Theory of quantum-circuit refrigeration by photon-assisted electron tunneling

We focus on a recently experimentally realized scenario of normal-metal–insulator–superconductor tunnel junctions coupled to a superconducting resonator. We develop a first-principles theory to describe the effect of photon-assisted electron tunneling on the quantum state of the resonator. Our results are in very good quantitative agreement with the previous experiments on refrigeration and heating of the resonator using the photon-assisted tunneling, thus providing a stringent verification of the developed theory. Importantly, our results provide simple analytical estimates of the voltage-tunable coupling strength and temperature of the thermal reservoir formed by the photon-assisted tunneling. Consequently, they are used to introduce optimization principles for initialization of quantum devices using such a quantum-circuit refrigerator. Thanks to the first-principles nature of our approach, extension of the theory to the full spectrum of quantum electric devices seems plausible.

Figure 1

(a) Schematic diagram of a superconducting coplanar waveguide (CPW) resonator which is connected through the capacitances $C_c$ and $C_g$ to a normal-metal island and a transmission line with an impedance $Z_{\mathrm{tr}}$. The superconductor (S)–insulator (I)–normal-metal (N) tunnel junctions defining the island are voltage biased through the superconducting electrodes. (b) Energy diagram of the photon-assisted electron tunneling. The blue arrows depict tunneling events leading to absorption of a photon from the coupled resonator and the red arrows correspond to emission. (c) Coupling strength ${\gamma }_T$ and temperature $T_T$ of the effective thermal reservoir formed by the photon-assisted tunneling as a function of the two-junction voltage bias. The parameters correspond to typical experimental values: see Fig. 3. In the highlighted region the thermal reservoir is cooler than the electrons of the normal-metal island.

Figure 2

Effective circuit diagram of the studied system. The fundamental mode of the coplanar waveguide resonator is modeled by the lumped-element capacitance $C$ and inductance $L$. The resonator couples to a normal-metal island through an input capacitance $C_c$. Tunneling is depicted by the gray symbol on the left representing the weak coupling between a normal-conducting and a superconducting electrode. The diagram shows only one of the NIS junctions with junction capacitance $C_j$ and voltage bias $V=V_B/2$. For tunneling through this junction, the other parallel junction acts as a capacitor, the capacitance of which is included in the capacitance $C_m$ of the metallic island to the ground. An output capacitor of the capacitance $C_g$ couples the resonator to a transmission line with a characteristic impedance $Z_{\mathrm{tr}}$. The node fluxes at the island and at the resonator are denoted by ${\mathrm{\Phi }}_N$ and $\mathrm{\Phi }$, respectively, and $Q_N$ and $Q$ are their conjugate charges.

Figure 3

Resonator transition rates ${\mathrm{\Gamma }}_{m,m^{\prime }}$ of Eq. (5.5) as functions of the single-junction bias voltage. Both the one-photon processes ${\mathrm{\Gamma }}_{1,0}$ (blue solid line) and ${\mathrm{\Gamma }}_{0,1}$ (red solid line) as well as the two-photon processes ${\mathrm{\Gamma }}_{2,0}$ (blue dotted line) and ${\mathrm{\Gamma }}_{0,2}$ (red dotted line) are shown. The elastic transition rate ${\mathrm{\Gamma }}_{0,0}$ (green dashed line) exceeds the inelastic rates. The used parameters correspond to a typical experiment [40, 41]: $\mathrm{\Delta }=200\mu \mathrm{eV}$, ${\gamma }_D={10}^{-4}$, $R_T=50\mathrm{k}\mathrm{\Omega }$, $T_N=100$ mK, $C_c=1.0$ pF, $C_{\Sigma m}=10$ fF, ${\omega }_r/2\pi =7.0$ GHz, and $Z_r=35\mathrm{\Omega }$.

Figure 4

(a) Spectral density of tunneling $S_T$ as a function of the angular frequency $\omega$ and single-junction bias voltage $V$. The parameters equal to those of Fig. 3. (b) Traces from panel (a) corresponding to $eV/\mathrm{\Delta }=0.1,0.5,0.9,1.3$ [vertical dashed lines in panel (a)]. The black dashed line is a fit to an ohmic behavior $S_T\propto \omega$ at $eV/\mathrm{\Delta }=1.3$. Near the gap voltage $|eV-\mathrm{\Delta }|\lesssim \hslash \omega$, the spectral density exhibits clear nonohmic, exponential behavior. In the region $|eV-\mathrm{\Delta }|\gtrsim \hslash \omega$, the spectral density $S_T\left(\omega \right)$ flattens to a linear, ohmic trend.

Figure 5

(a) The coupling strength ${\gamma }_T$ and (b) the temperature $T_T$ of the thermal reservoir by photon-assisted tunneling as a function of the single-junction bias voltage. The parameters equal to those of Fig. 3 except that the resonator frequencies are ${\omega }_r/2\pi =4.0,5.0,...,10.0$ GHz corresponding to line colors from light gray to black, respectively. The overall scaling of ${\gamma }_T$ comes from the scaling of ${\overline{\gamma }}_T$ introduced in Eq. (5.9) assuming that the characteristic impedance $Z_r$ is frequency independent. The coupling strength ${\overline{\gamma }}_T$ for ${\omega }_r/2\pi =4.0$ GHz is depicted by a magenta dash-dotted line. In the highlighted region the thermal reservoir is cooler than the island electrons with temperature $T_N$, depicted by a green dash-dotted line. The red dashed lines show the analytic results for ${\omega }_r/2\pi =4.0$ GHz in the deep subgap, thermal activation and above the gap regions, respectively, corresponding to the analytical results (6.2), (6.4), and (6.5).

Figure 6

Net output power $P_T$ as a function of the single-junction bias voltage. Comparison of Eq. (8.1) providing the results of our transition rate theory (solid line) with the measured values (filled circles) and results of the $P\left(E\right)$ theory (dashed line) from Ref. [41]. The parameters in the models correspond to the experiments: $\mathrm{\Delta }=220\mu \mathrm{eV}$, ${\gamma }_D=4\times {10}^{-4}$, $R_T=12.5\mathrm{k}\mathrm{\Omega }$, $C_c=840$ fF, $C_{\Sigma m}=12$ fF, ${\omega }_r/2\pi =4.55$ GHz, $Z_r=33.9\mathrm{\Omega }$, $C_g=72.0$ fF, $Z_{\mathrm{tr}}=53.0\mathrm{\Omega }$, and $T_{\mathrm{tr}}=180$ mK. For the electron temperature, we use the bias-voltage-dependent values $T_N\left(V\right)$ experimentally measured in Ref. [41] using an additional SINIS thermometer.

Figure 7

Effective temperature $T_Q$ of the charge distribution of the normal-metal island as a function of the single-junction bias voltage. The parameters equal to those of Fig. 3. The solid black line shows the numerical result based on Eqs. (A3) and (A4). The filled circles show the corresponding result based on Eqs. (A5, A6, A7) valid at $eV\gg E_N,k_B{T}_N$. In the deep-subgap regime, the temperature of the charge distribution $T_Q$ is close to the electron temperature $T_N=100$ mK depicted by a green dash-dotted line.