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Physical Review A

Rotating highly mixed Gaussian packets with minimal energy

V. V. Dodonov

Published Feb 8, 2016• 4 citations
Publisher

Abstract

We study two-dimensional quantum mixed Gaussian packets with a fixed value of mean angular momentum. Highly mixed packets, minimizing the mean energy of an isotropic oscillator with fixed mean angular momentum $\mathcal{L}$, are found under the condition that the inverse purity ${\mathcal{P}}^{\ensuremath{-}1}$ exceeds the sum $1+2\mathcal{L}$. These states can be considered rotating thermal states. Their statistical properties (such as variances of angular momentum and energy) are studied. The case of a charged particle in a homogeneous magnetic field is considered as well.

DOI
10.1103/physreva.93.022106

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We study two-dimensional quantum mixed Gaussian packets with a fixed value of mean angular momentum. Highly mixed packets, minimizing the mean energy of an isotropic oscillator with fixed mean angular momentum L\mathcal{L}L, are found under the condition that the inverse purity P\ensuremath−1{\mathcal{P}}^{\ensuremath{-}1}P\ensuremath−1 exceeds the sum 1+2L1+2\mathcal{L}1+2L. These states can be considered rotating thermal states. Their statistical properties (such as variances of angular momentum and energy) are studied. The case of a charged particle in a homogeneous magnetic field is considered as well.
We study two-dimensional quantum Gaussian packets with a fixed value of mean angular momentum. This value is the sum of two independent parts: the ‘external’ momentum related to the motion of the packet center and the ‘internal’ momentum due to quantum fluctuations. The packets minimizing the mean energy of an isotropic oscillator with the fixed mean angular momentum are found. They exist for ‘co-rotating’ external and internal motions, and they have nonzero correlation coefficients between coordinates and momenta, together with some (moderate) amount of quadrature squeezing. Variances of angular momentum and energy are calculated, too. Differences in the behavior of ‘co-rotating’ and ‘anti-rotating’ packets are shown. The time evolution of rotating Gaussian packets is analyzed, including the cases of a charge in a homogeneous magnetic field and a free particle. In the latter case, the effect of initial shrinking of packets with big enough coordinate-momentum correlation coefficients (followed by the well known expansion) is discovered. This happens due to a competition of ‘focusing’ and ‘de-focusing’ in the orthogonal directions.
We address the time evolution of two- and three-dimensional nonrelativistic Gaussian wave packets in the presence of a weak external potential of arbitrary functional form. The focus of our study is the phenomenon of rotation of a Gaussian wave packet around its center of mass, as quantified by mean angular momentum computed relative to the wave-packet center. Using a semiclassical approximation of the eikonal type, we derive an explicit formula for a time-dependent change of mean angular momentum of a wave packet induced by its interaction with a weak external potential. As an example, we apply our analytical approach to the scenario of a two-dimensional quantum particle crossing a tilted ridge potential barrier. In particular, we demonstrate that the initial orientation of the particle wave packet determines the sense of its rotation, and report a good agreement between analytical and numerical results.
We examine the long-term time dependence of Gaussian wave packets in a circular infinite well (billiard) system and find that there are approximate revivals. For the special case of purely m=0m=0m=0 states (central wave packets with no momentum) the revival time is ${T}_{\mathrm{rev}}^{(m=0)}=8\ensuremath{\mu}{R}^{2}/\ensuremath{\Elzxh}\ensuremath{\pi},$ where $\ensuremath{\mu}$ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that ${T}_{\mathrm{rev}}^{(m\ensuremath{\ne}0)}=({\ensuremath{\pi}}^{2}{/2)T}_{\mathrm{rev}}^{(m=0)}\ensuremath{\approx}{5T}_{\mathrm{rev}}^{(m=0)}$ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of $m\ensuremath{\ne}0$ states is increased. The dependence of the revival structure on the initial position, energy, and angular momentum of the wave packet and the connection to the energy spectrum is discussed in detail. The results are also compared to two other highly symmetrical two-dimensional infinite well geometries with exact revivals, namely, the square and equilateral triangle billiards. We also show explicitly how the classical periodicity for closed orbits in a circular billiard arises from the energy eigenvalue spectrum, using a WKB analysis.
We investigate the angular momentum of a two-dimensional non-interacting Fermi gas confined in a rotating harmonic trap. By employing a quantum statistical approach, analytical expressions for the angular momentum are derived and analyzed as functions of particle number and rotation frequency. The rotational motion introduces a shell structure in the energy spectrum, giving rise to oscillatory behavior analogous to the de Haas–van Alphen oscillations observed in electronic systems subjected to magnetic fields. Numerical calculations reveal that these oscillations are most pronounced for weak rotation and small particle numbers, while they become progressively damped as the particle number increases. The results emphasize the close analogy between ultracold rotating gases and charged particles in magnetic fields and provide further insight into quantum rotational effects in confined Fermi systems.
For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as 〈qi〉〈{q}_{i}〉〈qi​〉, 〈pi〉〈{p}_{i}〉〈pi​〉 and 〈qipj〉〈{q}_{i}{p}_{j}〉〈qi​pj​〉, 〈pipj〉〈{p}_{i}{p}_{j}〉〈pi​pj​〉, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.