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Journal of Applied Physics

Theory of signal and noise performance of superconducting quantum interferometers with two quantization loops

Ya. S. Greenberg, G.S. Krivoy, H. Koch

Published May 1, 1995• 4 citations
Publisher

Abstract

The superconducting quantum interferometer with two quantization loops, the so-called double SQUID, or D-SQUID, is a modification of the well-known rf SQUID. The modification consists of a replacement of the one junction of the rf SQUID by a superconducting loop with two Josephson junctions. This device can be used to transform the weak magnetic flux applied to the loop with junctions to a voltage across the tank circuit inductively coupled to the other loop. As has previously been shown in experiments [G. S. Krivoy and H. Koch, J. Appl. Phys. 74, 2925 (1993)], D-SQUIDs have an extremely large output signal swing up to 1 mV and a transfer function of several mV/(h/2e). In this paper, a D-SQUID theory using a numerical and an analytical approach is developed. The theoretical results are compared with the above-mentioned experiments.

DOI
10.1063/1.359556

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IEEE Transactions on Applied Superconductivity•Jun 1, 2007•2 citations
The superconducting quantum interferometer with two quantization loops, the so-called double SQUID, or D-SQUID, is a modification of the well-known rf SQUID. The modification consists of a replacement of the one junction of the rf SQUID by a superconducting loop with two Josephson junctions. This device can be used to transform the weak magnetic flux applied to the loop with junctions to a voltage across the tank circuit inductively coupled to the other loop. As has previously been shown in experiments [G. S. Krivoy and H. Koch, J. Appl. Phys. 74, 2925 (1993)], D-SQUIDs have an extremely large output signal swing up to 1 mV and a transfer function of several mV/(h/2e). In this paper, a D-SQUID theory using a numerical and an analytical approach is developed. The theoretical results are compared with the above-mentioned experiments.
Conventional rf SQUIDs (superconducting quantum interference devices) consist of a superconducting loop interrupted by one weak link, and when read out via a tank circuit they produce output signals in the order of a few tens of microvolts. When the single weak link is replaced by a dc SQUID (a superconducting loop with two junctions) a so-called double quantization loop SQUID is formed. In this paper it is shown theoretically, and supported experimentally with thin-film devices of three different designs, that the general behaviour can be explained by a simplified picture of the fixed critical current of the single junction of the rf SQUID being replaced by an effective critical current of a dc SQUID loop that is varied by the applied magnetic flux (i.e., we have a rf SQUID with a variable β parameter). A more detailed analysis of the experimental data and the theory of this novel device reveals its interesting nonlinear character and shows the deviations from this simple picture. It is demonstrated experimentally that output signals in the order of ≥1 mV can be obtained with a suitable read-out mode for the tank circuit, a clear advantage when compared to conventional rf SQUIDs. With respect to dc SQUIDs, the coupling to integrated planar input coils will be unproblematic since no microwave resonances should occur in the structures.
The dc SQUID had been presented for use in the rf SQUID instead of a weak link. The new device, referred to as “double SQUID”, possesses a large output signal (hundreds of microvolts) in operating in a hysteretic mode. For making an operating mode a double SQUID is coupled to a circuit traditional for rf SQUIDs containing a tank circuit and an rf current pumping generator. The magnetic flux being measured is recognized by the dc SQUID quantization loop which results in changing its critical current. As a result the height of the flat part of the tank circuit I–V characteristics coupled to a double SQUID is modulated. It is this modulation that is the SQUID output signal. The experimental investigations of the double SQUID showed the validity of the assumptions under consideration. Output signals up to 690 μV, noise spectral density ≈2×10−5ϕ0/ Hz 1/2 (ϕ0 is the flux quantum) and energy resolution ≈1.4×10−29 J/Hz have been obtained.
