Abstract
The anomalous lowtemperature properties of glasses arise from intrinsic excitable entities, socalled tunneling TwoLevelSystems (TLS), whose microscopic nature has been baffling solidstate physicists for decades. TLS have become particularly important for microfabricated quantum devices such as superconducting qubits, where they are a major source of decoherence. Here, we present a method to characterize individual TLS in virtually arbitrary materials deposited as thin films. The material is used as the dielectric in a capacitor that shunts the Josephson junction of a superconducting qubit. In such a hybrid quantum system the qubit serves as an interface to detect and control individual TLS. We demonstrate spectroscopic measurements of TLS resonances, evaluate their coupling to applied strain and DCelectric fields, and find evidence of strong interaction between coherent TLS in the sample material. Our approach opens avenues for quantum material spectroscopy to investigate the structure of tunneling defects and to develop lowloss dielectrics that are urgently required for the advancement of superconducting quantum computers.
Introduction
We are still lacking an explanation for the behaviour of amorphous materials at low temperatures <10 K^{1,2}. Why is it that even widely different materials ranging from biatomic glasses to polymers show quantitatively identical properties such as specific heat and thermal conductivity^{3}? The Standard Tunneling Model (STM)^{4,5} has been a first attempt to explain these universal anomalies on the basis of twolevel systems (TLS) believed to arise from the tunneling of atoms between two energetically similar configurations in the disordered lattice structure. While the STM neglects mutual TLS interactions and fails in the intermediate temperature range of 1–10 K, refined models include TLSTLS interactions^{6,7,8}, assume different types of TLS^{9}, or consider specific dependencies of TLS potential energies^{10,11}. Since insights from experiments on bulk materials were limited to observing the averaged response from large and inhomogeneous ensembles of TLS, their individual properties remained out of reach.
This situation has changed with the advent of superconducting qubits that realize wellcontrollable macroscopic quantum systems with customtailored energy spectra and couplings to the environment. Qubits are implemented from electric resonant circuits employing Josephson tunnel junctions that serve as nonlinear inductances to obtain anharmonic potential wells where discrete eigenstates can be selectively addressed. Driven by the desire to realize solidstate quantum information processors, intensive effort went into the development of advanced circuit designs^{12,13,14,15,16} and fabrication techniques^{17,18}, which resulted in a dramatic improvement of device coherence. The entry of commercial enterprises has further accelerated progress, culminating in the demonstration of machine learning algorithms^{19}, access to prototype quantum processors via the cloud^{20}, and the achievement of quantum supremacy by controlling a 53qubit system that could not anymore be simulated efficiently by classical supercomputers^{21}.
Despite these achievements, progress towards truly largescale quantum processors is still hindered by decoherence, of which the major part is due to losses in dielectric circuit materials^{22}. TLS residing in the tunnel barriers of Josephson junctions and in the native surface oxides of superconducting electrodes may couple via their electric dipole moments to the qubit’s oscillating Efield. When TLS are at resonance with the qubit, they can efficiently dissipate energy into the phonon^{23} or BCSquasiparticle^{24} bath which results in reduced qubit energy relaxation times T_{1}^{25} and, in the case of strong coupling, gives rise to avoided level crossings in qubit spectroscopy^{26}. Moreover, thermally activated TLS at low energies may interact with highenergy TLS that have frequencies near resonance with a qubit or resonator, and this causes temporal fluctuations of the device’s resonance frequency^{27,28} and energy relaxation rate^{29,30,31,32}.
Further progress with superconducting quantum processors based on current circuit architectures thus requires extensive material and fabrication process research to avoid the formation of TLS. Moreover, tools to verify the quality of metal films and junctions are required that are able to relate fabrication processes to TLS formation and to investigate the microscopic nature of the material defects. For these tasks, qubits themselves are well suited because of their sensitivity to TLS. In case of strong coupling, quantum state swapping between the qubit and TLS^{33} is possible, allowing one to characterize TLS’ coherence properties^{23,34,35}, and their coupling to the environment^{24,36,37}. A useful method for such studies is to control the TLS’ internal asymmetry energy and thus their resonance frequency by applied mechanical strain^{38} or DCelectric field^{39}. Operating qubits in electric fields allows one to distinguish defects in tunnel junction barriers from those on electrode surfaces^{40} and to obtain information on the positions of individual TLS in the quantum circuit^{41}.
