Transport properties of a superconducting single-electron transistor coupled to a nanomechanical oscillator

We investigate a superconducting single-electron transistor capacitively coupled to a nanomechanical oscillator and focus on the double Josephson quasiparticle resonance. The existence of two coherent Cooper-pair tunneling events is shown to lead to pronounced back action effects. Measuring the current and the shot noise provides a direct way of gaining information on the state of the oscillator. In addition to an analytical discussion of the linear-response regime, we discuss and compare results of higher-order approximation schemes and a fully numerical solution. We find that cooling of the mechanical resonator is possible and that there are driven and bistable oscillator states at low couplings. Finally, we also discuss the frequency dependence of the charge noise and the current noise of the superconducting single electron transistor.

Figure 1

(Color online) Schematic setup of the SSET-resonator system: two superconducting leads at voltages $V_L$ and $V_R$ are coupled by tunnel junctions to a superconducting island. Its chemical potential can be tuned by a gate voltage $V_G$. A nearby nanomechanical oscillator acts as an $x$-dependent gate.

Figure 2

(Color online) Illustration of the DJQP cycle: (i) Cooper-pair tunneling through the left junction, (ii) quasiparticle tunneling through the right junction, (iii) Cooper-pair tunneling through the right junction, and (iv) quasiparticle tunneling through the left junction.

Figure 3

(Color online) Oscillator energy in units of $\hslash \Omega$ in the Gaussian approximation as a function of the gate voltage $e{V}_G/\hslash \Omega$ and the bias voltage $eV/\hslash \Omega$ where (0, 0) denotes the resonance. The parameters used are ${\tilde{\Gamma }}_L={\tilde{\Gamma }}_R=10$, ${\tilde{J}}_L={\tilde{J}}_R=2,{\tilde{\gamma }}_{\text{ext}}={10}^{-4}$, ${\tilde{T}}_B=2.5$, and $\tilde{A}=0.02$. In the red-detuned area $\left(V<0,-V<V_G<V\right)$, cooling below the bath temperature is visible (blue region). Driving can be observed in the blue-detuned case $\left(V>0,-V<V_G<V\right)$. The highest energies are obtained in the yellow region.

Figure 4

(Color online) Oscillator energy in the Gaussian approximation in units of $\hslash \Omega$ for the Gaussian approximation as a function of the gate voltage $e{V}_G/\hslash \Omega$ and the bias voltage $eV/\hslash \Omega$ for increased coupling $\tilde{A}=0.03$. Cooling and driving effects are increased as compared to Fig. 3. In the black area two stable and one unstable solutions are found, i.e., the system becomes bistable. The area grows for stronger coupling.

Figure 5

(Color online) Distribution $P\left(n_{\text{osc}}\right)$ of the oscillator phonon number for different values of the bias voltage $eV/\hslash \Omega =\left\{-1,0,1,\dots ,7\right\}$ from left to right along the $x$ axis. The parameters used for this plot are $\tilde{A}=0.1$, ${\tilde{\Gamma }}_L={\tilde{\Gamma }}_R=12$, ${\tilde{J}}_L={\tilde{J}}_R=2.5$, ${\tilde{\gamma }}_{\text{ext}}=0.001$, and ${\tilde{T}}_B=3$. For negative $V$ and small positive $V$ we find an exponential decay corresponding to a thermal state. For larger $V>0$ the distribution develops a peak at $n\ne 0$ which indicates a driven state.

Figure 6

(Color online) Oscillator energy in arbitrary units as a function of bias voltage $eV/\hslash \Omega$ and coupling $\tilde{A}$ calculated in the Gaussian approximation. For $V<0$, back action leads to cooling of the oscillator (blue). On the contrary, strong driving (red/yellow) can be observed for $V>0$. Above a critical coupling $\tilde{A}\left(V\right)$, the system enters a bistable region (black). The parameters are the same as in Fig. 3 and $e{V}_G/\hslash \Omega =0$.

Figure 7

(Color online) Difference of the inverse current $1/⟨I⟩\left(\tilde{A}\right)-1/⟨I⟩\left(\tilde{A}=0\right)$ versus bias voltage $eV/\hslash \Omega$ for various coupling strengths $\tilde{A}$. From the intersection with the voltage axis, the shift $2\tilde{A}⟨\tilde{x}⟩$ of the oscillator position can be read out. Lines correspond to the numerical analysis and agree well with the Gaussian approximation (points) in this parameter regime. Inset: Average current $⟨I⟩$ through the SSET for the parameters ${\tilde{\Gamma }}_L={\tilde{\Gamma }}_R=10$, ${\tilde{J}}_L={\tilde{J}}_R=2$, ${\tilde{\gamma }}_{\text{ext}}={10}^{-4}$, and ${\tilde{T}}_B=3$.

Figure 8

(Color online) Average current $⟨I⟩$ versus bias voltage for $\tilde{A}=0.01$ and $\tilde{A}=0.032$, ${\tilde{T}}_B=2.5$, further parameters as in Fig. 7. For increased coupling the Lorentzian peak becomes distorted and two stable solutions emerge which result in two stable values for the current.

Figure 9

(Color online) Frequency-dependent charge noise of the charge $n$ on the SSET, symmetrized contribution $S_n^{\text{symm}}\left(\omega \right)$, versus $\omega /\Omega$ for different values of the bias voltage $eV/\hslash \Omega$ measured from the resonance. Parameters are identical to Fig. 5. While there is no structure in the case $V=0$, we can distinguish if the oscillator is driven or cooled by the symmetry of the peak at $\omega =\Omega$. Inset: $S_n^{\text{symm}}\left(\omega \right)$ in a larger parameter regime; numerical calculation $\tilde{A}=0.1$ compared to analytical result $\tilde{A}=0$.

Figure 10

(Color online) Difference in Fano factors $\Delta F=F\left(\tilde{A}\right)-F\left(\tilde{A}=0\right)$ in units of ${\tilde{A}}^2$ as a function of bias voltage $eV/\hslash \Omega$ for ${\tilde{\Gamma }}_L={\tilde{\Gamma }}_R=10$, ${\tilde{J}}_L={\tilde{J}}_R=2$, ${\tilde{\gamma }}_{\text{ext}}=0.0001$, and ${\tilde{T}}_B=2.5$. Lines depict results of the numerical approach. The results of the Gaussian approximation (points) fit quantitatively only for $\tilde{A}<0.01$. Inset: the Fano factor $F\left(\tilde{A}=0.02\right)$ decreases at $V\approx 0$ when the Cooper-pair tunneling is resonant.

Figure 11

(Color online) Logarithmic Fano factor $\text{ln}F$ (dashed line) and oscillator energy (solid line) $E/\hslash \Omega$ as a function of bias voltage $eV/\hslash \Omega$. The parameters are $e{V}_G/\hslash \Omega =2.5$, ${\tilde{\Gamma }}_L={\tilde{\Gamma }}_R=12$, ${\tilde{J}}_L={\tilde{J}}_R=2.5$, ${\tilde{\gamma }}_{\text{ext}}=0.001$, ${\tilde{T}}_B=3$, and $\tilde{A}=0.1$. The Fano factors shows a first peak when the system enters the driven state. At the onset of the bistability, the Fano factor increases drastically on the logarithmic scale, which we attribute to telegraph noise due to switching between the two stable configurations. Inset: frequency-dependent current noise in the bistable region $\left(eV/\hslash \Omega =12.8\right)$ which decays on the scale of the switching rates.