In the standard (Hermitian) formalism of quantum mechanics, usually when a potential parameter is varied the crossing of energy levels with the same symmetry is avoided. However, within the framework of the non-Hermitian formalism it is possible that two (or even more) complex eigenvalues with the same symmetry will cross. At the crossing point the eigenvalues are degenerate and this is accompanied by the coalescence of the eigenfunctions (or eigenvectors). Therefore, we may term this special situation as a non-Hermitian degeneracy. This special situation is associated with a branch point in the complex energy plane which is commonly termed an “exceptional point” in the spectrum of the non-Hermitian Hamiltonian. With respect to the c-product defined for non-Hermitian operators (matrices) in Chapter 6, the degenerate eigenstate is self-orthogonal. Since a branch point in the spectrum is removed by any infinitesimally small external perturbation, it seems to be inaccessible experimentally and may be considered just as a mathematical object rather than a physical one. However, as we will show here, by varying the potential parameters the existence of a branch point is reflected in the measurement of the geometrical phases also known as Berry phases. It should be stressed here that while in our case the geometrical phase results from a coalescence of eigenfunctions of a non-Hermitian Hamiltonian, the so-called Berry phase phenomenon occurs also within the Hermitian formalism of quantum mechanics when the eigenvalues of the molecular Hamiltonian in the Born–Oppenheimer approximation are degenerate for specific geometry of the poly-atomic molecule.
In the standard (Hermitian) formalism of quantum mechanics, usually when a potential parameter is varied the crossing of energy levels with the same symmetry is avoided. However, within the framework of the non-Hermitian formalism it is possible that two (or even more) complex eigenvalues with the same symmetry will cross. At the crossing point the eigenvalues are degenerate and this is accompanied by the coalescence of the eigenfunctions (or eigenvectors). Therefore, we may term this special situation as a non-Hermitian degeneracy. This special situation is associated with a branch point in the complex energy plane which is commonly termed an “exceptional point” in the spectrum of the non-Hermitian Hamiltonian. With respect to the c-product defined for non-Hermitian operators (matrices) in Chapter 6, the degenerate eigenstate is self-orthogonal. Since a branch point in the spectrum is removed by any infinitesimally small external perturbation, it seems to be inaccessible experimentally and may be considered just as a mathematical object rather than a physical one. However, as we will show here, by varying the potential parameters the existence of a branch point is reflected in the measurement of the geometrical phases also known as Berry phases. It should be stressed here that while in our case the geometrical phase results from a coalescence of eigenfunctions of a non-Hermitian Hamiltonian, the so-called Berry phase phenomenon occurs also within the Hermitian formalism of quantum mechanics when the eigenvalues of the molecular Hamiltonian in the Born–Oppenheimer approximation are degenerate for specific geometry of the poly-atomic molecule.
Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.
As one of the essential features in non-Hermitian systems coupled with environment, the exceptional point has attracted much attention in many physical fields. The phenomena that eigenvalues and eigenvectors of the system simultaneously coalesce at the exceptional point are also one of the important properties to distinguish from Hermitian systems. In non-Hermitian systems with parity-time reversal symmetry, the eigenvalues can be continuously adjusted in parameter space from all real spectra to pairs of complex-conjugate values by crossing the phase transition from the parity-time reversal symmetry preserving phase to the broken phase. The phase transition point is called an exceptional point of the system, which occurs in company with the spontaneous symmetry broken and many novel physical phenomena, such as sensitivity-enhanced measurement and loss induced transparency or lasing. Here, we focus on a two-qubit quantum system with parity-time reversal symmetry and construct an experimental scheme, prove and verify the features at its third-order exceptional point, including high-order energy response induced by perturbation and the coalescence of eigenvectors. We first theoretically study a two-qubit non-Hermitian system with parity-time reversal symmetry, calculate the properties of eigenvalues and eigenvectors, and prove the existence of a third-order exceptional point. Then, in order to study the energy response of the system induced by perturbation, we introduce an Ising-type interaction as perturbation and quantitatively demonstrate the response of eigenvalues. In logarithmic coordinates, three of the eigenvalues are indeed in the cubic root relationship with perturbation strength, while the fourth one is a linear function. Moreover, we study the eigenvectors around exceptional point and show the coalescence phenomenon as the perturbation strength becomes smaller. The characterization of the response of eigenvalues at high-order exceptional points is a quite diffcult task as it is in gneneral diffcult to directly measure eigenenergies in a quantum system composed of a few qubits. In practice, the time evolution of occupation on a particular state is used to indirectly fit the eigenvalues. In order to make the fitting of experimental data more reliable, we want to determine an accurate enough expressions for the eigenvalues and eigenstates. To this aim, we employ a perturbation treatment and show good agreement with the numerical results of states occupation obtained by direct evolution. Moreover, we find that after the system evolves for a long enough time, it will end up to one of the eigenstates, which gives us a way to demonstrate eigenvector coalescence by measuring the density matrix via tomography and parity-time reversal transformation. To show our scheme is experimentally applicable, we propose an implementation using trapped 171Yb+ ions. We can map the parity-time reversal symmetric Hamiltonian to a purely dissipative two-ion system:use microwave to achieve spin state inversion, shine a 370 nm laser to realize dissipation of spin-up state, and apply Raman operation for Mølmer-Sørensen gates to implement Ising interaction. By adjusting the corresponding microwave and laser intensity, the spin coupling strength, the dissipation rate and the perturbation strength can be well controlled. We can record the probability distribution of the four product states of the two ions and measure the density matrix by detecting the fluorescence of each ion on different Pauli basis.
