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arXiv:2209.13505 · Low Temperature Physics

Rate-equation approach for multi-level quantum systems

M. P. Liul, Sergey N Shevchenko

Published Jan 1, 2023• 5 citations
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Abstract

Strong driving of quantum systems opens opportunities for both controlling and characterizing their states. For theoretical studying of these systems properties we use the rate-equation formalism. The advantage of such approach is its relative simplicity. We used the formalism for description of a two-level system with further expanding it on a case of a multi-level system. Obtained theoretical results have good agreement with experiments. The presented approach can also be considered as one more way to explore properties of quantum systems and underlying physical processes such as, for instance, Landau–Zener–Stückelberg–Majorana transitions and interference.

DOI
10.1063/10.0016482

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Strong driving of quantum systems opens opportunities for both controlling and characterizing their states. For theoretical studying of these systems properties we use the rate-equation formalism. The advantage of such approach is its relative simplicity. We used the formalism for description of a two-level system with further expanding it on a case of a multi-level system. Obtained theoretical results have good agreement with experiments. The presented approach can also be considered as one more way to explore properties of quantum systems and underlying physical processes such as, for instance, Landau–Zener–Stückelberg–Majorana transitions and interference.
Since the pioneering works by Landau, Zener, Stückelberg, and Majorana (LZSM), it has been known that driving a quantum two-level system results in tunneling between its states. Even though the interference between these transitions is known to be important, it is only recently that it became both accessible, controllable, and useful for manipulating a growing number of quantum systems. Here, we systematically study various aspects of LZSM physics and review the relevant literature, significantly expanding the review article in Ref. [504].
Two-level systems are one of the most important quantum systems and they form the basis of quantum computers. We briefly look at the traditional approach to two-level systems with an external driving field as well as those subjected to noise. This project is aimed at studying two specific methods for obtaining analytic solutions for two-level systems. One of the methods enables us to obtain analytic solutions for driven time-dependent two-level systems while the other attempts to give exact solution of qubit decoherence using a transfer matrix method. A thorough study of both papers is done and results are reproduced. The latter method is generalized for a qutrit system as well as a two qubit system subjected to noise. A general method is formally derived for an N-dimensional quantum system and the difficulties in applying the method in real life systems is discussed.
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 investigate Landau-Zener-St\"uckelberg-Majorana (LZSM) interferometry in a nonlinear two-level system subjected to a sinusoidal driving field. A comprehensive analysis for the interference patterns is shown from which we can analytically obtain the condition of constructive and destructive interferences in some cases. In the presence of nonlinear interaction, the level structure is changed, which can lead to an asymmetric deformation of the interference patterns and a significant position shift of the interference fringes. Our findings suggest an application of the nonlinear LZSM interferometry in accurately calibrating the parameters characterizing a nonlinear two-level system, as well as its coupling with the driven fields.