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Laser Physics

Consideration of geometric phase and population transfer without using rotating-wave approximation

K. Berrada, Hichem Eleuch

Published Jan 9, 2015• 2 citations
Publisher

Abstract

Recently, Rostovtsev et al (2009 Phys. Rev. A 79 063833) established an analytical treatment of the detuned atom–field interaction without using the rotating-wave approximation (RWA). Using this result, here we study qualitatively the dynamics of geometric phase (GP) and population transfer in a two-level atom interacting with an external off-resonant classical field for arbitrary pulse shape without using RWA. We use this exact method (without RWA) to reexamine the variation of populations and GP and compare the result with the RWA considering two kinds of the external field. We show that it is possible to control the GP dynamics by choosing suitable parameters of the atom–field system. Interestingly, we find that the variation of the populations and GP in the time is unchanged and it differs as we get further from the resonance. Moreover, we show that the Rabi frequency plays a role in reduction or augmentation of the periodic time interval for the GP. The solution is very suitable for atom–field systems, especially when the system is far from the resonance so that the RWA method breaks down.

DOI
10.1088/1054-660x/25/2/025201

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Recently, Rostovtsev et al (2009 Phys. Rev. A 79 063833) established an analytical treatment of the detuned atom–field interaction without using the rotating-wave approximation (RWA). Using this result, here we study qualitatively the dynamics of geometric phase (GP) and population transfer in a two-level atom interacting with an external off-resonant classical field for arbitrary pulse shape without using RWA. We use this exact method (without RWA) to reexamine the variation of populations and GP and compare the result with the RWA considering two kinds of the external field. We show that it is possible to control the GP dynamics by choosing suitable parameters of the atom–field system. Interestingly, we find that the variation of the populations and GP in the time is unchanged and it differs as we get further from the resonance. Moreover, we show that the Rabi frequency plays a role in reduction or augmentation of the periodic time interval for the GP. The solution is very suitable for atom–field systems, especially when the system is far from the resonance so that the RWA method breaks down.
Abstract We investigate the effects of counterrotating terms on geometric phase and its relation to the resonance of the Rabi model. We apply the unitary transformation with a single parameter to the Rabi model and obtain the transformed Hamiltonian involving multiple harmonic terms. By combining the counter-rotating-hybridized rotating-wave method with time-dependent perturbation theory, we solve systematically time evolution operator and then obtain the geometric phase of the two-level system. Our results are beyond adiabatic approximation and rotating-wave approximation (RWA). Higher-order harmonic resonance happens when driving frequency is equal to higher-order subharmonic of the Rabi frequency. In comparison with numerically exact results, our calculated results are accurate over a wide range of parameters space, especially in higher-order harmonic resonance regimes. In these regimes we demonstrate geometric phases change dramatically while those of the RWA are smooth. The RWA is thoroughly invalid even if the driving strength is extremely weak. We find it is the higher-order harmonic terms that play an important role on the cyclic state and demonstrate the characters of geometric phase in higher-order harmonic resonance regime. We also present analytical formalism of the change rate of geometric phase and quasienergies, which agree well with numerically exact ones even in the strong driving case. The developed method can be applied to explore the dynamics of strongly driven qubits and physical properties of higher-order harmonic processes.
We considered the semiclassical Rabi model: a two-level atom fixed at a position and interacting with a classical, monochromatic electromagnetic field. In resonance and for an arbitrary initial state of the system, we obtained an approximate, analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/21/21/2 (1/41/41/4) of the angular frequency of the field and 111 (101010) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA. We found that the approximate solution is composed of the the solution in the RWA plus terms that oscillate at the angular frequency of the field and at the Rabi frequency. Moreover, these corrections are proportional to the Rabi frequency divided by two times the angular frequency of the field.
The trapping states of a quantum system occupy a very special place in quantum optics. Here, we propose a method to realize the population trapping (PT) in a two-level atom driven by a carrier envelope phase (CEP) locked laser pulse. We present detailed numerical results on the influence of CEP in different detuned atoms beyond the rotating wave approximation and find the population is strongly dependent on CEP, the detuning and the Rabi frequency. This allows us to choose proper parameters to realize three specific trapping states: upper trapping state, half trapping state and lower trapping state. To explain the fantastic influence of CEP, we present an analytical solution under weak-field approximation, which is consistent with the numerical solution. We further define the action quantity of the pulse acting on the atom as the integral for the interaction Hamiltonian and find the population varies approximatively linearly with the action quantity. This investigation provides directive significance for achieving PT at arbitrary quantum states and insightful schemes to regulate and control the quantum dynamics of atoms via accurately controlled optical field, and the results could be beneficial for applications to laser cooling, isotope separation and so on.
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.
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.
In the physics literature it is common to see the rotating wave approximation and the adiabatic approximation used to justify the use of chirped pulses for two-level quantum systems driven by one external field, in particular when the resonance frequency of the system is not known precisely. Both approximations need relatively long time and are essentially based on averaging theory of dynamical systems. Unfortunately, the two approximations cannot be done independently since, in a sense, the two time scales interact. The purpose of this paper is to study how the cascade of the two approximations can be justified and how large becomes the final time as the fidelity goes to one, while preserving the robustness of the adiabatic strategy. Our main result gives a precise quantification of the uncertainty interval of the resonance frequency for which the strategy works. We believe that this result, beside its mathematical interest, could be useful in experimental situations.