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arXiv:2401.09573

Application of the Schwinger Oscillator Construct of Angular Momentum to an Interpretation of the Superconducting Transmon Qubit

Erickson, R. P.

Published Jan 17, 2024• 0 citations
arXivHTMLPDF

Abstract

The Schwinger oscillator construct of angular momentum, applied to the superconducting transmon and its transmission-line readout, modeled as capacitvely coupled quantum oscillators, provides a natural and robust description of a qubit. The construct defines quantum-entangled, two-photon states that form an angular-momentum-like basis, with symmetry corresponding to physical conservation of total photon number, with respect to the combined transmon and readout. This basis provides a convenient starting point from which to study error-inducing effects of transmon anharmonicity, surrounding-environment decoherence, and random stray fields on qubit state and gate operations. Employing a Lindblad master equation to model dissipation to the surrounding environment, and incorporating the effect of weak transmon anharmonicity, we present examples of the utility of the construct. First, we calculate the frequency response associated with exciting the ground state to a Rabi resonance with the lowest-lying spin-1/2 moment, via a driving external voltage. Second, we calculate the frequency response between the three lowest two-photon states, within a ladder-type excitation scheme. The generality of the Schwinger angular-momentum construct allows it to be applied to other superconducting charge qubits.

DOI
10.48550/arxiv.2401.09573

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The Schwinger oscillator construct of angular momentum, applied to the superconducting transmon and its transmission-line readout, modeled as capacitvely coupled quantum oscillators, provides a natural and robust description of a qubit. The construct defines quantum-entangled, two-photon states that form an angular-momentum-like basis, with symmetry corresponding to physical conservation of total photon number, with respect to the combined transmon and readout. This basis provides a convenient starting point from which to study error-inducing effects of transmon anharmonicity, surrounding-environment decoherence, and random stray fields on qubit state and gate operations. Employing a Lindblad master equation to model dissipation to the surrounding environment, and incorporating the effect of weak transmon anharmonicity, we present examples of the utility of the construct. First, we calculate the frequency response associated with exciting the ground state to a Rabi resonance with the lowest-lying spin-1/2 moment, via a driving external voltage. Second, we calculate the frequency response between the three lowest two-photon states, within a ladder-type excitation scheme. The generality of the Schwinger angular-momentum construct allows it to be applied to other superconducting charge qubits.
In a quantum computer, the information carriers, which are bits in ordinary computers, are implemented as devices that exhibit coherent superpositions of physical states and entanglement. Such components, known as quantum bits or qubits, can be realized with various different types of two-state quantum systems. Quantum computers will be built for computational speed, with hoped for applications especially in cryptography and in other tasks where classical computers remain inefficient. Circuit quantum electrodynamics (cQED) is a quantum-computer architecture which employs superconducting electronic components and microwave photon fields as building blocks. Compared to cavity quantum electrodynamics (CQED), where atoms are trapped in physical cavities, cQED is more attractive in that its qubits are tunable and conveniently integrable with the electronics already in use. This architecture has shown some of the most promising qubit designs, despite their coherence times reaching tens of microseconds, are still below the state of the art with spin qubits, which reach milliseconds. Coherence times are historically the most relevant parameters describing the fitness of a qubit, although these days not necessarily the limiting factor. This thesis presents a comprehensive set of theoretical and experimental methods for measuring the characteristic parameters of superconducting qubits. We especially study transmission-line-shunted plasma oscillation qubits, or transmons, and presents experimental results for a single sample. Transmon capacitively couples a superconducting quantum interference device (SQUID) with a coplanar waveguide (CPW) resonator, often with added frequency tunability utilizing an external magnet. The number of superconducting charge carriers tunnelled through a junction in the SQUID are used as qubit degrees of freedom. Readout of the qubit state is carried out by measuring transmission through the CPW. A cryogenic setup is employed with measurement and driving pulses delivered from microwave sources. Steady-state spectroscopy is employed to determine the resonance frequencies of the qubit and the resonator, qubit-resonator coupling constants, and the energy parameters of the qubit. Pulse-modulated measurements are employed to determine the coherence times of the qubit. The related analysis- and simulation programs and scripts are collected togithub.com/patomaki.
