5.3 Higher Order Correlations in 1D
5.3.4 Higher Order Correlation Functions
The dramatic improvement in signal to noise for our g(2) plot in Fig. 5.7 compared to previous work [152, 159] suggests that enough correlated events are present in the thermal data to permit the calculation of higher order correlation functions. Using the same definitions of τ1 and τ2 as before, g(3)(0, τ1, τ2) is calculated following the same procedure used to plot Fig. 5.3. A square spatial bin of 5mm width (smaller than the x and y correlation lengths) and temporal bins of 10µs are used, with the resulting surface plot and diagonal plot (of τ1 = τ2 = τ) shown in Fig. 5.12. The graph is a considerable improvement on our previous g(3)(0, τ
1, τ2) plots (Fig. 5.3(a)), both in terms of absolute bunching (which reachesg(3)(0,0,0) = 5.11±0.28, compared to the theoretical maximum of 3! = 6) and SNR. It represents a factor of ∼ 70 improvement in bunching signal. The error is dominated by normalisation uncertainty, rather than statistical noise. Note that once again a gaussian plus an exponential is fitted as the theoretical form for the correlation function (red line).
While the g(2)(0, τ) and g(3)(0, τ
1, τ2) plots in Fig. 5.7 and Fig. 5.12 represent a significant improvement in the bunching signal, both correlation functions have previously been observed - albeit in a much reduced form. This higher bunching amplitude and SNR allows new physics to be elucidated along with greatly increasing any possibility of practical applications. However, no HBT style correlations have
8At the time of writing Karen Kheruntsyan from University of Queensland is working on the
0
1
2
3
4
5
x 10
−40.9
1
1.1
1.2
1.3
1.4
1.5
time bin (s)
g
(2)(0,0)
Figure 5.11: Spatial bunching amplitudesg(2)(0,0) vs time bins in the decoherent quan-
tum regime, along with a gaussian fit (red dashed line).
been observed at all for groups of 4 or more massive particles. Thus the next step was to attempt to observe these. The expected factorial values for the bunching amplitudes [17] are g(4)(0,0,0,0) = 4! = 24 and g(5)(0,0,0,0,0) = 5! = 120.
g(4)(0, τ
1, τ2, τ3) was calculated by extending the formalism and method used to calculate g(3)(0, τ
1, τ2). Thus the time intervals between each group of 4 particles arriving within the spatial correlation area τ1, τ2 and τ3 were recorded. τ1 and τ2 have the same definition as previously, while τ3 is defined as the time difference between the arrival times of particles 3 and 4. The normalised correlation function
g(4)(0, τ
1, τ2, τ3) is plotted in Fig. 5.13(a). Again, square spatial bins of 5mm width and temporal bins of 10µs were used. As mentioned above, there is no conceptually simple way to visually plot g(4)(0, τ
1, τ2, τ3) in entirety, due to the 4 dimensional rendering required. Hence, only the diagonal sliceτ1 =τ2 =τ3 =τ is plotted. The same theoretical form as in Fig. 5.8 is shown in red, while the bunching amplitude is measured to be g(4)(0,0,0,0) = 17.2 ± 1.4.
The process can be extended even further to calculateg(5)(0, τ
1, τ2, τ3, τ4). This is accomplished by histogramming in 10µs increments over all groups of 5 events within the same 5mm bins, definingτ1,τ2 andτ3 as above, along withτ4 as the arrival time difference between atoms 4 and 5. Once again the plot along the diagonal lineτ1=τ2 =τ3 =τ4 =τ is the most straightforward visualisation method and this is displayed in Fig. 5.13(b). Again, a gaussian and exponential combination fit is shown in red. The experimental bunching amplitude g(5)(0,0,0,0,0) = 78 ± 11. Consider that this shows that there is a 78 times higher likelihood of finding 5 particles in a close bunch than for any equivalent combination of spacings much longer than the
§5.3 Higher Order Correlations in 1D 113 0 1 2 3 4 5 6 x 10−4 0 1 2 3 4 5 6 x 10−4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 τ1 (s) τ2 (s) g (3) ( τ1 , τ2 ) 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 x 10−4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 τ (s) g (3) ( τ , τ )
Figure 5.12: Three particle correlation function plotted for our 1D thermal gas. The
top plot shows the full surface plot, while the lower graph is the diagonal sliceτ1 =τ2 =
τ. Experimental data is blue circles, while the sum of a gaussian and an exponential fit is the red line.
Figure 5.13: (a) Four particle correlation function plotted for our 1D thermal gas, only plotted along the diagonal sliceτ1 =τ2 =τ3=τ (for clarity). (b) Five particle correlation
function plotted for our 1D thermal gas, only plotted along the diagonal slice τ1 = τ2 =
τ3 =τ4 = τ (for clarity). In both cases experimental data is shown as blue circles, while
§5.3 Higher Order Correlations in 1D 115
coherence time. Also, our enhancement of 78 is nearly 3 orders of magnitude greater than any other bunching previously recorded for atoms.
