Time-dependent Unitary Transformation Research Articles

Efficient numerical approach for the simulations of high-power dispersive readout with time-dependent unitary transformation

Efficient numerical approach for the simulations of high-power dispersive readout with time-dependent unitary transformation

The time-dependent polaron frame

The single-photon emitters in solid-state photonic devices have become crucial for quantum technology implementations. The exciton–phonon interaction degrades the quality of the photons even when the coherent scattering is dominant. Some models describing an emitter’s trajectory in a phonon bath are the weak-coupling master equation and the standard and variational polaron master equations. The polaron theory based on a steady polaron transformation cannot capture the response of the phonon reservoir upon the exciton creation in its transformed Hamiltonian. Here, a time-dependent phonon displacement leads to the time evolution of the phonon-related energies and correlations. Variables as the phonon-induced energy shift and the phonon first-order correlation function converge to the standard polaron theory results. However, the expectation value of the phonon displacement operator differs. According to the time-dependent polaron master equation, the mitigation of the coupling between the single-photon emitter and the coherent driving field by the phonon reservoir evolves to be worse than predicted in the standard polaron theory. • A time-dependent polaron unitary transformation is derived • Time evolution of the exciton–phonon interaction • The path from the weak-coupling Hamiltonian to the polaron Hamiltonian

A Real Expectation Value of the Time-dependent Non-Hermitian Hamiltonians* *This paper is dedicated to the memory of our friends and colleagues Lahcene Alouani and Mebrouk Gherda died due to covid 19.

With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the −inner product, where is a obtained from the charge conjugation operator through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

Time-Dependent Unitary Transformation Method in the Strong-Field-Ionization Regime with the Kramers-Henneberger Picture.

Time evolution operators of a strongly ionizing medium are calculated by a time-dependent unitary transformation (TDUT) method. The TDUT method has been employed in a quantum mechanical system composed of discrete states. This method is especially helpful for solving molecular rotational dynamics in quasi-adiabatic regimes because the strict unitary nature of the propagation operator allows us to set the temporal step size to large; a tight limitation on the temporal step size () can be circumvented by the strict unitary nature. On the other hand, in a strongly ionizing system where the Hamiltonian is not Hermitian, the same approach cannot be directly applied because it is demanding to define a set of field-dressed eigenstates. In this study, the TDUT method was applied to the ionizing regime using the Kramers-Henneberger frame, in which the strong-field-dressed discrete eigenstates are given by the field-free discrete eigenstates in a moving frame. Although the present work verifies the method for a one-dimensional atom as a prototype, the method can be applied to three-dimensional atoms, and molecules exposed to strong laser fields.

Relativistic properties and invariants of the Du Fort–Frankel scheme for the one-dimensional Schrödinger equation

The Du Fort–Frankel scheme for the one-dimensional Schrödinger equation is shown to be equivalent, under a time-dependent unitary transformation, to the Ablowitz–Kruskal–Ladik scheme for the Klein–Gordon equation. The Schrödinger equation describes a non-relativistic quantum particle, while the Klein–Gordon equation describes a relativistic particle. The conditional convergence of the Du Fort–Frankel scheme to solutions of the Schrödinger equation arises because solutions of the Klein–Gordon equation only approximate solutions of the Schrödinger equation in the non-relativistic limit. The time-dependent unitary transformation is the discrete analog of the transformation that arises from seeking a non-relativistic limit using the interaction picture of quantum mechanics to decompose the Klein–Gordon Hamiltonian into the relativistic rest energy and a remainder. The Ablowitz–Kruskal–Ladik scheme is in turn decomposed into a quantum lattice gas automaton for the one-dimensional Dirac equation, which is also the one-dimensional discrete time quantum walk. This relativistic interpretation clarifies the origin of the known discrete invariant of the Du Fort–Frankel scheme as expressing conservation of probability for the 2-component wavefunction in the one-dimensional Dirac equation under discrete unitary evolution. It also leads to a second invariant, the matrix element of the evolution operator, whose imaginary part gives a discrete approximation to the expectation of the non-relativistic Schrödinger Hamiltonian.

