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Journal of Physics Conference Series

Leakage of the Josephson flux qubit

Daniel Domı́nguez, Ezequiel N. Pozzo

Published Mar 1, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1742-6596/150/5/052045

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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.
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 $\ensuremath{\alpha}$ 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.
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.
We study the dynamic behavior of a quantum two-level system with periodically varying parameters by solving numerically the master equation for the density matrix. Two limiting cases are considered: multiphoton Rabi oscillations and Landau–Zener transitions. The approach is applied to the description of the dynamics of superconducting qubits. In particular, the case of the interferometer-type charge qubit with periodically varying parameters (gate voltage or magnetic flux) is investigated. The time-averaged energy level populations are calculated as functions of the qubit’s control parameters.
It is well known that that superconducting qubits based on Josephson junctions have the advantages of scalability and the qubit states are easy to prepare and control. In addition, demonstration of Rabi oscillations in various superconducting circuits shows that superconducting qubits are promising for scalable quantum information processing. However, despite flexibility of design and fabrication, easier to scale up, and fast gate speed, superconducting qubits usually have much shorter decoherence time than trapped ions, NMR etc. due to the relatively strong interactions between qubits and environment. Recent experiments show that low frequency flux noise is the dominant mechanism of decoherence in superconducting flux qubits. However, despite extensive effort the origin of flux noise is still not well understood. The goal of this work is to identify the source and characterize the property of flux noisein rf SQUID flux qubit through various spectroscopy and time-resolved measurements. Our result show that one can determine all circuit parameters needed for reconstructing qubit Hamiltonian with high accuracy via measurement of macroscopic resonant tunneling (MRT) and photon assisted tunneling (PAT) and that the amount of flux noise in rf SQUID qubits scales linearly with self inductance of the qubits. In addition, we have investigated the dynamics of a three-level flux qubit in incoherent regime. The result demonstrates that treating a multi-level physical qubit, such as the superconducting flux qubit, as an ideal two-level quantum system may be inadequate under certain circumstances.
In this thesis measurements on single and coupled Josephson persistent current qubits are presented. The Josephson persistent current qubit consists of a superconducting loop intersected by three Josephson junctions. When half a flux quantum is applied to the loop this system acts as a quantum two state system. The two states are associated with a clock or anti clockwise current flowing in the loop and can be measured using a SQUID magnetometer. The research presented in this thesis aimed at achieving coupling between two Josephson persistent current qubits and to show conditional gate operations in such a coupled qubit system. Also it was tried to improve the coherence properties of a single qubit by using a gradiometer type configuration to minimize influence from fluxnoise which would cause a disturbance of the energy bias of the qubit system. The basic principles of the Josephson persistent current qubit are discussed, as well as design criteria to minimize decoherence due to the environment seen by the squid or microwave lines. Spectroscopy measurements have been performed on coupled qubit systems mapping out the energy level structure of these systems. For two coupled qubits the origin and magnitude of the coupling J is derived for coupling via the geometrical inductance between the qubits. Also it is shown that coupling via a shared junction or a shared line acts the same as the geometrical coupling. Spectroscopy measurements show coupling for qubits which are mainly geometrical coupled and spectroscopy measurements on an other sample show a coupling for qubits coupled via a large shared Josephson junction. Using such a shared junction allows for the coupling to be designed within a large range. For the coupled qubit system with coupling via a large junction spectroscopy measurements from both the ground state to higher states as well as from the partially populated first excited state to higher states have been performed and the level structure of this system is mapped out. The energy level structure is in good agreement with the 4 level approximation for two coupled qubits. Coherent Rabi oscillations between two states flipping either qubit of this system have been performed with an oscillation decay time of up to 70ns. Also conditional spectroscopy has been performed by first going coherently from the ground state to a higher state and then incoherently pumping the system to other states. Spectroscopy measurements on a gradiometer type Josephson persistent current qubit show trapped fluxoid biasing can be used to create a phase bias of pi over the qubit junctions. The results of this thesis show that coupling between Josephson persistent current qubits can be achieved and designed within the wanted range. Superpositions between the ground state and higher states in such a coupled qubit system have been created by performing Rabi oscillations. Also conditional spectroscopy has been performed.