At this point the extension to even higher orders breaks down, as the number of correlated events outside the correlation volume (necessary to ensure an accurate normalisation and meaningful correlation function plot) becomes too low to produce a plot with acceptable SNR. For theg(5) data there were less than 100 events in each of the bins at values of τ larger than 100µs. Calculation of g(6) reduces this number to a value so low that the normalisation procedure fails due to excessive noise. So
g(5) represents the limit of our current data.
Because all four of our correlation functions g(2)(0, τ), g(3)(0, τ, τ), g(4)(0, τ, τ, τ) and g(5)(0, τ, τ, τ, τ) were calculated with the same spatial and temporal bins, we can plot them on the same graph, shown in Fig. 5.14(a). The vertical axis is a logarithmic scale for ease of viewing, while the horizontal scale could be divided by a factor of√n−1 for the nth order correlation function to be equivalent correlation lengths. Rather the x axis units plotted represent the bin size of the individual
τ1, ..., τnbins, rather than the diagonal bins, which are larger. Note that the apparent decrease in correlation lengths for increasing n is an artifact previously discussed, due to our definition of τ1, ..., τn. Similarly, although the discrepancy between the fits and the data appears to deteriorate for higher n, along with the absolute noise significantly increasing, this is merely an illusion of the log scale.
For further comparison our four bunching amplitudes g(2)(0,0), g(3)(0,0,0),
g(4)(0,0,0,0) and g(5)(0,0,0,0,0) are also plotted in Fig. 5.14(b), along with the accompanying errors (which are dominated by normalisation uncertainty). To high- light the extremely sharp curve of the factorial dependence the factorial function
n! is also represented with the blue line. However, note that because the factorial function is only defined for integer values ofn, the points in between are represented by the gamma function [191]
Γ(n+ 1) =
Z ∞
0
tne−tdt (5.9)
which is the generalisation of the factorial to non-integer n, with Γ(n+ 1) = n! for
n ∈ Z+. A point worth noting about the gamma function plot in Fig. 5.14(b) is
that it does not represent a theoretical fit to our data. Rather, it is the absolute theoretical maximum which could ever be measured.
As with our earlier g(2) and g(3) measurement [159], an advantage of simultane- ously determining correlation functions of different orders is that we can calculate the ratios between the bunching amplitudes of the various orders. The six dis- tinct ratios g(n1)(0, ...,0)/g(n2)(0, ...,0) we can calculate, along with the unity ratios
g(n)(0, ...,0)/g(n)(0, ...,0) are shown in Fig. 5.15, with accompanying error bars. The- oretically (in the ideal case) the amplitudes are given by the factorial relationship between the different orders according to Wick’s theorem, so the gamma functions which describe these factorial curves are also plotted in Fig. 5.15. Our results nearly overlap the perfect theoretical ratios (within error), which is a further demonstration of how our data approaches the ideal case.
Figure 5.14: (a) Correlation functions from 2 to 5 particles for our 1D thermal gas all plotted as in the previous figures, on the same axis (log scale in y). (b) The measured bunching amplitudes of the correlation functions from 2 to 5 particles g(n) for our 1D thermal gas plotted against n. An ideal theoretical curve (Γ(n+ 1)) is plotted as the blue dashed line.
§5.4 Conclusion 117
Figure 5.15: The ratio of bunching amplitudes between the various order correlation
functions, along with theoretical curves (dashed lines).
5.4
Conclusion
5.4.1
Summary
We have performed simultaneous measurements of the second- and third-order cor- relation functions for atoms in a magnetic trap. Atom bunching in the arrival time for pairs and triplets of thermal atoms just above the Bose-Einstein condensation temperature was observed. At lower temperatures, we demonstrate conclusively (to 0.1%) the long-range coherence of the BEC for correlation functions to third order, in support of the prediction that, like coherent light, a BEC possesses long-range coherence to all orders.
When the thermal experiments were extended to a one-dimensional source in an optimal geometry, correlation functions were able to be measured up to 5th order. The large correlation length also allowed large bunching amplitudes approaching the theoretical maximums to be plotted. To demonstrate the dramatic improvement in measured bunching amplitude we were able to achieve compared to any previous measurement, Fig. 5.16 shows a plot of all bunching amplitudes previously measured for pure thermal bosons9.
These measurements provide strong confirmation of the quantum theory of boson
9There may appear to be three notable omissions from this plot. However, while Yasuda and
Shimizu also measured bunching in thermal bosons [23], they were unable to extract a bunching amplitude. Esslingeret. al. [28] were unable to measure bunching in ‘real’ thermal bosons (instead
using a pseudo-thermal source), while [32] measured bunching for a quasi-condensate rather than thermal atoms.
