Thermally pumped on-chip maser

We present a theoretical model of an on-chip three-level maser in a superconducting circuit based on a single artificial atom and pumped by a temperature gradient between thermal baths coupled to different interlevel transitions. We show that maser powers of the order of a few femtowatts, well exceeding the resolution of the sensitive bolometry, can be achieved with typical circuit parameters. We also demonstrate that population inversion in the artificial atom can be detected without measuring coherent radiation output of the maser. For that purpose, the system should operate as a three-terminal heat transport device. The hallmark of population inversion is the influx of heat power into the weakly coupled output terminal even though its temperature exceeds the temperatures of the two other terminals. The proposed method of on-chip conversion of heat into microwave radiation and control of energy-level populations by heating provide additional useful tools for circuit quantum electrodynamics experiments.

Figure 1

(a) Schematics of a three-level thermally driven maser. Energy levels of the system are selectively coupled to the two reservoirs via filters. (b) The circuit diagram of the proposed realization of the model shown in (a) with superconducting circuit components. (c) The potential of the loop with three junctions at $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.32$ is shown by the solid black line. The horizontal dashed lines show the positions of the first three energy levels, while the squares of the corresponding wave functions are shown by solid curves.

Figure 2

(a), (c), (e) Wigner functions of the output resonator, and (b), (d), (f) the corresponding photon number distributions (red solid lines). The coordinate $x$ and the momentum $p$ of the oscillator modeling the output resonator are normalized with their zero-point fluctuations $\tilde{x}$ and $\tilde{p}$. The ratio $T_c/T_h$ in the three sets takes the values 1.0, 0.4, 0.13, and $T_h=0.3$ K. The photon number distribution in (b) coincides with the Boltzmann distribution (blue dashed line). In (d) and (f) blue dashed lines show the Poissonian distribution. We have used the following parameters: $E_C/2\pi \hslash =380$ MHz, $E_J/2\pi \hslash =5.2$ GHz, $C_c=3$ fF, $C_h=18$ fF, $C_d=5$ fF, $R_c=80\mathrm{\Omega }$, $R_h=100\mathrm{\Omega }$, $C_c^{\prime }=136$ fF, $C_h^{\prime }=60$ fF, $C_d^{\prime }=4$ fF, ${\omega }_h/2\pi =7.09\mathrm{GHz}$, ${\omega }_d/2\pi =3.87\mathrm{GHz}$, ${\omega }_c/2\pi =3.20\mathrm{GHz}$, $Q_h\approx 44$, $Q_c\approx 53$, $Q_d\approx 66465$, $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.32$, ${\mathrm{\Delta }}_h=112$ MHz, ${\mathrm{\Delta }}_c=-45$ MHz, and ${\mathrm{\Delta }}_d=24.6$ MHz. (g) The lower red line shows the output power for $T_h=0.41$ K in a system with the parameters given above. The upper green line shows the power (17) in a system with $C_c=34$ fF, $C_h=76.5$ fF, $C_d=10$ fF, $C_c^{\prime }=289$ fF, $C_h^{\prime }=127$ fF, $C_d^{\prime }=4$ fF, ${\mathrm{\Delta }}_h=1.2$ MHz, ${\mathrm{\Delta }}_c=0.4$ MHz, $Q_h\approx 12$, and $Q_c\approx 10$, and the remaining parameters are the same as before. The approximation (19) is shown by the blue dashed line in both cases. (h) Red line, the efficiency found numerically, corresponding to the power represented with red line in (g); blue dashed line, the ratio of frequencies (${\omega }_{10}/{\omega }_{20}$); dashed pink line, Carnot efficiency.

Figure 3

(a) Power dissipated in the detector resistor versus applied magnetic flux; (b) the ratio $p_1/p_0$ as a function of external magnetic flux; (c) power dissipated in the detector resistor versus the detector temperature $T_d$ (blue line) and $T_d=T_h$ (red dashed line); (d) $p_1/p_0$ versus $T_d$ (blue line) and $T_d=T_h$ (red dashed line). In (a) and (b), the temperature of the the detector is fixed at $T_d=0.03$ K. In (c) and (d), $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.32$, and $T_d$ varies from 0.03 K to 0.5 K. The direction of heat flow does not change even when $T_d>T_h$. We have used the following parameter values: $C_d=1$ fF, $C_d^{\prime }=100$ fF, $R_d=35\mathrm{\Omega }$, $R_c=130\mathrm{\Omega }$, $C_c=4$ fF, $R_h=110\mathrm{\Omega }$, ${\omega }_h/2\pi =7.06$ GHz, ${\omega }_c/2\pi =3.21$ GHz, ${\omega }_d/2\pi =3.87$ GHz, $T_c=0.03$ K, and $T_h=0.38$ K. Other parameters are same as in the caption of Fig. 2. The quality factors of the resonators are $Q_d\approx 152$, $Q_c\approx 32$, and $Q_h\approx 40$. In the numerical simulation, we have kept the six lowest levels of the artificial atom and assumed equilibrium photon numbers in all the resonators for simplicity, and considered all the possible transitions due to all three of the resonators. The maximum population of the sixth level, achieved at $T_d=0.5$ K, is found to be $p_5\approx 0.05$.