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An information engine based on a two level system in contact with a thermalreservoir is studied analytically. The model incorporates delay time betweenthe measurement of the state of the system and the feedback. The engineefficiency and work extracted per cycle are studied as a function of delay timeand energy spacing between the two levels. It is found that the range of delaytime over which one can extract work from the information engine increases withtemperature. For delay times comparable to the relaxation time, efficiency andwork per cycle are both maximum when $k_B T \approx 2 U_0$, the energydifference between the levels. Generalized Jarzynski equality and thegeneralized integral fluctuation theorem are explicitly verified for the model.The results from the model are compared with the simulation results for afeedback engine based on a particle moving in a $1D$ square potential. Thevariation of efficiency, work per cycle and efficacy with the delay time iscompared using relaxation time in the two state model as the fitting parameterand leads to good fit.

Information engine with feedback delay based on a two level system