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arXiv:0711.4298 · Physical Review A

Steering quantum transitions between three crossing energy levels

Svetoslav S. Ivanov, Nikolay V. Vitanov

Published Feb 5, 2008• 19 citations

Abstract

We calculate the propagator and the transition probabilities for a coherently driven three-state quantum system. The energies of the three states change linearly in time, whereas the interactions between them are pulse shaped. We derive a highly accurate analytic approximation by assuming independent pairwise Landau-Zener transitions occurring instantly at the relevant avoided crossings, and adiabatic evolution elsewhere. Quantum interferences are identified, which occur due to different possible evolution paths in Hilbert space between an initial and a final state. A detailed comparison with numerical results for Gaussian-shaped pulses demonstrates a remarkable accuracy of the analytic approximation. We use the analytic results to derive estimates for the half-width of the excitation profile, and for the parameters required for creation of a maximally coherent superposition of the three states. These results are of potential interest in ladder climbing in alkali-metal atoms by chirped laser pulses, in quantum rotors, in transitions between Zeeman sublevels of a $J=1$ level in a magnetic field, and in control of entanglement of a pair of spin-$1∕2$ particles. The results for the three-state system can be generalized, without essential difficulties, to higher dimensions.

DOI: 10.1103/physreva.77.023406

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We calculate the propagator and the transition probabilities for a coherently driven three-state quantum system. The energies of the three states change linearly in time, whereas the interactions between them are pulse shaped. We derive a highly accurate analytic approximation by assuming independent pairwise Landau-Zener transitions occurring instantly at the relevant avoided crossings, and adiabatic evolution elsewhere. Quantum interferences are identified, which occur due to different possible evolution paths in Hilbert space between an initial and a final state. A detailed comparison with numerical results for Gaussian-shaped pulses demonstrates a remarkable accuracy of the analytic approximation. We use the analytic results to derive estimates for the half-width of the excitation profile, and for the parameters required for creation of a maximally coherent superposition of the three states. These results are of potential interest in ladder climbing in alkali-metal atoms by chirped laser pulses, in quantum rotors, in transitions between Zeeman sublevels of a

J=1

level in a magnetic field, and in control of entanglement of a pair of spin-

1∕2

particles. The results for the three-state system can be generalized, without essential difficulties, to higher dimensions.

The excitation amplitude after multiple crossings is not a mere product of Landau-Zener transition probabilities at each crossing, due to coherent evolution of the system in between crossings. In the three-level ladder system, the trapping of population by frequency modulated fields ensures coherent evolution, and inclusion of phase effects for population redistribution after multiple crossings becomes necessary. The relative phase accumulated by various adiabatic states as they evolve along different paths is tailored to show the existence of quantum interference effects. We present a method of inverting the population in a three-level system, without affecting the population in the intermediate state. We also present an all optical implementation of the three-level ladder system, where these effects can be realized.

Abstract Three analytic solutions to the Schrödinger equation for the time-dependent Landau–Zener (LZ) Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The LZ formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.

A model of the time evolution of two interacting Rydberg manifolds of energy levels subject to a linearly ramped electric field is solved exactly in the Landau-Zener (LZ) approximation. Each manifold's levels are treated as linear in time, parallel, equally spaced, and infinite in number. Their pairwise interactions produce a regular two-dimensional grid of isolated anticrossings. The time development of an initially populated state is then governed by two-level LZ transitions at avoided crossings and adiabatic evolution between them, parametrized by the LZ transition probability

D

and a dynamical phase unit \ensuremath{\varphi}. The resulting probability distributions of levels are given analytically in the form of recursion relations, generating functions, integral representations involving

D

and \ensuremath{\varphi}, and in certain limits by Bessel or Whittaker functions. Level populations are mapped out versus location on the grid for a range of cases. Interference effects lead to two principal types of probability distributions: a braiding adiabatic pattern with revivals for small

D

and a diabatic pattern for $D\ensuremath{\rightarrow}1$ in which only certain levels parallel to the initial one are appreciably populated. The sensitivity of the coherent evolution to \ensuremath{\varphi} is discussed, along with the relation of this model to others and to selective-field ionization.

The laser coupled Landau-Zener avoided crossing has been investigated with an aim towards obtaining the laser source parameters for precise controlling of the state dynamics in a two-level quantum system. The conventional Landau-Zener equation is modified for including the interaction of the system with a laser field during a bias energy sweep and the obtained Hamiltonian is numerically solved for the investigation of the two-state occupation probabilities. We have shown that in the Landau-Zener process, using an additional laser source with controlled amplitude, frequency, and phase, the system dynamics could be arbitrarily engineered. This is while, by synchronous frequency sweeping of a chirped-pulse laser, the system could be guided into a resonance condition, which again gives the remarkable possibility for precise tuning and controlling of the quantum system dynamics.

We study the

S

matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].