Studying phonon coherence with a quantum sensor

Nanomechanical oscillators offer numerous advantages for quantum technologies. Their integration with superconducting qubits shows promise for hardware-efficient quantum error-correction protocols involving superpositions of mechanical coherent states. Limitations of this approach include mechanical decoherence processes, particularly two-level system (TLS) defects, which have been widely studied using classical fields and detectors. In this manuscript, we use a superconducting qubit as a quantum sensor to perform phonon number-resolved measurements on a piezoelectrically coupled phononic crystal cavity. This enables a high-resolution study of mechanical dissipation and dephasing in coherent states of variable size (n ¯ ≃ 1 - 10 phonons). We observe nonexponential relaxation and state size-dependent reduction of the dephasing rate, which we attribute to TLS. Using a numerical model, we reproduce the dissipation signatures (and to a lesser extent, the dephasing signatures) via emission into a small ensemble (N = 5) of rapidly dephasing TLS. Our findings comprise a detailed examination of TLS-induced phonon decoherence in the quantum regime.

Giant Two-Level Systems in a Granular Superconductor

Disordered thin films are a common choice of material for superconducting, high impedance circuits used in quantum information or particle detector physics. A wide selection of materials with different levels of granularity are available, but, despite low microwave losses being reported for some, the high degree of disorder always implies the presence of intrinsic defects. Prominently, quantum circuits are prone to interact with two-level systems (TLS), typically originating from solid state defects in the dielectric parts of the circuit, like surface oxides or tunneling barriers. We present an experimental investigation of TLS in granular aluminum thin films under applied mechanical strain and electric fields. The analysis reveals a class of strongly coupled TLS having electric dipole moments up to 30  eÅ, an order of magnitude larger than dipole moments commonly reported for solid state defects. Notably, these large dipole moments appear more often in films with a higher resistivity. Our observations shed new light on granular superconductors and may have implications for their usage as a quantum circuit material.

Mechanically induced correlated errors on superconducting qubits with relaxation times exceeding 0.4 ms

Superconducting qubits are among the most advanced candidates for achieving fault-tolerant quantum computing. Despite recent significant advancements in the qubit lifetimes, the origin of the loss mechanism for state-of-the-art qubits is still subject to investigation. Furthermore, the successful implementation of quantum error correction requires negligible correlated errors between qubits. Here, we realize long-lived superconducting transmon qubits that exhibit fluctuating lifetimes, averaging 0.2 ms and exceeding 0.4 ms - corresponding to quality factors above 5 million and 10 million, respectively. We then investigate their dominant error mechanism. By introducing novel time-resolved error measurements that are synchronized with the operation of the pulse tube cooler in a dilution refrigerator, we find that mechanical vibrations from the pulse tube induce nonequilibrium dynamics in highly coherent qubits, leading to their correlated bit-flip errors. Our findings not only deepen our understanding of the qubit error mechanisms but also provide valuable insights into potential error-mitigation strategies for achieving fault tolerance by decoupling superconducting qubits from their mechanical environments.

Quantum heat valve and diode of strongly coupled defects in amorphous material

The mechanical strain can control the frequency of two-level atoms in amorphous material. In this work, we would like to employ two coupled two-level atoms to manipulate the magnitude and direction of heat transport by controlling mechanical strain to realize the function of a thermal switch and valve. It is found that a high-performance heat diode can be realized in the wide piezo voltage range at different temperatures. We also discuss the dependence of the rectification factor on temperatures and couplings of heat reservoirs. We find that the higher temperature differences correspond to the larger rectification effect. The asymmetry system-reservoir coupling strength can enhance the magnitude of heat transfer, and the impact of asymmetric and symmetric coupling strength on the performance of the heat diode is complementary. It may provide an efficient way to modulate and control heat transport's magnitude and flow preference. This work may give insight into designing and tuning quantum heat machines.

