Nanomechanical-resonator-induced synchronization in Josephson junction arrays

We show that a serial array of $N$ nonuniform, underdamped Josephson junctions coupled piezoelectrically to a nanoelectromechanical (NEM) oscillator results in phase locking of the junctions. Our approach is based on a semiclassical solution to a set of coupled differential equations that were generated by the Heisenberg operator equations, which in turn are based on a model Hamiltonian that includes the following effects: the charging and Josephson energies of the junctions, dissipation in the junctions, the effect of a dc bias current, an undamped simple harmonic oscillator (representing the NEM), and an interaction energy (due to the piezoelectric effect) between the NEM and the junctions. Phase locking of the junctions is signaled by a step in the current-voltage $\left(I\text{−}V\right)$ curve. We find the phase-locked states are (neutrally) stable at the bottom and top of the step but not for bias currents in the middle of the step. Using harmonic balance, we are able to calculate an analytical expression for the location of the resonance step, $v_{\mathit{\text{step}}}$, in the $I\text{−}V$ curve. Because of the multistability of the underdamped junctions, it is possible, with a judicious choice of initial conditions and bias current, to set a desired number $N_a⩽N$ of junctions on the resonance step, with $N-N_a$ junctions in the zero-voltage state. We are also able to show that, when $N_a$ junctions are in the phase-locked configuration, the time-averaged energy of the NEM oscillator scales like $N_a^2$.

Figure 1

A single Josephson junction and a dc current source $I_B$ in parallel with a NEM oscillator. The junction has critical current $I_c$, capacitance $C$, and shunt resistance $R$. The oscillator consists of a piezoelectric crystal sandwiched between the plates of a capacitor. The effective capacitance of the plates and crystal is $C_o$. The mechanical strain of the crystal, denoted by $U$, is taken to be along the $z$ axis and in a direction perpendicular to the disk.

Figure 2

A serial Josephson junction array and a dc current source $I_B$ in parallel with a NEM oscillator. Junction $i$ has critical current $I_{\mathit{ci}}$, capacitance $C_i$, and shunt resistance $R_i$. The oscillator consists of a piezoelectric crystal sandwiched between the plates of a capacitor. The effective capacitance of the plates and crystal is $C_o$.

Figure 3

(Color online) (a) $I\text{−}V$ curve for a serial array of $N=10$ underdamped junctions coupled to a NEM oscillator. Junction critical currents are assigned nonrandomly with $\Delta =0.05$ and ${\beta }_c=10$. The NEM’s natural frequency, scaled by the Josephson frequency, is $\Omega ={\omega }_o∕{\omega }_J=0.45$, and the coupling strength between the array and the NEM oscillator is $\Lambda =0.2$. The ratio of the average junction capacitance to the capacitance of the NEM is $⟨C⟩∕C_o=1$. A step due to resonance between the junctions and the NEM is evident at a voltage only slightly greater than $\Omega$, namely ${⟨v⟩}_{\tau }∕N=0.4683\pm 0.0007$. The error bars on the symbols represent the standard deviation of the mean of the time-averaged JJ voltages. As the bias current is decreased from its starting value of $i_B=1.2$, the error bars become smaller than the symbol size, signaling approximate frequency synchronization of the array. (b) Magnitude of the ten largest Floquet multipliers corresponding to on-step solutions for the same system parameters as in (a). For the stability analysis, the junction parameters were taken to be uniform, and $\mid \mu \mid$ was calculated as the bias current was ramped up while the system was on the step. For bias currents in the middle of the step, the system is linearly unstable. Near the top of the step, $i_B>0.85$, there is neutrally stable synchronization with two multipliers of magnitude unity. The next smallest multiplier has a magnitude reduced by 3 to 5% from unity. The system may also be neutrally stable in a very narrow bias current range at the bottom of the step.

