Resonators coupled to voltage-biased Josephson junctions: From linear response to strongly driven nonlinear oscillations

Motivated by recent experiments in which a voltage-biased Josephson junction is placed in series with a resonator, the classical dynamics of the circuit is studied in various domains of parameter space. This problem can be mapped onto the dissipative motion of a single degree of freedom in a nonlinear time-dependent potential, where in contrast to conventional settings the nonlinearity appears in the driving while the static potential is purely harmonic. For long times the system approaches steady states which are analyzed in the underdamped regime over the full range of driving parameters including the fundamental resonance as well as higher harmonics and subharmonics. Observables such as the dc-Josephson current and the radiated microwave power give direct information about the underlying dynamics covering phenomena such as bifurcations, irregular motion, and up- and down-conversion. Due to their tunability, present and future setups provide versatile platforms to explore the changeover from linear response to strongly nonlinear behavior in driven dissipative systems under well defined conditions.

Figure 1

Circuit diagram of a voltage-biased Josephson junction in series with a resonator with relevant mode frequency ${\omega }_0$ described by an effective impedance $Z\left(\omega \right)$ as specified in (1). Using a SQUID geometry [1] for the JJ allows for tuning the effective Josephson energy $E_J\left({\mathrm{\Phi }}_x\right)$ by an external magnetic flux ${\mathrm{\Phi }}_x$.

Figure 2

Tuning the dimensionless Josephson coupling $\lambda$ the steady-state dynamics at the fundamental resonance, $\mathrm{\Omega }={\omega }_J/{\omega }_0=1$, accesses different ranges discussed in Secs. 3a, 3b, 3c, which are characterized by typical phase-space portraits. Within the domain 3b, the dotted vertical line separates the range of regular dynamics from the range ${\lambda }_2<\lambda$, where the transition towards irregular dynamics occurs. The solid line illustrates the corresponding behavior of the dimensionless dc current through the JJ with the shadow indicating its standard deviation when starting the steady-state dynamics with thermal initial conditions. The specific data shown for $\gamma =0.01$ exemplify the generic behavior in the underdamped regime. In this regime, the parameters ${\lambda }_1/\gamma ,{\lambda }_1^{*}$, and ${\lambda }_2$ are not very sensitive to varying the friction strength.

Figure 3

Left: Mean dc current $\langle I_J\rangle$ through the JJ in steady state and on resonance $\mathrm{\Omega }=1$ as a function of the scaled driving amplitude $\lambda /\gamma$ for (from bottom to top) $\gamma =0.005$ (blue), 0.01 (black, cf. Fig. 2), 0.02 (red). Right: Power transfer $P_{\mathrm{JJ}\to \mathrm{HO}}$ from the JJ to the resonator during one period of the oscillations in the steady state at $t_{\infty }=n2\pi ,n\gg 1,n\in \mathbb{N}$. The power transfer is shown for $\gamma =0.01$ and at four different driving strengths (also marked in the left panel): $\lambda /\gamma =0.1$ (A), 2 (B), 3 (C), and 3.5 (D). While the harmonic-oscillator type-I solutions gain energy from the drive nearly during the whole oscillation cycle (A and B), for type-II oscillations the energy gained during part of the cycle is partly flowing back to the drive in other parts (C and D). In consequence, increasing the driving strength beyond ${\lambda }_1/\gamma \approx 2.9$ does not further increase oscillation amplitude and current.

Figure 4

Top: Dissipated power $P_{\mathrm{diss}}$ from the driven JJ+resonator system for $\gamma =0.01,\mathrm{\Omega }=1$, and various driving strengths ${\lambda }_1<\lambda <{\lambda }_2$ around the second bifurcation ${\lambda }_1^{*}$. Bottom: Snapshots of the effective time-dependent potential $V_{\mathrm{eff}}\left(\phi \right)$ (6) for various driving strengths (see top for the color code) beyond the first threshold $\lambda >{\lambda }_1$ and at times $\mathrm{\Omega }t=(2n+1)\pi ,n\in \mathbb{N}$.