Figure 5.16: Measured bunching amplitudes for thermal bosons as a function of tem- perature for various results in the literature. Blue triangles are the Wesbrook/Aspect magnetic trap measurements, green diamonds are our magnetic trap measurements and red squares are our optical dipole trap measurements (the result in this chapter and the guiding from chapter 6).
statistics first developed in [17], as well as the prediction that a BEC possesses long- range coherence of matter waves to all orders in the correlation functions, in direct analogy with the long-range coherence of laser light.
In the 1D DQ regime bunching was also shown in the far-field for the second order correlation function. This demonstrates the decoherence of the source due to thermal fluctuations, which in the far-field translates to density fluctuations and bunching.
5.4.2
Future Work
A further logical step would be to extend the higher order correlations to the DQ regime data. However, while this was attempted, at present it is not possible to perform reliably, due to normalisation difficulties associated with the unusual bulk profile of the DQ data. Normalisation of correlation functions can be an inherently challenging process [153]. The computation time required was also an issue, with the number of pairs requiring significantly longer run time than for the thermal data.
One minor limitation with the 1D higher order correlation data is that no direct comparison is available with coherent data, due to the DQ regime lacking coherence. A possible extension would be to use a crossed dipole trap to yield a 3D trap with tight trapping frequencies in all directions. Combined with the techniques to drive the atom number down, this could give similar correlation lengths, allowing direct comparison between thermal and BEC data. However, the increase in Tc accompanying the 3 tight trapping frequencies would mean the thermal cloud would
§5.4 Conclusion 119
have to be extremely low in atom number to give equivalent correlation lengths. Our current laser (60mW) also only just possesses enough power to split the beam into two to create a crossed dipole trap. This process would be difficult, as it would entail overlapping two beam waists of ∼20µm around 10cm away from their focusing lenses. A different approach would be to overlay the dipole trap over the magnetic trap, to use the tight vertical trapping frequency of our BiQUIC trap combined with the two tight radial trap frequencies of the dipole trap. However, the problem with using this approach in our current setup is the vastly different trap volumes of the optical and magnetic trap (the dipole trap is much smaller). This leads to the conclusion that a higher power laser used to form a crossed-dipole trap is the best method for the future, and would be needed before such a measurement was attempted.
A future goal is to extend the correlation functions to 2D thermal and condensed atoms. Such a system can readily be created in an optical trap using either a single light sheet [192, 193, 194] or an optical lattice [195]. In a 2D system at finite temperature the coherence of the condensate does not extend over the entire condensate [196, 108], thus bunching of a 2D condensate could be expected, similar to that which we have observed in the 1D regime. However, at present there is no complete theory to describe the 2nd order correlation function even in trap, much less in the far-field. A 2D gas undergoes several transitions from thermal to quasi-condensate to superfluid, the latter of which is referred to as the Berezinskii- Kosterlitz-Thouless (BKT) transition [195, 193]. Using second order correlations to map out the quantum properties of the gas across these transitions is an exciting prospect for future investigations.
Chapter 6
Matter-Wave Guiding: from
Single to Multi-Mode, Coherent to
Incoherent
Ultracold atoms whose de Broglie wavelength is of the same order as an extended confining potential can experience waveguiding along the potential. When the trans- verse kinetic energy of the atoms is sufficiently low, they can be guided in the lowest order mode of the confining potential in analogy with light guided by a single mode optical fibre. In the results presented in this chapter we investigate the transverse profiles for a spread of mode occupancies of guided matter waves ranging from single mode to highly multi-mode occupancies.
Second-order correlations (coherence), exhibited in phenomena such as photon bunching (the Hanbury Brown - Twiss (HBT) effect [18]), are a measure of quantum coherence [17]. To probe the coherence of our guided atoms, we investigate both the first order coherence using a diffraction grating and second order coherence using the HBT effect.
This chapter supplements work that has previously been published in:
• [197] R. G. Dall, S. S. Hodgman, M. T. Johnsson, K. G. H. Baldwin and A. G. Truscott,Transverse mode imaging of guided matter waves, Phys. Rev. A.
81, 011602 (2010)
• [198]: R. G. Dall, S. S. Hodgman, A. G. Manning, and A. G. Truscott, Ob- servation of the first excited transverse modes in guided matter waves, Optics Letters 36, 1131–1133 (2011)
• [199]: R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johnsson, K. G. H. Baldwin and A. G. Truscott, Observation of atomic speckle and Hanbury Brown - Twiss correlations in guided matter waves, Nature Communications, 2, 291 (2011).
6.1
Introduction
A major motivation for investigating matter wave guiding is the possibility of appli- cations to practical devices, such as interferometric sensors [9]. Such atomic inter- ferometers would be similar to their optical counterparts but could take advantage of the unique properties of atoms (sensitivity to gravity, inherent non-linearity of a BEC etc.) to improve upon the precision of corresponding optical devices in certain applications. Many proposed sensor designs require the transport (often coherently) of atoms over macroscopic distances. A promising technique to facilitate this is mat- ter wave guiding [200]. As is the case with optics, guiding in a predominately single mode (or low mode occupancy) along with maintaining the coherence of the atoms is desirable.