Initial state dependence in systems with multiple steady states: A Laplace transform approach*

Steady states with robust entanglement properties are essential resources in quantum metrology and quantum computation. These states can be engineered by fine-tuning the Liouvillian operator governing the system's dynamics. While many open quantum systems evolve toward a unique steady state, more general systems can support multiple steady states. Identifying which of these states exhibit useful entanglement and how they depend on the system's initial conditions is crucial for designing experimental protocols to generate entangled steady states.

In this work, we introduce a method based on the Laplace transform to determine the steady states of systems evolving under a Lindblad equation, with an explicit dependence on the initial state. We benchmark our approach analytically and numerically across various models relevant to dark-state engineering, demonstrating its effectiveness in selecting optimal initial states without requiring the integration of the dynamics. Additionally, we compare the computational efficiency of our method against widely used techniques, such as ODE solvers and matrix exponentiation. Our results show that the Laplace method can be significantly faster, particularly for systems where the time to reach the steady state scales rapidly with system size. This makes the Laplace method a practical alternative for studying steady states in many-body open quantum systems.