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New Journal of Physics

Superconducting flux qubit capacitively coupled to an LC resonator

Tsuyoshi Yamamoto, K. Inomata, Kazuki Koshino, P.-M. Billangeon, Yasunobu Nakamura, J. S. Tsai

Published Jan 10, 2014• 29 citations
Publisher

Abstract

We study the system where a superconducting flux qubit is capacitively coupled to an LC resonator. In three devices with different coupling capacitance, the magnitude of the dispersive shift is enhanced by the third level of the qubit and quantitatively agrees with the theory. We show by numerical calculation that the capacitive coupling plays an essential role for the enhancement in the dispersive shift. We investigate the coherence properties in two of these devices, which are in the strong-dispersive regime, and show that the qubit energy relaxation is currently not limited by the coupling. We also observe the discrete ac-Stark effect, a hallmark of the strong-dispersive regime, in accordance with the theory.

DOI
10.1088/1367-2630/16/1/015017

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We study the system where a superconducting flux qubit is capacitively coupled to an LC resonator. In three devices with different coupling capacitance, the magnitude of the dispersive shift is enhanced by the third level of the qubit and quantitatively agrees with the theory. We show by numerical calculation that the capacitive coupling plays an essential role for the enhancement in the dispersive shift. We investigate the coherence properties in two of these devices, which are in the strong-dispersive regime, and show that the qubit energy relaxation is currently not limited by the coupling. We also observe the discrete ac-Stark effect, a hallmark of the strong-dispersive regime, in accordance with the theory.
We demonstrate enhancement of the dispersive frequency shift in a coplanar waveguide resonator induced by a capacitively coupled superconducting flux qubit in the straddling regime. The magnitude of the observed shift, 80 MHz for the qubit-resonator detuning of 5 GHz, is quantitatively explained by the generalized Rabi model which takes into account the contribution of the qubit higher energy levels. By applying the enhanced dispersive shift to the qubit readout, we achieved 9090%90 contrast of the Rabi oscillations, which is mainly limited by the energy relaxation of the qubit.
This dissertation examines the design, fabrication, and characterization of a superconducting lumped-element tunable LC resonator that is used to vary the coupling between two superconducting qubits. Some level of qubit-qubit coupling is needed to perform gating operations. However, with fixed coupling, single qubit operations become considerably more difficult due to dispersive shifts in their energy levels transitions that depend on the state of the other qubit. Ideally, one wants a system in which the qubit-qubit coupling can be turned off to allow for single qubit operations, and then turned back on to allow for multi-qubit gate operations. I present results on a device that has two fixed-frequency transmon qubits capacitively coupled to a tunable thin-film LC resonator. The resonator can be tuned in situ over a range of 4.14 GHz to 4.94 GHz by applying an external magnetic flux to two single-Josephson junction loops, which are incorporated into the resonator’s inductance. The qubits have 0-to-1 transition frequencies of 5.10 GHz and 4.74 GHz. To isolate the system and provide a means for reading out the state of the qubit readout, the device was mounted in a 3D Al microwave cavity with a TE101 mode resonance frequency of about 6.1 GHz. The flux-dependent transition frequencies of the system were measured and fit to results from a coupled Hamiltonian model. With the LC resonator tuned to its minimum resonance frequency, I observed a qubit-qubit dispersive shift of 2χ_qq≈ 0.1 MHz, which was less than the linewidth of the qubit transitions. This dispersive shift was sufficiently small to consider the coupling “off”, allowing single qubit operations. The qubit-qubit dispersive shift varied with the applied flux up to a maximum dispersive shift of 2χ_qq≈ 6 MHz. As a proof-of-principle, I present preliminary results on performing a CNOT gate operation on the qubits when the coupling was “on” with 2χ_qq≈ 4 MHz. This dissertation also includes observations of the temperature dependence of the relaxation time T1 of three Al/AlOx/Al transmons. We found that, in some cases, T1 increased by almost a factor of two as the temperature increased from 30 mK to 100 mK. We found that this anomalous behavior was consistent with loss due to non-equilibrium quasiparticles in a transmon where one electrode in the tunnel junction had a smaller volume and slightly smaller superconducting energy gap than the other electrode. At sufficiently low temperatures, non-equilibrium quasiparticles accumulate in the electrode with a smaller gap, leading to an increased density of quasiparticles at the junction and a corresponding decrease in the relaxation time. I present a model of this effect, use the model to extract the density of non-equilibrium quasiparticles in the device, and find the values of the two superconducting energy gaps.
