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Journal of Applied Physics

Some analytical and intuitive results in the quantum theory of mixing

M. J. Feldman

Published Jan 1, 1982• 25 citations
Publisher

Abstract

This paper is an elaboration upon Tucker’s quantum theory of mixing, which has proven very valuable in interpreting recent experiments in superconducting quasiparticle mixing. Tucker’s formula for the conversion gain of a three-frequency quantum mixer in the low intermediate frequency (IF) limit is generalized to include arbitrary source reactance; one choice of source reactance is found to resonate out the quantum reactance in the conversion expression. The signal reflection gain is evaluated; it becomes infinite simultaneously with the IF conversion gain. Infinite gain occurs only within the region of negative dc differential resistance. A number of relationships among the complex elements of the small-signal admittance matrix is presented; the nonlinear reactive elements are shown to be small under certain conditions when the conversion gain is maximized. The local oscillator (LO) power required for quantum mixing is calculated and found to be related to the gain denominator. This implies that certain quantum mixers at high gain will exhibit a constant gain-powerwidth product and a gain-proportional noise temperature component. A particularly simple expression for the conversion gain of an optimized quantum mixer is developed, which agrees with previous experiments. The origin of nonclassical behavior, including conversion gain and the nonlinear quantum reactance, is discussed. We conclude that the nonlinear quantum reactance, though it certainly must exist, is not itself responsible for conversion gain and has a minor, perhaps insignificant, effect upon the properties of a quantum mixer.

DOI
10.1063/1.329923

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A quantum-statistical analysis of the signal and noise performance of the quasiparticle (SIS) heterodyne mixers is presented. In contrast with the earlier theories, the quantum properties of the signal source have been taken into consistent consideration. In the quantum range of sensitivity the noise properties of the mixer can be characterized by the noise parameter ΘNintroduced earlier for the parametric amplifiers. Using the conventional 3-port model of the mixer circuit and the microscopic theory of the superconducting tunnel junctions we have obtained a general expression for ΘNand evaluated it numerically for various quasiparticle current step widths, dc bias voltages, local oscillator powers, signal frequencies ωs, signal source admittances Ysand operation temperatures T. In a reasonable range of these parameters,\Theta_{N}^{DSB}has turned out to be less than the so-called quantum limith\omega_{s}/2. On the contrary, in the single sideband mode of the mixer performance, ΘNis always larger thanh\omega_{s}/2. The difference between the two modes of operation is discussed from the point of view of the quantum theory of measurements.

