The quantum $\mathrm{\Lambda }$ system is a ubiquitous structure in many quantum technologies, where two stable, lower-energy levels are coupled indirectly via an unstable, higher-energy level that is much more susceptible to decoherence and relaxation. In this paper, we solve the problem of finding a control driving field that leads to a desired final condition while minimizing the average population on the unstable level in finite time. We formulate the problem as an optimal control problem where the cost to be minimized is a combination of the energy of the control and the average population in the highest level. Our formulation applies to density matrix states, enabling control of mixed states and ensembles, which are essential for realistic settings such as thermal preparations, sensors, and scalable quantum information platforms. We utilize the Pontryagin Maximum Principle, symmetry reduction, and a number of auxiliary techniques to reduce the problem of finding the optimal control to a feasible numerical procedure. We prove several properties of the optimal control functions, such as normality and smoothness, and derive the system of differential equations that must be satisfied by the optimal pair of control and trajectory. We present a case study involving a Hadamard-like transformation for illustration. We discuss robustness of the control functions to the presence of unmodeled dynamics.

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• Open quantum systems & decoherence
• Quantum control
• Quantum state transfer