Computable upper bounds for the adiabatic approximation errors

For a given Hermitian Hamiltonian H(s)(s∈[0,1])with eigenvalues Ek(s)and the corresponding eigenstates|Ek(s)(1 k N),adiabatic evolution described by the dilated Hamiltonian HT(t):=H(t/T)(t∈[0,T])starting from any fixed eigenstate|En(0)is discussed in this paper.Under the gap-condition that|Ek(s)-En(s)|λ0 for all s∈[0,1]and all k n,computable upper bounds for the adiabatic approximation errors between the exact solution|ψT(t)and the adiabatic approximation solution|ψadi T(t)to the Schr¨odinger equation i|˙ψT(t)=HT(t)|ψT(t)with the initial condition|ψT(0)=|En(0)are given in terms of fidelity and distance,respectively.As an application,it is proved that when the total evolving time T goes to infinity,|ψT(t)-|ψadi T(t)converges uniformly to zero,which implies that|ψT(t)≈|ψadi T(t)for all t∈[0,T]provided that T is large enough.     

An upper bound for the generalized adiabatic approximation error with a superposition initial state

The classical adiabatic approximation theory gives an adiabatic approximate solution to the Schr¨odinger equation(SE)by choosing a single eigenstate of the Hamiltonian as the initial state.The superposition principle of quantum states enables us to mathematically discuss the exact solution to the SE starting from a superposition of two different eigenstates of the time-dependent Hamiltonian H(0).Also,we can construct an approximate solution to the SE in terms of the corresponding instantaneous eigenstates of H(t).On the other hand,any physical experiment may bring errors so that the initial state(input state)may be a superposition of different eigenstates,not just at the desired eigenstate.In this paper,we consider the generalized adiabatic evolution of a quantum system starting from a superposition of two different eigenstates of the Hamiltonian at t=0.A generalized adiabatic approximate solution(GAAS)is constructed and an upper bound for the generalized adiabatic approximation error is given.As an application,the fidelity of the exact solution and the GAAS is estimated.     

Effects of Pure Dzyaloshinskii-Moriya Interaction with Magnetic Field on Entanglement in Intrinsic Decoherence

We investigate the effects of pure Dzyaloshinskii-Moriya(DM) interaction with magnetic field on entanglement in intrinsic decoherence,assuming that the system is initially in four Bell states |φ± =(|00 ± |11)/2~(1/2) and|ψ± =(|01) ± |10)/2~(1/2) respectively.It is found that if the system is initially in the state ρ1(0) = |φ+φ+|,the entanglement can obtain its maximum when the DM interaction vector D is in the plane of XOZ and magnetic field B = B_y with the infinite time t,moreover the entanglement is independent of B_y and t when B_y is perpendicular to D.In addition,we obtain similar results when the system is initially in the states ρ2(0) =|φ-φ-| or ρ3(0) = |ψ+ψ+|.However,we find that if the system is initially in the state ρ4(0) =|ψ-ψ-|,the entanglement can obtain its maximum for infinite t,when the DM vector is in the plane of YOZ,XOZ,or XOY,with the magnetic field parallel to X,Y,or Z axis,respectively.Moreover,when the axial B is perpendicular to D for the initial state ρ4(0),the negativity oscillates with time t and reaches a stabie value,the larger the value of B is,the greater the stable value is,and the shorter the oscillation time of the negativity is.Thus we can adjust the direction and value of the external magnetic field to obtain the maximal entanglement,and avoid the adverse effects of external environment in some initial state.This is feasible within the current experimental technology.     

A generalized SHGI integrable hierarchy and its expanding integrable model

An anti-symmetric loop algebra \overline{A}_2 is constructed. It follows that an integrable system is obtained by use of Tu's scheme. The eminent feature of this integrable system is that it is reduced to a generalized Schr?dinger equation, the well-known heat-conduction equation and a Gerdjkov－Ivanov (GI) equation. Therefore, we call it a generalized SHGI hierarchy. Next, a new high-dimensional subalgebra \tilde{G} of the loop algebra ?_2 is constructed. As its application, a new expanding integrable system with six potential functions is engendered.     

