Measurement of the Boltzmann constant by Johnson noise thermometry using a superconducting integrated circuit

We report on our measurement of the Boltzmann constant by Johnson noise thermometry (JNT) using an integrated quantum voltage noise source (IQVNS) that is fully implemented with superconducting integrated circuit technology. The IQVNS generates calculable pseudo white noise voltages to calibrate the JNT system. The thermal noise of a sensing resistor placed at the temperature of the triple point of water was measured precisely by the IQVNS-based JNT. We accumulated data of more than 429 200 s in total (over 6 d) and used the Akaike information criterion to estimate the fitting frequency range for the quadratic model to calculate the Boltzmann constant. Upon detailed evaluation of the uncertainty components, the experimentally obtained Boltzmann constant was k=1.3806436×10−23 J K−1 with a relative combined uncertainty of 10.22×10−6. The value of k is relatively −3.56×10−6 lower than the CODATA 2014 value (Mohr et al 2016 Rev. Mod. Phys. 88 035009).


Introduction
According to the resolution of the 25th General Conference on Weights and Measures (CGPM), four of the SI (interna tional system of units) base units-namely the kilogram, the ampere, the kelvin and the mole are supposed to be redefined based on fixed numerical values of fundamental constants at the next CGPM in 2018 [1]. Therefore, toward the 'Revised SI', the determination of the Boltzmann constant k is an impor tant topic in the field of metrology and precision temperature measurements. There are several methods of determining the Boltzmann constant based on thermodynamic temper ature measurement, such as acoustic gas thermometry (AGT), dielectric constant gas thermometry (DCGT), Doppler broad ening thermometry (DBT) and Johnson noise thermometry (JNT) [2]. At the moment, the results obtained by AGT have the lowest uncertainty of 0.56 × 10 −6 [3]. Recently, the total uncertainty obtained by DCGT and JNT reached 1.9 × 10 −6 and 2.7 × 10 −6 , respectively [4,21], which satisfy the require ments for the determination of the Boltzmann constant [5].
Inspired by the pioneering works on JNT using a quantum voltage noise source [6][7][8][9], AIST has been trying to contribute to the determination of the Boltzmann constant with JNT based on an integrated quantum voltage noise source (IQVNS) [12][13][14][15][16][17][18]. The IQVNS fully consists of superconducting cir cuits including a pseudo random number generator based on a maximum length sequence algorithm, enabling us to eliminate concerns related to bit code generators such as inputoutput coupling, electromagnetic interference, and so on. Previously, the power spectral ratio was analyzed with a quadratic fitting model in [16], but the judgement of setting the optimal lower Metrologia Measurement of the Boltzmann constant by Johnson noise thermometry using a superconducting integrated circuit and upper cutoff frequencies were not discussed and the uncer tainty budget was not given. Therefore, the Boltzmann con stant was not determined in [16]. In this paper, we compared various fitting models using the Akaike information criterion (AIC) to estimate the lower and the uppercutoff frequencies for the quadratic fitting model, which, we believe, should be the bestdescribing model for the low frequency limit. The uncertainty of the Boltzmann constant accompanied by the uncertainty of the determination of the cutoff frequencies are evaluated for the quadratic model. Other uncertainty comp onents are also evaluated by exper imental data and available information. Finally, the Boltzmann constant obtained by JNT at NMIJ is officially shown for the first time with combined uncertainty. Figure 1 shows a schematic diagram of the NMIJ JNT system. This system includes two isolated measurement channels, A and B, for calculating the crosspower spectrum. Each mea surement channel consists of an ultralownoise preamplifier (PreAmp), a low pass filter (LPF), a buffer amplifier (BufAmp) and an analogtodigital (A/D) converter. The A/D converters are connected to a computer via optical fibers to eliminate ground loops and to reduce the noise from the computer. The calculation of the power spectra is done on the computer. During the JNT measurements, the sensing resistor and the IQVNS are alternately selected and connected to the measure ment channels by a mechanical switch board (SW) controlled by the computer. Optical fibers were also utilized for the con nection between the computer and the switch board. All the instruments except the microwave signal generator (SG) are driven by batteries to eliminate ground loops. The measure ment equipment was installed in a normal test lab. A shield room was not used for the experiments. Data was collected over 6 d from June 13 to 20, 2016.

