.The fact that the observed frequency difference between the two metastable states during run #5 is independent of the transition frequency of the qubit, suggests that the coupling is longitudinal.

Appendix I: Summary of circuit parameters in all cooldowns
In Tab.II we summarize the circuit parameters measured in each of the four cooldowns.The time intervals between subsequent runs, during which the sample is at room temperature and atmospheric pressure, therefore subjected to aging, are the following: run #1 to #2 -20 days, run #2 to #3 -50 days, run #3 to #4 -20 days, run #4 to #5 -176 days, run #5 to #6 -3 days, run #6 to #7 -102 days.

Figure 1 .
Figure1.Sample design.a) Photograph of a copperwaveguide sample holder equipped with a microwave port similar to Ref.[33], and, optionally, with a 2D vector magnet (cf.App.A).The vector magnet is schematically represented by the blue and red coils, oriented along the y and z directions.The sample is positioned in the center of the waveguide and couples to its electric field along the y direction.b) Optical image of the qubit sample, consisting of two Al pads forming a capacitor Cs ≈ 137 fF, connected by a grAl inductor L k ≈ 2.9 nH.We adjust the coupling of the sample to the waveguide by changing the gap w (cf.App.B). c) and d) Scanning electron microscope (SEM) image of the grAl inductor (false-colored in blue) with volume V grAl = 10×200×500 nm 3 and the Al leads (false-colored in red).The grainy surface structure is due to the antistatic Au layer used for imaging.The circuit is obtained in a single lithography step by performing a three-angle shadow evaporation.The Al layer shunts the grAl film in all areas, except for the volume V grAl in the center, which constitutes the source of non-linearity for the qubit[29].The geometric inductance is Ls = 0.45 nH, and the contacts contribute to the L k with 0.13 nH (cf.App.G).

Figure 2 .
Figure 2. Resonance fluorescence.a) Single-port reflection coefficient S11 measured around the qubit frequency f1 = 7.4887 GHz.For probe powers Pin well below the singlephoton regime (n ≪ 1), S11 closely resembles a circle in the quadrature plane (dark blue markers), from which, using Eq. 1, we extract the external and internal decay rates κ = 2π×40 kHz and γ = 2π×10 kHz, respectively.In b) andc) we show the real and imaginary part of the reflection coefficient Re(S11) and Im(S11), respectively, as a function of the detuning between the probe frequency f and the qubit frequency f1.When increasing the probe power Pin, the response becomes elliptic in the quadrature plane, which is the signature of resonance fluorescence of a two-level-system[34].The black lines indicate fits to the experimental data according to Eq. 1.The only fitting parameter is the Rabi-frequency ΩR; κ and γ are fixed by the fit to the low power response (cf.panel a).In d) we show Ω 2 R as a function of incident on-chip power Pin.For a two level system, given by the limit ΩR ≪ α, we expect a linear dependence, as confirmed by the black dashed line passing through the coordinate origin.

Figure 3 .
Figure 3. Energy spectrum.a) Phase of the measured reflection coefficient arg(S11) as a function of probe frequency f and incident on-chip power Pin (left panel).With increasing probe power we observe multi-photon transitions at frequencies fn, labeled (|0⟩ → |n⟩)/n where n denotes the level number, which are almost equidistant in frequency, as plotted in the right hand panel.For clarity we only highlight with arrows the odd n transitions; all indices n are listed on the right hand side of the 2D plot.From the first two points in the right hand panel (highlighted in red) we extract the qubit anharmonicity α = K(1) = 2π×4.48MHz, much larger than the total linewidth κ+γ = 2π×50 kHz (cf.Fig. 2).To highlight the change in K(n) with increasing level number n (see main text), the red line shows a linear extrapolation from the first two points.The extracted values K(n) are plotted in orange using the right hand axis.b) Three individual measurements (top panel) performed at different probe powers (Pin = −121, −106 and −101 dBm), as indicated by the vertical dashed lines in the 2D plot in a).Several multi-photon transitions are visible.With increasing power the linewidth of these transitions broadens, as shown in the bottom panels for n = 3 (right panel, Pin = −127 to −123 dBm) and n = 10 (left panel, Pin = −105.75 to −104.75 dBm).Here, δn = f −fn, with f3 = 7.4842 GHz and f10 = 7.4678 GHz, as indicated by the arrows in the top panel.In App.D we show that these experimental results can be quantitatively reproduced by a master-equation simulation.

