arXiv:1807.01499 · Physical review. B./Physical review. B
Kerr nonlinearity in a superconducting Josephson metamaterial
Y. Krupko, V. D. Nguyen, Thomas Weißl, Étienne Dumur, Javier Puertas, Rémy Dassonneville, Cécile Naud, F. W. J. Hekking, D. M. Basko, O. Buisson, Nicolas Roch, Wiebke Guichard
We present a detailed experimental and theoretical analysis of the dispersion and nonlinear Kerr frequency shifts of plasma modes in a one-dimensional Josephson junction chain containing 500 superconducting quantum interfence devices in the regime of weak nonlinearity. The measured low-power dispersion curve agrees perfectly with the theoretical model if we take into account the Kerr renormalization of the bare frequencies and the long-range nature of the island charge screening by a remote ground plane. We measured the self- and cross-Kerr shifts for the frequencies of the eight lowest modes in the chain. We compare the measured Kerr coefficients with theory and find good agreement.
We present a detailed experimental and theoretical analysis of the dispersion and nonlinear Kerr frequency shifts of plasma modes in a one-dimensional Josephson junction chain containing 500 superconducting quantum interfence devices in the regime of weak nonlinearity. The measured low-power dispersion curve agrees perfectly with the theoretical model if we take into account the Kerr renormalization of the bare frequencies and the long-range nature of the island charge screening by a remote ground plane. We measured the self- and cross-Kerr shifts for the frequencies of the eight lowest modes in the chain. We compare the measured Kerr coefficients with theory and find good agreement.
We present an experimental and theoretical analysis of the self- and cross-Kerr effect of extended plasma resonances in Josephson junction chains. We calculate the Kerr coefficients by deriving and diagonalizing the Hamiltonian of a linear circuit model for the chain and then adding the Josephson nonlinearity as a perturbation. The calculated Kerr coefficients are compared with the measurement data of a chain of 200 junctions. The Kerr effect manifests itself as a frequency shift that depends linearly on the number of photons in a resonant mode. By changing the input power on a low signal level, we are able to measure this shift. The photon number is calibrated using the self-Kerr shift calculated from the sample parameters. We then compare the measured cross-Kerr shift with the theoretical prediction, using the calibrated photon number.
Metamaterials are artificially designed materials to exhibit properties not available in natural materials by engineering the constituting elements at sub-wavelength scales, enabling unique interactions of matter with electromagnetic waves or coupling modes of electromagnetic waves. Examples of such novel optics phenomenons include negative index of refraction, cloaking, and super-resolution imaging. The study of superconducting metamaterials operating in the microwave frequency domain has recently generated significant interest in the framework of quantum technologies due to a wide range of direct applications, e.g., exploration of novel quantum optics phenomena, non-destructive quantum sensing, quantum-limited amplification, quantum information processing, and quantum illumination. In contrast to their normal metal and semiconducting counterparts, superconducting metamaterials offer some unique advantages. The most relevant of these for Josephson junction based metamaterials are low losses, flux quantization, and Josephson effects. The Josephson effect in a superconducting loop can also be exploited to exact more control over the nonlinearities of the meta-material.In this thesis, we present Josephson junction based metamaterials with in-situ tunability of second and third-order (Kerr) nonlinearities. We optimized the Josephson junction composites SNAILs, well-known in the superconducting circuits community, for the specific use case of building blocks for an impedance-matched nonlinear meta-material. In such devices, the strong third-order (Kerr) nonlinearity can be tuned from positive to negative values. To demonstrate the efficacy of this phenomenon, we have shown a use case of the meta-material as a TWPA with a new phase-matching mechanism: reversed Kerr phase matching, which utilizes the sign reversal of the Kerr nonlinearity to correct for the dispersion curvature of such devices arising due to its finite plasma. The meta-material operated as reversed Kerr TWPA exhibits up to 4 GHz wide combined amplification band with 8 GHz dynamic band tunability, -98 dBm saturation (1 dB compression point) around 20 dB gain, and amplifier added noise close to the standard quantum limit. The absence of any gaps in the transmission can provide a continuous amplification band with significantly lower gain ripples in contrast to the previously demonstrated TWPAs with dispersion engineering.We also present the generation of entanglement utilizing the metamaterials to demonstrate their versatility and further emphasize their potential in the general framework of nonlinear microwave quantum optics. Due to their large bandwidth, these devices have been identified as promising for generating two-mode squeezing and broadband entanglement. With a device optimized for the generation of broadband entanglement, we observed entanglement between two modes separated by up to 400 MHz. We measured logarithmic negativity between 0.27 and 0.51, and the collective quadrature squeezing below the vacuum limit between 1.5 and 2.1 dB. The presented proof-of-principle results can pave the way towards the implementation of Josephson junction based metamaterials in novel applications, including non-destructive single-photon detection, single-photon measurement and control in c-QED, photon-based quantum computing, precision sensing, the detection of dark matter, and quantum illumination.
