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arXiv:1112.5522 · Physical Review Letters

Multiple Schrödinger Pictures and Dynamics in Shortcuts to Adiabaticity

Sara Ibañez, Xi Chen, E. Torrontegui, Juan Gonzalo Muga, A. Ruschhaupt

Published Sep 7, 2012• 243 citations
arXivHTMLPDF

Abstract

A Schrödinger equation may be unitarily transformed into dynamical equations in different interaction pictures which describe a common physical process, i.e., the same underlying interactions and dynamics. In contrast to this standard scenario, other relations are also possible, such as a common interaction-picture dynamical equation corresponding to several Schrödinger equations that represent different physical processes. This may enable us to design alternative and feasible experimental routes for operations that are a priori difficult or impossible to perform. The power of this concept is exemplified by engineering Hamiltonians that improve the performance or make realizable several shortcuts to adiabaticity.

DOI
10.1103/physrevlett.109.100403

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A Schrödinger equation may be unitarily transformed into dynamical equations in different interaction pictures which describe a common physical process, i.e., the same underlying interactions and dynamics. In contrast to this standard scenario, other relations are also possible, such as a common interaction-picture dynamical equation corresponding to several Schrödinger equations that represent different physical processes. This may enable us to design alternative and feasible experimental routes for operations that are a priori difficult or impossible to perform. The power of this concept is exemplified by engineering Hamiltonians that improve the performance or make realizable several shortcuts to adiabaticity.
This chapter outlines the governing equations in non-relativistic quantum physics. The Schrödinger equation (the Schrödinger picture) looms large. Its intuitive simplicity and elegance render it particularly appealing to aspiring engineers and materials scientists. The other two equivalent pictures presented are the Heisenberg picture and the interaction picture. The Heisenberg picture emphasizes time-dependent operators, and the interaction picture splits the so-called interaction Hamiltonian from the total Hamiltonian of the system considered. Much attention has been paid in this chapter to establish relationships between the pictures, as meaningful as possible at this level of treatment.
Quantum adiabatic processes—that keep constant the populations in the instantaneous eigenbasis of a time-dependent Hamiltonian—are very useful to prepare and manipulate states, but take typically a long time. This is often problematic because decoherence and noise may spoil the desired final state, or because some applications require many repetitions. "Shortcuts to adiabaticity" are alternative fast processes which reproduce the same final populations, or even the same final state, as the adiabatic process in a finite, shorter time. Since adiabatic processes are ubiquitous, the shortcuts span a broad range of applications in atomic, molecular, and optical physics, such as fast transport of ions or neutral atoms, internal population control, and state preparation (for nuclear magnetic resonance or quantum information), cold atom expansions and other manipulations, cooling cycles, wavepacket splitting, and many-body state engineering or correlations microscopy. Shortcuts are also relevant to clarify fundamental questions such as a precise quantification of the third principle of thermodynamics and quantum speed limits. We review different theoretical techniques proposed to engineer the shortcuts, the experimental results, and the prospects.
This chapter demonstrates how one can find the time evolution of the operators from the knowledge of the time evolution of the state vector, and vice versa. It introduces a unitary operator and discusses the concept of a unitary transformation to show how one could transfer the time dependence from the state vectors to the operators. The possibility that either the state vectors or the operators can depend explicitly on time will lead to introduce the Schrodinger, Heisenberg, and interaction pictures. The chapter discusses the fundamental differences between these pictures. It also discusses the Ehrenfest theorem, which shows under which conditions quantum mechanics predicts the same results for measured physical quantities as classical physics. It illustrates unitary transformations of an arbitrary operator Â(t) which can depend explicitly on time. Quantum mechanics produces the same result as classical mechanics for a system in which particles are represented by well-localized wave functions.
Abstract We delve a little deeper into the construction of shortcuts to adiabatic passage for three-level systems by iterative interaction picture (multiple Schrödinger dynamics). As an application example, we use the deduced iterative based shortcuts to rapidly generate the Greenberger–Horne–Zeilinger (GHZ) state in a three-atom system with the help of quantum Zeno dynamics. Numerical simulation shows the dynamics designed by the iterative picture method is physically feasible and the shortcut scheme performs much better than that using the conventional adiabatic passage techniques. Also, the influences of various decoherence processes are discussed by numerical simulation and the results prove that the scheme is fast and robust against decoherence and operational imperfection.