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Physical Review A

Multiphoton resonances and Bloch-Siegert shifts observed in a classical two-level system

Marco W. Beijersbergen, R. J. C. Spreeuw, L. Allen, J. P. Woerdman

Published Feb 1, 1992• 39 citations
Publisher

Abstract

We report experimental observation of multiphoton resonances and Bloch-Siegert shifts in a strongly driven classical two-level system. The system is an optical ring resonator, and the levels are two orthogonal linear polarizations. The Rabi frequency and the Bloch-Siegert shift were measured for both the one- and three-photon cases. Satisfactory agreement was achieved with theory developed for a quantum two-level system. The experiment demonstrates that these coherence phenomena can be observed in a purely classical system.

DOI
10.1103/physreva.45.1810

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arXiv:0709.1958•Sep 12, 2007•1 citation
We report experimental observation of multiphoton resonances and Bloch-Siegert shifts in a strongly driven classical two-level system. The system is an optical ring resonator, and the levels are two orthogonal linear polarizations. The Rabi frequency and the Bloch-Siegert shift were measured for both the one- and three-photon cases. Satisfactory agreement was achieved with theory developed for a quantum two-level system. The experiment demonstrates that these coherence phenomena can be observed in a purely classical system.
Bloch-Siegert shifts and widths of resonances in two-level systems are evaluated within an adiabatic approximation applicable in the multiphoton regime. The level splitting is controlled by a static field. Most attention is paid to the case of a circular driving field with the static field in the plane of rotation. This nonconventional setup was used in a recent experiment on intrashell resonances in Rydberg atoms where shifts and widths for NNN-photon resonance with NNN as large as 23 were observed [D. Fregenal et al., Phys. Rev. A 69, 031401(R) (2004)]. For this in-plane configuration the resonance widths are drastically enhanced compared to their values in the conventional configuration (linearly polarized driving field perpendicular to static field), whereas the shifts are only weakly affected.
In the framework of the nonsecular perturbation theory based on the Bogoliubov averaging method, the coherent dynamics of multiphoton Raman transitions in a two-level spin system driven by an amplitude-modulated microwave field is studied. Closed-form expressions for the Rabi frequencies of these transitions are obtained beyond the rotating-wave approximation for the low-frequency driving component. It is shown that spin states dressed by the high-frequency component of the driving field are shifted due to the Bloch-Siegert-like effect caused by antiresonant interactions with the strong low-frequency driving. We predict that with increasing the order of the Raman transition, the Rabi frequency decreases and the contribution of the Bloch-Siegert shift to this frequency becomes dominant. It is found that the amplitude and phase of the Rabi oscillations strongly depend on the initial phase of the low-frequency field as well as on detuning from multiphoton resonance. The recent experimental data for the second- and third-order Raman transitions observed for the nitrogen-vacancy center in diamond (Z. Shu et al., arXiv:1804.10492) are well described in the frame of our approach. Our results may provide new possibilities for coherent control of quantum systems.
We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for a strong drive (classical phase) it presents an oscillatory behavior as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.
We study Rabi oscillations of the second-order Raman transition realized on dressed states of a qubit excited by an amplitude-modulated microwave field. The co-rotating component of the ultrastrong low-frequency modulation field excites virtual multiple photon processes between the dressed states and forms the Rabi frequency in the so-called rotating wave approximation. The counter-rotating modulation component also gives a significant contribution to the Rabi frequency owing to the Bloch-Siegert effect. It is shown that for properly chosen parameters of the modulation field and qubit, the Rabi oscillations in the rotating wave approximation vanish due to destructive interference of multiple photon processes. In this case the Rabi oscillation results exclusively from the Bloch-Siegert effect and is directly observed in the time-resolved coherent dynamics as the Bloch-Siegert oscillation. Correspondingly, in Fourier spectra of the coherent response, triplets are transformed into doublets with the splitting between the lines equal to twice the Bloch-Siegert shift. We demonstrate these features by calculations of the qubit’s evolution in the conditions of experiments with a NV center in diamond, where Raman transitions were observed. The direct observation of the Bloch-Siegert oscillation offers new possibilities for studying driven quantum systems in the ultastrong regime.
We study Rabi oscillations of the second-order Raman transition realized on dressed states of a qubit excited by an amplitude-modulated microwave field.The co-rotating component of the ultrastrong low-frequency modulation field excites virtual multiple photon processes between the dressed states and forms the Rabi frequency in the so-called rotating wave approximation (RWA).The counter-rotating modulation component also gives a significant contribution to the Rabi frequency owing to the Bloch-Siegert effect.It is shown that for properly chosen parameters of the modulation field and qubit, the Rabi oscillations in the RWA vanish due to destructive interference of multiple photon processes.In this case the Rabi oscillation results exclusively from the Bloch-Siegert effect and is directly observed in the time-resolved coherent dynamics as the Bloch-Siegert oscillation.Correspondingly, in Fourier spectra of the coherent response, triplets are transformed into doublets with the splitting between the lines equal to twice the Bloch-Siegert shift.We demonstrate these features by calculations of the qubit's evolution in the conditions of experiments with a NV center in diamond, where Raman transitions were observed.The direct observation of the Bloch-Siegert oscillation offers new possibilities for studying driven quantum systems in the ultastrong regime.