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 K 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 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.
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.
This thesis presents the results of the experimental studies on two kindof Superconducting circuits: one-dimensional Josephson junction chains andsuperconducting coplanar waveguide (CPW) resonators. One-dimensionalJosephson junction chains are constructed by connecting many Superconducting quantum interference devices (SQUIDs) in series. We have studied DC transport properties of the SQUID chains and model their nonlineardynamics with Thermally Activated Phase-Slips (TAPS). Experimental andsimulated results showed qualitative agreement revealing the existence of auniform phase-slipping and phase-sticking process which results in a voltage-independent current on the dissipative branch of the current-voltage char-acteristics (IVC). By modulating the effective Josephson coupling energy ofthe SQUIDs (EJ ) with an external magnetic field, we found that the ratio EJ /EC is a decisive factor in determining the qualitative shape of theIVC. A quantum phase transition between incoherent Quantum Phase Slip, QPS (supercurrent branch with a finite slope) to coherent QPS (IVC withwell-developed Coulomb blockade) via an intermediate state (supercurrentbranch with a remnant of Coulomb blockade) is observed as the EJ /EC ratio is tuned. This transition from incoherent QPS to the intermediate-statehappens around R0 ∼ RQ (RQ = h/4e^2 = 6.45kΩ). We also fabricated structured chains where a SQUID at the middle of the chain (central SQUID) has different junction size and loop area compared to other SQUIDs in the chain. Results showed that with these structured chains it is possible to localize andtune the amplitude of both TAPS and QPS at the central SQUID. The second part of the thesis describes the fabrication process and themeasurement results of superconducting CPW resonators. Resonators withdifferent design parameters were fabricated and measured. The transmissionspectra showed quality factors up to, Q ∼ 5 × 10^5 . We have observed bendingof the resonance curves to the lower frequencies due to existence of a nonlinear kinetic inductance. The origin of the nonlinear kinetic inductance isthe nonlinear relation between supercurrent density, Js, and superfluid veloc-ity, vs , of the charge carriers on the center line of the resonators. A simplemodel based on the Ginzburg-Landau theory is used in order to explain ob-served nonlinear kinetic inductance and estimates using this model showedgood agreement with the experimental results.
Josephson junction (JJ) is a key nonlinear element in superconducting devices such as qubits, amplifiers, and bolometers. Recently, gate-tunable JJs based on graphene and semiconductors have gained interest due to their rich Andreev physics and wide applications in circuit quantum electrodynamics devices. In addition to gate tunability, it offers many advantages over conventional JJs, such as exceptional thermal properties for bolometric sensors, magnetic-field compatibility, and operability at elevated temperatures above 1 K. Like conventional Al–AlOx–Al JJs, graphene JJs also act as nonlinear inductors, and at their heart lies the Kerr nonlinearity. Additionally, in graphene JJs, the nonlinearity is tunable via external knobs in a single device. However, a detailed experimental exploration of the tunable Kerr nonlinearity in graphene JJs has never been performed. In this work, we study the dependence of the Kerr nonlinearity on gate voltage, temperature, and DC bias—an interesting knob that has been less explored. Using these parameters, we show that the magnitude of the Kerr coefficient can be tuned over a wide range, from ∼200 kHz to 1.2 MHz. Our work will be a valuable resource for further understanding of the nonlinearity in graphene JJs and for the design of next-generation amplifiers and sensors, with enhanced performance.