In this letter, we present a quantum sensor that grants access to measurement and manipulation of individual TLS in virtually arbitrary materials. The device is based on a transmon qubit^{13,25}, which consists of a capacitively shunted DCSQUID formed by two Josephson junctions connected in parallel, as shown in Fig. 1a, c. Qubit readout is performed by measuring the dispersive resonance frequency shift of a coplanar resonator that is capacitively coupled to the qubit. The qubit resonance frequency can be tuned in a range of typically ≈ 1 GHz by an onchip coil providing magnetic flux, which frustrates the Josephson energy of the DCSQUID loop shown in Fig. 1d.
The material under test defines the dielectric in an additional "sample capacitor” shunting the transmon qubit. In this work, we use a capacitor having a plate or "overlap” geometry as shown in Fig. 1e. This allows one to study TLS in all dielectrics that can be deposited as thin films, e.g., by sputtering or evaporation. Alternatively, one can employ a socalled nanogap capacitor consisting of two coplanar electrodes (see Fig. 1f) that are separated by a few tens of nanometres, and then covered by the sample material. In this case, the coupling between TLS and the qubit occurs via the fringing electric field sketched in the inset of Fig. 1f. This provides the possibility to study TLS in pieces of bulk material by pressing it onto the nanogap capacitor. Moreover, the use of uncovered nanogap capacitors allows one to study single TLS residing in the native oxides of the electrode material and defects that are formed by surface adsorbates.
The STM describes TLS on the basis of a doublewell potential whose minima differ by the asymmetry energy ε, and transitions between wells occur at a tunneling energy Δ_{0}, resulting in the transition energy \(E=\sqrt{{{{\Delta }}}_{0}^{2}+{\varepsilon }^{2}}\). TLS in the sample material couple to the qubit at a strength \(\hslash g={\bf{pF}}=\bar{p} {\bf{F}}\), where F is the electric field inside the capacitor, which is induced by the qubit plasma oscillation, and \(\bar{p}={p}_{\parallel }\ ({{{\Delta }}}_{0}/E)\) is the projection of the TLS’ dipole moment p onto F, multiplied by the TLS’ matrix element^{39} Δ_{0}/E.
Results
Sample capacitor design
Single TLS can be detected if their energy exchange rate with the qubit (which equals their coupling strength g at resonance) is comparable to the energy decay rate 1/T_{1} of the isolated qubit. The criterium g ≈ 1/T_{1} togeher with the TLS’ abovementioned coupling strength \(\hslash g=\bar{p} {\bf{F}}\) define a suitable thickness d of the dielectric layer in overlap capacitors: \(d=\bar{p}\ {T}_{1}\ {V}_{{\rm{rms}}}/\hslash\), where the electric field ∣F∣ is substituted by V_{rms}/d. Here, \({V}_{{\rm{rms}}}=\sqrt{\hslash {\omega }_{10}/2C_{\rm{tot}}}\approx 4.5\ \mu {\rm{V}}\) is the vacuum voltage fluctuation on the qubit island at the designed plasma oscillation frequency ω_{10} ≈ 2π ⋅ 6.2 GHz when C_{tot} ≈ 100 fF is the sum of all capacitances shunting the qubit. To be able to detect a TLS dipole moment p_{∥} of minimum 0.1 eÅ^{42}, and assuming a rather conservative T_{1} ≈ 1μs, we arrive at a dielectric layer thickness d ≈ 70 nm. We chose d = 50 nm and a capacitor size of (0.25 × 0.3) μm^{2}, resulting in C_{s} ≈ 0.15 fF ≪ C_{tot} which ensures that the energy that is stored in the lossy sample capacitor remains limited to a small fraction of the qubit’s total energy, and coherence is preserved. A picture of the employed sample capacitor is shown in Fig. 1e while further details on the capacitor design are given in Supplementary Methods 1.