IEEE Transactions on Magnetics•Mar 1, 1985•19 citations
This paper is an elaboration upon Tucker’s quantum theory of mixing, which has proven very valuable in interpreting recent experiments in superconducting quasiparticle mixing. Tucker’s formula for the conversion gain of a three-frequency quantum mixer in the low intermediate frequency (IF) limit is generalized to include arbitrary source reactance; one choice of source reactance is found to resonate out the quantum reactance in the conversion expression. The signal reflection gain is evaluated; it becomes infinite simultaneously with the IF conversion gain. Infinite gain occurs only within the region of negative dc differential resistance. A number of relationships among the complex elements of the small-signal admittance matrix is presented; the nonlinear reactive elements are shown to be small under certain conditions when the conversion gain is maximized. The local oscillator (LO) power required for quantum mixing is calculated and found to be related to the gain denominator. This implies that certain quantum mixers at high gain will exhibit a constant gain-powerwidth product and a gain-proportional noise temperature component. A particularly simple expression for the conversion gain of an optimized quantum mixer is developed, which agrees with previous experiments. The origin of nonclassical behavior, including conversion gain and the nonlinear quantum reactance, is discussed. We conclude that the nonlinear quantum reactance, though it certainly must exist, is not itself responsible for conversion gain and has a minor, perhaps insignificant, effect upon the properties of a quantum mixer.
The quantum theory of mixing is essential for the interpretation of experimental superconducting quasiparticle (SIS) mixers. When the bias and tuning parameters are known, the theory is quite successful at predicting the mixer performance, gain and noise temperature. However, the theory usually predicts much higher conversion gain for some other set of bias and tuning parameters, and this is not seen in experiment. Thus the theory is less useful for engineering design of SIS mixers. We believe we have resolved this longstanding discrepancy. We find there are two distinct optimum operation modes for ideal SIS mixers. The high gain mode predicted by the theory will not be accessible in real experiments. To understand this we present the equations of a DSB SIS mixer in a novel form, in terms of the complex plane phase angle between the amplitude modulation input admittance vector and the local oscillator input admittance vector.
We have measured the properties of a superconductor-insulator-superconductor quasiparticle mixer which is operated in the quantum limit. Single sideband conversion gain larger than +4 dB was observed at 36 GHz with a mixer noise temperature T = 9±6 K, which is to be compared with the (Planck) quantum limit h/ω/(k ln 2)≃2.5 K. Complete three-port mixer calculations are presented which are in good agreement with the gain measurements. The dynamic resistance was observed to become infinite and then negative as the source conductance was decreased. This implies that arbitrarily large gain is available. The negative resistance is accompanied by IF oscillations.
We have investigated high-frequency limits of superconducting quasiparticle mixers from experiments on Al/Al-oxide-In/Al (or Ag) tunnel junctions, and from calculations based on Tucker’s quantum theory of mixing (QTM). The small superconducting energy gap of Al enabled us to study mixer performance at around the junction gap frequency, fg =(Δ1+Δ2)/h≊87 GHz, of superconductor-insulator-superconductor (SIS) mixers, and at twice the junction gap frequency for S-I-normal metal (SIN) mixers ( fg =Δ1/h) with available millimeter wave sources. Because of the low transition temperature of Al (Tc ≊1.3 K), a dilution refrigerator was used for providing temperatures below 1 K. The best mixer conversion efficiency was obtained for a low-resistance (∼34 Ω) six-junction array SIS mixer operated at f/fg ≊1 (in this case 73 GHz). The coupled single sideband conversion gain (with no image rejection) was −13.8 dB and the available conversion gain was estimated to −8.4 dB. Complicating nonequilibrium phenomena made it difficult to compare the performance of these relatively high-current-density mixers to the QTM. However, good agreement between calculated and corresponding measured conversion efficiencies and dynamic resistance values was obtained for the low-current-density mixers. Noise due to switching between rf-induced Josephson steps and interference from Josephson mixing could be quenched by a magnetic field. Several high-frequency limits are considered in this paper (such as Josephson switching noise, nonequilibrium phenomena, pair breaking, and energy gap limitations). Our results indicate that submillimeter wave SIS mixer receivers are likely to perform well, at least up to their gap frequencies, i.e., the THz region for large gap superconductors. Our QTM calculations showed that conversion gain larger than unity should be possible up to twice the gap frequency. Above this frequency the conversion loss will increase rapidly.
An analysis of the frequency dependence of superconductor-insulator-normal (SIN) and superconductor-insulator-superconductor (SIS) quasiparticle mixers is presented. Power-law expressions for conversion loss and mixer shot-noise temperature for the double-sideband SIN mixer are derived from the Tucker theory for the case of a source conductance which is small compared to the conductance of the junction. For an ideal SIN tunnel junction at T=0 the mixer conversion efficiency is shown to decrease approximately linearly with frequency up to f=Δ/h. From f=Δ/h to 2Δ/h the conversion efficiency remains approximately constant, while above f=2Δ/h it rolls off as the inverse square of the frequency. Expressions for the shot-noise contribution to the mixer noise temperature are also derived. At frequencies up to f=Δ/h the noise temperature rises as the square root of the frequency. From f=Δ/h to f=2Δ/h the noise temperature of the mixer increases linearly with frequency, whereas above f=2Δ/h it rises as the cube of the frequency. Conversion efficiency and noise temperature are also calculated numerically for the SIN mixer. Good agreement is found between the frequency dependencies calculated analytically in the limit of small local oscillator power and the numerical calculations for optimal pumping. The frequency dependence of the nonideal SIS junction is also analyzed, and shown to yield similar results, with the characteristic frequencies Δ/h and 2Δ/h for a SIN mixer transformed into 2Δ/h and 4Δ/h, respectively. Expressions also are derived for the saturation power as a function of frequency.
Calculations based on the quantum theory of mixing in single-particle tunnel junctions show that there is a fairly simple strategy for optimizing the performance of double-sideband superconductor-insulator-superconductor (SIS) quasiparticle mixers. The best mixer noise temperature is obtained when the signal source is matched to the local oscillator (LO) admittance of the junction. This applies over a very wide range of LO and DC bias conditions. These calculations support the contention that it is the energy dissipation in the device which is important in determining the noise performance, not the small signal admittance or the power gain. This appears to be another demonstration of the Callen and Welton fluctuation-dissipation theorem. which states that it is the dissipation of energy which is responsible for the noise generation in a wide range of devices, and it is this energy dissipation mechanism to which the signal should be coupled to minimize the noise.
A quantum-statistical analysis of the signal and noise performance of the quasiparticle (SIS) heterodyne mixers is presented. In contrast with the earlier theories, the quantum properties of the signal source have been taken into consistent consideration. In the quantum range of sensitivity the noise properties of the mixer can be characterized by the noise parameter ΘNintroduced earlier for the parametric amplifiers. Using the conventional 3-port model of the mixer circuit and the microscopic theory of the superconducting tunnel junctions we have obtained a general expression for ΘNand evaluated it numerically for various quasiparticle current step widths, dc bias voltages, local oscillator powers, signal frequencies ωs, signal source admittances Ysand operation temperatures T. In a reasonable range of these parameters,\Theta_{N}^{DSB}has turned out to be less than the so-called quantum limith\omega_{s}/2. On the contrary, in the single sideband mode of the mixer performance, ΘNis always larger thanh\omega_{s}/2. The difference between the two modes of operation is discussed from the point of view of the quantum theory of measurements.