Symplectic Schemes and the Shooting Method for Eigenvalues of the Schrodinger Equation

The one-dimensional time-independent Schr6dinger equation is transformed into a Hamiltonian canonical equation by means of the Legendre transformation, then the symplectic schemes and a new shooting method extended to the eigenvalues of the Schr6dinger equation. The method is applied to the calculations of one-dimensional harmonic oscillator, an anharmonic oscillator and the hydrogen atom. The numerical results are in good agreement with the exact ones.     

An upper bound for the adiabatic approximation error

In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution.As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.     

Symmetry groups and Gauss kernels of Schrdinger equations

The relationship between symmetries and Gauss kernels for the Schrdinger equation iu_t=u_xx+f（x）u is established.It is shown that if the Lie point symmetries of the equation are nontrivial,a classical integral transformations of the Gauss kernels can be obtained.Then the Gauss kernels of Schrdinger equations are derived by inverting the integral transformations.Furthermore,the relationship between Gauss kernels for two equations related by an equivalence transformation is identified.     

Schrdinger Equation of a Particle on a Rotating Curved Surface

We derive the Schrdinger equation of a particle constrained to move on a rotating curved surface S.Using the thin-layer quantization scheme to confine the particle on S,and with a proper choice of gauge transformation for the wave function,we obtain the well-known geometric potential V_g and an additive Coriolis-induced geometric potential in the co-rotational curvilinear coordinates.This novel effective potential,which is included in the surface Schrdinger equation and is coupled with the mean curvature of S,contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian.We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.     

Spontaneous U（1） symmetry breaking and Bose-Einstein condensation

Based on Bogoliubov's truncated Hamiltonian H_B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term $\sqrt{V}(\lambda a₀+\lambda^*a₀⁺) to H_B, we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose--Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schr\"{o}dinger eigenvalue equation and using the self-consistent condition.     

Memory effect in time fractional Schr?dinger equation

A significant obstacle impeding the advancement of the time fractional Schr?dinger equation lies in the challenge of determining its precise mathematical formulation. In order to address this, we undertake an exploration of the time fractional Schr?dinger equation within the context of a non-Markovian environment. By leveraging a two-level atom as an illustrative case, we find that the choice to raise i to the order of the time derivative is inappropriate. In contrast to the conventional approac...     

An Integrable Decomposition of the Derivative Nonlinear SchrSdinger Equation

The nonlinearization method of spectral problem is developed and applied to the derivative nonlinear Schr6dinger equation （DNLS）. As a result, an integrable decomposition of the DNLS equation is obtained.     

Envelope solitary waves and their reflection and transmission due to impurities in a granular material

A nonlinear Schrdinger equation in one-dimensional bead chain is first obtained and an envelope solitary wave of the system is verified numerically in this system. The reflection and the transmission of an incident envelope solitary wave due to impurities has also been investigated. It is found that the magnitudes of both the reflection and the transmission not only depend on the characters of impurity materials, the wave number, the incident wave amplitude, but also on the impurity number. This can be used to detect the character and the number of the impurity materials in the bead chain by measuring the reflection and the transmission of an incident pulse.     

Existence and construction of simultaneous cloning machines for mixed states

It is a well-known fact that the no-cloning theorem forbids the creation of identical copies of an arbitrary unknown quantum state.In other words,there does not exist a quantum cloning machine that can clone all quantum states.However,it is possible to clone given quantum states under certain conditions,for instance,k distinct pure states|ψ1,|ψ2,...,|ψk can be cloned simultaneously if and only if they are orthogonal.This paper discusses the existence and construction of simultaneous cloning machines for mixed states.It is proved that k distinct mixed statesρ1,ρ2,...,ρkof the n-dimensional quantum system Cncan be cloned simultaneously,that is,there exists a quantum channel Φ on MnMn and a state Σ in Mn,such that Φ(ρiΣ)=ρiρi for all i,if and only ifρiρj=0(i j).Also,the constructing procedure of the desired simultaneous cloning machine is given.     