It is furthermore necessary to be able to distinguish TLS in the sample material from those on electrode interfaces and from TLS in Josephson junctions. This is accomplished by probing the TLS’ response to a local electric field generated by voltagebiasing the sample capacitor’s electrode as indicated in Fig. 1a, where the additional capacitor C_{f} ~ 250 fF serves as a DCbreak. The bias voltage will not induce an electric field in the transmon’s shunt capacitor C_{q} nor inside the Josephson junctions’ tunnel barrier since the DCelectric potential difference of the transmon island and ground will be compensated by Cooperpair tunneling^{40}, so that only TLS in the sample capacitor respond to the applied voltage V_{s}. In addition, we can tune TLS residing at the perimeter of the qubit capacitor by a globally applied DCelectric field that is generated by an electrode installed above the qubit chip^{40} as shown in Fig. 1b. Moreover, all TLS including those residing inside the tunnel barriers of junctions can be tuned via physical strain by bending the chip with a piezo actuator^{38,43}, which is useful to enhance the number of observable TLS. The table in Fig. 2b summarizes how to identify the location of a TLS from its tunability characteristics.
We chose amorphous aluminum oxide AlO_{x} as the sample material for this work since it is well characterized and of general importance for superconducting quantum circuits where it is ubiquitously used as a reliable tunnel barrier material. The sample capacitor is patterned with electronbeam lithography, where the bottom electrode is deposited and connected to the qubit island in the same step as the qubit’s Josephson junctions, followed by a third lithography step depositing 50 nm of AlO_{x} by eBeamevaporation of Al in an oxygen atmosphere, and capping it by a top Al electrode. The filter capacitor C_{f} is formed simultaneously as a wider section in the top electrode. Here, we report results for samples employing small sample capacitors of size (0.25 × 0.3) μm^{2} which did not contribute significantly to decoherence. Two tested C_{s}shunted qubits reached T_{1}times of 3.3–4.2 μs, which is comparable with an isolated reference qubit (T_{1} ≈ 4.3μs) on the same chip. In another batch, we also tested larger sample capacitors (0.3 × 2.1) μm^{2}, which did limit the qubit’s T_{1} time^{44}. This allowed us to measure the loss tangent of the employed AlO_{x} dielectric as \(\tan {\delta }_{0}\approx (1.7\pm 0.2)\cdot 1{0}^{3}\), comparable with other reports^{42,45,46,47}.
TLS spectroscopy
To distinguish whether a TLS is located in a tunnel barrier, at the qubit’s film edges^{41}, or in the sample capacitor dielectric, we track its resonance frequency for a range of voltages applied to the global DC electrode (V_{g}), to the sample dielectric (V_{s}), and to the piezo (V_{p}). An example of such a measurement is presented in Fig. 2a, showing the frequency dependence of the qubit’s T_{1} time estimated by swap spectroscopy^{33,35,43}, where dark traces reveal enhanced qubit energy relaxation due to resonant TLS. These segmented hyperbolic traces are fitted to obtain the TLS’ coupling constants γ which determine their biasdependent asymmetry energy ε = ε_{i} + γ_{g}V_{g} + γ_{s}V_{s} + γ_{p}V_{p} up to an intrinsic offset ε_{i}. The fit also results in the value of the TLS’ tunneling energy Δ_{0} if it lies within the tunability range of the qubit’s resonance frequency.
Thanks to the wellspecified DCelectric field V_{s}/d in the sample capacitor, the coupling electric dipole moment p_{∥} = γ_{s}d/2 of TLS in the sample material is directly obtained from the identity 2p_{∥}V_{s}/d = γ_{s}V_{s}^{4,5}. In contrast, a measurement of the TLS’ coupling strength to a quantum circuit results in the effective dipole moment size \(\bar{p}\) where the matrix element (Δ_{0}/E) is often unknown. From measurements on two identical qubits in one cooldown, we characterized in total 138 TLS. Of those, 13 TLS were found inside the sample material, with a spectral density of 4.1 GHz^{−1} (see calculation details in Supplementary Methods 3), which results in a volume density of \({P}_{0}=4.1\ {({V}_{\text{d}}{\rm{GHz}})}^{1}=1800\ {(\mu {{\rm{m}}}^{3}\cdot {\rm{GHz}})}^{1}\). We estimated the fieldfree dielectric volume V_{d} = (0.15 × 0.3 × 0.05) μm^{3} by assuming that the global field penetrates the sample dielectric to a depth of about its thickness (50 nm) from the sides open to air.