Figure 4

Time average of the Kuramoto order parameter’s modulus for a serial array of $N=10$ junctions (${\beta }_c=10$ and nonrandomly assigned critical currents with $\Delta =0.05$) coupled to a NEM oscillator with scaled natural frequency $\Omega =0.45$. The coupling strength between the array and the NEM oscillator is $\Lambda =0.2$ and $⟨C⟩∕C_o=1$. (a) The bias current is decreased from a starting value of $i_B=1.2$. The array approaches phase synchronization, $r\to 1$, as $i_B$ is decreased toward the step. (b) Kuramoto order parameter versus bias current as $i_B$ is increased along the step. Phase synchronization is evident along a very narrow region at the bottom of the step as well as along a broader region at the top of the step.

Figure 5

(Color online) (a) Time dependence of ${\tilde{a}}_R$, which is proportional to the NEM oscillator’s displacement from equilibrium, for $i_B=0.9$ both on the resonance step (full curve) and on the Ohmic branch (dashed curve). Note the different scales on the two vertical axes. The junction and oscillator parameters are the same as in Fig. 3. (b) Time dependence of $v_1$, the voltage across junction 1, for $i_B=0.9$ both on the resonance step (full curve) and on the Ohmic branch (dashed curve). In both parts (a) and (b) the resonant behavior on the step is evident.

Figure 6

$I\text{−}V$ curve for a serial array of $N=10$ underdamped junctions coupled to a NEM oscillator. Junction critical currents are assigned nonrandomly with $\Delta =0.05$ and ${\beta }_c=10$. The NEM’s natural frequency is $\Omega =0.45$, and the coupling strength between the array and the NEM is $\Lambda =3$. The ratio of the average junction capacitance to the capacitance of the NEM is $⟨C⟩∕C_o=1$. A step due to resonance between the junctions and the NEM is evident at a voltage considerably greater than $\Omega$, namely ${⟨v⟩}_{\tau }∕N=0.6851\pm 0.0009$. The retrapping current is approximately equal to $\Omega$.

Figure 7

(Color online) Location of the voltage step as a function of the coupling strength $\Lambda$ between a serial array $(N=10,{\beta }_c=10)$ and NEM oscillator. The NEM’s natural frequency is $\Omega =0.45$ and $⟨C⟩∕C_o=10$. Junction critical currents are assigned nonrandomly with $\Delta =0.05$. Because the resonance step could not be seen for the nonuniform array at very small values of the coupling $\Lambda <0.1$, the results for $v_{\mathit{\text{step}}}$ for $\Lambda ⩽0.1$ were obtained for the array with uniform junctions (blue symbols on graph). The dashed horizontal line at $v_{\mathit{\text{step}}}=\Omega$ is shown for reference. The solid line is an analytic result for $v_{\mathit{\text{step}}}$ given by Eq. (61).

Figure 8

$I\text{−}V$ curve for a serial array of $N=50$ underdamped junctions coupled to a NEM oscillator. Junction critical currents are assigned nonrandomly with $\Delta =0.05$ and ${\beta }_c=10$. The NEM’s natural frequency, scaled by the Josephson frequency, is $\Omega ={\omega }_o∕{\omega }_J=0.45$, and the coupling strength between the array and the NEM oscillator is $\Lambda =0.01$. The ratio of the average junction capacitance to the capacitance of the NEM is $⟨C⟩∕C_o=50$. A step due to resonance between the junctions and the NEM is evident at a voltage only slightly greater than $\Omega$, namely ${⟨v⟩}_{\tau }∕N=0.5476\pm 0.0187$. The error bars on the symbols represent the standard deviation of the mean of the time-averaged JJ voltages. As the bias current is decreased from its starting value of $i_B=1.2$, the error bars become smaller than the symbol size, signaling approximate frequency synchronization of the array.

