Discrete time translation symmetry breaking in a Josephson junction laser

A Josephson junction laser is realized when a microwave cavity is driven by a voltage-biased Josephson junction. Through the ac Josephson effect, a dc voltage generates a periodic drive that acts on the cavity and generates interactions between its modes. A sufficiently strong drive enables processes that downconvert a drive resonant with a high harmonic into photons at the cavity fundamental frequency, breaking the discrete time translation symmetry set by the Josephson frequency. Using a classical model, we determine when and how this transition occurs as a function of the bias voltage and the number of cavity modes. We find that certain combinations of mode number and voltage tend to facilitate the transition which emerges via an instability within a subset of the modes. Despite the complexity of the system, there are cases in which the critical drive strength can be obtained analytically.

Figure 1

(a) Circuit model of the JJL. A JJ with Josephson energy, $E_J$, biased by a voltage, $V$, in series with a microwave cavity modeled as a series of LC oscillators with frequencies ${\omega }_1,{\omega }_2,\cdots$. Losses from the cavity lead to damping of the modes with rates ${\gamma }_1,{\gamma }_2,\cdots$. (b) DTTS breaking transition in the oscillations of the total cavity phase, $\mathrm{\Psi }\left(t\right)$: Above the transition (blue), these have the period of the fundamental mode, $T=2\pi /{\omega }_1$, but period $2\pi /{\omega }_J$ below (red). Here, $N=11$, ${\omega }_J=3{\omega }_1$, and the origin of the time axis has been displaced to display the long-time behavior clearly.

Figure 2

Long-time behavior of ${\tilde{\alpha }}_n$, the component of ${\alpha }_n$ oscillating at ${\omega }_n$, as a function of $\lambda$ with $p=3$ for $N=11$ (a), (b)—a continuous transition–and $N=12$ (c), (d)—a discontinuous transition. In both cases, ${\mathrm{\Delta }}_1=0.2$ and $\gamma ={\omega }_1/100$. (a), (c) show $|{\tilde{\alpha }}_n|$ for all modes; (b), (d) show just the fundamental, $|{\tilde{\alpha }}_1|$, which serves as an order parameter for the transition. See Appendix pp1 for details of the numerical method. Red and blue symbols in (b) indicate the $\lambda$ values illustrated in Fig. 1. The vertical dashed lines show the threshold predicted using linear stability analysis.

Figure 3

Critical drive strength, ${\lambda }_{\text{c}}$ at which a continuous transition takes place for a range of $N$ and $p$ values, calculated using linear stability analysis (detailed in Appendix pp2). Lines connect points with a specific relationship between $N$ and $p$ (labeled in each case).

Figure 4

Drive at which a DTTS breaking transition occurs in a given system, ${\lambda }_{\text{sb}}$, as a function of $N$ for $p=2,3,4$. Crosses: Discontinuous transitions. Solid (hollow) points: Continuous transitions originating within the $k=1$ ($k=2$) block.

Figure 5

Symmetry subspace, $k$, in which a continuous DTTSB transition emerges as a function of $N$ and $p$. Gray indicates no DTTSB transition was found because an instability in the $k=0$ space occurred first. White indicates an ($N,p$) combination that wasn't explored and black corresponds to cases where no transition is possible (see Appendix pp2-s4).

Figure 6

Threshold value ${\lambda }_{\text{th}}={\tilde{E}}_J{\mathrm{\Delta }}_1^2/\left(p^2\hslash {\gamma }_1\right)$ at which the symmetry unbroken solution becomes unstable with $p=2,3,4,5,6$, ${\gamma }_n=n^2{\gamma }_1$ and ${\mathrm{\Delta }}_n={\mathrm{\Delta }}_1/\sqrt{n}$. Note that ${\lambda }_{\text{th}}$ is defined in terms of $p^2{\gamma }_1={\gamma }_p$ rather than just ${\gamma }_1$.

Figure 7

Continuous and discontinuous transitions with soft cutoffs. The long-time behavior obtained from numerical integration of $|{\tilde{\alpha }}_n{|}^{1/2}$ (a),(c) and $|{\tilde{\alpha }}_1|$ (b),(d) is shown. In (a),(b) ${\gamma }_I=0$ leading to a continuous transition, while in (c),(d) with ${\gamma }_I=34{\mathrm{\Gamma }}_E$ it is discontinuous. (a),(b) dashed line: analytic prediction from stability analysis (see Fig. 6) (d) inset: zoom-in of the step-like discontinuous transition. Note that the number of modes that play a role at the transition clearly increases as we go from (a) to (b). In both cases, we set $p=2$.