Superconducting quantum circuit models are widely used to understand superconducting devices. This thesis consists of four studies wherein the superconducting quantum circuit is used to illustrate challenges related to quantum information encoding and processing, quantum simulation, quantum signal detection and amplification.The existence of scalar Aharanov-Bohm phase has been a controversial topic for decades. Scalar AB phase, defined as time integral of electric potential, gives rises to an extra phase factor in wavefunction. We proposed a superconducting quantum Faraday cage to detect temporal interference effect as a consequence of scalar AB phase. Using the superconducting quantum circuit model, the physical system is solved and resulting AB effect is predicted. Further discussion in this chapter shows that treating the experimental apparatus quantum mechanically, spatial scalar AB effect, proposed by Aharanov-Bohm, can't be observed. Either a decoherent interference apparatus is used to observe spatial scalar AB effect, or a quantum Faraday cage is used to observe temporal scalar AB effect.The second study involves protecting a quantum system from losing coherence, which is crucial to any practical quantum computation scheme. We present a theory to encode any qubit, especially superconducting qubits, into a universal quantum degeneracy point (UQDP) where low frequency noise is suppressed significantly. Numerical simulations for superconducting charge qubit using experimental parameters show that its coherence time is prolong by two orders of magnitude using our universal degeneracy point approach. With this improvement, a set of universal quantum gates can be performed at high fidelity without losing too much quantum coherence.Starting in 2004, the use of circuit QED has enabled the manipulation of superconducting qubits with photons. We applied quantum optical approach to model coupled resonators and obtained a four-wave mixing toolbox to operate photons states. The model and toolbox are engineered with a superconducting quantum circuit where two superconducting resonators are coupled via the UQDP circuit. Using fourth order perturbation theory one can realize a complete set of quantum operations between these two photon modes. This helps open a new field to treat photon modes as qubits. Additional, a three-wave mixing scheme using phase qubits permits one to engineer the coupling Hamiltonian using a phase qubit as a tunable coupler.Along with Feynman's idea using quantum to simulate quantum, superconducting quantum simulators have been studied intensively recently. Taking the advantage of mesoscopic size of superconducting circuit and local tunability, we came out the idea to simulate quantum phase transition due to disorder. Our first paper was to propose a superconducting quantum simulator of Bose-Hubbard model to do site-wise manipulation and observe Mott-insulator to superfluid phase transition. The side-band cooling of an array of superconducting resonators is solved after the paper was published. In light of the developed technology in manipulating quantum information with superconducting circuit, one can couple other quantum oscillator system to superconducting resonators in order manipulation of its quantum states or parametric amplification of weak quantum signal. A theory that works for different coupling schemes has been studied in chapter 5. This will be a platform for further research.
Superconducting quantum circuit models are widely used to understand superconducting devices. This thesis consists of four studies wherein the superconducting quantum circuit is used to illustrate challenges related to quantum information encoding and processing, quantum simulation, quantum signal detection and amplification.The existence of scalar Aharanov-Bohm phase has been a controversial topic for decades. Scalar AB phase, defined as time integral of electric potential, gives rises to an extra phase factor in wavefunction. We proposed a superconducting quantum Faraday cage to detect temporal interference effect as a consequence of scalar AB phase. Using the superconducting quantum circuit model, the physical system is solved and resulting AB effect is predicted. Further discussion in this chapter shows that treating the experimental apparatus quantum mechanically, spatial scalar AB effect, proposed by Aharanov-Bohm, can't be observed. Either a decoherent interference apparatus is used to observe spatial scalar AB effect, or a quantum Faraday cage is used to observe temporal scalar AB effect.The second study involves protecting a quantum system from losing coherence, which is crucial to any practical quantum computation scheme. We present a theory to encode any qubit, especially superconducting qubits, into a universal quantum degeneracy point (UQDP) where low frequency noise is suppressed significantly. Numerical simulations for superconducting charge qubit using experimental parameters show that its coherence time is prolong by two orders of magnitude using our universal degeneracy point approach. With this improvement, a set of universal quantum gates can be performed at high fidelity without losing too much quantum coherence.Starting in 2004, the use of circuit QED has enabled the manipulation of superconducting qubits with photons. We applied quantum optical approach to model coupled resonators and obtained a four-wave mixing toolbox to operate photons states. The model and toolbox are engineered with a superconducting quantum circuit where two superconducting resonators are coupled via the UQDP circuit. Using fourth order perturbation theory one can realize a complete set of quantum operations between these two photon modes. This helps open a new field to treat photon modes as qubits. Additional, a three-wave mixing scheme using phase qubits permits one to engineer the coupling Hamiltonian using a phase qubit as a tunable coupler.Along with Feynman's idea using quantum to simulate quantum, superconducting quantum simulators have been studied intensively recently. Taking the advantage of mesoscopic size of superconducting circuit and local tunability, we came out the idea to simulate quantum phase transition due to disorder. Our first paper was to propose a superconducting quantum simulator of Bose-Hubbard model to do site-wise manipulation and observe Mott-insulator to superfluid phase transition. The side-band cooling of an array of superconducting resonators is solved after the paper was published. In light of the developed technology in manipulating quantum information with superconducting circuit, one can couple other quantum oscillator system to superconducting resonators in order manipulation of its quantum states or parametric amplification of weak quantum signal. A theory that works for different coupling schemes has been studied in chapter 5. This will be a platform for further research.
