States Of Harmonic Oscillator Research Articles

Quantized Microcavity Polariton Lasing Based on InGaN Localized Excitons.

Exciton-polaritons, which are bosonic quasiparticles with an extremely low mass, play a key role in understanding macroscopic quantum effects related to Bose-Einstein condensation (BEC) in solid-state systems. The study of trapped polaritons in a potential well provides an ideal platform for manipulating polariton condensates, enabling polariton lasing with specific formation in k-space. Here, we realize quantized microcavity polariton lasing in simple harmonic oscillator (SHO) states based on spatial localized excitons in InGaN/GaN quantum wells (QWs). Benefiting from the high exciton binding energy (90 meV) and large oscillator strength of the localized exciton, room-temperature (RT) polaritons with large Rabi splitting (61 meV) are obtained in a strongly coupled microcavity. The manipulation of polariton condensates is performed through a parabolic potential well created by optical pump control. Under the confinement situation, trapped polaritons are controlled to be distributed in the selected quantized energy sublevels of the SHO state. The maximum energy spacing of 11.3 meV is observed in the SHO sublevels, indicating the robust polariton trapping of the parabolic potential well. Coherent quantized polariton lasing is achieved in the ground state of the SHO state and the coherence property of the lasing is analyzed through the measurements of spatial interference patterns and g(2)(τ). Our results offer a feasible route to explore the manipulation of macroscopic quantum coherent states and to fabricate novel polariton devices towards room-temperature operations.

Explicit n-particle harmonic oscillator states in products of SU(1,1) representations

In this paper, we present a simple algorithm for the explicit construction of [Formula: see text]-particle harmonic oscillator states simultaneously belonging to irreducible representations of [Formula: see text] (or [Formula: see text]) and [Formula: see text]. For degenerate representations, the construction can be done using generating functions or hyperspherical harmonics. The cases with [Formula: see text] and [Formula: see text] are investigated at greater length. The analysis is extended to nondegenerate states, although the states are not so easily obtained as in the degenerate case. Finally, we briefly touch on the [Formula: see text] structure of Bell states.

Ultra‐Narrow Linewidth Polariton Lasing in Optically Trapped Bose‐Einstein Condensates at Room Temperature

AbstractPolariton lasing via Bose‐Einstein condensation (BEC) provides a peculiar method to achieve low threshold coherent light sources. Until now, the cryogenic operating temperature required for polariton lasers has hampered the development of polaritonics. Here, a novel approach is first reported to realize the ultra‐narrow linewidth polariton lasing under quasi‐3D quantum confinement at room‐temperature (RT). The potential trap landscape is constructed by ring‐shaped optical excitation, wherein the continuum polariton dispersion is modulated into discrete simple harmonic oscillator (SHO) states. The coherent condensation lasing of trapped polaritons occurs beyond the threshold power. Benefiting from the horizontal quantum confinement, the trapped polariton lasing exhibits an ultra‐narrow linewidth (0.7 meV) and excellent quality factor (Q = 3510). Moreover, the second‐order quantum coherence properties of trapped polariton condensation are determined to reveal the quantum phase transition of Bosonic system. The results offer a feasible route for realizing a low‐threshold ultra‐narrow linewidth polariton laser at RT, which significantly facilitates the fabrication of polariton laser and switches on opportunities for future applications.

Coherent states in microwave-induced resistance oscillations and zero resistance states

We investigate irradiated high-mobility two-dimensional electron systems (2DES) under low or moderated magnetic fields. These systems present microwave-induced magnetoresistance oscillations (MIRO) which, as we demonstrate, reveal the presence of coherent states of the quantum harmonic oscillator. We also show that the principle of minimum uncertainty of coherent states is at the heart of MIRO and zero-resistance states (ZRS). Accordingly, we are able to explain, based on coherent states, important experimental evidence of these photo-oscillations, such as their physical origin, their periodicity with the inverse of the magnetic field and their peculiar oscillations minima and maxima positions in regards of the magnetic field. Thus, remarkably enough, we come to the conclusion that 2DES, under low magnetic fields, become a system of quasiclassical states or coherent states and MIRO would be the smoking gun of the existence of these peculiar states in these systems. Published by the American Physical Society 2024

