We report on the realization of a superinductor, a dissipationless element whose microwave impedance greatly exceeds the resistance quantum RQ. The design of the superinductor, implemented as a ladder of nanoscale Josephson junctions, enables tuning of the inductance and its nonlinearity by a weak magnetic field. The Rabi decay time of the superinductor-based qubit exceeds 1 µs. The high kinetic inductance and strong nonlinearity offer new types of functionality, including the development of qubits protected from both flux and charge noises, fault tolerant quantum computing, and high-impedance isolation for electrical current standards based on Bloch oscillations. PACS numbers: 74.50.+r, 74.81.Fa, 85.25.Am Superinductors are desired for the implementation of the electrical current standards based on Bloch oscillations [1,2], protection of Josephson qubits from the charge noise [3,4], and fault tolerant quantum computation [5,6]. The realization of superinductors poses a challenge. Indeed, the "geometrical" inductance of a wire loop is accompanied by a sizable parasitic capacitance, and the loop impedance Z does not exceed αR Q [7], where α = 1/137 is the fine structure constant and2 is the resistance quantum. This limitation does not apply to superconducting circuits whose kinetic inductance L K is associated with the inertia of the Cooper pair condensate [8].The kinetic inductance of a Josephson junction,2 /E J , scales inversely with its Josephson energy E J [8] (Φ 0 = h/2e is the flux quantum). The kinetic inductance can be increased by reducing the inplane junction dimensions and, thus, E J . However, this resource is limited: with shrinking the junction size, the charging energy E C = e 2 /2C (C is the junction capacitance) increases and the phase-slip rategrows exponentially, which leads to decoherence. Small Josephson energy (i.e., large kinetic inductance) can be realized in chains of dc SQUIDs frustrated by the magnetic field [9,10]. However, the phase-slip rate increases greatly with frustration, and the chains do not provide good isolation from the environment. For the linear chains of Josephson junctions with E J /E C ≫ 1, relatively large values of L K (up to 300 nH [3]) have been realized in the phase-slip-free regime; further increase of the impedance of these chains is hindered by the growth of their parasitic capacitance. Also, the linear chains, as well as the nanoscale superconducting wires with a kinetic inductance of ∼ 10 nH/µm [11,12], are essentially linear elements whose inductance is not readily tunable (unless large currents are applied).We propose a novel superinductor design that has several interesting features. This circuit can be continuously tuned by a weak magnetic field between the regimes characterized by a low linear inductance and a very large nonlinear inductance. Importantly, the large impedance Z ≫ R Q is realized when the decoherence processes associated with phase slips are fully suppressed. This combination of strong nonlinearity and low decoherence rate is an asset ...

We have studied the low-energy excitations in a minimalistic protected Josephson circuit which contains two basic elements (rhombi) characterized by the π periodicity of the Josephson energy. Novel design of these elements, which reduces their sensitivity to the offset charge fluctuations, has been employed. We have observed that the life time T 1 of the first excited state of this quantum circuit in the protected regime is increased up to 70µs, a factor of ∼ 100 longer than that in the unprotected state. The quality factor ω 01 T 1 of this qubit exceeds 10 6 . Our results are in agreement with theoretical expectations; they demonstrate the feasibility of symmetry protection in the rhombi-based qubits fabricated with existing technology.Quantum computing requires the development of quantum bits (qubits) with a long coherence time and the ability to manipulate them in a fault tolerant manner (see, e.g. [1] and references therein). Both goals can be achieved by the realization of a protected logical qubit formed by a collective state of an array of faulty qubits [2][3][4][5][6][7]. The building block (i.e. the faulty qubit) of the array is the Josephson element with an effective Josephson energy E(φ) = −E 2 cos(2φ), which is π -periodic in the phase difference φ across the element. In contrast to the conventional Josephson junctions with E(φ) = −E 1 cos(φ), this element supports the coherent transport of pairs of Copper pairs (the "4e" transport), whereas single Cooper pairs are localized and the "2e" transport is blocked [5,8,9]. Though this proposal has attracted considerable theoretical attention [10], the experimental realization of a protected qubit was lacking.In this Letter we make an essential step towards building a protected Josephson qubit by fabricating the simplest protected circuit and demonstrating that the first excited state of the circuit is protected from energy relaxation.The idea of protection is illustrated in Fig.1. Let us consider the simplest chain of two cos(2φ) elements. They share the central superconducting island whose charge is controlled by the gate. The Hamiltonian of this quantum circuit can be written as(1) where the energy E 2 describes the Josephson coupling of the central superconducting island to the current leads, E C is the effective charging energy of the island, n is the number of Cooper pairs on the island, n g is the charge induced on the island by the gate. The parity of n is preserved if the transfer of single Cooper pairs is blocked (E 1 = 0). In this case the states of the system can be characterized by the quantum number ℵ = n mod(2). The low energy states corresponding to number ℵ = 0, 1 are shown in Fig.1. The energy E 2 plays the role of the kinetic term that controls the "spread" of the wave functions along the n axis. Provided E 2 E C , the number of components with different n in these discrete Gaussian wavefunctions is large: n 2 = E 2 /E C 1, and the energy difference between the two states, E 01 = E |1 − E |0 , is exponentially small (see [11] and Suppleme...

