Effect of mechanical resonance on Josephson dynamics

We study theoretically dynamics in a Josephson junction coupled to a mechanical resonator looking at the signatures of the resonance in dc electrical response of the junction. Such a system can be realized experimentally as a suspended ultraclean carbon nanotube brought in contact with two superconducting leads. A nearby gate electrode can be used to tune the junction parameters and to excite mechanical motion. We augment theoretical estimations with the values of setup parameters measured in one of the samples fabricated. We show that charging effects in the junction give rise to a mechanical force that depends on the superconducting phase difference and can excite the resonant mode. We develop a model that encompasses the coupling of electrical and mechanical dynamics. We compute the mechanical response (the effect of mechanical motion) in the regime of phase and dc voltage bias. We thoroughly investigate the regime of combined ac and dc bias where Shapiro steps are developed and reveal several distinct regimes characteristic for this effect. Our results can be immediately applied in the context of experimental detection of the mechanical motion in realistic superconducting nanomechanical devices.

Figure 1

The setup. The sketch presents the mechanical resonator realized as a CNT suspended over two superconducting leads isolated from the back gate electrode. The CNT center is displaced in the $y$ direction by an electrostatic force produced by the gate voltage. The superconducting leads are parts of an electrical circuit characterized by an impedance $Z_e$. The setup can be biased by either voltage or current source.

Figure 2

(Left) Real part of the complex displacement amplitude $\mathrm{Re}\tilde{y}$ vs detuning. (Right) Imaginary part of the complex displacement amplitude $\mathrm{Im}\tilde{y}$ vs detuning. The three curves in each panel correspond to $|\tilde{F}|/F_c=0.5,1,2$.

Figure 3

The phase-dependent frequency shift for the case of weak driving. The curves give the linear response of dc current in units $I_a=2e{\left(\Delta {\omega }_0\right)}_{\max }{\left|\tilde{y}\right|}^2(M{\omega }_0/\hslash )$ as a function of frequency detuning for a set of phase bias values: from the lowermost to the uppermost curve the phase changes from $\varphi =\pi /8$ to $\varphi =15\pi /8$, with interval $\pi /8$. The curves are offset for clarity. Dashed lines give the positions of zero.

Figure 4

An example of Fano-type frequency dependence of the mechanical response (38). The curves correspond to the driving force values $|\tilde{F}|/F_c=0.2,0.6,1$ (top) and $|\tilde{F}|/F_c=2,6,10$ (bottom). The current is in units of $\frac{d^2{I}_1\left(q\right)}{d{q}^2}\frac{d{C}_g}{dy}{q}_0{\tilde{V}}_g{y}_c$, which assuming ${\tilde{V}}_g/V_{g0}\simeq {10}^{-2}$ amounts to $\simeq {10}^{-3}{I}_c$ for the parameter set chosen.

Figure 5

(Top left) the charge-dependent part of Josephson energy as a function of the gate-induced charge $q=C_g{V}_g$. The crosses indicate the values of $q$ that correspond to the values of Josephson force used in other panels. The blue dotted line indicates the value of $E_j$ where $F_j=F_c$. (Top right) the mechanical response as a function of detuning for relatively low values of the force $F_j/F_c<1$ at which the response increases with increasing the force. (Bottom) frequency dependence at the force values $F_j/F_c=3,5,7,10$ (from left to right ) where the response decreases with increasing the force.

Figure 6

The mechanical response at resonant driving $\Omega \simeq {\omega }_0$ and at the first Shapiro step $V_0=2e\Omega /\hslash$ vs the oscillating phase ${\varphi }_1$. The first (upper) plot gives the maximum current at the step. The second and fourth plots give the mechanical response defined as the extremum of the modification of this maximum current over the frequencies near the resonance, at maximum Josephson forces $\overline{F}=F_c$ and $\overline{F}=50F_c$, respectively. The actual amplitudes of the resonant Josephson forces are given at the lower plots, respectively third and fifth.

Figure 7

The effect of gate force. For all plots, the gate force is fixed to $F_g=F_c$. In the plots from top to bottom, the maximum Josephson force $\overline{F}$ assumes the values $\overline{F}/F_c=-5,-1,0,1,5$. We choose $\chi =0$ and ${\varphi }_1=1$. Dashed lines in the plots for $I^{-}$ give values opposite to the corresponding $I^{+}$, to stress the symmetry or asymmetry of the response.

Figure 8

The maximum current at the Shapiro step, the mechanical response and the force versus ${\varphi }_1$ for $\overline{F}=F_c$ and $\overline{F}=50F_c$ at the second Shapiro step $V_0=4e\Omega /\hslash$.

Figure 9

The same as in Fig. 8 at the fifth Shapiro step $V_0=10e\Omega /\hslash$.

Figure 10

The mechanical response at the nonresonant driving. Here, the ac driving frequency is $\Omega \simeq {\omega }_0/2$, corresponding to $N=2$. From left to right, the three columns correspond to Shapiro steps $m=1,2,3$. Plotted are the maximum current at the Shapiro step, the mechanical response and the amplitude of the force. The maximum of the Josephson force was set to $\overline{F}=50F_c$ for all plots.

Figure 11

The mechanical response at the nonresonant driving for $N=3$. Except this, all other parameters are the same as in Fig. 10.