The topic of this thesis is superconducting electric circuits. Technical advances have made possible the experimental study of Josephson junction based circuit elements which sustain quantum mechanical properties long enough to be denoted as quantum devices. The quantum state can be controlled with electronic variables and measured using standard electrical setups. The research is motivated by the possibility to examine quantum phenomena in circumstances that can be customized, prospects of new quantum devices, and the development of quantum information processing. This thesis presents theoretical studies on the nonlinear and stochastic driving of a superconducting quantum two-level system (qubit). We first investigate the energy level shifts a single-Cooper-pair transistor under large amplitude driving realized via the inherently nonlinear Josephson energy by using an external magnetic flux. The effective driving field substantially deviates from a circular polarization and linear coupling. The energy level shifts are compared to the cases of a vanishing and a weak driving field, measured as the Stark shift and the generalized Bloch-Siegert shift, respectively. We describe criteria for the natural basis of the analytical and the numerical calculations. In addition to that, we develop a formalism based on the Floquet method for the weak probe measurement of the strongly driven qubit. In the latter part of the thesis research, we study utilization of a stochastic driving field whose time evolution is not regular but follows probabilistic laws. We concentrate on the motional averaging phenomenon and show that it can be measured with an unparalleled accuracy by employing a flux-modulated transmon qubit. As the stochastically modulated qubit is simultaneously measured with a moderate driving field, we develop a theoretical description accounting the possible interference effects between the modulation and the drive. The comparison with experimental results shows good agreement. Motional averaging phenomenon can be applied to estimate the properties of fluctuation processes occurring in qubits, e.g., the quasiparticle tunneling or the photon shot noise. Resting on the motional averaging, we anticipate that the qubit dephasing times can be improved if one can accelerate the dynamics of two-level fluctuators. We apply a semiclassical formalism where the qubit is treated with quantum mechanical concepts whereas the driving fields are classical. In the solution procedure, the numerical results support the main analytical understanding. As the theoretical results are extensively compared to reflection measurements, we construct an explicit connection between the dynamics of the studied quantum devices and the measured reflection coefficient.
The topic of this thesis is superconducting electric circuits. Technical advances have made possible the experimental study of Josephson junction based circuit elements which sustain quantum mechanical properties long enough to be denoted as quantum devices. The quantum state can be controlled with electronic variables and measured using standard electrical setups. The research is motivated by the possibility to examine quantum phenomena in circumstances that can be customized, prospects of new quantum devices, and the development of quantum information processing. This thesis presents theoretical studies on the nonlinear and stochastic driving of a superconducting quantum two-level system (qubit). We first investigate the energy level shifts a single-Cooper-pair transistor under large amplitude driving realized via the inherently nonlinear Josephson energy by using an external magnetic flux. The effective driving field substantially deviates from a circular polarization and linear coupling. The energy level shifts are compared to the cases of a vanishing and a weak driving field, measured as the Stark shift and the generalized Bloch-Siegert shift, respectively. We describe criteria for the natural basis of the analytical and the numerical calculations. In addition to that, we develop a formalism based on the Floquet method for the weak probe measurement of the strongly driven qubit. In the latter part of the thesis research, we study utilization of a stochastic driving field whose time evolution is not regular but follows probabilistic laws. We concentrate on the motional averaging phenomenon and show that it can be measured with an unparalleled accuracy by employing a flux-modulated transmon qubit. As the stochastically modulated qubit is simultaneously measured with a moderate driving field, we develop a theoretical description accounting the possible interference effects between the modulation and the drive. The comparison with experimental results shows good agreement. Motional averaging phenomenon can be applied to estimate the properties of fluctuation processes occurring in qubits, e.g., the quasiparticle tunneling or the photon shot noise. Resting on the motional averaging, we anticipate that the qubit dephasing times can be improved if one can accelerate the dynamics of two-level fluctuators. We apply a semiclassical formalism where the qubit is treated with quantum mechanical concepts whereas the driving fields are classical. In the solution procedure, the numerical results support the main analytical understanding. As the theoretical results are extensively compared to reflection measurements, we construct an explicit connection between the dynamics of the studied quantum devices and the measured reflection coefficient.
