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Number-Operator-Based Inverse Engineering Technique for the Shortcut to Adiabaticity in Two Level Quantum Mechanical System

T. Kiran, M. Ponmurugan

Published Jan 7, 2025• 0 citations
Publisher

Abstract

In general, the conventional Lewis-Riesenfeld invariant-based inverse engineering is a nonadiabatic process that results in the adiabatic final states to achieve the shortcut to adiabaticity, which does not provide complete suppression of non-adiabaticity throughout the evolution. We propose a new method to accomplish the shortcut to adiabaticity through an entirely adiabatic path. This new method is developed using the number operator as an invariant of the Hamiltonian. This paper discusses the mathematical framework of the new method in two-level quantum system and analyzes its performance compared to the conventional Lewis-Riesenfeld invariant-based inverse engineering method.

DOI
10.20944/preprints202501.0389.v1

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In general, the conventional Lewis-Riesenfeld invariant-based inverse engineering is a nonadiabatic process that results in the adiabatic final states to achieve the shortcut to adiabaticity, which does not provide complete suppression of non-adiabaticity throughout the evolution. We propose a new method to accomplish the shortcut to adiabaticity through an entirely adiabatic path. This new method is developed using the number operator as an invariant of the Hamiltonian. This paper discusses the mathematical framework of the new method in two-level quantum system and analyzes its performance compared to the conventional Lewis-Riesenfeld invariant-based inverse engineering method.
Different methods have been recently put forward and implemented experimentally to inverse engineer the time-dependent Hamiltonian of a quantum system and accelerate slow adiabatic processes via nonadiabatic shortcuts. In the ``transitionless quantum driving'' proposed by Berry, shortcut Hamiltonians are designed so that the system follows exactly, in an arbitrarily short time, the approximate adiabatic path defined by a reference Hamiltonian. A different approach is based on first designing a Lewis-Riesenfeld invariant to carry the eigenstates of a Hamiltonian from specified initial to final configurations, again in an arbitrary time, and then constructing from the invariant the transient Hamiltonian that connects these boundary configurations. We show that the two approaches, apparently quite different in form and so far in results, are, in fact, strongly related and potentially equivalent, so that the inverse-engineering operations in one of them can be reinterpreted and understood in terms of the concepts and operations of the other one. We study, as explicit examples, expansions of time-dependent harmonic traps and the state preparation of two-level systems.
We apply the inversely-engineered control method based on Lewis-Riesenfeld invariants to control mixed states of a two-level quantum system. We show that the inversely-engineered control passages of mixed states - and pure states as special cases - can be made significantly faster than the conventional adiabatic control passages, which renders the method applicable to quantum computation. We devise a new type of inversely-engineered control passages, to be coined the antedated control passages, which further speed up the control significantly. We also demonstrate that by carefully tuning the control parameters, the inversely-engineered control passages can be optimized in terms of speed and energy cost.
This paper experimentally realize a new method for shortcuts to adiabaticity, number operator based inverse engineering method (NOBIE), using quantum computers built with transmon qubits. Digitized control pulses are programmed in an open source software development kit, qiskit, to execute the shortcut protocols. The obtained results shows the robustness of the NOBIE method, even though it is tested for an effective Hamiltonian of a qubit irrespective of the interaction with other qubits and noise associated with the control pulses.
Shortcuts to adiabaticity, originally developed in the context of quantum control, are powerful tools for the design of high coupling efficiency, robust, and short coupled-waveguide devices. The counterdiabatic protocol cancels the unwanted coupling in system evolution by the addition of a counterdiabatic term. The invariant-based inverse-engineering approach designs system evolution using the decoupled eigenstates of the Lewis-Riesenfeld invariant. The single-shot shaped-pulse technique directly parameterizes the solution of the coupled-mode equation. Starting from the counterdiabatic protocol, we show that these seemingly very different shortcuts to adiabaticity techniques are equivalent in the framework of coupled-waveguide systems. When combined with perturbation treatment of the coupled-mode equation, robust coupled-waveguide devices against errors can be obtained.
The ``fast-forward''approach by Masuda and Nakamura generates driving potentials to accelerate slow quantum adiabatic dynamics. First we present a streamlined version of the formalism that produces the main results in a few steps. Then we show the connection between this approach and inverse engineering based on Lewis-Riesenfeld invariants. We identify in this manner applications in which the engineered potential does not depend on the initial state. Finally we discuss more general applications exemplified by wave splitting processes.