Energy expectation values of a particle in nonstationary fields

We show that the origin of the nonequivalence of Hamiltonians in different representations is a change of the form of the time-derivative operator at a time-dependent unitary transformation. This nonequivalence does not lead to an ambiguity of the energy expectation values of a particle in nonstationary fields but assigns the basic representation. It has been explicitly or implicitly supposed in previous investigations that this representation is the Dirac one. We prove the alternative assertion about the basic role of the Foldy-Wouthuysen representation. We also derive the general equation for the energy expectation values in the Dirac representation. As an example, we consider a spin-1/2 particle with anomalous magnetic and electric dipole moments in strong time-dependent electromagnetic fields. We apply the obtained results to a spin-1/2 particle in a plane monochromatic electromagnetic wave and give an example of the exact Foldy-Wouthuysen transformation in the nonstationary case.

Change of the Pancharatnam phase under unitary transformation and application to a model three-level Γ-type system specified by a pair of dipole-coupled excited states

The interdependence of total, dynamic and geometric (Pancharatnam) phases of the state vector of a quantum system that are determined in different representations is considered. One finds that the phases calculated using two different bases of the Hilbert space, which are related by a time-dependent unitary transformation, are not invariant under this transformation; rather they are connected by non-trivial and well-defined relations. We determine the general relationships between the phases in different representations and then apply and verify these transformations in the case of the physically interesting model of the three-level Γ-type system specified by a pair of dipole-coupled excited states.

Time Evolution and Squeezing of Degenerate and Non-degenerate Coupled Parametric Down-Conversion with Driving Term

By properly selecting the time-dependent unitary transformation for the linear combination of the number operators, we construct a time-dependent invariant and derive the corresponding auxiliary equations for the degenerate and non-degenerate coupled parametric down-conversion system with driving term. By means of this invariant and the Lewis–Riesenfeld quantum invariant theory, we obtain closed formulae of the quantum state and the evolution operator of the system. We show that the time evolution of the quantum system directly leads to production of various generalized one- and two-mode combination squeezed states, and the squeezed effect is independent of the driving term of the Hamiltonian. In some special cases, the current solution can reduce to the results of the previous works.

Time evolution of a time-dependent harmonic oscillator in a static magnetic field

The analytic expression of the time evolution wavefunction of the two-dimensional harmonic oscillator with time-dependent mass and frequency in a static magnetic field is obtained using an operator-algebraic method slightly different from the usual Lie algebraic technique. The evolution operator of the one-dimensional harmonic oscillator with time-dependent mass and frequency is established first by forming an operator differential equation with the su(1, 1) Lie algebra, which is deduced from the time-dependent linear unitary transformation for boson operators (a, a†), and then by comparing this operator equation with the time evolution equation of the one-dimensional oscillator.

Critical property of the geometric phase in the Dicke model

We obtain the ground-state energy level and associated geometric phase in the Dicke model analytically by means of the Holstein-Primakoff transformation and the boson expansion approach in the thermodynamic limit. The nonadiabatic geometric phase induced by the photon field is derived with the time-dependent unitary transformation. It is shown that the quantum phase transition characterized by the nonanalyticity of the geometric phase is remarkably of the first order. We also investigate the scaling behavior of the geometric phase at the critical point, which can be measured in a practical experiment to detect the quantum phase transition.

Heisenberg Model in a Rotating Magnetic Field

We study the Heisenberg model under the influence of arotating magnetic field. By using a time-dependent unitarytransformation, the time evolution operator for the Schrödingerequation is obtained, which involves no chronological product. Thespin vectors (mean values of the spin operators) are obtained asexplicit functions of time in the most general case. A series ofcyclic solutions are presented. The nonadiabatic geometric phases ofthese cyclic solutions are calculated, and are expressed in terms ofthe solid angle subtended by the closed trace of the total spinvector, as well as in terms of those of the individual spins.

Kinetic Theory of Quantum Electrodynamic Plasma in a Strong Electromagnetic Field: I. The Covariant Formalism

We present a covariant approach to the kinetic theory of quantum electrodynamic plasma in a strong electromagnetic field. The method is based on the relativistic von Neumann equation for the nonequilibrium statistical operator defined on spacelike hyperplanes in Minkowski space. We use the canonical quantization of the system on hyperplanes and a covariant generalization of the Coulomb gauge. The condensate mode associated with the mean electromagnetic field is separated from the photon degrees of freedom by a time-dependent unitary transformation of the dynamic variables and the nonequilibrium statistical operator. This allows using expansions of correlation functions and of the statistical operator in powers of the fine structure constant even in the presence of a strong electromagnetic field. We present a general scheme for deriving kinetic equations in the hyperplane formalism.

