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Physical Review A

Coherence of atomic matter-wave fields

Elena V. Goldstein, O. Zobay, Pierre Meystre

Published Sep 1, 1998• 24 citations
Publisher

Abstract

In analogy to Glauber's analysis of optical coherence, we adopt an operational approach to introduce different classes of atomic coherence associated with different types of measurements. For the sake of concreteness we consider specifically fluorescence, nonresonant imaging, and ionization. We introduce definitions of coherence appropriate to them, which we call electronic, density, and field coherence, respectively. We illustrate these concepts in various descriptions of Bose-Einstein condensation, showing that each of these descriptions makes different implicit assumptions on the coherence of the system. We also study the impact of elastic collision on the field and density coherence properties of atom lasers.

DOI
10.1103/physreva.58.2373

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The central theme of this thesis is that of coherence in bosonic matter-wave fields. Analogous to optical coherence; the measure of the capacity of a photonic beam to produce fringes upon interference; matter-wave coherence is in many ways the defining characteristic of the BEC phase of identical massive bosons. Its study therefore plays a critical role in the quest to better understand the physics of ultra-cold quantum gases, which has been the goal of this research. This work approaches coherence from a number of different angles and is for this reason presented in two parts. The first of these delves into the physics of perfectly coherent matterwave fields which are adequately described by ‘mean field’ equations, while the second is devoted to origins and repercussions of the diminished coherence evident in actual atom laser experiments. The presentation of the original work of this project begins with a detailed study of the interaction between an atomic and a molecular condensate, coherently coupled at zero temperature via one of several possible mechanisms (a Feshbach resonance, for example) in three spatial dimensions. Despite the presence of repulsive scattering interaction between the atoms, the mean-field equations of motion are known to support bright solitary waves — time-independent solutions which remain localised even in the absence of a trapping potential. (These solutions are the matter-wave analogue of spatio-temporal solitons in quadratic nonlinear media, but can also represent stationary states of the Schr¨odinger-Newton equations describing bosons interacting via a gravitational field.) Ignoring all interactions save the repulsive inter-atomic scattering mentioned, a variational ansatz approach is employed to analytically approximate these stationary solutions using Gaussian density distributions of the atomic and molecular clouds. By evolving these approximate solutions numerically, a comprehensive map of the accessible parameter space is constructed, showing primarily stable localised propagation. Parameter space regions where the approximate Gaussian solutions display strong ‘breathing’ oscillations are identified, and a modified ansatz having exponentially decaying tails is found to dramatically reduce the amplitude of these oscillations. Finally, by seeding a numerical relaxation algorithm with our Gaussian solutions, we find that the only parameter space region in which the approximate solutions diffuse under propagation is also devoid of exact localised stationary solutions. The discussion then proceeds beyond the scope of mean field theory, and turns to a detailed treatment of the quantum mechanical observable corresponding to the centre of mass of systems of identical particles. Firstly, careful consideration is given to its operational definition in the presence of fluctuating particle numbers. Possible pitfalls due to naive approximations when the number fluctuations are large are also addressed. Attention is then given to the fundamental quantum noise associated with this variable, which provides a physical mechanism for diminished first order coherence in degenerate Bose gases, even at zero temperature. Characteristic lower bounds (standard quantum limits) to the resulting mean position uncertainty are identified both for systems of identical bosons and identical fermions with comparable density profiles. Interestingly, fermions are found to have an intrinsically lower mean-position uncertainty than bosons, a fact which is attributed to the action of Pauli repulsion between atoms confined to a fixed density distribution. The results of numerical simulations of the exact quantum evolution of mean position variance are then presented. These demonstrate firstly the quantum diffusion of the mean position of one dimensional matter-wave solitons and secondly the evolution of variance in the centre of mass of a Bose gas undergoing forced evaporative cooling. Finally, attention is paid to the fact that the Penrose-Onsager criterion, which essentially equates Bose condensation with perfect first-order matter-wave coherence, fails to correctly detect the presence of BEC when a priori knowledge of the exact condensate mode is imperfect or unavailable. Insight is gained by drawing analogies between this criterion and entropic measures of entanglement between particular system subspaces. This motivates employment of the concept of entanglement of formation as a means to providing a more accurate measure of condensation in the general case of mixed system states. Unfortunately, this new measure remains adversely sensitive to superpositions of condensate modes. For this reason, alternative criteria based on higher order particle number correlation functions are also proposed.

