Search for: All records

We present a generalization of the geometric phase to pure and thermal states in $$\mathcal{PT}$$-symmetric quantum mechanics (PTQM) based on the approach of the interferometric geometric phase (IGP). The formalism first introduces the parallel-transport conditions of quantum states and reveals two geometric phases, $$\theta^1$$ and $$\theta^2$$, for pure states in PTQM according to the states under parallel-transport. Due to the non-Hermitian Hamiltonian in PTQM, $$\theta^1$$ is complex and $$\theta^2$$ is its real part. The imaginary part of $$\theta^1$$ plays an important role when we generalize the IGP to thermal states in PTQM. The generalized IGP modifies the thermal distribution of a thermal state, thereby introducing effective temperatures. \textcolor{red}{At certain critical points, the generalized IGP may exhibit discrete jumps at finite temperatures, signaling a geometric phase transition. We illustrate the IGP of PTQM through two examples and compare their differences}. 

We first compare the geometric frameworks behind the Uhlmann andBerry phases in a fiber-bundle language and then evaluate the Uhlmannphases of bosonic and fermionic coherent states. The Uhlmann phases ofboth coherent states are shown to carry geometric information anddecrease smoothly with temperature. Importantly, the Uhlmann phasesapproach the corresponding Berry phases as temperature decreases.Together with previous examples in the literature, we propose acorrespondence between the Uhlmann and Berry phases in thezero-temperature limit as a general property except some special casesand present a conditional proof of the correspondence.