Charge noise at Cooper-pair resonances

We analyze the charge dynamics of a superconducting single-electron transistor (SSET) in the regime where charge transport occurs via Cooper-pair resonances. Using an approximate description of the system Hamiltonian, in terms of a series of resonant doublets, we derive a Born-Markov master equation describing the dynamics of the SSET. The average current displays sharp peaks at the Cooper-pair resonances and we find that the charge noise spectrum has a characteristic structure which consists of a series of asymmetric triplets of peaks. The strongest feature in the charge noise spectrum is the triplet of peaks centered at zero frequency which has a peak spacing equal to the level separation within the doublets and is similar to the triplet in the spectrum of a driven, damped, two-level system. We also explore the backaction that the SSET charge noise would have on an oscillator coupled to the island charge, measurement of which provides a way of probing the charge noise spectrum.

Figure 1

Circuit diagram of the SSET. The SSET island is linked to the leads by the Josephson junctions, $J_{L\left(R\right)}$, and is coupled to the voltage gate by the capacitance $C_g$. The bias voltage, $V$, and impedances in the circuit, $Z_0$, are taken to be distributed symmetrically.

Figure 2

(Color online) Energy levels of $H_{\text{ch}}$, (a) with $V=V_{\text{res}}^{\left(1\right)}$ at the $p=1$ resonance and (b) with $V\gtrsim V_{\text{res}}^{\left(0\right)}$ for $p=0$. The dashed lines enclose the doublets: the energy levels which are almost degenerate near the given resonance.

Figure 3

(Color online) Current-voltage characteristics of the SSET in the vicinity of (a) the $p=0$ resonance (b) the $p=1$ resonance. The current is scaled by $I_0=e\left({\Gamma }_L+{\Gamma }_R\right)$ in each case. The values of the other parameters are given in the text. In (a) we also show in the dashed (red) line the current calculated using the full voltage dependence of the decay rate ${\Gamma }_R$ and the Hamiltonian.

Figure 4

(Color online) Charge noise spectrum as a function of frequency and voltage around the (a) $p=0$ and (b) $p=1$ resonances. The resonances occur at the point where the central peak disappears.

Figure 5

(Color online) Charge noise, $S_{nn}$, as a function of $\Delta E$ for (a) $\Omega =100\text{MHz}$ and (b) $\Omega =4\text{GHz}$ oscillators. Solid (blue) lines are $S_{nn}\left(\Omega \right)$, dashed (red) lines are $S_{nn}\left(-\Omega \right)$. The inset in (a) shows the region around the minimum where the difference between the two curves is most pronounced.

Figure 6

[(a) and (b)] Backaction damping, ${\gamma }_{BA}$ and [(c) and (d)] effective occupation number, ${\overline{n}}_{BA}$ of an oscillator weakly coupled to the SSET. (a) and (c) are for $\Omega =100\text{MHz}$, (b) and (d) for $\Omega =4\text{GHz}$.

Figure 7

(Color online) $S_{n^{\left(1\right)}{n}^{\left(1\right)}}\left(\omega \right)$ in the vicinity of the $p=1$ resonance (a) and (b) show the spectrum at frequencies around $\hslash \omega =\pm 2eV$, respectively. A schematic illustration of the processes giving rise to the spectral features is shown in (c). This illustrates the interdoublet transitions and the corresponding frequencies of the features they give rise to.