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Time dependent quantum harmonic oscillator subject to a sudden change of mass: continuous solution

H. M. Moya-Cessa, M. Fernández‐Guasti

Published Jan 1, 2007• 11 citations
OpenAlex

Abstract

We show that a harmonic oscillator subject to a sudden change of mass produces squeezed states. Our study is based on an approximate analytic solution to the time-dependent harmonic oscillator equation with a subperiod function parameter. This continuous treatment differs from former studies that involve the matching of two time-independent solutions at the discontinuity. This formalism requires an ad hoc transformation of the original differential equation and is also applicable for rapid, although not necessarily instantaneous, mass variations.

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Time dependent quaritum harmonic oscillator subject to a sudden change of mass: continuous solution

H. M. Moya-Cessa, M. Fernández‐Guasti

We show that a harmonic oscillator subject to a sudden change of mas s produces squeezed states. Our study is based on an approximate analytic solution to the time-dependent harmonic oscillator equation with a subperiod function parameter. This continuous treatment differs from former studies that involve the matching of two time-independent solutions at the discontinuity. This formalism requires an ad hoc transformation of the original differential equation and isalso applicable for rapid, although not necessarily instantaneous, mass variations. Ke.ywords: Solutions in closed form; quantum mechanics; exact invariants. Mostramos que un oscilador armónico sujeto a un cambio repentino de masa produce estados comprimidos. Nuestro estudio está basado en una solución analítica aproximada para el oscilador armónico dependiente del tiempo. El tratamiento continuo que estudiamos difiere de estudios anteriores en los cuales se igualan las soluciones en la discontinuidad. Nuestro formalismo requiere una transformación ad hoc de la ecuación diferencial original y es aplicable también para variaciones de masa rápidas, no solo instantáneas. Descriptores: Soluciones analíticas; mecánica cuántica; invariantes exactos. PACS: 03.65.La; 03.65.Ge; 02.30.Ik 1. Introduction ing another transforrnation that produces a monotonic time-dependent parameter. The transformed equation can then be sol ved with the continuum approach, leading to squeezing of the momentum variables at certain times. An interesting ad-vantage of this method is that the change of mass need not be a step function in the strict mathematical sense. The require-ment, which is physically more plausible, is that the change in mass should take place in a time span much shorter than the characteristic period of the oscillator. The plan of the manuscript is the following: in Seco 2, we write the quadratic and linear invariants for a time-dependent mass and exhibit their relation with the Hamiltonian. In Seco 3, we discuss two transforrnations that translate the clas-sical time-dependent mass equation into the time dependent trequency problem. The analytic solution for a rapid varia-tion of the time dependent mass using the appropriate trans-formation is then presented. In Seco 4, we exhibit how a sudden change of mas s in the quantum oscillator produces squeezing. Section 5 is left for conclusions. 2. Time dependent mass

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Abstract In this work, we give a quantitative answer to the question: how sudden or how adiabatic is a frequency change in a quantum harmonic oscillator (HO)? We do that by studying the time evolution of a HO which is initially in its fundamental state and whose time-dependent frequency is controlled by a parameter (denoted by ϵ ) that can continuously tune from a totally slow process to a completely abrupt one. We extend a solution based on algebraic methods introduced recently in the literature that is very suited for numerical implementations, from the basis that diagonalizes the initial Hamiltonian to the one that diagonalizes the instantaneous Hamiltonian. Our results are in agreement with the adiabatic theorem and the comparison of the descriptions using the different bases together with the proper interpretation of this theorem allows us to clarify a common inaccuracy present in the literature. More importantly, we obtain a simple expression that relates squeezing to the transition rate and the initial and final frequencies, from which we calculate the adiabatic limit of the transition. Analysis of these results reveals a significant difference in squeezing production between enhancing or diminishing the frequency of a HO in a non-sudden way.

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Abstract The center of mass motion of trapped ions and neutral atoms is suitable for approximation by a time-dependent driven quantum harmonic oscillator whose frequency and driving strength may be controlled with high precision. We show the time evolution for these systems with continuous differentiable time-dependent parameters in terms of the three basic operations provided by its underlying symmetry, rotation, displacement, and squeezing, using a Lie algebraic approach. Our factorization of the dynamics allows for the intuitive construction of protocols for state engineering, for example, creating and removing displacement and squeezing, as well as their combinations, optimizing squeezing, or more complex protocols that work for slow and fast rates of change in the oscillator parameters.

