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arXiv:1005.3446 · Physical Review Letters

Stark Effect and Generalized Bloch-Siegert Shift in a Strongly Driven Two-Level System

Jani Tuorila, Matti Silveri, Mika A. Sillanpää, E. V. Thuneberg, Yuriy Makhlin, Pertti Juhani Hakonen

Published Dec 14, 2010• 105 citations
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Abstract

A superconducting qubit was driven in an ultrastrong fashion by an oscillatory microwave field, which was created by coupling via the nonlinear Josephson energy. The observed Stark shifts of the "atomic" levels are so pronounced that corrections even beyond the lowest-order Bloch-Siegert shift are needed to properly explain the measurements. The quasienergies of the dressed two-level system were probed by resonant absorption via a cavity, and the results are in agreement with a calculation based on the Floquet approach.

DOI
10.1103/physrevlett.105.257003

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A superconducting qubit was driven in an ultrastrong fashion by an oscillatory microwave field, which was created by coupling via the nonlinear Josephson energy. The observed Stark shifts of the "atomic" levels are so pronounced that corrections even beyond the lowest-order Bloch-Siegert shift are needed to properly explain the measurements. The quasienergies of the dressed two-level system were probed by resonant absorption via a cavity, and the results are in agreement with a calculation based on the Floquet approach.
We calculate a probe response of a homogeneously broadened two-level transition in the presence of a strong pump. The derivation is carried out for a generalized interaction Hamiltonian, which gives a uniform account of both the Bloch - Siegert shift and dynamic Stark splitting. It is shown that the Bloch - Siegert shift can be viewed as an extreme situation of the dynamic Stark splitting, commonly known as light shift in the literature. Finally using a perturbative approach an analytical expression for the Bloch - Siegert shift is obtained, which is compared with the previous results.
We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for a strong drive (classical phase) it presents an oscillatory behavior as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.
A cavity quantum electrodynamical (QED) system beyond the strong-coupling regime is expected to exhibit intriguing quantum phenomena. Here we report a direct measurement of the photon-dressed qubit transition frequencies up to four photons by harnessing the same type of state transitions in an ultrastrongly coupled circuit QED system realized by inductively coupling a superconducting flux qubit to a coplanar-waveguide resonator. This demonstrates a convincing observation of the photon-dressed Bloch-Siegert shift in the ultrastrongly coupled quantum system. Moreover, our results show that the photon-dressed Bloch-Siegert shift becomes more pronounced as the photon number increases, which is a characteristic of the quantum Rabi model.
Recent measurement on an LC resonator magnetically coupled to a superconducting qubit [ Phys. Rev. Lett. 105 (2010) 237001 ] shows that the system operates in the ultra-strong coupling regime and crosses the limit of validity for the rotating-wave approximation of the Jaynes-Cummings model. By using extended bosonic coherent states, we solve the Jaynes—Cummings model exactly without using the rotating-wave approximation. Our numerically exact results for the spectrum of the flux qubit coupled to the LC resonator are fully consistent with the experimental observations. The smallest Bloch—Siegert shift obtained is consistent with that observed in this experiment. In addition, the Bloch—Siegert shifts in arbitrary level transitions and for arbitrary coupling constants are predicted.
Recent measurement on an LC resonator magnetically coupled to a superconducting qubit[arXiv:1005.1559] shows that the system operates in the ultra-strong coupling regime and crosses the limit of validity for the rotating-wave approximation of the Jaynes-Cummings model. By using extended bosonic coherent states, we solve the Jaynes-Cummings model exactly without the rotating-wave approximation. Our numerically exact results for the spectrum of the flux qubit coupled to the LC resonator are fully consistent with the experimental observations. The smallest Bloch-Siegert shift obtained is consistent with that observed in this experiment. In addition, the Bloch-Siegert shifts in arbitrary level transitions and for arbitrary coupling constants are predicted.