This thesis consists of theoretical studies of superconducting qubits, and trapped bosons and fermions at ultracold temperature. In superconducting qubits I analyze the resonant properties and decoherence behavior of dc SQUID phase qubits, in which one junction acts as a phase qubit and the rest of the device provides isolation from dissipation and noise in the bias lead. Typically qubit states in phase qubits are detected by tunneling it to the voltage state. I propose an alternate non-destructive readout mechanism which relies on the difference in the magnetic flux through the SQUID loop due to state of the qubit. I also study decoherence effects in a dc SQUID phase qubit caused by the isolation circuit. When the frequency of the qubit is at least two times larger than the resonance frequency of the isolation circuit, I find that the decoherence time of the qubit is two orders of magnitude larger than the typical ohmic regime, where the frequency of the qubit is much smaller than the resonance frequency of the isolation circuit. This theory is extended to other similar superconducting quantum devices and has been applied to experiments from the group at the University of Maryland. I also demonstrate, theoretically, vacuum Rabi oscillations, analogous to circuit-QED, in superconducting qubits coupled to an environment with resonance. The result obtained gives an exact analytical expression for coherent oscillation of state between the system (the qubit) and the environment with resonance. Next I investigate ultracold atoms in harmonically confined optical lattices. They exhibit a `wedding cake structure' of alternating Mott shells with different number of bosons per site. In regions between the Mott shells, a superfluid phase emerges at low temperatures which at higher temperatures becomes a normal Bose liquid. Using finite-temperature quantum field theoretic techniques, I find analytically the properties of the superfluid, Bose liquid, and Mott insulating regions. This includes the finite temperature order parameter equation for the superfluid phase, excitation spectrum, Berezinskii-Kosterlitz-Thouless transition temperature and vortex-antivortex pair formation (in the two dimensional case), finite temperature compressibility and density - density correlation function. I also study interacting mixtures of ultracold bosonic and fermionic atoms in harmonically confined optical lattices. For a suitable choice of parameters I find emergence of superfluid and Fermi liquid (non-insulating) regions out of Bose-Mott and Fermi-band insulators, due to finite boson and fermion hopping. I also propose a possible experiment for the detection of superfluid and Fermi liquid shells through the use of Gauss-Laguerre and Gaussian beams followed by Bragg spectroscopy. Another area I explore is ultracold heteronuclear molecules such as KRb, RbCs and NaCs. I obtain the finite and zero-temperature phase diagram of bosons interacting via short range repulsive interactions and long-ranged isotropic dipolar interactions in two-dimensions. I build an analytical model for such systems that describes a first order quantum phase transition at zero temperature from a triangular crystalline phase (analogous to Wigner crystal phase of electrons) to superfluid phase. At finite temperature the crystalline phase melts, due to topological defects, to a hexatic phase where translational order is destroyed but hexagonal orientational order is preserved. Further temperature increase leads to the melting of the hexatic phase into a normal dipolar Bose liquid.
