Dynamics of a SQUID ratchet coupled to a nanomechanical resonator

We investigate the dynamics of a superconducting rectifying circuit, namely, a three-junction superconducting quantum interference device (SQUID), where one arm of the superconducting loop consists of a nanomechanical resonator. We find that the dc characteristic curve of the system displays features that are directly related to the frequency and amplitude of the mechanical oscillations; these effects can be further enhanced by biasing the SQUID with an ac current. We discuss potential future developments of this idea in the field of ultrasensitive position detection, and assess their feasibility in current setups.

Figure 1

(Color online) (a)–(b) A model and the corresponding scheme of the ratchet circuit considered here. Three Josephson junctions are coupled to a mechanical resonator oscillating in the plane of the circuit, threaded by a magnetic flux, and biased by a current. The junctions have equal critical currents and capacitances but different shunting resistances. (c) The potential $U\left(\gamma \right)$ in the absence of a moving part (solid line); the average slope in the decreasing part is larger in modulus than the average slope of the increasing part, as can be seen by comparing with a cosine curve (dashed line).

Figure 2

(Color online) (a) Dimensionless dc voltage drop $v_0=V_0/R{I}_c$ as a function of the dimensionless bias current $I_{\text{dc}}/I_c$, for different values of the flux. No oscillating mechanical part is present $\left(a=0\right)$. (b) The even part $v_0^e$ of the same curves plotted for positive values of $I_{\text{dc}}$ only. $v_0^e$ is zero for $\varphi =n\pi$ and $v_0^e\left(-\varphi \right)=-v_0^e\left(\varphi \right)$.

Figure 3

(Color online) $v_0^e\left(I_{\text{dc}}\right)$ curves for different values of the coupling parameter $a$. The two plots refer to different values of the flux: (a) $\varphi =\pi (n+1/2)$ and (b) $\varphi =\pi (n+1/20)$. While for small coupling one recovers the curves of Fig. 2, for high coupling the features are washed out. The effect is more pronounced for small flux bias.

Figure 4

(Color online) $v_0^e\left(I_{\text{dc}}\right)$ curves for $I_{\text{ac}}=I_c$, ${\Omega }_{\text{ac}}=2\Omega /3$ and different values of the coupling parameter $a$ (the same as in Fig. 3). The two plots refer to different values of the flux bias: (a) $\varphi =\pi (n+1/2)$ and (b) $\varphi =\pi (n+1/20)$. Again for small coupling the curves are indistinguishable from the uncoupled ones $\left(a=0\right)$; however for high couplings now a richer structure appear and its set on occurs at a smaller coupling for $\varphi \sim n\pi$.

Figure 5

(Color online) $v_0^e\left(I_{\text{dc}}\right)$ for zero bias flux and different values of the ratio ${\Omega }_{\text{ac}}/\Omega$. Here $a=1$, while for $a=0$ the curves should be flat $\left(v_0^e\equiv 0\right)$.

Figure 6

(Color online) (a) Exact $v_0\text{−}{I}_{\text{dc}}$ curves for different frequencies $\Omega$ of the mechanical resonator. Here $a=1$, $I_{\text{ac}}=0$ and $\varphi =n\pi$. Even though no ac bias current is supplied, the system exhibits Shapiro steps, whose heights are related to $\Omega$. (b) The same but for a fixed frequency $\Omega =1$ and different values of the coupling $a$. The width of Shapiro steps approaches zero for small values of $a$.

Figure 7

The even component $v_0^e$ of the voltage drop as a function of the dc bias current $I_{\text{dc}}$ at $\varphi =n\pi$ and $\Omega =1$ for different values of the coupling $a$ (the curves are displaced vertically). The amplitude of the features is the same for all couplings, but their widths become smaller for smaller couplings.

Figure 8

The even component $v_0^e$ of the voltage drop as a function of the dc bias current $I_{\text{dc}}$ at $\varphi =n\pi$ and $I_{\text{ac}}=I_c$, for different values of the coupling $a$. Here $\Omega =1$ and ${\Omega }_{\text{ac}}=0.67$.

Figure 9

(Color online) Some details of Fig. 8 for $a=0.1$, $I_{\text{ac}}=1$, $\Omega =1$, and ${\Omega }_{\text{ac}}=0.67$; the curves correspond to different temperatures: $\sqrt{\Theta }=0$ (solid line) and $\sqrt{\Theta }={10}^{-4}$ (dashed line).