We explore the role of phase fluctuations in a three-dimensional s-wave superconductor, NbN, as we approach the critical disorder for destruction of the superconducting state. Close to critical disorder, we observe a finite gap in the electronic spectrum which persists at temperatures well above T(c). The superfluid density is strongly suppressed at low temperatures and evolves towards a linear-T variation at higher temperatures. These observations provide strong evidence that phase fluctuations play a central role in the formation of a pseudogap state in a disordered s-wave superconductor.

The concept of mass-generation via the Higgs mechanism was strongly inspired by earlier works on the Meissner-Ochsenfeld effect in superconductors. In quantum field theory, the excitations of longitudinal components of the Higgs field manifest as massive Higgs bosons. The analogous Higgs mode in superconductors has not yet been observed due to its rapid decay into particle-hole pairs. Following recent theories, however, the Higgs mode should decrease below the pairing gap 2∆ and become visible in two-dimensional systems close to the superconductor-insulator transition (SIT). For experimental verification, we measured the complex terahertz transmission and tunneling density of states (DOS) of various thin films of superconducting NbN and InO close to criticality. Comparing both techniques reveals a growing discrepancy between the finite 2∆ and the threshold energy for electromagnetic absorption which vanishes critically towards the SIT. We identify the excess absorption below 2∆ as a strong evidence of the Higgs mode in two dimensional quantum critical superconductors.The Higgs mechanism, which has great implications to recent developments in particle physics [1], originates in Anderson's pioneering work on symmetry breaking with gauge fields in superconductors [2]. A superconductor spontaneously breaks continuous U (1) symmetry and acquires the well-known Mexican hat potential with a degenerate circle of minima described by the order parameter Ψ = Ae iϕ , see Fig. 1a. Excitations from the ground state can be classified as transverse Nambu-Goldstone (phase) modes and massive longitudinal Higgs (amplitude) modes (see blue and red lines in Fig. 1a). In particle physics, the latter manifest themselves as the Higgs boson which was recently discovered at CERN [3]. Indications of a Higgs mode in correlated many-body systems have been found in one-dimensional charge-densitywave systems [4], quantum antiferromagnets [5] and twodimensional superfluid to Mott transition in cold atoms [6]. An amplitude mode, also named Higgs mode, was theoretically predicted for superconductors [7] and recently reported to be measured by pump-probe spectroscopy [8]. This amplitude mode describes pairing fluctuations, which are qualitatively distinct from the purely bosonic mode expected from the O(2) field theory. The Higgs-amplitude mode analogous to the highenergy Higgs Boson has not yet been observed in superconductors. A partial reason is that in homogeneous, BCS superconductors the Higgs mode is short-lived and decays to particle hole (Bogoliubov) pairs [9,10]. Nevertheless, collective modes were recently predicted to be significant in strongly disordered superconductors [11], and, in particular it was shown [12][13][14] that the Higgs mode softens but remains sufficiently sharp near a quantum critical point (QCP) in two dimensions since it is found to be a critical energy scale of the quantum phase transition. Hence, the Higgs mass can be reduced below twice the pairing gap, 2∆, making this mode experimentally visible. Such a critical...

We have measured the magnetotransport properties and Hall effect of a series of epitaxial NbN films grown on (100) oriented single crystalline MgO substrate under different conditions using reactive magnetron sputtering. Hall effect measurements reveal that the carrier density in NbN thin films is sensitive to the growth condition. The carrier density increases by a factor of 3 between the film with highest normal state resistivity (ρ n~3 .83µΩ-m) and lowest transition temperature (T c~9 .99K) and the film with lowest normal state resistivity (ρ n~0 .94µΩm) and highest transition temperature (T c~1 6.11K) while the mobility of carriers does not change significantly. Our results show that the T c of NbN is governed primarily by the carrier density rather than disorder scattering. By varying the carrier concentration during growth we can vary the effective disorder (k F l) from the moderately clean limit to the dirty limit which makes this system ideal to study the interplay of carrier density and disorder on the superconducting properties of an s-wave superconductor.

We present an impedance engineered Josephson parametric amplifier capable of providing bandwidth beyond the traditional gain-bandwidth product. We achieve this by introducing a positive linear slope in the imaginary component of the input impedance seen by the Josephson oscillator using a λ/2 transformer. Our theoretical model predicts an extremely flat gain profile with a bandwidth enhancement proportional to the square root of amplitude gain. We experimentally demonstrate a nearly flat 20 dB gain over a 640 MHz band, along with a mean 1-dB compression point of -110 dBm and near quantum-limited noise. The results are in good agreement with our theoretical model.Josephson parametric amplifiers (JPAs) have become a crucial component of superconducting qubit 1 measurement circuitry, enabling recent studies of quantum jumps 2 , generation and detection of squeezed microwave field 3 , quantum feedback 4,5 , real-time tracking of qubit state evolution 6-8 and quantum error detection 9,10 . Although JPAs based on Josephson junctions embedded in a resonator 11-13 regularly achieve 20 dB power gain and quantum-limited noise, typical bandwidth is restricted to 10-50 MHz 11,14 , making them suitable for single qubit measurements only. The rapid progress towards multi-qubit architectures 9 for fault-tolerant quantum computing 15,16 demands an amplifier with much larger bandwidth to enable simultaneous readout of multiple qubits with minimal resources.There have been several attempts in this direction in recent years. One such attempt used a broadband impedance transformer 17 to lower the quality factor of a lumped-element Josephson oscillator which is the main component of a parametric amplifier. While the observed large bandwidth was qualitatively explained by a model consisting of a negative resistance 18 coupled to a frequency dependent impedance, no clear prescription on the design principle was provided. A different approach using Josephson non-linear transmission lines 19,20 was recently demonstrated 21,22 with nearly 4 GHz of bandwidth. However, this design requires fabrication of about 2000 nearly identical blocks of oscillator stages, demanding fairly sophisticated fabrication facilities. Multimode systems utilizing dissipative interactions have also been suggested theoretically as a route for enhancing bandwidth 23 , but have not been realized experimentally. In this Letter, we present a simple technique for enhancing the bandwidth of a JPA and beating the standard gain-bandwidth limit. It involves engineering the imaginary part of the environmental impedance: in particular, we introduce a positive linear slope in the imaginary component of the impedance shunting the JPA, while keeping the real part unchanged at the pump frequency. Our design uses a combination of a λ/4 and a λ/2 impedance transformers which are significantly easier to fabricate than a broadband impedance transformer 17 . Our theoretical model explains why the imaginary part of the impedance plays a crucial role in determining the amplifier bandw...

We present a phase diagram as a function of disorder in three-dimensional NbN thin films, as the system enters the critical disorder for the destruction of the superconducting state.The superconducting state is investigated using a combination of magnetotransport and tunneling spectroscopy measurements. Our studies reveal 3 different disorder regimes. At low disorder (k F l~10-4), the system follows the mean field Bardeen-Cooper-Schrieffer behavior where the superconducting energy gap vanishes at the temperature where electrical resistance appears. For stronger disorder ( k F l<4 ) a "pseudogap" state emerges where a gap in the electronic spectrum persists up to temperatures much higher than T c , suggesting that Cooper pairs continue to exist in the system even after the zero resistance state is destroyed. Finally, at even stronger disorder (k F l<1) the global superconducting ground state is completely destroyed, though superconducting correlations continue to survive as evidenced from a pronounced magnetoresistance peak at low temperatures. *