In the last few years, several groups have proposed and developed their own platforms demonstrating quantum-limited linear parametric amplification, with evident applications in quantum information and computation, electrical and optical metrology, radio astronomy and basic physics concerning axion detection. Here we propose a short review on the physics behind parametric amplification via metamaterials composed by coplanar wave-guides embedding several Josephson junctions. We present and compare different schemes that exploit the nonlinearity of the Josephson current-phase relation to mix the so-called signal, idler and pump tones. The chapter then presents and compares three different theoretical models, developed in the last few years, to predict the dynamics of these nonlinear systems in the particular case of a 4-Wave Mixing process and under the degenerate undepleted pump assumption. We will demonstrate that, under the same assumption, all the results are comparable in terms of amplification of the output fields.
The scattering through a Josephson junction interrupting a superconducting line is revisited including power leakage. We discuss also how to make tunable and broadband resonant mirrors by concatenating junctions. As an application, we show how to construct cavities using these mirrors, thus connecting two research fields: JJ quantum metamaterials and coupled cavity arrays. We finish by discussing the first non-linear corrections to the scattering and their measurable effects.
We have developed a recently proposed Josephson traveling-wave parametric amplifier with three-wave mixing [A. B. Zorin, Phys. Rev. Applied 6, 034006, 2016]. The amplifier consists of a microwave transmission line formed by a serial array of nonhysteretic one-junction SQUIDs. These SQUIDs are flux-biased in a way that the phase drops across the Josephson junctions are equal to 90 degrees and the persistent currents in the SQUID loops are equal to the Josephson critical current values. Such a one-dimensional metamaterial possesses a maximal quadratic nonlinearity and zero cubic (Kerr) nonlinear…
We report on the microwave characterization of a novel one-dimensional Josephson metamaterial composed of a chain of asymmetric superconducting quantum interference devices (SQUIDs) with nearest-neighbor coupling through common Josephson junctions. This metamaterial demonstrates a strong Kerr nonlinearity, with a Kerr constant tunable over a wide range, from positive to negative values, by a magnetic flux threading the SQUIDs. The experimental results are in good agreement with the theory of nonlinear effects in Josephson chains. The metamaterial is very promising as an active medium for Josep…
We report on experiments with superconducting metamaterials containing Josephson junctions. In these structures, split-ring resonators used in conventional metamaterials are replaced by superconducting loops that are interrupted by Josephson junctions, so called rf-SQUIDs. Like the split-ring resonators, these elements can be seen as LC-resonators that couple to the magnetic field. The advantage of superconducting thin-film metamaterials is that, due to the tunable intrinsic inductance of the Josephson junction, the resonance frequency of the rf-SQUID can be changed by applying an external dc …
We consider the propagation of a classical electromagnetic wave through a transmission line, formed by identical superconducting charge qubits inside a superconducting resonator. Since the qubits can be in a coherent superposition of quantum states, we show that such a system demonstrates interesting new effects, such as a ``breathing'' photonic crystal with an oscillating bandgap, and a ``quantum Archimedean screw'' that transports, at an arbitrary controlled velocity, Josephson plasma waves through the transmission line. The key ingredient of these effects is that the optical properties of t…
We demonstrate a low-dissipation dielectric metamaterial with tunable properties based on the Josephson effect. Superconducting wires loaded with regularly spaced Josephson junctions ($I_c \approx 0.25$ $μ$A) spanning a K-band waveguide and aligned with the microwave electric fields create a superconducting dielectric metamaterial. Applied dc current tunes the cutoff frequency and effective permittivity of this unique electric metamaterial. The results are in agreement with an analytical model for microwave transmission through the artificial dielectric medium.