Figure 4 .
Figure 4. Time-domain manipulation and measurements of the grAl transmon.a) Rabi-oscillations between the qubit's ground |0⟩ and first excited state |1⟩ as a function of the average drive power of the Gaussian shaped manipulation pulse Pm, and its duration τm.The color scale represents the qubit population inversion from equilibrium (red) to fully inverted (blue).With increasing drive power, the Rabi-frequency increases in agreement with the spectroscopy measurement (cf.Fig. 2 and App.C).The right-hand panel shows the Rabi-oscillation for Pm = −124.7 dBm, highlighted in the 2D plot by a black arrow.The black dashed lines indicate an exponentially decaying envelop.b) Measurement of the grAl transmon coherence times.The duration of the π-pulse is 723 ns at a manipulation power Pm = −131.4dBm.The markers in the top panel show the measured population inversion at a time τ after the π-pulse, and the black line indicates an exponential fit with a characteristic energy relaxation time T1 = 2.8 µs.The bottom panel shows the results of a Ramsey-fringes (circles) and an Hahn-echo (triangles) measurement.The observed frequency of the Ramsey fringes agrees within 5% with the frequency detuning ∆m = 2π×300 kHz of the π-pulse.From the fits indicated by solid lines, we extract coherence times T2 = 4.7 µs and T echo 2

Figure 5 .
Figure 5. Magnetic field dependence.a) Relative change in qubit frequency δf1 = f1(By)−f1 as a function of the applied in-plane magnetic field By (see App. F for field alignment).The experimental data was measured in two separate cooldowns: with (filled triangles) and without (open pentagons) an outer superconducting Al shield (cf.App.A).The field dependence can be fitted to a two-fluid model [43] (cf.App.G), indicated by the black dashed line, from which we extract the thin film Al critical flux density B c,Al = 150±5 mT, in agreement with Ref. [17].A more detailed analysis of the qubit transition frequency with much higher resolution is presented in App.H. b) Internal quality factor Qi for the shielded (filled triangles) and unshielded (open pentagons and triangles) case versus the in-plane magnetic field By.
Vool and W. Wulfhekel for fruitful discussions, and we acknowledge technical support from S. Diewald, A. Lukashenko and L. Radtke.Funding was provided by the Alexander von Humboldt foundation in the framework of a Sofja Kovalevskaja award endowed by the German Federal Ministry of Education and Research, and by the Initiative and Networking Fund of the Helmholtz Association, within the Helmholtz Future Project Scalable solid state quantum computing.PW and WW acknowledge support from the European Research Council advanced grant MoQuOS (N.741276).AVU acknowledges partial support from the Ministry of Education and Science of the Russian Federation in the framework of the Increase Competitiveness Program of the National University of Science and Technology MISIS (Contract No. K2-2018-015).Facilities use was supported by the KIT Nanostructure Service Laboratory (NSL).

Figure 6a (
Figure 6a (bottom right) depicts the numerically calculated magnetic flux density of the HH coils as a function of the lateral position x for a bias current I coil = 1 A and y = 0 mm.The center of the waveguide is the origin for x and y as indicated by the black dashed lines (top right panel).Since the magnetic flux density B y changes by only 3% in the region where the sample chip is mounted (−5 mm ≤ x ≤ 5 mm), we take the value b y = 80 mT/A to convert the HH bias current I coil into a magnetic flux density.For the compensation coil we find a conversion factor b z = 50 mT/A.

Figure 6 .
Figure 6.Photograph of the cryogenic setup used for the magnetic field measurements in Fig.5.The copper waveguide sample holder equipped with the 2D vector magnet (highlighted in color) is mounted at the dilution stage of a table-top Sionludi dilution refrigerator[57] (greyscaled), with a base temperature T base ≈ 20−30 mK.The flat copper cylinder visible in the lower part of the image is the lid of the outer shield (not shown), which consists of successive copper (Cu) and aluminum (Al) cylinders, similar to Ref.[58].The Cu shield was used in both measurements presented in Fig.5, while the Al shield was only used during the "shielded" cooldown.The inset shows the top view of the copper waveguide sample holder including the 2D vector magnet (top), and the numerically calculated Helmholtz field (bottom) By for a bias current I coil = 1 A as a function of lateral position x.The magnetic field of the two Helmholtz coils is aligned within machining precision with the in-plane direction y of the thin-films.The Bz coil is the compensation coil we use to align in-situ the in-plane field.b) Table summarizing the geometric parameters for the coils of the 2D vector magnet: number of winding layers nL, number of windings per layer nw, total number of windings Nw, inner coil radius R, coil length l, vertical distance between Helmholtz coils ∆y, magnetic flux density in y and z direction per 1 A of bias current, by and bz, respectively.

Figure 7 .
Figure 7. Finite-element method simulations of the linearized circuit.a) Simulated transition frequency f1 of the linearized circuit obtained using the eigenmode solver of a commercial finite-element method simulator (HFSS).For the simulation, the grAl volume is substituted with a lumped-element inductance L k which we sweep from 2 nH to 10 nH in order to fit the shunt capacitance Cs and additional geometric inductance Ls of our design using Eq.B1.We extract Cs = 137 fF and Ls = 450 pH.As indicated by the dotted cursors, the measured frequency f1 = 7.4887 GHz corresponds to a value L k = 2.9 nH.b) Simulated external quality factor Qc as a function of the gap w in the outer electrode for L k = 2.9 nH.Similarly to the design of Ref.[59], by closing the gap we can vary the external quality factor by three orders of magnitude.For clarity (cf.panel c), the color of the markers is related to the value of w.The colored dashed lines indicate the design value for the experiment and the black marker indicates the measured external quality factor Qc = 1.9×10 5 (cf.Fig.2). c) Simulated external quality factor as a function of the lateral chip position x and gap width w.The value of x is measured from the center of the waveguide to the symmetry axis of the circuit.Due to the large aspect ratio of the waveguide's cross section (30 mm×6 mm), the external quality factor varies by less than a factor of 2 along x.

Figure 8 .
Figure 8. frequency fR = ΩR/(2π) from spectroscopy and time-domain measurements.The blue markers (labeled spec.)indicate the Rabi frequency extracted from fits to the frequency dependence of the reflection coefficient according to Eq. C8, similar to Fig.2, while the red markers indicate the values extracted from the time-domain (TD) measurements shown in Fig.4a.Both spectroscopy and time-domain data sets were taken in the same cool down (run #7).The black solid line represents the expected Rabi frequency for an ideal qubit according to Eq. C7, provided drive power is Pin = Pm = ℏωq|αin| 2 .

Figure 9 .
Figure 9. Numerical simulations of resonance fluorescence.Complex reflection coefficient S11 (a), its real part (b) and imaginary part (c), numerically calculated for probe frequencies f around the resonance frequency f1 = 7.4887 GHz of the Kerr Hamiltonian in Eq.D1.For comparison, the input drive power Pin, the Kerr-coefficient K = 2π×4.5MHz, and the decay rates γ = 2π×10 kHz and κ = 2π×40 kHz, are set to be the same as in the experiment.The black lines indicate least-square fits using Eq. 1.The only fit parameter is the Rabi-frequency ΩR reported in d.The linear dependence of the Rabi frequency with drive amplitude and the quantitative agreement with the measured data shown in Fig. 2d confirm the validity of the qubit limit and the value of the input line attenuation A = 103 dB.

Figure 10 .
Figure 10.Numerical calculation of the energy spectrum.a) Phase of the numerically calculated reflection coefficient arg(S11) as a function of probe frequency f and probe power Pin (left panel) according to App.D and D. The multi-photon peaks are equally spaced in frequency, with the difference given by K/2 = 2π×2.25MHz (right panel).In contrast to the experiment, we observe a constant frequency shift K (orange markers, right hand panel), as expected, because the Kerr Hamiltonian in Eq.D1 only contains terms up to the 4th order.b) Three individual traces (top panel) calculated at distinct drive powers (P probe = −121, −106 and −101 dBm).Similarly to the experimental results shown in Fig. 3b, several multi-photon transitions are visible at any given drive power in this range.With increasing power, the linewidth of the transitions broadens as depicted by the zoom-ins (bottom panels) around the 3rd (right, P probe = −127 to −123 dBm) and 10th (left, P probe = −105.75 to −104.75 dBm) multi-photon transition.

Figure 11 .
Figure 11.Two-tone spectroscopy.a) Phase of the reflection coefficient arg(S11) measured with a weak probe tone of constant power Pin = −160 dBm in the vicinity of the qubit frequency f1, while an additional microwave drive is applied at ∆ drive /2π = f drive −f1 = 200 kHz detuning, indicated by the dashed white line.With increasing drive power P drive , the fundamental transition starts to split into two distinct transitions, corresponding to the side-bands of the Mollow-triplet, at frequencies f± = f1±ΩR/2π[34,65].Since the population of the first excited state increases with drive power, at P drive ≥ −150 dBm a second transition becomes visibile at 3.9 MHz below f1.This corresponds to the |1⟩ → |2⟩ transition.The measured anharmonicity α = 2π×3.9MHz, as well as the qubit frequency f1 = 7.6790 GHz are slightly different than the values reported in the main text, due to the fact that the measurements were taken in different cooldowns.In total we performed four cooldowns, summarized in Tab.II.Similar to the fundamental transition, the second transition also splits with increasing P drive , an effect denoted Autler-Townes[65].b) Extracted frequency splittings δf for the first two transitions.The black lines correspond to the theoretical predictions 2ΩR (Mollow triplet sidebands) and ΩR (Autler-Townes), respectively, with ΩR = √ ∆ 2 +Ω 2 and Ω = 4κP drive /hf drive .c) Expectation value for the photon number operator ⟨a † a⟩ as a function of the drive power P drive numerically calculated for a qubit (red line) and for an anharmonic multi-level oscillator (cf.App.D).

12 .
Qubit response to out-of-plane magnetic fields a) Change in qubit frequency δf1 = f1(Bz)−f1 as a function of the out-of-plane compensation field Bz for By = 0.The colored arrows indicate the order of the measurement cycle.b) Internal quality factor Qi as a function of Bz extracted from the same measurement sequence.The internal quality factor exhibits a maximum around Bz = −0.1 mT, reaching a factor of 2 higher than the initial Qi value in zero-field (cf.Fig. 5b, blue pentagons)

Figure 14 .
Figure 14. with other mesoscopic systems.a) Qubit transition frequency f1 as a function of the applied in-plane magnetic field By measured in two different cool downs.The black dashed lines indicate fits to Eq. G1.b) Frequency difference ∆f1 between the measured transition frequency f1,meas and the fit result f 1,fit (see Eq. G1) in three different field ranges.On this scale, discrete jumps in the qubit frequency become visible for the data taken in run #5.The jumps remain qualitatively unchanged over a 150 MHz span covered by the qubit frequency versus By, and in neither of the runs did we observe signatures of avoided level crossings.

Figure 15 .
Figure 15.Film height profile.Atomic force microscope image of the film height z in the area around the grAl inductor, measured on a sample fabricated on the same wafer as the sample presented in the main text.The inset shows the cross section along x (black) and y (grey), as indicated by the arrows and overlay lines in the 3D plot.The cross section along the grAl inductor (grey) confirms the t grAl = 10 nm grAl thickness and t Al = 40 nm for each Al layer.
Appendix J: Film height profile

Figure 15
Figure15shows an atomic force microscopy (AFM) image of the film height around the grAl volume for a sample fabricated in the same batch.The height profile shows several steps which originate from the three-angle evaporation process with angles 0 (grAl, t grAl = 10 nm) and ±35 • (Al, t Al = 40 nm each).The metal deposition parameters are summarized in Tab.III.The main features are the leads connecting the grAl volume.Close to the edge of the image, all three layers overlap and the total film height is around 90 nm.The inset in Fig.15shows a cross section along the short (black) and the long (grey) edge of the grAl volume.The AFM measurement confirms the grAl film thickness of t grAl = 10 nm.
Table summarizing the geometric parameters for the coils of the 2D vector magnet: number of winding layers nL, number of windings per layer nw, total number of windings Nw, inner coil radius R, coil length l, vertical distance between Helmholtz coils ∆y, magnetic flux density in y and z direction per 1 A of bias current, by and bz, respectively.