Integrated quantum voltage noise source
A simplified schematic diagram of the IQVNS is depicted in figure 2. The IQVNS is clocked at a frequency, f clk , using an external SG. The IQVNS contains a digital circuit block and a bipolar voltage multiplier (BPVM). The digital circuit block is fully implemented with superconducting electronics. The digital circuit block generates pseudo random noise sig nals based on the 21 bit maximum length sequence at a clock   rate of f LF = f clk /4096. The BPVM comprises a supercon ducting quantum interference device (SQUID) array. One of the remarkable features in the structure of the IQVNS is the inductive decoupling technique between grounds of the super conducting digital circuit block and the BPVM. Details of the IQVNS circuit design, fabrication, and its operation are given in [17]. Using the IQVNS design parameters of M (= 2 21 − 1, code length), N 1 (=74) and N 2 (=4), the power spectrum den sity (PSD) of the IQVNS at the DC limit is given by equa tion(1) [13] where h and e stand for the Planck constant and the elemen tary charge, respectively. By setting the clock frequency f clk at 8.192 GHz ( f LF = 2.000 MHz), we obtain a PSD IQ (0) ~ 1.499 × 10 −18 V 2 Hz −1 , closely matching the PSD of a 100 Ω sensing resistor at the temperature of the triple point of water (T TPW ). The tone spacing of the IQVNS, f LF /M, is 0.953 Hz.

Sensing resistor, resistor probe and triple point of water cell
A pair of custommade 50 Ω thickfilm resistors connected in series was used as a 100 Ω sensing resistor. The sensing resistor has two equivalent output channels for the crossspectrum measurement and was installed in a 5 mm × 5 mm × 1 mm ceramic package covered with a metallic cap. It was firmly connected to a copper block at the tip of a laboratory fabri cated probe, which was placed at the center of a commercial triple point of water (TPW) cell.

Amplifiers and filters
The low noise amplifier was composed of four stages: a low noise input amplifier, a highgain instrumentation amplifier, a first buffer amplifier and a second buffer amplifier, which were integrated on a printed circuit board. The amplifier was the same as, or very similar to, those used in the JNT systems at the National Institute of Standards and Technology (NIST), USA [8] and National Institute of Metrology (NIM), China [10]. An 11 pole Butterworth filter with a cutoff frequency of 650 kHz was connected between the first and second buffer stages to suppress aliasing signals. The signaltoaliasing noise ratio is suppressed to 70 dB at 600 kHz for a Nyquist frequency of 1 MHz, corresponding to 0.1 × 10 −6 in the Boltzmann con stant measurements. At higher frequencies, the contribution of the aliasing signals becomes significant and may lead to cru cial errors in the estimate of the Boltzmann constant. We define 600 kHz as the physical upper limit for analysis to avoid uncer tainty associated with the low pass filters.

Analog-to-digital converter
Commercial low distortion A/D converters were used for data acquisition. To make the tone spacing of equation (1) an integer multiple of the fast Fourier transform (FFT) bin spacing, D fs = f s /N s = 0.477 Hz, the sampling frequency, f s, was set to 2.000 MHz and the number of data points N s to 2.000 MHz and 2M = 2(2 21 − 1), respectively. According to the A/D converter's specifications, the resolution of the A/D converter was at least 20 bits below 2 MHz.

Calculation of cross power spectra
Crosspower spectra were calculated using FFTs of a set of data consisting of N s samples measured in each channel. We define a 'chop' as a set of 100 FFT results. The cross spectrum was calculated to eliminate uncorrelated noise picked up by the readout cables in two different channels and noise from the amplifiers. Figure 3 shows the ratio of spectral power densities of the thermal noise of the 100 Ω sensing resistor and the IQVNS waveform accumulated for 2146 chops [16]. Each data point of the ratio, P R /P IQ , is obtained by a mean average over 200 bins corresponding to 94.5 Hz. It is desirable to adjust the spectral ratio close to 1 to avoid the effect of nonlinearity. Due to operating limitations of the commercial A/D converters and the IQVNS, it was not possible to adjust the spectral ratio to 1.000 at the frequency of f = 0. However, the ratio of spec tral power densities always stays within 1 ± 0.006 over a wide frequency range below f 600 kHz.