Discussion
The average dipole moment of the observed sampleTLS was p_{∥} = (0.4 ± 0.2) eÅ (see Supplementary Methods 2 for calculations), which results in a loss tangent^{48} of the employed AlO_{x} (ε_{r} ≈ 10) of \(\tan{\delta}_{0}=\pi{P}_{0}{p}_{\parallel}^{2}{(3{\varepsilon}_{0}{\varepsilon}_{r})}^{1}\approx 1{0}^{3}\), comparable with the number quoted above. The statistics shown in Fig. 2c indicate that the qubits were mostly limited by TLS hosted inside the tunnel barrier of the stray Josephson junctions (light green in Fig. 1d), which are a fabrication artefact that could have been avoided by shorting them in an additional lithography step^{49,50}.
For the 1.5–2 nm thin^{51,52,53} and 17.17 μm^{2} large tunnel barriers of the two stray junctions shown in Fig. 1d, our measurements indicate a TLS volume density of P_{0,JJ} = 360 to \(270\ {(\mu {{\rm{m}}}^{3}\cdot {\rm{GHz}})}^{1}\), in good agreement to previous work^{40}. Notably, this is about six times smaller than the TLS density found in the thicker layer of deposited AlOx in the sample capacitor. This is probably due to the minimum detectable TLS dipole moment size, i.e., qubit’s sensitivity, which is smaller for sampleTLS due to stronger oscillating qubit fields (≈90 Vm^{−1}) inside the sample capacitor, compared to the field inside the tunnel barrier of the stray junctions (≈15 Vm^{−1}). We speculate that this notion might be dressed due to various reasons like a reduced dangling bond density due to facilitated atom diffusibility and selfannealing in the thin tunnel barrier^{54}, or enhanced shielding of TLS by the evanescent Cooperpair condensate^{44}, or reasons related to the material’s different growth conditions.
Efield spectroscopy also revealed coherent mutual interactions between TLS in the sample material, which manifest themselves in avoided level crossings as shown in Fig. 3. The coupling between the TLS is described by the interaction Hamiltonian \({H}_{{\rm{int}}}=\frac{\hslash }{2}({g}_{x}{\sigma }_{1}^{x}{\sigma }_{2}^{x}+{g}_{z}{\sigma }_{1}^{z}{\sigma }_{2}^{z})\), where \({\sigma }_{i}^{x}\) and \({\sigma }_{i}^{z}\) are the Pauli matrices of TLS i. As an advancement over earlier work^{43}, the combined control of strain and local Efield allowed us to mutually detune the TLS and shift the avoided level crossing through the symmetry point of the observed TLS as demonstrated by the lower panels of Fig. 3. Since the longitudinal coupling component \({g}_{z}\propto {\sigma }_{1}^{z}\) changes its sign when TLS 1 is tuned through its symmetry point, its effect can be well distinguished from the transversal component g_{x}. This enabled fitting of both components g_{x} = − 19 (μs)^{−1} and g_{z} = 25 (μs)^{−1}. More details on the description of coherently interacting TLS can be found in a previous work^{43} and in Supplementary Methods 4.
In conclusion, we demonstrated that superconducting qubits can serve as interfaces for studying quantum properties of individual atomicsize tunneling systems located in arbitrary materials deposited as thin films. Qubit swap spectroscopy in dependence on the applied electric field bias to the sample material enables precise measurement of the TLS’ coupling dipole moments and reveals avoided level crossings, which herald coherent interaction between TLS. The possibility to mutually detune interacting TLS by using mechanical strain as a second control parameter allows one to fully characterize the type of the interaction. The demonstrated approach has a large potential to provide further insights into the puzzling physics of amorphous solids. It may serve as a valuable tool in the search for lowloss materials urgently needed to advance nanofabricated devices and superconducting quantum processors where TLS play a detrimental role.
Methods
Sample fabrication
The qubit samples were fabricated and characterized at KIT. A microchip contained three independent Xmon qubits of whom two were shunted by a sample capacitor, and a third one served as a reference qubit. The qubit electrode, ground plane and resonators were patterned into a 100nm thick Al film with an inductively coupled Ar–Cl plasma. After Argonion milling^{55} of the optically patterned electrodes in a PLASSYS shadow evaporation device, the Josephson junctions were deposited in a subsequent electronbeam lithography step.
