We examine the effect of nonlinearity at a level crossing on the probability for nonadiabatic transitions $P$. By using the Dykhne-Davis-Pechukas formula, we derive simple analytic estimates for $P$ for two types of nonlinear crossings. In the first type, the nonlinearity in the detuning appears as a {\it perturbative} correction to the dominant linear time dependence. Then appreciable deviations from the Landau-Zener probability $P_{LZ}$ are found to appear for large couplings only, when $P$ is very small; this explains why the Landau-Zener model is often seen to provide more accurate results than expected. In the second type of nonlinearity, called {\it essential} nonlinearity, the detuning is proportional to an odd power of time. Then the nonadiabatic probability $P$ is qualitatively and quantitatively different from $P_{LZ}$ because on the one hand, it vanishes in an oscillatory manner as the coupling increases, and on the other, it is much larger than $P_{LZ}$. We suggest an experimental situation when this deviation can be observed.
This paper has been withdrawn by the author. Instead, more comprehensive and accurate works are placed in: 1) cond-mat/0610767: Universality in nonadiabatic behaviour of classical actions in nonlinear models with separatrix crossings, to appear in Phys.Rev.E; and 2) nlin/0611007: Change in the adiabatic invariant in a nonlinear two-mode model of Feshbach resonance passage, to appear in Physica D
We study finite-dimensional product Hilbert spaces, coupled spin systems, entanglement and energy level crossing. The Hamilton operators are based on the Pauli group. We show that swapping the interacting term can lead from unentangled eigenstates to entangled eigenstates and from an energy spectrum with energy level crossing to avoided energy level crossing.
The non-crossing rule for the energy levels of a parameter-dependent Hamiltonian is revisited and a flaw in a commonly accepted proof is revealed. Some aspects of avoided crossings are illustrated by means of simple models. One of them shows the close relationship between avoided crossings and exceptional points.
The Landau-Zener formula provides an analytical expression for the final excitation of a quantum system after passage of an avoided crossing of two energy levels. If the two levels correspond to a ground state, and to an excited state which is subject to radiative decay, the probability of exciting the system by adiabatic passage of the level crossing is reduced. In this article we use a stochastic master equation to study the level crossing dynamics when the system is subject to continuous probing of the emitted radiation. The measurement backaction on the system associated with the fluctuati…
The relation between level crossings, entanglement, and Berry phases is investigated for the Breit-Rabi Hamiltonian of hydrogen and sodium atoms, describing a hyperfine interaction of electron and nuclear spins in a magnetic field. It is shown that the entanglement between nuclear and electron spins is maximum at avoided crossings. An entangled state encircling avoided crossings acquires a marginal Berry phase of a subsystem like an instantaneous eigenstate moving around real crossings accumulates a Berry phase. Especially, the nodal points of a marginal Berry phase correspond to the avoided c…
A nonlinear Landau-Zener model was proposed recently to describe, among a number of applications, the nonadiabatic transition of a Bose-Einstein condensate between Bloch bands. Numerical analysis revealed a striking phenomenon that tunneling occurs even in the adiabatic limit as the nonlinear parameter $C$ is above a critical value equal to the gap $V$ of avoided crossing of the two levels. In this paper, we present analytical results that give quantitative account of the breakdown of adiabaticity by mapping this quantum nonlinear model into a classical Josephson Hamiltonian. In the critical r…
The global-state fidelity cannot characterize those quantum phase transitions (QPTs) induced by continuous level crossing due to its collapse around each crossing point. In this paper, we take the isotropic Lipkin-Meshkov-Glick (LMG) model and the antiferromagnetic one-dimensional Heisenberg model as examples to show that the partial-state fidelity can signal such level-crossing QPTs. Extending to the thermodynamic limit we introduce the partial-state fidelity susceptibility and study its scaling behavior. The maximum of the partial-state fidelity susceptibility goes like $N$ for the LMG model…
The (3+1)-dimensional Dirac equation with position dependent mass in 4-vector electromagnetic fields is considered. Using two over-simplified examples (the Dirac-Coulomb and Dirac-oscillator fields), we report energy-levels crossing as a spectral property or as an effect of the hidden supersymmetric quantum mechanical language and/or quasi-parity signatures. Under different settings of the related interactions' way-of-coupling into Dirac equation, it is observed that the two ultimate/effective descendents, Dirac-Coulomb and Dirac-oscillator, exhibit different conditions on the energy-levels cr…
Two-dimensional PT-symmetric quantum-mechanical systems with the complex cubic potential V_{12}=x^2+y^2+igxy^2 and the complex Henon-Heiles potential V_{HH}=x^2+y^2+ig(xy^2-x^3/3) are investigated. Using numerical and perturbative methods, energy spectra are obtained to high levels. Although both potentials respect the PT symmetry, the complex energy eigenvalues appear when level crossing happens between same parity eigenstates.
