Quantum correlations and atomic speckle

Quantum correlations and atomic speckle S. S. Hodgman R. G. Dall A. G. Manning M. T. Johnsson K. G. H. Baldwin A. G. Truscott ARC Centre of Excellence for Quantum-Atom Optics, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia Abstract Here we utilize the single-atom detection capability of metastable helium to measure the second-order and third-order correlation functions for ensembles of ultracold atoms. We then extend these measurements to characterize the quantum statistics of atoms guided in a dipole potential. By appropriately loading atoms into the guide we are able to populate a selected number of guided modes, ranging from the majority of atoms in the lowest order mode (a BEC), to multi-mode guiding (equivalent to a thermal source). The guided BEC was characterised by a smooth gaussian transverse spatial profile, and a second-order correlation value of unity consistent with a coherent source. For multi-mode guiding, the matter-wave equivalent of speckle was observed. Furthermore, at short arrival times, the second-order correlation function was greater than unity, corresponding to atom bunching characteristic of an incoherent (thermal) source. 1 Introduction In analogy to quantum optics, sources of atoms can be characterized by their coherence properties which can be viewed in either the wave or particle picture. First order coherence is a measure of amplitude fluctuations that determine, for example, the fringe visibility in an interferometer. Second order coherence measures intensity fluctuations, and is responsible for laser light speckle which arises from multimode interference between multiple independent sources. In a landmark experiment, Hanbury Brown and Twiss (HBT) demonstrated that in the particle picture, such incoherent sources are characterized by photon bunching [1], whereby the second order correlation function exceeds unity for short arrival times between pairs of photons (the coherence time). A coherent source by contrast, has a correlation function of unity for all times, and as shown by the quantum theory of Glauber [2], this is true to all orders of the correlation function. Previous HBT experiments for atoms [3 5] have employed metastable species because of their efficient single-atom detection capability [6]. Atom bunching was observed for thermal (incoherent) sources of bosonic atoms (anti-bunching for fermions [5]), while a second-order correlation function value of unity i.e. an equal probability for all arrival times, was observed for BECs in analogy with coherent optical sources. We have undertaken similar experiments using our metastable helium BEC apparatus [7] to extend these measurements to higher order correlation functions. Figure 1 shows an Kenneth.baldwin@anu.edu.au 231

experimental schematic for the third-order correlation measurement, which can be interpreted in the particle picture as the arrival probability at times 1 and 2 between triplets of particles. Figure 1: An ensemble of He* atoms (red spheres) falls under gravity onto the MCP detector creating a series of detection events (yellow) separated in space and time. The middle inset (blue box) shows the arrival of the atomic ensemble in more detail, whereas the right hand inset (green cylinder) shows the arrival of a triplet of atoms within our analysis region. The third-order correlation function g 3 (0, 1, 2 ) characterises the arrival time differences 1 and 2 between the three atoms. Adapted from ref. 9. We utilize a microchannel plate (MCP) coupled with a delay line detector to measure both the arrival time and position of atoms on the detector. Together with a new technique [8] for outcoupling multiple pulses of atoms from the ultracold atomic cloud, this efficiently creates very large data sets that enable the measurement of the third-order correlation function for atoms for the first time [9]. The results for these experiments are shown in figure 2, where the normalized third-order correlation function g 3 (0, 1, 2 ) is plotted on the left hand side in three dimensions for (A) a thermal ensemble and (B) a BEC. In the cross-sections shown on the right, we measure for g 2 (0, ) {g 3 (0, 1, 2 )} an atom bunching enhancement for the thermal case of ~ 2% {~ 6%} and a correlation time of ~ 90 s {120 s}, in good agreement with a theoretical model [9]. For the BEC, both the second- and third-order correlation functions have a uniform value of unity within 0.1%, demonstrating for the first time the coherence of a BEC to third order. While there may be a very small thermal component present at finite temperatures, the contribution of this to our BEC correlation function is not discernible within our experimental uncertainty. These measurements provide strong confirmation of the quantum theory of boson statistics first developed by Glauber [2], 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. 232

