A measurement method for responsivity of microwave kinetic inductance detector by changing power of readout microwaves

TL;DR

This paper introduces a novel method for calibrating the responsivity of microwave kinetic inductance detectors by varying readout microwave power to induce temperature changes, enabling accurate quasiparticle count and phase response measurement.

Contribution

The proposed method allows responsivity calibration of MKIDs through microwave power variation, providing a new approach that is consistent with traditional calibration techniques.

Findings

The method accurately measures quasiparticle numbers from phase response.

Responsivity calibration results are consistent with conventional methods.

The technique demonstrates effective calibration at 285mK.

Abstract

Superconducting detectors are a modern technology applied in various fields. The microwave kinetic inductance detector (MKID) is one of cutting-edge superconducting detector. It is based on the principle of a superconducting resonator circuit. A radiation entering the MKID breaks the Cooper pairs in the superconducting resonator, and the intensity of the radiation is detected as a variation of the resonant condition. Therefore, calibration of the detector responsivity, i.e., the variation of the resonant phase with respect to the number of Cooper-pair breaks (quasiparticles), is important. We propose a method for responsivity calibration. Microwaves used for the detector readout locally raise the temperature in each resonator, which increases the number of quasiparticles. Since the magnitude of the temperature rise depends on the power of readout microwaves, the number of quasiparticles also depends on the power of microwaves. By changing the power of the readout microwaves, we simultaneously measure the phase difference and lifetime of quasiparticles. We calculate the number of quasiparticles from the measured lifetime and by using a theoretical formula. This measurement yields a relation between the phase response as a function of the number of quasiparticles. We demonstrate this responsivity calibration using the MKID maintained at 285mK. We also confirm consistency between the results obtained using this method and conventional calibration methods in terms of the accuracy.

Tables

Table 1. Table 1: Measured results for each readout power change. Errors are dominated by systematic errors described in the text. Statistical errors are smaller than systematic errors by approximately one order of magnitude.

$P_H\to P_L$ [dB]	${\theta }_L-{\theta }_H$[rad]	${\tau }_{\mathrm{qp}}[\mu$s]	$N_{\mathrm{qp}}/{10}^6$
-11.0 $\to$ -17.5	$4.17\pm 0.31$	$41.6\pm 1.0$	$2.94\pm 0.07$
-12.0 $\to$ -17.5	$3.37\pm 0.30$	$43.4\pm 1.0$	$2.82\pm 0.06$
-13.0 $\to$ -17.5	$2.60\pm 0.30$	$48.1\pm 1.0$	$2.54\pm 0.05$
-14.0 $\to$ -17.5	$2.00\pm 0.30$	$54.9\pm 1.0$	$2.22\pm 0.04$
-15.0 $\to$ -17.5	$1.26\pm 0.30$	$60.7\pm 1.0$	$2.02\pm 0.03$

Equations

$$N_{\mathrm{qp}}=2N_{0}V\sqrt{2\pi k_{\mathrm{B}}T\Delta }\exp \left(-\frac{\Delta }{k_{\mathrm{B}}T}\right),$$

$$2\Delta =3.52k_{\mathrm{B}}{T}_c.$$

$$\delta \theta =\frac{\delta \theta }{\delta f_r}\delta f_r=-\frac{4Q_{r0}}{f_{r0}}\delta f_r,$$

$$S_{\theta }(f)=\frac{4N_{\mathrm{qp}}{\tau }_{\mathrm{qp}}}{(1+(2\pi f{\tau }_{\mathrm{qp}}{)}^2)(1+(2\pi f{\tau }_{\mathrm{res}}{)}^2)}{\left(\frac{\mathrm{d}\theta }{\mathrm{d}{N}_{\mathrm{qp}}}\right)}^2+X_{\mathrm{system}},$$

$$\theta (t)=\{\begin{array}{ll}{\theta }_{\mathrm{H}}& (t<t_0)\\ ({\theta }_{\mathrm{H}}-{\theta }_{\mathrm{L}})\exp \left(-\frac{t-t_0}{{\tau }_{\mathrm{qp}}}\right)+{\theta }_{\mathrm{L}}& (t\ge t_0),\end{array}$$

$$N_{\mathrm{qp}}=\frac{{\tau }_{0}V}{{\tau }_{\mathrm{qp}}}\frac{N_0(k_{\mathrm{B}}{T}_c{)}^3}{2{\Delta }^2},$$

