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arXiv:2008.10750 · Physical Review Research

Designing adiabatic time evolution from high-frequency bichromatic sources

Álvaro Gómez-León, Gloria Platero

Published Sep 15, 2020• 6 citations
arXivHTMLPDF

Abstract

We investigate the quantum dynamics of a two-level system driven by a bichromatic field, using a nonperturbative analysis.We make special emphasis in the case of two large frequencies, where the Magnus expansion can fail, and in the case of a large and a small frequency, where resonances can dominate.In the first case, we show that two large frequencies can be combined to produce an effective adiabatic evolution.In the second case, we show that high-frequency terms (which naturally arise as corrections to the adiabatic evolution obtained in the first case) can be used to produce a highly tunable adiabatic evolution over the whole Bloch sphere, controlled by multiphoton resonances.

DOI
10.1103/physrevresearch.2.033412

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We investigate the quantum dynamics of a two-level system driven by a bichromatic field, using a nonperturbative analysis.We make special emphasis in the case of two large frequencies, where the Magnus expansion can fail, and in the case of a large and a small frequency, where resonances can dominate.In the first case, we show that two large frequencies can be combined to produce an effective adiabatic evolution.In the second case, we show that high-frequency terms (which naturally arise as corrections to the adiabatic evolution obtained in the first case) can be used to produce a highly tunable adiabatic evolution over the whole Bloch sphere, controlled by multiphoton resonances.
We derive a Magnus expansion for a frequency chirped quantum two-level system. We obtain a time-independent effective Hamiltonian which generates a stroboscopic time evolution. At lowest order the according dynamics is identical to results from using a rotating wave approximation. We determine, furthermore, also the next higher-order corrections within our expansion scheme in correspondence to the Bloch-Siegert shifts for harmonically driven systems. Importantly, our scheme can be extended to more complicated systems, i.e., even many-body systems.
Excitation of an atomic system subjected to a slowly varying external electromagnetic field is studied using a two-level model. Time evolution of the system is found using an approach which is nonperturbative in the field strength. There is no constraint to small values of the applied field, that is, the field (in appropriate energy units) need not be small compared to the difference in energies of the two levels. Rather, we use the fact that the situation of interest to us is where the frequency of the exciting field is small compared to the frequency associated with the level difference. Transition probabilities and resonance conditions are found which circumscribe both the large- and small-field limits. In the weak-field limit the previous results of high-order perturbation theory are readily recovered. For a monochromatic field the characteristic features of resonance excitation at high harmonic number of the applied field are (a) extremely narrow resonance widths and (b) shifts in resonance positions which are strong functions of field intensity. Because of this sensitivity, we are able to demonstrate that when slow temporal evolution of the field amplitude is taken into account (e.g., due to finite pulse duration) the appropriate mean excitation rate is that due to the uncorrelated contribution of many resonances. The results of this analysis are used to estimate excitation rates in a specific atomic system, Cd12+{\mathrm{Cd}}^{12+}Cd12+, which are then compared to multiphoton-ionization rates. Our calculations suggest that the ionization rate exceeds the excitation rate by several orders of magnitude.
We study the behavior of a two-level atom that is driven by a bichromatic field consisting of a strong resonant component and a weaker tunable component. In addition to the splitting of the energy levels (the multiphoton AC Stark effect), we find that the weaker component also shifts the subharmonic resonances, an effect we attribute to a dynamic Stark shift. When the weaker component is tuned to a shifted resonance, no fluorescence occurs at either the frequency of the strong component or the three-photon mixing frequency. Results are obtained with numerical techniques and explained in terms of the dressed-atom model of the system.
We consider a two-level quantum system interacting with two classical time-periodic electromagnetic fields. The frequency of one of the fields far exceeds that of the other. The effect of the high-frequency field can be averaged out of the dynamics to realize an effective transition frequency of the field-dressed two-level system. We examine the linear response, second harmonic response and Stokes and anti-Stokes Raman response of the dressed two-level system, to the weak frequency field. The vibrational resonance enhancement in each case is demonstrated for optimal strength of the high-frequency field. Our theoretical scheme is corroborated by full numerical simulation of the two-level, two-field dynamics governed by loss-free Bloch equations. We suggest that quantum optics can offer an interesting arena for the study of the vibrational resonance. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.
The quantum trajectories method is illustrated for the resonance fluorescence of a two-level atom driven by a multichromatic field. We discuss the method for the time evolution of the fluorescence intensity in the presence of bichromatic and trichromatic driving fields. We consider the special case wherein one multichromatic field component is strong and resonant with the atomic transition whereas the other components are much weaker and arbitrarily detuned from the atomic resonance. We find that the phase-dependent modulations of the Rabi oscillations, recently observed experimentally [Q. Wu, D. J. Gauthier, and T. W. Mossberg, Phys. Rev. A 49, R1519 (1994)] for the special case when the weaker component of a bichromatic driving field is detuned from the atomic resonance by the strong-field Rabi frequency, appear also for detunings close to the subharmonics of the Rabi frequency. Furthermore, we show that for the atom initially prepared in one of the dressed states of the strong field component the modulations are not sensitive to the phase. We extend the calculations to the case of a trichromatic driving field and find that apart from the modulations of the amplitude there is a modulation of the frequency of the Rabi oscillations. Moreover, the time evolution of the fluorescence intensity depends on the phase regardless of the initial conditions and a phase-dependent suppression of the Rabi oscillations can be observed when the sideband fields are tuned to the subharmonics of the strong-field Rabi frequency.
We consider a quantum two-level system in bichromatic classical time-periodic fields, the frequency of one of which far exceeds that of the other. Based on systematic separation of timescales and averaging over the fast motion a reduced quantum dynamics in the form of a nonlinear forced Mathieu equation is derived to identify the stable oscillatory resonance zones intercepted by unstable zones in the frequency-amplitude plot. We show how this forcing of the dressed two-level system may generate the subharmonics and superharmonics of the weak field in the stable region, which can be amplified by optimization of the strength of the high frequency field. We have carried out detailed numerical simulations of the driven quantum dynamics to corroborate the theoretical analysis.