Frequencies-Selected Enhancement of the Extended High-Order Harmonic Generation Plateau from a United Two-Atom System Irradiated by a Combined Pulse

We present a high-order harmonic generation (HHG) spectrum from a united two-atom system exposed to a combined laser pulse, by numerically solving a one-dimensional time-dependent Schr6dinger equation. The combined laser pulse is composed of a low-frequency femtosecond pulse and a high-frequency attosecond one with respective appropriate amplitudes. For a suitable inter-nuclear separation, the harmonic emission efficiencies near the second cutoff of the extended plateau can be effectively enhanced by more than four orders of magnitude compared with the case of the low-frequency pulse alone. Such a combined pulse irradiating on the united twoatom system ionizes each atom, in a large rate (but not to a too large ionization yield), mainly at a particular time-interval. When the ionized electron from an atom reaches at the vicinity of the other atom and recombines with it, significant HHG enhancement is achieved for particular harmonics. This result, to our knowledge, is one of the best up to now in the endeavor for dramatically extending the width and simultaneously enhancing the height of the plateau.     

Two-step dissociation of a polaron in conjugated polymers

We have studied the electric-field-driven motion of a polaron by solving the time-dependent Schr\"{o}dinger equation nonadiabatically and the lattice equation of motion simultaneously. It is found that the polaron may experience two sequent transitions under high fields; one is the transition from the subsonic to the supersonic state, and the other from the supersonic to dissociated state. The acoustic mode is decoupled from the charge when the polaron moves at a speed faster than the sound speed, and then the optical mode is decoupled at the second transition to make the polaron dissociate completely.     

Chaos control in the nonlinear Schrodinger equation with Kerr law nonlinearity

The nonlinear Schr6dinger equation with Kerr law nonlinearity in the two-frequency interference is studied by the numerical method. Chaos occurs easily due to the absence of damping. This phenomenon will cause the distortion in the process of information transmission. We find that fiber-optic transmit signals still present chaotic phenomena if the control intensity is smaller. With the increase of intensity, the fiber-optic signal can stay in a stable state in some regions. When the strength is suppressed to a certain value, an unstable phenomenon of the fiber-optic signal occurs. Moreover we discuss the sensitivities of the parameters to be controlled. The results show that the linear term coefficient and the environment of two quite different frequences have less effects on the fiber-optic transmission. Meanwhile the phenomena of vibration, attenuation and escape occur in some regions.     

Abundant exact solutions for a strong dispersion-managed system equation

The generalized nonlinear Schr?dinger equation (NLSE), which governs the dynamics of dispersion-managed (DM) solitons, is considered. A novel transformation is constructed such that the DM fibre system equation with optical loss (gain) is transformed to the standard NLSE under a restricted condition. Abundant new soliton and periodic wave solutions are obtained by using the transformation and the solutions of standard NLSE. Further, we discuss their main properties and the interaction scenario between two neighbouring solitons by using direct computer simulation.     

Discrete energy transport in collagen molecules

The modulational instability in the three coupled α-polypeptide chains of a collagen molecule is investigated. Choosing symmetric and asymmetric solutions, and applying the so-called rotating-wave approximation, we describe the dynamics of the system by the discrete nonlinear Schrdinger(DNLS) equation. The linear stability analysis of the continuous wave solution is performed. The numerical simulations show the generation of trains of solitonic structures in the lattice with increasing amplitude as time progresses. The effect of damping and noise forces of the physiological temperature(T = 300 K) introduces an erratic behavior to the formed patterns, reinforcing the idea that the energy used in metabolic processes is confined to specific regions for efficiency.     

Ultraslow optical solitons via electromagnetically induced transparency: a density-matrix approach

We study the ultraslow optical solitons in a resonant three-level atomic system via electromagnetically induced transparency under a density-matrix (DM) approach. The results of linear and nonlinear optical properties are compared with those obtained by using an amplitude variable (AV) approach. It is found that the results for both approaches are the same in the linear regime if the corresponding relations between the population-coherence decay rates in the DM approach and the energy-level decay rates in the AV approach are appropriately imposed. However, in the nonlinear regime there is a small difference for the self-phase modulation coefficient of the nonlinear Schr\"{o}dinger equation that governs the time evolution of probe pulse envelope. All analytical predicts are checked by numerical simulations.