The perhaps most promising platform for quantum information processing is the circuit-QED architecture based on superconducting circuits representing quantum bits. These circuits must be made with low losses so that the quantum information is retained for as long as possible. We developed fabrication processes achieving state-of-the-art coherence times of over 100 µs. We identified the primary source of loss to be parasitic two-level systems by studying fluctuations of qubit relaxation times.Using our high-coherence circuits, we implemented a quantum processor built on fixed-frequency qubits and frequency-tunable couplers. The tunable couplers were lumped-element LC resonators, where the inductance came from a superconducting quantum interference device (SQUID). We achieved a controlled-phase gate with a fidelity of 99% by parametric modulation of the coupler frequency. Using this device, and another similar to it, we demonstrated two different quantum algorithms, the quantum approximate optimization algorithm, and density matrix exponentiation. We achieved high algorithmic fidelities, aided by our carefully calibrated gates.Additionally, we researched parametric oscillations using frequency-tunable resonators. Previously, degenerate parametric oscillations have been demonstrated by modulation of the resonant frequency at twice that frequency. We use this phenomenon to implement a readout method for a superconducting qubit with a fidelity of 98.7%. We demonstrated correlated radiation in nondegenerate parametric oscillations by modulating at the sum of two resonant frequencies of a multimode resonator. We showed an excellent quantitative agreement between the classical properties of the oscillations with a theoretical model. Moreover, we studied higher-order modulation at up to five times their resonant frequencies. These types of parametric oscillation states might be used as a quantum resource for continuous-variable quantum computing.
One of the key elements in circuit quantum electrodynamics (cQED) is the superconducting microwave resonator, which can enable strong coupling to and between nanoscale quantum systems, and as a result has emerged as an essential tool for both quantum information and quantum sensing. In this thesis, I will discuss new applications of superconducting resonators in both of these fields, which illustrates how quantum information and quantum sensing can be strongly tied together. As the field of quantum information continues to grow, various platforms for realizing a quantum computer have been proposed. One attractive platform is to control and manipulate the spin of an electron to form spin qubits, which have demonstrated a long coherence time, weak coupling to the environment, and high fidelity single- and two-qubit gates. However, scaling beyond two qubits has proven challenging, as has incorporating long-range coupling, although both are crucial for making spin qubits a scalable quantum computing platform. The confinement of electromagnetic fields in superconducting resonators enables strong coupling to small systems, such as the single electrons of a spin qubit. In the first part of this thesis, I will describe how a singlet-triplet spin qubit coupled to a high-impedance superconducting resonator constitutes a promising architecture for mediating long-ranged interactions based on a longitudinal coupling between the resonator and qubit. I will discuss the technical developments needed to successfully implement this novel type of interaction in a hybrid spin qubit-resonator device. This type of interaction has several advantages over the more commonly studied transverse coupling scheme: it does not require careful tuning of the qubit frequency to match that of the resonator, which facilitates scaling to a large number of qubits, and also avoids the loss of coherence, which is inevitable in resonant coupling. This type of interaction therefore opens promising paths for generating high-fidelity long-ranged two-qubit coupling by employing superconducting microwave resonators. By adapting techniques used in quantum information, we have also developed a new technique in quantum sensing, where we employ superconducting resonators as a probe of quantum materials such as mesoscopic systems, two-dimensional materials, and heterostructures, which are otherwise challenging to experimentally access with conventional techniques. In the second part of my thesis, I will discuss how implementing cQED techniques from quantum information allows us to probe the symmetry of the order parameter in unconventional superconductors. Unconventional superconductors are not readily available, and we therefore study an engineered system that comprises of interfacing a ferromagnet and superconductor, in which unconventional pairing can be induced into the ferromagnet. This platform allows us to simultaneously excite spin waves, a collective excitation of spins, in the ferromagnet, which opens new directions for studying interactions between coherently coupled magnons (quanta of spin waves) and unconventional superconducting states. Finally, by applying the same technique to a two-dimensional (2D) superconductor, tungsten ditelluride, we leverage this platform in an attempt to study the order parameter of its intrinsic superconducting phase. The nature of the order parameter in this material remains an open question, as both topological and superconducting phases can exist, possibly simultaneously. This work thus demonstrates a unique use of superconducting resonators beyond quantum information applications.
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.
