We obtain the quantized momentum eigenvalues, $P_n$ , and the momentum eigenstates for the space-like Schrödinger equation, the Feinberg-Horodecki equation, with the general potential which is constructed by the temporal counterpart of the spatial form of these potentials. The present work is illustrated with two special cases of the general form: time-dependent Wei-Hua Oscillator and time-dependent Manning-Rosen potential. We also plot the variations of the general molecular potential with its two special cases and their momentum states for few quantized states against the screening parameter.
We consider several problems that involve finding the eigenvalues and generating the eigenstates of unknown unitary gates. We first examine Controlled-U gates that act on qubits, and assume that we know the eigenvalues. It is then shown how to use singlet states to produce qubits in the eigenstates of the gate. We then remove the assumption that we know the eigenvalues and show how to both find the eigenvalues and produce qubits in the eigenstates. Finally, we look at the case where the unitary operator acts on qutrits and has eigenvalues of 1 and -1, where the eigenvalue 1 is doubly degenerat…
A growing number of dynamical situations involve the coupling of particles or singularities with physical waves. In principle these situations are very far from the wave-particle duality at quantum scale where the wave is probabilistic by nature. Yet some dual characteristics were observed in a system where a macroscopic droplet is guided by a pilot-wave it generates. Here we investigate the behaviour of these entities when confined in a two-dimensional harmonic potential well. A discrete set of stable orbits is observed, in the shape of successive generalized Cassinian-like curves (circles, o…
In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials this paper explains how to use the elementary procedure of Euler summation to sum these divergent s…
The quantum mechanical expression relating two commuting operators is reformulated such that the power method (also called method of moments) for iteratively calculating eigenvalues and eigenvectors becomes applicable. The new iterative scheme thus obtained allows to pre-select a quantum number of one of the two commuting operators and then calculate a corresponding eigenvalue of the other operator. The result is the common eigenvector of the eigenvalue pair. Among others the method may be used to calculate excited states, cope with degeneracy and/or accelerate convergence. Small example calcu…
We show that eigen-energies and energy eigenstates play different roles in the equilibration process of an isolated quantum system. Their roles are revealed numerically by exchanging the eigen-energies between an integrable model and a non-integrable model. We ?find that the structure of eigenenergies of a non-integrable model characterized by non-degeneracy ensures that quantum revival occurs rarely whereas the energy eigenstates of a non-integrable model suppress the fluctuations for the equilibrated quantum state. Our study is aided with a quantum entropy that describes how randomly a wave …
A new set of $ h(1) \oplus su(2)$ vector algebra eigenstates on the matrix domain is obtained by defining them as eigenstates of a generalized annihilation operator formed from a linear combination of the generators of this algebra which eigenvalues are distributed as the elements of a square complex normal matrix. A combined method is used to compute these eigenstates, namely, the method of exponential operators and that of a system of first-order linear differential equations. We compute these states for all possible combination of generators and classify them in different categories accordi…
The Born-Fock theorem is one of the most fundamental theorems of quantum mechanics and forms the basis for reliable and efficient navigation in the Hilbert space of a quantum system with a time-dependent Hamiltonian by adiabatic evolution. In the absence of level crossings, i.e. without degeneracies, and under adiabatic time evolution all eigenstates of the Hamiltonian keep their energetic order, labelled by a conserved integer quantum number. Thus controlling the eigenstates of the Hamiltonian and their energetic order in asymptotic limits allows to engineer a perfect adiabatic transfer betwe…
We show that the author of a recent paper [arXiv:2008.04929] put forward some false statements about the eigenstates of Hermitian and non-Hermitian systems. We conjecture that one of the non-Hermitian Hamiltonians for a one-dimensional lattice is similar to an Hermitian one and, consequently, exhibits real eigenvalues. Present theoretical analysis of the eigenvalue equation suggests that one of the sets of numerical results in the criticized paper may not be correct.
