arXiv subjects
Publications and source records attributed to Michael Martin Nieto.
A completely analytic description is given of the motion of a trapped ion which is in either an even or an odd squeezed state. Comparison is made to recent results on the even or odd coherent states, and possible experimental work is discussed.
As an aid to understanding the {\it displacement operator} definition of squeezed states for arbitrary systems, we investigate the properties of systems where there is a Holstein-Primakoff or Bogoliubov transformation. In these cases the {\it ladder-operator or minimum-uncertainty} definitions of squeezed states are equivalent to an extent displacement-operator definition. We exemplify this in a setting where there are operators satisfying $[A, A^{\dagger}] = 1$, but the $A$'s are not necessarily the Fock space $a$'s; the multiboson system. It has been previously observed that the ground state of a system often can be shown to to be a coherent state. We demonstrate why this must be so. We close with a discussion of an alternative, effective definition of displacement-operator squeezed states.
Using Baker-Campbell-Hausdorff relations, the squeeze and harmonic-oscillator time-displacement operators are given in the form $\exp[δI] \exp[α(x^2)]\exp[β(x\partial)] \exp[γ(\partial)^2]$, where $α$, $β$, $γ$, and $δ$ are explicitly determined. Applications are discussed.
Using the vehicle of resolving an apparent paradox, a discussion of quantum interference is presented. The understanding of a number of different physical phenomena can be unified, in this context. These range from the neutral kaon system to massive neutrinos, not to mention quantum beats, Rydberg wave packets, and neutron gravity.
Certain modern theories of gravity predict that antimatter will fall differently than matter in the Earth's gravitational field. However, no experimental tests of gravity on antimatter exist and all conclusions drawn from experiments on matter depend, at some level, on a specific model. We have proposed a direct measurement that would compare the gravitational acceleration of antiprotons to that of negatively charged hydrogen ions. Substantial progress towards the development of this experiment has been achieved. Based on our work a number of alternative proposals for measuring ``$g$" on both charged and neutral antimatter have been made. We summarize the present status of our experiment and also discuss the steps that would be necessary to produce antihydrogen in an environment suitable for gravity measurements.
For use in calculating higher-order coherent- and squeezed- state quantities, we derive generalized generating functions for the Hermite polynomials. They are given by $\sum_{n=0}^{\infty}z^{jn+k}H_{jn+k}(x)/(jn+k)!$, for arbitrary integers $j\geq 1$ and $k\geq 0$. Along the way, the sums with the Hermite polynomials replaced by unity are also obtained. We also evaluate the action of the operators $\exp[a^j(d/dx)^j]$ on well-behaved functions and apply them to obtain other sums.
We know that the generally accepted theories of gravity and quantum mechanics are fundamentally incompatible. Thus, when we try to combine these theories, we must beware of physical pitfalls. Modern theories of quantum gravity are trying to overcome these problems. Any ideas must confront the present agreement with general relativity, but yet be free to wonder about not understood phenomena, such as the dark matter problem and the anomalous spacecraft data which we announce here. This all has led some ``intrepid" theorists to consider a new gravitational regime, that of antimatter. Even more ``daring" experimentalists are attempting, or considering attempting, the measurement of the gravitational force on antimatter, including low-energy antiprotons and, perhaps most enticing, antihydrogen.
We describe the origin, development, and status of the Los Alamos antiproton catching trap. Originally designed for the antiproton gravity experiment, it now is clear that this device can be a source of low-energy antiprotons for a wide range of physics, both on site, at CERN, and also off site.
For zero energy, $E=0$, we derive exact, quantum solutions for {\it all} power-law potentials, $V(r) = -γ/r^ν$, with $γ> 0$ and $-\infty < ν< \infty$. The solutions are, in general, Bessel functions of powers of $r$. For $ν> 2$ and $l \ge 1$ the solutions are normalizable; they correspond to states which are bound by the angular-momentum barrier. Surprisingly, the solutions for $ν< -2$ are also normalizable, They are discrete states but do not correspond to bound states. For $2> ν\geq -2$ the states are unnormalizable continuum states. The $ν=2$ solutions are also unnormalizable, but are exceptional solutions. Finally, we find that by increasing the dimension of the \seq beyond 4 an effective centrifugal barrier is created, due solely to the extra dimensions, which is enough to cause binding. Thus, if $D>4$, there are $E=0$ bound states for $ν> 2$ even for $l=0$. We discuss the physics of the above solutions are compare them to the classical solutions of the preceding paper.
