Quantum Dynamics of a Resonator Driven by a Superconducting Single-Electron Transistor: A Solid-State Analogue of the Micromaser

We investigate the behavior of a quantum resonator coupled to a superconducting single-electron transistor (SSET) tuned to the Josephson quasiparticle resonance and show that the dynamics is similar in many ways to that found in a micromaser. Coupling to the SSET can drive the resonator into nonclassical states of self-sustained oscillation via either continuous or discontinuous transitions. Increasing the coupling further leads to a sequence of transitions and regions of multistability.

Figure 1

(a) Schematic diagram of the SSET-resonator system. The SSET consists of a superconducting island linked by tunnel junctions to superconducting leads, across which a voltage $V_{ds}$ is applied. (b) Charge processes involved in the JQP current cycle (details are given in the text).

Figure 2

Steady state properties of the resonator as a function of $\Delta E$ and resonator speed $\omega /\Gamma$: (a) average occupation number of the resonator, $⟨n⟩$; (b) Fano factor of the resonator, both plotted on a natural-log scale, with $\kappa =0.01$, ${\gamma }_{\mathrm{ext}}/\Gamma =0.002$, and $\overline{n}=0$.

Figure 3

(a) Average occupation number and Fano factor of the resonator together with the most probable number state in the density matrix, $n_{mp}$, for $\omega /\Gamma =0.1$, ${\gamma }_{\mathrm{ext}}/\Gamma =0.0005$, $\kappa =0.02$, and $\overline{n}=0$. (b) Examples of the resonator Wigner functions: I, fixed point state; II, bistable; III, limit cycle. The regions in (a) corresponding to the three basic Wigner function topologies are also marked as I (fixed point), II (bistable), and III (limit cycle).

Figure 4

Changes in the resonator state (a) and corresponding Wigner functions (b) as the coupling $\kappa$ is increased for $\Delta E/e{V}_{ds}=-0.1$, $\omega /\Gamma =1$, ${\gamma }_{\mathrm{ext}}/\Gamma =0.001$, and $\overline{n}=0$. After an initial continuous transition ($\kappa \simeq {0.025}^2$), the system undergoes discontinuous transitions at $\kappa \simeq {0.22}^2$ and $\kappa \simeq {0.44}^2$. The inset in (a) shows the region where the Fano factor drops below unity. From top to bottom, the Wigner functions show: a fixed point state, a single limit cycle, and two metastable limit cycles.