Modeling Classical Dynamics and Quantum Effects in Superconducting Circuit Systems

In recent years, superconducting circuits have come to the forefront of certain areas of physics. They have shown to be particularly useful in research related to quantum computing and information, as well as fundamental physics. This is largely because they provide a very flexible way to implement complicated quantum systems that can be relatively easily manipulated and measured. In this thesis we look at three different applications where superconducting circuits play a central role, and explore their classical and quantum dynamics and behavior. The first part consists of studying the Casimir [Proc. K. Ned. Akad. Wet (1948)] and Casimir-Polder like [Physical Review 73, 4 (1948)]effects. These effects have been discovered in 1948 and show that under certain conditions, vacuum field fluctuations can mediate forces between neutral objects. In our work, we analyze analogous behavior in a superconducting system which consists of a stripline cavity with a DC-SQUID on one of its boundaries, as well as, in a Casimir-Polder case, a charge qubit coupled to the field of the cavity. Instead of a force, in the system considered here, we show that the Casimir and Casimir-Polder like effects are mediated through a circulating current around the loop of the boundary DC-SQUID. Using detailed analysis, we examine how the values of these currents change as we vary different physical circuit parameters. We show that for the set of physical parameters that can be easily obtained experimentally, the Casimir and Casimir-Polder currents can be of the order of 10^(-8) A and 10^(-13) A respectively. In the second part, we theoretically model an experiment which was performed by Britton Plourde's group at Syracuse University, and which studied the transient dynamics of a nonlinear superconducting oscillator, based on a capacitively shunted DC-SQUID. Such DC-SQUID oscillators are used in many areas of physics and engineering, for example, as building blocks of amplifiers or qubits, qubit couplers, or as sensitive magnetic field detectors. In many of these situations, their steady state behavior is often considered, while in the experiment performed at Syracuse, of specific interest, was the response of a DC-SQUID oscillator to a short radiation that only briefly excited the system. In this thesis, we simulate this response at the experimental temperature, by numerically solving a set of classical stochastic differential equations that mimic the behavior of the circuit. This is done for different settings of the flux that is threaded through the DC-SQUID as well as different input pulse amplitudes. Furthermore, we briefly outline just how these kinds of brief excitations could be useful when applied in flux measurement protocols. We find that our simulations show good agreement with the experimentally obtained data. The final part considered in this thesis, looks at the dynamics of a qubit coupled to a measuring probe, which is modeled as a harmonic oscillator. An example superconducting circuit, that could be used to implement such a setup, consists of a flux qubit inductively coupled to a DC-SQUID. This measurement scenario has already been explored in [Phys. Rev. B 78, 5 (2008)], but there, the authors only consider very short interaction times between the DC-SQUID and the qubit. Here, in contrast, we concentrate our efforts on studying the evolution of qubit as the measurement takes place, by solving the corresponding Lindblad master equation, but over longer measurement times. This is done by calculating the measurement induced dephasing rate of the qubit, as well as, discussing its sometimes present effective relaxation, in regimes where the measurement is considered to not be quantum non-demolition (QND). Finally, we briefly explore how well a potentially complicated evolution of the qubit can be approximated as a very simple Kraus map.4 month

Surface Code Threshold Calculation and Flux Qubit Coupling

Building a quantum computer is a formidable challenge. In this thesis, we focus on two projects, which tackle very different aspects of quantum computation, and yet still share a common goal in hopefully getting us closer to implementing a quantum computer on a large scale. The first project involves a numerical error threshold calculation of a quantum error correcting code called a surface code. These are local check codes, which means that only nearest neighbour interaction is required to determine where errors occurred. This is an important advantage over other approaches, as in many physical systems, doing operations on arbitrarily spaced qubits is often very difficult. An error threshold is a measure of how well a given error correcting scheme performs. It gives the experimentalists an idea of which approaches to error correction hold greater promise. We simulate both toric and planar variations of a surface code, and numerically calculate a threshold value of approximately $6.0 \times 10^{-3}$, which is comparable to similar calculations done by others \cite{Raussendorf2006,Raussendorf2007,Wang2009}. The second project deals with coupling superconducting flux qubits together. It expands the scheme presented in \cite{Plourde2004} to a three qubit, two coupler scenario. We study L-shaped and line-shaped coupler geometries, and show how the coupling strength changes in terms of the dimensions of the couplers. We explore two cases, the first where the interaction energy between two nearest neighbour qubits is high, while the coupling to the third qubit is as negligible as possible, as well as a case where all the coupling energies are as small as possible. Although only an initial step, a similar scheme can in principle be extended further to implement a lattice required for computation on a surface code

