Thermally-fluctuated single-flux-quantum pulse intervals reflected in input-output characteristics of a double-flux-quantum amplifier

A double-flux-quantum amplifier (DFQA) is a voltage multiplier of quantum accuracy, which we have employed at the final stage of a single-flux-quantum (SFQ) digital-to-analog converter (DAC). We recently found that experimental input–output (IO) characteristics of DFQAs were always slightly different from numerical results assuming ideally-periodic SFQ pulse trains. That is, experimental IO characteristics obtained using an over-biasing method were gradually deteriorated near their maximum operation voltages. Numerical simulation including the over-biasing method at a finite temperature suggested that the difference was likely to be attributed to thermally-fluctuated intervals of input SFQ pulses.


Introduction
Over-biasing a Josephson junction beyond its critical current is a simple method for feeding SFQ pulse trains into Josephson circuits, which we refer to as an "over-biasing method." We have employed the over-biasing method for evaluation of double-flux-quantum amplifiers (DFQAs) [1], especially for determination of their maximum input voltage V IN−MAX and the corresponding Josephson frequency f IN−MAX = V IN−MAX /Φ 0 , where Φ 0 is a flux quantum [2,3]. We have recently found that there is always slight difference between the experimental and numerical input-output (IO) characteristics near V IN−MAX [4], for which the input SFQ pulses are fed by the over-biasing method in experiments and by periodic current pulses in simulation.
In this paper, we present experimental IO characteristics of a 100-fold DFQA and compare them with numerical results assuming different input methods of SFQ pulses. Numerical simulation including the over-biasing method at a finite temperature suggests that thermallyfluctuated pulse intervals gradually deteriorate IO characteristics of the DFQA near V IN−MAX when the over-biasing method is employed for feeding SFQ pulses.

Experiments
We designed a 100-fold DFQA including 99-stacked three-junction loops (3JLs). The circuit configuration with the measurement set-up is shown in figure 1(a). In experiments, the input SFQ pulse train was generated by over-biasing a Josephson junction in front of a 10-stage Josephson transmission line (JTL  Test chips were fabricated using the AIST 25-µA/µm 2 Nb standard process 2 (STP2). A photomicrograph of a fabricated circuit is shown in figure 1(b). In measurement, a test chip was cooled down in liquid helium.

Results
The experimental IO (V IN -V 100 ) relationships are presented in figure 2 as cross marks, where the ideal 100-fold voltage multiplication (V 100 = 100V IN ) is indicated by a solid straight line. It should be noted again that the over-biasing method was used for feeding SFQ input pulses. V IN−MAX and f IN−MAX are determined as 69.2 µV and 33.4 GHz, respectively, with multiplication errors less than 1%. These are typical values for our 100-fold DFQAs. It is confirmed that the experimental results gradually separated from the ideal Numerical simulation was executed assuming an ideally-periodic pulse current source for SFQ feeding. That is, periodic SFQ pulses were fed to the DFQA. The numerical IO characteristics are plotted as crosses combined with plus marks in figure 2. It is confirmed that numerical V 100 follows the ideal relationship up to 77 µV, of which the corresponding frequency is 37 GHz, and that the numerical V 100 values drop rapidly for V IN > 77 µV. Such rapid drop is not observed in the experimental results.

Discussion
To figure out the origin of such gradual degradation observed in the experimental IO characteristics, we employed the second simulation model where the over-biasing method was used for feeding the input SFQ pulses. Thermally-induced currents in resistors at 4.2 K were also included [5].  An example of input voltage waveforms generated using the over-biasing method is shown in figure 3(a). I IN was adjusted to realize SFQ pulse intervals of 93.1 ps on average, for which V IN was calculated to be 22.2 µV. It is confirmed that pulse intervals are fluctuated. Figure 3   It is found that the pulse intervals deviate between 19 and 34 ps, of which the corresponding frequencies are 52 and 29 GHz. As described in the previous section, the numerical simulation demonstrates that the DFQA operates for periodic SFQ pulses up to 37 GHz, and thus, pulse intervals shorter than 27 ps (= 1/37 GHz) are out of the correct operation range. The percentage of intervals shorter than 27 ps is as much as 22%, which cannot be ignored from the viewpoint of DFQA operation. That is, thermally-fluctuated SFQ pulse intervals should be a reason for the gradual degradation observed in experiments.

Conclusion
We presented experimental IO (V IN -V 100 ) characteristics of a 100-fold DFQA fabricated using the Nb integration technology. The voltage multiplication was gradually deteriorated near the maximum input voltage. Numerical simulation assuming periodic input SFQ pulses did not exhibit such gradual degradation, whereas another numerical simulation assuming the overbiasing method with thermal noise current demonstrated gradual degradation similar to the experimental results. The numerical results suggested that SFQ pulse trains generated by the over-biasing method contained non-negligible deviation of SFQ pulse intervals. Appendix A. Distribution of SFQ pulse intervals generated using the over-biasing method Although timing jitters induced by thermal noise have been investigated for two decades in the field of SFQ digital electronics [6], over-biased conditions are outside the scope of digital circuit operations. In this appendix, we describe the distributions of SFQ pulse intervals generated using the over-biasing method at 4.
, respectively. In figure A1(a), there is no significant difference in the both fitting results for the average interval of 27.4 ps. On the other hand, the pulse intervals for the average interval of 93.1 ps are fitted better by using the logarithmic normal distribution, as shown in figure A1(b). Another fitting for the average interval of 215 ps (figure A1(c)) clearly demonstrates the adequacy of the logarithmic normal distribution.
In addition, the variances σ 2 N and σ 2 LN increase rapidly as the average intervals increases. This feature should be related to nonlinear characteristics of a Josephson junction. The I-V curve of a resistively-shunted-junction (RSJ) model is presented in figure A1(d), where the (Since I c is assumed to be much larger than the thermal current 2πk B T /Φ 0 of 0.18 µA, the rounding effect around I = I c can be neglected [7].) dV /dI increases as I decreases down to I c , which means that the voltage fluctuation is enhanced as the voltage decreases to zero. The voltage fluctuation is proportional to the frequency fluctuation, resulting in the enhancement of the variances σ 2 N and σ 2 LN at low voltages. It should be noted that the large fluctuations at low voltages (long pulse intervals) do not modify the IO charcteristics of DFQAs unless they exceed the operation margins of DFQAs.