Isolated quantum systems evolve unitarily according to the Schrödinger equation, while an open quantum system, which is inevitably coupled to a heat bath, usually quickly loses all its quantum coherence. That is, all the off-diagonal terms of the density matrix of the system (in the energy representation) will decay to zero when the open system approaches the steady state. This phenomenon is called decoherence, and it is also believed that this is why our world appears as a classical one and no macroscopic superposition can exist stably in usual cases. It has been reported that if some non-vanishing steady quantum coherence exists in certain special environment, even with a quite small amount, it can lead to some novel physics, such as lasing without inversion, or extracting work from a single heat bath.

Recently, S. W. Li in CSRC, and his collaborators C.Y. Cai and C.P. Sun, studied the steady state of a three-level system in a non-equilibrium environment, which consists of two heat baths with different temperatures. They found that for the Λ-type and V-type systems, the interference between transitions can give rise to non-vanishing steady quantum coherence, if the two transitions couple to the two heat baths with different proportions of coupling strengths. The amount of the steady quantum coherence increases with the temperature difference of the two heat baths. If the two heat baths have the same temperature, all the quantum coherence vanishes and returns to the equilibrium case. These transition structures are quite common in natural and artificial quantum systems. The non-equilibrium environment can be implemented via current noises with different effective temperatures in quantum circuits, or electron leads with different chemical potentials.

The interference between transitions plays an essential role in the steady quantum coherence. But it was often omitted by secular approximation in previous literatures. They showed that indeed the secular approximation is consistent in the case of equilibrium environment, but for non-equilibrium environments, that would lead to the neglect of the steady quantum coherence.

They also showed that the quantum coherence has a clear physical meaning, i.e., it exactly reflects the internal non-equilibrium flux inside a composite system, which is an important characterization of non-equilibrium systems.

**Ref:** Sheng-Wen Li, C.Y. Cai, C.P. Sun, “Steady quantum coherence in non-equilibrium environment”, Annals of Physics 360 (2015) 19–32 (Available online 11 May 2015).

Fig.1. Numerical result for |ρ_{12} | in the non-equilibrium steady state of (a) Λ-type and (b) V-type system.