Chaos and hyperchaos in the chain of quantum coherent elements

We study the chain of coupled quantum coherent systems with energy pumping and dissipation. Examples of such systems include the coupled Rydberg atoms with laser driving and spontaneous emission or electromagnetically driven artificial qubits with decoherence and dissipation. We found out that such system is able to demonstrate spontaneous onset of chaotic and even hyperchaotic oscillations, which are characterized by one or several positive Lyapunov exponents. Remarkably, the number of the positive Lyapunov exponents grows with the number of the chain elements. Hence a large chain is able to demonstrate highly irregular behavior. We investigate transition from regular to chaotic behavior and reveal the related instabilities.