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arXiv:1508.04117

$N$-coherence vs. $t$-coherence: An alternative route to the Gross-Pitaevskii equation

Bettina Gertjerenken, Martin Holthaus

Published Aug 17, 2015• 1 citation
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Abstract

We show how a candidate mean-field amplitude can be constructed from the exact wave function of an externally forced $N$-Boson system. The construction makes use of subsidiary $(N-1)$-particle states which are propagated in time in addition to the true $N$-particle state, but does not involve spontaneous breaking of the $U(1)$ symmetry associated with particle number conservation. Provided the flow in Fock space possesses a property which we call maximum stiffness, or $t$-coherence, the candidate amplitude actually satisfies the time-dependent Gross-Pitaevskii equation, and then serves as macroscopic wave function of the forced $N$-particle system. The general procedure is illustrated in detail by numerical calculations performed for the model of a driven bosonic Josephson junction, which allows one to keep track of all contributions which usually are subject to uncontrolled assumptions. These calculations indicate that macroscopic wave functions can persist even under conditions of strong forcing, but are rapidly destroyed upon entering a regime of chaotic dynamics. Our results provide a foundation for future attempts to manipulate, and actively control, macroscopic wave functions by means of purposefully designed force protocols.

DOI
10.48550/arxiv.1508.04117

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NNN-coherence vs. ttt-coherence: An alternative route to the Gross-Pitaevskii equation
We show how a candidate mean-field amplitude can be constructed from the exact wave function of an externally forced NNN-Boson system. The construction makes use of subsidiary (N−1)(N-1)(N−1)-particle states which are propagated in time in addition to the true NNN-particle state, but does not involve spontaneous breaking of the U(1)U(1)U(1) symmetry associated with particle number conservation. Provided the flow in Fock space possesses a property which we call maximum stiffness, or ttt-coherence, the candidate amplitude actually satisfies the time-dependent Gross-Pitaevskii equation, and then serves as macroscopic wave function of the forced NNN-particle system. The general procedure is illustrated in detail by numerical calculations performed for the model of a driven bosonic Josephson junction, which allows one to keep track of all contributions which usually are subject to uncontrolled assumptions. These calculations indicate that macroscopic wave functions can persist even under conditions of strong forcing, but are rapidly destroyed upon entering a regime of chaotic dynamics. Our results provide a foundation for future attempts to manipulate, and actively control, macroscopic wave functions by means of purposefully designed force protocols.
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We introduce a model system which allows one to study the N-particle dynamics of an externally forced Bose-Einstein condensate at zero temperature numerically for reasonably large N, and gives rise to mean-field dynamics which can be either regular or chaotic. We illustrate by various examples the correspondence between the evolution of the N-particle system, governed by the linear many-body Schroedinger equation, and the mean-field dynamics described by the nonlinear Gross-Pitaevskii equation. This correspondence is strongly sensitive to whether or not the mean-field dynamics are chaotic.
The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number NNN of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator AAA for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the NNN particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of N\sqrt{N}N​, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different NNN. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for NNN particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed NNN is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of N\sqrt{N}N​ for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.
We investigate finite-size quantum effects in the dynamics of NNN bosonic particles which are tunneling between two sites adopting the two-site Bose-Hubbard model. By using time-dependent atomic coherent states (ACSs) we extend the standard mean-field equations of this bosonic Josephson junction, which are based on time-dependent Glauber coherent states. In this way we find 1/N1/N1/N corrections to familiar mean-field (MF) results: the frequency of macroscopic oscillation between the two sites, the critical parameter for the dynamical macroscopic quantum self-trapping (MQST), and the attractive critical interaction strength for the spontaneous symmetry breaking (SSB) of the ground state. To validate our analytical results we perform numerical simulations of the quantum dynamics. In the case of Josephson oscillations around a balanced configuration we also find that, for a few atoms, the numerical results are in good agreement with the predictions of time-dependent ACS variational approach, provided that the time evolution is not too long. Also, the numerical results of SSB are better reproduced by the ACS approach than with the MF approach. Instead, the onset of MQST is correctly reproduced by ACS theory only in the large NNN regime and, for this phenomenon, the 1/N1/N1/N correction to the MF formula is not reliable.
The concept of the macroscopic wave function is a key for understanding macroscopic quantum phenomena. The existence of this object reflects a certain order, as it is present in a Bose-Einstein condensate when a single-particle orbital is occupied by a macroscopic number of bosons. We extend these ideas to situations in which a condensate is acted on by an explicitly time-dependent force. While one might assume that such a force would necessarily degrade any pre-existing order, we demonstrate that macroscopic wave functions can persist even under strong forcing. Our definition of the time-dependent order parameter is based on a comparison of the evolution of N -particle states on the one hand, and of states with particles on the other. Our simulations predict the possibility of an almost instantaneous dynamical destruction of a macroscopic wave function under currently accessible experimental conditions.
We present a method to calculate the dynamics of very-low-temperature Bose-Einstein condensates in time-dependent traps. We consider a system with a well-defined number of particles, rather than a system in a coherent state with a well-defined phase. This preserves the U(1)U(1)U(1) symmetry of the problem. We use a systematic asymptotic expansion in the square root of the fraction of noncondensed particles. In lowest order we recover the time-dependent Gross-Pitaevskii equation for the condensate wave function. The next order gives the linear dynamics of noncondensed particles. The higher order gives corrections to the time-dependent Gross-Pitaevskii equation including the effects of noncondensed particles on the condensate. We compare this method with the Bogoliubov--de Gennes approach.