Non-KAM classical chaos topology for electrons in superlattice minibands determines the inter-well quantum transition rates

We investigate the quantum-classical correspondence for a particle tunnelling through a periodic superlattice structure with an applied bias voltage and an additional tilted harmonic oscillator potential. We show that the quantum mechanical tunnelling rate between neighbouring quantum wells of the superlattice is determined by the topology of the phase trajectories of the analogous classical system. This result also enables us to estimate, with high accuracy, the tunnelling rate between two spatially displaced simple harmonic oscillator states using a classical model, and thus gain new insight into this generic quantum phenomenon. This finding opens new directions for exploring and understanding the quantum-classical correspondence principle and quantum jumps between displaced harmonic oscillators, which are important in many branches of natural science.

Theory and modeling of molecular modes in the NMR relaxation of fluids

Traditional theories of the nuclear magnetic resonance (NMR) autocorrelation function for intra-molecular dipole pairs assume a single-exponential decay, yet the calculated autocorrelation of realistic systems displays a rich, multi-exponential behavior, resulting in anomalous NMR relaxation dispersion (i.e., frequency dependence). We develop an approach to model and interpret the multi-exponential intra-molecular autocorrelation using simple, physical models within a rigorous statistical mechanical development that encompasses both rotational diffusion and translational diffusion in the same framework. We recast the problem of evaluating the autocorrelation in terms of averaging over a diffusion propagator whose evolution is described by a Fokker–Planck equation. The time-independent part admits an eigenfunction expansion, allowing us to write the propagator as a sum over modes. Each mode has a spatial part that depends on the specified eigenfunction and a temporal part that depends on the corresponding eigenvalue (i.e., correlation time) with a simple, exponential decay. The spatial part is a probability distribution of the dipole pair, analogous to the stationary states of a quantum harmonic oscillator. Drawing inspiration from the idea of inherent structures in liquids, we interpret each of the spatial contributions as a specific molecular mode. These modes can be used to model and predict the NMR dipole–dipole relaxation dispersion of fluids by incorporating phenomena on the molecular level. We validate our statistical mechanical description of the distribution in molecular modes with molecular dynamics simulations interpreted without any relaxation models or adjustable parameters: the most important poles in the Padé–Laplace transform of the simulated autocorrelation agree with the eigenvalues predicted by the theory.

Physical interpretation of non-normalizable harmonic oscillator states and relaxation to pilot-wave equilibrium

Non-normalizable states are difficult to interpret in the orthodox quantum formalism but often occur as solutions to physical constraints in quantum gravity. We argue that pilot-wave theory gives a straightforward physical interpretation of non-normalizable quantum states, as the theory requires only a normalized density of configurations to generate statistical predictions. In order to better understand such states, we conduct the first study of non-normalizable solutions of the harmonic oscillator from a pilot-wave perspective. We show that, contrary to intuitions from orthodox quantum mechanics, the non-normalizable eigenstates and their superpositions are bound states in the sense that the velocity field vy→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v_y \\rightarrow 0$$\\end{document} at large ±y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm y$$\\end{document}. We argue that defining a physically meaningful equilibrium density for such states requires a new notion of equilibrium, named pilot-wave equilibrium, which is a generalisation of the notion of quantum equilibrium. We define a new H-function Hpw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{pw}$$\\end{document}, and prove that a density in pilot-wave equilibrium minimises Hpw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{pw}$$\\end{document}, is equivariant, and remains in equilibrium with time. We prove an H-theorem for the coarse-grained Hpw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{pw}$$\\end{document}, under assumptions similar to those for relaxation to quantum equilibrium. We give an explanation of the emergence of quantization in pilot-wave theory in terms of instability of non-normalizable states due to perturbations and environmental interactions. Lastly, we discuss applications in quantum field theory and quantum gravity, and implications for pilot-wave theory and quantum foundations in general.

Continuous-time quantum harmonic oscillator state engineering

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.

