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arXiv:1012.2243 · Journal of Experimental and Theoretical Physics Letters

Multiplication of qubits in a doubly resonant bichromatic field

A. P. Saǐko, Ryhor Fedaruk

Published Jun 1, 2010• 16 citations
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Abstract

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.

DOI
10.1134/s0021364010120131

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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.
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.
The coherent dynamics of relaxing spin qubits driven by a classical bichromatic field comprising a strong resonant component and a weaker component with a frequency close to the strong-field Rabi frequency is studied. The double dressing by the bichromatic field modifies dephasing and dissipation processes. We demonstrate that detuning of the weaker-field frequency from the strong-field Rabi frequency prolongs the decay of Rabi oscillations between some doubly dressed states. The sensitivity of Rabi oscillations to the modified detuning-dependent relaxation is illustrated for nitrogen-vacancy qubits in diamond. We discuss a steady-state population inversion of the doubly dressed-state levels.
The geometric phase in the dynamics of a spin qubit driven by transverse microwave (MW) and longitudinal radiofrequency (RF) fields is studied. The phase acquired by the qubit during the full period of the "slow" RF field manifests in the shift of Rabi frequency ω_{1} of a spin qubit in the MW field. We find out that, for a linearly polarized RF field, this shift does not vanish at the second and higher even orders in the adiabaticity parameter ω_{rf} / ω_{1}, where ω_{rf} is the RF frequency. As a result, the adiabatic (Berry) phases for the rotating and counter-rotating RF components compensate each other, and only the higher-order geometric phase is observed. We experimentally identify that phase in the frequency shift of the Rabi oscillations detected by a time-resolved electron paramagnetic resonance.
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.