A Time-Dependent Harmonic Oscillator with Two Frequency Jumps: an Exact Algebraic Solution
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A Time-Dependent Harmonic Oscillator with Two Frequency Jumps: an Exact Algebraic Solution D. M. Tibaduiza1 · L. Pires1 · D. Szilard1 · C. A. D. Zarro1 · C. Farina1 · A. L. C. Rego2 Received: 2 April 2020 © Sociedade Brasileira de F´ısica 2020
Abstract We consider a harmonic oscillator (HO) with a time-dependent frequency which undergoes two successive abrupt changes. By assumption, the HO starts in its fundamental state with frequency ω0 , then, at t = 0, its frequency suddenly increases to ω1 and, after a finite time interval τ , it comes back to its original value ω0 . Contrary to what one could naively think, this problem is quite a non-trivial one. Using algebraic methods, we obtain its exact analytical solution and show that at any time t > 0 the HO is in a vacuum squeezed state. We compute explicitly the corresponding squeezing parameter (SP) relative to the initial state at an arbitrary instant and show that, surprisingly, it exhibits oscillations after the first frequency jump (from ω0 to ω1 ), remaining constant after the second jump (from ω1 back to ω0 ). We also compute the time evolution of the variance of a quadrature. Last, but not least, we calculate the vacuum (fundamental state) persistence probability amplitude of the HO, as well as its transition probability amplitude for any excited state. Keywords Time-dependent harmonic oscillator · Squeezed states · Algebraic methods
1 Introduction The classical harmonic oscillator (HO) and its quantum counterpart are two of the most important systems in physics [1, 2]. Their relevance relies on the ubiquity of phenomena that can be modeled by them. As a remarkable example, in quantum electrodynamics (QED), quantum harmonic oscillators are the paradigm to describe the free electromagnetic field, hence being a cornerstone in quantum optics [3]. In fact, the description of a free bosonic field is frequently done by considering it as a set of HOs [4, 5]. The usual quantization of the HO leads directly to the so-called Fock states, which are eigenstates of the HO Hamiltonian. In quantum optics language, these Fock states correspond to n-photon states. Fock states of the HO are quite D. M. Tibaduiza
[email protected] 1
Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Avenida Athos da Silveira Ramos, 149, Centro de Tecnologia, Bloco A, Cidade Universit´aria, CEP: 21941-972 Caixa Postal 68528, Rio de Janeiro-RJ, Brazil
2
Instituto de Aplicac¸a˜ o Fernando Rodrigues da Silveira, Universidade do Estado do Rio de Janeiro, Rua Santa Alexandrina, 288, CEP: 20261-232, Rio de Janeiro-RJ, Brazil
non-classical states, as can be seen, for instance, if we take the quantum expectation value of the position operator in the Heisenberg picture (or momentum operator, or even any quadrature operator) in any Fock state, which is always zero. This is evidently in contrast to the oscillating behavior of the position of a classical HO. However, these states are very helpful and can be used as a convenient basis for describing other im
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