We point out that the Lyapunov exponent of the eigenstate places restrictions on the eigenvalue. Consequently, with regard to non-Hermitian systems, even without any symmetry, the non-conservative Hamiltonians can exhibit real spectra as long as Lyapunov exponents of eigenstates inhibit imaginary parts of eigenvalues. Our findings open up a new route to study non-Hermitian physics.
We observe that the many-body eigenstates of any quadratic, fermionic Hamiltonian with sublattice symmetry have quantized entanglement entropies between the sublattices: the entanglement comes in multiple singlets. Moreover, such systems always have a ground state that is maximally entangled between the two sublattices. In fact we also show that under the same assumptions there always exists a (potentially distinct) basis of energy eigenstates that do not conserve the particle number in which each energy eigenstate is maximally entangled between the sublattices. No additional properties, such …
We propose a quantum algorithm for solving the following problem: given the Hamiltonian of a physical system and one of its eigenvalues, how to obtain the corresponding eigenstate? The algorithm is based on the resonance phenomena. For a probe qubit coupled to a quantum system, the system exhibits a resonance dynamics when the frequency of the probe qubit matches a transition frequency in the system. Therefore the system can be guided to evolve to the eigenstate with known eigenvalue by inducing resonance between the probe qubit and a designed transition in the system. This algorithm can also …
We study the eigenvalue problem for a linear potential Hamiltonian and, by writing Airy equation in terms of momentum and position operators define Airy states. We give a solution of the Schrödinger equation for the symmetrical linear potential in terms of the squeeze and displacement operators. Finally, we write the unit operator in terms of Airy states and find a relation between them and position and momentum eigenstates.
We calculate the energy eigenvalues and eigenstates corresponding to coherent single and multiple excitations of an array of N identical qubits or two-level atoms (TLA's) arranged on the vertices of a regular polygon. We assume only that the coupling occurs via an exchange interaction which depends on the separation between the qubits. We include the interactions between all pairs of qubits, and our results are valid for arbitrary distances relative to the radiation wavelength. To illustrate the usefulness of these states, we plot the distance dependence of the decay rates of the n=2 (biexcito…
The scientific interest in the analytical solution of the quantum Rabi model is due to the widespread use of this simple model in quantum optics, quantum computing, cavity QED, and nanoelectromechanical systems. This interest is related to the need for the theoretical description of the interaction of a two-level system with a quantum oscillator in the case when the rotating wave approximation fails. In this Letter, we present an approach to the exact diagonalization of the quantum Rabi Hamiltonian. This approach is based on the properties of the Pauli operators and allows us to readily solve …
In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, ${\cal C}$, for this system. All calculations are performed analytically and without approximation.
Eigenstate Preparation Through Near-Optimal Eigenprobability Filtering
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Quantum Motion（量子运动）；HUN-REN Wigner Research Centre for Physics（匈牙利科学院维格纳物理研究中心）；Faculty of Informatics, Eötvös Loránd University（罗兰大学信息学院）
We demonstrate a method to systematically obtain eigenvalues and eigenstates of a many-body Hamiltonian describing collective neutrino oscillations. The method is derived from the Richardson-Gaudin framework, which involves casting the eigenproblem as a set of coupled nonlinear "Bethe Ansatz equations", the solutions of which can then be used to parametrize the eigenvalues and eigenvectors. The specific approach outlined in this paper consists of defining auxiliary variables that are related to the Bethe-Ansatz parameters, thereby transforming the Bethe-Ansatz equations into a different set of…
The factorization method was introduced by Schroedinger in 1940. Its use in bound-state problems is widely known, including in supersymmetric quantum mechanics; one can create a factorization chain, which simultaneously solves a sequence of auxiliary Hamiltonians that share common eigenvalues with their adjacent Hamiltonians in the chain, except for the lowest eigenvalue. In this work, we generalize the factorization method to continuum energy eigenstates. Here, one does not generically have a factorization chain -- instead all energies are solved using a "single-shot factorization," enabled b…
