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arXiv:2106.04137 · New Journal of Physics

Theoretical study of reflection spectroscopy for superconducting quantum parametrons

Shumpei Masuda, Aiko Yamaguchi, T. Yamaji, Tsuyoshi Yamamoto, T. Ishikawa, Yuichiro Matsuzaki, Shiro Kawabata

Published Aug 27, 2021• 11 citations
arXivHTMLPDF

Abstract

Abstract Superconducting parametrons in the single-photon Kerr regime, also called KPOs, have been attracting increasing attention in terms of their applications to quantum annealing and universal quantum computation. It is of practical importance to obtain information of superconducting parametrons operating under an oscillating pump field. Spectroscopy can provide information of a superconducting parametron under examination, such as energy level structure, and also useful information for calibration of the pump field. We theoretically study the reflection spectroscopy of superconducting parametrons, and develop a method to obtain the reflection coefficient. We present formulae of the reflection coefficient, the nominal external and the internal decay rates, and examine the obtained spectra. It is shown that the difference of the populations of energy levels manifests itself as a dip or peak in the amplitude of the reflection coefficient, and one can directly extract the coupling strength between the energy levels by measuring the nominal decay rates when the pump field is sufficiently large.

DOI
10.1088/1367-2630/ac21e1

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Abstract Superconducting parametrons in the single-photon Kerr regime, also called KPOs, have been attracting increasing attention in terms of their applications to quantum annealing and universal quantum computation. It is of practical importance to obtain information of superconducting parametrons operating under an oscillating pump field. Spectroscopy can provide information of a superconducting parametron under examination, such as energy level structure, and also useful information for calibration of the pump field. We theoretically study the reflection spectroscopy of superconducting parametrons, and develop a method to obtain the reflection coefficient. We present formulae of the reflection coefficient, the nominal external and the internal decay rates, and examine the obtained spectra. It is shown that the difference of the populations of energy levels manifests itself as a dip or peak in the amplitude of the reflection coefficient, and one can directly extract the coupling strength between the energy levels by measuring the nominal decay rates when the pump field is sufficiently large.
Abstract We report the spectroscopic characterization of a Kerr parametric oscillator (KPO) based on the measurement of its reflection coefficient under a two-photon drive induced by flux modulation. The measured reflection spectra show good agreement with numerical simulations in terms of their dependence on the two-photon drive amplitude. The spectra can be interpreted as changes in system’s eigenenergies, transition matrix elements, and the population of the eigenstates, although the linewidth of the resonance structure is not fully explained. We also show that the drive-amplitude dependence of the spectra can be explained analytically by using the concepts of Rabi splitting and the Stark shift. By comparing the experimentally obtained spectra with theory, we show that the two-photon drive amplitude at the device can be precisely determined, which is important for the application of KPOs in quantum information processing.
Kerr parametric oscillators (KPOs) implemented in the circuit quantum electrodynamic architecture can operate as qubits. Their applications to quantum annealing and universal quantum computation have been studied intensively. For these applications, the readout of the state of KPOs is of practical importance. We develop a scheme of state tomography for KPOs with reflection measurement. Although it is known that the reflection coefficient depends on the state of the KPO, it is unclear whether tomography of a qubit encoded into a KPO can be performed in a practical way mitigating decoherence during the measurement, and how accurate it is. We show that the reflection coefficient has a one-to-one correspondence with a diagonal element of the density matrix of the qubit when a probe frequency is properly chosen and an additional single-photon drive is introduced. Thus, our scheme offers a novel way to read out the qubit along an axis of the Bloch sphere, and therefore the reflection measurement and single-qubit gates can constitute state tomography.
Kerr parametric oscillators (KPOs), which can be implemented with superconducting parametrons possessing large Kerr nonlinearity, have been attracting much attention in terms of their applications to quantum annealing, universal quantum computation, and studies of quantum many-body systems. It is of practical importance for these studies to realize fast and accurate tunable coupling between KPOs in a simple manner. We develop a simple scheme of fast tunable coupling of KPOs with high tunability in speed and amplitude using the fast transitionless rotation of a KPO in the phase space based on the shortcuts to adiabaticity. Our scheme enables rapid switching of the effective coupling between KPOs, and can be implemented with always-on linear coupling between KPOs, by controlling the phase of the pump field and the resonance frequency of the KPO without controlling the amplitude of the pump field nor using additional drive fields and couplers. We apply the coupling scheme to a two-qubit gate, and show that our scheme realizes high gate fidelity compared with a purely adiabatic one, by mitigating undesired nonadiabatic transitions.
Abstract A Kerr parametric oscillator (KPO) is a Kerr-nonlinear resonator driven parametrically, which can serve as a qubit in quantum information processing. Spectroscopy using a weak probe field provides a practical method for extracting system parameters. In this work, we theoretically investigate reflection and transmission measurements of two-photon and four-photon KPOs, introducing interference effects between inter-level transitions. Because KPOs typically exhibit energy-level degeneracy, a probe field can simultaneously resonate with multiple inter-level transitions. We extend the previous theory of reflection measurements by incorporating the interaction between inter-level transitions and off-diagonal elements of the density matrix which had previously been neglected. We demonstrate that interference among these transitions substantially modifies the spectrum. We identify the conditions for the interference, as well as those under which the off-diagonal elements of the density matrix affect the spectrum. The theory is also generalized to transmission measurements, and is applicable to a broad class of systems beyond KPOs.
Pumped at approximately twice the natural frequency, a Josephson parametric oscillator called parametron or Kerr parametric oscillator shows self-oscillation. Quantum annealing and universal quantum computation using self-oscillating parametrons as qubits were proposed. However, controls of parametrons under the pump field are degraded by unwanted rapidly oscillating terms in the Hamiltonian, which is called counter rotating terms (CRTs) coming from the violation of the rotating wave approximation. Therefore, the pump field can be an intrinsic origin of the imperfection of controls of parameterons. Here, we theoretically study the effect of the CRTs on the accuracy of controls of a parametron: creation of a cat state and a single qubit gate along the xxx axis. It is shown that there is a trade-off relationship between the suppression of the nonadiabatic transitions and the validity of the rotating wave approximation in a conventional approach. We show that the tailored time dependence of the detuning of the pump field can suppress both of the nonadiabatic transitions and the disturbance of the state of the parametron due to the CRTs.
Pumped at approximately twice the natural frequency, a Josephson parametric oscillator called parametron or Kerr parametric oscillator shows self-oscillation. Quantum annealing and universal quantum computation using self-oscillating parametrons as qubits were proposed. However, controls of parametrons under the pump field are degraded by unwanted rapidly oscillating terms in the Hamiltonian, which we call non-resonant rapidly oscillating terms (NROTs) coming from the violation of the rotating wave approximation. Therefore, the pump field can be an intrinsic origin of the imperfection of controls of parametrons. Here, we theoretically study the influence of the NROTs on the accuracy of controls of a parametron: a cat-state creation and a single-qubit gate. It is shown that there is a trade-off relationship between the suppression of the nonadiabatic transitions and the validity of the rotating wave approximation in a conventional approach. We also show that the tailored time dependence of the detuning of the pump field can suppress both of the nonadiabatic transitions and the disturbance of the state of the parametron due to the NROTs.