$${\theta }_{\mathrm{measured}}={\theta }_{\mathrm{MKID}}+{\theta }_{\mathrm{system}}.$$

$$\theta =2\tan ({\theta }_{\mathrm{MKID}}/2).$$

Full text

††preprint: AIP/123-QED

Superconducting detectors have greater sensitivity compared with conventional technologies. This is because the gap energies of superconducting devices are lower than those of semiconductor devices by three orders of magnitude. Therefore, this technology is applied in various scientific fields. The microwave kinetic inductance detector (MKID) [MKIDs, ] is one of cutting-edge superconducting detector. It is based on the principle of superconductive resonator circuits, i.e., each detector consists of a simple inductor and a capacitor. These circuits are implemented in superconducting films on a substrate made of silicon or sapphire. The resonant frequency of each detector is defined by the geometry of the film. MKIDs naturally provide a multiplex readout in the frequency domain using microwaves. Therefore, compared with other types of superconducting detectors, there is an advantage in constructing a large format array. In particular, the field of radio astronomy has started using antenna-coupled MKID arrays [DESHIMA, ; NIKA, ; GB_nagasaki, ].

The detection mechanism of the MKIDs is simple. A radiation entering the detector breaks the Cooper pairs in the superconducting film, and changes the inductance of the resonator. It also changes the resonant condition of microwaves coupled to the inductors via a feedline. We measured the intensities of each detector signal based on the variations of each resonance. We used the resonant phase ($\theta$) for the measurements in this study. Broken Cooper pairs are called quasiparticles. The mean time for the recombination of the broken Cooper pairs is called the quasiparticle lifetime ($\tau_{\mathrm{qp}}$). The number of quasiparticles ($N_{\mathrm{qp}}$) was calculated with material properties, geometries, and the $\tau_{\mathrm{qp}}$ as described in [tauqp_Nqp, , pieter2, ].

In real applications, we have to calibrate the responsivity of the detector, i.e., $\mathrm{d}\theta/\mathrm{d}N_{\mathrm{qp}}$. It is one of important indicators to evaluate the sensitivity of MKID, and it is difficult to develope the MKID without this information. There are two conventional methods: a calibration by changing the physical temperature of a MKID device, and another calibration through a power spectral density. This paper proposes the third caliblation method for the responsivity, i.e., simultaneous measurements of $\theta$ and $\tau_{\mathrm{qp}}$ through the power change of readout microwaves.

Changing the physical temperature of a MKID device is the most popular calibration method [MKIDs, , gao2008, ]. We increase $N_{\mathrm{qp}}$ by warming up the device by using a heater set in the cryostat. The relation between the temperature $(T)$ and $N_{\mathrm{qp}}$ is given by the following formula [BCS, ]:

[TABLE]

where $N_{0}$ is the single spin density of the states at the Fermi energy (e.g., $N_{0}\sim 1.74\times 10^{10}\,\mathrm{eV}^{-1}/\mu\mathrm{m}^{3}$ for an aluminum [N0, ]), $V$ is the volume of the film, $k_{\mathrm{B}}$ is the Boltzmann constant, $T_{c}$ is the superconducting transition temperature (e.g., 1.2 K for the aluminum), and $\Delta$ is the gap energy,

[TABLE]

We change $T$ with several steps. In each step after temperature stabilization, we repeats the measurement of the resonant peak ($f_{r}$) through the transmittance spectrum as a function of the microwave frequency. The difference of the resonant peak ($\delta f_{r}$) from the original position ($f_{r0}$) is converted into the phase difference ($\delta\theta$) based on the following formula:

[TABLE]

where $Q_{r0}$ is the quality factor of the resonance at the minimum temperature. We obtain $\mathrm{d}\theta/\mathrm{d}N_{\mathrm{qp}}$ through the measured parameters: $Q_{r0},f_{r0}$, and $\delta f_{r}$ for each temperature.

However, this method has some issues for consideration. First, the variation of $T$ is too large compared with the conditions in real-world operations. It is typically $10\,\mathrm{mK}\sim 100\,\mathrm{mK}$ variation. Second, it is difficult to estimate the uncertainty of $T$ because we cannot directly probe the MKID temperature. Third, stray lights due to the emissions in the cryostat, e.g., the heater, and a readout microwave power also create bias toward $N_{\mathrm{qp}}$ [straylight, ; readout1, ; readout2, ; pieter, ]. They generate a large bias for the responsivity measurement, at least for a some factors. Fourth, disadvantage of this method is that it takes a long time to change the temperature and stabilize the system.

