The quantum system associated with a resistor-shunted weak-link junction circuit may be described in terms of a single Lagrangian. In this representation the degrees of freedom internal to the dissipative energy of the resistor can be systematically separated from the Hamiltonian formulation of the circuit. This allows us to solve the quantum states of the junction while the effects of the resistor on the circuit are included as a Lagrangian multiplier. In this manner the effects of the measuring apparatus or resistor on the quantum state of the junction may be calculable. The calculations are in very good agreement with experiments.
The quantum system associated with a resistor-shunted weak-link junction circuit may be described in terms of a single Lagrangian. In this representation the degrees of freedom internal to the dissipative energy of the resistor can be systematically separated from the Hamiltonian formulation of the circuit. This allows us to solve the quantum states of the junction while the effects of the resistor on the circuit are included as a Lagrangian multiplier. In this manner the effects of the measuring apparatus or resistor on the quantum state of the junction may be calculable. The calculations are in very good agreement with experiments.
We introduce a numerically exact and computationally feasible nonlinear-response theory developed for lossy superconducting quantum circuits based on a framework of quantum dissipation in a minimally extended state space. Starting from the Feynman-Vernon path-integral formalism for open quantum systems with the system degrees of freedom being the nonlinear elements of the circuit, we eliminate the temporally nonlocal influence functional of all linear elements by introducing auxiliary harmonic modes with complex-valued frequencies coupled to the nonlinear elements. In our work, we propose a concept of time-averaged observables, inspired by experiment, and provide an explicit formula for producing their quasiprobability distribution. We illustrate the consistency of our formalism with the well-established Markovian input-output theory by applying them the dispersive readout of a superconducting transmon qubit. For an important demonstration of our approach beyond weak coupling, we analyze the low-frequency linear response of a capacitively and resistively shunted Josephson junction and observe signatures of a much-debated quantum phase transition at a finite temperature. The developed framework enables a comprehensive fully quantum-mechanical treatment of nonlinear quantum circuits coupled to their environment, without the limitations of typical approaches to weak dissipation, high temperature, and weak drive. This versatile tool paves the way for accurate models of quantum devices and increased fundamental understanding of quanutm mechanics such as that of the quantum measurement.
We present a multilevel quantum theory of decoherence for a general circuit realization of a superconducting qubit. Using electrical network graph theory, we derive a Hamiltonian for the circuit. The dissipative circuit elements (external impedances, shunt resistors) are described using the Caldeira-Leggett model. The master equation for the superconducting phases in the Born-Markov approximation is derived and brought into the Bloch-Redfield form in order to describe multilevel dissipative quantum dynamics of the circuit. The model takes into account leakage effects, i.e., transitions from the allowed qubit states to higher excited states of the system. As a special case, we truncate the Hilbert space and derive a two-level (Bloch) theory with characteristic relaxation (T1) and decoherence (T2) times. We apply our theory to the class of superconducting flux qubits; however, the formalism can be applied for both superconducting flux and charge qubits.
This work concerns the study of Josephson junction circuits in the context of their usability for quantum computing. The zero-voltage state of a current-biased Josephson junction has a set of metastable quantum energy levels. If a junction is well isolated from its environment, it will be possible to use the two lowest states as a qubit in a quantum computer. I first examine the meaning of isolation theoretically. Using a master equation, I analyzed the effect of dissipation on escape rates and suggested a simple method, population depletion technique, to measure the relaxation time. Using a stochastic Bloch equation to analyze microwave resonance shapes, I found a relation between current noise induced decoherence and the noise spectrum. I then analyze and test a few qubit isolation schemes, including resistive isolation, inductor-capacitor (LC) isolation, and inductor-junction (LJ) isolation. I found the resistive isolation scheme has a severe heating problem. Macroscopic quantum tunneling and energy level quantization were observed in the LC isolated junction qubits at 25 mK. Relaxation times of 4-12 ns and spectroscopic coherence times of 1-3 ns were obtained for these LC isolated qubits. I measured a relaxation time of 50 ns and a spectroscopic coherence time of 5-8 ns for the LJ isolated junction qubit. Both times are much longer than those of the LC isolated qubits. Rabi oscillations were also observed on this sample with a decay time of around 10 ns. Using microwave spectroscopy techniques, I probed quantum phenomena in a coupled macroscopic three-qubit system that is comprised of two Nb/AlOx/Nb Josephson junctions and an LC resonator. The measured spectrum at 25 mK in the frequency range 4-15 GHz agrees well with quantum mechanical calculations, consistent with the existence of entangled states between the three degrees of freedom. These entangled states and a first-order strong coupling between two junction qubits open the possibility of using a resonator as a data bus for information storage and manipulation in a multi-qubit system. The measurements also demonstrate spectroscopy is a powerful tool and can be used to study a composite system with many qubits.
We report our time-resolved measurements of switching dynamics of grain-boundary, 5-μm-wide Y–Ba–Cu–O (YBCO) Josephson junctions, excited by 2-ps-wide electrical pulses optically generated by a voltage-biased, 5-μm-wide, 10-μm-long YBCO microbridge. The time-resolved transients were recorded at 20 K using a subpicosecond cryogenic electro-optic sampling system. The intrinsic junction response was separated from the circuit-related feedthrough, and we observed both experimentally and in numerical simulations a 0.8-ps-wide, single-flux-quantum (SFQ) pulse, generated by the switching of the junction with the IcRn product equal to ∼2 mV. At the same time, the measured turn-on delay time was almost three times shorter than that obtained from simulations, questioning applicability of the resistively shunted tunnel junction model to nonhysteretic, high-temperature superconducting weak links. Our test structure's power consumption associated with the single SFQ pulse generation was ∼0.1 μW, leading to a (switching time)×(dissipated power) product equal to 0.08 aJ.
The Josephson effect, as an example of a macroscopic quantum phenomenon, reveals itself in the three-layer heterostructures formed by two superconductors coupled by a weak link that usually consists of a 1-2 nm-thick insulating barrier.The traditional way of modelling such-systems' dynamics is based on an equivalent circuit that comprises three parallel elements: a pure superconducting element with a certain supercurrent-versus-Josephson phase difference dependence, a resistor R, and a capacitor C. In this short review, we analyse the practical problem of reducing the junction capacitance while maintaining or slightly impairing other characteristics.Some arguments are presented to explain why the capacitance should be suppressed and how it will affect performance of superconducting quantum interference devices (SQUIDs) and digital electronics circuits.As a solution for low-capacitance junctions, we propose a weak link made of an amorphous-silicon interlayer doped with nanoscale metallic drops between the two superconducting Mo-Re-alloy electrodes.
Thermodynamics and transport properties of resistance-shunted Josephson junctions are studied theoretically in the tight-binding limit E C / E J ≪1, where E C and E J are a charging energy and a Josephson coupling energy respectively. Based on a phenomenological harmonic-oscillator model, weak coupling region K = R Q / R ≪1 is analytically studied, where R and R Q = h /(2 e ) 2 are a shunt resistance and the quantum resistance. In addition to an effective bandwidth 2ℏΔeff, we find that this multi-level system genuinely has a novel crossover at lower energy KℏΔeff below which the density of states becomes strongly degenerate. These two energy scales control the linear DC responses, optical responses, and nonlinear I - V characteristics. The lower energy crossover indicates the existence of a new class of strongly-correlated phenomena beyond the framework of the Kondo problem.