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International Journal of Modern Physics C

NUMERICAL SIMULATION OF QUANTUM STATE REDUCTION IN BOSE–EINSTEIN CONDENSATES WITH ATTRACTIVE INTERACTIONS

Silvia Palpacelli, Sauro Succi A

Published May 1, 2010• 0 citations
Publisher

Abstract

Following an idea first proposed by Penrose in 1996 to explain the problem of quantum state reduction as a gravitational effect, Moroz, Penrose and Tod1 have shown that quantum state reduction due to gravitational interactions could take place in about one second for the case of 1011 nucleons. However, keeping 1011 nucleons together in a quantum macroscopic state does not appear to be feasible as yet. The closest physical system to such a situation is provided by Bose–Einstein condensates (BEC) with attractive interactions. We present numerical simulations of the Schrödinger–Newton equations, which show that an attractive BEC with 103 atoms would yield a decorrelation time of the order of 10-2 seconds. Hence, a "Penrose-like" reduction, induced by BEC attractive interaction instead of gravity, might be observable and possibly monitored in current BEC experiments with attractive interactions.

DOI
10.1142/s0129183110015385

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Following an idea first proposed by Penrose in 1996 to explain the problem of quantum state reduction as a gravitational effect, Moroz, Penrose and Tod1 have shown that quantum state reduction due to gravitational interactions could take place in about one second for the case of 1011 nucleons. However, keeping 1011 nucleons together in a quantum macroscopic state does not appear to be feasible as yet. The closest physical system to such a situation is provided by Bose–Einstein condensates (BEC) with attractive interactions. We present numerical simulations of the Schrödinger–Newton equations, which show that an attractive BEC with 103 atoms would yield a decorrelation time of the order of 10-2 seconds. Hence, a "Penrose-like" reduction, induced by BEC attractive interaction instead of gravity, might be observable and possibly monitored in current BEC experiments with attractive interactions.
We analyze Penrose's ideas concerning the role of gravity in the quantum-mechanical state reduction from the perspective of analogue-gravity models. We first review a line of work which argues that it is not enough to find a satisfactory quantization scheme for gravity in the description of a matter-gravity system. Instead of that; the standard rules of quantum mechanics as they are actually applied to matter systems need to be modified when the gravitational interaction is relevant. Among these ideas we focus on Penrose's argument for the absence of macroscopic quantum superpositions. He argues that such phenomenon has its roots in the idiosyncratic aspects of gravity, which become relevant just when the system is macroscopic and hence able to generate a sufficiently strong gravitational field, and not in the environmental decoherence. We then present an analogue model consisting of a Bose-Einstein condensate in a double well potential, where it is possible to build states that would correspond to putative superpositions of spacetimes. Such configurations are unstable and the source of instability from a microscopic point of view can be related to the abscence of a well-defined causal structure in the effective geometric description. We also discuss some additional lessons that can be taken from these instabilities.
We consider Bose-Einstein condensates (BECs) in situations where the density undergoes a symmetry breaking, either in real space or in momentum space. This occurs for a suitable number of condensed atoms in a double well potential, obtained in real space for a BEC of lithium atoms by adding a standing wave light field to the trap potential, or in momentum space for rubidium atoms by Raman coupling two BECs in different magnetic states with two tilted light fields. Evidence of bistability results from the solution of the Gross-Pitaevskii equation. By second quantization, we show that the classical bistable situation is in fact a Schrödinger cat (SC) and we evaluate the macroscopic quantum coherence between the two SC states.
The Bose-Einstein condensate (BEC) of excited states, provides a different platform to explore the interplay between gravity and quantum physics. In this Letter, we study the response of excited-state BECs to an external gravitational field and their dynamics under gravity when space is expanding. We reveal the anomalous response of the center-of-mass of the BEC to the gravitational field and the exotic gravity-induced accelerating expansion phenomena. We demonstrate that these effects result from the interplay among gravity, space and quantum effects. We also propose related experiments to observe these anomalies.