A power spectral density (PSD) contains the responisivity in its formula [pieter, ],

[TABLE]

where $S_{\theta}$ is the phase response, $f$ is the frequency of the detector response, $\tau_{\mathrm{res}}$ is the resonator ring time given by $\tau_{\mathrm{res}}=Q_{r}/\pi f_{r}$, and $X_{\mathrm{system}}$ is the noise of the readout system. As described in [pieter, ], the fitting for the measured PSD is in the way to obtain the parameters $\mathrm{d}\theta/\mathrm{d}N_{\mathrm{qp}}$, $\tau_{\mathrm{qp}}$ and $X_{\mathrm{system}}$. The $Q_{r}$ and $f_{r}$ should be measured in advance through the resonant spectrum as a function of the microwave frequency.

The biggest concern with this method is the effects of a two-level system (TLS). The TLS creates a frequency-dependent noise structure, i.e., 1/f noise, which degrades the accuracy of fitting parameters unless we suppress it. Another consideration is the need to construct a low-noise setup for the suppression of $X_{\mathrm{system}}$.

It is known that $N_{\mathrm{qp}}$ depends on the power of readout microwaves [readout1, ; readout2, ; pieter, ]. This is because a tiny loss of microwaves locally warms up the resonator device. Based on this knowledge, we propose a method for the responsivity calibration. Figure 1 illustrates the response of $N_{\mathrm{qp}}$ and $\theta$ to change in readout power from high to lower power at time $t=t_{0}$. The $N_{\mathrm{qp}}$ decreases with this change in the time scale of the detector response. The phase response as a function of time is formulated as follows:

[TABLE]

where $\theta_{\mathrm{H}}$ and $\theta_{\mathrm{L}}$ are the phases before and after stabilizing the power change, respectively. This method allows us to measure the response, i.e., $\theta_{\mathrm{L}}-\theta_{\mathrm{H}}$, and $\tau_{\mathrm{qp}}$. As described in [tauqp_Nqp, , pieter2, ], we obtain the $N_{\mathrm{qp}}$ using the following formula:

[TABLE]

where $\tau_{0}$ is the interaction time between electrons and photons (approximately 450 ns for the aluminum [tauqp_Nqp, , pieter2, ]). By using various sets of readout power changes, we obtain the phase response as a function of $N_{\mathrm{qp}}$, i.e., the responsivity. In this paper, we demonstrate this methodology by using hybrid-type MKIDs with aluminum film and niobium film. Note that this methodology should work in case of different types (e.g., multi-layer processing [vissers, ]) unless the quasiparticle lifetime is longer than several $\mu\mathrm{sec}$.

We applied the proposed method in real measurements. The diagram of our setup is shown in Figure 2. Our cryostat consists of 4 K, 40 K, and 300 K thermal shields [GB_nagasaki, ], and is cooled using a pulse tube refrigerator (PT415, Cryomech Co. Ltd). Magnetic shields (MS-FR, Hitachi material) were set outside of the 40 K and 300 K thermal shields; three sheets were set outside of the 40 K shield, three sheets (four sheets) were set outside the wall (bottom plate) of the 300 K shield. The MKID device was set in a light-tight box made of aluminum to mitigate the effects of stray light. The average temperature at the mounting plate of MKID was 285 mK and was cooled by a helium-sorption refrigerator (Gas-Light type, Simon chase Co. LTD).

Our MKID device was fabricated at RIKEN for astrophysical applications [GB_nagasaki, ]. This device consists of 10 hybrid-type MKIDs. The region for the input signal deposition is made of aluminum film, and other circuits are made of niobium film. The volume of the aluminum film is 920$\,\mu\mathrm{m}^{3}$ (the width is 4$\,\mu$m, the length is 2300$\,\mu$m, and the thickness is 100 nm). The MKID used in this demonstration is not coupled to any antenna. Under a dark condition, its resonant frequency and quality factor are $f_{r0}=6.07\,$GHz and $Q_{r0}=4.78\times 10^{4}$, respectively. Our readout electronics measured the resonance based on a direct down-conversion logic with 200 MHz sampling speed [RHEA_ishitsuka, , RHEA_suzuki, ], and the data was down-sampled to 1 MHz step. This step is sufficient for this demonstration.