We investigate competition between one-and two-dimensional topological excitations-phase slips and vortices-in the formation of resistive states in quasi-two-dimensional superconductors in a wide temperature range below the mean-field transition temperature T C0 . The widths w = 100 nm of our ultrathin NbN samples are substantially larger than the Ginzburg-Landau coherence length = 4 nm, and the fluctuation resistivity above T C0 has a two-dimensional character. However, our data show that the resistivity below T C0 is produced by one-dimensional excitations-thermally activated phase slip strips ͑PSSs͒ overlapping the sample cross section. We also determine the scaling phase diagram, which shows that even in wider samples the PSS contribution dominates over vortices in a substantial region of current and/or temperature variations. Measuring the resistivity within 7 orders of magnitude, we find that the quantum phase slips can only be essential below this level.The nature of the resistive state in superconductors attracts much attention from the physics community, since it involves fundamental phenomena and advanced concepts 1,2 such as the mechanisms of high-T c superconductivity, 3 thermal fluctuations, 4,5 macroscopic quantum tunneling, 6-8 coherence, 9 topological excitations, 10 and phase disordering. 11 Resistive states are used in a number of quantum nanodevices, such as logic elements, 12 ultrasensitive detectors of radiation, single-photon counters, and nanocalorimeters. 13,14 Understanding resistive states in nanoscale superconductors is critical for the advancement of fundamental science and the development of novel applications.Phase slips and vortices are elementary topological excitations which create resistive states. 1,2,10 Wires with radius less than the coherence length or stripes with w Ͻ are one-dimensional ͑1D͒ superconductors. In 1D structures, the resistive state is produced by phase slips and is well described by the Langer-Ambegaokar-McCumber-Halperin ͑LAMH͒ theory of thermally activated phase slips ͑TAPSs͒. 1,2,4,5 At low enough temperatures, the quantum phase slips ͑QPSs͒ should be important, 6 but the magnitude of this effect and the characteristic resistance at the transition from TAPSs to QPSs are still under debate. 7,8 In two-dimensional ͑2D͒ superconductors, the resistive state is formed by moving vortices. Above the BerezinskiiKosterlitz-Thouless ͑BKT͒ transition temperature T C , there is a nonzero concentration of free vortices due to thermal unbinding of vortex-antivortex pairs ͑VAPs͒. 15,16 Below T C in an infinite 2D superconductor, VAPs are tightly bound and only a significant bias current can unbind the pairs, resulting in a nonlinear flux flow resistance. In finite size samples, free vortices can exist below T C and produce a linear resistance at low bias currents. 15 It is commonly believed that the transition from 1D phase slip excitations to 2D vortex physics takes place at w / ϳ 1. 1,2,7,12,17 However, despite thorough studies of phase slip and vortex mechanisms, an in...

We develop and characterize a symmetry-protected superconducting qubit that offers simultaneous exponential suppression of energy decay from charge and flux noises, and dephasing from flux noise. The qubit consists of a Cooper-pair box (CPB) shunted by a superinductor, forming a superconducting loop. Provided the offset charge on the CPB island is an odd number of electrons, the qubit potential corresponds to that of a cos (φ/2) Josephson element, preserving the parity of fluxons in the loop via Aharonov-Casher interference. In this regime, the logical-state wavefunctions reside in disjoint regions of Hilbert space, thereby ensuring protection against energy decay. By switching the protection on, we observe a tenfold increase of the decay time, reaching up to 100 μs. Though the qubit is sensitive to charge noise, the sensitivity is much reduced in comparison with the charge qubit, and the charge-noise-induced dephasing time of the current device exceeds 1 μs. Implementation of full dephasing protection can be achieved in the next-generation devices by combining several cos(φ/2) Josephson elements in a small array.

We have observed the effect of the Aharonov-Casher (AC) interference on the spectrum of a superconducting system containing a symmetric Cooper pair box (CPB) and a large inductance. By varying the charge ng induced on the CPB island, we observed oscillations of the device spectrum with the period ∆ng = 2e. These oscillations are attributed to the charge-controlled AC interference between the fluxon tunneling processes in the CPB Josephson junctions. The measured phase and charge dependences of the frequencies of the |0 → |1 and |0 → |2 transitions are in good agreement with our numerical simulations. Almost complete suppression of the tunneling due to destructive interference has been observed for the charge ng = e(2n + 1). The CPB in this regime enables fluxon pairing, which can be used for the development of parity-protected superconducting qubits.The Aharonov-Casher (AC) effect is a non-local topological effect: the wave function of a neutral particle with magnetic moment moving in two dimensions around a charge acquires a phase shift proportional to the charge [1]. This effect has been observed in experiments with neutrons, atoms, and solid-state semiconductor systems (see, e.g., [2][3][4] and references therein). Similar effects have been predicted for superconducting networks of nanoscale superconducting islands coupled by Josephson junctions. For example, the wave function of the flux vortices (fluxons) moving in such a network should acquire a phase that depends on the charge on superconducting islands [5]. Indeed, oscillations of the network resistance in the flux-flow regime have been observed as a function of the gate-induced island charge [6]; these oscillations have been attributed to the interference associated with the AC phase. However, this attribution is not unambiguous, because qualitatively similar phenomena can be produced by the Coulomb-blockade effect due to the quantization of charge on the superconducting islands [7].More recently, indirect evidence for the AC effect in superconducting circuits has been obtained in the study of suppression of the macroscopic phase coherence in onedimensional (1D) chains of Josephson junctions by quantum fluctuations [8]. The quantum phase slips (QPS) in the junctions can be viewed as the charge-sensitive fluxon tunneling [9,10] provided the conditions discussed below are satisfied. Microwave experiments [11] have demonstrated that dephasing of a fluxonium, a small Josephson junction shunted by a 1D Josephson chain, can be due to the effect of fluctuating charges on the QPS in the chain. Applications of the AC effect in classical Josephson devices have been discussed in Refs. [7,12].In this Letter we describe microwave experiments which provide direct evidence for the charge-dependent interference between the amplitudes of fluxon tunneling. We have studied the microwave resonances of the device consisting of two nominally identical Josephson junctions separated by a nanoscale superconducting island (the socalled Cooper-pair box, CPB) and a large inductanc...