The model of a two-level system (TLS) coupled to a harmonic oscillator has found a widespread application in physics and chemistry. In this thesis we focus on the field of circuit quantum electrodynamics where the TLS stands for a quantum bit (qubit), which is the basic building unit of a potential quantum computer, and the oscillator represents a transmission-line resonator, which can be used, e.g., to store information contained on the qubit or for communication between several qubits. Furthermore, the oscillator can model a superconducting quantum interference device (SQUID) determining the state of the qubit. In first experimental realizations the coupling between the qubit and the oscillator was small compared to characteristic frequencies of the two devices. In such a situation the Jaynes-Cummings model provides a realistic and completely analytical description of the energy spectrum and the dynamics of the setup. However, it relies on a rotating-wave approximation (RWA) which neglects counter-rotating terms in the full qubit-oscillator Hamiltonian. These counter-rotating terms become important under increase of the coupling strength, as it has been achieved in recent experiments. We present two analytical approaches to take these additional contributions into account: The first one treats the qubit-oscillator Hamiltonian perturbatively in the coupling strength and predicts a frequency shift in its energy spectrum, the Bloch-Siegert shift. The second one considers the qubit's tunneling splitting as small parameter and thus treats the problem to all orders in the coupling allowing to enter the ultrastrong coupling regime, where the coupling strength becomes comparable to the qubit's and/or oscillator's frequency. For both cases we give a thorough analysis of the qubit's dynamics. In order to perform operations the qubit is usually probed by an external driving force. We model this situation by coupling the TLS to a classical oscillator and examine the resulting time dependent Hamiltonian using a combination of Floquet theory and Van Vleck perturbation theory. Thus, we provide an analysis of the qubit's energy spectrum and dynamics which is nonperturbative in the driving amplitude. We investigate effects like coherent destruction of tunneling and driving-inducing tunneling oscillations. By coupling the driven TLS to a quantized oscillator we give for the first time an analytical treatment of the qubit-oscillator system being simultaneously exposed to both ultrastrong coupling and extreme driving. We observe interesting phenomena in the dynamics like the suppression of specific frequencies under a variation of the coupling strength. To mimic environmental influences on the qubit we apply the Caldeira-Leggett master equation approach to the driven TLS and to the qubit-oscillator system and provide for both cases analytical expressions for the relaxation and dephasing rates.
The field of superconducting quantum computing, based on Josephson junctions, has recently seen remarkable strides in scaling the number of logical qubits. In particular, the fidelities of one- and two-qubit gates are close to the breakeven point with the novel error mitigation and correction methods. Parallel to these advances is the effort to expand the Hilbert space within a single device by employing high-dimensional qubits, otherwise known as qudits. Research has demonstrated the possibility of driving higher-order transitions in a transmon or designing innovative multimode superconducting circuits, termed multimons. These advances can significantly expand the computational basis while simplifying the interconnects in a large-scale quantum processor. This thesis provides a detailed introduction to the superconducting qudit and demonstrates a comprehensive analysis of decoherence in an artificial atom with more than two levels using Lindblad master equations and stochastic master equations (SMEs). After extending the theory of the design, control, and readout of a conventional superconducting qubit to that of a qudit, the thesis focuses on modeling the dispersive measurement of a transmon qutrit in an open quantum system using quadrature detections. Under the Markov assumption, master equations with different levels of abstraction are proposed and solved; in addition, both the ensemble-averaged and the quantum-jump approach of decoherence analysis are presented and compared analytically and numerically. The thesis ends with a series of experimental results on a transmon-type qutrit, verifying the validity of the stochastic model.
Quantum computers utilize qubits to store and process quantum information. In superconducting quantum computers, qubits are implemented as quantum superconducting resonant circuits. The circuits are operated only at the two energy states, which form the computational basis for the qubit. To suppress leakage to uncomputational states, superconducting qubits are designed to be anharmonic oscillators, which is achieved using one or more Josephson junctions, a nonlinear superconducting element. One of the main challenges in developing quantum computers is minimizing the decoherence caused by environmental noise. Decoherence is characterized by two coherence times, T1 for depolarization processes and T2 for dephasing. This thesis reviews and investigates the decoherence properties of superconducting qubits. The main goal of the thesis is to analyze the tradeoff between anharmonicity and dephasing in a qubit unimon. Recently developed unimon incorporates a single Josephson junction shunted by a linear inductor and a capacitor. Unimon is tunable by external magnetic flux, and at the half flux quantum bias, the Josephson energy is partially canceled by the inductive energy, allowing unimon to have relatively high anharmonicity while remaining fully protected against low-frequency charge noise. In addition, at the sweet spot with respect to the magnetic flux, unimon becomes immune to first-order perturbations in the flux. The sweet spot, however, is relatively narrow, making unimon susceptible to dephasing through the quadratic coupling to the flux noise. In the first chapter of this thesis, we present a comprehensive look into the basic theory of superconducting qubits, starting with two-state quantum systems, followed by superconductivity and superconducting circuit elements, and finally combining these two by introducing circuit quantum electrodynamics (cQED), a framework for building superconducting qubits. We follow with a theoretical discussion of decoherence in two-state quantum systems, described by the Bloch-Redfield formalism. We continue the discussion by estimating decoherence using perturbation theory, with special care put into the dephasing due to the low-frequency 1/f noise. Finally, we review the theoretical model of unimon, which is used in the numerical analysis. As a main result of this thesis, we suggest a design parameter regime for unimon, which gives the best ratio between anharmonicity and T2.
