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The dc SQUID near the quantum noise limit

D. J. VanHarlingen, R. H. Koch, Jim A. Clarke

Published Jan 1, 1981• 0 citations
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Abstract

A series of dc SQUIDs has been fabricated in an attempt to obtain quantum-limited sensitivity. Typically, the SQUID inductance, L, is 2 pH, the capacitance and critical current of each tunnel junction are 0.5 pF and 0.5 mA, and the shunt resistance for each junction is 1Ω. The measured spectral density of the voltage noise Sv contains a l/f component that extends typically to 100 kHz or higher. When the l/f component is subtracted out, the best energy sensitivity achieved to date is ε1Hz = Sv2L(θVθΦ)2 less than or approximately equal to 2h, where θV/θΦ PHI is the transfer coefficient.

DOI
10.1016/0378-4363%2881%2990845-7

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A series of dc SQUIDs has been fabricated in an attempt to obtain quantum-limited sensitivity. Typically, the SQUID inductance, L, is 2 pH, the capacitance and critical current of each tunnel junction are 0.5 pF and 0.5 mA, and the shunt resistance for each junction is 1Ω. The measured spectral density of the voltage noise Sv contains a l/f component that extends typically to 100 kHz or higher. When the l/f component is subtracted out, the best energy sensitivity achieved to date is ε1Hz = Sv2L(θVθΦ)2 less than or approximately equal to 2h, where θV/θΦ PHI is the transfer coefficient.
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.
Noise characteristics of dc SQUIDs in large β regimes have been studied experimentally in the case when a SQUID inductance is resistively shunted. Here, β=2L I 0/Φ0 is the modulation parameter, L is the SQUID inductance, I 0 is the critical current of the Josephson junction and Φ0 is the flux quantum. It is shown experimentally that the performance of the SQUID does not degrade significantly even for large β values; for example, the minimum energy resolution of the SQUID is as small as ε=25h and the flux region where ε<50h is as wide as 0.1Φ0, even for the case β=12, where h is the Planck constant. The experimental results agree quantitatively with the theoretical ones.
Tunnel-junction dc SQUIDs are being fabricated from Nb films using electron-beam lithography to define linewidths as small as 0.5 μm. The barriers in the Nb-NbOx-Nb junctions are grown in an oxygen plasma discharge, and the Pd2Si resistive shunts are formed by annealing Pd on the Si substrates. These devices have excellent storage and recycling properties. At a measurement frequency of 40 kHz, a SQUID with an inductance of about 1 nH, a critical current of about 4 μA, and a parallel shunt resistance of about 15 Ω had an intrinsic energy resolution of 2.5×10−32 J Hz−1 at 4.2 K and 1.1×10−32 J Hz−1 at 1.6 K, in good agreement with a model of the dc SQUID.
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.
The operation and noise limitations of de and RF SQUID's are outlined, and recent advances in their sensitivity are discussed. A model of the dc SQUID predicts an energy noise level per hertz referred to the SQUID of approximately8 k_{B}TL/R \approx 8 k_{B}T(\piLC)^{1/2}, whereL, R,andCare the SQUID inductance and the shunt resistance and capacitance of each Josephson junction. Some examples of dc SQUID's are described to show that their performance is generally in reasonable agreement with the model. The noise energy has improved from about 2 × 10-30J. Hz-1for a device withL = 1nH and a tunnel junction area of 104µm2to about 2 × 10-33J . Hz-1for a device withL = 0.1nH and a microbridge resistance of 40 Ω. Further improvements axe expected in the near future. The model of the RF SQUID predicts a noise energy per hertz referred to the SQUID of[(\pi\alpha^{2}\phi\min{0}\max{2}/2L) + 2 \pi \alpha k_{B}T\min{a}\max{(eff)}]/\omega_{RF}, where α is the intrinsic width of the distribution of flux transitions,T\min{a}\max{(eff)}is an effective amplifier noise temperature, and ωRFis the pump frequency. With one exception, the performance of the seven types of RF SQUID listed is in reasonable agreement with the model. The noise energy ranges from about 1.5 × 10-29J . Hz-1for a 20-MHz toroidal SQUID to 3.5 × 10-31J . Hz-1for 9-GHz reentrant toroidal SQUID; a somewhat better sensitivity has been reported for a 430-MHz device, apparently in conflict with the theory. In both dc and RF SQUID's, 1/fnoise (fis frequency) is likely to extend to higher frequencies as the white-noise level is decreased.