Avoided Level Crossing and Entangled States of Interacting Hydrogen Molecules Detected by the Quantum Superposition Microscope

Pump-probe measurements by ultrashort THz pulses can be used to excite and follow the coherence dynamics in the time domain of single hydrogen molecules (H2) in the junction of a scanning tunneling microscope (STM). By tailoring the resonance frequency through the sample bias, we identified two spectral signatures of the interactions among multiple H2 molecules. First, the avoided level crossing featured by energy gaps ranging from 20 to 80 GHz was observed because of the level repulsion between two H2 molecules. Second, the tip can sense the signal of H2 outside the junction through the projective measurement on the H2 inside the junction, owing to the entangled states created through the interactions. A dipolar-type interaction was integrated into the tunneling two-level system model of H2, enabling accurate reproduction of the observed behaviors. Our results obtained by the quantum superposition microscope reveal the intricate quantum mechanical interplay among H2 molecules and additionally provide a 2D platform to investigate unresolved questions of amorphous materials.

Experimentally revealing anomalously large dipoles in the dielectric of a quantum circuit

Quantum two-level systems (TLSs) intrinsic to glasses induce decoherence in many modern quantum devices, such as superconducting qubits. Although the low-temperature physics of these TLSs is usually well-explained by a phenomenological standard tunneling model of independent TLSs, the nature of these TLSs, as well as their behavior out of equilibrium and at high energies above 1 K, remain inconclusive. Here we measure the non-equilibrium dielectric loss of TLSs in amorphous silicon using a superconducting resonator, where energies of TLSs are varied in time using a swept electric field. Our results show the existence of two distinct ensembles of TLSs, interacting weakly and strongly with phonons, where the latter also possesses anomalously large electric dipole moment. These results may shed new light on the low temperature characteristics of amorphous solids, and hold implications to experiments and applications in quantum devices using time-varying electric fields.

Soft-Clamped Silicon Nitride String Resonators at Millikelvin Temperatures

We demonstrate that soft-clamped silicon nitride strings with a large aspect ratio can be operated at mK temperatures. The quality factors (Q) of two measured devices show consistent dependency on the cryostat temperature, with soft-clamped mechanical modes reaching Q>10^{9} at roughly 46 mK. For low optical readout power, Q is found to saturate, indicating good thermalization between the sample and the stage it is mounted on. Our best device exhibits a calculated force sensitivity of 9.6  zN/sqrt[Hz] and a thermal decoherence time of 0.38 s, which bode well for future applications such as nanomechanical force sensing.

Two-level systems in superconducting quantum devices due to trapped quasiparticles

A major issue for the implementation of large-scale superconducting quantum circuits is the interaction with interfacial two-level system (TLS) defects that lead to qubit parameter fluctuations and relaxation. Another major challenge comes from nonequilibrium quasiparticles (QPs) that result in qubit relaxation and dephasing. Here, we reveal a previously unexplored decoherence mechanism in the form of a new type of TLS originating from trapped QPs, which can induce qubit relaxation. Using spectral, temporal, thermal, and magnetic field mapping of TLS-induced fluctuations in frequency tunable resonators, we identify a highly coherent subset of the general TLS population with a low reconfiguration temperature ∼300 mK and a nonuniform density of states. These properties can be understood if the TLS are formed by QPs trapped in shallow subgap states formed by spatial fluctutations of the superconducting order parameter. This implies that even very rare QP bursts will affect coherence over exponentially long time scales.

Intrinsic Photon Loss at the Interface of Superconducting Devices

We present a quantum theory of dielectric energy loss arising from the piezoelectric coupling between photons and phonons in superconducting devices. Photon loss is shown to occur predominantly at the interface, where the piezoelectric effect is nonzero even when the materials are perfectly crystalline (epitaxial) and free of two-level system defects. We present explicit numerical calculations for the value of the intrinsic loss tangent at several interfaces to conclude that the T_{1} of superconducting qubits may reach over 10^{4}  μs if the device is made with defect-free interfaces.

Why Phonon Scattering in Glasses is Universally Small at Low Temperatures

We present a novel view of the standard model of tunneling two level systems (TLSs) to explain the puzzling universal value of a quantity, C∼3×10^{-4}, that characterizes phonon scattering in glasses below 1 K as reflected in thermal conductivity, ultrasonic attenuation, internal friction, and the change in sound velocity. Physical considerations lead to a broad distribution of phonon-TLS couplings that (1) exponentially renormalize tunneling matrix elements, and (2) reduce the TLS density of states through TLS-TLS interactions. We find good agreement between theory and experiment for a variety of individual glasses.