We present a detailed experimental and theoretical analysis of the dispersion and non-linear Kerr frequency shifts of plasma modes in a one-dimensional Josephson junction chain containing 500 SQUIDs 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 renormalisation 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 measur…
We report low temperature, microwave transmission measurements on a new switchable and tunable class of nonlinear metamaterials. A wireless two dimensional array of Josephson junctions (JJ) is probed as a metamaterial where each plaquette in the array is considered as a meta-atom. In the presence of microwaves, this compact metamaterial of 30,000 connected meta-atoms synchronizes the flow of Cooper pairs to yield a single robust resonant signal with a quality factor of 2800 at the lowest temperature for our measurements. The transmission signal is switched on and off and its amplitude and freq…
A quantum model for Josephson-based metamaterials working in the Three-Wave Mixing (3WM) and Four-Wave Mixing (4WM) regimes at the single-photon level is presented. The transmission line taken into account, namely Traveling Wave Josephson Parametric Amplifier (TWJPA), is a bipole composed by a chain of rf-SQUIDs which can be biased by a DC current or a magnetic field in order to activate the 3WM or 4WM nonlinearities. The model exploits a Hamiltonian approach to analytically determine the time evolution of the system both in the Heisenberg and interaction pictures. The former returns the analy…
Transparency-engineered SQUID cells for Kerr-free three-wave-mixing Josephson metamaterials
Università degli Studi di Salerno（萨莱诺大学）；INFN, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；Consiglio Nazionale delle Ricerche, CNR-SPIN（意大利国家研究委员会 SPIN 研究所）；QuantWare（QuantWare 公司）；University of Sannio（桑尼奥大学）
We present results of numerical simulations of a metamaterial element (meta-atom) for planar negative media based on cuprate Josephson junctions. The proposed circuit consists of two coupled resonators: "magnetic dipole" and "electric dipole". The first one is a superconducting spiral resonator with multijunction superconducting structure - SQIF (SQIF - superconducting quantum interference filter), and the second one is a comb-shape resonator made from normal metal film. We show, that in comparison with passive metal resonators, the superconducting resonator with Josephson junctions (magnetic …
Manipulating the propagation of electromagnetic waves through sub-wavelength sized artificial structures is the core function of metamaterials. Resonant structures, such as split ring resonators, play the role of artificial "atoms" and shape the magnetic response. Superconducting metamaterials moved into the spotlight for their very low ohmic losses and the possibility to tune their resonance frequency by exploiting the Josephson inductance. Moreover, the nonlinear nature of the Josephson inductance enables the fabrication of truly artificial atoms. Arrays of such superconducting quantum two-l…
When a soliton propagates in a discrete lattice it excites small-amplitude linear waves in its wake. In a dc current-biased Josephson-junction (JJ) array these manifest as electromagnetic (EM) waves excited by a (magnetic field induced) chain of propagating magnetic vortices. When the vortex velocity and the phase velocity of one of the excited EM waves match, phase-locking occurs. This produces resonant steps in the current-voltage characteristics where amplification of EM radiation occurs. We report the first observation of phase-locking-induced amplification of EM radiation at 77K and above…
We develop the concept of fractal metamaterials which consist of arrays of nano and micron sized rings containing Josephson junctions which play the role of "atoms" in such artificial materials. We show that if some of the junctions have $π$-shifts in the Josephson phases that the "atoms" become magnetic and their arrays can have tuned positive or negative permeabilty. Each individual "$π$-ring" - the Josephson ring with one $π$-junction - can be in one of two energetically degenerate magnetic states in which the supercurrent flows in the clockwise or counter-clockwise direction. This results …
We have realized a microwave quantum-limited amplifier that is directional and can therefore function without the front circulator needed in many quantum measurements. The amplification takes place in only one direction between the input and output ports. Directionality is achieved by multi-pump parametric amplification combined with wave interference. We have verified the device noise performances by using it to readout a superconducting qubit and observed quantum jumps. With an improved version of this device, qubit and preamplifer could be integrated on the same chip.