For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-γ/r^ν$, with $γ>0$ and $-\infty <ν<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits $\r(t)= [\cos μ(þ(t)-þ_0(t))]^{1/μ}$, for all $μ\equiv ν/2-1 \ne 0$. When $ν>2$, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of $þ(t)$ and $þ_0(t)$, as functions of $t$, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special $ν= 2$ case.
For zero energy, $E=0$, we derive exact, classical and quantum solutions for {\em all} power-law oscillators with potentials $V(r)=-γ/r^ν$, $γ>0$ and $-\infty <ν<\infty$. When the angular momentum is non-zero, these solutions lead to the classical orbits $\r(t)= [\cos μ(þ(t)-þ_0(t))]^{1/μ}$, with $μ=ν/2-1 \ne 0$. For $ν>2$, the orbits are bound and go through the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. The unbound orbits are also discussed in detail. Quantum mechanically, this system is also exactly solvable. We find that when $ν>2$ the solutions are normalizable (bound), as in the classical case. Further, there are normalizable discrete, yet {\it unbound}, states. They correspond to unbound classical particles which reach infinity in a finite time. Finally, the number of space dimensions of the system can determine whether or not an $E=0$ state is bound. These and other interesting comparisons to the classical system will be discussed.
After reviewing the general properties of zero-energy quantum states, we give the explicit solutions of the \seq with $E=0$ for the class of potentials $V=-|γ|/r^ν$, where $-\infty < ν< \infty$. For $ν> 2$, these solutions are normalizable and correspond to bound states, if the angular momentum quantum number $l>0$. [These states are normalizable, even for $l=0$, if we increase the space dimension, $D$, beyond 4; i.e. for $D>4$.] For $ν<-2$ the above solutions, although unbound, are normalizable. This is true even though the corresponding potentials are repulsive for all $r$. We discuss the physics of these unusual effects.
Both the coherent states and also the squeezed states of the harmonic oscillator have long been understood from the three classical points of view: the 1) displacement operator, 2) annihilation- (or ladder-) operator, and minimum-uncertainty methods. For general systems, there is the same understanding except for ladder-operator and displacement-operator squeezed states. After reviewing the known concepts, I propose a method for obtaining generalized minimum-uncertainty squeezed states, give examples, and relate it to known concepts. I comment on the remaining concept, that of general displacement-operator squeezed states.
We point out that Rydberg wave packets (and similar ``coherent" molecular packets) are, in general, squeezed states, rather than the more elementary coherent states. This observation allows a more intuitive understanding of their properties; e.g., their revivals.
We propose a ladder-operator method for obtaining the squeezed states of general symmetry systems. It is a generalization of the annihilation-operator technique for obtaining the coherent states of symmetry systems. We connect this method with the minimum-uncertainty method for obtaining the squeezed and coherent states of general potential systems, and comment on the distinctions between these two methods and the displacement-operator method.
Proposed close-encounter solar probes (Vulcan and Solar Probe) are planned to have highly eccentric orbits, with a perihelion of about $4R_S$ and an inclination close to $90^\circ$ out of the plane of the ecliptic. We show this could allow at least an order-of-magnitude improvement in the present directly measured limit on the photon mass.
After reviewing the role of phase in quantum mechanics, I discuss, with the aid of a number of unpublished documents, the development of quantum phase operators in the 1960's. Interwoven in the discussion are the critical physics questions of the field: Are there (unique) quantum phase operators and are there quantum systems which can determine their nature? I conclude with a critique of recent proposals which have shed new light on the problem.
A general technique is outlined for investigating supersymmetry properties of a charged spin-$\half$ quantum particle in time-varying electromagnetic fields. The case of a time-varying uniform magnetic induction is examined and shown to provide a physical realization of a supersymmetric quantum-mechanical system. Group-theoretic methods are used to factorize the relevant Schrödinger equations and obtain eigensolutions. The supercoherent states for this system are constructed.