We derive the Hamiltonian of a superconducting circuit that comprises a single-Josephson-junction flux qubit inductively coupled to an LC oscillator, and we compare the derived circuit Hamiltonian with the quantum Rabi Hamiltonian, which describes a two-level system coupled to a harmonic oscillator. We show that there is a simple, intuitive correspondence between the circuit Hamiltonian and the quantum Rabi Hamiltonian. While there is an overall shift of the entire spectrum, the energy level structure of the circuit Hamiltonian up to the seventh excited states can still be fitted well by the quantum Rabi Hamiltonian even in the case where the coupling strength is larger than the frequencies of the qubit and the oscillator, i.e., when the qubit-oscillator circuit is in the deep-strong-coupling regime. We also show that although the circuit Hamiltonian can be transformed via a unitary transformation to a Hamiltonian containing a capacitive coupling term, the resulting circuit Hamiltonian cannot be approximated by the variant of the quantum Rabi Hamiltonian that is obtained using an analogous procedure for mapping the circuit variables onto Pauli and harmonic oscillator operators, even for relatively weak coupling. This difference between the flux and charge gauges follows from the properties of the qubit Hamiltonian eigenstates.
We have experimentally studied a superconducting flux qubit strongly coupled to a single mode of a high-quality transmission-type coplanar-waveguide resonator. The qubit-resonator coupling is revealed in the resonator transmission spectrum as a vacuum Rabi splitting. In the dispersive regime the qubit energy levels are spectroscopically probed. We observe a shift in the qubit transition frequency that linearly depends on the number of photons in the resonator, which is attributed to the ac-Zeeman shift. The observations are in agreement with the theoretical predictions.
We have performed spectroscopy of a superconducting charge qubit coupled nonresonantly to a single mode of an on-chip resonator. The strong coupling induces a large ac Stark shift in the energy levels of both the qubit and the resonator. The dispersive shift of the resonator frequency is used to nondestructively determine the qubit state. Photon shot noise in the measurement field induces qubit level fluctuations leading to dephasing which is characteristic for the measurement backaction. A crossover in line shape with measurement power is observed and theoretically explained. For weak measurement a long intrinsic dephasing time of T2>200 ns of the qubit is found.
Metamaterial resonant structures made from arrays of superconducting lumped-circuit elements can exhibit microwave-mode spectra with left-handed dispersion for the standing-wave resonances, resulting in a high density of modes in the same frequency range where superconducting qubits are typically operated, as well as a bandgap at lower frequencies that extends down to dc. Using this regime for multimode circuit quantum electrodynamics, we perform a series of measurements of such a superconducting metamaterial resonator coupled to a flux-tunable transmon qubit. Through microwave measurements of the metamaterial, we observe the coupling of the qubit to each of the modes that it passes through. Using a separate readout resonator, we probe the qubit dispersively and characterize the qubit energy relaxation as a function of frequency, which is strongly affected by the Purcell effect in the presence of the dense mode spectrum. Additionally, we investigate the ac Stark shift of the qubit as the photon number in the various metamaterial modes is varied. Through numerical simulations, we explore designs based on this scheme with enhanced coupling so that the coupling energy between the qubit and the metamaterial modes can exceed the intermode spacing in future devices with achievable circuit parameters. The ability to tailor the dense mode spectrum through the choice of circuit parameters and manipulate the photonic state of the metamaterial through interactions with qubits makes this a promising platform for analog quantum simulation with microwave photons and quantum memories.