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Applied Physics Letters

Quantum noise theory for the dc SQUID

R. H. Koch, Dale J. Van Harlingen, John L. Clarke

Published Mar 1, 1981• 93 citations
Publisher

Abstract

The noise temperature of a dc superconducting quantum interference device (SQUID) coupled to a tuned input circuit is computed using the complete quantum expression for the equilibrium noise in the shunt resistance of each junction. At T = 0, where the noise reduces to zero-point fluctuations, the noise temperature for an optimized system is hn/kB ln2, where n is the signal frequency. The computation is extended to nonzero temperatures, and it is shown that a SQUID operated at 1K can approach the quantum limit.

DOI
10.1063/1.92345

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IEEE Transactions on Magnetics•Mar 1, 1991•2 citations
The noise temperature of a dc superconducting quantum interference device (SQUID) coupled to a tuned input circuit is computed using the complete quantum expression for the equilibrium noise in the shunt resistance of each junction. At T = 0, where the noise reduces to zero-point fluctuations, the noise temperature for an optimized system is hn/kB ln2, where n is the signal frequency. The computation is extended to nonzero temperatures, and it is shown that a SQUID operated at 1K can approach the quantum limit.
A low-noise coupled DC superconducting quantum interference device (SQUID) especially optimized for low frequency is discussed. Using large Josephson junctions and a low loop inductance the contribution of the critical current fluctuation to 1/f noise can be minimized. To minimize the inductance the Josephson junctions are placed in the center of the washer close to the SQUID loop. A scaled-up copper model of the SQUID and the signal coil is used to analyze the impedance of the SQUID loop affected by the signal coil. An equivalent circuit model describing the effective inductance of the SQUID loop as a function of the frequency is used to design appropriate damping of the resonances. The DC SQUID characteristics are smooth and the noise performance of the SQUID does not markedly suffer from resonances. The contribution of the 1/f noise at 1 Hz is found to be about 5*10/sup -7/ Phi /sub 0// square root Hz.>
The design, fabrication, and noise properties of an ultra-low-noise tunnel junction dc SQUID (superconducting quantum interference device) are presented. The two resistively shunted Pb-alloy tunnel junctions are each 3.6 μm diam. The SQUID has an inductance L of ≈11 pH, a critical current per junction, I0 of ≈100 μA, and a resistance per shunt R of ≈2 Ω. At 2.3 K, eI0R≈kBT, and simple theory predicts that the minimum intrinsic energy sensitivity should be ≈2h. The actual minimum value of the measured high-frequency (70 kHz) flux noise at 4.2 K is φn=1.3×10−7φ0 Hz−1/2. This corresponds to an intrinsic energy sensitivity, φ2n/2L, of 3.2×10−33 J Hz−1 or 5h. Lowering the bath temperature to 1.8 K gave no improvement in performance.
The sensitivity limitations due to thermal, quantum and shot noise are calculated for dc and ac SQUIDs. Two approximations are used for the Josephson junctions: i. RSJ model with the equilibrium source of thermal-and-quantum fluctuations and ii. microscopic theory of the tunnel junctions with small capacitance (the latter theory takes into account the shot noise as well). The ultimate "output" sensitivity ε_{V}=/2L\Deltaf is shown to be substantially less than the Planck's constant h, for the both types of SQUIDs. This result does not contradict to the uncertainty principle, because the "total" sensitivity ε=(ε_{V}ε_{I}-εmin{IV}\max{2})^{1/2} rather than ε_{V} is an adequate figure of merit of SQUID as a linear amplifier. The total sensitivity is shown to be, in fact, limited by the quantum fluctuations, ε>h/2. These fluctuations originate in the shunting resistances of the dc SQUID, while in the ac SQUID they are due to quantum noise of the following stage of amplification. Nevertheless, even the total sensitivity ε can be made less than h/2, if the "degenerate" type of SQUID is used, being sensitive to the only one of the quadrature components of the input signal.
We have measured the low-frequency noise in the critical current Ic of six dc superconducting quantum interference devices (SQUIDs) with resistively shunted Nb–NbOx–PbIn Josephson junctions in the temperature range T=0.09–4.2K. Each device is voltage biased, the applied flux is an integer number of flux quanta, and the current fluctuations are measured with a second dc SQUID. At low frequencies f, there is a component of the power spectrum of the critical current fluctuations given approximately by SIc(f)=CIc2T2∕Af, where A is the area of both junctions, and C≈(3.9±0.4)×10−23m2∕K2. For quantum bits based on Josephson junctions, the scaling of SIc(f) with T2 implies that the dephasing time limited by critical current l∕f noise should scale as 1∕T for temperatures down to at least 0.09K.
The impedance and noise of a resistive superconductive quantum interference device (R-SQUID) have been measured as a function of the dc and rf currents applied to it. The Josephson junction was adjustable so that data were also taken for several values of the junction critical current. The results were compared with the predictions of a resistively shunted junction (RSJ) model which takes into account the influence of the Josephson junction on the impedance and noise. The agreement was found to be quite good and demonstrates that the noise in the circuit is well understood. Use of the R-SQUID as a noise thermometer below 1 K is assessed in terms of corrections due to the RSJ model. It is demonstrated how the dc and rf currents may be adjusted so that the total noise of the R-SQUID is reduced to within 0.1% of the Johnson noise generated by the resistor alone. Under these conditions, the R-SQUID may be used as a noise thermometer to determine thermodynamic temperature to this inaccuracy from 6 to 700 mK.
A small signal analysis for the DC SQUID (superconducting quantum interference device) is presented. This analysis involves the linearization of the SQUID equations for a given flux and current bias. Considering the SQUID to be a two-port device, an impedance matrix can be determined for a SQUID coupled to a general input circuit. The two intrinsic noise sources referred to the input are computed. One advantage of this analysis over other models of the DC SQUID is that an analytic expression can be determined for the cross-correlation power spectral density of the noise sources. These results should be useful in determining the signal-to-noise ratio of low-noise SQUIDs for a given application. Other differences and similarities of the small signal analysis to other DC SQUID models are discussed.