We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, H= p^2/2 + v(x), in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1+1)-dimensions. For the case that v(x) is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v(x)=i εx^3. This reveals an infinite class of previously unknown CPT- as well as non-CPT-inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in real line that is consistent with the usual choice for the quantum Hamiltonian. Finally we discuss subtleties involved with higher dimensions and systems having a fixed length scale.
Exceptional points (EPs) have recently attracted considerable attention in the study of non-Hermitian systems and in applications such as sensors and mode switching. In particular, nontrivial topological structures of EPs have been studied intensively in relation to encircling EPs. Thus, EP generation is currently an important issue in several fields. To generate multiple EPs, multiple levels or composite physical systems have been employed with Hermitian couplings. In this study, we generate multiple EPs on two-level systems in a single microcavity by adopting the non-Hermitian coupling of a …
We study a general class of PT-symmetric tridiagonal Hamiltonians with purely imaginary interaction terms in the quasi-hermitian representation of quantum mechanics. Our general Hamiltonian encompasses many previously studied lattice models as special cases. We provide numerical results regarding domains of observability and exceptional points, and discuss the possibility of explicit construction of general metric operators (which in turn determine all the physical Hilbert spaces). The condition of computational simplicity for the metrics motivates the introduction of certain one-parametric sp…
Imaginary eigenvalues of Hermitian Hamiltonian with an inverted potential well and transition to the real spectrum at exceptional point by a non-Hermitian interaction
We investigate a uniformly coupled non-Hermitian system with asymmetric coupling amplitude. The asymmetric coupling equals to a symmetric coupling threaded by an imaginary gauge field. In a closed configuration, the imaginary gauge field leads to an imaginary magnetic flux, which induces a non-Hermitian phase transition. For an open boundary, the imaginary gauge field results in an eigenstate localization. The eigenstates under Dirac and biorthogonal norms and the scaling laws are quantitatively investigated to show the affect of asymmetric coupling induced one-way amplification. However, the …
In this paper, we study the topological properties of the non-Hermitian Su-Schrieffer-Heeger (SSH) lattice by periodically introducing onsite imaginary potentials in the manner of ($iγ_1$, $-iγ_2$, $-iγ_1$, $iγ_2$) where $γ_1$ and $γ_2$ are the imaginary potential strengths. Results show that by changing the lattice to a tetramerized non-Hermitian system, such imaginary potentials induce the nontrivial transition of the topological properties of the SSH system. First, the topologically-nontrivial region is extended, followed by the non-Hermitian spontaneous breaking of the anti-$\cal PT$ symme…
Arbitrarily Configurable Wavefunctions via Imaginary Gauge Phase Imprint in Non-Hermitian Lattices
South China Normal University（华南师范大学）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心（广东））；Hefei National Laboratory（合肥国家实验室）
One of the most important applications of quantum mechanics is the thermodynamic description of quantum gases. Despite the fundamental importance of this topic, a comprehensive description of the thermodynamic properties of non-Hermitian quantum gases is still lacking. Here, we investigate the properties of bosonic and fermionic non-Hermitian systems at finite temperatures. We show that non-Hermitian systems exihibit oscillations both in temperature and imaginary time. As such, they can be a possible platform to realize an imaginary time crystal (iTC) phase. The Hatano-Nelson model is identifi…
Scattering phase shift in quantum mechanics on quantum computers: non-Hermitian systems and imaginary-time simulations
College of Arts and Sciences, Dakota State University（达科他州立大学文理学院）；Department of Physics, George Washington University（乔治华盛顿大学物理系）；Physics Division, Argonne National Laboratory（阿贡国家实验室物理部）
A nonadiabatic and robust method of excitation transfer in a non-Hermitian tight-binding linear chain, assisted by an imaginary gauge field, is theoretically proposed. The gauge field undergoes a linear gradient in time, from a negative to a positive value, which results in an effective transfer of excitation between the two edge sites of the chain. An imaginary (gain and loss) gradient of site energy potentials is introduced to exactly cancel nonadiabatic effects, thus providing an effective shortcut to adiabaticity and pseudo-Hermitian dynamics. Numerical simulations indicate that the non-He…
Imaginary gauge potentials in a non-Hermitian spin-orbit coupled quantum gas
Joint Quantum Institute, University of Maryland and National Institute of Standards and Technology（马里兰大学与国家标准技术研究院联合量子研究所）；Department of Physics, Harvard University（哈佛大学物理系）
Anomalous Localization and Mobility Edges in Non-Hermitian Quasicrystals with Disordered Imaginary Gauge Fields
Sanli Honors College, School of Physics and Electronics Engineering, Shanxi University（山西大学物理与电子工程学院三立荣誉学院）；Institute of Theoretical Physics and State Key Laboratory of Quantum Optics Technologies and Devices, Shanxi University（山西大学理论物理研究所量子光学技术与器件国家重点实验室）；State Key Laboratory of Quantum Optics Technologies and Devices, Institute of Laser Spectroscopy, Shanxi University（山西大学激光光谱研究所量子光学技术与器件国家重点实验室）
Localized states of the non-Hermitian Schr{ö}dinger equation with a complex $1/r^{2}$ potential: from Efimov states to bipolarons