This thesis discusses a series of experimental and theoretical studies on fast control and decoherence in superconducting quantum circuits. These two subjects are complementary aspects of the problem of improving the gate fidelities in quantum information processing devices based on superconducting circuits. \n \nIn the first part of this thesis, we present experiments on fast control of a superconducting flux qubit using strong driving. We explore driving with short pulses and a driving strength significantly exceeding the transition frequency. The observed dynamics is described in terms of quasienergies and quasienergy states, in agreement with Floquet theory. We observe the role of pulse shaping in the dynamics, as determined by non-adiabatic transitions between Floquet states. In addition, based on adiabatic perturbation theory in the Floquet picture, we propose a control scheme for fast and high-fidelity single-qubit operations with strong driving. These results pave the way to quantum control with a fidelity beyond the threshold of fault-tolerant quantum computation. \n \nIn the second part of this thesis, we discuss experimental methods and present the characterization of decoherence in superconducting resonators and superconducting qubits. Firstly, we present an analysis method on the measurements of photon loss in superconducting resonators. We use this method to characterize the microwave loss of aluminum oxide, an important material in superconducting circuits. The power and temperature dependence of the measured loss is consistent with loss due to two-level defects in amorphous materials. Secondly, we present experimental results on decoherence and noise spectroscopy of a superconducting flux qubit. Using the qubit as a noise spectrometer, we perform detailed measurements on the power spectral density of the flux noise in superconducting circuits over a wide range of frequencies, from 10^-3 to 10^8 Hz, and over a temperature range from 35 to 130 mK. We also present measurements of frequency and temperature dependence of the qubit energy relaxation.
The work of this thesis focuses on stochastic methods for the simulation of superconducting charge qubits, also known as Cooper pair boxes, which are promising candidates for large scale quantum information processing. To aid non-physicists, a brief outline of the structure of quantum mechanics is provided using the· Dirac formalism. Using the so called 'Backreaction effect', we consider if any information can be obtained regarding the qubit behaviour, through detecting the changes observed in the frequency spectrum of the coupled biasing circuitry, modelled as a dissipative oscillator circuit. The process of modelling dissipative quantum systems is described, however an alternative approach called 'quantum trajectories' is used rather than the traditional Calderia-Leggett model, as the time evolution of a single 'trajectory' represents the evolution of an individual system coupled to a noisy environment. Through the noise generated by an excited qubit, the energy level structure of the qubit can be probed with a microwave drive field, by observing the noise power within the biasing circuit. We consider a biasing circuit of unusually high resonant frequency which can drive the qubit, this creates frequency splitting features that would not normally be observed. Weak measurement is also examined as this is closely related to the stochastic 'quantum trajectories', where the measurement is recorded by the observer rather than lost to an environment. Weakly measuring a qubit does not completely collapse it and therefore 'quantum feedback' may be employed to alter the qubit controls favourably. In particular we consider the problem of purifying a weakly measured system rapidly; given a qubit in the completely mixed state what is the best feedback to become confident in the actual qubit state quickly. There are two optimal feedback protocols proposed by Jacobs [1] and by Wiseman and Ralph [2] for purifying qubits that have ideal controls. However, we adapt these protocols for the charge qubit, whose finite Hamiltonian resources and non-zero ax tunnelling term means the Bloch vector can not be easily held in the optimal location.
The research work discussed in this thesis deals with the study of superconducting Josephson qubits. Superconducting qubits are solid-state artificial atoms which are based on lithographically defined Josephson tunnel junctions properties. When sufficiently cooled, these superconducting devices exhibit quantized states of charge, flux or junction phase depending on their design parameters. This allows to observe coherent evolutions of their states. The results presented can be divided into two parts. In a first part we investigate operations of superconducting qubits based on the quantum coherence in superconducting quantum interference devices (SQUID). We explain experimental data which has been observed in a SQUID subjected to fast, large-amplitude modifications of its effective potential shape. The motivations for this work come from the fact that in the past few years there have been attempts to interpret the supposed quantum behavior of physical systems, such as Josephson devices, within a classical framework. Moreover, we analyze the possibility of generating GHZ states, namely maximally entangled states, in a quantum system made out of three Josephson qubits. In particular, we investigate the possible limitations of the GHZ state generation due to coupling to bosonic baths. In the second part of the thesis we address a particular cause of decoherence of flux qubits which has been disregarded until now: thermal gradients, which can arise due to accidental non equilibrium quasiparticle distributions. The reason for these detrimental effects is that heat currents flowing through Josephson tunnel junctions in response to a temperature gradient are periodic functions of the phase difference between the electrodes. The phase dependence of the heat current comes from Andreev reflection, namely an interplay between the quasiparticles which carry heat and the superconducting condensate which is sensitive to the superconducting phase difference. Generally speaking, the flux qubit states are characterized by different values of the phase difference through their Josephson junctions. Consequently, the phase-dependent thermal current through a device subject a temperature gradient is related to the phase-dependent qubit states. We study how the thermal currents change according to the state of the qubits hence yielding a measurement of the qubit state. This in turn leads to an impact of temperature gradient on the dynamics of the system. We show that flux qubits in the Delft qubit design can have limitations of the decoherence time to the order of microseconds as a result of this newly discovered source of decoherence. In contrast, the fluxonium qubit is found to be well protected due to its superinductance.