Determination of fitting model and fitting frequency range
The physical model for the spectral ratio of the sensing resistor and the IQVNS can generally be expressed by equa tion (2) [10,19] The Boltzmann constant is calculated using the constant term a 0 of this fitting model. It is considered that the highorder terms are mainly due to the difference between the transfer functions of the wirings of the two different noise sources. In general, the quadratic model (d = 2) should best describe the physics in the low frequency limit [7,8]. However, there is always the possibility to have small contributions from higher order terms (d > 2) than f 2 even if the transfer functions of the wirings in both of the probes are expressed with up to 2nd order. The influences of the higher order terms cannot be ignored as the frequency increases. Therefore, it is necessary to quantitatively determine the frequency range that can be described with the quadratic model, if we employ a quadratic model to calculate the Boltzmann constant.
To find the optimal frequency range for fitting is indeed a complicated issue. Qu et al and Mohr et al [10,19] deter mined the optimal fitting model using the crossvalidation method. The fitting bandwidth was selected by an uncertainty minimization criterion. In this paper, we introduce various fit ting models up to d = 10 to find the optimal fitting frequency range described by the quadratic model (d = 2), which is describing the low frequency limit.
For this purpose, the AIC was used to compare various order fitting models [20]. In our case, the AIC value is defined as follows, where r( f i ) denotes the experimentally obtained spectral ratio at f = f i and r d ( f i ) is the spectral ratio calculated with the dth order fitting model. n stands for the total number of data points. In this equation, the first term decreases with increasing d.
The AIC values for the models with d = 2, 4, 6, 8 and 10 were calculated as a function of the lower cutoff frequency, F LC . To find the optimal and the widest frequency range with the model with d = 2, first, we changed the lower cutoff fre quency, F LC , from 2.5 kHz to 200 kHz, and fit the data from the F LC up to the cutoff frequency of the Butterworth filter of 650 kHz. Figure 4 shows the frequency range bestfitted by the model with d = 2 as a function of F LC . Note that figure 4 shows the result only for F LC ⩽ 100 kHz and the result with higher F LC showed a narrower bestfitting frequency range. We found that the data is bestfitted with the model with d = 2 and with the widest frequency range when we set F LC = 40 kHz [16] and F UC = 380 kHz. Figure 5(a) depicts the optimal fitting order d as a function of F UC obtained by fixing F LC = 40 kHz. The quadratic model with d = 2 is optimal for F UC 380 kHz, while the model with d = 4 is the best one for F UC 460 kHz, i.e. the order d of the optimal model was changed from d = 2 to d = 4 at F UC = 380 kHz. The optimal order is unstable in the frequency range of 380 kHz < F UC < 460 kHz. This widest and maximized fre quency range led us to minimize the statistical uncertainty of the Boltzmann constant determination.
The uncertainty of the Boltzmann constant associated with the uncertainty of the determination of F LC and F UC for the quadratic model are discussed in the next chapter.
As shown in figure 5(b), the statistical uncertainty of the esti mate of a 0 , σ a0 , monotonically decreases along the model with d = 2 down to σ a0 = 9.85 × 10 −6 at F UC = 380 kHz. After showing unstable behavior in 380 kHz < F UC < 460 kHz, the value of σ a0 decreases monotonically along the model with d = 4 at F UC 460 kHz. As the order of the optimal d increases, statistical uncertainty increases due to increase of fitting parameters. For this reason, in spite of the increasing fitting bandwidth, the statistical uncertainty obtained for the model with d = 4 at the maximum F UC is almost the same as the best result for the model with d = 2.
Here, we define a 2014 as a 2014 = 4k 2014 T TPW R/PSD IQ , where k 2014 is the 2014 CODATA value of the Boltzmann con stant, k 2014 = 1.380 648 52 × 10 −23 J K −1 [11]. As dis played in figure 5(c), (a 0 − a 2014 ) gradually approaches to zero with increasing F UC . (a 0 − a 2014 ) is −3.56 × 10 −6 at F UC = 380 kHz, which is the upper limit for the model with d = 2. At the frequency range of 380 kHz < F UC < 600 kHz, the value of (a 0 − a 2014 ) is within the range of ±5 × 10 −6 . The change is well within the statistical uncertainty at F UC = 380 kHz, as discussed above.
In summary, there is no significant difference between the best result obtained for the quadratic model and the result obtained for maximum frequency range, at least at the cur rent statistical uncertainty level. We adopt the simplest quad ratic model, which is supposed to best describe the behavior of the spectral ratio at the low frequency limit. We obtained (a 0 − a 2014 ) = − 3.56 × 10 −6 and σ a0 = 9.85 × 10 −6 for the quadratic fitting model in the frequency from F LC = 40 kHz to F UC = 380 kHz. The values of a 0 and the statistical uncer tainty of the estimate of a 0 , σ a0 , are listed in tables 1 and 2, respectively.

