Quantum properties of a strongly driven Josephson junction

A Josephson junction embedded in a dissipative circuit can be externally driven to induce nonlinear dynamics of its phase. Classically, under sufficiently strong driving and weak damping, dynamic multistability emerges, associated with dynamical bifurcations so that the often-used modeling as a Duffing oscillator, which can exhibit bistability at most, is insufficient. The present work analyzes in this regime corresponding quantum properties by mapping the problem onto a highly nonlinear quasienergy operator in a rotating frame. This allows us to identify in detail parameter regions where simplifications such as the Duffing approximation are valid, to explore classical-quantum correspondences, and to study how quantum fluctuations impact the effective junction parameters as well as the dynamics around higher-amplitude classical fixed points.

Figure 1

Parameter regimes of the Josephson Hamiltonian and regions of validity of different approximations in a rotating frame. The rotating-wave approximation fails for strong driving, $F/E_J\gtrsim 1$ (red). For weak driving, $F/E_J\ll \tilde{\gamma }$, the Bessel function in (2) can be expanded to the lowest-order nonlinear term, so that $H_{\text{JJ}}^{\text{RF}}\approx H_{\text{JJ,2}}^{\text{RF}}$ (blue, plotted for $\tilde{\gamma }=0.2$). Conventional descriptions employ the Duffing Hamiltonian (4), which further requires $\kappa \ll 1$. With increasing $\kappa$ the system moves from a classical towards a quantum regime.

Figure 2

Classical limit of the Hamiltonian $g(Q,P=0)$ for $\epsilon =17.3$, $f=0.56$ (black solid line) and for $\epsilon =2.05$ with ($f=0.8$; blue solid line) and without ($f=0$; red dashed line) driving. Stable orbits (for small damping) are found for all minima of quasienergies at $Q\ge 0$ and for all maxima at $Q<0$ (labeled by odd numbers in increasing order of amplitude; cf. Fig. 3). All maxima (minima) of $g(Q,P)$ for $Q>0$ ($Q<0$) correspond to unstable solutions (even labels). The case $\epsilon \gg 1$ corresponds locally to a Duffing oscillator, while for $\epsilon \gtrsim 2$ the full impact of the Bessel function is seen. In this regime multiple stable solutions can be found even without driving. Driving breaks the $Q\text{−}P$ symmetry and allows us to experimentally access higher dynamical stable solutions.

Figure 3

Steady-state amplitude $A=\sqrt{Q^2+P^2}/2\kappa$ for (a) $\epsilon =17.3$, $\kappa =0.1$ and (b) $\epsilon =2.05$, $\kappa =1$, both including dissipation $\tilde{\gamma }=0.05$. For $\epsilon \gg 1$ the bifurcation pattern known from the Duffing oscillator is recovered. For $\epsilon \gtrsim 2$ replicas of the Duffing orbits appear at larger amplitudes. Red segments indicate stable orbits located at the minima of the quasienergies for $Q\ge 0$ (labels 1 and 5; see Fig. 2), while stable orbits corresponding to maxima at $Q<0$ (labeled 3 and 7) are shown in blue. Unstable solutions (green, even labels) refer to maxima (minima) at $Q>0$ ($Q<0$).

Figure 4

Quasienergy spectrum for the Josephson Hamiltonian $g$ in the rotating frame for $\kappa =0.1$, $\epsilon =17.3$, and increasing $n$ (a) without driving and (b) for changing driving amplitude. The eigenenergies form a parabola with $n$ resembling the classical amplitude dependence. With growing $f$ an ordering of the eigenvalues according to Fock state numbers is no longer possible. In the bottom row Wigner transforms of eigenstates for (c) $n=0$, (d) $n=25$, and (e) $n=43$ are plotted (for a minute but finite $f=0.01$). The weak driving localizes the state $n=25$ corresponding to the top of the energy parabola at $Q<0$, while the $n=0$ and $n=43$ states remain centered around $Q=P=0$.

Figure 5

Quasienergy spectrum for the Josephson Hamiltonian $g$ for $\kappa =1$, $\epsilon =2.05$, and increasing $n$ (a) without driving and (b) for changing driving amplitude. Instead of the simple parabola shape of Fig. 4 multiple extrema in the quasienergy spectrum appear for increasing $n$. Wigner transforms of eigenvectors for (c) $n=3$, (d) $n=24$, and (e) $n=62$ ($f=0.01$) show for small $n$ a Fock-state-like distribution around $Q=P=0$ for the first maximum. Beyond the first energy maximum strongly displaced and squeezed states localized around the maxima at $Q<0$ appear.