Resolving the positions of defects in superconducting quantum bits

Solid-state quantum coherent devices are quickly progressing. Superconducting circuits, for instance, have already been used to demonstrate prototype quantum processors comprising a few tens of quantum bits. This development also revealed that a major part of decoherence and energy loss in such devices originates from a bath of parasitic material defects. However, neither the microscopic structure of defects nor the mechanisms by which they emerge during sample fabrication are understood. Here, we present a technique to obtain information on locations of defects relative to the thin film edge of the qubit circuit. Resonance frequencies of defects are tuned by exposing the qubit sample to electric fields generated by electrodes surrounding the chip. By determining the defect's coupling strength to each electrode and comparing it to a simulation of the field distribution, we obtain the probability at which location and at which interface the defect resides. This method is applicable to already existing samples of various qubit types, without further on-chip design changes. It provides a valuable tool for improving the material quality and nano-fabrication procedures towards more coherent quantum circuits.

Towards understanding two-level-systems in amorphous solids: insights from quantum circuits

Amorphous solids show surprisingly universal behaviour at low temperatures. The prevailing wisdom is that this can be explained by the existence of two-state defects within the material. The so-called standard tunneling model has become the established framework to explain these results, yet it still leaves the central question essentially unanswered-what are these two-level defects (TLS)? This question has recently taken on a new urgency with the rise of superconducting circuits in quantum computing, circuit quantum electrodynamics, magnetometry, electrometry and metrology. Superconducting circuits made from aluminium or niobium are fundamentally limited by losses due to TLS within the amorphous oxide layers encasing them. On the other hand, these circuits also provide a novel and effective method for studying the very defects which limit their operation. We can now go beyond ensemble measurements and probe individual defects-observing the quantum nature of their dynamics and studying their formation, their behaviour as a function of applied field, strain, temperature and other properties. This article reviews the plethora of recent experimental results in this area and discusses the various theoretical models which have been used to describe the observations. In doing so, it summarises the current approaches to solving this fundamentally important problem in solid-state physics.

Correlating Decoherence in Transmon Qubits: Low Frequency Noise by Single Fluctuators

We report on long-term measurements of a highly coherent, nontunable superconducting transmon qubit, revealing low-frequency burst noise in coherence times and qubit transition frequency. We achieve this through a simultaneous measurement of the qubit's relaxation and dephasing rate as well as its resonance frequency. The analysis of correlations between these parameters yields information about the microscopic origin of the intrinsic decoherence mechanisms in Josephson qubits. Our results are consistent with a small number of microscopic two-level systems located at the edges of the superconducting film, which is further confirmed by a spectral noise analysis.

Fluctuations of Energy-Relaxation Times in Superconducting Qubits

Superconducting qubits are an attractive platform for quantum computing since they have demonstrated high-fidelity quantum gates and extensibility to modest system sizes. Nonetheless, an outstanding challenge is stabilizing their energy-relaxation times, which can fluctuate unpredictably in frequency and time. Here, we use qubits as spectral and temporal probes of individual two-level-system defects to provide direct evidence that they are responsible for the largest fluctuations. This research lays the foundation for stabilizing qubit performance through calibration, design, and fabrication.

Correlating the nanostructure of Al-oxide with deposition conditions and dielectric contributions of two-level systems in perspective of superconducting quantum circuits

This work is concerned with Al/Al-oxide(AlO_x)/Al-layer systems which are important for Josephson-junction-based superconducting devices such as quantum bits. The device performance is limited by noise, which has been to a large degree assigned to the presence and properties of two-level tunneling systems in the amorphous AlO_x tunnel barrier. The study is focused on the correlation of the fabrication conditions, nanostructural and nanochemical properties and the occurrence of two-level tunneling systems with particular emphasis on the AlO_x-layer. Electron-beam evaporation with two different processes and sputter deposition were used for structure fabrication, and the effect of illumination by ultraviolet light during Al-oxide formation is elucidated. Characterization was performed by analytical transmission electron microscopy and low-temperature dielectric measurements. We show that the fabrication conditions have a strong impact on the nanostructural and nanochemical properties of the layer systems and the properties of two-level tunneling systems. Based on the understanding of the observed structural characteristics, routes are suggested towards the fabrication of Al/AlO_x/Al-layers systems with improved properties.

