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arXiv:quant-ph/0406084

Bremsstrahlung from Charged Bose-Einstein Condensates

Mark P. Davidson

Published Jun 13, 2004• 0 citations
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Abstract

Low energy bremsstrahlung formulae are derived for a particle beam of charged bosons forming a Bose-Einsten condensate. The expression for energy radiated consists of two terms in this case. One of them, the larger one in the limit of large numbers of Bosons in the state, is proportional to the number of bosons squared and has the same form as one obtains in a hydrodynamic model of quantum wave mechanics. This term is sensitive to the size and shape of the wave packet especially when the force field causing acceleration is localized in extent. The second term, identical to the single particle scattering formula, is less sensitive to the wave packet size and shape and is proportional to the number of bosons in the condensate. The conclusion is that for a Bose-Einstein condensate the radiated bremsstrahlung is a sensitive function of the wave packet shape which is quite different than for a beam of incoherent particles which do not show very much dependence on the wave packet. Only lowest order radiation is calculated. It is also found that to lowest order there does not exist any situation in which the radiation loss from a coherent state of bosons vanishes completely, so that there is no analog of superconductivity in a particle beam of this type at least within the net of assumptions made in this paper. The amount of radiation can be greatly suppressed however, as it is a sensitive function of the form of the wave function.

DOI
10.48550/arxiv.quant-ph/0406084

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Low energy bremsstrahlung formulae are derived for a particle beam of charged bosons forming a Bose-Einsten condensate. The expression for energy radiated consists of two terms in this case. One of them, the larger one in the limit of large numbers of Bosons in the state, is proportional to the number of bosons squared and has the same form as one obtains in a hydrodynamic model of quantum wave mechanics. This term is sensitive to the size and shape of the wave packet especially when the force field causing acceleration is localized in extent. The second term, identical to the single particle scattering formula, is less sensitive to the wave packet size and shape and is proportional to the number of bosons in the condensate. The conclusion is that for a Bose-Einstein condensate the radiated bremsstrahlung is a sensitive function of the wave packet shape which is quite different than for a beam of incoherent particles which do not show very much dependence on the wave packet. Only lowest order radiation is calculated. It is also found that to lowest order there does not exist any situation in which the radiation loss from a coherent state of bosons vanishes completely, so that there is no analog of superconductivity in a particle beam of this type at least within the net of assumptions made in this paper. The amount of radiation can be greatly suppressed however, as it is a sensitive function of the form of the wave function.
It is shown that the hydrodynamic interpretation of a charged quantum particle leads to a different theoretical prediction for low energy bremsstrahlung than does quantum electrodynamics (QED). In the calculations, the electromagnetic fields are treated classically in the hydrodynamic case, but are quantized in QED. Calculations show the hydrodynamic model to have a different and more sensitive dependence on the size and shape of the radiating particle's wave packet then does QED. In particular it is shown that bremsstrahlung is sometimes greatly reduced when the force acting on the particle is localized to a volume small compared to the particle's wave packet. QED exhibits no such reduction. Therefore it is possible to test this effect experimentally. An experiment is proposed. It involves an electron microscope with a Wien filter for producing monochromatic beam electrons and an accurate energy measurement of the particle after passing through a local force field.
We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.
It is shown that the effective interaction strength of three bosons at small collision energies can be extracted from their wave function at zero energy. Asymptotic expansions of this wave function at large interparticle distances are derived, from which is defined a quantity DDD named three-body scattering hypervolume, which is an analog of the two-body scattering length. Given any finite-range interactions, one can thus predict the effective three-body force from a numerical solution of the Schr\"odinger equation. In this way, the constant DDD for hard-sphere bosons is computed, leading to the first complete result for the ground-state energy per particle of a dilute Bose-Einstein condensate (BEC) of hard spheres to order ${\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density. Effects of DDD are also demonstrated in the three-body energy in a finite box of size LLL, which is expanded to the order ${L}^{\ensuremath{-}7}$, and in the three-body scattering amplitude in vacuum. The three-body scattering amplitude calculated in this paper disagrees with an earlier calculation in the literature, because of the omission of the two-body effective range in that earlier work. Another key prediction is the condensate fractions of dilute BECs, which also disagree with an earlier work in the literature based on the effective field theory (EFT), as a result of short-range physics. An EFT prediction of the BEC ground-state energy, however, is corroborated.
The mathematical expression for the electromagnetic current of a quantum particle describes an extended charge in the sense that it is non-vanishing over the volume of the wave packet or the Schroedinger wave for the particle. In the hydrodynamic model this charge current and associated density are interpreted as a physical reality. Coupling such a current to a classical electromagnetic field in the soft photon limit then results in a theory for radiation. It is shown that this predicts that bremsstrahlung is sometimes greatly suppressed when the force acting on the particle is due to a classical potential field and when the volume over which the force is active is small compared to the volume of the particle's wave packet. Solving the same problem using conventional quantum radiation theory gives a different result. Therefore it is possible to test this effect experimentally and either confirm or rule out a hydrodynamic model for Schroedinger wave mechanics. An experiment is proposed to make such a test. It involves an electron microscope with a Wien filter for producing monochromatic beam electrons and an accurate energy measurement of the particle after passing through a local force field.
The properties of the many-particle Schr\"odinger wave function \ensuremath{\Psi} are examined in the presence of Bose-Einstein condensation (BEC). It is shown that it is possible to define, in terms of \ensuremath{\Psi}, a function $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s}),$ which can be regarded as the single-particle wave function of an arbitrary particle for a fixed configuration $\stackrel{\ensuremath{\rightarrow}}{s}$ of all other particles. It is shown that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ plays an analogous role to the field operator of standard field-theoretical treatments of superfluidity. It is shown that in the presence of a Bose-Einstein condensate fraction f, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ must be nonzero and phase coherent within at least a fraction f of the total volume of the N-particle system for essentially all $\stackrel{\ensuremath{\rightarrow}}{s}.$ Examination of the form of variational many-particle wave functions shows that in liquid 4He,{}^{4}\mathrm{He},4He, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ extends throughout the spaces left between the hard cores of the other atoms at $\stackrel{\ensuremath{\rightarrow}}{s}.$ By contrast, in the absence of BEC, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ in the ground state must be nonzero only over a localized region of space. It is shown that in order for long-range phase coherence in $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ to be maintained in the presence of velocity fields, any circulation must be quantized over macroscopic length scales. Some numerical calculations of the properties and fluctuations of liquid helium are presented. These suggest that the approach outlined in this paper may have significant advantages for the numerical calculations of the properties of Bose-Einstein condensed systems. The properties of $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ are used to show that there is no general connection between the static structure factor and the size of the Bose-Einstein condensate fraction in a Bose fluid. It is suggested that the observed connection in liquid 4He{}^{4}\mathrm{He}4He is due to the creation of vacancies in the liquid structure, which are required so that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ can delocalize, in the presence of hard-core interactions. It is shown that this suggestion is quantitatively consistent with observations on liquid 4He.{}^{4}\mathrm{He}.4He. It is suggested that the superfluid transition in 4He{}^{4}\mathrm{He}4He occurs when the vacancies in the liquid structure form a connected network.
This chapter will focus on the application of kinetic and hydrodynamic approaches to Bose–Einstein condensates. A Bose–Einstein condensate is a state of matter, formed of identical bosons (usually atoms behaving like bosons), so that at a sufficiently low temperature, many of the particles making up the system occupy the lowest energy state (ground state). This is only possible for bosons (particles with an integer spin) unlike fermions (particles with a half-integer spin) due to the Pauli exclusion principle.