Qubit samples with large and small sample capacitors were studied. The bottom electrode of large sample capacitors consisted a narrow extension of the qubit island. The bottom layer of the small sample capacitor (see Fig. 1e) was made simultaneously with the Josephson junctions. Sample dielectric and top electrodes of both capacitor types were formed in the PLASSYS device using an MMA/PMMA copolymer mask patterned in an electronbeam lithography step. After removing the native oxide of the bottom electrode with the Ar milling process, the sample dielectric (here 50 nm AlO_{x}) was formed during a perpendicular deposition of Al at a rate of 0.2 nm s^{−1} in an oxygen atmosphere (chamber pressure of 3 × 10^{−4} mBar, oxygen flow of 5 sccm). The dielectric was in situ covered by perpendicularly deposited 100nm thick layer of Al that formed the top electrode. Further details are reported in the PhD thesis by AB, Chap. 3.2.3^{44}.
Experimental setup
The sample was measured in an Oxford Kelvinox 100 wet dilution refrigerator at a temperature of 30 mK. The qubit chip was installed in a lighttight aluminium housing protected by a cryoperm magnetic shield. The coaxial control lines were heavily attenuated, filtered, and equipped with custommade infrared filters. The qubit state was detected via the dispersive shift^{12} of a notchtype readout resonator, which was capacitively coupled to the qubit, and probed in a standard homodyne microwave detection setup.
The DCgate for tuning the surfacedefects consisted of a copperfoil/Kapton foil stack that was glued to the lid of the sample box. It was connected via a twisted pair equipped with an RClowpass filter (cutoff ca 10 kHz) at the 1Kstage, and a custommade copper powder lowpass filter (1 MHz cutoff) at the 30 mK stage. The top electrode of the sample capacitor was controlled via an attenuated microwave line, as further detailed in the Supplementary Methods 2.
Data availability
Data are available upon reasonable request.
References
Zeller, R. C. & Pohl, R. O. Thermal conductivity and specific heat of noncrystalline solids. Phys. Rev. B 4, 2029–2041 (1971).
Leggett, A. J. & Vural, D. C. "Tunneling twolevel systems” model of the lowtemperature properties of glasses: are "smokinggun” tests possible? J. Phys. Chem. B 117, 12966–71 (2013).
Freeman, J. J. & Anderson, A. C. Thermal conductivity of amorphous solids. Phys. Rev. B 34, 5684–5690 (1986).
Anderson, P. W., Halperin, B. I. & Varma, C. M. Anomalous lowtemperature thermal properties of glasses and spin glasses. Philos. Magazine 25, 1–9 (1972).
Phillips, W. A. Tunneling states in amorphous solids. J. Low Temp. Phys. 7, 351 (1972).
Burin, A. & Kagan, Y. On the nature of the universal properties of amorphous solids. Phys. Lett. A 215, 191–196 (1996).
Lubchenko, V. & Wolynes, P. G. The microscopic quantum theory of low temperature amorphous solids. Adv. Chem. Phys. 136, 95–206 (2007).
Carruzzo, H. M. & Yu, C. C. Why phonon scattering in glasses is universally small at low temperatures. Phys. Rev. Lett. 124, 075902 (2020).
Schechter, M. & Stamp, P. C. E. Inversion symmetric twolevel systems and the lowtemperature universality in disordered solids. Phys. Rev. B 88, 174202 (2013).
Karpov, V., Klinger, I. & Ignat’Ev, F. Theory of the lowtemperature anomalies in the thermal properties of amorphous structures. Zh. eksp. teor. Fiz 84, 760–775 (1983).
Buchenau, U., Galperin, Y. M., Gurevich, V. L. & Schober, H. R. Anharmonic potentials and vibrational localization in glasses. Phys. Rev. B 43, 5039–5045 (1991).
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162 (2004).
Koch, J. et al. Chargeinsensitive qubit design derived from the cooper pair box. Phys. Rev. A 76, 042319 (2007).
Wang, C. et al. Surface participation and dielectric loss in superconducting qubits. Appl. Phys. Lett. 107, 162601 (2015).
Lin, Y.H. et al. Demonstration of protection of a superconducting qubit from energy decay. Phys. Rev. Lett. 120, 150503 (2018).