Generator coordinate method for 1D contacting bosons in harmonic trap

We propose a new method, termed generator coordinate method (GCM)-correlated pair wave function (CPWF), for studying one-dimensional bosons confined in harmonic potentials with contact repulsive interactions. Our approach involves using the effective CPWF as a basis, combined with the GCM to handle complex many-particle correlations accurately. We demonstrate the reliability of our GCM-CPWF wave functions by comparing ground energy and one-body density with those obtained by other numerical methods. Moreover, we present the energy spectrum up to six particles and the occupation number on the harmonic oscillator state. Utilizing these GCM-CPWF wave functions, we explore the properties of the ground and excited states of the many-particle system. Our GCM-CPWF framework is highly flexible and can be generalized to investigate more complex many-particle systems.

Dynamics-Based Entanglement Witnesses for Non-Gaussian States of Harmonic Oscillators.

We introduce a family of entanglement witnesses for continuous variable systems, which rely on the sole assumption that their dynamics is that of coupled harmonic oscillators at the time of the test. Entanglement is inferred from the Tsirelson nonclassicality test on one of the normal modes, without any knowledge about the state of the other mode. In each round, the protocol requires measuring only the sign of one coordinate (e.g., position) at one among several times. This dynamic-based entanglement witness is more akin to a Bell inequality than to an uncertainty relation: in particular, it does not admit false positives from classical theory. Our criterion detects non-Gaussian states, some of which are missed by other criteria.

Stein’s method and approximating the multidimensional quantum harmonic oscillator

AbstractStein’s method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction of finitely many classical ‘worlds’, with the role of sample size played by the number of worlds. Each approximation arises as the ground state of a Hamiltonian involving a particular interworld potential function. Our approach, framed in terms of spherical coordinates, provides the rate of convergence of the discrete approximation to the ground state in terms of Wasserstein distance. Applying a novel Stein’s method technique to the radial component of the ground state solution, the fastest rate of convergence to the ground state is found to occur in three dimensions.

Truncated generalized coherent states

A generalization of the canonical coherent states of a quantum harmonic oscillator has been performed by requiring the conditions of normalizability, continuity in the label, and resolution of the identity operator with a positive weight function. Relying on this approach, in the present scenario, coherent states are generalized over the canonical or finite dimensional Fock space of the harmonic oscillator. A class of generalized coherent states is determined such that the corresponding distributions of the number of excitations depart from the Poisson statistics according to combinations of stretched exponential decays, power laws, and logarithmic forms. The analysis of the Mandel parameter shows that the generalized coherent states exhibit (non-classical) sub-Poissonian or super-Poissonian statistics of the number of excitations, based on the realization of determined constraints. Mittag-Leffler and Wright generalized coherent states are analyzed as particular cases.

Investigation of spectroscopic factors of deeply-bound nucleons in drip-line nuclei with the Gamow shell model

Spectroscopic factors involving well bound nucleons in light nuclei are calculated with standard shell model, no-core shell model and Gamow shell model. Continuum coupling is included exactly in the Gamow shell model, due to the use of the Berggren basis, which contains bound, resonance and scattering states. Conversely, it is absent from standard and no-core shell models, where a basis of harmonic oscillator states is used. As the A−1 nuclei for which spectroscopic factors are calculated are either weakly bound or unbound, coupling to continuum is prominent, even though the A nuclei are well bound. It is then showed that Gamow shell model can properly reproduce experimental data and is a predictive tool for detailed nuclear structure at drip-line, contrary to standard and no-core shell model.

Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory

We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution to a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.

Shannon information entropy for a quantum nonlinear oscillator on a space of non-constant curvature

The so-called Darboux III oscillator is an exactly solvable N-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. This oscillator can be interpreted as a smooth (super)integrable deformation of the usual N-dimensional harmonic oscillator in terms of a non-negative parameter λ which is directly related to the curvature of the underlying space. In this paper, a detailed study of the Shannon information entropy for the quantum version of the Darboux III oscillator is presented, and the interplay between entropy and curvature is analysed. In particular, analytical results for the Shannon entropy in the position space can be found in the N-dimensional case, and the known results for the quantum states of the N-dimensional harmonic oscillator are recovered in the limit of vanishing curvature λ→0. However, the Fourier transform of the Darboux III wave functions cannot be computed in exact form, thus preventing the analytical study of the information entropy in momentum space. Nevertheless, we have computed the latter numerically both in the one and three-dimensional cases and we have found that by increasing the absolute value of the negative curvature (through a larger λ parameter) the information entropy in position space increases, while in momentum space it becomes smaller. This result is indeed consistent with the spreading properties of the wave functions of this quantum nonlinear oscillator, which are explicitly shown. The sum of the entropies in position and momentum spaces has been also analysed in terms of the curvature: for all excited states such total entropy decreases with λ, but for the ground state the total entropy is minimized when λ vanishes, and the corresponding uncertainty relation is always fulfilled.

Optical Schrödinger cats with generalized coherent states

Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.

Quadrature squeezing and temperature estimation from the Fock distribution

We present a method to estimate the amount of squeezing and temperature of a single-mode Gaussian harmonic oscillator state based on the weighted least-squares estimator applied to measured Fock state populations. Squeezing and temperature, or equivalently the quadrature variances, are essential parameters states used in various quantum communication and sensing protocols. They are often measured with homodyne-style detection, which requires a phase reference such as a local oscillator. Our method allows estimation without a phase reference, by using for example a photon-number-resolving detector. To evaluate the performance of our estimator, we simulated experiments with different values of squeezing and temperature. From 10,000 Fock measurement events, we produced estimates for states whose fidelities to the true state are greater than 99.99% for small squeezing (\(r<1.0\)), and for high squeezing (\(r=2.5\)), we obtain fidelities greater than 99.9%. We also report confidence intervals and their coverage probabilities.

Oblate Distorted Harmonic Oscillator Symmetry

Abstract: Harmonic oscillator states have shown advantageous for nuclear structure. Nuclear physics experts have developed sophisticated group theory-based mathematical techniques to handle n-particle states in the harmonic oscillator (ho) potential as a result of this. The deformed harmonic oscillator (HO)'s oblate shell closure has been studied, and the results demonstrate that the accompanying magic numbers are connected to a universal sequence of the triangular numbers that identically characterize oblate, spherical, and prolate. This offers a distinct framework for comprehending the distorted HO. There is discussion of the effects on oblate nuclei in the present study. The current work examines the symmetries of the deformed HO on the oblate side of deformation and demonstrates that these spherical degeneracies are equally important on this side of deformation as they are on the prolate side. The significance of these symmetries for comprehending nuclear structure is highlighted.

Born’s rule in multiqubit Bohmian systems

We study the Bohmian trajectories of a 4-qubit system made of coherent states of the quantum harmonic oscillator. We find, as in the 2-qubit and 3-qubit cases, that all the chaotic Bohmian trajectories are also ergodic. However, the chaotic trajectories are effectively ordered for longer times than in the 2 and 3-qubit cases. Then we study the long time dynamical establishing of Born’s rule with arbitrary initial distributions. We find that the higher the dimensionality of the system, the larger the proportion of the chaotic trajectories within the Born distribution, for any non-zero value; additionally, the proportion becomes practically total for a larger range of entanglements than in the 2-qubit and 3-qubit cases. Thus we conclude that in multiqubit systems Born’s rule will be accessible by practically all initial distributions.

Quantum error correction using squeezed Schrödinger cat states

Bosonic quantum codes redundantly encode quantum information in the states of a quantum harmonic oscillator, making it possible to detect and correct errors. Schr\"odinger cat codes -- based on the superposition of two coherent states with opposite displacements -- can correct phase-flip errors induced by dephasing, but they are vulnerable to bit-flip errors induced by particle loss. Here, we develop a bosonic quantum code relying on squeezed cat states, i.e. cat states made of a linear superposition of displaced-squeezed states. Squeezed cat states allow to partially correct errors caused by particle loss, while at the same time improving the protection against dephasing. We present a comprehensive analysis of the squeezed cat code, including protocols for code generation and elementary quantum gates. We characterize the effect of both particle loss and dephasing and develop an optimal recovery protocol that is suitable to be implemented on currently available quantum hardware. We show that with moderate squeezing, and using typical parameters of state-of-the-art quantum hardware platforms, the squeezed cat code has a resilience to particle loss errors that significantly outperforms that of the conventional cat code.