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Physical Review A

Classical analog of Stückelberg interferometry in a two-coupled-cantilever–based optomechanical system

Hao Fu, Zhi-cheng Gong, Tian-hua Mao, Chang-Pu Sun, Su Yi, Yong Li, Gengyu Cao

Published Oct 31, 2016• 19 citations
Publisher

Abstract

We present the realization of a St\"uckelberg interferometer with two mechanical modes in a two-coupled-cantilever--based optomechanical system. By optically tuning the frequency of one cantilever inside a cavity, we show that nonadiabatic crossing of the mechanical modes twice creates the interference fringes of the mechanical oscillations as a result of coherent splitting and subsequent recombination of the oscillations. By measuring the nonadiabatic phase acquired at the transition, we unveil an in-depth analog between the two-mode and quantum two-level systems. With this interferometry-based technique, transferring of oscillation between spatially separated cantilevers is achieved with 97% efficiency.

DOI
10.1103/physreva.94.043855

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We present the realization of a St"uckelberg interferometer with two mechanical modes in a two-coupled-cantilever--based optomechanical system. By optically tuning the frequency of one cantilever inside a cavity, we show that nonadiabatic crossing of the mechanical modes twice creates the interference fringes of the mechanical oscillations as a result of coherent splitting and subsequent recombination of the oscillations. By measuring the nonadiabatic phase acquired at the transition, we unveil an in-depth analog between the two-mode and quantum two-level systems. With this interferometry-based technique, transferring of oscillation between spatially separated cantilevers is achieved with 97% efficiency.
Coupled nanomechanical resonators have recently attracted much attention for both fundamental studies in physics and broad applications in high-precession detection or sensing. By studying the Landau-Zener transitions and Rabi oscillation of two coupled resonators, it has been shown that such a two-mode system acts as a classical two-level system bearing the analog to a quantum mechanical two-level one. Here we construct a Stückelberg interferometer with two coupled cantilevers by driving the system through the avoided crossing twice. By measuring the non-adiabatic phase acquired at the Landau-Zener transition, we unveil an in-depth analog between the two-mode and quantum two-level systems. Our study opens up new opportunities for constructing interferometers with classical devices.
St\"uckelberg interferometry is a phenomenon that has been well established for quantum-mechanical two-level systems. Here, we present classical two-mode interference of a nanomechanical two-mode system, realizing a classical analog of St\"uckelberg interferometry. Our experiment relies on the coherent energy exchange between two strongly coupled, high-quality factor nanomechanical resonator modes. Furthermore, we discuss an exact theoretical solution for the double-passage St\"uckelberg problem by expanding the established finite-time Landau-Zener single-passage solution. For the parameter regime explored in the experiment, we find that the St\"uckelberg return probability in the classical version of the problem formally coincides with the quantum case, which reveals the analogy of the return probabilities in the quantum-mechanical and the classical version of the problem. This result qualifies classical two-mode systems at large to simulate quantum-mechanical interferometry.
Systems of coupled mechanical resonators are useful for quantum information processing and fundamental tests of physics. Direct coupling is only possible with resonators of very similar frequency, but by using an intermediary optical mode, non-degenerate modes can interact and be independently controlled in a single optical cavity. Here we demonstrate coherent optomechanical state swapping between two spatially and frequency separated resonators with a mass ratio of 4. We find that, by using two laser beams far detuned from an optical cavity resonance, efficient state transfer is possible. Although the demonstration is classical, the same technique can be used to generate entanglement between oscillators in the quantum regime.Coupled mechanical resonators where each mode can be separately controlled are a promising system for quantum information processing. Here, Weaver et al. demonstrate coherent swapping of optomechanical states between two separate resonators.
Abstract Systems of coupled mechanical resonators are useful for quantum information processing and fundamental tests of physics. Direct coupling is only possible with resonators of very similar frequency, but by using an intermediary optical mode, non-degenerate modes can interact and be independently controlled in a single optical cavity. Here we demonstrate coherent optomechanical state swapping between two spatially and frequency separated resonators with a mass ratio of 4. We find that, by using two laser beams far detuned from an optical cavity resonance, efficient state transfer is possible. Although the demonstration is classical, the same technique can be used to generate entanglement between oscillators in the quantum regime.
We show that an atom can be coupled to a mechanical oscillator via quantum vacuum fluctuations of a cavity field enabling energy transfer processes between them. In a hybrid quantum system consisting of a cavity resonator with a movable mirror and an atom, these processes are dominated by two pair-creation mechanisms: the counterrotating (atom-cavity system) and dynamical Casimir interaction terms (optomechanical system). Because of these two pair-creation mechanisms, the resonant atom-mirror coupling is the result of high-order virtual processes with different transition paths well described in our theoretical framework. We perform a unitary transformation to the atom-mirror system Hamiltonian, exhibiting two kinds of multiple-order transitions of the pair creation. By tuning the frequency of the atom, we show that photon frequency conversion can be realized within a cavity of multiple modes. Furthermore, when involving two atoms coupled to the same mechanical mode, a single vibrating excitation of the mechanical oscillator can be simultaneously absorbed by the two atoms. Considering recent advances in strong and ultrastrong coupling for cavity optomechanics and other systems, we believe our proposals can be implemented using available technology.
Stückelberg interferometry describes the interference of two strongly coupled modes during a double passage through an avoided energy level crossing. In this work, we investigate finite-time effects in Stückelberg interferometry and discuss the exact analytical solution of the double passage Stückelberg problem by expanding the finite-time solution of the Landau–Zener problem. Approximating the return probability amplitudes of the double passage in distinct limits reveals uncharted parameter regimes of Stückelberg interferometry where finite-time effects affect the coherent exchange of energy. We find the long-time limit of the exact solution to formally coincide with the well-established adiabatic impulse model which is, to the best of our knowledge, the only regime of Stückelberg interferometry reported so far. Experimentally, we study all predicted regimes using a purely classical, strongly coupled nanomechanical two-mode system of high quality factor. The classical two-mode system consists of the in-plane and out-of-plane fundamental flexural mode of a high stress silicon nitride string resonator, coupled via electric gradient fields. We exploit our experimental and theoretical findings by studying the onset of Stückelberg interference in dependence of the characteristic system control parameters and obtain characteristic excitation oscillations between the two modes even without the explicit need of traversing the avoided crossing. The presented findings are not limited to classical mechanical two-mode systems but can be applied to every strongly coupled (quantum) two-level system, for example a spin-1/2 system or superconducting qubit.