Earnest, N. et al. Realization of a λ system with metastable states of a capacitively shunted fluxonium. Phys. Rev. Lett. 120, 150504 (2018).
Nersisyan, A. et al. Manufacturing low dissipation superconducting quantum processors. in 2019 IEEE International Electron Devices Meeting (IEDM), 31.1.1–31.1.4 (2019).
P. M. Place, A. et al. New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. arXiv preprint arXiv:2003.00024 (2020).
Otterbach, J. et al. Unsupervised machine learning on a hybrid quantum computer. arXiv preprint arXiv:1712.05771 (2017).
Amico, M., Saleem, Z. H. & Kumph, M. Experimental study of shor’s factoring algorithm using the ibm q experience. Phys. Rev. A 100, 012305 (2019).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).
Müller, C., Cole, J. H. & Lisenfeld, J. Towards understanding twolevelsystems in amorphous solids: insights from quantum circuits. Rep. Prog. Phys. 82, 124501 (2019).
Lisenfeld, J. et al. Measuring the temperature dependence of individual twolevel systems by direct coherent control. Phys. Rev. Lett. 105, 230504 (2010).
Bilmes, A. et al. Electronic decoherence of twolevel systems in a josephson junction. Phys. Rev. B 96, 064504 (2017).
Barends, R. et al. Coherent josephson qubit suitable for scalable quantum integrated circuits. Phys. Rev. Lett. 111, 080502 (2013).
Simmonds, R. W., Lang, K. M., Hite, D. A., Pappas, D. P. & Martinis, J. M. Decoherence in Josephson qubits from junction resonances. Phys. Rev. Lett. 93, 077003 (2004).
Burnett, J. et al. Evidence for interacting twolevel systems from the 1/f noise of a superconducting resonator. Nat. Commun. 5, 4119 (2014).
Schlör, S. et al. Correlating decoherence in transmon qubits: low frequency noise by single fluctuators. Phys. Rev. Lett. 123, 190502 (2019).
Faoro, L. & Ioffe, L. B. Internal loss of superconducting resonators induced by interacting twolevel systems. Phys. Rev. Lett. 109, 157005 (2012).
Klimov, P. et al. Fluctuations of energyrelaxation times in superconducting qubits. Phys. Rev. Lett. 121, 090502 (2018).
Burnett, J. J. et al. Decoherence benchmarking of superconducting qubits. npj Quant. Inform. 5, 54 (2019).
Müller, C., Lisenfeld, J., Shnirman, A. & Poletto, S. Interacting twolevel defects as sources of fluctuating highfrequency noise in superconducting circuits. Phys. Rev. B 92, 035442 (2015).
Cooper, K. B. et al. Observation of quantum oscillations between a josephson phase qubit and a microscopic resonator using fast readout. Phys. Rev. Lett. 93, 180401 (2004).
Neeley, M. et al. Process tomography of quantum memory in a josephsonphase qubit coupled to a twolevel state. Nat. Phys. 4, 523 (2008).
Shalibo, Y. et al. Lifetime and coherence of twolevel defects in a Josephson junction. Phys. Rev. Lett. 105, 177001 (2010).
Lisenfeld, J. et al. Decoherence spectroscopy with individual tls. Sci. Rep. 6, 23786 (2016).
Meißner, S. M., Seiler, A., Lisenfeld, J., Ustinov, A. V. & Weiss, G. Probing individual tunneling fluctuators with coherently controlled tunneling systems. Phys. Rev. B 97, 180505 (2018).
Grabovskij, G. J., Peichl, T., Lisenfeld, J., Weiss, G. & Ustinov, A. V. Strain tuning of individual atomic tunneling systems detected by a superconducting qubit. Science 338, 232 (2012).
Sarabi, B., Ramanayaka, A. N., Burin, A. L., Wellstood, F. C. & Osborn, K. D. Projected dipole moments of individual twolevel defects extracted using circuit quantum electrodynamics. Phys. Rev. Lett. 116, 167002 (2016).
Lisenfeld, J. et al. Electric field spectroscopy of material defects in transmon qubits. npj Quant. Inform. 5, 1–6 (2019).
Bilmes, A. et al. Resolving the positions of defects in superconducting quantum bits. Sci. Rep. 10, 1–6 (2020).
