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arXiv:1109.2490 · New Journal of Physics

Nonlinear spectroscopy of superconducting anharmonic resonators

David Peter Divincenzo, John A Smolin

Published Jan 24, 2012• 8 citations
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Abstract

We formulate a model for the steady state response of a nonlinear quantum oscillator structure, such as those used in a variety of superconducting qubit experiments, when excited by a steady, but not necessarily small, ac tone. We show that this model can be derived directly from a circuit description of some recent qubit experiments in which the state of the qubit is read out directly, without a superconducting quantum interference device (SQUID) magnetometer. The excitation profile has a rich structure depending on the detuning of the tone from the small-signal resonant frequency, on the degree of damping and on the excitation amplitude. We explore two regions in detail. Firstly, at high damping there is a trough in the excitation response as a function of detuning, near where the classical Duffing bifurcation occurs. This trough has been understood as a classical interference between two metastable responses with opposite phase. We use Wigner function studies to show that while this picture is roughly correct, there are also more quantum mechanical aspects to this feature. Secondly, at low damping we study the emergence of sharp, discrete spectral features from a continuum response. We show that these the structures, associated with discrete transitions between different excited-state eigenstates of the oscillator, provide an interesting example of a quantum Fano resonance. The trough in the Fano response evolves continuously from the 'classical' trough at high damping.

DOI
10.1088/1367-2630/14/1/013051

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We formulate a model for the steady state response of a nonlinear quantum oscillator structure, such as those used in a variety of superconducting qubit experiments, when excited by a steady, but not necessarily small, ac tone. We show that this model can be derived directly from a circuit description of some recent qubit experiments in which the state of the qubit is read out directly, without a superconducting quantum interference device (SQUID) magnetometer. The excitation profile has a rich structure depending on the detuning of the tone from the small-signal resonant frequency, on the degree of damping and on the excitation amplitude. We explore two regions in detail. Firstly, at high damping there is a trough in the excitation response as a function of detuning, near where the classical Duffing bifurcation occurs. This trough has been understood as a classical interference between two metastable responses with opposite phase. We use Wigner function studies to show that while this picture is roughly correct, there are also more quantum mechanical aspects to this feature. Secondly, at low damping we study the emergence of sharp, discrete spectral features from a continuum response. We show that these the structures, associated with discrete transitions between different excited-state eigenstates of the oscillator, provide an interesting example of a quantum Fano resonance. The trough in the Fano response evolves continuously from the 'classical' trough at high damping.
We formulate a model for the steady state response of a nonlinear quantum oscillator structure, such as those used in a variety of superconducting qubit experiments, when excited by a steady, but not necessarily small, ac tone. We show that this model can be derived directly from a circuit description of some recent qubit experiments in which the state of the qubit is read out directly, without a superconducting quantum interference device (SQUID) magnetometer. The excitation profile has a rich structure depending on the detuning of the tone from the small-signal resonant frequency, on the degree of damping and on the excitation amplitude. We explore two regions in detail. Firstly, at high damping there is a trough in the excitation response as a function of detuning, near where the classical Duffing bifurcation occurs. This trough has been understood as a classical interference between two metastable responses with opposite phase. We use Wigner function studies to show that while this picture is roughly correct, there are also more quantum mechanical aspects to this feature. Secondly, at low damping we study the emergence of sharp, discrete spectral features from a continuum response. We show that these the structures, associated with discrete transitions between different excited-state eigenstates of the oscillator, provide an interesting example of a quantum Fano resonance. The trough in the Fano response evolves continuously from the 'classical' trough at high damping.
Superconducting quantum interference devices (SQUIDs) are used across physics and engineering, from amplification to detecting tiny magnetic fields to quantum computing. In many applications the SQUID's steady-state response is of interest, but here the authors measure and model the $t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t$ response to short bursts of radiation. The resulting ringdown oscillations in voltage decay over time, with a rich structure that may be particularly useful in discriminating weak magnetic fluxes---in qubits, for example.
We develop a theory of parametric resonance in tunable superconducting cavities. The nonlinearity introduced by the superconducting quantum interference device (SQUID) attached to the cavity and damping due to connection of the cavity to a transmission line are taken into consideration. We study in detail the nonlinear classical dynamics of the cavity field below and above the parametric threshold for the degenerate parametric resonance, featuring regimes of multistability and parametric radiation. We investigate the phase-sensitive amplification of external signals on resonance, as well as amplification of detuned signals, and relate the amplifier performance to that of linear parametric amplifiers. We also discuss applications of the device for dispersive qubit readout. Beyond the classical response of the cavity, we investigate small quantum fluctuations around the amplified classical signals. We evaluate the noise power spectrum both for the internal field in the cavity and the output field. Other quantum-statistical properties of the noise are addressed such as squeezing spectra, second-order coherence, and two-mode entanglement.
We discuss the theory of linear and nonlinear spectroscopy of an rf superconducting quantum interference device (rf-SQUID) coupled to a Josephson spectrometer. Recent experimental measurements on this system have shown a strongly nonlinear absorption line shape, whose current peak maximum undergoes a forward-backward bending transition depending on the value of the rf-SQUID phase. We show that this transition can be qualitatively understood by mapping the dynamics of the driven rf-SQUID onto a generalized Duffing oscillator, with tunable drive and nonlinearity, undergoing a bifurcation. Finally we show that in order to quantitatively reproduce the experimental data reported by Griesmar et al. [Phys. Rev. Research 3, 043078 (2021)], it is crucial to include the feedback from the load line, leading to an additional source of nonlinearity.
Superconducting qubit networks (SQNs) embedded in a low-dissipative resonator is a promising device allowing one not only to establish the collective quantum dynamics on a macroscopic scale but also to greatly enhance the sensitivity of detectors of microwave photons. A quantum ac Stark effect provided by coupling between an SQN and microwave photons of a resonator, leads to a strong nonlinear interaction between photons. Here, we present a two-tone spectroscopy experiment in which a set of 10 superconducting flux qubits is coupled to the input R- resonator and the output T- transmission line. An external microwave pump field close to the resonance frequency populates macroscopically the resonator mode as a Bose-Einstein condensate, while a second probe beam scans the resonances referred also as Bogoliubov-like excitations. The corresponding excitation frequency measured from the transmission coefficient, |S21(f)| displays an abrupt change of the resonant dip position once the power of the pump field overcomes a critical value Pcr. This sharp shift occurs in a narrow region of pump frequencies, and can be tuned by an applied magnetic field. It is a signature of bistability of the photon number inside the resonator, in agreement with theory.
We report microwave transmission measurements of superconducting Fabry-Perot resonators, having a superconducting nanowire placed at a supercurrent antinode. As the plasma oscillation is excited, the supercurrent is forced to flow through the nanowire. The microwave transmission of the resonator-nanowire device shows a nonlinear resonance behavior, significantly dependent on the amplitude of the supercurrent oscillation. We show that such amplitude-dependent response is due to the nonlinearity of the current-phase relationship of the nanowire. The results are explained within a nonlinear oscillator model of the Duffing oscillator, in which the nanowire acts as a purely inductive element, in the limit of low temperatures and low amplitudes. The low-quality factor sample exhibits a ``crater'' at the resonance peak at higher driving power, which is due to dissipation. We observe a hysteretic bifurcation behavior of the transmission response to frequency sweep in a sample with a higher quality factor. The Duffing model is used to explain the Duffing bistability diagram. We also propose a concept of a nanowire-based qubit that relies on the current dependence of the kinetic inductance of a superconducting nanowire.