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Characterisation and rapid purification of a superconducting charge qubit

Elias J. Griffith

Published Jan 1, 2020• 0 citations
Publisher

Abstract

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.

DOI
10.17638/03069606

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In this work we consider measurements on quantum systems emphasizing\non read-out of superconducting qubits. The qubit is always coupled to an\nenvironment, containing a macroscopic number of degrees of freedom. This\nperturbs the qubit and destroys the phase coherence of the quantum state.\nOne tries to minimize this destructive effect by encoding the quantum information in quantum states which do not couple strongly to the dominant\nnoise sources. It is therefore necessary to design read-out methods that can\ndistinguish these states.\n\nWe focus primarily on the single Cooper-pair box (SCB) charge qubit, whose\nproperties are based on the Josephson effect and Coulumb blockade in superconducting electrical circuits. By encoding the information in the qubit\nenergy eigenstates, it is possible to protect it from the most dangerous decoherence source - charge fluctuations in the environment. We investigate\nthe measurement setup, where the SCB is coupled to a lumped element LCoscillator whose resonance frequency is orders of magnitude smaller than the qubit energy splitting. In this regime the SCB will act as an effective positive or negative capacitance, depending on the quantum state. This quantum capacitance then shifts the oscillator resonance frequency, which can be detected in a homodyne phase measurement.\n\nWe have found that reading out a charge qubit by measuring its quantum\ncapacitance in this manner is quantum limited, independently of the oscillator\nquality factor Q. Moreover, we have found that the low-Q oscillator\nis advantageous for the measurement in two ways: first, such an oscillator\nbetter protects the qubit from thermal dephasing, and, second, it allows to\nincrease the speed of the read-out, making it possible to distinguish the qubit states in a single shot measurement.

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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 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 topic of this thesis is superconducting electric circuits. Technical advances have made possible the experimental study of Josephson junction based circuit elements which sustain quantum mechanical properties long enough to be denoted as quantum devices. The quantum state can be controlled with electronic variables and measured using standard electrical setups. The research is motivated by the possibility to examine quantum phenomena in circumstances that can be customized, prospects of new quantum devices, and the development of quantum information processing. This thesis presents theoretical studies on the nonlinear and stochastic driving of a superconducting quantum two-level system (qubit). We first investigate the energy level shifts a single-Cooper-pair transistor under large amplitude driving realized via the inherently nonlinear Josephson energy by using an external magnetic flux. The effective driving field substantially deviates from a circular polarization and linear coupling. The energy level shifts are compared to the cases of a vanishing and a weak driving field, measured as the Stark shift and the generalized Bloch-Siegert shift, respectively. We describe criteria for the natural basis of the analytical and the numerical calculations. In addition to that, we develop a formalism based on the Floquet method for the weak probe measurement of the strongly driven qubit. In the latter part of the thesis research, we study utilization of a stochastic driving field whose time evolution is not regular but follows probabilistic laws. We concentrate on the motional averaging phenomenon and show that it can be measured with an unparalleled accuracy by employing a flux-modulated transmon qubit. As the stochastically modulated qubit is simultaneously measured with a moderate driving field, we develop a theoretical description accounting the possible interference effects between the modulation and the drive. The comparison with experimental results shows good agreement. Motional averaging phenomenon can be applied to estimate the properties of fluctuation processes occurring in qubits, e.g., the quasiparticle tunneling or the photon shot noise. Resting on the motional averaging, we anticipate that the qubit dephasing times can be improved if one can accelerate the dynamics of two-level fluctuators. We apply a semiclassical formalism where the qubit is treated with quantum mechanical concepts whereas the driving fields are classical. In the solution procedure, the numerical results support the main analytical understanding. As the theoretical results are extensively compared to reflection measurements, we construct an explicit connection between the dynamics of the studied quantum devices and the measured reflection coefficient.
This paper discusses the use of continuous weak measurement and quantum feedback for the rapid purification of the quantum state of a model solid state qubit: a superconducting Cooper pair box. The feedback algorithm uses Jacobs' rapid purification protocol, which starts with a completely mixed state and applies controls to rotate the qubit Bloch vector onto the plane orthogonal to the measurement axis. This rotation maximises the rate of increase of the average purity of the state but can require large changes in the control fields to produce the required rotation. Since solid state qubits have finite controls and feedback channels have limited bandwidth, such rotations may not be practical. This paper applies Jacobs' protocol to the Cooper pair box with realistic control fields.
During this thesis, we have used superconducting circuits with Josephson junctions, derived from the Cooper pair box, in order to implement quantum bits (qubits). To implement two-qubit gates, we have developed a new circuit, the quantroswap, which consists in two capacitively coupled Cooper pair box, each of them being manipulated and read separately. We have demonstrated coherent exchange of energy between them, but we have also observed a problem of qubit instability. In order to avoid this spurious effect, we have implemented another circuit based on a charge insensitive split Cooper pair box coupled to a non-linear resonator for readout-out purpose. We have measured large coherence time, and obtained large readout fidelity (90%) using the bifurcation phenomenon. For metrological purpose, microwave reflectometry measurement on a quantronium also allowed us to relate an applied current I to the frequency f=I/2e of induced Bloch oscillations.
In this work we consider measurements on quantum systems emphasizing\non read-out of superconducting qubits. The qubit is always coupled to an\nenvironment, containing a macroscopic number of degrees of freedom. This\nperturbs the qubit and destroys the phase coherence of the quantum state.\nOne tries to minimize this destructive effect by encoding the quantum information in quantum states which do not couple strongly to the dominant\nnoise sources. It is therefore necessary to design read-out methods that can\ndistinguish these states.\n\nWe focus primarily on the single Cooper-pair box (SCB) charge qubit, whose\nproperties are based on the Josephson effect and Coulumb blockade in superconducting electrical circuits. By encoding the information in the qubit\nenergy eigenstates, it is possible to protect it from the most dangerous decoherence source - charge fluctuations in the environment. We investigate\nthe measurement setup, where the SCB is coupled to a lumped element LCoscillator whose resonance frequency is orders of magnitude smaller than the qubit energy splitting. In this regime the SCB will act as an effective positive or negative capacitance, depending on the quantum state. This quantum capacitance then shifts the oscillator resonance frequency, which can be detected in a homodyne phase measurement.\n\nWe have found that reading out a charge qubit by measuring its quantum\ncapacitance in this manner is quantum limited, independently of the oscillator\nquality factor Q. Moreover, we have found that the low-Q oscillator\nis advantageous for the measurement in two ways: first, such an oscillator\nbetter protects the qubit from thermal dephasing, and, second, it allows to\nincrease the speed of the read-out, making it possible to distinguish the qubit states in a single shot measurement.