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arXiv:1105.3446 · Physical Review B

Jaynes-Cummings treatment of superconducting resonators with dielectric loss due to two-level systems

Mishkatul Bhattacharya, Kevin D. Osborn, Ari Mizel

Published Sep 13, 2011• 9 citations
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Abstract

We perform a quantum mechanical analysis of superconducting resonators subject to dielectric loss arising from charged two-level systems. We present numerical and analytical descriptions of the dynamics of energy decay from the resonator within the Jaynes-Cummings model. Our analysis allows us to distinguish the strong and weak coupling regimes of the model and to describe within each regime cases where the two-level system is unsaturated or saturated. We find that the quantum theory agrees with the classical model for weak coupling. However, for strong coupling the quantum theory predicts lower loss than the classical theory in the unsaturated regime. Also, in contrast to the classical theory, the photon number at which saturation occurs in the strong coupling quantum theory is independent of the coupling between the resonator and the two-level system.

DOI
10.1103/physrevb.84.104517

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We perform a quantum mechanical analysis of superconducting resonators subject to dielectric loss arising from charged two-level systems. We present numerical and analytical descriptions of the dynamics of energy decay from the resonator within the Jaynes-Cummings model. Our analysis allows us to distinguish the strong and weak coupling regimes of the model and to describe within each regime cases where the two-level system is unsaturated or saturated. We find that the quantum theory agrees with the classical model for weak coupling. However, for strong coupling the quantum theory predicts lower loss than the classical theory in the unsaturated regime. Also, in contrast to the classical theory, the photon number at which saturation occurs in the strong coupling quantum theory is independent of the coupling between the resonator and the two-level system.
Superconducting quantum devices have gained increasing interest over the past 15 years because of their role as promising candidates for implementing a large-scale quantum computer and as ultra-sensitive detectors. Superconducting resonators are one type of superconducting device which are used in a variety of applications including quantum computing and radioastronomy. They can achieve large quality factors which are important for these applications and a variety of others. Specific examples of these applications include the readout and coupling of superconducting qubits and also microwave kinetic inductance detector arrays for sub-mm wavelength astronomical measurements. High quality factors also enable the study of very subtle features in the electromagnetic response of the resonator. This feature in turn offers provides a method to characterize loss mechanisms which affect the performance of superconducting qubits, by measuring the quality factor of a resonator. Dielectric loss is an example of such a loss mechanism, and is considered to be one of, if not \\it the \\rm major obstacle to overcome in order to construct scalable quantum technologies based on superconducting devices. This type of loss arises from the coupling of the electric field in the device to spurious degrees of freedom which arise naturally the dielectric environment of the device. The fast-growing interest in the physics and applications of superconducting resonators has renewed interest in an improving our fundamental understanding of dielectric loss. This interest involves the study of a much bigger and older question in condensed matter physics; namely, what is the microscopic physics which governs the behaviour and of an amorphous material at low-temperatures? \n \nIn this thesis, we present a systematic study of dielectric loss in amorphous aluminum oxide using superconducting microwave resonators. Measurements of the intrinsic loss tangent over a large temperature range of 40~mK to 1~K are presented and analyzed in detail. Some aspects of the data are well understood in terms of the standard tunnelling two-level system model, while others require a deeper analysis of the electrodynamics of the resonator and insights from more advanced theories. We employ a consistent approach using numerical simulations of the resonator to understand loss mechanisms arising from both an inhomogeneous electric field and multiple dielectric layers. We measure saturation effects which provide insight into coherent properties of two-level system ensembles and the mutual interaction between them. Excellent agreement with the two-level system model is found for intrinsic losses, however the strong dependence on temperature which we observe for saturation effects is not predicted in the standard form of the model. We find that this dependence can be understood in terms of the theory of spectral diffusion which is consistent with a strong mutual interaction between two-level systems contributes to broadening of their spectral linewidths.