It has long been recognized that a control current $I_a$ injected into the section of a two-junction superconducting quantum interference device (SQUID) is able to produce a change of its critical current $I_c$, so that a current gain $g=|dI_c/dI_a|$ can be identified. We investigate the circumstances under which large gains can be achieved by using vertical Josephson interferometers which are characterized by small loop inductances. We discuss the theory of operation of such a novel device, its performances and its advantages with respect to planar interferometers used in the previous works. …
We investigate the response of a stack of intrinsic Josephson junctions (IJJs) to terahertz (THz) electromagnetic (EM) irradiation. A significant amplification of the EM wave can be achieved by the IJJs stack when the incident frequency equals to one of the cavity frequencies. The irradiation excites pi phase kinks in the junctions, which stimulate the cavity resonance when the bias voltage is tuned. A large amount of dc energy is then pumped into the Josephson plasma oscillation, and the incident wave gets amplified. From the profound current step in IV characteristics induced at the cavity r…
We report the efficient coupling of a $50\,Ω$ microwave circuit to a high impedance conductor. We use an impedance transformer consisting of a $λ/4$ co-planar resonator whose inner conductor contains an array of superconducting quantum interference devices (SQUIDs), providing the resonator with a large and tunable lineic inductance $\mathcal{L}\sim 80 μ_0$, resulting in a large characteristic impedance $Z_C\sim 1\,\mathrm{k}Ω$. The impedance matching efficiency is characterized by measuring the shot noise power emitted by a dc biased high resistance tunnel junction connected to the resonator. …
Recently Yaakobi and co-workers [Phys. Rev. B 87, 144301 (2013)] theoretically studied four-wave mixing and parametric amplification in a nonlinear transmission line consisting of capacitively shunted Josephson junctions. By deriving and solving the coupled-mode equations, they have arrived at the conclusion that in a wide frequency range around the pump frequency exponential parametric gain (in which the signal grows exponentially with distance) can be achieved. However, we have found a mathematical error in their derivation of the coupled-mode equations (Equation (A13)), which leads to the w…
We observe parametric amplification of electromagnetic (EM) waves produced by a flux-flow oscillator made of YBa2Cu3O7 Josephson junctions arrays coupled to the resonant modes of a millimeter wave Fabry-Perot resonator at a pump frequency fP=45 GHz. For temperatures in the range (30-45) K the frequency fS of the EM signal to be amplified could be tuned continuously in the range (1-25) GHz by an applied B-field induced flux with a one-flux-quantum periodicity. Consequently, we measured a significant parametric gain that is almost frequency independent, with a maximum of (8-10.4) dB reached at 4…
Josephson parametric converters (JPCs) are superconducting devices capable of performing nondegenerate, three-wave mixing in the microwave domain without losses. One drawback limiting their use in scalable quantum architectures is the large footprint of the auxiliary circuit needed for their operation, in particular, the use of off-chip, bulky, broadband hybrids and magnetic coils. Here, we realize a JPC which eliminates the need for these bulky components. The pump drive and flux bias are applied in the new device through an on-chip, lossless, three-port power divider and on-chip flux line, r…
Recent progress in solid state quantum information processing has stimulated the search for ultra-low-noise amplifiers and frequency converters in the microwave frequency range, which could attain the ultimate limit imposed by quantum mechanics. In this article, we report the first realization of an intrinsically phase-preserving, non-degenerate superconducting parametric amplifier, a so far missing component. It is based on the Josephson ring modulator, which consists of four junctions in a Wheatstone bridge configuration. The device symmetry greatly enhances the purity of the amplification p…
It has recently become possible to encode the quantum state of superconducting qubits and the position of nanomechanical oscillators into the states of microwave fields. However, to make an ideal measurement of the state of a qubit, or to detect the position of a mechanical oscillator with quantum-limited sensitivity requires an amplifier that adds no noise. If an amplifier adds less than half a quantum of noise, it can also squeeze the quantum noise of the electromagnetic vacuum. Highly squeezed states of the vacuum serve as an important quantum information resource. They can be used to gener…
Photon-pressure coupling between two superconducting circuits is a promising platform for investigating radiation-pressure coupling in novel parameter regimes and for the development of radio-frequency (RF) quantum photonics and quantum-limited RF sensing. So far, the intrinsic Josephson nonlinearity of photon-pressure coupled circuits has not been considered a potential resource for enhanced devices or novel experimental schemes. Here, we implement photon-pressure coupling between a RF circuit and a microwave cavity containing a superconducting quantum interference device (SQUID) which can be…
We review progress in the development and applications of superconducting metamaterials. The review is organized in terms of several distinct advantages and unique properties brought to the metamaterials field by superconductivity. These include the low-loss nature of the meta-atoms, their compact structure, their extraordinary degree of nonlinearity and tunability, magnetic flux quantization and the Josephson effect, quantum effects in which photons interact with quantized energy levels in the meta-atom, as well as strong diamagnetism.