Uncertainty evaluation and the Boltzmann constant
Evaluation of uncertainty of the JNT using quantum voltage noise source has been extensively investigated by preceding studies [8,10,21,22]. In the case of JNT using IQVNS the Boltzmann constant is expressed using equation (1) as Therefore, the uncertainty of JNT using a quantum voltage noise source can be divided into four categories: the uncer tainty of the measured spectral ratio u r (P R /P IQ ), the uncer tainty of realization of TPW u r (T TPW ), the uncertainty of the   figure 6. This suggests that the influence of electromagnetic interference (EMI) is consider ably lower than 10 × 10 −6 .

Model ambiguity.
As discussed in section 3, there are possibilities that other fitting models were selected depending on the upper cutoff frequency, resulting in a different answer. We assume the model ambiguity is not so different from the  of a 0 , a 0 was calculated by fixing one of the cutoff frequencies, F LC = 40 kHz or F UC = 380 kHz, and varying the other. The ambiguity of determination of the cutoff frequency is 5 kHz, because the frequency resolution of the calculation shown in figure 5 was 10.0 kHz. This ambiguity of ±5 kHz for F LC and F UC can lead to ambiguity of 1.9 × 10 −6 for a 0 in both cases. We account for this ambiguity by assigning a rectangu lar probability distribution of halfwidth 1.9 × 10 −6 for each case. Combining the uncertainties of F LC and F UC , we assign 1.5 × 10 −6 for the cutoff frequency dependence (table 2).

Electromagnetic interference, EMI.
In order to discuss the influences of EMI on the correlated signals of the IQVNS, we compared the noise floor of the IQVNS waveform appear ing at odd bins to the IQVNS tones observed at even bins. The noise floor of the IQVNS waveform is expected to conv erge to zero with accumulation of data, since it is equivalent to measuring the correlated signal of a superconductor whose thermal noise is zero. It was found that the average value of the noise floor of the IQVNS waveform over the IQVNS tones was at most 0.1 × 10 -6 . In order to see the influences of EMI in the resistance probe, the 100 Ω sensing resistor was removed and short circuited instead, and the correlated sig nal was accumulated for two days. We assign 0.5 × 10 −6 in total as the influence of EMI on the correlated signals of the IQVNS probe and the resistance probe (table 2).