Topological Maxwell Metal Bands in a Superconducting Qutrit

We experimentally explore the topological Maxwell metal bands by mapping the momentum space of condensed-matter models to the tunable parameter space of superconducting quantum circuits. An exotic band structure that is effectively described by the spin-1 Maxwell equations is imaged. Threefold degenerate points dubbed Maxwell points are observed in the Maxwell metal bands. Moreover, we engineer and observe the topological phase transition from the topological Maxwell metal to a trivial insulator, and report the first experiment to measure the Chern numbers that are higher than one.

Two-color second-order sideband generation in an optomechanical system with a two-level system

Second-order sideband generation in an optomechanical system with the coupling between a mechanical resonator and a two-level system is discussed beyond the conventional linearized description of optomechanical interactions. The features of two-color second-order sideband generation are demonstrated in this hybrid system. We discovery that the switch between one- and two-color second-order sideband generation is easily realized by shifting the detuning between the control field and the cavity field or the transition frequency of the two-level system. The efficiency of two-color second-order sideband generation increases monotonously with the control field strength. The coupling strength between the mechanical resonator and the two-level system plays a decisive role in the appearance of the two-color second-order sidebands. The two-color second-order sideband generation may provide measurement with higher precision in new degrees of freedom.

Probing nanocrystalline grain dynamics in nanodevices

Dynamical structural defects exist naturally in a wide variety of solids. They fluctuate temporally and hence can deteriorate the performance of many electronic devices. Thus far, the entities of these dynamic objects have been identified to be individual atoms. On the other hand, it is a long-standing question whether a nanocrystalline grain constituted of a large number of atoms can switch, as a whole, reversibly like a dynamical atomic defect (that is, a two-level system). This is an emergent issue considering the current development of nanodevices with ultralow electrical noise, qubits with long quantum coherence time, and nanoelectromechanical system sensors with ultrahigh resolution. We demonstrate experimental observations of dynamic nanocrystalline grains that repeatedly switch between two or more metastable coordinate states. We study temporal resistance fluctuations in thin ruthenium dioxide (RuO₂) metal nanowires and extract microscopic parameters, including relaxation time scales, mobile grain sizes, and the bonding strengths of nanograin boundaries. These material parameters are not obtainable by other experimental approaches. When combined with previous in situ high-resolution transmission electron microscopy, our electrical method can be used to infer rich information about the structural dynamics of a wide variety of nanodevices and new two-dimensional materials.

Observation and coherent control of interface-induced electronic resonances in a field-effect transistor

Electronic defect states at material interfaces provide highly deleterious sources of noise in solid-state nanostructures, and even a single trapped charge can qualitatively alter the properties of short one-dimensional nanowire field-effect transistors (FET) and quantum bit (qubit) devices. Understanding the dynamics of trapped charge is thus essential for future nanotechnologies, but their direct detection and manipulation is rather challenging. Here, a transistor-based set-up is used to create and probe individual electronic defect states that can be coherently driven with microwave (MW) pulses. Strikingly, we resolve a large number of very high quality (Q ∼ 1 × 10⁵) resonances in the transistor current as a function of MW frequency and demonstrate both long decoherence times (∼1 μs-40 μs) and coherent control of the defect-induced dynamics. Efficiently characterizing over 800 individually addressable resonances across two separate defect-hosting materials, we propose that their properties are consistent with weakly driven two-level systems.

Dissipation induced by phonon elastic scattering in crystals

We demonstrate that the phonon elastic scattering leads to a dominant dissipation in crystals at low temperature. The two-level systems (TLSs) should be responsible for the elastic scattering, whereas the dissipation induced by static-point defects (SPDs) can not be neglected. One purpose of this work is to show how the energy splitting distribution of the TLS ensemble affects the dissipation. Besides, this article displays the proportion of phonon-TLS elastic scattering to total phonon dissipation. The coupling coefficient of phonon-SPD scattering and the constant P₀ of the TLS distribution are important that we estimate their magnitudes in this paper. Our results is useful to understand the phonon dissipation mechanism, and give some clues to improve the performance of mechanical resonators, apply the desired defects, or reveal the atom configuration in lattice structure of disordered crystals.