At high level density, two states avoid usually crossing at the critical value $a_{\rm cr}$ of the parameter $a$ by which the system is controlled. The wavefunctions of the two states are mixed in a finite parameter range around $a_{\rm cr}$. This holds true for discrete states as well as for narrow resonance states which are coupled via the environment of scattering wavefunctions. We study the influence of avoided level crossings onto four overlapping complex eigenvalues of a symmetric non-Hermitian operator. The mixing of the two wavefunctions around $a_{\rm cr}$ is simulated, in each case, …
We study the dynamics of a nonlinear two-level crossing model with a cubic modification of the linear Landau-Zener diabatic energies. The solutions are expressed in terms of the bi-confluent Heun functions --- the generalization of the confluent hypergeometric functions. We express the finial transition probability as a convergent series of the parameters of the nonlinear laser detuning, and derive analytical approximations for the state populations in terms of parabolic cylinder and Whittaker functions. Tractable closed-form expressions are derived for a large part of the parameter space. We …
We address parameter estimation in two-level systems exhibiting level anti-crossing and prove that universally optimal strategies for parameter estimation may be designed, that is, we may find a parameter independent measurement scheme leading to the ultimate quantum precision independently on the nature and the value of the parameter of interest. Optimal estimation may be achievable also at high temperature depending on the structure of the two-level Hamiltonian. Finally, we show that no improvement is achievable by dynamical strategies and discuss examples of applications.
The anomalous dynamical evolution and the crossing of nonadiabatic energy levels are investigated for exactly solvable time-dependent quantum systems through a reverse-engineering scheme. By exploiting a typical driven model, we elucidate the peculiarities of its dynamics with anomalous behavior: the evolution of the adiabatic states and of the nonadiabatic ones exhibits opposite behavior with their representative vectors evolving from a parallel state to an antiparallel state: the nonadiabatic level crossing is identified as a necessary consequence since the crossing point corresponds exactly…
We study the energy level crossing behavior in two-dimensional quantum well with the Rashba and Dresselhaus spin-orbit couplings (SOCs). By mapping the SOC Hamiltonian onto an anisotropic Rabi model, we obtain the approximate ground state and its quantum Fisher information (QFI) via performing a unitary transformation. We find that the energy level crossing can occur in the quantum well system within the available parameters rather than in cavity and circuit quantum eletrodynamics systems. Futhermore, the influence of two kinds of SOCs on the QFI is investigated and an intuitive explanation fr…
The ground state of a one-dimensional spin-1/2 chain with periodical boundary condition in the Heisenberg XY model is investigated. We consider the spatial correlation and concurrence between any nearest-neighbor pair of spins under the conditions of different coupling strength, anisotropic parameter and magnitude of a transverse field. Quantum phase transitions due to the competition between coupling and alignment which cause the abrupt changes of the correlation and concurrence are observed. The transition are direct results of the level crossing.
