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Journal of Experimental and Theoretical Physics Letters

Effect of the Bloch-Siegert shift on the frequency responses of Rabi oscillations in the case of nutation resonance

A. P. Saǐko, Г. Г. Федорук

Published Apr 1, 2008• 25 citations
Publisher

Abstract

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 microwave field. It is shown theoretically that Rabi oscillations between dressed-spin states with the frequency ɛ are accompanied by higher-frequency oscillations at frequencies n ω rf and n ω rf ± ɛ, 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 ɛ on 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 + ɛ increases, while the amplitude at the difference frequency ω rf − ɛ decreases. The predicted effects are confirmed by observations of the nutation signals of the electron paramagnetic resonance (EPR) of E ′ 1 centers in quartz and should be taken into account to realize qubits with a low Rabi frequency in solids.

DOI
10.1134/s002136400803003x

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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 microwave field. It is shown theoretically that Rabi oscillations between dressed-spin states with the frequency ɛ are accompanied by higher-frequency oscillations at frequencies n ω rf and n ω rf ± ɛ, 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 ɛ on 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 + ɛ increases, while the amplitude at the difference frequency ω rf − ɛ decreases. The predicted effects are confirmed by observations of the nutation signals of the electron paramagnetic resonance (EPR) of E ′ 1 centers in quartz and should be taken into account to realize qubits with a low Rabi frequency in solids.
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.
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 multiphoton transitions in a two-level spin system excited by transverse microwave and longitudinal RF fields with the frequencies ω mw and ω rf , respectively, is analyzed. The effective time-independent Hamiltonian describing the “dressed” spin states of the “spin + bichromatic field” system is obtained by using the Krylov-Bogoliubov-Mitropolsky averaging method. The direct detection of the time behavior of the spin system by the method of nonstationary nutations makes it possible to identify the multiphoton transitions for resonances ω 0 = ω mw + r ω rf (ω 0 is the central frequency of the EPR line, r = 1, 2), to measure the amplitudes of the effective fields of these transitions, and to determine the features generated by the inhomogeneous broadening of the EPR line. It is shown that the Bloch-Siegert shifts for multiphoton resonances at the inhomogeneous broadening of spectral lines reduce only the nutation amplitude but do not change their frequencies.
Multiplication of spin qubits arises at double resonance in a bichromatic field when the frequency of the radio-frequency (rf) field is close to that of the Rabi oscillation in the microwave field, provided its frequency equals the Larmor frequency of the initial qubit. We show that the operational multiphoton transitions of dressed qubits can be selected by the choice of both the rotating frame and the rf phase. In order to enhance the precision of dressed qubit operations in the strong-field regime, the counter-rotating component of the rf field is taken into account.
EPR transient nutation spectroscopy is used to measure the effective field (Rabi frequency) for multiphoton transitions in a two-level spin system bichromatically driven by a transverse microwave (MW) field and a longitudinal radio-frequency (RF) field. The behavior of the effective field amplitude is examined in the case of a relatively strong MW field, when the derivation of the effective Hamiltonian cannot be reduced to first-order perturbation theory in ω1/ωrf(ω1 is the microwave Rabi frequency, ωrf is the RF frequency). Experimental results are consistently interpreted by taking into account the contributions of second and third order in ω1/ωrf evaluated by Krylov-Bogolyubov-Mitropolsky averaging. In the case of inhomogeneously broadened EPR line, the third-order correction modifies the nutation frequency, while the second-order correction gives rise to a change in the nutation amplitude due to a Bloch-Siegert shift.