Quantum Harmonic Oscillators with Nonlinear Effective Masses in the weak density approximation

We study the eigen-energy and eigen-function of a quantum particle acquiring the probability density-dependent effective mass (DDEM) in harmonic oscillators. Instead of discrete eigen-energies, continuous energy spectra are revealed due to the introduction of a nonlinear effective mass. Analytically, we map this problem into an infinite discrete dynamical system and obtain the stationary solutions in the weak density approximation, along with the proof on the monotonicity in the perturbed eigen-energies. Numerical results not only give agreement to the asymptotic solutions stemmed from the expansion of Hermite-Gaussian functions, but also unveil a family of peakon-like solutions without linear counterparts. As nonlinear Schr ¨odinger wave equation has served as an important model equation in various sub-fields in physics, our proposed generalized quantum harmonic oscillator opens an unexplored area for quantum particles with nonlinear effective masses.

Identifying optimal cycles in quantum thermal machines with reinforcement-learning

The optimal control of open quantum systems is a challenging task but has a key role in improving existing quantum information processing technologies. We introduce a general framework based on reinforcement learning to discover optimal thermodynamic cycles that maximize the power of out-of-equilibrium quantum heat engines and refrigerators. We apply our method, based on the soft actor-critic algorithm, to three systems: a benchmark two-level system heat engine, where we find the optimal known cycle; an experimentally realistic refrigerator based on a superconducting qubit that generates coherence, where we find a non-intuitive control sequence that outperforms previous cycles proposed in literature; a heat engine based on a quantum harmonic oscillator, where we find a cycle with an elaborate structure that outperforms the optimized Otto cycle. We then evaluate the corresponding efficiency at maximum power.

The noncommutative geometry of the Landau Hamiltonian: Differential aspects

In this work we study the differential aspects of the noncommutative geometry for the magnetic C*-algebra which is a 2-cocycle deformation of the group C*-algebra of R2. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic C*-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic C*-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

Time-dependent quantum harmonic oscillator: a continuous route from adiabatic to sudden changes

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.

Thermodynamics of dissipative coherent states

Almost all traditional physical formalisms are developed by using conservative forces, and the microscopic implementation of dissipation involves a sort of unusual process, mainly in quantum systems. In this work, we study the quantum harmonic model endowed with a non-Hermitian term responsible for dissipation. In addition, we also include an oscillating field that drives the model to a coherent state, which is dominated by fluctuation in a specific frequency, while regular thermal states are lowlily occupied. The usual coherent state formalism at zero temperature is extended to treat dissipative models at finite temperature. We define a generating function that is used in the evaluation of the most relevant statistical averages, such as the particle distribution. Then, we successfully employ the developed formalism to discuss two well-known applications; the damped quantum harmonic oscillator, and the precession magnetization in a ferromagnetic sample.

A New Framework for the Determination of the Eigenvalues and Eigenfunctions of the Quantum Harmonic Oscillator

This study was designed to obtain the energy eigenvalues and the corresponding Eigenfunctions of the Quantum Harmonic oscillator through an alternative approach. Starting with an appropriate family of solutions to a relevant linear di erential equation, we recover the Schr¨odinger Equation together with its eigenvalues and eigenfunctions of the Quantum Harmonic Oscillator via the use of Gram Schmidt orthogonalization process in the usual Hilbert space. Significantly, it was found that there exists two separate sequences arising from the Gram Schmidt Orthogonalization process; one in respect of the even eigenfunctions and the other in respect of the odd eigenfunctions.

Effective mass of the discrete values as a hidden feature of the one-dimensional harmonic oscillator model: Exact solution of the Schrödinger equation with a mass varying by position

In this paper, exactly solvable model of the quantum harmonic oscillator is proposed. Wave functions of the stationary states and energy spectrum of the model are obtained through the solution of the corresponding Schrödinger equation with the assumption that the mass of the quantum oscillator system varies with position. We have shown that the solution of the Schrödinger equation in terms of the wave functions of the stationary states is expressed by the pseudo Jacobi polynomials and the mass varying with position depends from the positive integer [Formula: see text]. As a consequence of the positive integer [Formula: see text], energy spectrum is not only non-equidistant, but also there are only a finite number of energy levels. Under the limit, when [Formula: see text], the dependence of effective mass from the position disappears and the system recovers known non-relativistic quantum harmonic oscillator in the canonical approach where wave functions are expressed by the Hermite polynomials.