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arXiv:2306.05577 · Physica Scripta

Squeezing equivalence of quantum harmonic oscillators under different frequency modulations

Stanley Santiago Coelho, Lucas Queiroz, Danilo T. Alves

Published Jun 11, 2024• 2 citations

Abstract

Abstract The papers by Janszky and Adam [Phys. Rev. A 46, 6091 (1992)] and Chen et al [Phys. Rev. Lett. 104, 063 002 (2010)] are examples of works where one can find the following equivalences: quantum harmonic oscillators subjected to different time-dependent frequency modulations, during a certain time interval τ, exhibit exactly the same final null squeezing parameter (r f = 0). In the present paper, we discuss a more general case of squeezing equivalence, where the final squeezing parameter can be non-null (r f ≥ 0). We show that when the interest is in controlling the forms of the frequency modulations, but keeping free the choice of the values of r f and τ, this in general demands numerical calculations to find these values leading to squeezing equivalences (a particular case of this procedure recovers the equivalence found by Jansky and Adams). On the other hand, when the interest is not in previously controlling the form of these frequencies, but rather r f and τ (and also some constraints, such as minimization of energy), one can have analytical solutions for these frequencies leading to squeezing equivalences (particular cases of this procedure are usually applied in problems of shortcuts to adiabaticity, as done by Chen et al). In this way, this more general squeezing equivalence discussed here is connected to recent and important topics in the literature as, for instance, generation of squeezed states and the obtaining of shortcuts to adiabaticity.

DOI: 10.1088/1402-4896/ad56d6

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Abstract The papers by Janszky and Adam [Phys. Rev. A 46, 6091 (1992)] and Chen et al [Phys. Rev. Lett. 104, 063 002 (2010)] are examples of works where one can find the following equivalences: quantum harmonic oscillators subjected to different time-dependent frequency modulations, during a certain time interval τ, exhibit exactly the same final null squeezing parameter (r f = 0). In the present paper, we discuss a more general case of squeezing equivalence, where the final squeezing parameter can be non-null (r f ≥ 0). We show that when the interest is in controlling the forms of the frequency modulations, but keeping free the choice of the values of r f and τ, this in general demands numerical calculations to find these values leading to squeezing equivalences (a particular case of this procedure recovers the equivalence found by Jansky and Adams). On the other hand, when the interest is not in previously controlling the form of these frequencies, but rather r f and τ (and also some constraints, such as minimization of energy), one can have analytical solutions for these frequencies leading to squeezing equivalences (particular cases of this procedure are usually applied in problems of shortcuts to adiabaticity, as done by Chen et al). In this way, this more general squeezing equivalence discussed here is connected to recent and important topics in the literature as, for instance, generation of squeezed states and the obtaining of shortcuts to adiabaticity.

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.

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 Using operator ordering techniques based on Baker-Campbell-Hausdorff (BCH) relations of the su(1,1) Lie algebra and a time-splitting approach, we present an alternative method of solving the dynamics of a time-dependent quantum harmonic oscillator for any initial state. We find an iterative analytical solution given by simple recurrence relations that are very well suited for numerical calculations. We use our solution to reproduce and analyse some results from the literature to prove the usefulness of our method. We also discuss the efficiency in squeezing by comparing the parametric resonance modulation with the so-called Janszky-Adam scheme.

Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.

The concept of squeezing has so far been applied mainly to light, as is evidenced by the number of research works on the subject of squeezed light. Since, in quantum mechanics, both light and the simple harmonic oscillator are described within the same mathematical framework, there is of course no difficulty in applying the concept to the simple harmonic oscillator as well. In fact, the theoretical development of squeezed states and squeezed light owes much to the physical insights that one obtains as the analogy between light and the harmonic oscillator is exploited. The example presented shows clearly that two states with different phases in general have different degrees of squeezing, even if they have the same state distribution. This means that, even if one considers collision processes that produce the same state distribution, the degree of squeezing obtained during and after the collisions can be quite different, depending on how the phases phi(sub n) of the probability amplitudes develop in time as the collisions proceed. It is therefore evident that, for a detailed study of collision-induced squeezing, further study on the time development of the phases in collisions and its relation to collision parameters such as potential energy surfaces and collision energy is needed.

Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.