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Письма в Журнал экспериментальной и теоретической физики

POSSIBILITY OF DIRECT OBSERVATION OF THE BLOCH-SIEGERT SHIFT IN COHERENT DYNAMICS OF MULTIPHOTON RAMAN TRANSITIONS

A. P. Saǐko, Siarhei A. Markevich, Ryhor Fedaruk

Published Jan 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s0370274x19190019

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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.
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.
Abstract We study the coherent dynamics of a qubit excited by an amplitude-modulated electromagnetic field under the Rabi resonance when the frequency of the low-frequency modulation field matches the Rabi frequency in the high-frequency field. Due to the destructive interference of multiple photon processes within the ultrastrong coupling between the qubit and the low-frequency driving field, Rabi oscillations result exclusively from the Bloch–Siegert effect. It is directly observed in the time-resolved coherent dynamics as the Bloch–Siegert oscillation. In this case, triplets in the Fourier spectra of the coherent response are transformed into doublets with the splitting between the lines equal to twice the Bloch–Siegert shift. These unusual properties are demonstrated in the conditions of experiments with a nitrogen vacancy center in the diamond.
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.
The dynamics of a two-level spin system dressed by bichromatic radiation is studied under the conditions of double resonance when the frequency of one (microwave) field is equal to the Larmor frequency of the spin system and the frequency of the other (radio-frequency) field ω_{rf} is close to the Rabi frequency ω_{1} in a micro-wave field. It is shown theoretically that Rabi oscillations between dressed-spin states with the frequency εare accompanied by higher-frequency oscillations at frequencies nω_{rf} and nω_{rf}\pm ε, where n = 1, 2, .... The most intense among these are the signals corresponding to n = 1. The counter-rotating (antiresonance) components of the RF field give rise to a shift of the dressed-state energy, i.e., to a frequency shift similar to the Bloch-Siegert shift. In particular, this shift is manifested as the dependence of the Rabi-oscillation frequency εon the sign of the detuning ω{1} -ω{rf} from resonance. In the case of double resonance, the oscillation amplitude is asymmetric; i.e., the amplitude at the sum frequency ω_{rf} +εincreases, while the amplitude at the difference frequency ω_{rf} -εdecreases. The predicted effects are confirmed by observations of the nutation signals of the electron paramagnetic resonance (EPR) of centers in quartz and should be taken into account to realize qubits with a low Rabi frequency in solids.
Using electron paramagnetic resonance (EPR) at Rabi frequencies, we study the Bloch-Siegert effect in the coherent dynamics of electron spin states driven by a bichromatic radiation consisting of a transverse microwave (MW) and longitudinal radio frequency (RF) fields. The Rabi oscillations on the electron spin states dressed by the MW field are observed at the double resonance, when the MW frequency equals the Larmor frequency of the spin system and the frequency ωrf of the RF field is close to the Rabi frequency ω1 in the MW field. The dispersive (|ωrf - ω1| ~ ωrf, ω1) and near-resonant (ωrf ≈ ω1) regimes are investigated. We demonstrate that the Bloch-Siegert shift increases with increas- ing RF amplitude and at the fixed RF amplitude it has the largest values in the low-frequency (ωrf < ω1) range of the dispersive regime. Experimental results were obtained in the time-resolved EPR for E' centres in quartz.
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.