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arXiv:0910.5640 · Physical Review B

Intrinsic leakage of the Josephson flux qubit and breakdown of the two-level approximation for strong driving

Alejandro Ferrón, Daniel Domı́nguez

Published Mar 8, 2010• 25 citations
arXivHTMLPDF

Abstract

Solid-state devices for quantum-bit computation (qubits) are not perfect isolated two-level systems since additional higher energy levels always exist. One example is the Josephson flux qubit, which consists on a mesoscopic superconducting quantum interference device loop with three Josephson junctions operated at or near a magnetic flux of half quantum. We study intrinsic leakage effects, i.e., direct transitions from the allowed qubit states to higher excited states of the system during the application of pulses for quantum computation operations. The system is started in the ground state and rf-magnetic field pulses are applied at the qubit resonant frequency with pulse intensity ${f}_{p}$. A perturbative calculation of the average leakage for small ${f}_{p}$ is performed for this case, obtaining that the leakage is quadratic in ${f}_{p}$, and that it depends mainly on the matrix elements of the supercurrent. Numerical simulations of the time-dependent Schr\"odinger equation corresponding to the full Hamiltonian of this device were also performed. From the simulations we obtain the value of ${f}_{p}$ above which the two-level approximation breaks down, and we estimate the maximum Rabi frequency that can be achieved. We study the leakage as a function of the ratio $\ensuremath{\alpha}$ among the Josephson energies of the junctions of the device, obtaining the best value for minimum leakage $(\ensuremath{\alpha}\ensuremath{\approx}0.85)$. The effects of flux noise on the leakage are also discussed.

DOI
10.1103/physrevb.81.104505

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Solid-state devices for quantum-bit computation (qubits) are not perfect isolated two-level systems since additional higher energy levels always exist. One example is the Josephson flux qubit, which consists on a mesoscopic superconducting quantum interference device loop with three Josephson junctions operated at or near a magnetic flux of half quantum. We study intrinsic leakage effects, i.e., direct transitions from the allowed qubit states to higher excited states of the system during the application of pulses for quantum computation operations. The system is started in the ground state and rf-magnetic field pulses are applied at the qubit resonant frequency with pulse intensity fp{f}_{p}fp​. A perturbative calculation of the average leakage for small fp{f}_{p}fp​ is performed for this case, obtaining that the leakage is quadratic in fp{f}_{p}fp​, and that it depends mainly on the matrix elements of the supercurrent. Numerical simulations of the time-dependent Schr"odinger equation corresponding to the full Hamiltonian of this device were also performed. From the simulations we obtain the value of fp{f}_{p}fp​ above which the two-level approximation breaks down, and we estimate the maximum Rabi frequency that can be achieved. We study the leakage as a function of the ratio \ensuremathα\ensuremath{\alpha}\ensuremathα among the Josephson energies of the junctions of the device, obtaining the best value for minimum leakage (\ensuremathα\ensuremath≈0.85)(\ensuremath{\alpha}\ensuremath{\approx}0.85)(\ensuremathα\ensuremath≈0.85). The effects of flux noise on the leakage are also discussed.
We study the dynamics of the Josephson flux qubit, which consists on a SQUID loop with three Josephson junctions operated at or near a magnetic flux of half quantum. We perform simulations of the time dependent Schrödinger equation when the system is started in the ground state. The leakage from the two lowest energy levels of the qubit computational subspace to higher energy levels is calculated, when pulses in the magnetic field are applied. We obtain, as expected, that constant field pulses have a higher leakage than rf field pulses at the resonant Rabi frequency. In the case of the constant field pulse we study the average leakage as a function of the pulse intensity, f p . For low f p we obtain that the leakage is quadratic in f p and we compare with a perturbative calculation.
Superconducting flux qubits with three Josephson junctions are promising candidates for the building blocks of a quantum computer. We have applied the imaginary time evolution method to study the model of this qubit accurately by calculating its wavefunctions and eigenenergies. Because such qubits are manipulated with magnetic flux microwave pulses, they might be irradiated into non-computational states, which is called the leakage effect. By the evolution of the density matrix of the qubit under either hard-shaped π-pulse or Gaussian-shaped π-pulse to carry out quantum NOT operation, it has been demonstrated that the leakage effect for a flux qubit is very small even for hard-shaped microwave pulses while Gaussian-shaped pulses may suppress the leakage effect to a negligible level.
At low temperature and low damping, the magnetic flux of a rf Superconducting QUantum Interference Device (SQUID) acts as a macroscopic quantum variable whose dynamic is described by a double well potential. A rf SQUID consists of a superconducting loop interrupted by a thin insulating barrier, known as a Josephson junction. The modified rf SQUID design discussed in this thesis has independent, in situ, controls that allow varying the barrier height between the potential wells and their relative tilt. This high level of control of the potential together with the capability to implement an experimental setup optimized to reduce coupling to external sources of noise, made this system ideal not only for tests of quantum transition at a macroscopic level, but also as a scalable solid state device for quantum computation. A variety of measurements of quantum phenomena are reported in this thesis, including microwave level spectroscopy, coherent oscillations, and macroscopic resonant tunneling (MRT). These measurements are compared with theoretical models to study the possible sources of decoherence in a rf SQUID qubit. The analysis confirmed that the main source of decoherence is not due to coupling to the external environment, but it can be linked to intrinsic low frequency flux noise. Work from many groups over the last several years has shown that the performance of a superconductive qubit is limited by coupling to spurious defects in the materials used for the device fabrication. In this work we analyze the possibility of using the MRT as a tool for future studies of effects of materials and fabrication process in the realization of improved qubits, and we also explore the feasibility for measurements of quantum phenomena that involve fast pulsing of the potential barrier height.
The three-junction flux qubit (quantum bit) consists of three Josephson junctions connected in series on a superconducting loop. We present a numerical treatment of this device for the general case in which the ratio ${\ensuremath{\beta}}_{Q}$ of the geometrical inductance of the loop to the kinetic inductance of the Josephson junctions is not necessarily negligible. Relatively large geometric inductances allow the flux through each qubit to be controlled independently with on-chip bias lines, an essential consideration for scalability. We derive the three-dimensional potential in terms of the macroscopic degrees of freedom, and include the possible effects of asymmetry among the junctions and of stray capacitance associated with them. To find solutions of the Hamiltonian, we use basis functions consisting of the product of two plane wave states and a harmonic oscillator eigenfunction to compute the energy levels and eigenfunctions of the qubit numerically. We present calculated energy levels for the relevant range of ${\ensuremath{\beta}}_{Q}$. As ${\ensuremath{\beta}}_{Q}$ is increased beyond 0.50.50.5, the tunnel splitting between the ground and first excited states decreases rapidly, and the device becomes progressively less useful as a qubit.
We analyze the coupling of two flux qubits with a general many-body projector into the low-energy subspace. Specifically, we extract the effective Hamiltonians that controls the dynamics of two qubits when they are coupled via a capacitor and/or via a Josephson junction. While the capacitor induces a static charge coupling tunable by design, the Josephson junction produces a magnetic-like interaction easily tunable by replacing the junction with a superconducting quantum interference device. Those two elements allow to engineer qubits Hamiltonians with XX, YY, and ZZ interactions, including ultrastrongly coupled ones. We present an exhaustive numerical study for two three-Josephson junctions flux qubit that can be directly used in experimental work. The method developed here, namely, the numerical tool to extract qubit effective Hamiltonians at strong coupling, can be applied to replicate our analysis for general systems of many qubits and any type of coupling.
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.