Quantum Superposition of Two Temperatures

In the classical world, temperature is a measure of how hot or cold a physical object is. We never find a physical system which can be both hot and cold at the same time. Here, we show that for a quantum system, it is possible to have superposition of two temperatures which can lead to a situation that it can be found both in hot and cold state. We propose a physical mechanism for how to create a quantum state which is superposition of two temperatures. Furthermore, we define an operator for the inverse temperature and show that the thermal state is, in fact, an eigenstate of this operator. The quantum state which represents superposition of two temperatures is not an eigenstate of the inverse temperature operator. Our findings can have new applications in quantum thermodynamics, quantum nano scale devices and quantum statistical mechanics.