The recent advancements in large language models (LLMs) have sparked a growing apprehension regarding the potential misuse. One approach to mitigating this risk is to incorporate watermarking techniques into LLMs, allowing for the tracking and attribution of model outputs. This study examines a crucial aspect of watermarking: how significantly watermarks impact the quality of model-generated outputs. Previous studies have suggested a trade-off between watermark strength and output quality. However, our research demonstrates that it is possible to integrate watermarks without affecting the output probability distribution with appropriate implementation. We refer to this type of watermark as an unbiased watermark. This has significant implications for the use of LLMs, as it becomes impossible for users to discern whether a service provider has incorporated watermarks or not. Furthermore, the presence of watermarks does not compromise the performance of the model in downstream tasks, ensuring that the overall utility of the language model is preserved. Our findings contribute to the ongoing discussion around responsible AI development, suggesting that unbiased watermarks can serve as an effective means of tracking and attributing model outputs without sacrificing output quality.