We investigate the eigenstate thermalization properties of the spin-1/2 $XXZ$ model in two-dimensional rectangular lattices of size $L_1\times L_2$ under periodic boundary conditions. Exploiting the symmetry property, we can perform an exact diagonalization study of the energy eigenvalues up to system size $4\times 7$ and of the energy eigenstates up to $4\times 6$. Numerical analysis of the Hamiltonian eigenvalue spectrum and matrix elements of an observable in the Hamiltonian eigenstate basis supports that the two-dimensional $XXZ$ model follows the eigenstate thermalization hypothesis. When…
Quantum computers have been shown to have tremendous potential in solving difficult problems in quantum chemistry. In this paper, we propose a new classical quantum hybrid method, named as power of sine Hamiltonian operator (PSHO), to evaluate the eigenvalues of a given Hamiltonian (H). In PSHO, for any reference state, the normalized energy of the sine Hamiltonian power state can be determined. With the increase of the power, the initial reference state can converge to the eigenstate with the largest absolute eigenvalue in the coefficients of the expansion of reference state, and the normaliz…
Quantum simulation of complex quantum systems and their properties often requires the ability to prepare initial states in an eigenstate of the Hamiltonian to be simulated. In addition, to compute the eigenvalues of a Hamiltonian is in general a non-trivial problem. Here, we propose a hybrid quantum-classical probabilistic method to compute eigenvalues and prepare eigenstates of Hamiltonians which are simulatable with a trapped-ion quantum processor.
Besides perturbation theory, which requires, of course, the knowledge of the exact unperturbed solution, variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H usually can be located with acceptable precision whereas the trial-subspace vectors corresponding to these eigenvalues approximate, in general, the exact eigenstates of H with much less accuracy. Accordingly, various measures for the accuracy of t…
The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. Here, we introduce the concept of an "eigenstate witness" and through it provide a new quantum approach which combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled-unitary operations, and proj…
Measurement of the energy eigenvalues (spectrum) of a multi-qubit system has recently become possible by qubit tunneling spectroscopy (QTS). In the standard QTS experiments, an incoherent probe qubit is strongly coupled to one of the qubits of the system in such a way that its incoherent tunneling rate provides information about the energy eigenvalues of the original (source) system. In this paper, we generalize QTS by coupling the probe qubit to many source qubits. We show that by properly choosing the couplings, one can perform projective measurements of the source system energy eigenstates …
The original model of the infinite square well contains a vague notation infinity and therefore results some ambiguities. We investigate to obtain a functional form for the potential energy V(x). This is done by substituting back the original energy eigenstates and eigenvalues into the Schrodinger equation. We then obtain a precise functional form of the V(x). From this reformed model, we show that energy eigenstates and eigenvalues can directly be obtained without the need of imposing boundary condition, Ehrenfest's theorem can directly be confirmed, and ambiguities in the original model can …
An adiabatic cycle of parameters in a quantum system can yield the quantum anholonomies, nontrivial evolution not just in phase of the states, but also in eigenvalues and eigenstates. Such exotic anholonomies imply that an adiabatic cycle rearranges eigenstates even without spectral degeneracy. We show that an arbitrarily large quantum circuit generated by recursive extension can also exhibit the eigenvalue and eigenspace anholonomies.
Consider a path of non-degenerate eigenstates of unitary operators or Hamiltonians with minimum eigenvalue gap G. The eigenpath traversal problem is to transform one or more copies of the initial to the final eigenstate. Solutions to this problem have applications ranging from quantum physics simulation to optimization. For Hamiltonians, the conventional way of doing this is by applying the adiabatic theorem. We give ``digital'' methods for performing the transformation that require no assumption on path continuity or differentiability other than the absence of large jumps. Given sufficient in…
Many body models undergoing a quantum phase transition to a broken-symmetry phase that survives up to a critical temperature must possess, in the ordered phase, symmetric as well as non-symmetric eigenstates. We predict, and explicitly show in the fully-connected Ising model in a transverse field, that these two classes of eigenstates do not overlap in energy, and therefore that an energy edge exists separating low-energy symmetry-breaking eigenstates from high-energy symmetry-invariant ones. This energy is actually responsible, as we show, for the dynamical phase transition displayed by this …
The flow equation approach investigated by Wegner et al. is applied to an unbounded Hamiltonian system with a generalization. We show that a well-known quantized complex energy eigenvalues which is related to decay widths can be given with this approach.