A technique for the analysis of complex superconducting circuits containing one or more current biased inductively coupled Josephson junction interferometers is described. The equations of motion for the system are shown to separate into a set of non-linear junction equations relating the voltage drops across the junctions to the currents passing through the junctions, and a set of linear equations expressing the quantization of the fluxoid around the superconducting loops and the corresponding linear circuit equations for the normal loops. In many cases, the linear equations can be integrated, resulting in an effective elimination of the normal circuit loops from the problem. The circuit is thus reduced to that of an equivalent set of coupled SQUIDs whose device parameters are a function of the original circuit parameters and the inductive coupling strengths. The dc SQUID tuned linear amplifier is analyzed to display the method. As a result of this analysis, the optimal noise temperature of a tuned SQUID amplifier is derived as a function of the current and voltage noise spectral densities, Svand Sj, and forward transfer function, Vφ. The results obtained demonstrate that the behavior of the SQUID cannot be isolated from that of the rest of the circuit in the manner assumed in previous analyses. The existence of a quantum noise limit on the noise temperature of any phase preserving linear amplifier is used to infer that the noise figures must satisfy the constraint[(S_{v}/2LV_{\phi}^{2})(LS_{j}/2)-S_{vj}^{2}]^{1/2}>\hbarThe behavior of several simple circuits containing a pair of inductively coupled current biased interferometers is discussed to display the extension of the method to circuits containing more than one interferometer.
We present an improved analytic theory, numerical simulation, and analysis of noise characteristics of a bi-SQUID in comparison with those of a dc SQUID in an open loop configuration. The analytic theory which had been developed earlier, neglecting a pulse component of the difference of the phases of Josephson junctions, is now completed taking into account the pulse component. In the bi-SQUID, the additional Josephson junction introduces another source of fluctuations and changes its transfer function, nonlinear dynamics, and the noise spectrum transformation. Some increase in the reduced-to-input noise at low values of applied magnetic flux comes from the nonlinear flux to phase difference transformation that was introduced in bi-SQUID as a way to linearize its voltage response.
Relaxation oscillation superconducting quantum interference devices (SQUIDs) (ROSs) and double relaxation oscillation SQUIDs (DROSs) have been fabricated and characterized. The SQUIDs are based on hysteretic Nb/Al,AlOx/Al/Nb Josephson tunnel junctions. The relaxation oscillations were simulated by computer. By these simulations the minimum and maximum values of the shunt inductance at which the relaxation oscillations persist have been determined for different values of the shunt resistance and of the critical current and capacitance of the junctions. The transfer from flux to dc voltage in ROSs increases linearly with the shunt resistance. The operation range of the shunt inductance, however, becomes smaller with increasing shunt resistance. In ROSs with a SQUID inductance of 20 pH, a maximum voltage modulation width of 400 μV and a flux-to-voltage transfer δV/δφ of 4 mV/φ0 have been obtained. Two ROSs have been operated in a simple flux-locked loop with direct voltage readout. The experimental white flux noise spectral density of about 3μφ0/√Hz is in good agreement with the theoretical model. The sensitivity of these SQUIDs is limited by so-called switching noise, which is noise due to the interaction between the relaxation oscillations and the Josephson frequency. In DROSs typical flux-to-voltage transfer values of 10–30 mV/φ0 and maximum values up to 80 mV/φ0 have been obtained. This transfer is determined by thermal noise on the critical currents of the SQUIDs. The effective flux noise density of DROSs in flux-locked-loop operation with direct voltage readout is in good agreement with the theoretical noise level which is mainly determined by thermal noise on the critical currents. In one of the DROSs with a SQUID inductance of 20 pH, a white flux noise of 1μφ0/√Hz has been obtained at an estimated relaxation frequency of about 80 MHz. This corresponds to an energy sensitivity of about 160 h.
We report on experiments performed on a system consisting of a double SQUID (Superconducting Quantum Interference Device) built with gradiometer geometry. Two single-turn coils provide two independent control fluxes: one of these allows biasing the device and tilting the potential, while the other changes the barrier height of the potential. When the dynamics of the inner dc SQUID can be neglected, the free energy of the double SQUID, as a function of the internal magnetic flux, is just the corrugated parabola of an rf SQUID whose local minima represent metastable states for the system. Our analysis instead is substantially concerned with the interesting phenomenology generated by the static configurations of an internal two-junction interferometer and by the tunability of the internal loop inductance. Two readout systems are employed to thoroughly characterize the dynamics of our system. We investigate the dynamical response at temperatures low enough (tens of mK) to minimize the effects of thermal fluctuations concentrating the analysis on the aspects that could be relevant for macroscopic quantum coherence and computing. The results indicate that from the finite inductance of the inner loop originates a potential well generating competing processes with the tunneling between the two main wells of the rf-SQUID potential.