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We show that a harmonic oscillator subject to a sudden change of mass produces squeezed states. Our study is based on an approximate analytic solution to the time-dependent harmonic oscillator equation with a subperiod function parameter. This continuous treatment differs from former studies that involve the matching of two time-independent solutions at the discontinuity. This formalism requires an ad hoc transformation of the original differential equation and is also applicable for rapid, although not necessarily instantaneous, mass variations.
We show that a harmonic oscillator subject to a sudden change of mas s produces squeezed states. Our study is based on an approximate analytic solution to the time-dependent harmonic oscillator equation with a subperiod function parameter. This continuous treatment differs from former studies that involve the matching of two time-independent solutions at the discontinuity. This formalism requires an ad hoc transformation of the original differential equation and isalso applicable for rapid, although not necessarily instantaneous, mass variations. Ke.ywords: Solutions in closed form; quantum mechanics; exact invariants. Mostramos que un oscilador armónico sujeto a un cambio repentino de masa produce estados comprimidos. Nuestro estudio está basado en una solución analítica aproximada para el oscilador armónico dependiente del tiempo. El tratamiento continuo que estudiamos difiere de estudios anteriores en los cuales se igualan las soluciones en la discontinuidad. Nuestro formalismo requiere una transformación ad hoc de la ecuación diferencial original y es aplicable también para variaciones de masa rápidas, no solo instantáneas. Descriptores: Soluciones analíticas; mecánica cuántica; invariantes exactos. PACS: 03.65.La; 03.65.Ge; 02.30.Ik 1. Introduction ing another transforrnation that produces a monotonic time-dependent parameter. The transformed equation can then be sol ved with the continuum approach, leading to squeezing of the momentum variables at certain times. An interesting ad-vantage of this method is that the change of mass need not be a step function in the strict mathematical sense. The require-ment, which is physically more plausible, is that the change in mass should take place in a time span much shorter than the characteristic period of the oscillator. The plan of the manuscript is the following: in Seco 2, we write the quadratic and linear invariants for a time-dependent mass and exhibit their relation with the Hamiltonian. In Seco 3, we discuss two transforrnations that translate the clas-sical time-dependent mass equation into the time dependent trequency problem. The analytic solution for a rapid varia-tion of the time dependent mass using the appropriate trans-formation is then presented. In Seco 4, we exhibit how a sudden change of mas s in the quantum oscillator produces squeezing. Section 5 is left for conclusions. 2. Time dependent mass
The quantum harmonic oscillator with time-dependent frequency is a paradigmatic model of driven quantum dynamics and one of the few nontrivial systems that admits an exact analytical solution. In this review paper, we present a unified treatment of the time-dependent oscillator based on the Lewis-Riesenfeld invariant method, Bogoliubov transformations and the Ermakov-Pinney equation. We show how these approaches naturally connect to squeezing for the description of excitations production, and to the breakdown of adiabaticity under generic frequency protocols. Exact results for sudden quenches and smooth ramps are discussed in detail. By explicitly bridging invariant methods and squeezing formalism, this review is meant to provide a comprehensive framework for understanding nonequilibrium dynamics in quadratic potentials, with applications ranging from thermodynamics and condensed matter to quantum control theory.
Abstract In this work, we give a quantitative answer to the question: how sudden or how adiabatic is a frequency change in a quantum harmonic oscillator (HO)? We do that by studying the time evolution of a HO which is initially in its fundamental state and whose time-dependent frequency is controlled by a parameter (denoted by ϵ ) that can continuously tune from a totally slow process to a completely abrupt one. We extend a solution based on algebraic methods introduced recently in the literature that is very suited for numerical implementations, from the basis that diagonalizes the initial Hamiltonian to the one that diagonalizes the instantaneous Hamiltonian. Our results are in agreement with the adiabatic theorem and the comparison of the descriptions using the different bases together with the proper interpretation of this theorem allows us to clarify a common inaccuracy present in the literature. More importantly, we obtain a simple expression that relates squeezing to the transition rate and the initial and final frequencies, from which we calculate the adiabatic limit of the transition. Analysis of these results reveals a significant difference in squeezing production between enhancing or diminishing the frequency of a HO in a non-sudden way.
In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.
In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.
Abstract The center of mass motion of trapped ions and neutral atoms is suitable for approximation by a time-dependent driven quantum harmonic oscillator whose frequency and driving strength may be controlled with high precision. We show the time evolution for these systems with continuous differentiable time-dependent parameters in terms of the three basic operations provided by its underlying symmetry, rotation, displacement, and squeezing, using a Lie algebraic approach. Our factorization of the dynamics allows for the intuitive construction of protocols for state engineering, for example, creating and removing displacement and squeezing, as well as their combinations, optimizing squeezing, or more complex protocols that work for slow and fast rates of change in the oscillator parameters.