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arXiv:0705.1768 · Physical Review B

Pseudo-Rabi oscillations in superconducting flux qubits in the classical regime

A. N. Omelyanchouk, Sergey N Shevchenko, Alexandre M. Zagoskin, Evgeni V. Il’ichev, Franco Nori

Published Aug 15, 2008• 12 citations
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Abstract

Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux qubit in the classical regime can produce low-frequency oscillations in the presence of an external field in resonance with the (high-frequency) harmonic mode $\ensuremath{\omega}$ of the system. Like in the case of quantum Rabi oscillations, the frequency of these pseudo-Rabi oscillations is much smaller than $\ensuremath{\omega}$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external drive not only at the resonance frequency $\ensuremath{\omega}$ and its subharmonics $(\ensuremath{\omega}/n)$, but also at its overtones, $n\ensuremath{\omega}$.

DOI
10.1103/physrevb.78.054512

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Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux qubit in the classical regime can produce low-frequency oscillations in the presence of an external field in resonance with the (high-frequency) harmonic mode \ensuremathω\ensuremath{\omega}\ensuremathω of the system. Like in the case of quantum Rabi oscillations, the frequency of these pseudo-Rabi oscillations is much smaller than \ensuremathω\ensuremath{\omega}\ensuremathω and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external drive not only at the resonance frequency \ensuremathω\ensuremath{\omega}\ensuremathω and its subharmonics (\ensuremathω/n)(\ensuremath{\omega}/n)(\ensuremathω/n), but also at its overtones, n\ensuremathωn\ensuremath{\omega}n\ensuremathω.
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 present an analytical and computational study of resonances and transient responses in a classical Josephson junction system. A theoretical basis for resonances in a superconducting loop with three junctions is presented, outlining both the direct relationship between the dynamics of single- and multijunction systems and the direct relationships between observations of the classical counterparts to Rabi oscillations, Ramsey fringes, and spin-echo oscillations in this class of systems. We show simulation data along with analytical analyses of the classical model, and the results are related to previously reported experiments conducted on three junction loops. We further investigate the effect of off-resonant microwave perturbations to, e.g., the Rabi-type response of the Josephson system, and we relate this response back to the nonlinear and multivalued resonance behavior previously reported for a single Josephson junction. The close relationships between single and multijunction behaviors demonstrate the underlying dynamical mechanism for a whole class of classical counterparts to expected quantum-mechanical observations in a variety of systems, namely, the resonant and transient behavior of a particle in an anharmonic potential well with subsequent escape.
We predict theoretically the enhancement of quantum coherence in a superconducting flux qubit by a classical external noise. First, the off-diagonal components of the qubit density matrix are increased. Second, in the presence of both ac drive and noise, the resulting Rabi oscillations survive ``in perpetuity,'' i.e., for times greatly exceeding the Rabi decay time in a noiseless system. The coherence-enhancing effects of the classical noise can be considered as a manifestation of quantum stochastic resonance and are relevant to experimental techniques, such as Rabi spectroscopy.
Rabi oscillations have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (qubits) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase qubit (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi frequency, and two-photon Rabi frequency are compared to measurements made on a dc SQUID phase qubit with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment.
We infer the high-frequency flux noise spectrum in a superconducting flux qubit by studying the decay of Rabi oscillations under strong driving conditions. The large anharmonicity of the qubit and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi frequencies up to 1.7 GHz were achieved, approaching the qubit's level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide frequency range decreases with increasing frequency up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f1/f1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins.