Resolving Phonon Fock States in a Multimode Cavity with a Double-Slit Qubit

Resolving Phonon Fock States in a Multimode Cavity with a Double-Slit Qubit

L. R. Sletten lucas.sletten@colorado.edu JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, Colorado 80309, USA Department of Physics, University of Colorado, Boulder, Colorado 80309, USA    B. A. Moores JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, Colorado 80309, USA Department of Physics, University of Colorado, Boulder, Colorado 80309, USA    J. J. Viennot JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, Colorado 80309, USA Department of Physics, University of Colorado, Boulder, Colorado 80309, USA    K. W. Lehnert JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, Colorado 80309, USA Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
July 30, 2019
Abstract

We resolve phonon number states in the spectrum of a superconducting qubit coupled to a multimode acoustic cavity. Crucial to this resolution is the sharp frequency dependence in the qubit-phonon interaction engineered by coupling the qubit to surface acoustic waves in two locations separated by acoustic wavelengths. In analogy to double-slit diffraction, the resulting self-interference generates high-contrast frequency structure in the qubit-phonon interaction. We observe this frequency structure both in the coupling rate to multiple cavity modes and in the qubit spontaneous emission rate into unconfined modes. We use this sharp frequency structure to resolve single phonons by tuning the qubit to a frequency of destructive interference where all acoustic interactions are dispersive. By exciting several detuned yet strongly-coupled phononic modes and measuring the resulting qubit spectrum, we observe that, for two modes, the device enters the strong dispersive regime where single phonons are spectrally resolved.

Quantum control over mechanical degrees of freedom promises insight into fundamental physics as well as the development of innovative quantum technologies. As mechanical resonators are massive and macroscopic, they can probe quantum theories at large scales Arndt and Hornberger (2014); Viennot et al. (2018); Chu et al. (2018), while the ability of mechanical motion to couple to a variety of quantum systems has inspired numerous mechanics-based transduction schemes Andrews et al. (2014); Teissier et al. (2014); Schuetz et al. (2015); Noguchi et al. (2017); Forsch et al. (2018); Arrangoiz-Arriola et al. (2018); Whiteley et al. (2019). Additionally, mechanical elements are compact compared to their electromagnetic counterparts, enabling the on-chip fabrication of many wavelength microwave structures such as high-performance filters and multi-mode resonators Morgan (2007); Aref et al. (2016); Renninger et al. (2018). High fidelity control over the large number of modes achievable in acoustic platforms would be a powerful resource for quantum information processing Pechal et al. (2018).

The field of circuit quantum electrodynamics (cQED) has provided both guidance and tools for achieving quantum control over mechanical excitations. In cQED, the state of a photonic mode is measured and manipulated using superconducting qubits. These qubits can also interact with mechanical systems using piezoelectric materials. Two seminal works leveraged this fact to couple a qubit to a dilatational resonator O’Connell et al. (2010) and to propagating surface acoustic waves (SAWs) Gustafsson et al. (2014). Both surface and bulk acoustic waves can be confined to form high-overtone resonators Manenti et al. (2016); Kharel et al. (2018), leading to demonstrations of qubit-phonon coupling in multi-mode cavities Manenti et al. (2017); Chu et al. (2017); Moores et al. (2018); Bolgar et al. (2018); Kervinen et al. (2018). Most recently, a pair of experiments used resonant interactions to create number and superposition states of an acoustic cavity mode, thereby demonstrating basic quantum control of acoustic phonons Satzinger et al. (2018); Chu et al. (2018). Following the example of cQED, strong dispersive interactions in acoustic systems would lead to improved quantum control through quantum non-demolition phonon measurement Schuster et al. (2007); Johnson et al. (2010) and qubit mediated phonon-phonon interactions Wang et al. (2016); Naik et al. (2017), enabling the processing of quantum information encoded in acoustic modes Heeres et al. (2017).

If quantum control were extended to multi-mode acoustic cavities, quantum technology could capitalize on their large mode density. However, combining a high mode density with dispersive operation imposes an apparent stringent constraint on the qubit-cavity coupling rate . In a cavity with free spectral range , the qubit transition can be detuned by at most from a cavity mode. Dispersive operation constrains to be much less than the mode spacing, , effectively limiting the number of modes controllable with a single qubit. This constraint could be circumvented if the coupling strength were not uniform across all modes but instead varied with frequency.

In fact, acoustic platforms excel at realizing strongly frequency-dependent couplings. In SAW devices, an interdigitated transducer (IDT) converts between electrical and acoustic signals with a frequency dependence determined by the Fourier transform (FT) of the IDT geometry Frisk Kockum et al. (2014); Aref et al. (2016). A desired frequency response can be engineered by computing its inverse FT and shaping the IDT accordingly. Moreover, the slow speed of sound ( m/s on GaAs) implies MHz frequency resolution can be realized with millimeter geometries. Indeed, the utility of on-chip SAW filters has led to their widespread commercial use Morgan (2007). Combining superconducting qubits with the the filtering capabilities of SAWs enables frequency-tailored interaction strengths and the exploration of spatially extended qubit-field interactions Frisk Kockum et al. (2014); Guo et al. (2017); Kockum et al. (2018); Moores et al. (2018); Andersson et al. (2018).

In this Article, we engineer a frequency-dependent coupling between a transmon qubit and a multi-mode SAW cavity to realize together with dispersive operation. The qubit couples to phonons through an IDT that is bisected to create a pair of interaction regions separated by a long travel time,  ns [Fig. 1(a)]. In close analogy to double-slit interference, the many-wavelength separation between interaction regions creates sharp fringes in the frequency dependence of the qubit-phonon interaction strength Frisk Kockum et al. (2014). We observe the designed frequency dependence as a high-contrast modulation of both the coherent exchange rate between the qubit and cavity modes and the qubit spontaneous emission rate into unconfined phonons. This frequency dependence greatly reduces the coupling to certain modes to create frequency windows for dispersive operation. We tune the qubit transition to such a window and observe the single-phonon Stark shift from three strongly coupled modes of the cavity by populating these modes while measuring the qubit spectrum. For two of these modes, we enter the strong dispersive regime where the single-phonon Stark shift exceeds the qubit and acoustic linewidths, demonstrating that spatially extended coupling can be leveraged to take full advantage of multi-mode acoustic systems.

Figure 1: Double-slit qubit concept and device. (a) Schematic of the device. A transmon qubit resembling a double-slit interferometer is formed by shunting two halves of an interdigitated transducers (IDT, red/blue) with a pair of Josephson junctions (yellow). Two distributed Bragg arrays (green) confine phonons in a narrow frequency band to create acoustic resonances (gray curve). (b-e) False color SEM micrographs of a representative device at three different scales. (b) The qubit IDT (purple) is shunted by Josephson junctions and located within an acoustic cavity of width m. The imaged device has half the cavity length of the measured device. Insets show enlarged images (c) of the Bragg mirror and (d) the IDT, whose innermost fingers are extended to shunt the junctions. (e) Antenna paddles couple the qubit to a copper waveguide cavity at 5.9 GHz for readout and control. (f) When the qubit is tuned outside the mirror bandwidth (unshaded), the IDT launches phonons with a frequency dependence determined by the Fourier transform of the IDT geometry, creating fringes in the qubit loss rate . (g) Inside the mirror band, the IDT geometry modifies the qubit coupling strength to the evenly spaced cavity modes. By tuning the qubit frequency to a zero in the coupling at , coupling rates comparable to the mode spacing can be achieved with dispersive operation, provided the slope near (red dotted line) is sufficiently small.

The device we study comprises a tunable transmon qubit on a piezoelectric GaAs surface with two IDT halves embedded in a multi-mode SAW cavity [Fig. 1(a)]. The cavity is formed between two Bragg mirrors made of aluminum strips that reflect surface waves over a 100 MHz bandwidth to form a phononic Fabry-Perot cavity [Fig. 1(b,c)]. The effective cavity length extends beyond the mirror separation m by 20 m of phonon penetration into the mirrors to create a mode spacing  MHz. The mirrors and IDT were designed with periodicity  nm, which corresponds to a center frequency near  GHz. The IDT halves, each 8 periods long, are mirror images of each other reflected across the center of the cavity and separated by m. The mechanical loading effects on the resonator from the IDT are minimized by using thin metal (30 nm) and a split electrode design [Fig. 1(d)] Morgan (2007). Qubit readout and control are enabled by attaching antenna paddles [Fig. 1(e)] that strongly couple the qubit to a copper waveguide cavity (see appendix A).

An IDT split in half achieves a mode-selective coupling by creating a frequency profile analogous to the spatial profile of double-slit diffraction. The IDT is split into two regions of length separated by distance . The Fourier transform about the symmetry point between these two regions is real and the product of two factors: a slow sinc envelope centered on with period and a fast sinusoidal modulation with period ,

(1)

Outside the mirror bandwidth, the qubit loses energy to propagating phonons at a rate . [Fig 1(f)] Frisk Kockum et al. (2014); Moores et al. (2018). Within the mirror band, the qubit coherently exchanges excitations with confined acoustic modes [Fig 1(g)]. If the IDT is symmetric about the cavity center, the coupling strength between the qubit and confined acoustic mode has the form where the slowly varying sinc is approximated as unity and is the frequency of cavity mode . With the designed separation between IDT halves, the coupling varies with a periodicity approximately equal to the mirror bandwidth (  MHz).

Dispersive operation with all modes can be achieved by tuning the qubit to a zero in at frequency . If a mode exists with a small detuning from the qubit, the mode couples with rate that is bounded above by its detuning multiplied by the slope of the coupling strength near [Fig.  1(g)]. If this slope is sufficiently small, as in this device, then will be much less than the detuning, and the interaction is dispersive for any mode regardless of the mode density.

Figure 2: Avoided crossings spectroscopy Tuning the qubit across the cavity modes measures the mode frequencies and qubit-cavity coupling strengths. (a) Within the mirror bandwidth, nine avoided crossings appear with varying coupling strength and a nearly even mode spacing  MHz. The strong avoided crossings have a spurious mode on their high-frequency shoulder, which we attribute to a resonance with non-zero transverse mode number. (b) The coupling strengths to each mode are extracted from the measured crossings. The qubit couples more strongly to modes at the center of the mirror band with characteristic strength  MHz. The IDT is symmetric about the cavity center, resulting in strong coupling to even modes (green) and weak coupling due to odd modes (blue). Coupling to transverse modes (orange) is a factor of 5 smaller. An inference of the frequency-dependence of the qubit coupling strength (gray dotted line) made from measurements of the qubit spontaneous emission rate (see main text) agrees well with the measured coupling rates.

To confirm the designed frequency structure in the device, we measure the qubit spectrum as an applied magnetic flux tunes its frequency. We begin by tuning the qubit across the mirror bandwidth to investigate the frequency region where phonons are confined. We observe pronounced avoided crossings in the qubit spectrum where the qubit coherently exchange energy with cavity phonons [Fig. 2(a)]. The extracted coupling rates [Fig. 2(b), see appendix C] vary between the modes, with several strongly coupled modes in close proximity to crossing-free regions wider than . Three main effects explain the observed behavior. First, the split-IDT modulates the coupling proportional to , coupling the qubit strongly to modes near 4.25 GHz with  MHz while decoupling it from modes roughly above or below. Second, the coupling strength of neighboring modes alternates between weak and strong because, near the cavity center, the acoustic standing wave pattern alternates between having nodes and antinodes spatially aligned with the IDT. Lastly, resonant exchange between the qubit and cavity modes at the edge of the mirror bandwidth is unresolved as the coupling rate is much less than the loss rate of these weakly confined modes.

Figure 3: Interaction with propagating phonons (a) Tuning the qubit across the  GHz IDT bandwidth probes its interaction with the continuum of propagating phonons outside the mirror band. Subtracting the expected flux tuning without acoustic interactions makes clear an oscillatory feature in both linewidth and frequency. Zig-zag features bordering the central coherent avoided crossings are well described as resonant interactions with lossy acoustic modes. A strongly coupled system is present at 4.41 GHz that is not a SAW-cavity mode. (b) The decay rate is found by measuring the qubit decay time. The measured energy loss rate varies by more than a factor of 25 within a 55 MHz span. A model closely fits the measurement except in the regions with the largest , where increased uncertainty from the short decay times compounds with effects from the semi-reflective mirrors that are not included in the model. The measured frequency-dependent Lamb shift agrees well with the calculations from the IDT model (inset).

To study A(f) outside the mirror band, we tune the qubit over a 1 GHz span and examine the influence of propagating phonons on the qubit linewidth and transition frequency. In contrast to the discrete cavity modes, propagating modes form a continuum, enabling a dense sampling of over a broad frequency range and affording a clear picture of how effectively the split IDT tailored the qubit-phonon interaction. In the measured qubit spectra [Fig. 3(a)], the features arising from acoustic interactions are emphasized by subtracting the flux dependence expected from an acoustically uncoupled qubit (see appendix B). At frequencies detuned from the central avoided crossings, the qubit linewidth oscillates with a period of 110 MHz that is consistent with the expected delay time and an amplitude that decays as the qubit tunes out of the IDT bandwidth. Additionally, the qubit frequency deviates from the uncoupled flux dependence with a similarly enveloped oscillation with matching 110 MHz periodicity. Both of these effects can be understood by modelling the IDT phonon emission as a frequency-dependent resisitance, which must be accompanied by a frequency-dependent reactance from Kramers-Kronig relations Morgan (2007); Aref et al. (2016). We observe this reactance as a modulation of the qubit frequency compared to its uncoupled flux tuning, an effect describable as a phononic Lamb shift Wang et al. (2015); Frisk Kockum et al. (2014).

We determine the qubit energy decay rate with increased precision by measuring qubit excited state lifetime () in the time domain. With the qubit tuned away from the resonant modes, we observe oscillating in frequency with large amplitude; the loss increases by a factor of 25 above its minimal value within a  MHz span [Fig. 3(b)]. A simple model that combines a prediction for the phonon emission rate from the IDT and a constant internal quality factor closely fits the measured qubit loss rate, giving and  ns (see appendix D). The nulls in arise from destructive interference between the two IDT halves, an effect with close parallels to an atom interfering with its mirror image Eschner et al. (2001); Hoi et al. (2015). As the depth of these nulls is approximately uniform across the IDT bandwidth, phonon loss from imperfect destructive interference is less than 75 kHz. Additionally, the extracted IDT parameters from the qubit loss rate can be used to calculate the frequency-dependent phononic Lamb shift, showing agreement with the measured qubit frequency [inset, Fig. 3(b)].

Our measurement of the qubit interaction with propagating modes also provides an independent inference of the interaction strength between the qubit and cavity modes. The best-fit model from Fig. 3(b) determines using propagating modes and can be extended to frequencies inside the mirror band, where it closely follows the measured coupling rates [Fig. 2(b)].

Figure 4: Phonon-number splitting The qubit is tuned to  GHz where no resonant interactions occur and the first two transmon transitions straddle the acoustic modes. (a) To excite and measure the acoustic modes, the drive power is increased relative to the power used near the qubit transition (shaded, +19 dB or +28 dB). The acoustic modes are all dispersive, with , , and . (b) While driving acoustic modes 3, 5, or 7, we observe the qubit transition broaden and shift up in frequency with increasing phonon population. The spectra are fit to a model assuming a coherent state with average phonons to determine . For modes 5 and 7, the measured single-phonon Stark shift exceeds both the qubit and acoustic linewidths. Traces are vertically offset to aid clarity. For the trace in each mode with the highest phonon occupancy, the contributions to the fit from each phonon number state are plotted (shaded purple).

Having characterized the qubit interaction with both confined and propagating phonons, we turn to resolving the Stark shift of individual cavity phonons. This resolution requires dispersive operation with all modes, i.e. for all modes , where . Tuning the qubit to  GHz realizes dispersive operation; the least dispersive mode is with . Furthermore, the Stark shift must exceed both the qubit and acoustic loss rates. In the dispersive regime, a single phonon shifts the qubit frequency by , given by

(2)

where  MHz is the anharmonicity of the transmon qubit. With the qubit at , the qubit transition frequency is above the acoustic modes while the transition is below such that and for all modes . With this level ordering, the two terms in Eq. 2 add constructively to create large and positive Stark shifts Koch et al. (2007).

To populate a target cavity mode with phonons, we drive the qubit at a frequency far detuned from its own transition but resonant with the cavity mode Chu et al. (2018). In Fig. 4a, spectroscopy shows the qubit transition at 4.318 GHz and, with much higher drive power, three acoustic resonances at lower frequencies. The measured qubit linewidth  kHz is only marginally larger than the sum of contributions from , intrinsic dephasing, and expected power broadening, confirming that residual phonon loss does not contribute significantly. The acoustic linewidths are measured to be  kHz for all three modes, only slightly larger than expected 200 kHz of diffraction loss from the flat-flat mirror design of the cavity Manenti et al. (2016); Moores et al. (2018); Aref et al. (2016).

We measure the single-phonon Stark shift by varying the population in three modes and measuring the qubit spectrum. A 3 µs drive pulse at creates a coherent state in mode with average phonons Schuster et al. (2007). The resulting Stark-driven qubit spectrum, measured with a spectroscopy pulse concurrent with the acoustic drive, consists of a sum of Lorentzians that each correspond to a phonon number state in the cavity [Fig. 4(b)]. These Lorentzians are spaced by and broaden with higher phonon number in proportion to . Sweeping the drive power at one of the three strongly coupled modes, the measured qubit spectrum broadens and shifts up in frequency. Crucially, several resolved peaks appear for modes 5 and 7. We fit the measured spectra assuming a phonon coherent state to determine the average phonon number in each trace as well as the acoustic linewidths  kHz and single-phonon Stark shifts  kHz (see appendix E). As the single-phonon Stark shifts for modes 5 and 7 exceeds both the qubit and acoustic linewidths, we confirm that the device enters the strong dispersive regime for two acoustic modes.

Resolving phonon Fock states in a multi-mode cavity through spatial engineering suggests multiple future directions. For the measured device, the dominant source of phonon loss was likely diffraction and could be eliminated by using curved reflectors to form a stable cavity. Combing improved phonon lifetimes with the demonstrated coupling strengths would enable quantum non-demolition phonon detection and qubit-mediated interactions between phonon modes. Furthermore, the number of modes accessible to the qubit can be increased simply by elongating the cavity, highlighting the promise of SAW systems for multi-mode quantum information processing Naik et al. (2017); Chu et al. (2018). More generally, the engineering of time-delayed self-interactions not only enables a wide range of frequency structures but can also fundamentally alter quantum dynamics Andersson et al. (2018), providing an avenue with theoretical promise for quantum computation Pichler et al. (2017).

See related work in Arrangoiz-Arriola et al. (2019).

Acknowledgements

We thank Xizheng Ma for insightful discussions as well as Daniel Palken and Maxime Malnou for providing the quantum-limited amplifier. This work was supported by NSF grant 1734006.

Appendix A Qubit readout

The qubit state is measured through its dispersive interaction with a 5.9 GHz copper waveguide cavity. The qubit has a large electric dipole moment, coupling it to the readout cavity with strength  MHz. Different readout techniques were used to probe the qubit state depending on the measurement details.

We used bright-state readout Reed et al. (2010) to measure the qubit decay rate as a function of frequency [Fig. 3(b)]. This type of readout is well suited for measuring fast decays as the cavity can persist in the bright state for a time that exceeds the natural qubit lifetime.

For qubit spectroscopy, we used single quadrature dispersive readout backed by a flux-pumped Josephson parametric amplifier. To measure the Stark-driven qubit spectra, we used a pulsed readout scheme that minimized qubit dephasing from readout phonons [Fig. 4]. Continuous readout was used for qubit spectroscopy as a function of flux [Fig. 2(a) and Fig. 3(a)]. In the broad qubit spectroscopy, we compensate for the varying excited state contrast resulting from frequency-dependent qubit loss by adjusting the qubit drive power. This power level is independently determined from the measured times.

Appendix B Qubit flux dependence

The qubit transition frequency is tuned using an off-chip coil to thread magnetic flux through the 50 loop formed by the two Josephson junctions. Omitting acoustic interactions, we model the qubit frequency as a function of coil current as

where is the zero-field qubit frequency, is the coil current required to thread a half flux quantum through the qubit loop, is the current offset required to offset ambient fields, and is the normalized difference between the junction critical currents. From fitting the measured qubit frequency [Fig. 5(a)], we find  GHz,  mA,  µA, and .

The qubit flux dependence is weakly modified by its interaction with propagating phonons. We model this phononic Lamb shift as

where is the maximal loss rate to phonons, is the center frequency of the IDT, is the number of finger periods in each IDT, and is the intra-IDT delay Frisk Kockum et al. (2014). The measured phonon loss rate (see appendix D) independently determines , and , allowing the Lamb shift to be calculated with no free parameters. This calculated Lamb shift closely matches the residual from the flux fit [inset Fig. 3(b), Fig. 5(b)] except near avoided crossings.

Figure 5: Qubit flux dependence (a) The qubit frequency is measured and fit (white line) over a large range of applied flux. Regions near avoided crossings are ignored in the fit (gray). (b) The residual between the measured transitions and the best-fit uncoupled flux dependence matches the expected acoustic Lamb shift (black).

Appendix C Acoustic cavity characterization

Extracting the coupling strengths from the closely spaced avoided crossings requires a multi-mode formalism. The eigenmodes of the system are found by diagonalizing the interaction Hamiltonian,

including 9 purely longitudinal modes and 5 transverse modes. The eigenvalues of the matrices as a function of flux are fit to the measured avoided crossing spectrum [Fig. 6(a)].

Figure 6: Avoided crossings analysis (a) The measured qubit-cavity avoided crossings are fit by a multi-mode interaction model (white) [Fig. 2] (b) The spacings between the modes are well matched by a mirror model with a single-element reflectivity of (green).

The general properties of the mirrors can be inferred from the precise measurement of the mode spacings. Near the center of the mirror bandwidth, the modes are spaced by  MHz, but they become more closely spaced near the edge of the mirror bandwidth due to deeper phonon propagation into the mirror stack [Fig. 6(b)]. We find a simple mirror model matches the measurements with a single-element reflectivity , which corresponds to a mirror bandwidth of 100 MHz.

Appendix D Phonon emission rate

The qubit lifetime is measured over a wide frequency range to directly probe the qubit spontaneous emission rate into unconfined phonons. The qubit loss rate as a function of qubit frequency is modelled by

where is the qubit internal quality factor, is the maximal loss rate to phonons, is the center frequency of the IDT, is the number of finger periods in each IDT, and is the intra-IDT delay time. We find ,  MHz,  GHz, and  ns. The best-fit is close to the expected value of 12.5 MHz calculated using room temperature GaAs properties Gustafsson et al. (2014).

Figure 7: Qubit spontaneous emission (a) Each measured qubit decay is shown normalized by its maximum value. (b) The extracted times match the model for the qubit phonon emission rate. A small fraction of decays displayed non-exponential features with timescales inconsistent with the intra-IDT delay time (gray).

The qubit studied constitutes a giant atom where the intra-IDT delay time approaches the phonon-limited qubit lifetime. Deep in this regime, the qubit fully decays before a phonon can travel between the IDT halves, leading to a host of effects such as non-exponential decay. The transition to this regime occurs when the product reaches 1 Frisk Kockum et al. (2014); Andersson et al. (2018). For this device, . However, evidence of non-Markovian physics was obscured by the presence of mirrors and the short time scale (9 ns) associated with the non-exponential decays. A small fraction of the measured time traces display non-exponential features but with time scales far exceeding the intra-IDT delay time. These decays are excluded from the reported qubit energy decay rates [Fig 7].

Appendix E Number splitting analysis

Figure 8: Phonon number power dependence The average phonon number extracted from the measurements depends linearly on the power applied to each mode.

The measured Stark-driven spectra are fit to a sum of unit-area Lorentzians with weights assumed to be Poissonian distributed with mean ,

where is the spectroscopy frequency, is a constant offset, is an overall amplitude, and is a cutoff phonon number. The two factors in the sum are given by

where is the zero-phonon qubit linewidth, is the zero-phonon qubit frequency, is the loss rate of mode , and is the single-phonon Stark shift from mode . Fits of the average phonon number show a linear dependence on applied drive power for the three measured modes [Fig. 8]. The strong drive used to populate the acoustic modes also weakly excites the qubit, causing the trace offset to increase with . Additionally, the bare qubit frequency pulls weakly up with off-resonant drive power at a rate of about 150 kHz per phonon, an unexplained effect that is included in the fits.

The qubit coherence times at are measured to be  ns and  ns. The time is almost twice , and we calculate a intrinsic dephasing rate of kHz. The spectroscopic qubit linewidth was measured to be  kHz at . Together, frequency-independent energy loss (360 kHz), intrinsic dephasing (30 kHz), the effective Rabi rate from the drive tone (100 kHz), and the finite duration of the drive pulse (50 kHz) sum to a 540 kHz qubit linewidth. marginally smaller than the measured value.

Additionally, an unstable avoided crossing appeared intermittently between 4.312 GHz and 4.322 GHz with sub-MHz coupling rate, fluctuating with a several hour time-scale. We reject data when the defect was present by interleaving independent diagnostics with the Stark-driven spectra and removing defect-present data in post-processing.

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