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arXiv:1801.01863 · Scientific Reports

Simulating quantum dynamical phenomena using classical oscillators: Landau-Zener-Stückelberg-Majorana interferometry, latching modulation, and motional averaging

O. V. Ivakhnenko, Sergey N Shevchenko, Franco Nori

Published Aug 9, 2018• 31 citations
arXivHTMLPDF

Abstract

A quantum system can be driven by either sinusoidal, rectangular, or noisy signals. In the literature, these regimes are referred to as Landau-Zener-Stückelberg-Majorana (LZSM) interferometry, latching modulation, and motional averaging, respectively. We demonstrate that these pronounced and interesting effects are also inherent in the dynamics of classical two-state systems. We discuss how such classical systems are realized using either mechanical, electrical, or optical resonators. In addition to the fundamental interest of such dynamical phenomena linking classical and quantum physics, we believe that these are attractive for the classical analogue simulation of quantum systems.

DOI
10.1038/s41598-018-28993-8

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A quantum system can be driven by either sinusoidal, rectangular, or noisy signals. In the literature, these regimes are referred to as Landau-Zener-Stückelberg-Majorana (LZSM) interferometry, latching modulation, and motional averaging, respectively. We demonstrate that these pronounced and interesting effects are also inherent in the dynamics of classical two-state systems. We discuss how such classical systems are realized using either mechanical, electrical, or optical resonators. In addition to the fundamental interest of such dynamical phenomena linking classical and quantum physics, we believe that these are attractive for the classical analogue simulation of quantum systems.
Quantum mechanics increasingly penetrates modern technologies but, due to its non-deterministic nature seemingly contradicting our classical everyday world, our comprehension often stays elusive. Arguing along the correspondence principle, classical mechanics is often seen as a theory for large systems where quantum coherence is completely averaged out. Surprisingly, it is still possible to reconstruct the coherent dynamics of a quantum bit (qubit) by using a classical model system. This classical-to-quantum analogue is based on wave mechanics, which applies to both, the classical and the quantum world. In this spirit we investigate the dynamics of macroscopic physical pendula with a modulated coupling. As a proof of principle, we demonstrate full control of our one-to-one analogue to a qubit by realizing Rabi oscillations, Landau-Zener transitions and Landau-Zener-Stückelberg-Majorana interferometry. Our classical qubit demonstrator can help comprehending and developing useful quantum technologies.
We demonstrate amplification (and attenuation) of a probe signal by a driven two-level quantum system in the Landau-Zener-St\"uckelberg-Majorana regime by means of an experiment, in which a superconducting qubit was strongly coupled to a microwave cavity, in a conventional arrangement of circuit quantum electrodynamics. Two different types of flux qubit, specifically a conventional Josephson junctions qubit and a phase-slip qubit, show similar results, namely, lasing at the working points where amplification takes place. The experimental data are explained by the interaction of the probe signal with Rabi-like oscillations. The latter are created by constructive interference of Landau-Zener-St\"uckelberg-Majorana (LZSM) transitions during the driving period of the qubit. A detailed description of the occurrence of these oscillations and a comparison of obtained data with both analytic and numerical calculations are given.
We review the physical phenomena that arise when quantum mechanical energy levels are modulated in time. The dynamics resulting from changes in the transition frequency is a problem studied since the early days of quantum mechanics. It has been of constant interest both experimentally and theoretically since, with the simple two-state model providing an inexhaustible source of novel concepts. When the transition frequency of a quantum system is modulated, several phenomena can be observed, such as Landau-Zener-Stückelberg-Majorana interference, motional averaging and narrowing, and the formation of dressed states with the appearance of sidebands in the spectrum. Adiabatic changes result in the accumulation of geometric phases, which can be used to create topological states. In recent years, an exquisite experimental control in the time domain was gained through the parameters entering the Hamiltonian, and high-fidelity readout schemes allowed the state of the system to be monitored non-destructively. These developments were made in the field of quantum devices, especially in superconducting qubits, as a well as in atomic physics, in particular in ultracold gases. As a result of these advances, it became possible to demonstrate many of the fundamental effects that arise in a quantum system when its transition frequencies are modulated. The purpose of this review is to present some of these developments, from two-state atoms and harmonic oscillators to multilevel and many-particle systems.
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.
Experimental demonstrations of coherent dynamics for the driven quantum system are usually limited by their shorter coherent times (due to the inevitable environment noises). Given the Maxwell equation for the electromagnetic waves (EMWs) prorogating in the optical waveguides takes the similar form of the Schr\"odinger equation for the driven quantum system, the quantum dynamics in time domain can be simulated by the light propagating along the waveguide in spatial domain. To simulate the fast time evolutions of quantum states during the Landau-Zener (LZ) transitions and Landau-Zenner-St\"uckelberg (LZS) interferences for the periodically driven two-level systems, we develop an invariant method to design the compacted curved wavguides and demonstrate the LZ transitions and LZS interferometery in the spatial domain. Due to significantly long coherent lengths of the EMWs propagating along the waveguides, the proposed optical-quantum analogies should be feasible with the current integrated optical devices.
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.