Spectral properties of a resonator driven by a superconducting single-electron transistor

We analyze the spectral properties of a resonator coupled to a superconducting single-electron transistor (SSET) close to the Josephson quasiparticle resonance. Focusing on the regime where the resonator is driven into a limit-cycle state by the SSET, we investigate the behavior of the resonator linewidth and the energy relaxation rate which control the widths of the main features in the resonator spectra. We find that the linewidth becomes very narrow in the limit-cycle regime, where it is dominated by a slow phase-diffusion process, as in a laser. The overall phase-diffusion rate is determined by a combination of direct phase diffusion and the effect of amplitude fluctuations which affect the phase because the resonator frequency is amplitude dependent. For sufficiently strong couplings we find that a regime emerges where the phase diffusion is no longer minimized when the average resonator energy is maximized. Finally we show that the current noise of the SSET provides a way of measuring both the linewidth and energy relaxation rate.

Figure 1

(a) Superconducting single-electron transistor: two tunnel junctions and a gate capacitor $C_g$ form the SSET island to which the resonator is coupled capacitively. A drain source voltage, $V_{\text{ds}}$, and gate voltage, $V_{\text{g}}$, are used to tune the operating point of the device. (b) JQP cycle: Cooper-pair tunneling at the left-hand junction between island states $|0⟩$ and $|2⟩$ is interrupted by two quasiparticle tunneling events which take the island charge back to $|0⟩$ via the $|1⟩$ state.

Figure 2

(Color online) (a) Average resonator energy $⟨n⟩$ and (b) Fano factor $F_n$, for varying $\Delta E$ and $\kappa$, with $\Omega =10$ and ${\gamma }_{\text{ext}}=3\times {10}^{-4}$ (we adopt units such that $\Gamma =1$). Within the inner dashed line the resonator is in a limit-cycle state, between the two lines it is in the transition region (as defined in the text) and elsewhere it is in a fixed-point state.

Figure 3

(Color online) Resonator spectrum $S_{a,a^{†}}\left(\omega \right)$ for $\kappa =0.003$ (other parameters are the same as in Fig. 2). (a) Off-resonance $\Delta E=-1.75e{V}_{\text{ds}}$, ${\Omega }_{\text{SSET}}=1.1\Omega$ on a ${\text{log}}_{10}$ scale, (b) on-resonance $\Delta E=-1.59e{V}_{\text{ds}}$, ${\Omega }_{\text{SSET}}=\Omega$ on a ${\text{log}}_{10}$ scale. The insets show the behavior around the peak at $\omega \simeq \Omega$ without logarithmic scaling (full lines) along with Lorentzian fits to the peaks (dashed lines).

Figure 4

(Color online) Width of zero-frequency peak in the $S_{n,n}\left(\omega \right)$ spectrum (upper curves) as determined from a Lorentzian fit (${\gamma }_n/2$, solid line) and from the smallest nonzero eigenvalue ($-{\lambda }_1/2$, dashed line). The lower curves show the linewidth of the resonator as determined from a Lorentzian fit to $S_{a,a^{†}}\left(\omega \right)$ (${\gamma }_{\Omega }$, solid line) and from the eigenvalue ($-\mathfrak{R}{\lambda }_{\Omega }$, dashed line). $\kappa =0.003$ with other parameters the same as in Fig. 2. Vertical dashed lines indicate the transition region.

Figure 5

(Color online) Widths of spectral peaks calculated from the eigenvalues as a function of $\kappa$ and $\Delta E$. (a) ${\text{log}}_{10}\left[{\gamma }_{\Omega }\right]$, the logarithm of the linewidth of $S_{a,a^{†}}\left(\omega \right)$. (b) ${\gamma }_n$, the width of the zero-frequency peak in $S_{n,n}\left(\omega \right)$, (other parameters and lines are as in Fig. 2).

Figure 6

(Color online) Comparison of the width of the zero-frequency peak in $S_{n,n}\left(\omega \right)$, as given by the eigenvalue ${\gamma }_n\left(\Delta E\right)$ (solid line), with the energy relaxation rate ${\gamma }_{\text{lin}}\left(\Delta \tilde{E}\right)$ (dashed line). Here $\kappa =0.03$, with other parameters the same as in Fig. 2.

Figure 7

(Color online) Comparison of the linewidth ${\gamma }_{\Omega }$ of $S_{a,a^{†}}\left(\omega \right)$ obtained from the ${\lambda }_{\Omega }$ eigenvalue (plotted as a function of $\Delta E$) with the analytic expression ${\gamma }_{\varphi }={\gamma }_{\varphi }^{\varphi }+{\gamma }_{\varphi }^n$ (plotted as a function of $\Delta \tilde{E}$). Also shown are the direct $\left({\gamma }_{\varphi }^{\varphi }\right)$ and indirect $\left({\gamma }_{\varphi }^n\right)$ phase-diffusion contributions. Here $\kappa =0.03$ with other parameters the same as in Fig. 2.

Figure 8

(Color online) Finite-frequency current noise spectrum for $\kappa =0.003$ with the other parameters the same as in Fig. 2. (a) Off-resonance $\Delta E=-1.75e{V}_{\text{ds}}$, (b) on-resonance for $\Delta E=-1.59e{V}_{\text{ds}}$. The insets show expansions of the narrow peaks at $\omega \simeq \Omega$ without logarithmic scaling (full lines) along with fits to the peaks (dashed lines) with the form of Eq. (13), inset (a), and Lorentzian form, inset (b).