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arXiv:1202.3240 · New Journal of Physics

Networks of nonlinear superconducting transmission line resonators

Martin Leib, Frank Deppe, Achim Marx, R Gross, Michael J. Hartmann

Published Jul 27, 2012• 81 citations
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Abstract

We investigate a network of coupled superconducting transmission line resonators, each of them made nonlinear with a capacitively shunted Josephson junction coupling to the odd flux modes of the resonator. The resulting eigenmode spectrum shows anticrossings between the plasma mode of the shunted junction and the odd resonator modes. Notably, we find that the combined device can inherit the complete nonlinearity of the junction, allowing for a description as a harmonic oscillator with a Kerr nonlinearity. Using a dc SQUID instead of a single junction, the nonlinearity can be tuned between 10 kHz and 4 MHz while maintaining resonance frequencies of a few gigahertz for realistic device parameters. An array of such nonlinear resonators can be considered a scalable superconducting quantum simulator for a Bose–Hubbard Hamiltonian. The device would be capable of accessing the strongly correlated regime and be particularly well suited for investigating quantum many-body dynamics of interacting particles under the influence of drive and dissipation.

DOI
10.1088/1367-2630/14/7/075024

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We investigate a network of coupled superconducting transmission line resonators, each of them made nonlinear with a capacitively shunted Josephson junction coupling to the odd flux modes of the resonator. The resulting eigenmode spectrum shows anticrossings between the plasma mode of the shunted junction and the odd resonator modes. Notably, we find that the combined device can inherit the complete nonlinearity of the junction, allowing for a description as a harmonic oscillator with a Kerr nonlinearity. Using a dc SQUID instead of a single junction, the nonlinearity can be tuned between 10 kHz and 4 MHz while maintaining resonance frequencies of a few gigahertz for realistic device parameters. An array of such nonlinear resonators can be considered a scalable superconducting quantum simulator for a Bose–Hubbard Hamiltonian. The device would be capable of accessing the strongly correlated regime and be particularly well suited for investigating quantum many-body dynamics of interacting particles under the influence of drive and dissipation.
We present the Josephson junction intersected superconducting transmission line resonator. In contrast to the Josephson parametric amplifier, Josephson bifurcation amplifier and Josephson parametric converter we consider the regime of few microwave photons. We review the derivation of eigenmode frequencies and zero point fluctuations of the nonlinear transmission line resonator and the derivation of the eigenmode Kerr nonlinearities. Remarkably these nonlinearities can reach values comparable to Transmon qubits rendering the device ideal for accessing the strongly correlated regime. This is particularly interesting for investigation of quantum many-body dynamics of interacting particles under the influence of drive and dissipation. We provide current profiles for the device modes and investigate the coupling between resonators in a network of nonlinear transmission line resonators.
We provide a general method to find the Hamiltonian of a linear circuit in the presence of a nonlinearity. Focusing on the case of a Josephson junction embedded in a transmission-line resonator, we solve for the normal modes of the system by taking into account exactly the effect of the quadratic (i.e., inductive) part of the Josephson potential. The nonlinearity is then found to lead to self and cross-Kerr effects, as well as beam-splitter-type interactions between modes. By adjusting the parameters of the circuit, the Kerr coefficient KKK can be made to reach values that are weak ($K<\ensuremath{\kappa}$), strong ($K>\ensuremath{\kappa}$), or even very strong ($K\ensuremath{\gg}\ensuremath{\kappa}$) with respect to the photon-loss rate $\ensuremath{\kappa}$. In the latter case, the resonator+junction\mathrm{resonator}+\mathrm{junction}resonator+junction circuit corresponds to an in-line version of the transmon. By replacing the single junction by a SQUID, the Kerr coefficient can be tuned in situ, allowing, for example, the fast generation of Schr\"odinger cat states of microwave light. Finally, we explore the maximal strength of qubit-resonator coupling that can be reached in this setting.
In its infancy circuit quantum electrodynamics (cQED) has quickly started reproducing fundamental quantum optical experiments, e.g. observation of vacuum rabi oscillations in frequency and time domain, with unprecedented cooperativity. This was possible because of the large coupling of the quasi one-dimensional microwave field of the superconducting transmission line resonators to the macroscopic dipole moment of superconducting qubits. cQED has matured to a discipline of experimental physics capable of performing fundamental quantum information tasks and is currently on the verge of crossing the border between few- to many body physics [1]. This opens up a exciting realm of completely new physical phenomena. Because of the ubiquitous influence of the electromagnetic environment however the number of microwave photons is not conserved which separates cQED systems from other quantum simulators involving atoms, e.g. cold atoms in optical latices. Instead cQED is ideally suited for exploring quantum many-body physics in the driven dissipative regime