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.
Highlights
The problem of controlling the quantum state of a system is investigated using a time varying potential
Since the development of Quantum Theory, the study of time dependent quantum systems has been of wide interest[1,2,3,4], and in particular, the control the quantum phenomena has been an implicit task
In this letter we revisit the problem of a particle in a one-dimensional infinite square-well potential, but with a moving wall which we use to drive the system systematically from one quantum state to another
Summary
The problem of controlling the quantum state of a system is investigated using a time varying potential. As Dong and Petersen wrote, “One of the main goals in quantum control theory is to establish a firm theoretical footing and develop a series of systematic methods for the active manipulation and control of quantum systems” Such capability could be used to store information in the system, or in an array of them, using many states to store information as the experimental limits allow us[5]. With this in mind, in this letter we revisit the problem of a particle in a one-dimensional infinite square-well potential, but with a moving wall which we use to drive the system systematically from one quantum state to another. The problem has been discussed in several publications using different approaches, such as the adiabatic approximation, sudden expansion or contraction of the wall, rapidly varying square well width, etc.[1,2,3,4]
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