Figure 2: Left: Normalized third-order correlation functions g 3 (0, 1, 2 ) for (A) ensembles of thermal atoms and (B) the BEC. Right: Sections of normalized third-order correlation functions. (A) The diagonal section ( 1 = 2 ) of g 3 (0, 1, 2 ) for both thermal atoms (red circles) and the BEC (blue squares). (B) Second-order correlation function g 2 (0, ) (i.e. g 3 (0, 1, 2 ) averaged over both 1 and 2 > 200 μs), for thermal atoms (red circles) and the BEC (blue squares). The dashed red and solid blue lines correspond to two-dimensional gaussian surface fits to the left-hand thermal and BEC plots, respectively. Adapted from ref. 9. 2 Atom Waveguide Experiments We have further extended the above experiments by using correlation measurements to characterize the coherence properties of atoms guided in an optical potential, which doubles as a diagnostic of the number of modes sustained in the effective de Broglie waveguide. Atoms coherently output-coupled from a Bose-Einstein condensate (BEC) form a coherent beam of matter waves an atom laser. Most condensates are confined in a magnetic potential, where to achieve maximum flux, the atom laser beam is outcoupled from the centre of the BEC. These atoms in the atom laser beam originate from the highest density region of the BEC and experience a large repulsive force (mean field repulsion) via s-wave interactions. These interactions strongly distort the atom laser beam, yielding a non-ideal spatial profile with a double-peaked structure [10,11]. One method to alleviate this problem is to use an optically trapped BEC. An atom laser is then created by simply reducing the optical power, and letting the atoms fall out of the spatial minimum of the trap where the atomic density is low. Further, by not extinguishing the optical trap completely, the atom laser beam experiences a weak confining potential that acts like an optical fibre to guide the atomic de Broglie waves. Here we demonstrate near-single-mode guiding of a metastable helium (He*) atom laser in an optical dipole potential using a far-detuned laser beam [12]. Atoms cooled to ~ 1 μk in a magnetic trap are transferred to an optical trap aligned in the vertical direction (figure 3), where evaporative cooling takes place and a BEC is achieved. Subsequent lowering of the optical 233

potential releases the atoms into the guide, and they fall under gravity until they strike a multichannel plate (MCP) and are imaged. The process is adiabatic, allowing the atoms in the BEC to transfer smoothly from the ground state of the trap to the ground state of the guide. As the atoms progress down the guide, they are further adiabatically cooled before eventually falling out of the guide and onto the MCP. Figure 3: A thermal cloud/bec is confined in an optical trap, produced by a single focused laser beam in the vertical direction. By reducing the power in the optical beam the cloud is no longer supported against gravity and falls towards the single atom detector. The transverse potential is still strong enough to confine the atoms in that direction so the atoms are effectively guided. Adapted from ref. 14. The laser intensity ramp shown by the red (solid) line in figure 4 yields a BEC with no discernible thermal fraction. The BEC then equilibrates for a period before finally lowering the potential slightly to outcouple the BEC into the guide. By rapidly decreasing the guide laser intensity (blue dotted line), an ensemble of predominantly thermal atoms populates the trap, the proportion of which is controlled by varying the height of the final intensity step. The ensemble then equilibrates for a period before again lowering the potential, this time sufficiently to couple thermal atoms into the guide, but not as far as the BEC ramp in order to continue to trap any condensed atoms. The parameters of the intensity ramp can be varied to populate almost a pure (65%) lowest-order BEC guided mode [12], predominantly the first excited mode [13], or multiple modes [14]. An example of selective excitation of predominantly the first excited mode is shown in the MCP image of the transverse spatial profile in figure 5. A cross-section of this image is shown in figure 6, along with a mode composition fit to the profile. The fit incorporates contributions up to the fifth excited mode of the guide, with almost half the atoms being present in the first excited mode [13]. This is the first time to our knowledge that the first excited matter wave mode has selectively been excited in an optical waveguide. 234

Figure 4: Intensity ramp for the dipole beam for the single-mode BEC guide (solid red line) and the multimode thermal guide (dotted blue line). Adapted from ref. 14. These experiments demonstrate that mode selectivity can be achieved which enables a desired mode occupancy (in the single-figure range) to be populated, starting with the lowest (TEM00) mode populated by the BEC. This development of a few-mode matter waveguide may enable future sensors based upon matter wave interferometers that promise improved sensitivity over their current state-of-the-art optical counterparts. Entanglement of the lower-order atomic modes may also be possible in a direct analogy with multimode light entanglement, with potential applications for quantum information and quantum imaging. Figure 5: False color image of the transverse spatial profile of guided atoms dominated by the first excited mode of the waveguide. Adapted from ref. 13. 235

Figure 6: Modal fit to the transverse profile of the guided atomic beam incorporating the lowest six supported modes, dominated by the first excited mode 1. Adapted from ref. 13 3 Speckle and Second Order Correlation Results The ability to determine the mode occupancy then allows a quantum statistical study of the transition from a thermal to a coherent source of guided matter waves. When a thermal atomic cloud is loaded into the guide, a multimode transverse spatial profile results. Interference between these modes causes intensity variations to be measured as shown in figure 7. This corresponds to the speckle observed in a multimode guided laser beam, and we believe that this is the first spatial observation of an atomic speckle pattern arising from interfering matter waves. The speckle pattern changes (as expected) between different realizations of the experiment (figures 7 a c). When the speckle patterns are averaged over a large number of shots (figure 7d) then the transverse spatial structure is removed to yield a smooth profile, consistent with the random independent nature of the modes that create the speckle pattern. It is noticeable that this averaged profile is much broader in the transverse plane than the almost purely singlemode pattern created by a guided BEC (figure 7e), which has a smooth but much narrower gaussian profile. To further test the speckle hypothesis, we measured the second-order correlation function for the guided atoms using the same delay line detector as we employed previously to measure the second- and third-order correlation functions for ultracold He* atoms [9]. Figure 8 shows that when thermal atoms are loaded into the dipole guiding potential, clear atom bunching is detected (the HBT effect), indicating that multimode guiding is occurring and is associated with matter-wave speckle. When a BEC is loaded into the guide, the atom bunching disappears, consistent with propagation of a coherent matter wave in the lowest-order mode of the guide. 236

