PT symmetry and spontaneous symmetry breaking in a microwave billiard

We demonstrate the presence of parity-time (PT) symmetry for the non-Hermitian two-state Hamiltonian of a dissipative microwave billiard in the vicinity of an exceptional point (EP). The shape of the billiard depends on two parameters. The Hamiltonian is determined from the measured resonance spectrum on a fine grid in the parameter plane. After applying a purely imaginary diagonal shift to the Hamiltonian, its eigenvalues are either real or complex conjugate on a curve, which passes through the EP. An appropriate basis choice reveals its PT symmetry. Spontaneous symmetry breaking occurs at the EP.     

Complex extension of quantum mechanics

Requiring that a Hamiltonian be Hermitian is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry C of the Hamiltonian. Using C, an inner product whose associated norm is positive definite can be constructed. The procedure is general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalization of it.     

Anti-parity-time symmetric phase transition in diffusive systems

Parity-time (PT) symmetry/anti-parity-time (APT) symmetry in non-Hermitian systems reveal profound physics andspawn intriguing effects. Recently, it has been introduced into diffusive systems together with the concept of exceptionalpoints (EPs) from quantum mechanics and the wave systems. With the aid of convection, we can generate complex thermalconductivity and imitate various wavelike dynamics in heat transfer, where heat flow can be “stopped” or moving against thebackground motion. Non-Hermitian diffusive systems offer us a new platform to investigate the heat wave manipulation.In this review, we first introduce the construction of APT symmetry in a simple double-channel toy model. Then we showthe phase transition around the EP. Finally, we extend the double-channel model to the four-channel one for showing thehigh-order EP and the associated phase transition. In a general conclusion, the phase difference of adjacent channels isalways static in the APT symmetric phase, while it dynamically evolves or oscillates when the APT symmetry is broken.     

Optical resonance in inhomogeneous parity-time symmetric systems

We show that inhomogeneous waveguides of slowly varied parity-time(PT) symmetry support localized optical resonances.The resonance is closely related to the formation of exceptional points separating exact and broken PT phases. Salient features of this kind of non-Hermitian resonance, including the formation of half-vortex flux and the discrete nature,are discussed. This investigation highlights the unprecedented uniqueness of field dynamics in non-Hermitian systems with many potential adaptive applications.     

$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Generalized PT\mathcal{PT} symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT\mathcal{PT}-symmetric Hamiltonian. There is a corresponding class of PT\mathcal{PT}-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit PT\mathcal{PT} symmetry and also the ANNNI model, which has a hidden PT\mathcal{PT} symmetry. For both quantum and classical models, the class of models with generalized PT\mathcal{PT} symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT\mathcal{PT} symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT\mathcal{PT} symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT\mathcal{PT} symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with PT\mathcal{PT} symmetry which can be simulated for all parameter values, including cases where PT\mathcal{PT} symmetry is broken.     

共轭线性对称性及其对PT-对称量子理论的应用

传统量子系统的哈密顿是自伴算子,哈密顿的自伴性不仅保证系统遵循酉演化和保持概率守恒,而且也保证了它自身具有实的能量本征值,这类系统称为自伴量子系统.然而,确实存在一些物理系统(如PT-对称量子系统),其哈密顿不是自伴的,这类系统称为非自伴量子系统.为了深入研究PT-对称量子系统,并考虑到算子PT的共轭线性性,首先讨论了共轭线性算子的一些性质,包括它们的矩阵表示和谱结构等;其次,分别研究了具有共轭线性对称性和完整共轭线性对称性的线性算子,通过它们的矩阵表示,给出了共轭线性对称性和完整共轭线性对称性的等价刻画;作为应用,得到了关于PT-对称及完整PT-对称算子的一些有趣性质,并通过一些具体例子,说明了完整PT-对称性对张量积运算不具有封闭性,同时说明了完整PT-对称性既不是哈密顿算子在某个正定内积下自伴的充分条件,也不是必要条件.     

Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean–Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schrödinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy spectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.     

Finding Short-Range Parity-Time Phase-Transition Points with a Neural Network

The non-Hermitian PT-symmetric system can live in either unbroken or broken PT-symmetric phase. The separation point of the unbroken and broken PT-symmetric phases is called the PT-phase-transition point.Conventionally, given an arbitrary non-Hermitian PT-symmetric Hamiltonian, one has to solve the corresponding Schrodinger equation explicitly in order to determine which phase it is actually in. Here, we propose to use artificial neural network(ANN) to determine the PT-phase-transition points for non-Hermitian PT-symmetric systems with short-range potentials. The numerical results given by ANN agree well with the literature, which shows the reliability of our new method.     