We make use of a recently developed method to, not only obtain the exactly known eigenstates and eigenvalues of a number of quasi-exactly solvable Hamiltonians, but also construct a convergent approximation scheme for locating those levels, not amenable to analytical treatments. The fact that, the above method yields an expansion of the wave functions in terms of corresponding energies, enables one to treat energy as a variational parameter, which can be effectively used for the identification of the eigenstates. It is particularly useful for the quasi-exactly solvable systems, where the groun…
We explore the spectral properties and behaviour of confining superexponential potentials. Several prototypes of these highly nonlinear potentials are analyzed in terms of the eigenvalues and eigenstates of the underlying stationary Schrödinger equation up to several hundreds of excited states. A generalization of the superexponential self-interacting oscillator shows a scaling behaviour of the spacing of the eigenvalues which turns into an alternating behaviour for the power law modified oscillator. Superexponential potentials with an oscillating power show a very rich spectral structure with…
The appearance of topological singularities, namely exceptional points (EPs) is an intriguing feature of parameter-dependent open quantum or wave systems. EPs are the special type of nonHermitian degeneracies where two (or more) eigenstates of the underlying system coalesce. In this paper, we present a three level non-Hermitian Hamiltonian which hosts three interacting eigenstates. The matrix elements are optimized in such a way that the intermediate eigenstate interacts with both the other states and the underlying system hosts at least two different second order EPs. The impact of quasi-stat…
Besides perturbation theory (which clearly requires the knowledge of the exact unperturbed solution), variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H usually can be located with acceptable precision whereas the trial-subspace vectors corresponding to these eigenvalues approximate, in general, the exact eigenstates of H with much less accuracy. Accordingly, various measures for the accuracy of the …
The soliton effect is defined in nonlinear physics by the transformation of a nonlinear time-dependent dynamical system into an equivalent linear spectral eigenproblem whose invariant eigenvalues unambiguously define all the dynamical properties of the original system. We point out the existence of such an effect in a non-relativistic isotropic two-electron mean-field quantum-dot model. It yields the prediction of observable modulation of radiation absorption between its two lowest-energy zero-angular-momentum nonlinear eigenstates (i.e. eigenstates which include electron-electron interaction:…
We introduce a class of non-Hermitian Hamiltonians that offers a dynamical approach to short-cut to adiabaticity (DASA). In particular, in our proposed 2 * 2 Hamiltonians, one eigenvalue is absolutely real and the other one is complex. This specific form of the eigenvalues helps us to exponentially decay the population in an undesired eigenfunction or amplify the population in the desired state while keeping the probability amplitude in the other eigenfunction conserved. This provides us with a powerful method to have a diabatic process with the same outcome as its corresponding adiabatic proc…
Quantum dynamics controlled by a time-dependent coupling constant are studied. It is proven that an energy eigenstate expectation value of work done by the system in a quench process cannot exceed the work in the corresponding quasi-static process, if and only if the energy eigenvalue is a concave function of the coupling constant. We propose this concavity of energy eigenvalues as a new universal criterion for quantum dynamical systems to satisfy the second law of thermodynamics. We argue simple universal conditions on quantum systems for the concavity, and show that every energy eigenvalue i…
We show that the subregion entanglement Hamiltonians of excited eigenstates of a quantum many-body system are approximately linear combinations of subregionally (quasi)local approximate conserved quantities, with relative commutation errors $\mathcal{O}\left(\frac{\text{subregion boundary area}}{\text{subregion volume}}\right)$. By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigen-operators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is c…
We obtain the quantized momentum eigenvalues, Pn, and the momentum eigenstates for the space-like Schrodinger equation, the Feinberg-Horodecki equation, with the improved deformed exponential-type potential which is constructed by temporal counterpart of the spatial form of these potentials. We also plot the variations of the improved deformed exponential-type potential with its momentum eigenvalues for few quantized states against the screening parameter.
