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Scientific Reports

Controlling the Quantum State with a time varying potential

Sebastian Camilo Carrasco, Jose Antonio Rogan, J. A. Valdivia

Published Oct 10, 2017• 10 citations
Publisher

Abstract

The problem of controlling the quantum state of a system is investigated using a time varying potential. As a concrete example we study the problem of a particle in a box with a periodically oscillating infinite square-well potential, from which we obtain results that can be applied to systems with periodically oscillating boundary conditions. We derive an analytic expression for the frequencies of resonance between states, and against standard intuition, we show how to use this behavior to control the quantum state of the system at will. In particular, we offer as an example the transition from the ground state to the first excited state of the square well potential. At first sight, it may be counter intuitive that we can control the state of such a quantum Hamiltonian, as the Schrödinger equation conserves the norm of the wave function. In this manuscript, we show how that can be achieved.

DOI
10.1038/s41598-017-13313-3

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The problem of controlling the quantum state of a system is investigated using a time varying potential. As a concrete example we study the problem of a particle in a box with a periodically oscillating infinite square-well potential, from which we obtain results that can be applied to systems with periodically oscillating boundary conditions. We derive an analytic expression for the frequencies of resonance between states, and against standard intuition, we show how to use this behavior to control the quantum state of the system at will. In particular, we offer as an example the transition from the ground state to the first excited state of the square well potential. At first sight, it may be counter intuitive that we can control the state of such a quantum Hamiltonian, as the Schrödinger equation conserves the norm of the wave function. In this manuscript, we show how that can be achieved.
A fast and robust approach to controlling the quantum state of a multi-level quantum system is investigated using a twofrequency time-varying potential. A comparison with other related approaches in the context of a two-level system is also presented, showing similar times and fidelities. As a concrete example, we study the problem of a particle in a box with a periodically oscillating infinite square-well potential, from which we obtain results that can be applied to systems with periodically oscillating boundary conditions. We show that the transition between the ground and first excited state is about 20 times faster than the one performed using a single frequency, both with fidelity of 99.97%. The transition time is about 3.5 times the minimum allowed by quantum mechanics. A test of the robustness of the approach is presented, concluding that, counter-intuitively, it is not only faster but also easier to tune up two frequencies than one. This robustness makes the approach suitable for real applications.
The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.
Abstract The work continues systematic applications of the method developed by Di Martino and Facchi (2015) for obtaining, for systems with variable boundary conditions, an equivalent Schrödinger equation with constant boundary conditions. From the resulting equation with rapidly varying terms, an effective equation is then derived for the averaged (slowly varying) wave functions. In this work, we obtain this effective equation for rapidly driven systems in the interaction picture (in first order in reverse frequency of external driving). This is the most suitable picture for systems with external driving. As an example, an extended two-state fast driven system is considered. As a second example, we consider the one-dimensional motion of a particle in a potential box with high-frequency oscillations of its width. Oscillations of width lead to additional terms in the effective equation proportional not only to the second derivative of the potential, but to the first derivative as well. In the case of width oscillations, compared to the oscillations of the box as a whole, the changes are even more significant. In our examples, phenomena such as slow averaged oscillations, resonance, the appearance of volcano potentials are observed, as is often the case in driven systems.