One of the central challenges in the development of parametric amplifiers is the control of the dynamic range relative to its gain and bandwidth, which typically limits quantum limited amplification to signals which contain only a few photons per inverse bandwidth. Here, we discuss the control of the dynamic range of Josephson parametric amplifiers by using Josephson junction arrays. We discuss gain, bandwidth, noise, and dynamic range properties of both a transmission line and a lumped element based parametric amplifier. Based on these investigations we derive useful design criteria, which ma…
We have developed a Josephson parametric amplifier, comprising a superconducting coplanar waveguide resonator terminated by a dc SQUID (superconducting quantum interference device). An external field (the pump, $\sim 20$ GHz) modulates the flux threading the dc SQUID, and, thereby, the resonant frequency of the cavity field (the signal, $\sim 10$ GHz), which leads to parametric signal amplification. We operated the amplifier at different band centers, and observed amplification (17 dB at maximum) and deamplification depending on the relative phase between the pump and the signal. The noise tem…
We have prepared meta-atoms based on radio frequency superconducting quantum interference devices (RF SQUIDs) and examined their tunability with dc magnetic field, rf current, and temperature. RF SQUIDs are superconducting split ring resonators in which the usual capacitance is supplemented with a Josephson junction, which introduces strong nonlinearity in the rf properties. We find excellent agreement between the data and a model which regards the Josephson junction as the resistively and capacitively-shunted junction. A magnetic field tunability of 80 THz/Gauss at 12 GHz is observed, a total…
We present a lumped-element Josephson parametric amplifier designed to operate with strong coupling to the environment. In this regime, we observe broadband frequency dependent amplification with multi-peaked gain profiles. We account for this behaviour using the "pumpistor" model which allows for frequency dependent variation of the external impedance. Using this understanding, we demonstrate control over gain profiles through changes in the environment impedance at a given frequency. With strong coupling to a suitable external impedance we observe a significant increase in dynamic range, and…
We report the parametric amplification of a microwave signal in a Kerr medium formed from superconducting qubits. Two mutually coupled flux qubits, embedded in the current antinode of a superconducting coplanar waveguide resonator, are used as a nonlinear element. Shared Josephson junctions provide the qubit-resonator coupling, resulting in a device with a measured gain of about 20 dB. We argue, that this arrangement represents a unit cell which can be straightforwardly extended to a quasi one-dimensional quantum metamaterial with a large tunable Kerr nonlinearity.