Nonlinearity of noise thermometer.
Ideally, the ratio of the power spectral density of thermal noise of the sens ing resistor to that of the IQVNS is close to unity, in order to avoid the influence of the nonlinearity of the amplifier. As shown in figure 3, the spectral ratio is always within 1 ± 0.006 in the wide frequency range below 600 kHz. Instead of per forming the Boltzmann constant measurement with several mismatches of the power spectral ratio of the sensing resistor to QVNS [10], the IQVNS waveforms with different PSDs were directly compared by supplying different reference clock frequencies in order to eliminate the effect of the dif ference in the transfer functions of the wirings in the sensing resistance probe and the IQVNS probe. As a result, the ratio of power spectrum densities becomes flat against frequency. This comparison is possible for the IQVNS, because PSD can   be switched quickly by changing microwave frequency and power at the same time. During the measurement switchboards were turned off so that the amplifiers are always connected to the IQVNS. The microwave frequency and power for the IQVNS were changed every time after accumulating spectra for one chop (100 s) like a comparison between the sensing resistor and IQVNS. Spectral ratios, R Exp = P IQ1 /P IQ2 , for various nominal values were obtained by half day measure ments. Figure 7 displays nominal spectral ratio, R nom , ver sus relative deviation of R exp from the nominal values R nom , (R exp − R nom ) /R nom . The nominal spectral ratio is equivalent to the ratio of clock frequencies to the IQVNS. The error bar is negligibly small for the nominal ratio of unity, because all the even chops and odd chops of the IQVNS collected so far were used. The slope obtained by a weighted fitting is −2 × 10 −5 , which is of the same order as [10], although the sign is oppo site. The influence of the nonlinearity is estimated to be about 0.1 × 10 −6 for the nominal PSD ratio of 1.0057 at around f = 0. An uncertainty of 0.1 × 10 −6 is assigned to the non linearity effect of the noise thermometer (table 2).

Dielectric losses.
It is not practical to actually measure dielectric losses of insulating materials of wirings and printed circuit boards (PCB) in the experimental equipment of the JNT system. Instead of measuring those quantities, we roughly esti mated the influence of dielectric loss, tan δ, on power spectral densities. The calculations were based on the physical dimen sions and properties of the insulating materials of the wirings in the noise sources and the PCB of the switch boards. The insulating materials of the wirings and the PCB are PTFE and FR4, respectively. We assigned 1 ×10 −6 as an influence on the spectrum intensity due to the dielectric loss (table 2).

Triple point of water temperature u r (T TPW )
The thermowell of the TPW cell, in which the resistance probe is inserted, is filled with water so that the surface of the water in the thermowell has the same level as that of the TPW cell. The TPW temperature, T TPW , is defined as 273.16 K in the present SI. In practice, however, there is always a dif ference from the definition in the actual TPW cell. The TPW cell was calibrated to be traceable to the national standard of temper ature maintained at NMIJ/AIST. The temperature of the bottom of the thermowell was 273.159 74 K. Using hydro static pres sure correction, we determined the temperature of the cell: 273.159 93 K (table 1), which is −0.07 mK lower than the definition of T TPW = 273.16 K. The relative uncer tainty of calibration that includes the uncertainty of the TPW realization itself, such as the isotopic and chemical effects, is 0.08 mK for k = 1 (table 2). We also assigned uncertainty due to hydrostatic pressure correction as 0.06 × 10 −6 based on the uncertainties asso ciated with the package size of the sensing resistor and its position, that is 30 mm from the bottom (see section 2, and table 2).
Thermal inflow through the body of the resistance probe and the wires for the sensing resistor may result in change in the resistance value. To evaluate the immersion effect, a small platinum thermometer was fixed just next to the sensing resistor in the body of the resistance probe, and temperature at the sensing resistor was measured as a function of posi tion in the thermowell. The result is shown in figure 8. The temperature is stable within around 0.2 mK in the region up to 100 mm from the bottom. The slope below 100 mm is esti mated to be 1.6 mK m −1 , which is about twice as large as hydrostatic pressure correction coefficient of 0.73 mK m −1 . We assigned 1 × 10 −7 as the uncertainty for the temperature of the sensing resistor placed at 30 mm from the bottom of the thermowell due to immersion effect (table 2).

