5.2 Third Order Correlation Function
5.2.4 Theoretical Model
In order to quantitatively account for our results, we have applied a theoretical approach described below which is similar to that of [89] (which in turn was based on the work in [102]). These theoretical studies assume an ideal gas model, which is applicable in our experiments because the particle spacing covers a large range of distances (>100µm), all of which are much greater than the inter-particle scattering length (7.512(5) nm [75]).
The model calculations were performed as follows4. In three dimensions (r = {x, y, z =vt}) for impact velocity v and arrival time (t), the third order correlation function can be described using the creation and annihilation operators ˆΨ† and ˆΨ for pure, stationary boson modes:
4The majority of the model calculations were performed by fellow PhD student Andrew
g(3)(r1,r2,r3) = D ˆ Ψ†(r1) ˆΨ†(r2) ˆΨ†(r3) ˆΨ(r3) ˆΨ(r2) ˆΨ(r1))E D ˆ Ψ†(r1) ˆΨ(r1)E DΨˆ†(r2) ˆΨ(r2)E DΨˆ†(r3) ˆΨ(r3)E (5.5) By applying Wick’s theorem, we can rewrite this in terms of the first or- der correlation function G(1)(r
1,r2) =
D
ˆ
Ψ†(r1) ˆΨ(r2))E noting that density ρ(r) =
D ˆ Ψ†(r) ˆΨ(r))E, so that g(3)(r1,r2,r3) = 1 + G(1)(r1,r2) 2 ρ(r1)ρ(r2) + G(1)(r2,r3) 2 ρ(r2)ρ(r3) + G(1)(r1,r3) 2 ρ(r1)ρ(r3) (5.6) +2ℜ G(1)(r 1,r2)G(1)(r2,r3)G(1)(r1,r3) ρ(r1)ρ(r2)ρ(r3) (5.7) A similar model has been developed by Gomes et al. [89] to describe a HBT experiment which measured the second-order correlation function [24]. Using the above equation,g(3)(r
1,r2,r3) can readily be evaluated under ideal conditions, given the relevant experimental parameters such as temperature (∼1.3(2)µK for the ther- mal atoms) and trapping frequencies (565, 52 and 565 Hz).
However, a major complication is the experimental resolution. Gomes et al. [89] showed that the peak correlation amplitudeg(2)(0,0) reduced from 2! towards unity as a result of decreasing resolution. To model our experiment accurately we need to carefully simulate the effects due to both the detector characteristics and the data processing.
This reduction in the correlation function enhancement factor due to finite detec- tor resolution has been found in the previous HBT measurements of the second order correlation function using He∗ with similar detectors [24, 25]. In our experiment, the detector spatial resolution (determined primarily by the detection electronics) was measured using a transmission mask to be ∼ 130 µm. However, we have chosen to analyze the data by measuring events within 700 µm in the x-y plane (as indicated in Fig. 5.2), significantly larger than the detector resolution. This binning has the effect of further reducing the enhancement factor, but any reduction is more than compensated by a significant improvement in the signal-to-noise ratio. Smaller bins - while producing a more pronounced bunching signal due to the greater average wavefunction overlap - yield a comparatively smaller sample set. Finally, local de- tector saturation will result in a decreased likelihood of measuring particles at small spatial separations. After a single MCP pore has recorded an atom, depleting the charge within it, there is a dead time as the charge replenishes exponentially with a time constant of order 500µs [185]. During this period no further hits will be recorded, and since this time is greater than our correlation time we can assume a maximum of one measured event per pore. The surrounding pores will also be depleted for a similar length of time. To model these effects we follow the approach of [186], setting the probability of detecting more than one particle to zero within a circle of radius equal to the MCP pore centre-spacing of 60µm (effectively assuming
§5.2 Third Order Correlation Function 103 Experiment Theory g(2)(0, τ) max. 1.022(2) 1.025(5) g(2)(0, τ) width (τ c, µs) 90(10) 80(20) g(3)(0, τ 1, τ2) max. 1.061(6) 1.075(15) g(3)(0, τ1, τ2) width (µs) 120(10) 100(20) g(3)(0,0,0)−1 g(2)(0,0)−1 2.8(3) 3.0(3)
Table 5.1: Comparison of experimental and model values forg(2)(0, τ) and g(3)(0, τ1, τ2)
with their respective uncertainties.
100% depletion for the centre and all adjacent pores). The artificially increased elec- tronic dead time (implemented to remove mis-triggers - see chapter 4) also prevents pairs separated by less than ∼100 ns from being detected.
Our model explicitly includes these factors by averaging g(3)(r
1,r2,r3) over co- ordinates (r1,r2,r3) for a uniform distribution of triplets which abide by the above restrictions. We employ 20 µs temporal increments for τ1 and τ2, to generate the equivalent of the experimental data plotted in Figure 5.3 forg(3)(0, τ1, τ2). The sec- ond order correlation function g(2)(0, τ) corresponding to our experimental data is derived by repeating this process for pairs within the generated triplets. In this case, we require that the third particle is detected long after the correlation time of the other two, thereby yielding the average of g(2)(0, τ) where
g(2)(r1,r2) = 1 + G(1)(r1,r2) 2 ρ(r1)ρ(r2) (5.8) Although there are a large number of experimental parameters to consider for this model, most of them are well known. By varying their values within experimentally realistic ranges, they were shown to have a relatively minor effect on the simulated correlation function. The largest contribution to our theoretical uncertainty was the effect of local detector saturation, which clearly has a significant impact on our ability to measure groups of particles in close proximity, and thus restricts the contrast of the bunching signal. We use the known detector spatial and temporal resolution (∼130µm and∼2 ns respectively), the radial bin size (700µm), the local charge depletion around the 30 µm diameter MCP pores, and the electronic dead time (∼100 ns).
Thus the model effectively contains no free parameters. The predicted g(2)(0,0) and g(3)(0,0,0) enhancements and correlation widths are shown in Table 5.1 and are in excellent agreement with experiment within the combined uncertainties. The model reproduces both the measured amplitude and width of g(2)(0, τ) and
g(3)(0, τ
1, τ2) as well as the dramatic reduction in bunching signal resulting from the finite resolution.