Non-reciprocal devices, which have different transmission coefficients for propagating waves in opposite directions, are crucial components in many low noise quantum measurements. In most schemes, magneto-optical effects provide the necessary non-reciprocity. In contrast, the proof-of-principle device presented here, consists of two on-chip coupled Josephson parametric converters (JPCs), which achieves directionality by exploiting the non-reciprocal phase response of the JPC in the trans-gain mode. The non-reciprocity of the device is controlled in-situ by varying the amplitude and phase diffe…
The Bloch-oscillating transistor (BOT) is an amplifier that utilizes semiclassical dynamics of states in energy bands under traveling quasimomentum. In a BOT, a single quasiparticle tunneling across a base tunnel junction switches the state of a superconducting tunnel junction to a lower Bloch band, triggering a series of resonant Cooper-pair tunnelings through an emitter Josephson junction (Bloch oscillations). Here, we investigate experimentally and theoretically an alternative realization of this device, based only on superconducting tunnel junctions. We discover new amplification schemes, …
Microwave parametric amplifiers based on Josephson junctions have become a key component of many quantum information experiments. One key limitation which has not been well predicted by theory is the gain saturation behavior which determines its ability to process large amplitude signals. The typical explanation for this behavior in phase-preserving amplifiers based on three-wave mixing is pump depletion, in which the consumption of pump photons to produce amplification results in a reduction in gain. However, in this work we present experimental data and theoretical calculations showing that …
The advent of ultra-low noise microwave amplifiers revolutionized several research fields demanding quantum-limited technologies. Exploiting a theoretical bimodal description of a linear phase-preserving amplifier, in this contribution we analyze some of the intrinsic properties of a model architecture (i.e., an rf-SQUID based Josephson Traveling Wave Parametric Amplifier) in terms of amplification and noise generation for key case study input states (Fock and coherents). Furthermore, we present an analysis of the output signals generated by the parametric amplification mechanism when thermal …
Evanescent wave amplification has been predicted under the ideal condition n = -1 + i 0 precisely, in the absence of retardation effects, but is difficult to observe in practice because current metamaterial designs suffer from high losses. We present results on a new metamaterial design that employs superconducting Niobium metals and low-loss dielectric materials to reduce the losses. We present transmission data on a wire medium, a split-ring resonator medium, and a combination of the two at temperatures between 4.2 K and room temperature. Evidence of negative effective permittivity, permeabi…
The low-noise amplification of weak microwave signals is crucial for countless protocols in quantum information processing. Quantum mechanics sets an ultimate lower limit of half a photon to the added input noise for phase-preserving amplification of narrowband signals, also known as the standard quantum limit (SQL). This limit, which is equivalent to a maximum quantum efficiency of $0.5$, can be overcome by employing nondegenerate parametric amplification of broadband signals. We show that, in principle, a maximum quantum efficiency of 1 can be reached. Experimentally, we find a quantum effic…
We present the driven response at T=30mK of 6 GHz superconducting resonators constructed from capacitively-shunted three dimensional (3D) aluminum nanobridge superconducting quantum interference devices (nanoSQUIDs). We observe flux modulation of the resonant frequency in quantitative agreement with numerical calculation and characteristic of near-ideal short weak link junctions. Under strong microwave excitation, we observe stable bifurcation in devices with coupled quality factor (Q) ranging from ~30-3500. Near this bias point, parametric amplification with > 20dB gain, 40 MHz bandwidth, and…
Parametrically amplified Josephson plasma waves in YBa_2Cu_3O_(6+x): evidence for local superconducting fluctuations up to the pseudogap temperature $T^*$
研究通过强太赫兹脉冲作用于欠掺杂YBa₂Cu₃O₆₊x并在Tc和伪间隙尺度T*之间温度下，发现类似超导态的反射率边缘和探针脉冲的二次谐波产生，提出通过参数放大低约瑟夫森等离子体模式解释，但本文提出另一种解释，假设在Tc < T < T*时存在局部配对幅度和相位，从而解释反射率数据。
We realize and characterize a quantum-limited, directional Josephson amplifier suitable for qubit readout. The device consists of two nondegenerate, three-wave-mixing amplifiers that are coupled together in an interferometric scheme, embedded in a printed circuit board. Nonreciprocity is generated by applying a phase gradient between the same-frequency pumps feeding the device, which plays the role of the magnetic field in a Faraday medium. Directional amplification and reflection-gain elimination are induced via wave interference between multiple paths in the system. We measure and discuss th…
We have constructed a new type of amplifier whose primary purpose is the readout of superconducting quantum bits. It is based on the transition of an RF-driven Josephson junction between two distinct oscillation states near a dynamical bifurcation point. The main advantages of this new amplifier are speed, high-sensitivity, low back-action, and the absence of on-chip dissipation. Using pulsed microwave techniques, we demonstrate bifurcation amplification in nanofabricated Al junctions and verify that the performance predicted by theory is attained.