Martinis, J. M. et al. Decoherence in Josephson qubits from dielectric loss. Phys. Rev. Lett. 95, 210503 (2005).
Lisenfeld, J. et al. Observation of directly interacting coherent twolevel systems in an amorphous material. Nat. Commun. 6, 6182 (2015).
Bilmes, A. Resolving locations of defects in superconducting transmon qubits. Dissertation. https://doi.org/10.5445/KSP/1000097557. (2019)
Pappas, D. P., Vissers, M. R., Wisbey, D. S., Kline, J. S. & Gao, J. Two level system loss in superconducting microwave resonators. IEEE Trans. Appl. Superconduct. 21, 871–874 (2011).
Deng, C., Otto, M. & Lupascu, A. Characterization of lowtemperature microwave loss of thin aluminum oxide formed by plasma oxidation. Appl. Phys. Lett. 104, 043506 (2014).
Brehm, J. D., Bilmes, A., Weiss, G., Ustinov, A. V. & Lisenfeld, J. Transmissionline resonators for the study of individual twolevel tunneling systems. Appl. Phys. Lett. 111, 112601 (2017).
Gao, J. The physics of superconducting microwave resonators. Phd thesis, (2008).
Quintana, C. et al. Characterization and reduction of microfabricationinduced decoherence in superconducting quantum circuits. Appl. Phys. Lett. 105, 062601 (2014).
Osman, A. et al. Simplified josephsonjunction fabrication process for reproducibly highperformance superconducting qubits. arXiv preprint arXiv:2011.05230 (2020).
Kang, X. et al. Measurements of tunneling barrier thicknesses for nb/alalox/nb tunnel junctions. Phys. C Superconduct. Appl. 503, 29–32 (2014).
Zeng, L. J. et al. Direct observation of the thickness distribution of ultra thin AlOxbarriers in al/AlOx/al josephson junctions. J. Phys. D. 48, 395308 (2015).
Fritz, S., Radtke, L., Schneider, R., Weides, M. & Gerthsen, D. Optimization of al/alox/allayer systems for josephson junctions from a microstructure point of view. J. Appl. Phys. 125, 165301 (2019).
MolinaRuiz, M. et al. Origin of mechanical and dielectric losses from twolevel systems in amorphous silicon. arXiv preprint arXiv:2008.07489v1 (2020).
Grünhaupt, L. et al. An argon ion beam milling process for native alox layers enabling coherent superconducting contacts. Appl. Phys. Lett. 111, 072601 (2017).
Acknowledgements
A.B. acknowledges support from the Helmholtz International Research School for Teratronics (HIRST) and the LandesgraduiertenförderungKarlsruhe (LGF). J.B. was financially supported by Studienstiftung des Deutschen Volkes. J.L. gratefully acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG), grant LI24461/2. A.V.U acknowledges support provided by the Ministry of Education and Science of the Russian Federation in the framework of the Program to Increase Competitiveness of the NUST MISIS (contract No. K22020017). The work was also supported by the Initiative and Networking Fund of the Helmholtz Association and by Google LLC. We acknowledge support by the KITPublication Fund of the Karlsruhe Institute of Technology. We acknowledge Johannes Rotzinger and Ioan Pop for fruitful discussions, Silvia Diewald and Patrice Brenner for their technical assistance with electronbeam devices, and Lucas Radtke for his indispensable assistance in the cleanroom.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Contributions
The qubit samples were designed and fabricated by A.B. Experiments were devised and performed by J.L. in a setup implemented by A.B. and J.L. S.V. performed calculations for the mutually coupled TLS system, and J.B. simulated the electricfield distribution of nanogap capacitors. The manuscript was written by J.L. and A.B. with contributions from all authors.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bilmes, A., Volosheniuk, S., Brehm, J.D. et al. Quantum sensors for microscopic tunneling systems. npj Quantum Inf 7, 27 (2021). https://doi.org/10.1038/s4153402000359x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4153402000359x
This article is cited by

Experimentally revealing anomalously large dipoles in the dielectric of a quantum circuit
Scientific Reports (2022)

Probing defect densities at the edges and inside Josephson junctions of superconducting qubits
npj Quantum Information (2022)