We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schrödinger equation follows from the Liouville equation, with $\hbar$ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. Interestingly, it is now classical mecha…
We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter, $Λ$. The behaviors apply equally well to few- and many-body systems, e.g.\ interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems as long as their subsystems are quantum chaotic, and not localized …
We investigate the transition probabilities for the "flavor" eigenstates in the two-level quantum system, which is described by a non-Hermitian Hamiltonian with the parity and time-reversal (PT) symmetry. Particularly, we concentrate on the so-called PT-broken phase, where two eigenvalues of the non-Hermitian Hamiltonian turn out to be a complex conjugate pair. In this case, we find that the transition probabilities will be unbounded in the limit of infinite time $t \to +\infty$. However, after performing a connection between a non-Hermitian system, which exhibits passive PT-symmetry and globa…
We introduce aspects of quantum chaos by analyzing the eigenvalues and the eigenstates of quantum many-body systems. The properties of quantum systems whose classical counterparts are chaotic differ from those whose classical counterparts are not chaotic. The spectrum of the first exhibits repulsion of the energy levels. This is one of the main signatures of quantum chaos. We show how level repulsion develops in one-dimensional systems of interacting spins 1/2 which are devoid of random elements and involve only two-body interactions. In addition to the statistics of the eigenvalues, we analyz…
The classical Liouville density on the constant energy surface reveals a number of interesting features when the initial density has no directional preference. It has been shown (Physical Review Letters, 93 (2004) 204102) that the eigenvalues and eigenfunctions of the momentum-projected density evolution operator have a correspondence with the quantum Neumann energy eigenstates in billiard systems. While the classical eigenfunctions are well approximated by the quantum Neumann eigenfunctions, the classical eigenvalues are of the form f(\sqrt{E_n} vt) where {E_n} are close to the quantum Neuman…
Inspired by the idea of mimicking the measurement on a quantum system through a decoherence process to target specific eigenstates based on Born's law, i.e. the hiearchy of probabilities instead of the hierarchy of eigenvalues, we transform a Lindblad equation for the reduced density operator into a nonlinear Schrödinger equation to obtain a computationally feasible simulation of the decoherent dynamics in the open quantum system. This gives the opportunity to target the eigenstates which have the largest $L^2$ overlap with an initial superposition state and hence more flexibility in the selec…
We demonstrate that the time operator that measures the time of arrival of a quantum particle into chosen state can be defined as a self-adjoint quantum-mechanical operator using periodic boundary conditions on applied to wavefuncions in energy representation. The time becomes quantized into discreet eigenvalues and the eigenstates of the time operator, the stroboscopic wavepackets introduced recently [Phys. Rev. Lett. 101, 046402 (2008).] form orthogonal system of states. The formalism provides simple physical interpretation of the time-measurement process and direct construction of normalize…
Eigenvalues are defined for any element of an algebra of observables and do not require a representation in terms of wave functions or density matrices. A systematic algebraic derivation based on moments is presented here for the harmonic oscillator, together with a perturbative treatment of anharmonic systems. In this process, a collection of inequalities is uncovered which amount to uncertainty relations for higher-order moments saturated by the harmonic-oscillator excited states. Similar saturation properties hold for anharmonic systems order by order in perturbation theory. The new method,…
At the exceptional point where two eigenstates coalesce in open quantum systems, the usual diagonalization scheme breaks down and the Hamiltonian can only be reduced to Jordan block form. Most of the studies on the exceptional point appearing in the literature introduce a phenomenological effective Hamiltonian that essentially reduces the problem to that of a finite non-Hermitian matrix for which it is straightforward to obtain the Jordan form. In this paper, we demonstrate how the Hamiltonian of an open quantum system reduces to Jordan block form at an exceptional point in an exact manner tha…