University of Electro-Communications（电气通信大学）；Osaka Metropolitan University（大阪公立大学）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP)（南部阳一郎理论与实验物理研究所）；Institute for Advanced Science（高等科学研究所）
Asymmetric coupling amplitudes effectively create an imaginary gauge field, which induces a non-Hermitian Aharonov-Bohm (AB) effect. Nonzero imaginary magnetic flux invalidates the bulk-boundary correspondence and leads to a topological phase transition. However, the way of non-Hermiticity appearance may alter the system topology. By alternatively introducing the non-Hermiticity under symmetry to prevent nonzero imaginary magnetic flux, the bulk-boundary correspondence recovers and every bulk state becomes extended; the bulk topology of Bloch Hamiltonian predicts the (non)existence of edge sta…
We consider non-Hermitian dynamics of a quantum particle hopping on a one-dimensional tight-binding lattice made of $N$ sites with asymmetric hopping rates induced by a time-periodic oscillating imaginary gauge field. A deeply different behavior is found depending on the lattice topology. While in a linear chain (open boundary conditions) an oscillating field can lead to a complex quasi energy spectrum via a multiple parametric resonance, in a ring topology (Born-von Karman periodic boundary conditions) an entirely real quasi energy spectrum can be found and the dynamics is pseudo-Hermitian. I…
We investigate the quantum irreversibility and quantum diffusion in a non-Hermitian kicked rotor model for which the kicking strength is complex. Our results show that the exponential decay of Loschmidt echo gradually disappears with increasing the strength of the imaginary part of non-Hermitian driven potential, demonstrating the suppress of the exponential instability by non-Hermiticity. The quantum diffusion exhibits the dynamical localization in momentum space, namely, the mean square of momentum increases to saturation with time evolution, which decreases with the increase of the strength…
Transport of quantum or classical waves in open systems is known to be strongly affected by non-Hermitian terms that arise from an effective description of system-enviroment interaction. A simple and paradigmatic example of non-Hermitian transport, originally introduced by Hatano and Nelson two decades ago [N. Hatano and D.R. Nelson, Phys. Rev. Lett. 77, 570 (1996)], is the hopping dynamics of a quantum particle on a one-dimensional tight-binding lattice in the presence of an imaginary vectorial potential. The imaginary gauge field can prevent Anderson localization via non-Hermitian delocaliza…
Imaginary Gauge Field and Non-Hermitian Topological Transition Emerging Through Attenuation-Gauge Duality in Conservative Systems
The Chinese University of Hong Kong（香港中文大学）；Shenzhen Campus of Sun Yat-sen University（中山大学深圳校区）；Shenzhen Research Institute, The Chinese University of Hong Kong（香港中文大学（深圳）研究院）
A topologically equivalent tight binding model is proposed to study the quantum phase transitions of dimer chain driven by an imaginary ac field. I demonstrate how the partner Hamiltonian is constructed by a similarity transformation to fulfil the $\mathcal{PT}$ symmetry. The $\mathcal{PT}$ symmetry of the partner model allows us to study the topological properties of the original non-Hermitian model as the Bloch bands of the Hermitian system. The quantum phase transitions are discussed in different frequency regime. The approach has the potential applications to investigate the topological st…
Extended imaginary gauge transformation in a general nonreciprocal lattice
Tsinghua University（清华大学）；North China University of Technology（华北理工大学）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；Frontier Science Center for Quantum Information（量子信息前沿科学中心）；Beijing National Research Center for Information Science and Technology（北京信息科学与技术国家研究中心）
An imaginary gauge transformation is at the core of the non-Hermitian skin effect. Here we show that such a transformation can be performed in momentum space as well, which reveals that certain gain and loss modulated systems in their parity-time (PT) symmetric phases are equivalent to Hermitian systems with real potentials. Our analysis in momentum space also distinguishes two types of exceptional points (EPs) in the same system. Besides the conventional type that leads to a PT transition upon the continuous increase of gain and loss, we find real-valued energy bands connected at a Dirac EP i…
We take non-Hermitian Aubry-André-Harper models and quasiperiodic Kitaev chains as examples to demonstrate the equivalence and superposition of real and imaginary quasiperiodic potentials (QPs) on inducing localization of single-particle states. We prove this equivalence by analytically computing Lyapunov exponents (or inverse of localization lengths) for systems with purely real and purely imaginary QPs. Moreover, when superposed and with the same frequency, real and imaginary QPs are coherent on inducing the localization, under a way which is determined by the relative phase between them. Th…
Chiral symmetry provides the symmetry protection for a large class of topological edge states. It exists in non-Hermitian systems as well, and the same anti-commutation relation between the Hamiltonian and a linear chiral operator, i.e., $\{H,Π\}=0$, now warrants a symmetric spectrum about the origin of the complex energy plane. Here we show two general approaches to identify and generate chiral symmetry in non-Hermitian systems, with an emphasis on lattices with detuned on-site potentials that can vary in both their real and imaginary parts. One approach utilizes the Clifford algebra satisfie…