Quantum electromechanics on silicon nitride nanomembranes

Radiation pressure has recently been used to effectively couple the quantum motion of mechanical elements to the fields of optical or microwave light. Integration of all three degrees of freedom—mechanical, optical and microwave—would enable a quantum interconnect between microwave and optical quantum systems. We present a platform based on silicon nitride nanomembranes for integrating superconducting microwave circuits with planar acoustic and optical devices such as phononic and photonic crystals. Using planar capacitors with vacuum gaps of 60 nm and spiral inductor coils of micron pitch we realize microwave resonant circuits with large electromechanical coupling to planar acoustic structures of nanoscale dimensions and femtoFarad motional capacitance. Using this enhanced coupling, we demonstrate microwave backaction cooling of the 4.48 MHz mechanical resonance of a nanobeam to an occupancy as low as 0.32. These results indicate the viability of silicon nitride nanomembranes as an all-in-one substrate for quantum electro-opto-mechanical experiments.

Preparation and detection of mechanical objects at the quantum zero-point level has been achieved in both the optical and microwave regimes. Here, the authors develop silicon nitride nanomembranes that are suitable for integrating nanophotonic, nanomechanical and superconducting microwave circuits together.

Projected Dipole Moments of Individual Two-Level Defects Extracted Using Circuit Quantum Electrodynamics

Material-based two-level systems (TLSs), appearing as defects in low-temperature devices including superconducting qubits and photon detectors, are difficult to characterize. In this study we apply a uniform dc electric field across a film to tune the energies of TLSs within. The film is embedded in a superconducting resonator such that it forms a circuit quantum electrodynamical system. The energy of individual TLSs is observed as a function of the known tuning field. By studying TLSs for which we can determine the tunneling energy, the actual p_{z}, dipole moments projected along the uniform field direction, are individually obtained. A distribution is created with 60 p_{z}. We describe the distribution using a model with two dipole moment magnitudes, and a fit yields the corresponding values p=p_{1}=2.8±0.2  D and p=p_{2}=8.3±0.4  D. For a strong-coupled TLS the vacuum-Rabi splitting can be obtained with p_{z} and tunneling energy. This allows a measurement of the circuit's zero-point electric-field fluctuations, in a method that does not need the electric-field volume.

Decoherence spectroscopy with individual two-level tunneling defects

Recent progress with microfabricated quantum devices has revealed that an ubiquitous source of noise originates in tunneling material defects that give rise to a sparse bath of parasitic two-level systems (TLSs). For superconducting qubits, TLSs residing on electrode surfaces and in tunnel junctions account for a major part of decoherence and thus pose a serious roadblock to the realization of solid-state quantum processors. Here, we utilize a superconducting qubit to explore the quantum state evolution of coherently operated TLSs in order to shed new light on their individual properties and environmental interactions. We identify a frequency-dependence of TLS energy relaxation rates that can be explained by a coupling to phononic modes rather than by anticipated mutual TLS interactions. Most investigated TLSs are found to be free of pure dephasing at their energy degeneracy points, around which their Ramsey and spin-echo dephasing rates scale linearly and quadratically with asymmetry energy, respectively. We provide an explanation based on the standard tunneling model, and identify interaction with incoherent low-frequency (thermal) TLSs as the major mechanism of the pure dephasing in coherent high-frequency TLS.

Observation of directly interacting coherent two-level systems in an amorphous material

Parasitic two-level tunnelling systems originating from structural material defects affect the functionality of various microfabricated devices by acting as a source of noise. In particular, superconducting quantum bits may be sensitive to even single defects when these reside in the tunnel barrier of the qubit's Josephson junctions, and this can be exploited to observe and manipulate the quantum states of individual tunnelling systems. Here, we detect and fully characterize a system of two strongly interacting defects using a novel technique for high-resolution spectroscopy. Mutual defect coupling has been conjectured to explain various anomalies of glasses, and was recently suggested as the origin of low-frequency noise in superconducting devices. Our study provides conclusive evidence of defect interactions with full access to the individual constituents, demonstrating the potential of superconducting qubits for studying material defects. All our observations are consistent with the assumption that defects are generated by atomic tunnelling.