•Oct 1, 2007•0 citations
In analogy to Glauber's analysis of optical coherence, we adopt an operational approach to introduce different classes of atomic coherence associated with different types of measurements. For the sake of concreteness we consider specifically fluorescence, nonresonant imaging, and ionization. We introduce definitions of coherence appropriate to them, which we call electronic, density, and field coherence, respectively. We illustrate these concepts in various descriptions of Bose-Einstein condensation, showing that each of these descriptions makes different implicit assumptions on the coherence of the system. We also study the impact of elastic collision on the field and density coherence properties of atom lasers.
In analogy to Glauber's analysis of optical coherence, we adopt an operational approach to introduce different classes of atomic coherence associated with different types of measurements. For the sake of concreteness we consider specifically fluorescence, nonresonant imaging, and ionization. We introduce definitions of coherence appropriate to them, which we call electronic, density, and field coherence, respectively. We illustrate these concepts in various descriptions of Bose-Einstein condensation, showing that each of these descriptions makes different implicit assumptions on the coherence of the system. We also study the impact of elastic collision on the field and density coherence properties of atom lasers.
This review discusses progress in the new field of coherent matter waves, in particular with respect to Bose-Einstein condensates. We give a short introduction to Bose-Einstein condensation and the theoretical description of the condensate wavefunction. We concentrate on the coherence properties of this new type of matter wave as a basis for fundamental physics and applications. The main part of this review treats various measurements and concepts in the physics with coherent matter waves. In particular we present phase manipulation methods, atom lasers, nonlinear atom optics, optical elements, interferometry and physics in optical lattices. We give an overview of the state of the art in the respective fields and discuss achievements and challenges for the future.
The availability of coherent matter waves from Bose-Einstein condensates had led to the extension of atom optics from the linear to the nonlinear regime, in analogy with the developments in optics following the invention of the laser. We provide a tutorial introduction to this emerging field, emphasizing the role of collisions as a “nonlinear medium” in the focusing and defocusing of matter waves, the launching of atomic solitons, the generation of vortices, and the demonstration of four-wave mixing both among matter waves and between optical and matter waves.
The central theme of this thesis is that of coherence in bosonic matter-wave fields. Analogous to optical coherence; the measure of the capacity of a photonic beam to produce fringes upon interference; matter-wave coherence is in many ways the defining characteristic of the BEC phase of identical massive bosons. Its study therefore plays a critical role in the quest to better understand the physics of ultra-cold quantum gases, which has been the goal of this research. This work approaches coherence from a number of different angles and is for this reason presented in two parts. The first of these delves into the physics of perfectly coherent matterwave fields which are adequately described by ‘mean field’ equations, while the second is devoted to origins and repercussions of the diminished coherence evident in actual atom laser experiments. The presentation of the original work of this project begins with a detailed study of the interaction between an atomic and a molecular condensate, coherently coupled at zero temperature via one of several possible mechanisms (a Feshbach resonance, for example) in three spatial dimensions. Despite the presence of repulsive scattering interaction between the atoms, the mean-field equations of motion are known to support bright solitary waves — time-independent solutions which remain localised even in the absence of a trapping potential. (These solutions are the matter-wave analogue of spatio-temporal solitons in quadratic nonlinear media, but can also represent stationary states of the Schr¨odinger-Newton equations describing bosons interacting via a gravitational field.) Ignoring all interactions save the repulsive inter-atomic scattering mentioned, a variational ansatz approach is employed to analytically approximate these stationary solutions using Gaussian density distributions of the atomic and molecular clouds. By evolving these approximate solutions numerically, a comprehensive map of the accessible parameter space is constructed, showing primarily stable localised propagation. Parameter space regions where the approximate Gaussian solutions display strong ‘breathing’ oscillations are identified, and a modified ansatz having exponentially decaying tails is found to dramatically reduce the amplitude of these oscillations. Finally, by seeding a numerical relaxation algorithm with our Gaussian solutions, we find that the only parameter space region in which the approximate solutions diffuse under propagation is also devoid of exact localised stationary solutions. The discussion then proceeds beyond the scope of mean field theory, and turns to a detailed treatment of the quantum mechanical observable corresponding to the centre of mass of systems of identical particles. Firstly, careful consideration is given to its operational definition in the presence of fluctuating particle numbers. Possible pitfalls due to naive approximations when the number fluctuations are large are also addressed. Attention is then given to the fundamental quantum noise associated with this variable, which provides a physical mechanism for diminished first order coherence in degenerate Bose gases, even at zero temperature. Characteristic lower bounds (standard quantum limits) to the resulting mean position uncertainty are identified both for systems of identical bosons and identical fermions with comparable density profiles. Interestingly, fermions are found to have an intrinsically lower mean-position uncertainty than bosons, a fact which is attributed to the action of Pauli repulsion between atoms confined to a fixed density distribution. The results of numerical simulations of the exact quantum evolution of mean position variance are then presented. These demonstrate firstly the quantum diffusion of the mean position of one dimensional matter-wave solitons and secondly the evolution of variance in the centre of mass of a Bose gas undergoing forced evaporative cooling. Finally, attention is paid to the fact that the Penrose-Onsager criterion, which essentially equates Bose condensation with perfect first-order matter-wave coherence, fails to correctly detect the presence of BEC when a priori knowledge of the exact condensate mode is imperfect or unavailable. Insight is gained by drawing analogies between this criterion and entropic measures of entanglement between particular system subspaces. This motivates employment of the concept of entanglement of formation as a means to providing a more accurate measure of condensation in the general case of mixed system states. Unfortunately, this new measure remains adversely sensitive to superpositions of condensate modes. For this reason, alternative criteria based on higher order particle number correlation functions are also proposed.