通过量子轨迹的后选择连接非Hermitian Hamilton和Liouville例外点的Hybrid-Liouville形式
Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories
摘要：
例外点（EP）是经典和量子开放系统的简并，在许多物理学领域包括光学，光电子学，等离子体和凝聚态物理中进行了研究。在半经典体系中，开放系统可以用唯象的有效非厄米哈密顿量（NHH）来描述，这些哈密顿量用虚场来描述增益和损耗的影响。表征这种哈密顿量（HEP）的谱的EP描述了没有量子跳跃的系统的时间演化。众所周知，描述更一般动力学的完整量子处理必须关键地考虑这种量子跳跃。在最近的一篇论文[F。Minganti $et$ $al.$，Phys. Rev. A $\mathbf{100}$，$062131$（$2019$）]，我们将EP的概念推广到Lindblad主方程描述的Liouvillian超算子的谱.有趣的是，我们发现，在经典到量子对应存在的情况下，两种类型的动力学可以产生不同的EP。在最近的一项实验工作中[M. Naghiloo $et$ $al.$，Nat. Phys. $\mathbf{15}$，$1232$（$2019$）]，表明可以通过在某些量子跳跃轨迹上进行后选择来设计量子极限中的非厄米哈密顿量。这就提出了一个有趣的问题，即关于哈密顿和林德布拉德EP与量子轨道之间的关系。我们通过引入一个混合Liouvillian超算子来讨论这些连接，该超算子能够描述从NHH（当一个人只选择那些没有量子跳跃的轨迹时）到一个真正的Liouvillian（包括量子跳跃）（没有后选择）的通道。除了它的根本利益，我们的方法可以直观地将后选择和有限效率检测器的影响。
英文摘要：
Exceptional points (EPs) are degeneracies of classical and quantum open systems, which are studied in many areas of physics including optics, optoelectronics, plasmonics, and condensed matter physics. In the semiclassical regime, open systems can be described by phenomenological effective non-Hermitian Hamiltonians (NHHs) capturing the effects of gain and loss in terms of imaginary fields. The EPs that characterize the spectra of such Hamiltonians (HEPs) describe the time evolution of a system without quantum jumps. It is well known that a full quantum treatment describing more generic dynamics must crucially take into account such quantum jumps. In a recent paper [F. Minganti $et$ $al.$, Phys. Rev. A $\mathbf{100}$, $062131$ ($2019$)], we generalized the notion of EPs to the spectra of Liouvillian superoperators governing open system dynamics described by Lindblad master equations. Intriguingly, we found that in situations where a classical-to-quantum correspondence exists, the two types of dynamics can yield different EPs. In a recent experimental work [M. Naghiloo $et$ $al.$, Nat. Phys. $\mathbf{15}$, $1232$ ($2019$)], it was shown that one can engineer a non-Hermitian Hamiltonian in the quantum limit by postselecting on certain quantum jump trajectories. This raises an interesting question concerning the relation between Hamiltonian and Lindbladian EPs, and quantum trajectories. We discuss these connections by introducing a hybrid-Liouvillian superoperator, capable of describing the passage from an NHH (when one postselects only those trajectories without quantum jumps) to a true Liouvillian including quantum jumps (without postselection). Beyond its fundamental interest, our approach allows to intuitively relate the effects of postselection and finite-efficiency detectors.
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