Superconducting qubits are fabricated "loss-free" electrical circuits on a chip with size features of tens of nanometers. If cooled to cryogenic temperatures below -273 °C they behave as quantum elements, similar to atoms and molecules. Such a qubit can be manipulated by fast-oscillating magnetic fields and its resulting quantum state can be measured using a proper magnetic detector. These qubits could be the basic elements for a quantum processor. To that order, multiple qubits have to controllably interact, and to that purpose very high quality resonators are fabricated on the same chip. Effectively the interaction results from the phase-coherent exchange of a single photon between qubit and resonator. In this way it is highly analogous to the so-called cavity QED in atomic physics where single atoms couple to a single trapped photon, all inside a cavity resonator. In this thesis, a superconducting flux qubit has been coupled to an LC resonator and coherent exchange of a single photon between qubit and resonator has been achieved. A striking option in these superconducting circuits, not attainable in "common" atomic cavity-QED, is that the interaction between qubit and photon can be made very large, up to the limit that one no longer can talk about the qubit and resonator as such. This limit requires a fully different description and it leads to novel effects, e.g. in the energy spectrum of the qubit-resonator composite. The Bloch-Siegert spectral shift is experimentally demonstrated in this thesis. As a small but critical technical aside a new type of control line filter has been developed, which allows strong suppression of excess electrical noise that forms a threat for qubit coherence. Having two modes of operation, it shorts all low-frequency noise if control is idle while being transparent if control is active, where short or open is determined by the control line signal level itself.
In this thesis, experimentally relevant aspects and open questions of quantum information processing and measurement in circuit quantum electrodynamics have been investigated theoretically. Circuit quantum electrodynamics is a relatively young field combining superconducting transmission line resonators on-chip with superconducting quantum bits which serve as artificial atoms. Remarkable experiments have underlined the prospects of circuit QED as a possible architecture for quantum information processing as well as a framework within which quantum optics experiments can be performed. In contrast to their optical counterparts, these experiments reach the ultra-strong coupling limit and allow e.g. the generation of Fock states, the study of decoherence and the observation of quantum jumps. We present the physics of superconducting circuits based on the Josephson junction as a non-linear circuit element. We discuss the various types of quantum bits (qubits) before turning to an introduction on cavity QED and conclude the introduction with a brief presentation of the most relevant experiments and a short introduction to the principles and prospects of quantum information processing. Building on the pioneering theoretical and experimental work in which up to three qubits have been integrated on a chip and successfully coupled via a cavity mode, we propose a scheme to overcome the natural limits concerning scalability in these systems. Usually, scalability is severely limited by the resonance width of the qubit transitions and the attainable frequency range for both qubit transition and cavity resonance frequencies. This limits the present setups to at most ten qubits. As a possible solution, we propose and investigate a cross-bar grid layout of cavities with qubits at the intersections. It turns out that this setup, allowing for up to about 1000 qubits, offers significant advantages over e.g. nearest-neighbor coupling schemes. In addition, it can serve as a building block for a truly scalable scheme for fault-tolerant quantum information processing with superconducting circuits by combining 7x7 arrays in a staircase manner. Furthermore, we discuss and simulate different possibilities to improve upon the suggested scheme by e.g. using resonant gates and optimal control theory. Another focus of the present thesis is measurement physics. In order to investigate e.g. the possibilities of using a circuit QED system to dispersively detect single microwave photons or to generate multi-qubit entangled states, we introduce in a pedagogical manner the basics of stochastic master equations which generate quantum trajectories before turning to concrete applications. Quantum trajectories have proven to be a state-of-the-art tool to analyze measurement situations in a very realistic manner by giving access to both the measurement record as well as the internal quantum dynamics conditioned on this record. Using the method of quantum trajectories as generated by stochastic master equations, we discuss a scheme to efficiently generate multi-qubit entangled states in a very flexible way. The method is based on a collective, dispersive qubit readout. By this measurement, a suitable product state of N qubits can be reliably converted into e.g. an N-qubit Greenberger-Horne-Zeilinger or W state. We propose and discuss the scheme and investigate the effects of decoherence and parameter spread as they would be encountered in experiment. Last but not least, we propose and fully analyze a scheme to dispersively detect single itinerant microwave photons. While experiments to generate single microwave photons have already been successfully performed, their single-shot quantum non-demolition detection has not been possible so far. The scheme presented in this thesis closes this gap in the circuit QED toolbox and reaches detection efficiencies of 30% only limited by fundamental quantum mechanics in this case the Quantum Zeno effect. This detection scheme could be used in quantum communication or quantum cryptography applications.