Quantum–Mechanical Fields

Luigi Accardi, Igor Vasil'evich Volovich, Yun Gang Lu

Jennifer Harlow

In the past several years, the field of optomechanics has progressed from proof-of-principle experiments to the realization of mechanical oscillators and measurements in the quantum regime. Mechanical oscillators are of great interest because they can have small dissipation rates, can couple to many different systems of interest, and are the fundamental elements of ultrasensitive force detectors. Coupling these mechanical oscillators to microwave or optical fields provides a twofold advantage. Firstly, information about mechanical position can be encoded in the interrogating field, enabling sensitive readout of the mechanical oscillator. Secondly, the radiation pressure force of that field can be used to control the state of the mechanical oscillator. Including a high-quality microwave or optical cavity enhances both of these effects, as the field strength is resonantly increased.\nThe major questions in the field of optomechanics in the last several years have dealt with using mechanical oscillators for ultrasensitive measurements and as tools for quantum information. Both of these goals have the prerequisite that we be able to read out the motion of the mechanical oscillator in a quantum efficient manner. To that end, we developed a nearly shot-noise limited microwave interferometer capable of measuring mechanical motion with an imprecision below that at the standard quantum limit. This achievement is not only a critical improvement for the electromechanical experiments we do, but is also an important tool for any experiment that encodes the information of interest in microwave fields. In order to use mechanical oscillators as tools for quantum information, the mechanical oscillator must also be cooled into the quantum regime and fully controllable by the interrogating fields. To this end, we used the radiation pressure of microwave fields to cool our macroscopic mechanical oscillator to less than one phonon. We also demonstrated coherent transfer between itinerant microwave states and the mechanical oscillator, even for incident fields with less than one photon of energy.\nThese accomplishments have set the foundation for further experiments to extend the quantum information abilities of optomechanical systems, couple diverse quantum systems via a mechanical intermediary, and potentially explore the foundations of quantum mechanics at macroscopic scales.

Hans-Peter Eckle

Abstract The connection between statistical mechanics on the one side and quantum mechanics and quantum field theory on the other side is based on the analogy between thermal and quantum fluctuations. Formally, the connection is expressed through the mathematical equivalence between the partition function in statistical mechanics and the propagator in quantum field theory. This chapter explores the equivalence between statistical mechanics and quantum mechanics or quantum field theory in general terms using the Feynman path integral and with the example of the equivalence between the classical XY Heisenberg model and the sigma model of quantum field theory. Invoking the concepts of the partition function and the transfer matrix, an example demonstrates the passage from the quantum mechanics of a single degree of freedom, a zero-dimensional system, to the statistical mechanics of a one-dimensional system represented by classical variables.

Philippe A. Martin, FRANÇOIS ROTHEN

Philippe A. Martin, FRANÇOIS ROTHEN

James P. Finley

A generalized Euler equation of fluid dynamics is derived for describing many-body states of quantum mechanics. The Eulerian Eq. can be viewed as representing the interaction of two substates, where each substate has its own velocity and pressure fields. These field quantities are given by maps of the wavefunction. For one-body systems, the Eulerian Eq. can model either a fluid or particle description of quantum states. The generalized Euler Eq. is shown to be the gradient of an equation representing the total-energy of the two substates, having two energy fields. This total-energy Eq. is a generalization of the Bernoulli Eq. of fluid dynamics. The total-energy Eq., along with a continuity-equation, is equivalent to the time-dependent Schroedinger Eq. An equation is also derived that is equivalent to the main equation of Bohmian mechanics with additional identifications: The quantum potential of Bohmian mechanics is given as a sum of a kinetic energy and pressure fields. Also, the time derivative of the wavefunction phase is replaced by an energy field. In the formalism, field quantities are identified from their placement in equations of classical mechanics and separately, by definitions that involve the wavefunction and operators of quantum mechanics. This approach yields, unintended, and unknown energy and pressure fields. These fields, however, are shown to satisfy a continuity Eq., an equation that is equivalent to the other equation of Bohmian mechanics. It is also demonstrated that energy conservation holds for both of these energy fields, if the wavefunction is a linear-combination of eigenvectors, where the eigenvectors can be nondegenerate. A detailed investigation is given on the possible behavior, or source, of an electron that has one of the velocity fields. Alternate formulae for this velocity fields are also considered.