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arXiv:0912.5360 · Physical Review Letters

Beyond Spontaneously Broken Symmetry in Bose-Einstein Condensates

William J. Mullin, Franck Laloë

Published Apr 14, 2010• 8 citations
arXivHTMLPDF

Abstract

Spontaneous symmetry breaking (SSB) for Bose-Einstein condensates cannot treat phase off-diagonal effects, and thus cannot explain Bell inequality violations. We describe another situation that is beyond a SSB treatment: an experiment where particles from two (possibly macroscopic) condensate sources are used for conjugate measurements of the relative phase and populations. Off-diagonal phase effects are characterized by a "quantum angle" and observed via "population oscillations," signaling quantum interference of macroscopically distinct states.

DOI
10.1103/physrevlett.104.150401

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Spontaneous symmetry breaking (SSB) for Bose-Einstein condensates cannot treat phase off-diagonal effects, and thus cannot explain Bell inequality violations. We describe another situation that is beyond a SSB treatment: an experiment where particles from two (possibly macroscopic) condensate sources are used for conjugate measurements of the relative phase and populations. Off-diagonal phase effects are characterized by a "quantum angle" and observed via "population oscillations," signaling quantum interference of macroscopically distinct states.
Quantum systems in Fock states do not have a phase. When two or more Bose-Einstein condensates are sent into interferometers, they nevertheless acquire a relative phase under the effect of quantum measurements. The usual explanation relies on spontaneous symmetry breaking, where phases are ascribed to all condensates and treated as unknown classical quantities. However, this image is not always sufficient: when all particles are measured, quantum mechanics predicts probabilities that are sometimes in contradiction with it, as illustrated by quantum violations of local realism. In this Rapid communication, we show that interferometers can be used to demonstrate a large variety of violations with an arbitrarily large number of particles. With two independent condensates, we find violations of the Bell-Clauser-Horne-Shimony-Holt inequalities, as well as NNN-body Hardy impossibilities. With three condensates, we obtain Greenberger-Horne-Zeilinger-type contradictions.
Quantum systems in Fock states do not have a phase. When two or more Bose-Einstein condensates are sent into interferometers, they nevertheless acquire a relative phase under the effect of quantum measurements. The usual explanation relies on spontaneous symmetry breaking, where phases are ascribed to all condensates and treated as unknown classical quantities. However, this image is not always sufficient: when all particles are measured, quantum mechanics predicts probabilities that are sometimes in contradiction with it, as illustrated by quantum violations of local realism. In this letter, we show that interferometers can be used to demonstrate a large variety of violations with an arbitrarily large number of particles. With two independent condensates, we find violations of the BCHSH inequalities, as well as new N-body Hardy impossibilities. With three condensates, we obtain new GHZ (Greenberger, Horne and Zeilinger) type contradictions.
Why does spontaneous symmetry breaking occur? Why is a state breaking symmetry realized? We explore an idea that measurement selects such a state even if a system is given in a state respecting the symmetry of the system. We point out that the spectrum of the relevant observable is important, and simply apply the projection postulate for quantum measurement. We first show that this approach correctly describes the well-known interference of Bose-Einstein condensates. We then examine a fermionic system and prove that superconducting states with a definite relative phase are selected by the measurement of the current flowing between two superconductors, eliminating the need to assume the presence of an a priori phase to explain the Josephson effect.
The question of whether Bose-Einstein condensation involves spontaneous symmetry breaking is surprisingly controversial. We review the theory of spontaneous symmetry breaking in ferromagnets, compare it with the theory of symmetry breaking in condensates, and discuss the different viewpoints on the correspondence to experiments. These viewpoints include alternative perspectives in which we can treat condensates with fixed particle numbers, and where coherence arises from measurements. This question relates to whether condensates of quasiparticles such as polaritons can be viewed as “real” condensates. Introduction Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles. It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Mølmer [1]).
The realisation of Bose-Einstein condensation under grand-canonical conditions has provided the experimental evidence for the simultaneous occurrence of macroscopic fluctuations and phase coherence of the condensate. The observation of these two features, against a consolidated tradition which wants the fluctuations to be pathological (grand-canonical catastrophe) and incompatible with spontaneous symmetry braking, calls for a comprehensive rethinking of the approach to the problem. In this paper we consider the uniform ideal gas in a box and we present an alternative conceptual framework. We show that the usually-employed Bogoliubov quasi-average construction fails to reproduce the broken-symmetry state. The observed features are accounted for by a different pattern of spontaneous symmetry breaking, characterised by condensation of fluctuations and long-range correlations of the order parameter.
We show that, when two Bose-Einstein condensates are combined, a spatial interference pattern ensues even if the condensates are in number states with no spontaneously broken gauge symmetry. In our measurement-theoretical approach the interference pattern arises from correlations between the detected positions of the atoms. We also simulate numerically experiments intended to distinguish number-state condensates and spontaneous symmetry breaking, the latter being modelled by coherent-state condensates. To detect the difference in the pair correlation functions, the experiment has to be repeated about a thousand times even for condensates with as few as five atoms each.