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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Generalized condensation of an interacting bose gas with pair Hamiltonian

Antonio Coniglio A, Ramabhadra Vasudevan

Published Nov 1, 1970• 3 citations
Publisher
DOI
10.1007/bf02712492

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We study the Evans-Imry pairing phase of dilute homogeneous bose gas with attractive interaction. We find that it is generally unstable against fluctuations in pairing. Therefore in addition to the well-known mechanical collapse, the system can also collapse due to the pairing instability. Under certain conditions the critical temperature of pairing instability can be higher than that of mechanical collapse. 1 The system of bose gas with attractive interaction has received a lot of attention recently due to the success in cooling 7 Li [1] and 85 Rb [2] systems. In these trapped systems, Bose Einstein condensation can be realized at zero temperature if the number of atoms in the condensate is below a critical value [3] due to finite size effect. In 7 Li system, the condensate experiences repetitive growth and collapse as the system is cooled down [1]. In 85 Rb system, the interaction can be tuned from being repulsive to being attractive by Feshbach resonance technique. During this process, the system loses particles until the critical number is reached [2].

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Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode $(\mathbf{k}=0)$ Hamiltonians and on the analysis of the associated microcanonical ensembles.

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Bose condensation of weakly interacting massLESS particles

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We study the Evans-Imry pairing phase of dilute homogeneous bose gas with attractive interaction. We find that it is generally unstable against fluctuations in pairing. Therefore in addition to the well-known mechanical collapse, the system can also collapse due to the pairing instability. Under certain conditions the critical temperature of pairing instability can be higher than that of mechanical collapse. 1 The system of bose gas with attractive interaction has received a lot of attention recently due to the success in cooling 7 Li [1] and 85 Rb [2] systems. In these trapped systems, Bose Einstein condensation can be realized at zero temperature if the number of atoms in the condensate is below a critical value [3] due to finite size effect. In 7 Li system, the condensate experiences repetitive growth and collapse as the system is cooled down [1]. In 85 Rb system, the interaction can be tuned from being repulsive to being attractive by Feshbach resonance technique. During this process, the system loses particles until the critical number is reached [2].
Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode (k=0)(\mathbf{k}=0)(k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.