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arXiv:1604.00209

Classical analog of Stückelberg interferometry with two coupled mechanical resonators

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

Published Apr 1, 2016• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1604.00209

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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.
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.
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.
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.
The studies of mechanical resonators in the quantum regime not only provide insight into the fundamental nature of quantum mechanics of massive objects, but also introduce promising platforms for novel hybrid quantum technologies. Here we demonstrate a configurable interaction between a superconducting qubit and many acoustic modes in the quantum regime. Specifically, we show how consecutive Landau-Zener-Stückelberg (LZS) tunneling type of transitions, which take place when a system is tuned through an avoided crossing of the coupled energy levels, interfere in a multimode system. The work progresses experimental LZS interference to cover a new class of systems where the coupled levels are those of a quantum two-level system interacting with a multitude of mechanical oscillators. The work opens up applications in controlling multiple acoustic modes via parametric modulation.
Time evolution of spin-orbit-coupled cold atoms in an optical lattice is studied, with a two-band energy spectrum having two avoided crossings. A force is applied such that the atoms experience two consecutive Landau-Zener tunnelings while transversing the avoided crossings. St\"uckelberg interference arises from the phase accumulated during the adiabatic evolution between the two tunnelings. This phase is gauge field dependent and thus provides new opportunities to measure the synthetic gauge field, which is verified via calculation of spin transition probabilities after a double-passage process. Time-dependent and time-averaged spin probabilities are derived, in which resonances are found. We also demonstrate chiral Bloch oscillation and rich spin-momentum locking behavior in this system.
Two basic physical models, a two-level system and a harmonic oscillator, are realized on the mesoscopic scale as coupled qubit and resonator. The realistic system includes moreover the electronics for controlling the distance between the qubit energy levels and their populations and to read out the resonator’s state, as well as the unavoidable dissipative environment. Such rich system is interesting both for the study of fundamental quantum phenomena on the mesoscopic scale and as a promising system for future electronic devices. We present recent results for the driven superconducting qubit–resonator system, where the resonator can be realized as an LC circuit or a nanomechanical resonator. Most of the results can be described by the semiclassical theory, where a qubit is treated as a quantum two-level system coupled to the classical driving field and the classical resonator. Application of this theory allows to describe many phenomena for the single and two coupled superconducting qubits, among which are the following: the equilibrium-state and weak-driving spectroscopy, Sisyphus damping and amplification, Landau–Zener–Stückelberg interferometry, the multiphoton transitions of both direct and ladder-type character, and creation of the inverse population for lasing.