Sensing resistor u r (R)
The resistance across the pair of 50 Ω resistors, that form the sensing resistor, was found to be 99.903 95 Ω (table 1). The resistance value was measured every day before measure ment of the thermal noise. The statistical uncertainty was 0.7 × 10 −7 (table 2). The resistance of the thermometer was measured using a commercial resistance thermometry bridge, Fluke 1594A. This bridge measures the resistance ratio of a device under test with respect to a 100 Ω reference standard resistor. The reference resistor was calibrated using a resistance calibration system traceable to the quantized Hall resistance standard (the national standard of resistance) of NMIJ/AIST. The uncertainty of resistance calibration is 5.6 × 10 −8 (k = 2). We assigned 2.8 × 10 −8 (k = 1) as the uncertainty associated with the longterm drift of the sensing resistor (table 2). The ratio error of the resistance ther mometry bridge was investigated using two calibrated 100 Ω standard resistors. The uncertainty due to the ratio error of the resistance thermometry bridge was found to be 6 × 10 −8 (table 2). The uncertainty due to leakage resistance was found to be of the order of 1 × 10 −10 (table 2). The temperature dependence of the resistance was measured in the temper ature range from 273.6 K to 303.2 K. Extrapolating the result to 273.16 K, the temperature coefficient of the sensing resistor was estimated to be 2 × 10 −6 K −1 . Since temperature dis tribution in the thermowell is about 0.2 mK, the upper limit of the uncertainty due to the temperature dependence of the sensing resistor is 2 × 10 −10 (table 2).
Precise measurement of the frequency dependence of the sensing resistor is not straightforward, especially when the relative resistance change caused by frequency change is below the 1 × 10 −6 level. Instead of actually measuring the frequency dependence of the sensing resistor installed on the resistance probe, we estimated the uncertainty associated with frequency dependence of the sensing resistor using data on the specification sheet of a similar thickfilm resistor. The resist ance change of a 100 Ω thickfilm resistor is 2% at 500 MHz. Assuming that the frequency dependence is quadratic, we estimated that the resistance change is estimated to be of the order of 0.02 × 10 −6 at 500 kHz. We assigned 0.02 × 10 −6 to the uncertainty associated with frequency dependence of the sensing resistor (table 2).

IQVNS waveform u r (PSD IQ )
The clock frequency of the pseudo random number generator in the superconducting digital circuit block f LF is related to the clock frequency supplied to the IQVNS from an external SG, f clk as f LF = f clk /4096. Since the uncertainty of the external SG is governed by the uncertainty of the 10 MHz reference signal from GPS, which is 1 × 10 −11 , relative uncertainty of PSD IQ given by equation (1) is also expected to be of the order of 10 −11 , which is negligibly small compared to other uncertainty components (table 2).
The IQVNS does not utilize any modulation techniques with quantization error. The pseudo random number sequence is generated based on a primitive polynomial. The SQUID arrays in the BPVM guarantee quantization of voltage pulses. We have investigated bit error probability of the IQVNS, and estimated that the error in the spectrum density of the IQVNS is as small as 1 × 10 −7 (table 2).

The Boltzmann constant and combined uncertainty
The Boltzmann constant k 90 determined by the actual experiment is obtained to be k 90 = 1.3806434 × 10 −23 J K −1 using the parameters, a 0 (=P R /P IQ ), T TPW , R and PSD IQ summarized in table 1. The value of k 90 and the Boltzmann constant k to be evaluated are related as k/h = k 90 /h 90 , where the Planck constant h and h 90 ≡ 4/(K 2 J-90 R K-90 ) are defined as h = 6.626 070 040(81) × 10 −34 J · s and h 90 = 6.626 068 854 . . . × 10 −34 J · s, respectively [11]. Therefore, k is finally determined as k = 1.380 6436 × 10 −23 J K −1 . The value of k is relatively −3.56 × 10 −6 lower than the CODATA 2014 value [11]. The combined relative uncer tainty in the measurement of k is calculated as 10.22 × 10 −6 using equation (5). The details of the uncertainties are listed in table 2.

Conclusion
We evaluated the Boltzmann constant k by JNT calibrated with an IQVNS. To obtain the Boltzmann constant, the ratio of power spectral densities of the sensing resistor over that of the IQVNS were fitted with a quadratic fitting function. The Akaike information criterion was employed to roughly estimate the upper and the lowercutoff frequency for the quadratic fitting model. Based on experiments and theor etical considerations, the uncertainty budget was given. The Boltzmann constant evaluated for 2146 chops is determined to be k = 1.380 6436 × 10 −23 J K −1 with a relative combined uncertainty of 10.22 × 10 −6 . Considering that the noise gen eration employing the IQVNS differs from the established QVNS, our result reported here has significance as an inde pendent result for the JNT.