Hydrogen-free amorphous silicon with no tunneling states

The ubiquitous low-energy excitations, known as two-level tunneling systems (TLSs), are one of the universal phenomena of amorphous solids. Low temperature elastic measurements show that e-beam amorphous silicon (a-Si) contains a variable density of TLSs which diminishes as the growth temperature reaches 400 °C. Structural analyses show that these a-Si films become denser and more structurally ordered. We conclude that the enhanced surface energetics at a high growth temperature improved the amorphous structural network of e-beam a-Si and removed TLSs. This work obviates the role hydrogen was previously thought to play in removing TLSs in the hydrogenated form of a-Si and suggests it is possible to prepare "perfect" amorphous solids with "crystal-like" properties for applications.

Coherent Josephson qubit suitable for scalable quantum integrated circuits

We demonstrate a planar, tunable superconducting qubit with energy relaxation times up to 44 μs. This is achieved by using a geometry designed to both minimize radiative loss and reduce coupling to materials-related defects. At these levels of coherence, we find a fine structure in the qubit energy lifetime as a function of frequency, indicating the presence of a sparse population of incoherent, weakly coupled two-level defects. We elucidate this defect physics by experimentally varying the geometry and by a model analysis. Our "Xmon" qubit combines facile fabrication, straightforward connectivity, fast control, and long coherence, opening a viable route to constructing a chip-based quantum computer.

"Tunneling two-level systems" model of the low-temperature properties of glasses: are "smoking-gun" tests possible?

Following a brief review of the "two-level (tunneling) systems" model of the low-temperature properties of amorphous solids ("glasses"), we ask whether it is in fact the unique explanation of these properties as is usually assumed, concluding that this is not necessarily the case. We point out that (a) one specific form of the model is already experimentally refuted and (b) that a definitive test of the model in its most general form, while not yet carried out, would appear to be now experimentally feasible.

Nuclear quadrupole moments as a microscopic probe to study the motion of atomic tunneling systems in amorphous solids

The properties of amorphous solids below 1 K are dominated by atomic tunneling systems. A basic description is given by the standard tunneling model. Despite its success, the standard tunneling model still remains phenomenological and little is known about the microscopic nature of tunneling systems in amorphous solids. We present dielectric polarization echo experiments on partially deuterated amorphous glycerol. Nuclear quadrupoles, introduced by the deuteration, influence the echo amplitude in a characteristic way and allow us to draw for the first time detailed conclusions about the microscopic nature of the tunneling processes in amorphous glycerol.

Nonlinear quantum optomechanics via individual intrinsic two-level defects

We propose to use the intrinsic two-level system (TLS) defect states found naturally in integrated optomechanical devices for exploring cavity QED-like phenomena with localized phonons. The Jaynes-Cummings-type interaction between TLS and mechanics can reach the strong coupling regime for existing nano-optomechanical systems, observable via clear signatures in the optomechanical output spectrum. These signatures persist even at finite temperature, and we derive an explicit expression for the temperature at which they vanish. Further, the ability to drive the defect with a microwave field allows for realization of phonon blockade, and the available controls are sufficient to deterministically prepare non-classical states of the mechanical resonator.

Delocalized oxygen as the origin of two-level defects in Josephson junctions

One of the key problems facing superconducting qubits and other Josephson junction devices is the decohering effects of bistable material defects. Although a variety of phenomenological models exist, the true microscopic origin of these defects remains elusive. For the first time we show that these defects may arise from delocalization of the atomic position of the oxygen in the oxide forming the Josephson junction barrier. Using a microscopic model, we compute experimentally observable parameters for phase qubits. Such defects are charge neutral but have nonzero response to both applied electric field and strain. This may explain the observed long coherence time of two-level defects in the presence of charge noise, while still coupling to the junction electric field and substrate phonons.