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arXiv:quant-ph/0503197

Analytic pulse design for selective population transfer in many-level quantum systems: optimizing the pulse duration

Duje Bonacci

Published Mar 24, 2005• 0 citations
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Abstract

In the previous paper on this topic it was shown how, for a pulse of arbitrary shape and duration, the drive frequency can be analytically optimized to maximize the amplitude of the population oscillations between the selected two levels in a general many level quantum system. It was shown how the standard Rabi theory can be extended beyond the simple two-level systems. Now, in order to achieve the quickest and as complete as possible population transfer between two pre-selected levels, driving pulse should be tailored so that it produces only a single half-oscillation of the population. In this paper, this second (and final) step towards the controlled population transfer using modified (i.e. many level system) Rabi oscillations is discussed. The results presented herein can be regarded as an extension of the standard $π$-pulse theory - also strictly valid only in the two level systems - to the coherently driven population oscillations in general many level systems.

DOI
10.48550/arxiv.quant-ph/0503197

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In the previous paper on this topic it was shown how, for a pulse of arbitrary shape and duration, the drive frequency can be analytically optimized to maximize the amplitude of the population oscillations between the selected two levels in a general many level quantum system. It was shown how the standard Rabi theory can be extended beyond the simple two-level systems. Now, in order to achieve the quickest and as complete as possible population transfer between two pre-selected levels, driving pulse should be tailored so that it produces only a single half-oscillation of the population. In this paper, this second (and final) step towards the controlled population transfer using modified (i.e. many level system) Rabi oscillations is discussed. The results presented herein can be regarded as an extension of the standard πππ-pulse theory - also strictly valid only in the two level systems - to the coherently driven population oscillations in general many level systems.
State-selective preparation and manipulation of discrete-level quantum systems such as atoms, molecules, or quantum dots is the ultimate tool for many diverse fields such as laser control of chemical reactions, atom optics, high-precision metrology, and quantum computing. Rabi oscillations are one of the simplest, yet potentially quite useful mechanisms for achieving such manipulation. Rabi theory establishes that in the two-level systems resonant drive leads to the periodic and complete population oscillations between the two system levels. In this paper an analytic optimization algorithm for producing Rabi-like oscillations in the general discrete many-level quantum system is presented.
The quantum dynamics of a pulse-driven finite-dimensional quantum chain with only nearest-neighbor coupling is studied. We extend the concept of Rabi oscillations from two-level quantum systems to the multi-level quantum chains. The time-dependent quantum dynamics and solutions producing perfect population transfer are obtained for up to five-level quantum chains. The Gröbner basis analysis technique is used to generalize the results and get all analytical solutions for perfect population transfer. Explicit formulas for the solutions up to nine levels are presented. These results could be used to design control strategies for general finite-dimensional quantum systems.
The maximal population transfer in a two-level system induced by an ultrashort, linearly polarized laser pulse and the optimal pulse shape for maximal population transfer depend critically on the pulse duration and phase. In the case of temporally symmetric pulses, half-cycle pulses are optimal and are capable of inverting the population completely. For antisymmetric pulses, bimodal pulses are optimal but cannot invert the population completely. Deviations from the Rabi formula for the pulse parameters do not occur until the pulses become shorter than a few optical cycles.
We present a simple approach allowing one to obtain analytical expressions for laser pulses that can drive a two-level system in an arbitrarily chosen way. The proposed scheme relates every desired population-evolution path to a single resonant laser pulse. It allows one to drive the system from any initial superposition of the two states to a final state having the desired distribution of the populations. We exemplify the scheme with a concrete example, where the system is driven from a nonstationary superposition of states to one of its eigenstates. We argue that the proposed approach may have interesting applications for designing pulses that can control ultrafast charge-migration processes in molecules. Although focused on laser-driven population control, the results obtained are general and could be applied for designing other types of control fields.
We study the dynamics of a two-level system driven by an off-resonance few-cycle pulse which has a phase jump $\ensuremath{\phi}$ at t=t0t={t}_{0}t=t0​, in contrast to many-cycle pulses, under the nonrotating-wave approximation (NRWA). We give a closed form analytical solution for the evolution of the probability amplitude ∣Ca(t)∣|{C}_{a}(t)|∣Ca​(t)∣ for the upper level. Using the appropriate pulse parameters like the phase jump $\ensuremath{\phi}$, jump time t0{t}_{0}t0​, pulse width $\ensuremath{\tau}$, frequency $\ensuremath{\nu}$, and Rabi frequency ${\ensuremath{\Omega}}_{0}$ the population transfer after the pulse is gone can be optimized and, for the pulse considered here, an enhancement factor of 106−−108{10}^{6}--{10}^{8}106−−108 was obtained.
Coherent population transfer in quantum systems is of fundamental importance in many fields such as atomic and molecular collision dynamics, and information processing for qubit systems. Stimulated Raman nonadiabatic passage technique, when implemented in an externally driven three-level system, provides an efficient approach for realizing accelerated population transfer while maintaining robust quantum coherence, with no need of rotating wave approximation. However, previous protocols employ multiple pulses and imply that Rabi frequencies have a few oscillations during dynamical evolution. In this work, under the condition of two-photon resonance, we use a gauge transformation method to inversely design a Λ-configuration three-level system that can be solved exactly. By using SU(3) transformation, we establish the relationship between Schrödinger representation and gauge representation, where the effective Hamiltonian is an Abelian operator. Subsequently, we construct the desired Hamiltonian and further investigate its dynamic behavior. The result shows that by imposing appropriate boundary conditions on the control parameters, high-fidelity population transfer can be achieved in ideal evolution. In addition, for the practical case with pulse truncation and intermediate state decay, the fidelities of specific models can reach about 99.996% and 99.983%, respectively. Compared with other existing nonadiabatic quantum control schemes, the present scheme has the distinctive advantages. We achieve the required quantum control by applying only a few sets of Stokes and pump pulses without introducing any additional microwave field. This method does not exhibit Rabi oscillations in the dynamic process, nor does it produce singularities in the pulse itself.