Figure 7: Same scale: (a-c) Guided thermal atom speckle patterns for 3 data runs. (d) Average over 20 runs yielding a smooth pattern. (e) BEC guided in the lowest order mode, with a gaussian profile. Adapted from ref. 14. Figure 8: Temporal second-order correlation function for a guided multimode thermal cloud. Adapted from ref. 14. Furthermore, we can measure the bunching enhancement of the second order correlations as a function of the average mode occupancy of the guide (figure 9). Our ability to selectively determine the mode number by varying the intensity ramp enables a continuous variation of the average mode occupancy as shown in the figure. As the temperature of the trapped and guided 237

ensemble decreases, so does the mode occupancy, and at the same time the transverse correlation length becomes larger. Since the transverse spatial resolution of the detector is finite (~150 m), the bunching signal is enhanced at lower temperatures (larger de Broglie wavelengths) as the coherence length becomes larger. This is demonstrated in figure 9 where the experimental data match a simple theoretical model [14]. The maximum atom-bunching enhancement (~20%) is the highest value measured thus far. Eventually, when a BEC is formed at low temperatures and the atoms are mostly in the lowest order mode of the guide, the bunching enhancement collapses and the correlation function attains a value of unity. This indicates that the atoms guided in the lowest order mode are coherent, as would be expected when a BEC is coupled into a matching de Broglie waveguide. Such a pure coherent guided mode would be useful for applications which require a highly coherent beam such as atom interferometry. These experiments indicate that measurement of the correlation functions can be a useful diagnostic for determining the coherence properties of guided matter waves for such applications. Figure 9: Graph of peak bunching amplitude as a function of average mode occupancy. A simulation of our experiment yields the theory curve shown as a dashed light blue line, in comparison to our multimode data points shown as blue circles. Also shown is the unity bunching amplitude when a BEC is guided (red point). Adapted from ref. 14. 4 Conclusions and Future Work We have demonstrated in these experiments the correspondence between the quantum statistical properties of matter waves and the transverse mode occupancy and structure (speckle) for atoms guided in an optical potential. The mode occupancy can be controlled in a selective fashion as a function of temperature by varying the intensity ramp parameters for the guiding potential. 238

For the future, at very low transverse temperatures, the transverse coherence length can be made much larger than the detector resolution, enabling large bunching enhancements to be achieved that may approach the theoretical maximum (n!) for the n th -order correlation function. Further, this opens up the possibility of measuring fourth- and fifth-order correlation functions, beyond the second- and third-order correlation measurements presented here. This topic will be the subject of future publications. Acknowledgements This work is supported by the Australian Research Council Centre of Excellence for Quantum- Atom Optics. References [1] R. Hanbury Brown and R.Q. Twiss, Nature 177, 27 (1956). [2] R.J. Glauber, Physical Review 130, 2529 (1963). [3] M. Yasuda and F. Shimizu, Physical Review Letters 77, 3090 (1996). [4] M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, C. I. Westbrook, Science 310, 648 (2005). [5] T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect and C. I. Westbrook, Nature 445, 402 (2007). [6] K. G. H. Baldwin, Contemporary Physics 46, 105 (2005). [7] R. G. Dall and A. G. Truscott, Optics Commun. 270, 255 (2007). [8] A. G. Manning, S. S. Hodgman, R. G. Dall, M. T. Johnsson, and A. G. Truscott, Optics Express 18, 18712 (2010). [9] S.S. Hodgman, R.G. Dall, A.G. Manning, K. G. H. Baldwin, and A.G. Truscott, Science 331, 1046 (2011). [10] J.-F. Riou, W. Guerin, Y. Le Coq, M. Fauquembergue, V. Josse, P. Bouyer, and A. Aspect, Phys. Rev. Lett., 96, 070404 (2006). [11] R. G. Dall, L. J. Byron, A. G. Truscott, G. R. Dennis, M. T. Johnsson, M. Jeppesen and J. J. Hope, Optics Express 15, 17673 (2007). [12] R.G. Dall, S.S. Hodgman, M.T. Johnsson, K. G. H. Baldwin, and A.G. Truscott, Physical Review A 81, 011602(R) (2010). [13] R. G. Dall, S. S. Hodgman, A. G. Manning, and A. G. Truscott, Optics Letters 36, 1131 (2011). [14] R.G. Dall, S.S. Hodgman, A.G. Manning, M.T. Johnsson, K.G.H. Baldwin and A.G. Truscott, Nature Communications 2, article 291 (2011). 239