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AIP conference proceedings

Superconducting Qubits and Environmental Noise

Elias J. Griffith

Published Jan 1, 2004• 1 citation
Publisher

Abstract

We examine the energy level structure of a Josephson charge qubit (Cooper pair box) coupled to its environment, utilising a backreaction technique originally proposed for characterising persistent current qubits. The system consists of a model charge qubit coupled to three electromagnetic modes: a low frequency bias field, a higher frequency microwave to excite the qubit into higher states, and a lossy reservoir representing the cavity containing the qubit and control fields. Specifically, the qubit is a small superconducting island connected to a bulk superconductor via a weak Josephson junction, allowing Cooper pairs to tunnel on and off the island coherently. The classical bias field experiences a backreaction from the qubit and the noise power increases when the qubit is operating near a transition.

DOI
10.1063/1.1834462

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•Feb 2, 2005•3 citations
We examine the energy level structure of a Josephson charge qubit (Cooper pair box) coupled to its environment, utilising a backreaction technique originally proposed for characterising persistent current qubits. The system consists of a model charge qubit coupled to three electromagnetic modes: a low frequency bias field, a higher frequency microwave to excite the qubit into higher states, and a lossy reservoir representing the cavity containing the qubit and control fields. Specifically, the qubit is a small superconducting island connected to a bulk superconductor via a weak Josephson junction, allowing Cooper pairs to tunnel on and off the island coherently. The classical bias field experiences a backreaction from the qubit and the noise power increases when the qubit is operating near a transition.
In this paper, we propose a technique to characterise the energy level structure of a superconducting charge qubit. The technique relies on the backreaction of a solid-state qubit on its environment and the incoherent transfer of energy from a high frequency mode to a low frequency mode due to the stochastic transitions of the qubit between energy eigenstates. We consider a coupled system consisting of a model charge qubit and several classical degrees of freedom. The qubit is coupled to three electromagnetic modes: a low frequency bias field, a higher frequency mode (which is used to pump the qubit from the ground state to an excited state), and a lossy reservoir (which represents the cavity that contains the qubit and control fields). The reservoir provides a mechanism to allow the qubit to dissipate energy and to induce spontaneous decays from an excited state into the ground state. We show that these spontaneous decays can have a significant effect on the noise in the classical bias field, and that this noise can be used to characterise the energy level structure of the qubit.
The work of this thesis focuses on stochastic methods for the simulation of superconducting charge qubits, also known as Cooper pair boxes, which are promising candidates for large scale quantum information processing. To aid non-physicists, a brief outline of the structure of quantum mechanics is provided using the· Dirac formalism. Using the so called 'Backreaction effect', we consider if any information can be obtained regarding the qubit behaviour, through detecting the changes observed in the frequency spectrum of the coupled biasing circuitry, modelled as a dissipative oscillator circuit. The process of modelling dissipative quantum systems is described, however an alternative approach called 'quantum trajectories' is used rather than the traditional Calderia-Leggett model, as the time evolution of a single 'trajectory' represents the evolution of an individual system coupled to a noisy environment. Through the noise generated by an excited qubit, the energy level structure of the qubit can be probed with a microwave drive field, by observing the noise power within the biasing circuit. We consider a biasing circuit of unusually high resonant frequency which can drive the qubit, this creates frequency splitting features that would not normally be observed. Weak measurement is also examined as this is closely related to the stochastic 'quantum trajectories', where the measurement is recorded by the observer rather than lost to an environment. Weakly measuring a qubit does not completely collapse it and therefore 'quantum feedback' may be employed to alter the qubit controls favourably. In particular we consider the problem of purifying a weakly measured system rapidly; given a qubit in the completely mixed state what is the best feedback to become confident in the actual qubit state quickly. There are two optimal feedback protocols proposed by Jacobs [1] and by Wiseman and Ralph [2] for purifying qubits that have ideal controls. However, we adapt these protocols for the charge qubit, whose finite Hamiltonian resources and non-zero ax tunnelling term means the Bloch vector can not be easily held in the optimal location.
We have analyzed and measured the quantum coherent dynamics of a circuit containing two coupled superconducting charge qubits. Each qubit is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two qubits are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation ofthe qubit circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each qubit by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup the measured pulse-induced probe currents are proportional to the probability for each qubit to have an extra Cooper pai1r after the manipulation. As expected from theory and observed experimentally the measured pulse-induced current in each probe has two frequency components whose position on the frequency axis can be externally controlled. This is a result ofthe inter-qubit coupling which is also responsible for the avoided level crossing that we observed in the qubits' spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape the system remains entangled most ofthe time reaching maximally entangled states at certain instances.
After a brief review of recent proposals for using small superconductors whose charging energy is comparable to their superconducting gap as physical qubits in quantum computation, the ‘Cooper-pair box’ experiments of Nakamura et al. are analysed. In these, the gate voltage on a small superconducting island is perturbed by a pulse of height δV and duration Δt and coherent oscillations between the charge states with n̄ and n̄ +1 Cooper pairs are detected. The effects of noise in δV and Δt on their measurements is studied and it is found that the two kinds of noise have qualitatively different signatures in the experimental data.
The number of Cooper pairs on a small superconducting island coupled to superconducting leads by a Josephson junction fluctuates due to tunneling of Cooper pairs through the junction. On the other hand, the impedance of the circuit surrounding the island tends to suppress these fluctuations and to restore charge quantization. We consider the influence of the electromagnetic environment within a two-state approximation for the charge on the island. The average charge and the slope of the steps of the Coulomb staircase are calculated for a realistic environmental impedance and finite temperatures in the range of current experimental studies. It is shown that even a low-frequency environment may lead to a considerable reduction of charge fluctuations provided the integrated total impedance is large enough.
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.