The coalescence of two eigenfunctions with the same energy eigenvalue is not possible in Hermitian Hamiltonians. It is, however, a phenomenon well known from non-hermitian quantum mechanics. It can appear, e.g., for resonances in open systems, with complex energy eigenvalues. If two eigenvalues of a quantum mechanical system which depends on two or more parameters pass through such a branch point singularity at a critical set of parameters, the point in the parameter space is called an exceptional point. We will demonstrate that exceptional points occur not only for non-hermitean Hamiltonians but also in the nonlinear Schroedinger equations which describe Bose-Einstein condensates, i.e., the Gross-Pitaevskii equation for condensates with a short-range contact interaction, and with additional long-range interactions. Typically, in these condensates the exceptional points are also found to be bifurcation points in parameter space. For condensates with a gravity-like interaction between the atoms, these findings can be confirmed in an analytical way.
Non-Hermitian quantum physics is used successfully for the description of different puzzling experimental results, which are observed in open quantum systems. Mostly, the influence of exceptional points on the dynamical properties of the system is studied. At these points, two complex eigenvalues Ei≡Ei+iΓi/2 of the non-Hermitian Hamiltonian H coalesce (where Ei is the energy and Γi is the inverse lifetime of the state i ). We show that also the eigenfunctions Φi of the two states play an important role, sometimes even the dominant one. Besides exceptional points, other critical points exist in non-Hermitian quantum physics. At these points a=acr in the parameter space, the biorthogonal eigenfunctions of H become orthogonal. For illustration, we show characteristic numerical results.
A main distinguishing feature of non-Hermitian quantum mechanics is the presence of exceptional points (EPs). They correspond to the coalescence of two energy levels and their respective eigenvectors. Here, we use the Lipkin-Meshkov-Glick (LMG) model as a test bed to explore the strong connection between EPs and the onset of excited state quantum phase transitions (ESQPTs). We show that for finite systems, the exact degeneracies (EPs) obtained with the non-Hermitian LMG Hamiltonian continued into the complex plane are directly linked with the avoided crossings that characterize the ESQPTs for the real (physical) LMG Hamiltonian. The values of the complex control parameter $\ensuremath{\alpha}$ that lead to the EPs approach the real axis as the system size $N\ensuremath{\rightarrow}\ensuremath{\infty}$. This happens for both the EPs that are close to the separatrix that marks the ESQPT and also for those that are far away, although in the latter case, the convergence rate is smaller. With the method of Pad\'e approximants, we can extract the critical value of $\ensuremath{\alpha}$.
Although the standard formalism of quantum mechanics is based on the requirement of the physical operators to be Hermitian, the use of non-Hermitian operators in the study of different types of phenomena is not uncommon. One of the most well-known non-Hermitian potentials is the optical potential where for any given choice of N channels the exact eigenvalue is a solution of a single-channel problem. The optical potential is a non-Hermitian, non-local and energy-dependent operator. In his book on scattering theory Taylor writes: “In practice, the optical potential is far too complicated for exact use in actual calculations”. It is often believed that the complex energies which are obtained by the use of optical potentials result from the approximations in the calculations. However, this is not true. The complex energy obtained by solving the one-channel problem with an optical potential is the exact eigenvalue of the original N -channel problem which is obtained by imposing outgoing boundary conditions on the eigenfunctions of the time-independent Schrödinger equation. In this chapter we wish to show that the study of the resonances in multi-channel problems (so-called Feshbach resonances) can be considered as the point where quantum mechanics branches into the standard (Hermitian) and non-standard (non-Hermitan) formalisms. Feshbach resonances Quite a long time ago Feshbach showed that the exact energy spectrum of the full physical problem can be obtained by solving two different self-energy problems.