Phonons in expanding Bose-Einstein condensates with wavelengths much larger than the healing length behave in the same way as quantum fields within a universe undergoing an accelerated expansion. This analogy facilitates the application of many tools and concepts known from general relativity (such as horizons) and the prediction of the corresponding effects such as the freezing of modes after horizon crossing and the associated amplification of quantum fluctuations. Basically the same amplification mechanism is (according to our standard model of cosmology) supposed to be responsible for the generation of the initial inhomogeneities -and hence the seeds for the formation of structures such as our galaxy -during cosmic inflation (i.e., a very early epoch in the evolution of our universe). After a general discussion of the analogy (analogue cosmology), we calculate the frozen and amplified density-density fluctuations for quasi-two dimensional (Q2D) and three dimensional (3D) condensates which undergo a free expansion after switching off the (longitudinal) trap.PACS numbers: 03.75.Kk, 04.62.+v.
We present a formal derivation of the mean-field expansion for dilute Bose-Einstein condensates with two-particle interaction potentials which are weak and finite-range, but otherwise arbitrary. The expansion allows for a controlled investigation of the impact of microscopic interaction details (e.g., the scaling behavior) on the mean-field approach and the induced higher-order corrections beyond the s-wave scattering approximation.
For a graph G and a related symmetric matrix M , the continuous-time quantum walk on G relative to M is defined as the unitary matrix U (t) = exp(−itM ), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time τ if the (u, v)-entry of U (τ ) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer:• If a n-vertex graph has perfect state transfer at time τ relative to the Laplacian, then so does its complement if nτ ∈ 2πZ. As a corollary, the double cone over any m-vertex graph has perfect state transfer relative to the Laplacian if and only if m ≡ 2 (mod 4). This was previously known for a double cone over a clique (S. Bose, A. Casaccino, S. Mancini, S. Severini, Int. J. Quant. Inf., 7:11, 2009).• If a graph G has perfect state transfer at time τ relative to the normalized Laplacian, then so does the weak product G × H if for any normalized Laplacian eigenvalues λ of G and µ of H, we have µ(λ − 1)τ ∈ 2πZ. As a corollary, a weak product of P 3 with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of P 3 has perfect state transfer relative to the adjacency matrix.As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (C. Godsil, Discrete Math., 312:1, 2011).
For many physical systems which can be approximated by a classical background
field plus small (linearized) quantum fluctuations, a fundamental question
concerns the correct description of the backreaction of the quantum
fluctuations onto the dynamics of the classical background. We investigate this
problem for the example of dilute atomic/molecular Bose-Einstein condensates,
for which the microscopic dynamical behavior is under control. It turns out
that the effective-action technique does not yield the correct result in
general and that the knowledge of the pseudo-energy-momentum tensor ${<\hat
T_{\mu\nu}>}$ is not sufficient to describe quantum backreaction.Comment: 8 pages of RevTex4; extended discussion with additional sections, to
be published in Physical Review
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