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arXiv:1507.07533 · Europhysics Letters (EPL)

Emergence and destruction of macroscopic wave functions

Bettina Gertjerenken, Martin Holthaus

Published Aug 1, 2015• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/0295-5075/111/30006

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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.
It is shown that any Bose-Einstein-condensed fluid in its ground state will exhibit macroscopic single-particle quantum behavior (MSPQB). That is, (1) the many-particle wave function $\ensuremath{\Psi}({\mathbf{r}}_{1},\dots{},{\mathbf{r}}_{n})$ factors into a single-particle product ${\ensuremath{\prod}}_{n}\ensuremath{\eta}({\mathbf{r}}_{n})$; (2) the function $\ensuremath{\eta}(\mathbf{r})$ extends over macroscopic length scales and obeys the usual quantum equations for particle flux in a single-particle system; and (3) $\ensuremath{\eta}(\mathbf{r})$ obeys a nonlinear single-particle Schr\"odinger equation. The latter equation reduces to the Gross-Pitaevskii equation when interactions are weak and determines the density distribution of the fluid and the time development of this distribution. The arguments used rely only on elementary concepts of probability theory and many-particle wave mechanics and are valid even in strongly interacting fluids such as superfluid 4He^{4}\mathrm{He}4He. It is shown that Bose-Einstein condensation implies that the NNN-particle wave function $\ensuremath{\Psi}$ is delocalized. That is, if one considers a single-particle coordinate r\mathbf{r}r, then for all values that occur of the other $N\ensuremath{-}1$ coordinates, $\ensuremath{\Psi}$ is a nonzero function of r\mathbf{r}r over a region of space proportional to VVV, where VVV is the total volume within which the fluid is contained. MSPQB is a consequence of this delocalization and the absence of long-range correlations between particle positions in fluids. The results are accurate provided that only averages over regions of space containing many particles are considered. For averages over volumes of space containing ${N}_{\ensuremath{\Omega}}$ particles, inaccuracies due to quantum fluctuations are $\ensuremath{\sim}1∕\sqrt{{N}_{\ensuremath{\Omega}}}$.
The decay of Bose-Einstein condensates from a metastable ground state into collapse due to macroscopic quantum tunneling is investigated using a semiclassical approximation to Feynman’s path integral formalism. Applying a variational ansatz of a single Gaussian to the wave function determined by the Gross-Pitaevskii equation, a special choice of the Gaussian width parameters yields a mean-field energy functional in Hamiltonian form. The temporal spatial extension of the condensate then is described in the picture of a particle moving in an external potential. In this picture the decay of the condensate wave function is investigated using the bounce trajectory method, accounting for the action of the tunneling orbit in imaginary time and quadratic fluctuations around it, the latter described by the Gelfand-Yaglom differential equation. Additionally, the tunneling formalism is used for describing the condensate wave function by a trial function of a superposition of several Gaussians. Using this ansatz a Hamiltonian form of the mean-field energy does not exist and the formalism of describing the tunneling process has to be extended. The bounce trajectory is described in the parameter space of the Gaussian width parameters and the equations of motion for the imaginary time evolution are obtained by applying a time-dependent variational principle. The action of the bounce trajectory is investigated as a function of the number of Gaussians taken into account and the contribution of the fluctuations is evaluated using the monodromy matrix.
Bose-Einstein Condensates (BECs) represent a unique phase of matter, predicted by Einstein and Bose in 1924 and experimentally realized in 1995. When atoms are cooled near absolute zero, they coalesce into a single quantum state, exhibiting macroscopic quantum phenomena. This paper provides a high-school-level exploration of BEC formation, the distinction between bosons and fermions, and the emergence of superfluidity and superconductivity. We introduce the concept of matter-wave interference in BECs and highlight practical applications in quantum computing and precision measurement. Additionally, recent advancements such as dipolar condensates and quantum turbulence are discussed to showcase the evolving nature of BEC research.
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