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IEEE Transactions on Magnetics

Quantum noise in Josephson-junction parametric amplifiers

L. S. Kuzmin, K. K. Likharev, V. V. Migulin, Alexander B Zorin

Published May 1, 1983• 9 citations
Publisher

Abstract

Crossover between thermal and quantum noise in the externally-pumped Josephson-junction parametric amplifiers is theoretically followed. Two different approaches are used for the analysis : i. Josephson junction is replaced with its RSJ model with the equilibrium source of fluctuations; ii. microscopic theory of the tunnel junctions is used, which takes into account not only thermal and quantum but also shot noise. Because of inconvenience of the noise temperature TNfor the amplifier sensitivity characterization in the quantum limit, we express our results in terms of the figure ΘNwhich is essentially the amplifier output noise energy, reduced to its input (in the classical limit,\Theta_{N}=k_{B}T_{N}). For the nondegenerate amplifier, the minimum value of ΘN. equals hω/2 and is due to quantum noise (zero-point oscillations) in the idle-frequency resonator of the amplifier. For the degenerate amplifier, sensitive to only one quadrature component of the input signal, ΘNcan be made much less than the above "quantum limit" even at operation temperatures much higher thanh\omega/k_{B}, if the Josephson junction characteristic frequency ωcis high enough,h\omega_{c}\gsimk_{B}T. Our analysis of the experimental situatlon shows that the values\Theta_{N}<h\omega/2can be achieved using two-dimensional arrays of a large number (N\simeq100) single-tunnel-junction interferometers with low inductances, inductively coupled to a microwave cavity.

DOI
10.1109/tmag.1983.1062472

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Crossover between thermal and quantum noise in the externally-pumped Josephson-junction parametric amplifiers is theoretically followed. Two different approaches are used for the analysis : i. Josephson junction is replaced with its RSJ model with the equilibrium source of fluctuations; ii. microscopic theory of the tunnel junctions is used, which takes into account not only thermal and quantum but also shot noise. Because of inconvenience of the noise temperature TNfor the amplifier sensitivity characterization in the quantum limit, we express our results in terms of the figure ΘNwhich is essentially the amplifier output noise energy, reduced to its input (in the classical limit,\Theta_{N}=k_{B}T_{N}). For the nondegenerate amplifier, the minimum value of ΘN. equals hω/2 and is due to quantum noise (zero-point oscillations) in the idle-frequency resonator of the amplifier. For the degenerate amplifier, sensitive to only one quadrature component of the input signal, ΘNcan be made much less than the above "quantum limit" even at operation temperatures much higher thanh\omega/k_{B}, if the Josephson junction characteristic frequency ωcis high enough,h\omega_{c}\gsimk_{B}T. Our analysis of the experimental situatlon shows that the values\Theta_{N}<h\omega/2can be achieved using two-dimensional arrays of a large number (N\simeq100) single-tunnel-junction interferometers with low inductances, inductively coupled to a microwave cavity.
The signal and noise properties of two basic types of externally pumped parametric amplifiers, using Josephson junctions with low capacitance, are calculated. The first type is a single- frequency nondegenerate amplifier and second one is a two-frequency degenerate amplifier, referred to sometimes as SUPARAMP. Calculations are based on the conventional RSJ model of the Josephson junction. In contrast to the works published earlier, the most realistic open circuit approximation is used for idler combinational frequencies, as well as for the pumping frequency harmonics. A single-frequency amplifier is shown to have relatively low noise temperatures (T_{N} \approx 25T) at signal frequencies of the order of the Josephson junction characteristic frequency ωc. On the other hand, SUPARAMP can have much lower noise but at lower frequencies, and it behaves approximately as a usual parametric amplifier with a critical frequency\omega_{cr} \approx 0.2 \omega_{c}.
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.
Gain and noise properties of a non-degenerate one-frequency parametric amplifier using a self-pumped Josephson junction are investigated both theoretically and experimentally. The amplifier uses the phenomenon of microwave negative resistance of the junction. Theoretical analysis made in usual resistively shunted junction (RSJ) model shows that the minimum noise temperature is equal to approximately 42T. It takes place at a signal frequency f close to the junction characteristic frequency f0. Maximum effectivity (the product of the square root of power gain and relative bandwidth) can reach 0.4 at f≃f0. Gain up to 20 dB was obtained at X-band using amplifier with Nb-Ta point-contact junction. Minimum noise temperature at T = 4.2K was about 200K.
We report the observation of zero-point noise squeezing by means of a Josephson- junction parametric amplifier, which has been used previously to squeeze 4.2 K thermal noise1. To observe quantum noise squeezing, a number of improvements were made on the apparatus; these include the in stallation of two low-noise cryogenic high- electron-mobility-transistor (HEMT) amplifiers, which boosted the detector sensitivity an order of magnitude to 215 K. The Josephson- junction parametric amplifier was cooled to 30 mK and was operated in the degenerate mode with the signal carrier frequency at 19.16 GHz. At this frequency the vacuum noise floor is hv/2k = 0.46 K. In the deamplified quadrature a drop in the noise of 0.223 K was observed relative to the pump-off noise floor. The pump-off noise floor was established to be within 3.4 + 2.0% of the vacuum noise floor by varying the temperature of the cold termination at the parametric amplifier's input from 30 mK to 1K. A probe signal at the input of the first HEMT amplifier was used to establish that the detector system saturation was less than 7 × 10−4 dB. We have thus observed squeezing 47% ± 8% below the vacuum noise floor.
A comprehensive account of the theory of the singly quasidegenerate Josephson junction parametric amplifier is given. In this mode the signal and idler frequencies are both approximately equal to half the pump frequency, and hence the signal and idler channels have a common termination. It is shown that the performance of the amplifier characterized by properties such as gain-bandwidth product, noise temperature, and saturation depends strongly on the embedding microwave circuit. In particular any uncompensated series reactance may deteriorate the overall performance. Internal shot and Johnson noise as well as incident thermal radiation is included in the discussion. It is shown that the noise temperature is virtually independent of noise injected from external sources and that minimum noise temperature is automatically achieved if the amplifier is tuned to maximize the gain-bandwidth product. The analytical results are illustrated by self-consistent numerical solutions of the circuit equations.
Parametric amplifiers have become a workhorse in superconducting quantum computing; however, research and development of these devices has been hampered by inconsistent and, sometimes, misleading noise performance characterization methodologies. The concepts behind noise characterization are deceptively simple, and there are many places where one can make mistakes, either in measurement or in interpretation and analysis. In this article, we cover the basics of noise performance characterization and the special problems it presents in parametric amplifiers with limited power handling capability. We illustrate the issues with three specific examples: a high-electron mobility transistor amplifier, a Josephson traveling-wave parametric amplifier, and a Josephson parametric amplifier. We emphasize the use of a 50-Ω shot noise tunnel junction (SNTJ) as a broadband noise source, demonstrating its utility for cryogenic amplifier amplifications. These practical examples highlight the role of loss as well as the additional parametric amplifier "idler" input mode.