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arXiv:cond-mat/0407080 · Low Temperature Physics

Rabi oscillations in systems with small anharmonicity

M. H. S. Amin

Published Mar 1, 2006• 13 citations
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Abstract

When a two-level quantum system is irradiated with a microwave signal in resonance with the energy difference between the levels, it starts Rabi oscillations between those states. If there are other states close, in energy, to the first two, the microwave signal will also induce transitions to those. Here we study the probability of transition to the third state, in a three-level system, while Rabi oscillations between the first two states are performed. We investigate the effect of pulse shaping on the probability and suggest methods for optimizing the pulse shapes to reduce the transition probability.

DOI
10.1063/1.2178467

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When a two-level quantum system is irradiated with a microwave signal in resonance with the energy difference between the levels, it starts Rabi oscillations between those states. If there are other states close, in energy, to the first two, the microwave signal will also induce transitions to those. Here we study the probability of transition to the third state, in a three-level system, while Rabi oscillations between the first two states are performed. We investigate the effect of pulse shaping on the probability and suggest methods for optimizing the pulse shapes to reduce the transition probability.
We demonstrate Rabi oscillation of a resonant two-photon transition in a three-level ladder-type quantum system of gaseous rubidium atoms induced by a single femtosecond laser pulse. For this, we shape the flat-top spatial profile of the laser pulse and perform the ultrafast population cycling of the atoms as a function of pulse energy. The experimental result confirms that the Rabi frequency of the transition from a ground state to a final state depends linearly on the pulse area, although the transition is a two-photon process.
Rabi oscillations are coherent transitions in a quantum two-level system under the influence of a resonant drive, with a much lower frequency dependent on the perturbation amplitude. These serve as one of the signatures of quantum coherent evolution in mesoscopic systems. It was shown recently (Grønbech-Jensen N and Cirillo M 2005 Phys. Rev. Lett. 95 067001) that in phase qubits (current-biased Josephson junctions) this effect can be mimicked by classical oscillations arising due to the anharmonicity of the effective potential. Nevertheless, we find qualitative differences between the classical and quantum effects. Firstly, while the quantum Rabi oscillations can be produced by the subharmonics of the resonant frequency ω 10 (multiphoton processes), the classical effect also exists when the system is excited at the overtones, n ω 10 . Secondly, the shape of the resonance is, in the classical case, characteristically asymmetric, whereas quantum resonances are described by symmetric Lorentzians. Thirdly, the anharmonicity of the potential results in the negative shift of the resonant frequency in the classical case, in contrast to the positive Bloch–Siegert shift in the quantum case. We show that in the relevant range of parameters these features allow us to distinguish confidently the bona fide Rabi oscillations from their classical Doppelgänger.
We have observed Rabi-like oscillations in a current-biased dc superconducting quantum interference device presenting enhanced coherence times compared to our previous realization [J. Claudon et al., Phys. Rev. Lett. 93, 187003 (2004)]. This Josephson device behaves as an anharmonic oscillator which can be driven into a coherent superposition of quantum states by resonant microwave flux pulses. Increasing the microwave amplitude, we study the evolution of the Rabi frequency from the two-level regime to the regime of multilevel dynamics. When up to three levels are involved, the Rabi frequency is a clear signature of a quantum behavior. At higher excitation amplitude, classical and quantum predictions for the Rabi frequency converge. This result is discussed in the light of a calculation of the Wigner function. In particular, our analysis shows that pronounced quantum interferences always appear in the course of the Rabi-like oscillations.
The coupling between microwave fields and atoms (or atomlike systems) is inherently weaker than for optical fields, making microwave signal manipulation for applications like quantum information processing technically challenging. In order to better understand this coupling and to develop tools for measuring it, we explore the microwave coupling to atoms using the ``atomic candle'' technique, and push it beyond the bounds of the small-signal regime in order to deliver a larger signal. In familiar two-level systems, responses beyond the usual Rabi oscillations can arise when a single-tone drive is phase modulated, causing the steady-state populations to oscillate at integer harmonics of the modulation frequency. Resonant behavior of the first two harmonics for frequencies near the Rabi frequency, known as $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ Rabi resonances, is widely used for microwave-field magnetometry and as a power standard known as the atomic candle. Here, we explore Rabi resonances beyond the small-signal approximation and report upon experimental observations of higher-order-harmonic population response for microwave hyperfine transitions in cold 87Rb^{87}\mathrm{Rb}87Rb atoms, which we compare to numerical simulations.