Metamaterials (MMs), i.e. artificial media designed to achieve properties not available in natural materials, have been the focus of intense research during the last two decades. Many properties have been discovered and multiple designs have been devised that lead to multiple conceptual and practical applications. Superconducting MMs have the advantage of ultra low losses, a highly desirable feature. The additional use of the Josephson effect and SQUID configurations produce further specificity and functionality. SQUID-based MMs are both theoretically investigated but also fabricated and analy…
We demonstrate that parametric driving of suitable collective modes in cuprate superconductors results in a reflectivity $R>1$ for frequencies in the low terahertz regime. We propose to exploit this effect for the amplification of coherent terahertz radiation in a laser-like fashion. As an example, we consider the optical driving of Josephson plasma oscillations in a monolayer cuprate at a frequency that is blue-detuned from the Higgs frequency. Analogously, terahertz radiation can be amplified in a bilayer cuprate by driving a phonon resonance at a frequency slightly higher than the upper Jos…
We present the design, measurement and analysis of a current sensor based on a process of Josephson parametric upconversion in a superconducting microwave cavity. Terminating a coplanar waveguide with a nanobridge constriction Josephson junction, we observe modulation sidebands from the cavity that enable highly sensitive, frequency-multiplexed output of small currents for applications such as transition-edge sensor array readout. We derive an analytical model to reproduce the measurements over a wide range of bias currents, detunings and input powers. Tuning the frequency of the cavity by mor…
Josephson tunnel junctions are widely used as nonlinear elements in superconducting circuits such as low noise amplifiers and quantum bits. However, microscopic defects in the oxide tunnel barrier can produce low and high frequency noise which can potentially limit the coherence times and quality factors of resonant circuits. Weak link Josephson junctions are an attractive alternative provided that sufficient nonlinearity can be engineered. We compute the current phase relation for superconducting weak links, with dimensions comparable to the zero temperature coherence length, connected to two…
We present experimental results on a Josephson parametric amplifier tailored for readout of ultra-sensitive thermal microwave detectors. In particular, we discuss the impact of fabrication details on the performance. We show that the small volume of deposited dielectric materials enabled by the side-wall passivated spacer niobium junction technology leads to robust operation across a wide range of operating temperatures up to 1.5 K. The flux-pumped amplifier has gain in excess of 20 dB in three-wave mixing and its center frequency is tunable between 540 MHz and 640 MHz. At 600 MHz, the amplifi…
Many applications in photonics require all-optical manipulation of plasma waves, which can concentrate electromagnetic energy on sub-wavelength length scales. This is difficult in metallic plasmas because of their small optical nonlinearities. Some layered superconductors support weakly damped plasma waves, involving oscillatory tunneling of the superfluid between capacitively coupled planes. Such Josephson plasma waves (JPWs) are also highly nonlinear, and exhibit striking phenomena like cooperative emission of coherent terahertz radiation, superconductor-metal oscillations and soliton format…
Natural atomic superlattices are formed in certain strongly anisotropic layered compound. Here we study interlayer transport in single crystals of Bi2Sr2CaCu2O8 cuprates, which represent stacks of atomic scale intrinsic Josephson junctions. A series of resonant dips in conductance is observed at condition when bremsstrahlung and recombination bands in non-equilibrium spectrum of Josephson junctions overlap. The phenomenon is explained in terms of self-detection of a new type of collective strongly non-equilibrium state in stacked Josephson junctions, bearing certain resemblance with operation …
