Emergent ac Effect in Nonreciprocal Coupled Condensates

Abstract

We report an emergent ac Josephson-like effect arising without external bias, driven by the interplay between nonreciprocity and nonlinearity in coupled condensates. Using a minimal model of three mutually nonreciprocally coupled condensates, we uncover a rich landscape of dynamical phases governed by generalized Josephson equations. This goes beyond the Kuramoto framework owing to inherent nonreciprocity and dynamically evolving effective couplings, leading to static and dynamical ferromagnetic and (anti)vortex states with nontrivial phase winding. Most strikingly, we identify an ac phase characterized by the emergence of two distinct frequencies, which spontaneously break the time-translation symmetry: one associated with the precession of the global U(1) Goldstone mode and the other with a stabilized limit cycle in a five-dimensional phase space. This phase features bias-free autonomous oscillatory currents beyond conventional Josephson dynamics. We further examine how instabilities develop in the ferromagnetic and vortex states, and how they drive transitions into the ac regime. Interestingly, the transition is hysteretic: phases with different winding numbers destabilize under distinct conditions, reflecting their inherently different nonlinear structures. Our work lays the foundation for exploring nonreciprocity-driven novel dynamical phases in a broad class of condensate platforms.

Figures (4)

• Figure 1: (a) Schematic of three mutually nonreciprocally coupled condensates, with asymmetric couplings $J_L$ and $J_R$. (b) Chiral ferromagnetic phase, where all condensates share the same phase and undergo collective precession at a constant frequency. (c) Chiral vortex phase, characterized by a phase winding of $2\pi$ across the three sites; relative phases are locked while the global phase rotates uniformly. (d) Spontaneous ac phase, featuring two emergent frequencies: one associated with global phase rotation and the other with oscillations of the relative phases. Dashed red: relative phase evolution $\theta_{ij}$; solid lines: individual phases $\theta_i(t)$fig1parameter.
• Figure 2: (a) Phase diagram showing the ferromagnetic state and its transition to other phases upon instability for $J=0$. It is obtained by numerically solving the generalized Josephson equations starting from initial conditions near the ferromagnetic fixed point Fig2parameter. (b) Steady-state oscillation amplitude of $\theta_{21}$ as a function of dissipative coupling $G$ for $D/\gamma = 1$ and $J = 0$ obtained Fig2parameter.
• Figure 3: (a) Steady-state oscillation amplitude of $\theta_{21}$ as a function of coupling $J$ for $D/\gamma = 1.2$ and $G/\gamma = 0.1$. (b) Projected trajectories of persistent oscillations on the reduced phase space $\mathcal{M}$ at $J=0$, shown on the $\mathbb{T}^2$ submanifold of relative phases, for varying values of $G$ (with $\gamma = 3$, $D = 5$). Inset: oscillation frequency as a function of $D$, for $G = 0$ (red), $G = 1$ (green), and $G = 3$ (blue), with $\gamma = 3$ fixed. Both plots are produced from long-time numerical solutions of the generalized Josephson equation, starting near the ferromagnetic fixed point Fig2parameter.
• Figure 4: (a) Instability boundary of the vortex phase for different $J$. Red: $J = 0$; orange: $J/\gamma = -\sqrt{3}$; green: $J/\gamma = \sqrt{3}$. The shaded region marks the parameter space where the vortex state is unstable for $J = 0$. (b) Upper panel: In the nonreciprocal Kuramoto model limit_cycle_sm, both the ferromagnetic state (green) and the vortex state (purple) lose stability exactly at $G=0$, switching directly into each other. Lower panel: For the full dynamics, the ferromagnetic phase (green) first destabilises at a positive $G$, entering an intermediate ac state (red) before reaching the vortex phase (purple); conversely, the vortex phase becomes unstable at a negative $G$. The separation of these thresholds evidences hysteresis. (c) Phase diagram showing the vortex phase and its transition to other phases for $J=0$, which is obtained numerically. The yellow curve shows the analytically obtained instability boundary given by Eq. \ref{['vortexcondition']}, which agrees well with the numerics. (d) Frequency $\omega$ of the steady state as a function of $D$ for $J=0$ and $G/\gamma =-0.98$. (e) Limit‑cycle trajectories projected onto the relative‑phase torus $\mathbb{T}^2$ for $J=0$ and $G/\gamma=-0.98$. From inner to outer trajectories, we set $D/\gamma=2.7$, $2.6$, $1.5$. Inset: oscillation amplitude versus $D$. Plots (c)-(e) are obtained from numerical solutions of the generalized Josephson equations, starting near the vortex fixed point fig4parameter.