We study analytically and numerically the properties of the Jaynes-Cummings model under monochromatic driving. The analytical results allow us to understand the regime of two branches of multiphoton excitation in the case of close resonance between resonator and driven frequencies. The rotating wave approximation allows us to reduce the description of the original driven model to an effective Jaynes-Cummings model with strong coupling between photons and qubit. The analytical results are in good agreement with the numerical ones even if there are certain deviations between the theory and numerics in the close vicinity of the resonance. We argue that the rich properties of the driven Jaynes-Cummings model represent a new area for experimental investigations with superconducting qubits and other systems.
Increasing and stabilizing the coherence of superconducting quantum circuits and resonators is of utmost importance for various technologies, ranging from quantum information processors to highly sensitive detectors of low-temperature radiation in astrophysics. A major source of noise in such devices is a bath of quantum two-level systems (TLSs) with broad distribution of energies, existing in disordered dielectrics and on surfaces. Here we study the dielectric loss of superconducting resonators in the presence of a periodic electric bias field, which sweeps near-resonant TLSs in and out of resonance with the resonator, resulting in a periodic pattern of Landau–Zener transitions. We show that at high sweep rates compared to the TLS relaxation rate, the coherent evolution of the TLS over multiple transitions yields a significant reduction in the dielectric loss relative to the intrinsic value. This behavior is observed both in the classical high-power regime and in the quantum single-photon regime, possibly suggesting a viable technique to dynamically decouple TLSs from a qubit.
Superconducting resonators are widely used in many applications such as qubit readout for quantum computing, and kinetic inductance detectors. These resonators are susceptible to numerous loss and noise mechanisms, especially the dissipation due to two-level systems (TLS) which become the dominant source of loss in the few-photon and low temperature regime. In this study, capacitively-coupled aluminum half-wavelength coplanar waveguide resonators are investigated. Surprisingly, the loss of the resonators is observed to decrease with a lowering temperature at low excitation powers and temperatures below the TLS saturation. This behavior is attributed to the reduction of the TLS resonant response bandwidth with decreasing temperature and power to below the detuning between the TLS and the resonant photon frequency in a discrete ensemble of TLS. When response bandwidths of TLS are smaller than their detunings from the resonance, the resonant response and thus the loss is reduced. At higher excitation powers, the loss follows a logarithmic power dependence, consistent with predictions from the generalized tunneling model (GTM). A model combining the discrete TLS ensemble with the GTM is proposed and matches the temperature and power dependence of the measured internal loss of the resonator with reasonable parameters.
Abstract Increasing and stabilizing the coherence of superconducting quantum circuits and resonators is of utmost importance for various technologies, ranging from quantum information processors to highly sensitive detectors of low-temperature radiation in astrophysics. A major source of noise in such devices is a bath of quantum two-level systems (TLSs) with broad distribution of energies, existing in disordered dielectrics and on surfaces. Here we study the dielectric loss of superconducting resonators in the presence of a periodic electric bias field, which sweeps near-resonant TLSs in and out of resonance with the resonator, resulting in a periodic pattern of Landau–Zener transitions. We show that at high sweep rates compared to the TLS relaxation rate, the coherent evolution of the TLS over multiple transitions yields a significant reduction in the dielectric loss relative to the intrinsic value. This behavior is observed both in the classical high-power regime and in the quantum single-photon regime, possibly suggesting a viable technique to dynamically decouple TLSs from a qubit.
Summary Recent research in superconducting quantum circuits operating close to the quantum limit results in the need of a quantum mechanical treatment of losses. Of special interest is the dynamic behaviour of an open quantum system. As an example, a negative‐resistance Josephson parametric amplifier is treated. The DC bias voltage is chosen such that a strong interaction between the Josephson junction and the two resonant circuits, the signal and the idler circuit, is achieved. Power exchange occurs between the two considered resonator modes and also between the resonator modes and the DC power supply. Losses in the resonators are modeled by the quantum Langevin method, which describes the losses by coupling the resonators to a heat bath representing a photon gas in thermal equilibrium. The derived dynamic behaviour does not provide signal energy saturation, like classically expected for parametric amplifiers. Introducing a phenomenological multi‐photon coupling approach, saturation of the amplified signal is ensured. The time evolution of the signal and noise energy is calculated and numerically evaluated for a specific example in cases of both, the quantum Langevin method with and without the phenomenological multi‐photon coupling approach. Copyright © 2017 John Wiley & Sons, Ltd.