Space-time correlation function representation of reactive scattering by large systems

Measurement of One-Particle Correlations and Momentum Distributions for Trapped 1D Gases

“…The derivation closely parallels our previous work [11]. The exact transition rate from a specified initial state to a specified final state is ͑2p͞h͒ j͗ fjT 1 2 k 2 ͒ 2 ͑h 2 k 2 ͞2m͒ 2 D, the energy transferred to the BEC; the momentum transferred ishq h͑k 1 2 k 2 2 k͒.…”

“…Furthermore, the impulse approximation relating the scattering cross section to the dynamic structure factor fails in this case since particle impenetrability prevents free recoil of the struck particle. Here we introduce another approach based on a generalization [10,11] of the classic and extremely general Van Hove theory [12] of scattering of probe particles by a macroscopic target. We apply this approach to stimulated Raman transitions between two different hyperfine levels of a 1D trapped gas, for example, a Bose-Einstein condensate (BEC).…”

“…Derivation.-In order to probe one-particle correlations one should use a reactive scattering process which effectively removes one particle from the target system; this leads to the hole propagator and a related one-particle momentum-time correlation function [11]. We use the following simple choice of asymptotic initial and final states: ji͘ jlaser͘ jB i ͘ ÛjB i ͘ and j f͘ â y k,m0 jlaser͘ jB f ͘ â y k,m0Û jB f ͘.…”

“…1). T y ͑t͒ is the Heisenberg operator T y ͑t͒ e it͑Ĥ B 1Ĥ 0ph 1Ĥ 0a ͒͞hT y e 2it͑Ĥ B 1Ĥ 0ph 1Ĥ 0a ͒͞h , (2) and the BEC final states have been eliminated by an exact closure summation P Bf jB f ͘ ͗B f j 1 after introducing the integral representation of the energy delta function [10][11][12].…”

“…For example, it follows from the Fermi-Bose mapping theorem [4] that the dynamic structure factor of a one-dimensional (1D) system of hard core bosons (Tonks gas [5,6]) in a tightly confined atom waveguide [7][8][9] is identical with that of the corresponding ideal Fermi gas, although their momentum distributions are quite different. Here we introduce another approach based on a generalization [10,11] of the classic van Hove theory [12] of scattering of probe particles by a macroscopic target. We apply this very general approach to stimulated Raman transitions between two different hyperfine levels of a 1D trapped gas, for example, a Bose-Einstein condensate (BEC).…”