Philip B. WalczakBack-reaction of perturbations on gray solitons in dilute Bose gases | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
ISBN: | 978-3-8440-1213-2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Series: | Physik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Keywords: | back-reaction; gray soliton; perturbations | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Type of publication: | Thesis | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Language: | English | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Pages: | 156 pages | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Figures: | 22 figures | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Weight: | 231 g | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Format: | 24,0 x 17,0 cm | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Binding: | Paperback | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Price: | 48,80 € / 61,00 SFr | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Published: | August 2012 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Abstract: | The main topic of this work is the back-reaction from perturbations of small amplitude on gray solitons in weakly interacting quasi-one-dimensional dilute Bose gases. These systems are of special interest, since they raise physical questions connected with the emergence of collective degrees of freedom. An essential part of the present work constitutes the result of a linearization of the Gross-Pitaevskii equation (GPE) about a gray-soliton background solution. The linearization corresponds to a systematic expansion of the Heisenberg equation of motion in the dilute gas parameter up to and including first order. The solutions to the linearized equation of motion are derived with a method similar to the factorization from scattering theory. To this end the field amplitude is expanded with respect to normal modes with real amplitude and harmonic time evolution, which represent the elementary excitations of the system. The results are exact for the infinite system without boundary conditions, but may be transferred to finite systems with periodic boundary conditions up to exponentially small errors. Furthermore it is confirmed that the normal mode solutions indeed form a complete and orthonormal set of functions. The advantage of the real amplitude representation is that the modes with natural frequency zero, the so-called zero modes, are straightforwardly found from the quadratic Hamilton function, and they join at the bottom of a (quasi-)continuous spectrum of normal modes with finite frequency. In particular, a discrete zero mode with negative mass is obtained, which is associated with a spatial translation of the soliton. However, it is clarified that this assignment is not as unambiguous as commonly assumed, since a mixture of the zero modes is allowed by canonical transformation although their masses have different signs. From the analytical form of the normal mode solutions one can immediately conclude that gray solitons are in general transparent to excitations, yet the crossing of the excitations through the soliton is accompanied with a well-defined wavelength-dependent phase shift. This phase shift represents the foundation of the interaction between gray solitons and excitations of any kind. In order to study the interaction, wave-packet excitations of finite wavelength are considered that are constructed as a certain superposition of the normal modes. Here a complex amplitude representation is used. By asymptotic considerations it becomes apparent that wave packets, due to the phase shift when passing the soliton, experience a finite spatial displacement in comparison to propagation in constant background. Therefore, the presence of the soliton has a nontrivial effect on the wave excitation. Since wave packets are density perturbations, their displacement at first order in the systematic expansion in the dilute gas parameter necessarily means a displacement of the center of mass, but at second order of the expansion. To determine further effects, such as the back-reaction on the soliton, one is forced to solve the equation of motion at second order. Combining the asymptotic second order solution for the wave perturbation and the exact conservation laws of total linear momentum and total particle number, an analytical formula for the back-reaction effect on the soliton is derived. Upon interaction with the wave perturbation the soliton is displaced in space, which we verify by numerical integration of the GPE. The back-reaction of solitons under the influence of hydrodynamic perturbations, which are characterized by wavelengths much larger than the healing length of the condensate, are studied within a multiple scale boundary layer theory. In this respect, the aforementioned wave packets of the same wavelength represent the special case of small amplitude excitations within this theory. One can derive an equation of motion for the soliton, whose solution yields a closed expression for the soliton displacement. Furthermore the expression for the soliton shift may be interpreted as the limit of infinite wavelength of the previous result. The formula is explicitly evaluated for perturbation pulses with Gaussian density profile and subsequently confirmed by numerical integration of the GPE. In either case the results for the back-reaction can be explained qualitatively in terms of the involved velocities. The smaller the difference between the group velocity of the excitation and the soliton speed is, the longer the interaction time and the larger the back-reaction of the soliton is. |