The power of the readout microwave was controlled using a variable attenuator (LDA-602E, Vaunix Co. LTD). It takes several $\,\mu$s to change the power after we set the attenuation value. For the responsivity calibration, we used five attenuation sets to change the microwave power from $P_{\mathrm{H}}$ to $P_{\mathrm{L}}$: -11.0 dB $\rightarrow$ -17.5 dB, -12.0 dB $\rightarrow$ -17.5 dB, -13.0 dB $\rightarrow$ -17.5 dB, -14.0 dB $\rightarrow$ -17.5 dB, and -15.0 dB $\rightarrow$ -17.5 dB. The feed power into the MKID is approximately $-70\,\mathrm{dBm}$ in the condition of $P_{\mathrm{L}}$.

The measured phase response ($\theta_{\mathrm{measured}}$) is the sum of the MKID signal ($\theta_{\mathrm{MKID}}$) and phase offset due to cables in the system ($\theta_{\mathrm{system}}$).

[TABLE]

The $\theta_{\mathrm{system}}$ is a constant value unless the attenuation is changed. We measured it in the transmittance spectrum as a function of the microwave frequency for each power state, $P_{\mathrm{H}}$ and $P_{\mathrm{L}}$. In this responsivity calibration, we subtracted them in the analysis. We conservatively estimated the systematic error as 0.3 rad. We also corrected the nonlinear effect of the response using the formula described in Appendix E of [gao, ],

[TABLE]

Figure 3 shows phase responses as a function of time. We reset the attenuation value at $t=100\,\mu$s with a precision of 1$\,\mu$s. We observe the response in the phase. We fit the data to Eq. (5), and extract parameters. We did not use the data in the short period $t=100\,\mu\mathrm{s}\sim 110\,\mu\mathrm{s}$ because of the uncontrolled state of attenuation after the reset, as mentioned above. We repeat this measurement 40  times for each set of power change. The results are summarized in Table 1.

Figure 4 shows the relation between the phase response as a function of $N_{\mathrm{qp}}$. This is based on values listed in Table 1. We obtain $\mathrm{d}\theta/\mathrm{d}N_{\mathrm{qp}}=(2.8\pm 0.3)\times 10^{-6}\,$rad from this plot, assuming the linear relation.

For comparison, we also perform conventional methods: the heater control method and PSD method. With the heater control method, we changed the temperature of the mounting plate for the MKID from 285 mK to 300 mK, and obtained $\mathrm{d}\theta/\mathrm{d}N_{\mathrm{qp}}=(9.9\pm 0.3)\times 10^{-7}\,$rad, here we only assign the statistical error. The systematic uncertainty of this result is a few factors, as mentioned above. With the PSD method, we obtained $\mathrm{d}\theta/\mathrm{d}N_{\mathrm{qp}}=(2.4\pm 0.2)\times 10^{-6}\,$rad. The bias due to the 1/f noise was not evaluated and it was not included. However, the 1/f noise tends to generate small bias. Comparing to systematic uncertainties in each method, discrepancies among the three results are not significant.

In summary, MKID is one of cutting-edge superconducting detector based on the principle of superconducting resonator in the microwave range. The responsivity calibration, i.e., understanding the relation between the response in the resonant phase and number of quasiparticles, is important for both its application and development. We proposed a calibration method, i.e., simultaneous measurements of the phase difference and the quasiparticle lifetime following the power change of the readout microwave. The number of quasiparticles was calculated using the measured lifetimes and geometrical parameters of the detector device. We demonstrated this method, and confirmed its simplicity and ease-of-use. We also performed conventional methods: the heater control method and PSD method. We confirmed the consistency between the results obtained using our method and conventional methods in terms of accuracy. Therefore, our method is promising for the responsivlity calibration.

This work was supported by the JRA program in RIKEN, and Grants-in-Aid for Scientific Research from The Ministry of Education, Culture, Sports, Science and Technology, Japan (KAKENHI Grants. 15H05743, 16J09435, 18H05539, 15K13491, 16H00874, and R2804). We thank Shunsuke Honda, Taketo Nagasaki, and Junya Suzuki for useful discussions. We thank Mr. Noboru Furukawa, Advanced Technology Center of National Astronomical Observatory of Japan, and Advanced Manufacturing Support Team of RIKEN.