Dirac particles in a rotating magnetic field

We study a relativistic charged Dirac particle moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Dirac equation with the time-dependent Hamiltonian can be reduced to a Dirac-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Dirac equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explictly available. For a slowly rotating magnetic field, the eigenvalue problem of the effective Hamiltonian is solved approximately and the geometric phases are calculated. The same problem for a charged or neutral Dirac particle with an anomalous magnetic moment is discussed briefly.

Charged particles in a rotating magnetic field

We study the valence electron of an alkaline atom or a general charged particle with arbitrary spin and with magnetic moment moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Schr\"odinger equation with the time-dependent Hamiltonian can be reduced to a Schr\"odinger-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Schr\"odinger equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explicitly available. For the alkaline atomic electron and a strong magnetic field, the eigenvalue problem of the effective Hamiltonian is completely solved, and the geometric phase turns out to be a linear combination of two solid angles. For a weak magnetic field, the same problem is solved partly. For a general charged particle, the problem is solved approximately in a slowly rotating magnetic field, and the geometric phases are also calculated.

Unitary transformation approach to the cyclic evolution of SU(1, 1) and SU(2) time-dependent systems and geometrical phases

Using the time-dependent unitary transformation instead of the invariant operator, the solutions of SU(1, 1) and SU(2) time-dependent quantum systems are obtained. It is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the Hamiltonian and the other correspond to the Floquet operator which gives the cyclic states and their associated phases over one period of the evolution. The non-adiabatic Berry's (or Aharonov-Anandan) phases are determined totally by such a unitary transformation.

Inverting a time-dependent harmonic oscillator potential by a unitary transformation and a class of exactly solvable oscillators

A time-dependent unitary (canonical) transformation is found which maps the Hamiltonian for a harmonic oscillator with time-dependent real mass and real frequency to that of a generalized harmonic oscillator with time-dependent real mass and imaginary frequency. The latter may be reduced to an ordinary harmonic oscillator by means of another unitary (canonical) transformation. A simple analysis of the resulting system leads to the identification of a previously unknown class of exactly solvable time-dependent oscillators. Furthermore, it is shown how one can apply these results to establish a canonical equivalence between some real and imaginary frequency oscillators. In particular it is shown that a harmonic oscillator whose frequency is constant and whose mass grows linearly in time is canonically equivalent with an oscillator whose frequency changes from being real to imaginary and vice versa repeatedly.

Quantum states for a Paul-trapped particle in an intense laser field.

The eigenstates are constructed for a Paul-trapped particle in an intense laser field. This is done by using a time-dependent unitary transformation. The application of these results is discussed.

THE QUADRATIC TIME-DEPENDENT HAMILTONIAN: EVOLUTION OPERATOR, SQUEEZING REGIONS IN PHASE SPACE AND TRAJECTORIES

We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a+, which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.

Nonuniqueness of the current operators and gauge invariance

For a point-like Dirac particle we describe a class of time-dependent unitary transformations which bring about an explicitly gauge invariant one-body current operator. We do not resort to the 1/ m expansion scheme but retain all orders in 1/ m. We exploit the nonuniqueness (unitary equivalence) inherent to the time-dependent unitary transformation method, and investigate its consequences on the nuclear electromagnetic interactions. We show that the nonuniqueness of the one-body current should influence the exchange current up to the quantity which is expressed in terms of the commutator of the nuclear interaction hamiltonian with the generator of the unitary transformation. We apply the result to the electromagnetic interaction of a nucleon in a relativistic mean field.

Evolution of a molecular exciton on a Davydov lattice at T=0.

We develop a theory of exciton evolution on a zero-temperature Davydov lattice which is free of certain deficiencies found in the standard Davydov theory. The approach makes use of a time-dependent unitary transformation on a Davydov Hamiltonian parametrized by a dimensionless lattice constant a and a dimensionless exciton-phonon coupling constant \ensuremath{\alpha}. The transformation generator is expanded in a normal-ordered series of multiphonon operators with expansion coefficients chosen to eliminate various terms in the transformed Schr\odinger equation. At the one-phonon level, we obtain equations of motion which differ from those of Davydov. In the small-polaron transportless limit (infinite a) the equations are exact. In the large-polaron continuum limit (vanishing a) the equations become field equations whose stationary solutions are those of Gross's interpolation theory. For a one-spine model of an \ensuremath{\alpha}-helix (a=2.7) we find that soliton formation during evolution from a localized initial state requires a significantly larger value of \ensuremath{\alpha} than is required by Davydov theory.