Abstract
The central problem of this chapter is temporal coherence of a three-dimensional spatially homogeneous Bose-condensed gas, initially prepared at finite temperature and then evolving as an isolated interacting system. A first theoretical tool is a number-conserving Bogoliubov approach that allows to describe the system as a weakly interacting gas of quasi-particles. This approach naturally introduces the phase operator of the condensate: a central actor since loss of temporal coherence is governed by the spreading of the condensate phase-change. A second tool is the set of kinetic equations describing the Beliaev-Landau processes for the quasi-particles. We find that in general the variance of the condensate phase-change at long times t is the sum of a ballistic term ∝t 2 and a diffusive term ∝t with temperature and interaction dependent coefficients. In the thermodynamic limit, the diffusion coefficient scales as the inverse of the system volume. The coefficient of t 2 scales as the inverse volume squared times the variance of the energy of the system in the initial state and can also be obtained by a quantum ergodic theory (the so-called eigenstate thermalization hypothesis).
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- 1.
Particle losses are not discussed in this chapter. Their effect on temporal coherence is weak at relevant times as explicitly shown in [10] for one-body losses in the canonical ensemble.
- 2.
This results from the formula \(g_{0}^{-1}=g^{-1} - \int_{\mathrm{FBZ}} \frac{d^{3}k}{(2\pi)^{3}} \frac {m}{\hbar^{2} k^{2}}\).
- 3.
One may wonder in 2D about the value of μ 0=g 0 ρ, since g 0 logarithmically depends on the lattice spacing b [27], and dimensionality reasons prevent from forming a coupling constant g (such that gρ is an energy) from the quantities ħ, m and a, where a is now the 2D scattering length, given in [26, 28]. According to [27] one simply has to take for μ 0 the gas chemical potential μ(T).
- 4.
- 5.
The expectation values 〈…〉pulse differ from the original ones 〈…〉 in the absence of pulse by O(η 2): To first order in η, the perturbation of \(\hat{\psi}_{a}\) due to the pulse is linear in \(\hat{\psi}_{b}(0^{-})\) and has a zero contribution to the expectation values since component b is initially in vacuum.
- 6.
Here we have neglected the non-commutation of \(\hat{\theta }(t)\) and \(\hat{\theta}(0)\). From the Baker-Campbell-Hausdorff formula, and to zeroth order in the non-condensed fraction, see (15.45), the correction is a factor \(e^{-\frac{it}{2\hbar}\mu'(N)+O(N^{-2})}\) which is irrelevant for our discussion.
- 7.
We have neglected oscillating terms in \(\hat{b}\hat{b}\) and \(\hat{b}^{\dagger}\hat{b}^{\dagger}\): after time integration of \(\dot{\hat{\theta}}\) they give a negligible contribution to \(\hat{\theta}(t)-\hat{\theta}(0)\).
- 8.
- 9.
For a large system the level-spacing δE vanishes exponentially with the system size, and one may fear that an exponentially long time t>ħ/δE is needed to reach the limit (15.51). However, the corresponding off-diagonal matrix elements of \(\dot{\hat{\theta}}\) also vanish exponentially with the system size in the eigenstate thermalization hypothesis [47].
- 10.
- 11.
For an infinite system, the stationary solution of (15.58) is ensemble independent and corresponds to the Bose formula \(\bar{n}_{\mathbf{k}}(E) =1/(\exp{\beta \varepsilon _{k}}-1)\), where β is adjusted to give the mean energy E. Finite size effects on the \(\bar{n}_{\mathbf{k}}\), that can be calculated from (61) of [8], are here not relevant.
- 12.
This is true to leading order in the system size since our linearized kinetic approach cannot access the subleading terms.
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Castin, Y., Sinatra, A. (2013). Spatial and Temporal Coherence of a Bose-Condensed Gas. In: Bramati, A., Modugno, M. (eds) Physics of Quantum Fluids. Springer Series in Solid-State Sciences, vol 177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37569-9_15
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