In this thesis, we study two topics on superconducting circuits. \nThe first topic is the tunable coupler for resonators. \nThis coupler is implemented with a superconducting circuit coupling with resonators in the dispersive regime. \nIt possesses a high on/off ratio and the coupling between two degenerate resonators can be turned off by tuning the frequency of the coupler via the biasing flux. \nWe present the design, the simulation, \nand the fabrication of the chip implementing this coupler. \nWe derive the Hamiltonian of the chip via formal circuit quantization. \nWe also present numerical results of the eigenenergies and the eigenstates of the chip. \nFrom the numerical results, \nwe show that the tunable coupling and the turn-off point can be realized on the chip. \n \nThe second topic is impedance engineering for enhancing the bandwidth of a Josephson parametric amplifier. \nBy transforming the input impedance seen by the Josephson parametric amplifier, \nthe frequency dependence of the gain function can be decreased and the bandwidth of the Josephson parametric amplifier can be enhanced. \nWe study impedance engineering for two different types of flux pumping Josephson parametric amplifier. \nFor each type of amplifier, \nwe develop a corresponding method of impedance engineering, \nand study implementing the method with transmission lines.
In this thesis, we study two topics on superconducting circuits. \nThe first topic is the tunable coupler for resonators. \nThis coupler is implemented with a superconducting circuit coupling with resonators in the dispersive regime. \nIt possesses a high on/off ratio and the coupling between two degenerate resonators can be turned off by tuning the frequency of the coupler via the biasing flux. \nWe present the design, the simulation, \nand the fabrication of the chip implementing this coupler. \nWe derive the Hamiltonian of the chip via formal circuit quantization. \nWe also present numerical results of the eigenenergies and the eigenstates of the chip. \nFrom the numerical results, \nwe show that the tunable coupling and the turn-off point can be realized on the chip. \n \nThe second topic is impedance engineering for enhancing the bandwidth of a Josephson parametric amplifier. \nBy transforming the input impedance seen by the Josephson parametric amplifier, \nthe frequency dependence of the gain function can be decreased and the bandwidth of the Josephson parametric amplifier can be enhanced. \nWe study impedance engineering for two different types of flux pumping Josephson parametric amplifier. \nFor each type of amplifier, \nwe develop a corresponding method of impedance engineering, \nand study implementing the method with transmission lines.
Broadband quantum-limited amplifiers play a critical role in the single-shot readout of superconducting qubits, but a popular implementation, the traveling wave parametric amplifier, involves a complex design and fabrication process. Here, we present a simple design for a Josephson parametric amplifier, using a lumped element resonator comprising a superconducting quantum interference device whose useful bandwidth is enhanced with an on-chip impedance-matching circuit. We demonstrate a flux-coupling geometry that maximizes the coupling to the Josephson loop and minimizes spurious excitation of the amplifier resonant circuit. The amplifier, which operates in a flux-pumped mode, is demonstrated with a power gain of more than 20 dB over a bandwidth of about 300 MHz, where approximate noise measurements indicate quantum-limited performance. A procedure is given for optimizing the bandwidth for this kind of amplifier, using a linearized circuit simulation while minimizing non-linearities.
In this licentiate thesis, I present the design, fabrication, and characterization of superconducting parametric resonators, for use in quantum information processing. These devices are quarter-wavelength coplanar waveguide resonators (~5 GHz), terminated at one end by a non-linear inductance provided by a SQUID (superconducting quantum interference device). The SQUID acts as a flux-controlled boundary condition, which effectively changes the electrical length of the resonator. This enables the modulation of the resonant frequency by coupling microwave magnetic flux into the SQUID, using on-chip transmission lines. The modulation occurs on a timescale much faster than the photon loss out of the resonator. The non-linearity provides several different regimes for operating this device. I focused on the parametric regime, in which the non-linear element was being modulated (pumped) at or near twice the resonant frequency. This pumping can lead to amplification and frequency conversion of an incoming signal. It can also give rise to instabilities and self-oscillations "parametric oscillations" - that is, the generation of an intense electric field in the resonator (the creation of photons). Parametric oscillations set in when the pumping strength exceeds a threshold value, and also depends on the pump-frequency detuning from twice the resonance and on the static flux through the SQUID. I characterized the devices over a wide parameter range by doing homodyne detection of the reflected signal or of the emitted field oscillations. In particular, I investigated the damping and two leading nonlinearities, dominating in different operating regimes, which influence the dynamics of the parametric resonator. First, I extracted the Duffing nonlinearity by studying the line width of the resonator as a function of the input signal (without pumping the SQUID). Second, with high pump amplitude, I extracted the pump-induced nonlinearity by determining the threshold parameters (pump amplitude and detuning) for the onset of parametric oscillations; this nonlinearity leads to the generation of higher-order pump terms that complicate the system dynamics, but can be mitigated now that they have been understood. My results validate a recent theoretical model for the classical, non-linear dynamics of parametric resonance, and helps determine workable design parameters. Finally, I helped develop a linearized model − the "pumpistor" - describing this device in terms of a flux-controlled impedance. This will be useful when designing more complex circuits. One goal of this work is to demonstrate single-shot measurements on superconducting qubits (quantum bits of information). We will use the parametric resonator as a threshold discriminator, associating the qubit's two energy eigenstates with the parametric oscillations either turning on or not. This would be a very useful device in the quantum engineer's toolbox when designing systems for quantum information processing and communication.
