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arXiv:1212.5957 · Physical Review B

Magnetic resonance in slowly modulated longitudinal field: Modified shape of the Rabi oscillations

Rachel Glenn, Mark E. Limes, Brendan Pankovich, Brian T. Saam, M. É. Raǐkh

Published Apr 15, 2013• 19 citations
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Abstract

The sensitivity of the Rabi oscillations of a resonantly driven spin-$\frac{1}{2}$ system to a weak and slow modulation of the static longitudinal magnetic field ${B}_{0}$ is studied theoretically. We establish the mapping of a weakly driven two-level system with modulation onto a strongly driven system without modulation. The mapping suggests that different regimes of spin dynamics, known for a strongly driven system, can be realized under common experimental conditions of weak driving (driving field ${B}_{1}\ensuremath{\ll}{B}_{0}$) upon proper choice of the domains of modulation frequency ${\ensuremath{\omega}}_{m}$ and amplitude ${B}_{2}$. Fast modulation ${\ensuremath{\omega}}_{m}\ensuremath{\gg}{\ensuremath{\Omega}}_{R}$, where ${\ensuremath{\Omega}}_{R}$ is the Rabi frequency, emulates the regime of driving frequency much bigger than the resonant frequency. Strong modulation ${B}_{2}\ensuremath{\gg}{B}_{1}$ emulates the regime ${B}_{1}\ensuremath{\gg}{B}_{0}$. Resonant modulation ${\ensuremath{\omega}}_{m}\ensuremath{\approx}{\ensuremath{\Omega}}_{R}$ gives rise to an envelope of the Rabi oscillations. The shape of this envelope is highly sensitive to the detuning of the driving frequency from the resonance. Theoretical predictions for different domains of ${B}_{2}$ and ${\ensuremath{\omega}}_{m}$ were tested experimentally using NMR of protons in water, where, without modulation, the pattern of Rabi oscillations could be observed over many periods. We present experimental results which reproduce the three predicted modulation regimes, and agree with theory quantitatively.

DOI
10.1103/physrevb.87.155128

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We study theoretically the Rabi oscillations of a resonantly driven two-level system linearly coupled to a harmonic oscillator (vibrational mode) with frequency ${\ensuremath{\omega}}_{0}$. We show that for weak coupling, ${\ensuremath{\omega}}_{p}\ensuremath{\ll}{\ensuremath{\omega}}_{0}$, where ${\ensuremath{\omega}}_{p}$ is the polaronic shift, Rabi oscillations are strongly modified in the vicinity of the Rabi-vibronic resonance ${\ensuremath{\Omega}}_{R}={\ensuremath{\omega}}_{0}$, where ${\ensuremath{\Omega}}_{R}$ is the Rabi frequency. The width of the resonance is $({\ensuremath{\Omega}}_{R}\ensuremath{-}{\ensuremath{\omega}}_{0})\ensuremath{\sim}{\ensuremath{\omega}}_{p}^{2/3}{\ensuremath{\omega}}_{0}^{1/3}\ensuremath{\gg}{\ensuremath{\omega}}_{p}$. Within this domain of ${\ensuremath{\Omega}}_{R}$ the actual frequency of the Rabi oscillations exhibits a bistable behavior as a function of ${\ensuremath{\Omega}}_{R}$. Importantly, within the resonant domain, the oscillator is highly excited, which allows us to treat it classically.
Coherent control of two-level quantum systems is typically achieved using resonant driving fields, forming the basis for qubit operations. Here, we report a mechanism for inducing complete Rabi oscillations in monochromatically driven two-level quantum systems, when the drive frequency is much larger than the Larmor frequency of the qubit. This effect$\unicode{x2015}$Rabi oscillations induced by nonresonant gyrating drive (RING)$\unicode{x2015}$requires a gyrating control field, i.e., an elliptic drive trajectory that encloses a nonzero area per cycle. We illustrate the effect with numerical simulations, and provide an analytical understanding via a simple effective Hamiltonian obtained from Floquet theory and perturbation theory. We show that RING enables coherent oscillations without relying on resonant energy exchange, allows for high-pass noise filtering, provides access to non-Abelian phases in finite magnetic fields. We detail a realization in electrically driven spin-orbit qubits and argue that the RING mechanism enables amplification of the Rabi frequency using the same gate voltage amplitudes at higher drive frequencies. Our results broaden the landscape of quantum control techniques, by highlighting a pathway to achieving coherent oscillations under off-resonant driving conditions.
Abstract Recent electron spin resonance experiments on CaWO 4_44​ 4 :Gd 3+^{3+}3+ 3 + and other magnetic impurities have demonstrated that sustained Rabi oscillations can be created by driving a magnetic moment with a microwave field frequency slightly larger than the Larmor frequency and tuned to the Floquet resonance, together with another microwave field (image drive) with a frequency smaller than the Larmor frequency. These observations are confirmed by the new experimental results reported in this paper. We use numerical and analytical techniques to study the interplay between the microwave drives and three different mechanisms of relaxation. The first model describes a magnetic moment subject to microwave fields, interacting with a bath of two-level systems which acts as a source of decoherence and dissipation. The second model describes identical, interacting magnetic moments, subject to the same microwave fields. The decay of the Rabi oscillations is now due to the interactions. Third, we study Rabi oscillation decay due to the inhomogeneity of the microwave radiation. We show that under appropriate conditions, and in particular at the Floquet resonance, the magnetization exhibits sustained Rabi oscillations, in some cases with additional beatings. Although these two microscopic models separately describe the experimental data well, a simulation study that simultaneously accounts for both types of interactions is currently prohibitively costly. To gain further insight into the microscopic dynamics of these two different models, we study the time dependence of the bath and system energy and of the correlations of the spins, data that are not readily accessible experimentally. Graphic abstract
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.