The ground state of Bose-Einstein condensates with attractive particle interactions is metastable. One of the decay mechanisms of the condensate is a collapse by macroscopic quantum tunneling, which can be described by the bounce trajectory as a solution of the time-dependent Gross-Pitaevskii equation in imaginary time. For condensates with an electromagnetically induced gravity-like interaction, the bounce trajectory is computed with an extended variational approach using coupled Gaussian functions and is simulated numerically exact within the mean-field approach on a spacetime lattice. It is shown that the variational computations converge very rapidly to the numerically exact result with increasing number of Gaussians. The tunneling rate of the condensate is obtained from the classical action and additional parameters of the bounce trajectory. The converged variational and numerically exact results drastically improve by several orders of magnitude the decay rates obtained previously with a simple variational approach using a single Gaussian-type orbital for the condensate wave function.
We study the collapse of a self-gravitating Bose-Einstein condensate with attractive self-interaction. Equilibrium states in which the gravitational attraction and the attraction due to the self-interaction are counterbalanced by the quantum pressure (Heisenberg's uncertainty principle) exist only below a maximum mass ${M}_{\mathrm{max}}=1.012\ensuremath{\hbar}/\sqrt{Gm|{a}_{s}|}$ where as<0{a}_{s}<0as​<0 is the scattering length of the bosons and mmm is their mass [P. H. Chavanis, Phys. Rev. D 84, 043531 (2011)]. For M>MmaxM>{M}_{\mathrm{max}}M>Mmax​ the system is expected to collapse and form a black hole. We study the collapse dynamics by making a Gaussian ansatz for the wave function and reducing the problem to the study of the motion of a particle in an effective potential. We find that the collapse time scales as $(M/{M}_{\mathrm{max}}\ensuremath{-}1{)}^{\ensuremath{-}1/4}$ for $M\ensuremath{\rightarrow}{M}_{\mathrm{max}}^{+}$ and as ${M}^{\ensuremath{-}1/2}$ for $M\ensuremath{\gg}{M}_{\mathrm{max}}$. Other analytical results are given above and below the critical point corresponding to a saddle-node bifurcation. We apply our results to QCD axions with mass $m={10}^{\ensuremath{-}4}\text{ }\text{ }\mathrm{eV}/{c}^{2}$ and scattering length ${a}_{s}=\ensuremath{-}5.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}53}\text{ }\text{ }\mathrm{m}$ for which ${M}_{\mathrm{max}}=6.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}{M}_{\ensuremath{\bigodot}}$ and $R=3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{R}_{\ensuremath{\bigodot}}$. We confirm our previous claim that bosons with attractive self-interaction, such as QCD axions, may form low mass stars (axion stars or dark matter stars) but cannot form dark matter halos of relevant mass and size. These mini axion stars could be the constituents of dark matter. They can collapse into mini black holes of mass $\ensuremath{\sim}{10}^{\ensuremath{-}14}{M}_{\ensuremath{\bigodot}}$ in a few hours. In that case, dark matter halos would be made of mini black holes. We also apply our results to ultralight axions with mass $m=1.93\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ }\text{ }\mathrm{eV}/{c}^{2}$ and scattering length ${a}_{s}=\ensuremath{-}8.29\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}60}\text{ }\text{ }\mathrm{fm}$ for which ${M}_{\mathrm{max}}=0.39\ifmmode\times\else\texttimes\fi{}1{0}^{6}{M}_{\ensuremath{\bigodot}}$ and R=33  pcR=33\text{ }\text{ }\mathrm{pc}R=33  pc. These ultralight axions could cluster into dark matter halos. Axionic dark matter halos with attractive self-interaction can collapse into supermassive black holes of mass $\ensuremath{\sim}1{0}^{6}{M}_{\ensuremath{\bigodot}}$ (similar to those reported at the center of galaxies) in about one million years. We point out the limitations of the Gaussian ansatz to describe the late stages of the collapse dynamics. We also mention the possibility that, instead of forming a black hole, the collapse may be accompanied by a burst or relativistic axions (bosenova) leading to a cycle of collapses and explosions as observed for nongravitational Bose-Einstein condensates with attractive self-interaction.