An open-domain question answering (QA) system usually follows a retrieve-then-read paradigm, in which a retriever is used to retrieve relevant passages from a large corpus, and then a reader generates answers based on the retrieved passages and the original question. In this paper, we propose a simple and novel mutual learning framework to improve the performance of retrieve-then-read-style models via an intermediate module named the knowledge selector, which we train with reinforcement learning. The key benefits of our proposed intermediate module are: 1) no requirement for additional annotated question-passage pairs; 2) improvements in both retrieval and QA performance, as well as computational efficiency, compared to prior competitive retrieve-then-read models; 3) with no finetuning, improvement in the zero-shot performance of large-scale pre-trained language models, e.g., ChatGPT, by encapsulating the input with relevant knowledge without violating the input length constraint.