The performance of a great variety of electronic devices-ranging from semiconductor transistors to superconducting qubits-is hampered by low-frequency noise with spectra proportional to 1/f. The ubiquity and negative impact of 1/f noise has motivated intensive research into its cause, and it is now believed to originate from a bath of fluctuating two-level defect states (TLSs) embedded in the material. This phenomenon is commonly described by the long-established standard tunnelling model (STM) of independent TLS. A key prediction of STM is that the noise should vanish at low temperatures. Here we report measurements on superconducting microresonators over previously unattainable, very long time scales that show an increase in 1/f noise at low temperatures and low microwave power, contrary to the STM. We propose a new generalised tunnelling model that includes significant interaction between multiple TLSs, which fully describes these observations, as well as recent studies of individual TLS lifetimes in superconducting qubits.

Little-Parks oscillations are observed in a system of 110 series-connected aluminum rings 2 µm in diameter with the use of measuring currents from 10 nA to 1 µA. The measurements show that the amplitude and character of the oscillations are independent of the relation between the measuring current and the amplitude of the persistent current. By using asymmetric rings, it is demonstrated that the persistent current has clockwise or contra-clockwise direction. This means that the total current in one of the semi-rings may be directed against the electric field at measurement of the Little-Parks oscillations. The measurements at zero and low measuring current have revealed that the persistent current, like the conventional circulating current, causes a potential difference on the semi-rings with different cross sections in spite of the absence of the Faraday's voltage.

The obtained periodic magnetic-field dependencies of the critical current Ic+(Φ/Φ0), Ic−(Φ/Φ0), measured in opposite directions on asymmetric superconducting aluminum rings, allow to explain observed earlier quantum oscillations of a dc voltage as a result of alternating current rectification. It is found, that the high efficiency of the rectification of both individual rings and ring systems is connected to a hysteresis of the current-voltage characteristics. The asymmetry of the currentvoltage characteristics providing the rectification effect is due to the relative shifts of the magnetic dependencies Ic−(Φ/Φ0) = Ic+(Φ/Φ0 + ∆φ) of the critical current measured in opposite directions. This shift means that position of Ic+(Φ/Φ0) and Ic−(Φ/Φ0) minimums does not correspond to n+0.5 magnetic flux quantum Φ0 which is in the direct contradiction with measurement results of the Little-Parks resistance oscillations. Despite of this contradiction the amplitude of the critical current anisotropy oscillations Ic,an(Φ/Φ0) = Ic+(Φ/Φ0) − Ic−(Φ/Φ0) and its variations with temperature correspond to expected amplitude of the persistent current oscillations and to its variations with temperature.

Magnetic field dependences of critical current, resistance, and rectified voltage of asymmetric (half circles of different widths) and symmetrical (half circles of equal widths) aluminum rings were measured in the temperature region close to the superconducting transition. All these dependences are periodic magnetic field functions with periods corresponding to the flux quantum in the ring. The periodic dependences of critical current measured in opposite directions were found to be close to each other for symmetrical rings and shifted with respect to each other by half the flux quantum in asymmetric rings with ratios between half circle widths of from 1.25 to 2. This shift of the dependences by a quarter of the flux quantum as the ring becomes asymmetric makes critical current anisotropic, which explains the rectification effect observed for asymmetric rings. Shifts of the extrema of the periodic dependences of critical current by a quarter of the flux quantum contradict directly to the results obtained by measuring asymmetric ring resistance oscillations, whose extrema are, as for symmetrical rings, observed at magnetic fluxes equal to an integer and a half of flux quanta.

A possibility to observe the persistent voltage in a superconducting ring of different widths of the arms is experimentally investigated. It was earlier found that switching of the arms between superconducting and normal states by ac current induces the dc voltage oscillation in magnetic field with a period corresponding to the flux quantum inside the ring. We use systems with a large number of asymmetric rings connected in series in order to investigate the possibility to observe this quantum phenomenon near the superconducting transition where thermal fluctuations switch ring segments without external influence and the persistent current is much smaller than in the superconducting state.