Level Spacing Distributions and Quantum Chaos in Hermitian and non-Hermitian Systems

The correspondence between quantum level spacing distribu tions and classical motion of 1-D P T symmetric non-Hermitian systems is investigated using two PT symmetric complex potentials: complex rational power potential V1 (x) = (ix)(2n 1)/m and general polynomial potential V2(x) = x2M ib1x2M-1 b2x2M-2 ... ib2M-1x. The level spacing distribution of V1 has two forms. When 2n 1 - 2m is positive, the level spacing distribution of real eigen values assumes a decreasing power function, while it behaves as an increasing power function when 2n 1 - 2m is negative.The PT symmetry of this system is spontaneously broken as 2n 1 - 2m becomes negative. This change manifests itself in classical mechanics as it is found by Bender et al. However, it was found that the change in the form of level spacing distribution mentioned above is not due to the spontaneous breaking down of PT symmetry. Level spacing distribution of V2 assumes an increasing power function when order of the polynomial is greater than two.     

囚禁单离子的量子阻尼运动

用包含偶极和四极虚势能项的非厄米哈密顿算符来描述Paul阱中囚禁阻尼单离子在静电场下的量子运动.通过导出和分析系统的精确解,得到在PT对称和不对称情形下的不同实能谱与稳定量子态,以及PT不对称情形的虚能谱和衰减量子态,同时给出相应于不同态的参数区域和存活概率.结果发现该非厄米系统外场参数能惟一确定量子稳定态并导致波函数形态变化,据此提出非相干操控相应量子跃迁的方法.让量子态衰减导致的离子位置期待值的衰减与经典阻尼谐振子的衰减一致,得到虚势能参数与经典阻尼参数的对应关系.所得结果将进一步丰富具有广泛应用背景的囚禁离子动力学.     

Chiral State Conversion in a Levitated Micromechanical Oscillator with In Situ Control of Parameter Loops

Physical systems with gain and loss can be described by a non-Hermitian Hamiltonian, which is degenerated at the exceptional points(EPs).Many new and unexpected features have been explored in the non-Hermitian systems with a great deal of recent interest.One of the most fascinating features is that chiral state conversion appears when one EP is encircled dynamically.Here, we propose an easy-controllable levitated microparticle system that carries a pair of EPs and realize slow evolution of the Hamiltonian along loops in the parameter plane.Utilizing the controllable rotation angle, gain and loss coefficients, we can control the structure, size and location of the loops in situ.We demonstrate that, under the joint action of topological structure of energy surfaces and nonadiabatic transitions, the chiral behavior emerges both along a loop encircling an EP and even along a straight path away from the EP.This work broadens the range of parameter space for the chiral state conversion, and proposes a useful platform to explore the interesting properties of exceptional points physics.     

Investigation of PT-symmetric Hamiltonian Systems from an Alternative Point of View

Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones.Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart,this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged.In order to give the positive definite inner product for the PT-symmetric systems,a new operator V,instead of C,can be introduced.The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics,however,it can be constructed,as an advantage,directly in terms of Hamiltonians.The spectra of the two non-Hermitian PT-symmetric systems are obtained,which coincide with that given in literature,and in particular,the Hilbert spaces associated with positive definite inner products are worked out.     

Effect of the qubit relaxation on transport properties of microwave photons

In this work, using the non-Hermitian Hamiltonian method, the transmission of a single photon in a one-dimensional waveguide interacting with the cavity containing an arbitrary number of photons and the two-level artificial atom is studied with allowance for the relaxation of the latter. For transport factors, analytical expressions which explicitly take into account the qubit relaxation parameter have been obtained. The form of the transmission (reflection) coefficient when there is more than one photon in the cavity qualitatively differs from the single-photon cavity and contains the manifestation of the photon blockade effect. The qubit lifetime depends on the number of photons in the cavity.     

The superspin approach to a disordered quantum wire in the chiral-unitary symmetry class with an arbitrary number of channels

We use a superspin Hamiltonian defined on an infinite-dimensional Fock space with positive definite scalar product to study localization and delocalization of noninteracting spinless quasiparticles in quasi-one-dimensional quantum wires perturbed by weak quenched disorder. Past works using this approach have considered a single chain. Here, we extend the formalism to treat a quasi-one-dimensional system: a quantum wire with an arbitrary number of channels coupled by random hopping amplitudes. The computations are carried out explicitly for the case of a chiral quasi-one-dimensional wire with broken time-reversal symmetry (chiral-unitary symmetry class). By treating the space direction along the chains as imaginary time, the effects of the disorder are encoded in the time evolution induced by a single site superspin (non-Hermitian) Hamiltonian. We obtain the density of states near the band center of an infinitely long quantum wire. Our results agree with those based on the Dorokhov–Mello–Pereyra–Kumar equation for the chiral-unitary symmetry class.     

