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IEEE Conference on Decision and Control (CDC)

Operational techniques for the Floquet Hamiltonians

Bogdan Mielnik, David J. Fernández C.

Published Nov 11, 2002• 1 citation
Publisher

Abstract

The control operations on a charged non-relativistic particle by variable external fields are examined. Contrasting with the static theory, the effective Hamiltonians for the sequences of field pulses have not necessarily lower bounds. In particular, the Floquet systems of rotating fields can generate the harmonic oscillators turned upside down, with infinite ladders of negative levels. The Floquet Hamiltonians reduced to pure negative kinetic energy can also be achieved; they permit to invert the free evolution of Schrodinger's particle, opening new perspectives in the control techniques. The manipulation of Hamiltonians without lower bounds imply instability at the level of the quantum field theory.

DOI
10.1109/cdc.2000.912897

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The control operations on a charged non-relativistic particle by variable external fields are examined. Contrasting with the static theory, the effective Hamiltonians for the sequences of field pulses have not necessarily lower bounds. In particular, the Floquet systems of rotating fields can generate the harmonic oscillators turned upside down, with infinite ladders of negative levels. The Floquet Hamiltonians reduced to pure negative kinetic energy can also be achieved; they permit to invert the free evolution of Schrodinger's particle, opening new perspectives in the control techniques. The manipulation of Hamiltonians without lower bounds imply instability at the level of the quantum field theory.
The techniques of controlling the free evolution of a nonrelativistic charged particle by time-dependent magnetic fields are proposed and the possibility of more general operations upon the Schrödinger wave packet is discussed. It is found that a properly programmed sequence of magnetic pulses can invert the free evolution process, forcing an arbitrary wave packet to ‘‘go back in time’’ to recover its past shape. Our manipulation prescriptions hold also for nonrelativistic particles of arbitrary spin.
In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.
Floquet engineering of isolated systems is often based on the concept of the effective time-independent Floquet Hamiltonian, which describes the stroboscopic evolution of a periodically driven quantum system in steps of the driving period and which is routinely obtained analytically using high-frequency expansions. The generalization of these concepts to open quantum systems described by a Markovian master equation of Lindblad type turns out to be nontrivial: On the one hand, already for a two-level system two different phases can be distinguished, where the effective time-independent Floquet generator (describing the stroboscopic evolution) is either again Markovian and of Lindblad type or not. On the other hand, even though in the high-frequency regime a Lindbladian Floquet generator (Floquet Lindbladian) is numerically found to exist, this behavior is, curiously, not correctly reproduced within analytical high-frequency expansions. Here, we demonstrate that a proper Floquet Lindbladian can still be obtained from a high-frequency expansion, when treating the problem in a suitably chosen rotating frame. Within this approach, we can then also describe the transition to a phase at lower driving frequencies, where no Floquet Lindbladian exists, and show that the emerging non-Markovianity of the Floquet generator can entirely be attributed to the micromotion of the open driven system.
Parametric gates and processes engineered from the perspective of the static effective Hamiltonian of a driven system are central to quantum technology. However, the perturbative expansions used to derive static effective models may not be able to efficiently capture all the relevant physics of the original system. In this work, we investigate the conditions for the validity of the usual low-order static effective Hamiltonian used to describe a Kerr oscillator under a squeezing drive. This system is of fundamental and technological interest. In particular, it has been used to stabilize Schrödinger cat states, which have applications for quantum computing. We compare the states and energies of the effective static Hamiltonian with the exact Floquet states and quasi-energies of the driven system and determine the parameter regime where the two descriptions agree. Our work brings to light the physics that is left out by ordinary static effective treatments and that can be explored by state-of-the-art experiments.
Abstract We present a non-adiabatic approach to transfer a quantum state between different superpositions of the eigenstates within a degenerate subspace, namely between eigenstates of various spatial orientations, while avoiding excitation. This is achieved by inducing different accumulated dynamical phases for the left- and right-rotating components of the state, in analogy with optical rotation. The control of the dynamical phase is based on the asymmetric Floquet energy shift induced by a monochromatic circularly polarized external laser field. In contrast to adiabatic control methods demanding an extremely slowly varying carrier wave of the laser field for excitation-free control, our approach requires only moderately slow variation in the envelope. This methodology promises fast quantum-state manipulation and control of electron dynamics in biological and chemical reactions.