We discuss dynamics of approximate adiabatic invariants in several nonlinear models being related to physics of Bose-Einstein condensates (BEC). We show that nonadiabatic dynamics in Feshbach resonance passage, nonlinear Landau-Zener (NLZ) tunnelling, and BEC tunnelling oscillations in a double-well can be considered within a unifying approach based on the theory of separatrix crossings. The separatrix crossing theory was applied previously to some problems of classical mechanics, plasma physics and hydrodynamics, but has not been used in the rapidly growing BEC-related field yet. We derive ex…
The ground electronic, vibrational and rotational state of the OH molecule is currently of interest as it can be manipulated by electric and magnetic fields for experimental studies in ultracold chemistry and quantum degeneracy. Based on our recent exact solution of the corresponding effective Stark-Zeeman Hamiltonian, we present an analytical study of the crossings and avoided crossings in the spectrum. These features are relevant to non-adiabatic transitions, conical intersections and Berry phases. Specifically, for an avoided crossing employed in the evaporative cooling of OH, we compare ou…
Here, we show that, although quantum fidelity can truly identify two quantum phase transitions of a one-dimensional spin-1/2 quantum Ising model with competing nearest and next-nearest neighbour interactions in a transverse magnetic field, it may not be a suitable approach for analyzing its ground-state phase diagram.
Due to the absence of degeneracy in one dimension, when a parameter, $λ$, of a potential is varied slowly the discrete energy eigenvalue curves, $E_n(λ)$, cannot cross but they are allowed to come quite close and diverge from each other. This phenomena is called avoided crossing of energy levels. The parametric evolution of eigenvalues of the generally known one dimensional potentials do not display avoided crossings and on the other hand some complicated and analytically unsolvable models do exhibit this. Here, we show that this interesting spectral property can be found in simple one-dimensi…
In this paper I study the dynamics of a two-level atom interacting with a standing wave field. When the atom is subjected to a weak linear force, the problem can be turned into a time dependent one, and the evolution is understood from the band structure of the spectrum. The presence of level crossings in the spectrum gives rise to Bloch oscillations of the atomic motion. Here I investigate the effects of the atom-field detuning parameter. A variety of different level crossings are obtained by changing the magnitude of the detuning, and the behaviour of the atomic motion is strongly affected d…
This paper is devoted to energy-spectral analysis for the system of a two-level atom coupled with photons in a cavity. It is shown that the Dicke-type energy level crossings take place when the atom-cavity interaction of the system undergoes changes between the weak coupling regime and the strong one. Using the phenomenon of the crossings we develop the idea of cavity-induced atom cooling proposed by the group of Ritsch, and we lay mathematical foundations of a possible mechanism for another superradiant cooling in addition to that proposed by Domokos and Ritsch. The process of our superradian…
Avoided crossings are common in the quasienergy spectra of strongly driven nonlinear quantum wells. In this paper we examine the sinusoidally driven particle in a square potential well to show that avoided crossings can alter the structure of Floquet states in this system. Two types of avoided crossings are identified: on type leads only to temporary changes (as a function of driving field strength) in Floquet state structure while the second type can lead to permanent delocalization of the Floquet states. Radiation spectra from these latter states show significant increase in high harmonic ge…
Nitrogen-vacancy (NV) centers in diamond have become an important tool for quantum technologies. All of these applications rely on long coherence times of electron and nuclear spins associated with these centers. Here, we study the energy level anti-crossings of an NV center in diamond coupled to a first-shell 13C nuclear spin in a small static magnetic field. These level anti-crossings occur for specific orientations of the static magnetic field due to the strong non-secular components of the Hamiltonian. At these orientations we observe decoherence-free subspaces, where the electron spin coh…
