Emergent synchronization mode in coupled Rydberg atomic chains

Abstract

We report a new oscillatory form in the two coupled dissipative Rydberg atomic chains by modulating its spacing. Such oscillation has $π$-phase difference between two neighboring sites, which distinguishes itself from antiferromagnetic-type synchronization in the previous studies. Theoretically, we find a phase with coexisting two types of continuous time crystals, and recognize that the transition belongs to Hopf and pitchfork bifurcation. Furthermore, we generalize the conclusion to multiple chains and verify the uniqueness of the new synchronization mode. We also discuss its experimental feasibility.

Figures (5)

• Figure 1: (a) Sketch map of two coupled Rydberg chains. The region colored by red is the specified unit cell in the MF calculation, and the lattice sites are denoted as $1A, 1B, 2A, 2B$. The interaction considered in Eq. \ref{['eq4']} are divided into three class $V$, $V_i$ and $V_{i2}$, which are represented by bonds with different thicknesses. (b) Time-evolution of Rydberg populations at decoupled case $V_i=0$. Two oscillations of both chains have same period but an arbitrary phase difference, determined by its initial conditions. (c,d) Time-evolution at $V_i = 1$ for different initial conditions. (c) shows the traditional AF-type oscillation pattern, and (d) shows the emergent $\text{AF}_2$-type oscillation.
• Figure 2: (a) The phase diagram of two coupled Rydberg chains with respect to the inter-chain interaction $V_i$. The blue, pink and purple regions denote phase 1, 2, 3, respectively. The transition point are labeled by white circle. (b) Time-evolution of Rydberg population for different phases over a sufficiently long time. The red and blue lines represents two different initial conditions, corresponding two coexisting and distinguishable trajectories, which flows to AF-type and $\text{AF}_2$-type synchronization modes, respectively. The solid (hollow) circle is the associated stable (unstable) solution. The blue limit cycle (AF-type) is always existing, and encircles its unstable solution. The red limit cycle ($\text{AF}_2$-type) only exists in (b2). Note that (b1) shows the decoupled case, where only AF-type limit cycle survives. The sub-figure in (b4) is a clearer presentation of the limit cycle. (c) Stabilization analysis by varying $V_i$ in three-dimensional diagram. We begin at small $V_i = 0.05$ and end at $V_i = V = 5$. We select $V_i$ values at transition points, and the solutions are represented by the dots on the cross section. (d) Stabilization analysis showing the behavior near transition points in two-dimensional diagram and its sketch map. In the sketch map, the circles are the solutions from stabilization analysis, and the ovals indicate the oscillatory solutions. The wells denote the BOAs with associated colors. Note in (d1), the region covered by red is the oscillatory range for different $V_i$.
• Figure 3: (a) The sketch map of two-dimensional anisotropy model. The solutions for $4 \times 2$ and $2 \times 4$ rectangle region correspond to figure (b1) and (b2), respectively. (b) Time-evolution for eight sites region by MF approach. The initial Rydberg population for each sites are distinct, and both evolutions finally collapse to $\text{AF}_2$-type oscillation.
• Figure S1: The time-dependent solution of Rydberg population with different $V$ and $r$. All solutions are obtained by the same initial condition.
• Figure S2: The time-dependent solution of Rydberg population with different $V_t$ and existence of the next-nearest neighbor term for region shape $2 \times 4$. Here, we only display the results of the $1A, 1B, 2A, 2B$ lattice site. In particular for (c,d), we find the Hopf-bifurcation point decreases when we consider the next-nearest neighbor interaction. All solutions are obtained by the same initial condition.