This research investigates the realization of parametric amplification in superconducting circuits and structures where nonlinearity is provided by Josephson junction (JJ) elements. We aim to develop a systematic analysis over JJ-based devices toward design of novel traveling-wave Josephson parametric amplifiers (TW-JPA). Chapters of this thesis fall into three categories: lumped JPA, superconducting periodic structures and discrete Josephson transmission lines (DJTL). \nThe unbiased Josephson junction (JJ) is a nonlinear element suitable for parametric amplification through a four-photon process. Two circuit topologies are introduced to capture the unique property of the JJ in order to efficiently mix signal, pump and idler signals for the purpose of signal amplification. Closed-form expressions are derived for gain characteristics, bandwidth determination, noise properties and impedance for this kind of parametric power amplifier. The concept of negative resistance in the gain formulation is observed. A design process is also introduced to find the regimes of operation for gain achievement. Two regimes of operation, oscillation and amplification, are highlighted and distinguished in the result section. Optimization of the circuits to enhance the bandwidth is also carried out. \nMoving toward TW-JPA, the second part is devoted to modelling the linear wave propagation in a periodic superconducting structure. We derive closed-form equations for dispersion and s-parameters of infinite and finite periodic structures, respectively. Band gap formation is highlighted and its potential applications in the design of passive filters and resonators are discussed. The superconducting structures are fabricated using YBCO and measured, illustrating a good correlation with the numerical results. \nA novel superconducting Transmission Line (TL), which is periodically loaded by Josephson junctions (JJ) and assisted by open stubs, is proposed as a platform to realize a traveling-wave parametric device. Using the TL model, this structure is modeled by a system of nonlinear partial differential equations (PDE) with a driving source and mixed-boundary conditions at the input and output terminals, respectively. This model successfully emulates parametric and nonlinear microwave propagation when long-wave approximation is applicable. The influence of dispersion to sustain three non-degenerate phased-locked waves through the TL is highlighted. \nA rigorous and robust Finite Difference Time Domain (FDTD) solver based on the explicit Lax-Wendroff and implicit Crank-Nicolson schemes has been developed to investigate the device responses under various excitations. Linearization of the wave equation, under small-amplitude assumption, dispersion and impedance analysis is performed to explore more aspects of the device for the purpose of efficient design of a traveling-wave parametric amplifier. \nKnowing all microwave characteristics and identifying different regimes of operation, which include impedance properties, cut-off propagation, dispersive behaviour and shock-wave formation, we exploit perturbation theory accompanied by the method of multiple scale to derive the three nonlinear coupled amplitude equations to describe the parametric interaction. A graphical technique is suggested to find three waves on the dispersion diagram satisfying the phase-matching conditions. Both cases of perfect phase-matching and slight mismatching are addressed in this work. The incorporation of two numerical techniques, spectral method in space and multistep Adams-Bashforth in time domain, is employed to monitor the unilateral gain, superior stability and bandwidth of this structure. Two types of functionality, mixing and amplification, with their requirements are described. These properties make this structure desirable for applications ranging from superconducting optoelectronics to dispersive readout of superconducting qubits where high sensitivity and ultra-low noise operation is required.
We propose and realize a multi-resonator coupled impedance-matched Josephson parametric amplifier. Parallel plate capacitors serve as couplers to achieve coupling between the nonlinear resonator containing the Josephson loop and the half-wavelength resonator, and between the half-wavelength resonators themselves. By precisely designing each coupling strength, we effectively extend the amplifier's bandwidth. Through a combination of circuit-level and electromagnetic field simulations, we optimize the structural parameters of the resonators and capacitors. The two measured devices have 15 dB gain with a bandwidth of 1.3 GHz and 20 dB gain with a bandwidth of 1.1 GHz, respectively, with near-quantum-limited noise temperatures.
We demonstrate a lumped-element Josephson parametric amplifier, using a single-ended design that includes an on-chip, high-bandwidth flux bias line. The amplifier can be pumped into its region of parametric gain through either the input port or through the flux bias line. Broadband amplification is achieved at a tunable frequency ω/2π between 5 and 7 GHz with quantum-limited noise performance, a gain-bandwidth product greater than 500 MHz, and an input saturation power in excess of −120 dBm. The bias line allows fast frequency tuning of the amplifier, with variations of hundreds of MHz over time scales shorter than 10 ns.
We present a fishbone-like lumped-element artificial transmission line to overcome impedance mismatch in a reflection-type Josephson parametric amplifier between a nonlinear resonator and an external transmission line. Using this easily prepared architecture, we design and fabricate a broadband Josephson parametric amplifier, which has gain in an excess of 20 dB with a bandwidth of hundreds of MHz. Furthermore, by varying the working point of the device, the operating frequency of amplification can be tuned in a wide frequency range of 1 GHz while the amplifier operates in the mode of either three-wave mixing or four-wave mixing. Such a parametric amplifier is suitable for engineering applications of superconducting circuit quantum electrodynamics.