Symmetry and Topology of the Topological Counterparts in Non-Hermitian Systems

The symmetries and topological properties of the topological counterparts in 1D non-Hermitian systems are investigated. It is found that, after applying the non-unitary similarity transformation, the non-unitary topological counterpart in real space exhibits completely different global symmetries except for the sublattice symmetry and reveals many brand new local symmetries. Due to the abundant symmetries of non-unitary topological counterparts, it is also found that the unique overlapping projections about the unit sphere vector representing the eigenstates appear in the nontrivial regions, and the triviality of the point-gap topology of non-unitary topological counterpart completely eliminate the intrinsic skin effect in non-Hermitian systems. It is also shown that the unitary topological counterpart never arises any changes for the original symmetries and topological structures even in real space. Unitary topological counterparts are further summarized about the two-band Bloch Hamiltonian, which can expand the definition of non-Bloch winding number. Furthermore, it is demonstrated theoretically that the Bloch Hamiltonian would still hold time-reversal symmetry, abnormal particle-hole symmetry, and sublattice symmetry even suffering from the non-unitary transformation. This work provides a new way to understand the roles of symmetry and topology in non-Hermitian systems from the perspective of topological counterparts.     

Quantum signal transmission through a single-qubit chain

A system of a two-level atom of an impurity (qubit) inserted into a periodic chain coupled to the continuum is studied with the use of the effective non-Hermitian Hamiltonian. Exact solutions are derived for the quasistationary eigenstates, their complex energies, and transport properties. Due to the presence of the qubit, two long-lived states corresponding to the ground and excited states of the qubit emerge outside the Bloch energy band. These states remain essentially localized at the qubit even in the limit of sufficiently strong coupling between the chain and the environment when the super-radiant states are formed. The transmission through the chain is studied as a function of the continuum coupling strength and the chain-qubit coupling; the perfect resonance transmission takes place through isolated resonances at weak and strong continuum coupling, while the transmission is lowered in the intermediate regime.     

Solving the time-dependent Schrödinger equation by combining smooth exterior complex scaling and Arnoldi propagator

Abstract We develop a highly efficient scheme for numerically solving the three-dimensional time-dependent Schr?dinger equation of the single-active-electron atom in the field of laser pulses by combining smooth exterior complex scaling(SECS)absorbing method and Arnoldi propagation method.Such combination has not been reported in the literature.The proposed scheme is particularly useful in the applications involving long-time wave propagation.The SECS is a wonderful absorber,but its application results in a non-Hermitian Hamiltonian,invalidating propagators utilizing the Hermitian symmetry of the Hamiltonian.We demonstrate that the routine Arnoldi propagator can be modified to treat the non-Hermitian Hamiltonian.The efficiency of the proposed scheme is checked by tracking the time-dependent electron wave packet in the case of both weak extreme ultraviolet(XUV)and strong infrared(IR)laser pulses.Both perfect absorption and stable propagation are observed.     

Exceptional points in a microwave billiard with time-reversal invariance violation

We report on the experimental study of an exceptional point (EP) in a dissipative microwave billiard with induced time-reversal invariance (T) violation. The associated two-state Hamiltonian is non-Hermitian and nonsymmetric. It is determined experimentally on a narrow grid in a parameter plane around the EP. At the EP the size of T violation is given by the relative phase of the eigenvector components. The eigenvectors are adiabatically transported around the EP, whereupon they gather geometric phases and in addition geometric amplitudes different from unity.     

非厄米哈密顿量的物理意义

分析了一个脉冲激光与原子相互作用的四能级系统，并考虑最上层能级的自电离过程，从而引入非厄米哈密顿量.在缀饰原子模型下，通过直接求解此哈密顿量的本征值与本征函数，得到系统布居的演化函数.与数值方法所得演化函数的对比表明二者相当符合，从而肯定了非厄米哈密顿量在量子力学框架中的地位，并得到其本征值虚部的物理意义.这将使传统量子力学中力学量的定义得以拓展. 关键词： 非厄米哈密顿量 缀饰原子模型     

再谈“PT对称的非厄米体系的能谱性质”

本文详细讨论了具有PT对称的一维量子系统虚数势对能谱的影响,补充了《大学物理》2018年第3期《PT对称的非厄米体系的能谱性质》一文在系统维度为奇数情况下的结论.