In the circuit quantum electrodynamics architecture, both the resonance frequency and the coupling of superconducting qubits to microwave field modes can be controlled via external electric and magnetic fields to explore qubit-photon dynamics in a wide parameter range. Here, we experimentally demonstrate and analyze a scheme for tuning the coupling between a transmon qubit and a microwave resonator using a single coherent drive tone. We treat the transmon as a three-level system, with the qubit subspace defined by the ground and the second excited states. If the drive frequency matches the difference between the resonator and the qubit frequencies, a Jaynes-Cummings-type interaction is induced, which is tunable in both amplitude and phase. We show that coupling strengths of about 10 MHz can be achieved in our setup, limited only by the anharmonicity of the transmon qubit. This scheme has been successfully used to generate microwave photons with a controlled temporal shape [M. Pechal et al., Phys. Rev. X 4, 041010 (2014)] and can be directly implemented with superconducting quantum devices featuring larger anharmonicity for higher coupling strengths.
This paper deals with the features of quantum computing for a system containing a relatively large number of qubits n, which is characterized by a half-integer effective spin S in the Schwinger representation of paired bosons. It was shown that for n = 70, the number of boson states corresponding to lowest 2S + 1 excited energy levels of each of the two harmonic oscillators implementing the two-boson representation of the effective spin S is N = 1070 = 1.18 x 1021. Under such conditions, it can be assumed with great precision that for n ≥ 70 all boson states of both harmonic oscillators participate in the implementation of the two-boson representation of the effective spin S. This allows one to apply the quantum field theory methods when performing quantum computations for multi-qubit systems. Another advantage of the Schwinger representation of paired bosons compared to the spinor representation when performing quantum computations for multi-qubit systems is that in the first case the form of the spin projection operators of the effective spin S does not depend on the number of qubits n.
Using circuit QED, we consider the measurement of a superconducting transmon qubit via a coupled microwave resonator. For ideally dispersive coupling, ringing up the resonator produces coherent states with frequencies matched to transmon energy states. Realistic coupling is not ideally dispersive, however, so transmon-resonator energy levels hybridize into joint eigenstate ladders of the Jaynes--Cummings type. Previous work has shown that ringing up the resonator approximately respects this ladder structure to produce a coherent state in the eigenbasis (a dressed coherent state). We numerically investigate the validity of this coherent-state approximation to find two primary deviations. First, resonator ring-up leaks small stray populations into eigenstate ladders corresponding to different transmon states. Second, within an eigenstate ladder the transmon nonlinearity shears the coherent state as it evolves. We then show that the next natural approximation for this sheared state in the eigenbasis is a dressed squeezed state and derive simple evolution equations for such states by using a hybrid phase--Fock-space description.
We measured the relaxation and decoherence rates of a superconducting transmon qubit in a resonator-free setting. In our experiments, the qubit is coupled to an open coplanar waveguide such that the transmission of microwaves through this line depends on the qubit's state. To determine the occupation of the first excited qubit energy level, we introduced a two-pulse technique. The first applied pulse, at a frequency close to the eigenfrequency of the qubit, serves to excite the qubit. A second pulse is then used for probing the transition between the first and second excited energy levels. Utilizing this measurement technique allowed for the reconstruction of the relaxation dynamics and Rabi oscillations. Furthermore, we demonstrate the consistency between the extracted parameters and the corresponding estimations from frequency-domain measurements.
The dynamics of a weakly anharmonic superconducting qubit in a complex electromagnetic environment is generally well described by an effective multimode Kerr Hamiltonian at sufficiently weak excitation. This Hamiltonian can be embedded in a master equation with losses determined by the details of the electromagnetic environment. Recent experiments indicate, however, that when a superconducting circuit is driven with microwave signals populating the system with sufficiently high excitations, the observed relaxation rates appear to be substantially different from expectations based on the electromagnetic environment of the qubit alone. This issue is a limiting factor in the optimization of superconducting qubit readout schemes. We claim here that an effective master equation with drive-power-dependent parameters is an efficient approach to model such quantum dynamics. In this sequence of papers, we derive effective master equations, whose parameters exhibit nonlinear dependence on the excitation level of the circuit as well as the electromagnetic environment of the qubit. We show that the number nonconserving terms in the qubit nonlinearity generally lead to a renormalization of dissipative parameters of the effective master equation, while the number conserving terms give rise to a renormalization of the system frequencies. Here, in Paper I, we consider the excitation-relaxation dynamics of a transmon qubit that is prepared in a certain initial state, but is not driven otherwise. A unitary transformation technique is introduced to study the renormalization of (i) qubit relaxation due to coupling to a generic bath and (ii) Purcell decay. Analytic expressions are provided for the dependence of the nonlinear dissipative terms on the details of the electromagnetic environment of the qubit. The perturbation technique based on unitary transformations developed here is generalized to the continuously driven case in Paper II [A. Petrescu et al., Phys. Rev. B 101, 134510 (2020)].