Imprecise Maxwell’s demon with feedback delay: An exactly solvable information engine model

A finite cycle time information engine based on a two-level system in contact with a thermal bath is studied analytically. The model for the engine incorporates an error in measuring the system's state and time delay between the measurement and the feedback process. The efficiency and power of the engine in steady state are derived as a function of level spacing, feedback delay time, engine cycle time, and measurement error. For a fixed value of level spacing and feedback delay, there is an upper bound on measurement error such that the engine can extract positive work. This threshold value of error is found to be independent of the cycle time. For a range of values of level spacing, feedback delay time and cycle time, efficiency has a non-monotonic dependence on the measurement error, implying that there is an optimal measurement error for the information engine to operate efficiently. The range of feedback delays within which the system functions as an engine, extracting positive work, depends on the measurement error and the level spacing.