We investigate a two-body quantum system with hard-core interaction potential in a two-dimensional harmonic trap. We provide the exact analytical solution of the problem. The energy spectrum of this system as a function of the range of the interaction reveals level crossings which can be explained examining the potential and kinetic part of the energy, as well as the localization and oscillatory properties of the wavefunctions quantified by the Fisher information. The latter is sensitive to the wavefunction localization between nodes promoting this property as a resource of quantum correlation…
We present a specific constant-amplitude periodic level-crossing model of the semi-classical quantum time-dependent two-state problem that belongs to a general Heun class of field configurations. The exact analytic solution for the probability amplitude, generally written for this class in terms of the general Heun functions, in this specific case admits series expansion in terms of the incomplete Beta functions. Terminating this series results in an infinite hierarchy of finite-sum closed-form solutions each standing for a particular two-state model, which generally is only conditionally inte…
We present a method for achieving complete population transfer in a two-state quantum system via adiabatic time evolution in which, contrary to conventional rapid adiabatic passage produced by chirped pulses, there occurs no crossing of diabatic energy curves: there is no sign change of the detuning. Instead, we use structured pulses, in which, in addition to satisfying conditions for adiabatic evolution, there occurs a sign change of the Rabi frequency when the detuning is zero. We present simulations that offer simple geometrical interpretation of the two-dimensional motion of the Bloch vect…
We discuss adiabatic quantum phenomena at a level crossing. Given a path in the parameter space which passes through a degeneracy point, we find a criterion which determines whether the adiabaticity condition can be satisfied. For paths that can be traversed adiabatically we also derive a differential equation which specifies the time dependence of the system parameters, for which transitions between distinct energy levels can be neglected. We also generalize the well-known geometric connections to the case of adiabatic paths containing arbitrarily many level-crossing points and degenerate lev…
We study the quantum phase transition mechanisms that arise in the Interacting Boson Model. We show that the second-order nature of the phase transition from U(5) to O(6) may be attributed to quantum integrability, whereas all the first-order phase transitions of the model are due to level repulsion with one singular point of level crossing. We propose a model Hamiltonian with a true first-order phase transition for finite systems due to level crossings.
In a recent publication [Phys. Rev. A 79, 065602 (2009)] it was shown that an avoided-crossing resonance can be defined in different ways, according to level-structural or dynamical aspects, which do not coincide in general. Here a simple $3$-level system in a $Λ$ configuration is discussed, where the difference between both definitions of the resonance may be observed. We also discuss the details of a proposed experiment to observe this difference, using microwave fields coupling hyperfine magnetic sublevels in alkali atoms.
We consider the Born-Oppenheimer problem near conical intersection in two dimensions. For energies close to the crossing energy we describe the wave function near an isotropic crossing and show that it is related to generalized hypergeometric functions 0F3. This function is to a conical intersection what the Airy function is to a classical turning point. As an application we calculate the anomalous Zeeman shift of vibrational levels near a crossing.
The standard Born Oppenheimer theory does not give an accurate description of the wave function near points of level crossing. We give such a description near an isotropic conic crossing, for energies close to the crossing energy. This leads to the study of two coupled second order ordinary differential equations whose solution is described in terms of the generalized hypergeometric functions of the kind 0F3(;a,b,c;z). We find that, at low angular momenta, the mixing due to crossing is surprisingly large, scaling like μ^(1/6), where μis the electron to nuclear mass ratio.
Unavoided crossings of the energy levels due to a variation of a real parameter are studied. It is found that after the quantum system in question passes through one of its energy-crossing points {\it alias} Kato's exceptional points (EP), its physical interpretation may {\em dramatically} change even when the crossing energies themselves do not complexify. The anomalous physical phase-transition mechanism of the change is revealed, attributed to the EP-related mathematics and illustrated via several exactly solvable matrix toy models.