Figure 9

(Color online) $I\text{−}V$ curve for a serial array of $N=50$ underdamped junctions coupled to a NEM oscillator showing different numbers of junctions, $N_a$, biased on the resonance step. $N-N_a$ junctions are in the zero-voltage state. The curves represent $N_a=25$, 30, 35, 40, 45, and 50. Junction critical currents were uniform and ${\beta }_c=10$. The NEM’s natural frequency, scaled by the Josephson frequency, is $\Omega ={\omega }_o∕{\omega }_J=0.45$, and the coupling strength between the array and the NEM oscillator is $\Lambda =0.1$. The ratio of the average junction capacitance to the capacitance of the NEM is $⟨C⟩∕C_o=50$. Each curve was produced by increasing the bias current from a starting value between 0.41 to 0.45. The dotted lines represent the expected location of the step, $(N_a∕N)v_{\mathit{\text{step}}}$, where the value of $v_{\mathit{\text{step}}}$ is obtained from Eq. (61).

Figure 10

(Color online) Numerical solutions of Eqs. (38, 39, 40, 41) for $\gamma \left(\tau \right)$ and $d\gamma ∕d\tau$ versus time and fits to these numerical results. Parameters are $N=10,{\beta }_c=10,⟨C⟩∕C_o=1,\Lambda =0.5,\Omega =0.45$. For simplicity, the junction critical currents were taken to be uniform. (a) Josephson phase $\gamma$ with the linearly growing term, $v_{\mathit{\text{step}}}\tau$, subtracted out, leaving oscillatory behavior. The crosshairs represent a regression fit to a function $f\left(\tau \right)=\varphi +A\sin (v_{\mathit{\text{step}}}\tau +\delta )$. From the fit, we find $A=1.3051$ and $v_{\mathit{\text{step}}}=0.4978$. (b) The voltage across an individual junction, $d\gamma ∕d\tau$. The “$\times$” symbols represent a regression fit to a function $g\left(\tau \right)=v_{\mathit{\text{step}}}+v_{\mathit{\text{step}}}A\cos (v_{\mathit{\text{step}}}\tau +\delta )$. As a consistency check, note that the amplitude of $d\gamma ∕d\tau$ can be estimated from the graph and is in good agreement with the numerical value of $v_{\mathit{\text{step}}}A$ as specified by Eq. (49), where the values of $v_{\mathit{\text{step}}}$ and $A$ are taken from the fit in part (a).

Figure 11

(Color online) Numerical solution for ${\tilde{a}}_I$ versus time. For convenience, the linearly growing (in time) contribution is subtracted out, and the vertical axis is scaled by a factor of ${10}^5$. Parameters are $N=10,{\beta }_c=10,⟨C⟩∕C_o=1,\Lambda =0.5,\Omega =0.45$. For simplicity, the junction critical currents were taken to be uniform. The numerical value of the constant $F$ was determined from the fact that $F=-N{v}_{\mathit{\text{step}}}\sqrt{\Lambda }∕2$. From the fit, we find $D=20.6987$, which agrees with the expression $D=\{N{\Omega }^2\sqrt{\Lambda }∕\left[2(v_{\mathit{\text{step}}}^2-{\Omega }^2)\right]\}A$ [see Eq. (51)], if we use the value $A=1.3051$ from Fig. 10.

Figure 12

Square of the amplitude of the sinusoidal contribution to the Josephson phase versus the dc bias current. Parameters are $N=10$, ${\beta }_c=10$, $⟨C⟩∕C_o=1$, and $\Omega =0.45$. The triangle symbols denote numerical results for the squared amplitude of the Josephson phase oscillations, denoted by $A^2$, for a coupling $\Lambda =0.5$. The solid line is the result $A^2=2(i_B∕v_{\mathit{\text{step}}}-1)$ for $v_{\mathit{\text{step}}}=0.4945$ versus $i_B$. The square symbols denote numerical results for $A^2$ for $\Lambda =3$. The dotted line is the result $A^2=2(i_B∕v_{\mathit{\text{step}}}-1)$ for $v_{\mathit{\text{step}}}=0.68$. The numerical results are obtained only for those applied currents on the step where phase locking occurs.