Figure 6

Wigner transforms of eigenvectors for (b) $m=0$, (c) $m=14$, and (d) $m=48$ (counted in decreasing order of energy) including finite driving $f=0.3$ for $\kappa =1$, $\epsilon =2.05$. Finite driving breaks the $Q\text{−}P$ symmetry and thus induces additional displacement and squeezing. The eigenstates chosen correspond to classically stable solutions at stationary points in the (classical) quasienergy landscape shown in (a). For instance, the near-Gaussian state for $m=48$ corresponds to the classical solution in the central well at $Q=P=0$ [see (d)].

Figure 7

Steady-state population of the dissipative dynamics described by the dimensionless Lindblad master equation (9) for $f=0.56$, $\kappa =0.1$, $\epsilon =17.3$, $\tilde{\gamma }=0.05$. For these parameters eigenstate populations of (a) the Josephson Hamiltonian ($\langle n\rangle =30.77$, maximum at $n=31$) are very close to the results of (b) the pure Duffing system ($\langle n\rangle =29.29$, maximum at $n=29$). Both show a bimodal distribution with contributions from the two classically stable orbits (see Fig. 3).

Figure 8

Full density $\mathrm{Re}\left\{{\rho }_{n,m}\right\}$ in the steady state for $f=0.56$, $\kappa =0.1$, $\epsilon =17.3$, $\tilde{\gamma }=0.05$ for (a) the Josephson Hamiltonian and (b) the Duffing system. The large-amplitude solution is scaled down by 0.05. We find two isolated domains in Fock space with the only off-diagonal contributions (coherences) concentrated close to the diagonal. The absence of coherences between the two domains indicates a classical mixture of the two classically stable orbits. The imaginary part of the density shows a similar decrease of coherences with increasing distance from the diagonal.

Figure 9

Steady-state population of the dissipative dynamics for $f=0.8$, $\kappa =1$, $\epsilon =2.05$, $\tilde{\gamma }=0.05$. For these parameters, specifically for $\epsilon \sim 2$, full Josephson and Duffing-approximation results differ drastically. (a) The full Josephson population distribution contains more than two maxima, while (b) for the Duffing case only monostability is seen. In that case, the bistability classically expected in this domain is completely washed out by quantum fluctuations. The Josephson population distribution exhibits maxima at $n=0$, $n=4$ (see inset), $n=26$, and $n=66$. The higher maxima are related to the maxima in the bare quasienergy spectrum [Fig. 5]. Comparing the maxima with the steady-state amplitudes in Figs. 2 and 3 reveals that only stable orbits located at $Q\approx 0$ (labeled 1 with $A^2\approx 0.04$) and at $Q\le 0$ (labeled 3 and 7 with $A^2\approx 4$ and 26, respectively) can be associated with maxima in ${\rho }_{n,n}$.

Figure 10

Full density $\mathrm{Re}\left\{{\rho }_{n,m}\right\}$ in the steady state for $f=0.8$, $\kappa =1$, $\epsilon =2.05$, $\tilde{\gamma }=0.05$ for the Josephson Hamiltonian. The Fock space density shows distinct domains according to the three pronounced peaks in Fig. 9 with no coherences between them. (b) The right section shows the coherences concentrated close to the diagonal for the largest peak. (a) The left section of small $n$ reveals substantial coherences between the weakly pronounced peak at $n=4$ and the central $n=0$ contribution.

Figure 11

Number fluctuations for $\kappa =1$, $\epsilon =2.05$, $\tilde{\gamma }=0.05$ around the semiclassical solution with $n\approx 0$ (red) and $n\approx 26$ (blue) for different driving strengths. The red (blue) solid line can be associated with the red branch of smallest amplitude (the branch with $A\approx 5$) in Fig. 3. The number fluctuations for $n\approx 0$ take the harmonic value, $\delta n_2=1$, for weak driving. With a stronger driving the number fluctuations grow and eventually diverge at the bifurcation point. Number fluctuations for $n\approx 26$ grow for a weak driving and at a stronger driving amplitude because on both sides bifurcation points are present. However, between these regions a narrow domain is found, where the system reduces again to a coherent state with $ln\left(\delta n_2\right)=0$.