The dynamics of classical and quantum systems which are driven by a high frequency (ω) field is investigated. For classical systems the motion is separated into a slow part and a fast part. The motion for the slow part is computed perturbatively in powers of ω −1 to order ω −4 and the corresponding time independent Hamiltonian is calculated. Such an effective Hamiltonian for the corresponding quantum problem is computed to order ω −4 in a high frequency expansion. Its spectrum is the quasienergy spectrum of the time dependent quantum system. The classical limit of this effective Hamiltonian is the classical effective time independent Hamiltonian. It is demonstrated that this effective Hamiltonian gives the exact quasienergies and quasienergy states of some simple examples as well as the lowest resonance of a non trivial model for an atom trap. The theory that is developed in the paper is useful for the analysis of atomic motion in atom traps of various shapes.

The classical and quantum dynamics in a high frequency field are found to be described by an effective time independent Hamiltonian. It is calculated in a systematic expansion in the inverse of the frequency (omega) to order omega(-4). The work is an extension of the classical result for the Kapitza pendulum, which was calculated in the past to order omega(-2). The analysis makes use of an implementation of the method of separation of time scales and of a quantum gauge transformation in the framework of Floquet theory. The effective time independent Hamiltonian enables one to explore the dynamics in the presence of rapidly oscillating fields, in the framework of theories that were developed for systems with time independent Hamiltonians. The results are relevant, in particular, for exploring the dynamics of cold atoms.

We show here that due to non-adiabatic couplings in decaying systems applying the same time-dependent protocol in the forward and reverse direction to the same mixed initial state leads to different final pure states. In particular, in laser driven molecular systems applying a specifically chosen positively chirped laser pulse or an equivalent negatively chirped laser pulse yields entirely different final vibrational states. This phenomenon occurs when the laser frequency and intensity are slowly varied around an exceptional point (EP) in the laser intensity and frequency parameter space where the non-hermitian spectrum of the problem is degenerate. The protocol implies that a positively chirped laser pulse traces a counter-clockwise loop in time in the laser parameters' space whereas a negatively chirped pulse follows the same loop in the clockwise direction. According to this protocol one can choose the final pure state from any initial state. The obtained results imply the intrinsic non-adiabaticity of quantum transport around an EP, and offer a way to observe the EP experimentally in time-dependent quantum systems.For open quantum systems where the effective Hamiltonian is non-hermitian (NH) the non crossing rule [1] is replaced by an intersection of two complex energy levels associated with two eigenfunctions of the NH Hamiltonian that have the same symmetry. Let us consider 2 × 2 Hamiltonian matrix H which depends on potential parameters q 1 and q 2 . These can be for instance the laser frequency and intensity when light interacts with two normal modes of a molecule. In open quantum system where the effective Hamiltonian is NH all matrix elements can attain complex values. The complex diagonal matrix elements are associated with metastable (resonance) states, such that −2ImH 11 and −2ImH 22 are the decay rates of the meta-stable states. The eigenvalues of this NH Hamiltonian are degenerate when ∆ = (H 11 − H 22 ) 2 + 4H 12 H 21 = 0 even though all matrix elements are different from zero. This situation is very different from the hermitian (standard) case where crossing requires H 12 = H 21 = 0 and H 11 = H 22 . At the crossing point a nonhermitian degeneracy (NHD) is obtained when the following two equations are satisfied:NHD is very different in its nature from hermitian degeneracy. NHD is obtained at the crossing point denoted by (q EP 1 , q EP 2 ), where the two eigenvalues coalesce and form a branch point (BP) in the complex energy spectrum. At the BP the first order derivatives of the eigenvalues with respect to q 1 or q 2 acquire infinitely large values (see for example Chapter 9 in Ref. [2]). This BP is also known as an exceptional point (EP) in the energy spectrum [3,4]. Moreover, at the BP (EP) not only the eigenvalues coalesce but also the corresponding eigenvectors. Such a phenomenon can never occur in standard QM. In NHQM as q 1 → q EP 1 and q 2 → q EP 2

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Here we show that the adiabatic theorem, originally derived for bound molecular systems, does not necessarily hold for dissociative molecular systems. In particular, we demonstrate here the asymmetric adiabatic evolution of the vibrational states of H+2 under the influence of slowly varying chirped laser pulses. There are some laser parameters for which non-Hermitian degeneracies occur in the spectrum of the problem. These are known as exceptional points. When one slowly cycles the laser intensity and wavelength in a loop in parameter space around an exceptional point, the initial tenth vibrational state of H+2 flips into the ninth vibrational state by the end of the loop. However, within the same excursion around an exceptional point in the laser parameters space, the ninth vibrational state of H+2 does not flip into the tenth vibrational state but winds up unchanged by the end of the loop.