Squeezed superradiance enables robust entanglement-enhanced metrology even with highly imperfect readout

Quantum metrology protocols using entangled states of large spin ensembles attempt to achieve measurement sensitivities surpassing the standard quantum limit (SQL), but in many cases they are severely limited by even small amounts of technical noise associated with imperfect sensor readout. Amplification strategies based on time-reversed coherent spin-squeezing dynamics have been devised to mitigate this issue, but are unfortunately very sensitive to dissipation, requiring a large single-spin cooperativity to be effective. Here, we propose a new dissipative protocol that combines amplification and squeezed fluctuations. It enables the use of entangled spin states for sensing well beyond the SQL even in the presence of significant readout noise. Further, it has a strong resilience against undesired single-spin dissipation, requiring only a large collective cooperativity to be effective.Comment: 6+9 pages, 3+3 figures; equivalent to published version; a reference to a previously unpublished manuscript has been update

Simple master equations for describing driven systems subject to classical non-Markovian noise

Driven quantum systems subject to non-Markovian noise are typically difficult to model even if the noise is classical. We present a systematic method based on generalized cumulant expansions for deriving a time-local master equation for such systems. This master equation has an intuitive form that directly parallels a standard Lindblad equation, but contains several surprising features: the combination of driving and non-Markovianity results in effective time-dependent dephasing rates that can be negative, and the noise can generate Hamiltonian renormalizations even though it is classical. We analyze in detail the highly relevant case of a Rabi-driven qubit subject to various kinds of non-Markovian noise including $1/f$ fluctuations, finding an excellent agreement between our master equation and numerically-exact simulations over relevant timescales. The approach outlined here is more accurate than commonly employed phenomenological master equations which ignore the interplay between driving and noise.Comment: 12+4 pages, 6+4 figure

Analytic Design of Accelerated Adiabatic Gates in Realistic Qubits: General Theory and Applications to Superconducting Circuits

Shortcuts to adiabaticity is a general method for speeding up adiabatic quantum protocols, and has many potential applications in quantum information processing. Unfortunately, analytically constructing shortcuts to adiabaticity for systems having complex interactions and more than a few levels is a challenging task. This is usually overcome by assuming an idealized Hamiltonian [e.g., only a limited subset of energy levels are retained, and the rotating-wave approximation (RWA) is made]. Here we develop an $analytic$ approach that allows one to go beyond these limitations. Our method is general and results in analytically derived pulse shapes that correct both nonadiabatic errors as well as non-RWA errors. We also show that our approach can yield pulses requiring a smaller driving power than conventional nonadiabatic protocols. We show in detail how our ideas can be used to analytically design high-fidelity single-qubit "tripod" gates in a realistic superconducting fluxonium qubit.Comment: 14 pages + 8 pages of Appendix, 13 figures (Published version

Transient dynamics of a superconducting nonlinear oscillator

We investigate the transient dynamics of a lumped-element oscillator based on a dc superconducting quantum interference device (SQUID). The SQUID is shunted with a capacitor forming a nonlinear oscillator with resonance frequency in the range of several GHz. The resonance frequency is varied by tuning the Josephson inductance of the SQUID with on-chip flux lines. We report measurements of decaying oscillations in the time domain following a brief excitation with a microwave pulse. The nonlinearity of the SQUID oscillator is probed by observing the ringdown response for different excitation amplitudes while the SQUID potential is varied by adjusting the flux bias. Simulations are performed on a model circuit by numerically solving the corresponding Langevin equations incorporating the SQUID potential at the experimental temperature and using parameters obtained from separate measurements characterizing the SQUID oscillator. Simulations are in good agreement with the experimental observations of the ringdowns as a function of applied magnetic flux and pulse amplitude. We observe a crossover between the occurrence of ringdowns close to resonance and adiabatic following at larger detuning from the resonance. We also discuss the occurrence of phase jumps at large amplitude drive. Finally, we briefly outline prospects for a readout scheme for superconducting flux qubits based on the discrimination between ringdown signals for different levels of magnetic flux coupled to the SQUID.Comment: 15 pages, 9 figure