Single-atom maser with an engineered circuit for population inversion

We present a blueprint for a maser with a single three-level transmon superconducting artificial atom. The system can be pumped coherently via a two-photon process, and to achieve high population inversion, the relaxation rate of the metastable state is increased via an auxiliary low-$Q$ cavity coupled to a transition between the transmon excited states. We show numerically that such a maser can operate both in the intermediate-coupling regime with super-Poissonian photon statistics and in the strong-coupling regime, where the statistics is sub-Poissonian. For the former the maser exhibits thresholdless behavior, and for the latter there is a well-defined pumping threshold. A useful side effect of the auxiliary resonator is that it allows to overcome the photon blockade effect for the pump, which would otherwise prevent a high photon population. Finally, we observe the bistability of the steady-state Wigner function and the self-quenching effect for some parameters.

Figure 1

(a) Schematic configuration of the correctly aligned energy levels without coupling between subsystems. The reservoir is resonant with the $|g\rangle \leftrightarrow |e\rangle$ transition, and the auxiliary cavity is resonant with the $|e\rangle \leftrightarrow |f\rangle$ transition. $|g\rangle ,|e\rangle ,|f\rangle$ are in blue (medium gray), orange (light gray), and red (dark gray), respectively. (b) Schematic of the circuit. To the left is the auxiliary, and to the right is the reservoir cavity. $C_g^{ge}$ ($C_g^{ef}$) is a capacitor that couples the transmon with the reservoir (auxiliary) cavity, and $C_{\kappa }^{ge}$ ($C_{\kappa }^{ef}$) is a capacitor that provides the desired external quality factor.

Figure 2

(a) Population of the transmon levels under a two-photon drive, $\mathrm{\Omega }/2\pi =25$ MHz. Upper panel: $\gamma =0, g_a=0$; middle panel: $\gamma =1\mu {\text{s}}^{-1}, g_a=0$; lower panel: $\gamma =1\mu {\text{s}}^{-1}, g_a/2\pi =8$ MHz, ${\kappa }_a=138\mu {\text{s}}^{-1}$. (b) Low-energy subspace of the system spectrum. Dashed arrows show two-photon pumping, wavy arrows show relaxation processes. Blue (medium gray) levels correspond to $|g\rangle$, orange (light gray) correspond to $|e\rangle$, and red (dark gray) correspond to $|f\rangle$. (c) A single pumping stage showing the transition from $N-1$ to $N$ photons in the reservoir and the participating eigenstates.

Figure 3

(a) Estimating the effective relaxation rate ${\kappa }_a^{\text{eff}}$ due to the auxiliary cavity. (b) Estimating the effective pumping rate due to the two-photon drive ${\mathrm{\Omega }}_{2\text{ph}}$ and the effective relaxation ${\kappa }_a^{\text{eff}}$. (c) Estimating the effective relaxation rate due to the reservoir with $N$ photons inside. (d) $|g\rangle$ and $|e\rangle$ levels of transmon with effective $|e\rangle \to |g\rangle$ dissipation, which act as a two-level laser.

Figure 4

(a,b) Average number of photons in the reservoir depending on the drive frequency $\omega$ for $g_r/2\pi =g_a/2\pi =15$ MHz, ${\kappa }_r=0.2\mu {\text{s}}^{-1}, {\kappa }_a=128\mu {\text{s}}^{-1}, \mathrm{\Omega }/2\pi =20$ MHz [point A in (e,f)] in the steady state (a) and in 1 and $2\mu \mathrm{s}$ after the pumping was started (b). (c,e) Pumped photons and (d,f) logarithm of Fano factor vs coupling strengths for ${\kappa }_a=138\mu {\text{s}}^{-1}, {\kappa }_r=0.2\mu {\text{s}}^{-1}$. (c,d) $\mathrm{\Omega }/2\pi =15$ MHz, (e,f) $\mathrm{\Omega }/2\pi =20$ MHz. The letters are different sets of the parameters used in the other subsections (O depicts the optimal parameters). The orange dashed line marks the transition to the threshold-bearing regime, and the blue dashed line denotes the thresholdless regime.

Figure 5

Reservoir population (a) and the Fano-factor (b) for the thresholdless and threshold-bearing regimes (points B–F in Fig. 4, values for $g_{a,r}/2\pi$ in the legend are in MHz). For all panels, ${\kappa }_a=138\mu {\text{s}}^{-1}, {\kappa }_r=0.2\mu {\text{s}}^{-1}$. (c,d) Reservoir Fock state probability for various $\mathrm{\Omega }$: (c) point G, (d) point F. (e)–(h) Steady-state reservoir Wigner functions: (e) for (c), $\mathrm{\Omega }/2\pi =19.5$ MHz; (f) for (c), $\mathrm{\Omega }/2\pi =21$ MHz; (g) for (d), $\mathrm{\Omega }/2\pi =23$ MHz; (h) for (d), $\mathrm{\Omega }/2\pi =26$ MHz.

Figure 6

Visualized reservoir states for the system with the following parameters: ${\kappa }_a=138\mu {\text{s}}^{-1}, {\kappa }_r=0.2\mu {\text{s}}^{-1}, g_r/2\pi =8$ MHz, $g_a=11$ MHz [point F in Figs. 4 and 4], for different $\mathrm{\Omega }$. (a) With an ordinary interaction Hamiltonian, (b) with a modified interaction Hamiltonian. (c,d) Number of photons in the reservoir (c) and Fano factor (d) for ordinary and modified interaction Hamiltonian. (e) Number of photons in the reservoir depending on the detuning of the drive frequency $\omega$ from a frequency of reservoir resonator ${\omega }_{ge}$. The system is modified in such a way that allows only one-photon direct reservoir pumping: the number of transmon states is truncated to two when $g_r/2\pi =8$ MHz, $g_a=0$, and the other parameters are the same as in Fig. 5.