## Introduction

We are still lacking an explanation for the behaviour of amorphous materials at low temperatures <10 K1,2. Why is it that even widely different materials ranging from biatomic glasses to polymers show quantitatively identical properties such as specific heat and thermal conductivity3? The Standard Tunneling Model (STM)4,5 has been a first attempt to explain these universal anomalies on the basis of two-level 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 TLS-TLS interactions6,7,8, assume different types of TLS9, or consider specific dependencies of TLS potential energies10,11. 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 well-controllable macroscopic quantum systems with custom-tailored 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 solid-state quantum information processors, intensive effort went into the development of advanced circuit designs12,13,14,15,16 and fabrication techniques17,18, 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 algorithms19, access to prototype quantum processors via the cloud20, and the achievement of quantum supremacy by controlling a 53-qubit system that could not anymore be simulated efficiently by classical supercomputers21.

Despite these achievements, progress towards truly large-scale quantum processors is still hindered by decoherence, of which the major part is due to losses in dielectric circuit materials22. 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 E-field. When TLS are at resonance with the qubit, they can efficiently dissipate energy into the phonon23 or BCS-quasiparticle24 bath which results in reduced qubit energy relaxation times T125 and, in the case of strong coupling, gives rise to avoided level crossings in qubit spectroscopy26. Moreover, thermally activated TLS at low energies may interact with high-energy TLS that have frequencies near resonance with a qubit or resonator, and this causes temporal fluctuations of the device’s resonance frequency27,28 and energy relaxation rate29,30,31,32.

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 TLS33 is possible, allowing one to characterize TLS’ coherence properties23,34,35, and their coupling to the environment24,36,37. A useful method for such studies is to control the TLS’ internal asymmetry energy and thus their resonance frequency by applied mechanical strain38 or DC-electric field39. Operating qubits in electric fields allows one to distinguish defects in tunnel junction barriers from those on electrode surfaces40 and to obtain information on the positions of individual TLS in the quantum circuit41.

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 qubit13,25, which consists of a capacitively shunted DC-SQUID 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 on-chip coil providing magnetic flux, which frustrates the Josephson energy of the DC-SQUID 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 so-called 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 double-well potential whose minima differ by the asymmetry energy ε, and transitions between wells occur at a tunneling energy Δ0, 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 element39 Δ0/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/T1 of the isolated qubit. The criterium g ≈ 1/T1 togeher with the TLS’ above-mentioned 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 Vrms/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 ω10 ≈ 2π 6.2 GHz when Ctot ≈ 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 eÅ42, and assuming a rather conservative T1 ≈ 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) μm2, resulting in Cs ≈ 0.15 fF Ctot 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 voltage-biasing the sample capacitor’s electrode as indicated in Fig. 1a, where the additional capacitor Cf ~ 250 fF serves as a DC-break. The bias voltage will not induce an electric field in the transmon’s shunt capacitor Cq nor inside the Josephson junctions’ tunnel barrier since the DC-electric potential difference of the transmon island and ground will be compensated by Cooper-pair tunneling40, so that only TLS in the sample capacitor respond to the applied voltage Vs. In addition, we can tune TLS residing at the perimeter of the qubit capacitor by a globally applied DC-electric field that is generated by an electrode installed above the qubit chip40 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 actuator38,43, 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 AlOx 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 electron-beam 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 AlOx by eBeam-evaporation of Al in an oxygen atmosphere, and capping it by a top Al electrode. The filter capacitor Cf 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) μm2 which did not contribute significantly to decoherence. Two tested Cs-shunted qubits reached T1-times of 3.3–4.2 μs, which is comparable with an isolated reference qubit (T1 ≈ 4.3μs) on the same chip. In another batch, we also tested larger sample capacitors (0.3 × 2.1) μm2, which did limit the qubit’s T1 time44. This allowed us to measure the loss tangent of the employed AlOx dielectric as $$\tan {\delta }_{0}\approx (1.7\pm 0.2)\cdot 1{0}^{-3}$$, comparable with other reports42,45,46,47.

### TLS spectroscopy

To distinguish whether a TLS is located in a tunnel barrier, at the qubit’s film edges41, or in the sample capacitor dielectric, we track its resonance frequency for a range of voltages applied to the global DC electrode (Vg), to the sample dielectric (Vs), and to the piezo (Vp). An example of such a measurement is presented in Fig. 2a, showing the frequency dependence of the qubit’s T1 time estimated by swap spectroscopy33,35,43, 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 bias-dependent asymmetry energy ε = εi + γgVg + γsVs + γpVp up to an intrinsic offset εi. The fit also results in the value of the TLS’ tunneling energy Δ0 if it lies within the tunability range of the qubit’s resonance frequency.

Thanks to the well-specified DC-electric field Vs/d in the sample capacitor, the coupling electric dipole moment p = γsd/2 of TLS in the sample material is directly obtained from the identity 2pVs/d = γsVs4,5. 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 (Δ0/E) is often unknown. From measurements on two identical qubits in one cool-down, 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 field-free dielectric volume Vd = (0.15 × 0.3 × 0.05) μm3 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 sample-TLS was p = (0.4 ± 0.2) eÅ (see Supplementary Methods 2 for calculations), which results in a loss tangent48 of the employed AlOx (εr ≈ 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 step49,50.

For the 1.5–2 nm thin51,52,53 and 17.17 μm2 large tunnel barriers of the two stray junctions shown in Fig. 1d, our measurements indicate a TLS volume density of P0,JJ = 360 to $$270\ {(\mu {{\rm{m}}}^{3}\cdot {\rm{GHz}})}^{-1}$$, in good agreement to previous work40. 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 sample-TLS 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 self-annealing in the thin tunnel barrier54, or enhanced shielding of TLS by the evanescent Cooper-pair condensate44, or reasons related to the material’s different growth conditions.

E-field 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 work43, the combined control of strain and local E-field 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 gx. This enabled fitting of both components gx = − 19 (μs)−1 and gz = 25 (μs)−1. More details on the description of coherently interacting TLS can be found in a previous work43 and in Supplementary Methods 4.

In conclusion, we demonstrated that superconducting qubits can serve as interfaces for studying quantum properties of individual atomic-size 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 low-loss materials urgently needed to advance nano-fabricated 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 100-nm thick Al film with an inductively coupled Ar–Cl plasma. After Argon-ion milling55 of the optically patterned electrodes in a PLASSYS shadow evaporation device, the Josephson junctions were deposited in a subsequent electron-beam 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 electron-beam lithography step. After removing the native oxide of the bottom electrode with the Ar milling process, the sample dielectric (here 50 nm AlOx) 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 100-nm thick layer of Al that formed the top electrode. Further details are reported in the PhD thesis by AB, Chap. 3.2.344.

### 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 light-tight aluminium housing protected by a cryoperm magnetic shield. The coaxial control lines were heavily attenuated, filtered, and equipped with custom-made infrared filters. The qubit state was detected via the dispersive shift12 of a notch-type readout resonator, which was capacitively coupled to the qubit, and probed in a standard homodyne microwave detection setup.

The DC-gate for tuning the surface-defects consisted of a copper-foil/Kapton foil stack that was glued to the lid of the sample box. It was connected via a twisted pair equipped with an RC-lowpass filter (cutoff ca 10 kHz) at the 1K-stage, and a custom-made 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.