Microwave bifurcation of a Josephson junction: Embedding-circuit requirements

A Josephson tunnel junction which is rf driven near a dynamical bifurcation point can amplify quantum signals. However, the bifurcation point will exist robustly only if the electrodynamic environment of the junction meets certain criteria. We develop a general formalism for dealing with the nonlinear dynamics of a Josephson junction embedded in an arbitrary microwave circuit. We find sufficient conditions for the existence of the bifurcation regime: (a) the embedding impedance of the junction needs to present a resonance at a particular frequency ${\omega }_R$, with the quality factor $Q$ of the resonance and the participation ratio $p$ of the junction satisfying $Qp⪢1$, and (b) the drive frequency should be low frequency detuned away from ${\omega }_R$ by more than $\sqrt{3}{\omega }_R∕\left(2Q\right)$.

Figure 1

(Color) Upper panels: response of the Duffing $\text{linear}+\text{cubic}$ oscillator [see Eq. (2)] as a function of the dimensionless detuning $\Omega$, for different values of the dimensionless driving amplitude $f$. The three panels correspond to three different values of the small oscillation quality factor $Q=20$, 50, and 200 and are shown over the same reduced frequency range $0.9<w<1.05$. The response for the critical drive amplitude $f_c$, where the curve presents a single point with diverging susceptibilities $\partial x∕\partial \Omega$ and $\partial x∕\partial f$, is shown in green. From bottom to top, the increments in the drive amplitude $f$ are $2\mathrm{dB}$ (left), $2\mathrm{dB}$ (center), and $3\mathrm{dB}$ (right). In the gray region, the oscillation amplitude is taking place in the strongly nonlinear regime (see text) and the curves, which are calculated within the weak non-linear hypothesis, are not to be trusted. Lower panels: stability diagram for the dynamical states corresponding to the system response shown in the upper panels. The $y$ axis is the drive amplitude $f$, scaled by the critical amplitude $f_c$. The blue $\left[f_B\left(\Omega \right)\right]$ and red $\left[f_{\overline{B}}\left(\Omega \right)\right]$ lines delimit the region of bistability for the system and correspond to the points of diverging susceptibility which are visible in the upper panels. The two curves meet at the critical point shown by a green dot, whose position is determined by the critical detuning ${\Omega }_c=-\sqrt{3}$ and drive amplitude $f_c$. The dashed line $\left[f_{ms}\left(\Omega \right)\right]$ “continues” the blue and red lines and corresponds to the points of maximal susceptibility with respect to driving force. The gray regions correspond to that in the upper panels. Note that the “trusworthy” domain of bistability increases monotonously with the quality factor $Q$.

Figure 2

(a) Schematic of a Josephson junction rf biased through an arbitrary electric quadrupole containing only linear components. The Thevenin (b) and Norton (c) representations of the circuit of (a) show the dipole seen by the junction. Note that the new source amplitude may now depend on frequency.

Figure 3

Two possible ways of separating the linear and nonlinear contributions of a Josephson junction to a microwave circuit. Different symbols were chosen for the nonlinear elements to suggest their different current-voltage relations. The Josephson inductance in both cases has the usual value $L_J={\phi }_0∕I_0$.

Figure 4

Equivalent Norton (a) and Thevenin (b) representations of a linear circuit driving a PNL and SNL Josephson element, respectively. The appropriate bias source is shown as either a parallel current source or a series voltage source.

Figure 5

Josephson junction biased by (a) a parallel LCR circuit and (b) a series LCR circuit. In the first case, $L_p^{-1}=L_{pe}^{-1}+L_J^{-1}$ and in the second case, $L_s=L_{se}+L_J$ where $L_{pe}$ and $L_{se}$ are the inductances contributed by the environment.

Figure 6

Sketch of the graphical determination of whether a circuit characterized by impedance $Z\left(\omega \right)$ (see text) will allow a Josephson junction with critical current $I_0$ to bifurcate. The procedure consists in (i) plotting simultaneously $\mathrm{Re}\left[Z\left(\omega \right)\right]$, $-\frac{1}{\sqrt{3}}\mathrm{Im}\left[Z\left(\omega \right)\right]$ and $\mid Z_J\left(\omega \right)\mid =\omega {\phi }_0∕I_0$, (ii) finding the critical frequency ${\omega }_c$ (if it exists) at the eventual intersection $\mathrm{Re}\left[Z\left(\omega \right)\right]=-\frac{1}{\sqrt{3}}\mathrm{Im}\left[Z\left(\omega \right)\right]$, and (iii) verifying that the condition for weak nonlinearity is satisfied at ${\omega }_c$: namely, $\mid Z_J\left({\omega }_c\right)\mid ⪢\mathrm{Re}\left[Z\left({\omega }_c\right)\right]$.