Wigner flow of a non-Hermitian system with PT symmetry

We present a phase-space study of a non-Hermitian system deriving a continuity equation for the Wigner distribution and arbitrary complex potential, defining a Wigner function flow accordingly. In particular, we reveal how a PT-symmetry breaking manifests itself in the phase-space representation. A quantitative measure on the circulation value for the Wigner function flow shows that the phase transition in the vicinity of the exceptional point (EP) is a continuous function of the system parameter. Our study in phase-space representation indicates that a PT-symmetric phase transition is a secondorder phase transition. Instead of conventional quantum mechanics with Hermitian Hamiltonians, non-Hermitian parity-time (PT) symmetry was initially introduced to generalize quantum mechanics, which was first established by Bender and his colleagues in 1998 [1]. Here, P is parity operator and T is time-reversal operator. Since then, lots of work have been done to investigate. With the additional degree of freedom from a non-conservative Hamiltonian, as well as the existence of exceptional points between unbroken and broken PT-symmetries, optical PT-symmetric devices have been demonstrated with many useful applications [2]. Although calling for more caution on physical interpretations, especially on the consistency problem of local PT-symmetric operation and the no-signaling principle [3], PT -symmetric quantum mechanics has been stimulating our understanding on many interesting problems such as spectral equivalence, quantum brachistochrone, Riemann hypothesis, and the manifestation of superposition and coherence [4], as well as week measurement [5]. In this talk [6], with the introduction of Wigner function flow, we derive the corresponding continuity equation. Moreover, through the Gauss-Ostrogradsky theorem, we show that the phase transition in the vicinity of exceptional point (EP), i.e., from an unbroken PT-symmetry phase to a broken one, is a continuous function of the system parameter, which indicates that a PT-symmetric phase transition is a second-order phase transition. Fig. 1. Wigner function currents for the (upper panel) firstand (lower panel) second-excited states. Here, arrows denote the direction of Jψ , while the color of the arrows depends on the norm NJ with dark gray denoting the highest value. References: [1] C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80, 5243 (1998). [2] Y. -C. Lee, J. Liu, Y. -L. Chuang, M. -H. Hsieh, and R. -K. Lee, Passive PT-symmetric couplers without complex optical potentials, Phys. Rev. A 92, 053815 (2015). [3] Y. -C. Lee, M. -H. Hsieh, S. T. Flammia, and R. -K. Lee, Local PT Symmetry Violates the No-Signaling Principle, Phys. Rev. Lett. 112, 130404 (2014). [4] M. Huang, R. -K. Lee, and J. Wu, Manifestation of superposition and coherence in PT-symmetry through the η-inner product, J. Phys. A: Math. Theor. 51, 414004 (2018). [5] M. Huang, R. -K. Lee, L. Zhang, S. -M. Fei, and J. Wu, Simulating broken PT-symmetric Hamiltonian systems by weak measurement, arXiv: 1809.09918 (2018). [6] L. Praxmeyer, P. Yang, and R. -K. Lee, Phase-space representation of a non-Hermitian system with PT symmetry, Phys. Rev. A 93, 042122 (2016). This abstract was submitted to PQE-2019, the 49th Winter Colloquium on the Physics of Quantum Electronics. Submitted by Ray-Kuang Lee on Tue Dec 4 17:38:10 CST 2018. See the program for placement of this presentation.