where the interplay of constantly injecting microwave photons and the unpreventable loss of microwave photons into the electromagnetic environment generates a whole new class of steady- but not equilibrium states. We propose an array of capacitively coupled superconducting resonators each coupled to an superconducting charge qubit and show that this device can be considered as an simulator for Bose Hubbard physics combined with the high flexibility of network topologies unique to cQED systems[2]. We further investigate the driven dissipative dynamics of this device. We refine this approach and calculate the eigenmodes and nonlinearities of a Josephson junction intersected transmission line resonator[3](c.f. fig.1a). This spectrum of eigenmodes shows the ultrastrong coupling of the Josephson junction to the resonator modes and nonlinearities are attainable that are significant on the single photon limit. We now proceed one step further and and consider the incorporation of multiple Josephson junctions. A novel transfer matrix technique has to be employed to investigate the eigenmode spectrum of the device.
The high frequency properties of the one-dimensional transmission line consisting of a parallel array of resistively shunted Josephson tunnel junctions have been studied in the limit of relatively low damping where this nonlinear system exhibits new and interesting phenomena. Here we report on experimental and numerical investigations of a resonant step observed at a voltage corresponding to 600 GHz in the dc current-voltage characteristic of a parallel array of 20 identical small NbAl2O3Nb Josephson junctions interconnected by short sections of superconducting microstrip line. The junctions are mutually phase locked due to collective interaction with the line sections excited close to the half wavelength resonance. The phase locking range can be adjusted by means of an external dc magnetic field and the step size varies periodically with the magnetic field. The largest step corresponds to a superconducting phase difference of π between neighboring junctions.
We investigated parallel arrays of superconducting Nb/AlOx/Nb tunnel junctions nonevenly distributed in a superconducting Nb/SiO/Nb microstrip transmission line. Such devices are discretized Josephson transmission lines (DJTLs) in which, from theory, magnetic flux quanta (“fluxons”) can travel as solitonic waves when a dc current bias and a dc magnetic field are applied. We observed a reproducible series of resonant branches in each device’s I−V curve, at Josephson submillimeter-wave frequencies (from 240 to 720 GHz) matching the resonances predicted using a transmission line analysis, where the loading of the N=5 junctions is fully taken into account. The nonperiodic distribution was optimized to provide rf matching over a large bandwidth (450–650 GHz typically), implying that the plasma resonance of junctions is inductively tuned out over a similar band by the array. A confirmation of this comes from the observation, at frequencies higher than the untuned junctions plasma frequency, of several Josephson phenomena reported in this article: Fiske-like resonances, phase-locking of the n=3 resonance to an external 600 GHz microwave source, rf-induced zero crossing, and resonances at fractional harmonics of the rf signal. These experimental results are all compatible with a fluxon-based resonances interpretation, as in the extensively studied long Josephson junctions yet at higher frequencies. As reported elsewhere, we could detect ∼500 GHz microwave radiation emitted by our devices in the n=1 and n=3 modes. In light of these unique properties, we propose nonuniform DJTLs as a promising type of Josephson device for submillimeter-wave oscillators and fast fluxon-based electronics.
Precise control of mechanical modes in the quantum regime is a key resource for quantum technologies, offering promising pathways for quantum sensing with macroscopic systems and scalable architectures for quantum simulation. In this work, we realize a multimode mechanical cavity coupled to a superconducting Kerr resonator, which induces nonlinearity in the mechanical modes. The Kerr mode is realized by a flux-tunable superconducting quantum interference device (SQUID) array resonator, while the mechanical modes are implemented by a surface acoustic wave cavity. Both mechanical and electromagnetic modes are individually addressable via dedicated measurement lines, enabling full spectroscopic characterization. We introduce a straightforward protocol to measure the SQUID array resonator’s participation ratio in the hybrid acoustic modes, quantifying the degree of hybridization. The participation ratio reveals that our device operates at the onset of the multimode coupling regime, where multiple acoustic modes simultaneously interact with the nonlinear superconducting element. Furthermore, this platform allows controllable Kerr-type nonlinearities in multiple acoustic modes, with the participation ratio serving as the key parameter determining both the dissipation rates and nonlinear strengths of these hybridized modes. Close to the resonant regime, we measure a cross-Kerr interaction between seven pairs of mechanical modes, which is controllable via the SQUID array resonator detuning. Finally, we apply a two-photon parametric drive to the SQUID array resonator and observe the resulting parametric down-conversion and metastable state switching in a mechanical mode. These results establish a platform for engineering nonlinear multimode mechanical interactions, offering potential for future integration with superconducting qubits and implementation of multiple mechanical qubits.