Figure 5

Various classes of steady-state orbits (outer panels) depending on the initial conditions (inner panels) for $\gamma =0.01$ and strong $\lambda =1$ (left) and very strong driving $\lambda =2.455$ (right). The color code for the initial conditions in phase space corresponds to a particular type of steady-state orbit in phase space. Red dots result from Poincaré plots and indicate points of return for steady-state orbits after one period $2\pi /\mathrm{\Omega }$.

Figure 6

(a) Phase-space portraits of steady-state orbits with Poincaré points (red) in the regime of extremely strong driving with $\lambda =30\gtrsim {\lambda }_3$ and $\gamma =0.01$ (top left), 0.025 (top right), 2.5 (bottom right), 5 (bottom left). Sketched in (b) is the particle dynamics in the effective potential $V_{\mathrm{eff}}\left(\phi \right)$, see Eq. (6), corresponding to phase-space orbits in the right panels of (a). For moderate damping the particle is trapped in one of the local wells, “elevated” upwards in $V_{\mathrm{eff}}\left(\phi \right)$ [the motion towards large negative $\phi$ values with small $\dot{\phi }$ in (a)] until its escape and consequent retrapping.

Figure 7

(a) Power spectrum, $ln\left\{\right|\tilde{\phi }\left(\omega \right){|}^2\}$, of steady-state orbits $\phi \left(t\right)$ for a range of driving frequencies $\mathrm{\Omega }$ in the underdamped regime $\gamma =0.1$ and at moderate driving amplitude $\lambda =0.2$. For the sake of presentation the spectrum is numerically broadened by taking the Fourier transform of a finite-time signal. (b) Dominant components $|{\phi }_k{|}^2$ of $\phi \left(t\right)$ [see (7)], corresponding to lines $\omega =k\mathrm{\Omega }$ in (a). Panel (c) shows a cut of the power spectrum at the (shifted) subharmonic resonance $\mathrm{\Omega }=0.365\approx {\mathrm{\Omega }}_{\frac{1}{3}}$ at a larger driving amplitude $\lambda =0.7$, where the system dominantly responds with $\omega =3\mathrm{\Omega }\approx 1$.

Figure 8

The dc current through the JJ vs driving frequency $\mathrm{\Omega }$ and driving amplitude $\lambda$ for $\gamma =0.1$. For weak driving the system is nearly linear and dominated by the fundamental resonance, $\mathrm{\Omega }={\omega }_0=1$, while for stronger driving the subharmonic resonances at $\mathrm{\Omega }={\mathrm{\Omega }}_{\frac{1}{n}}={\omega }_0/n$ and the multiphoton resonances at $\mathrm{\Omega }={\mathrm{\Omega }}_n=n{\omega }_0$ become apparent. Note the sharp onset of two-photon processes at ${\lambda }_c\approx \gamma \mathrm{\Omega }=0.2$ described in Sec. 4a, while subharmonic resonances increase smoothly and shift with increasing $\lambda$ as discussed in Sec. 4b (cf. also Fig. 9).

Figure 9

Time-averaged steady-state energy in the resonator for $\gamma =0.1$ and at different driving strengths: $\lambda =0.1$ (black), 0.15 (blue), 0.2 (green), 0.5 (red). For the two-photon resonance, $\mathrm{\Omega }=2$, there is a rather sharp threshold at $\lambda \gtrsim {\lambda }_c=2\gamma$, while subharmonic resonances, $\mathrm{\Omega }\approx 1/n$, gradually increase and shift with increased driving strength.

Figure 10

Mean steady-state oscillation amplitude, ${\phi }_{\max }=\max \left[{\langle \phi (t\to \infty )\rangle }_{\xi }\right]$, vs driving strength averaged over 10 000 realizations of thermal noise at temperatures $k_{\mathrm{B}}T=0$ (black), 0.01 (green dashed), 0.1 (blue), 0.25 (red) for friction parameter $\gamma =0.01$.

Figure 11

Mean dissipated power at temperature $k_{\mathrm{B}}T=0$ (top two panels) and $k_{\mathrm{B}}T=0.1$ (bottom) with $\gamma =0.1$ and $\mathrm{\Omega }=2$ for various driving strengths $\lambda =0.15$ (black), 0.2 (blue), 0.5 (green). The threshold for the parametric amplification in the absence of noise is ${\lambda }_c\approx 2\gamma =0.2$.