Experimental results obtained last years corroborate a prediction made by I.O. Kulik forty years ago that the energy dissipation does not result in disappearance of equilibrium circular current observable in the normal state of superconductor rings and normal metal rings. Contrary interpretations of the persistent current as a Brownian motion or a dissipationless current are compared in the point of view of the observations of this phenomenon at presence of an electric potential difference. Distinctions between the quantum phenomena at atomic and mesoscopic levels are accentuated. In connection of the quantum oscillations in magnetic field of potential difference observed on asymmetric rings with the persistent current, it is pointed out that an experimental check of such phenomenon at thermodynamic equilibrium is possible.Key words: mesoscopic quantum phenomena, a persistent current IntroductionThe experimental results obtained last years show that predictions of the persistent current circulating in a ring with nonzero resistance, made by I.O. Kulik forty years ago [1,2], may have fundamental importance. In the first work [1], obviously initiated by the well-known AslamazovLarkin theory [3] of fluctuation superconductivity, it has been shown that the persistent current can be observed not only in a superconducting state when the electric resistance is equal to zero, but also in the normal state when the resistance is not equal to zero. It was shown in the second work [2] that the persistent current state is possible without the superconducting long-rang order and consequently this quantum phenomenon can be observed in normal metal.The possibility of the persistent current state is connected with the quantization rp = nh of the angular momentum rp, postulated by Bohr as far back as 1913 for the description of stability of electron orbits in atom. The permitted states of a free (not dissipating) electron being in an onedimensional (with small section of the circle s) ring with radius r should be discrete as well as in atomic orbits. Because of the relation p = mv + qA between velocity v and canonical momentum p in the presence of a magnetic vector potential A, the permitted velocity

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We study operation of a new device, the superconducting differential double contour interferometer (DDCI), in application for the ultra sensitive detection of magnetic flux and for digital read out of the state of the superconducting flux qubit. DDCI consists of two superconducting contours weakly coupled by Josephson Junctions. In such a device a change of the critical current and the voltage happens in a step-like manner when the angular momentum quantum number changes in one of the two contours. The DDCI may outperform traditional Superconducting Quantum Interference Devices when the change of the quantum number occurs in a narrow magnetic field region near the half of the flux quantum due to thermal fluctuations, quantum fluctuations, or the switching a loop segment in the normal state for a while by short pulse of an external current. Higher sensitivity of DDCI compared to the conventional SQUID is provided by strong discreteness of the energy spectrum of the continuous superconducting loop [6]. According to the conventional theory [7] the total energy of the persistent currentin a loop with small cross section s ≪ λ 2 L (T ) is determined mainly by the kinetic energy [8]:L /s)L is the kinetic inductance of the loop of side l; L ≈ µ 0 l is the magnetic inductance; s is the cross section of superconducting wires; n s is the density of the Cooper pairs;is the London penetration depth. Two permitted states n and n + 1 have equal energy at Φ = (n + 0.5)Φ 0 according to (2) and thus equal probability P ∝ exp(−E k /k B T ). The probability of other permitted states is negligible and P (n+1) ≈ 1−P (n) at Φ ≈ (n+0.5)Φ 0 when Φ 2 0 /2L k = I p,A Φ 0 ≫ k B T . The probability of the n state may be described with the relationat the magnetic flux Φ = (n + 0.5)Φ 0 + δΦ, when δΦ ≪ Φ 0 . Here I p,A = Φ 0 /2L k is the persistent current (1) at |n − Φ/Φ 0 | = 1/2 and ǫ = I p,A Φ 0 /k B T . The probability (3) changes from P (n) ≈ 1 to P (n) ≈ 0 in a region of the magnetic flux from δΦ ≈ −Φ 0 /2ǫ to δΦ ≈ Φ 0 /2ǫ. This region may be very narrow due to a big value ǫ = I p,A Φ 0 /k B T ≫ 1 equal, for example ǫ ≈ 1500 at the temperature of measurement T ≈ 1 K and a typical value I p,A = 10 µA measured, for example in [9]. Measurements [10] of flux qubit (superconducting loop with three