The dynamics of a two-state system whose energies undergo a real crossing at some instant of time is studied. At this instant, both the coupling and the detuning vanish simultaneously, which leads to an exact degeneracy of the eigenenergies of the system. It is found that the dynamics of the system is primarily determined by the manner in which the degeneracy occurs. This interesting behavior is reminiscent of a symmetry breaking process, since the totally symmetric situation occurring at the crossing is significantly altered by infinitesimal quantities, which remove the degeneracy, with very …
We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations --- the generalization of the confluent hypergeometric equations. We obtain analytical approximations for the state populations in terms of Airy and Bessel functions. Applicable expressions are derived for a large part of the parameter space. We also provide simple formulas which connect local solution in different time regimes. The validity of the analytical approximations is shown by comparing them to numerical simu…
Recent research dealing with magnetocaloric effect (MCE) study of antiferromagnetic (AFM) low dimensional spin systems have revealed a number of fascinating ground-state crossover characteristics upon application of external magnetic field. Herein, through MCE investigation we have explored field-induced quantum level-crossing characteristics of one such spin system: NH4CuPO4.H2O (NCP), an AFM spin 1/2 dimer. Experimental magnetization and specific heat data are presented and the data have been employed to evaluate entropy, magnetic energy and magnetocaloric properties. We witness a sign chang…
An infinite family of exact solutions of the two-photon Rabi model was found by investigating the differential algebraic properties of the Hamiltonian. This family corresponds to energy level crossings not covered by the Juddian class, which is given by elemetary functions. In contrast, the new states are expressible in terms of parabolic cylinder or Bessel functions. We discuss three approaches for discovering this hidden structure: factorization of differential equations, Kimura transformation, and a doubly-infinite, transcendental basis of the Bargmann space.
We show that a simple approximation based on concepts underlying the Kibble-Zurek theory of second order phase transition dynamics can be used to treat avoided level crossing problems. The approach discussed in this paper provides an intuitive insight into quantum dynamics of two level systems, and may serve as a link between the theory of dynamics of classical and quantum phase transitions. To illustrate these ideas we analyze dynamics of a paramagnet-ferromagnet quantum phase transition in the Ising model. We also present exact unpublished solutions of the Landau-Zener like problems.
We consider a \textit{PT}-symmetric cubic oscillator with an imaginary double well. We prove the existence of an infinite number of level crossings with a definite selection rule. Decreasing the positive parameter $\hbar$ from large values, at a value $\hbar_n$ we find the crossing of the pair of levels $(E_{2n+1}(\hbar),E_{2n}(\hbar))$ becoming the pair of levels $(E_n^+(\hbar),E_n^-(\hbar))$. For large parameters, a level is a holomorphic function $E_m(\hbar)$ with different semiclassical behaviors, $E_j^\pm(\hbar),$ along different paths. The corresponding $m$-nodes delocalized state $ψ_m(\…
We generalize the Brundobler-Elser hypothesis in the multistate Landau-Zener problem to the case when instead of a state with the highest slope of the diabatic energy level there is a band of states with an arbitrary number of parallel levels having the same slope. We argue that the probabilities of counterintuitive transitions among such states are exactly zero.
We discuss common properties and reasons for integrability in the class of multistate Landau-Zener (MLZ) models with all diabatic levels crossing at one point. Exploring the Stokes phenomenon, we show that each previously solved model has a dual one, whose scattering matrix can be also obtained analytically. For applications, we demonstrate how our results can be used to study conversion of molecular into atomic Bose condensates during passage through the Feshbach resonance, and provide purely algebraic solutions of the bowtie and special cases of the driven Tavis-Cummings model (DTCM).
The spectral branching behavior of the 2x2 operator matrix of the magneto-hydrodynamic alpha^2-dynamo is analyzed numerically. Some qualitative aspects of level crossings are briefly discussed with the help of a simple toy model which is based on a Z_2-graded-pseudo-Hermitian 2x2 matrix. The considered issues comprise: the underlying SU(1,1) symmetry and the Krein space structure of the system, exceptional points of branching type and diabolic points, as well as the algebraic and geometric multiplicity of corresponding degenerate eigenvalues.
