Collective Dipole-Dipole Theory
What is collective behaviour?
When light is shone on an atom, if the frequency of the light closely matches the energy difference between two energy levels in the atom, the light can create an electric dipole moment in the atom. This electric dipole radiates more light, which can interfere with the original light to produce a variety of scattering effects, such as reflection and modified refractive index.
If there are more atoms nearby, then the light scattered from the first dipole can create more dipoles in the neighbouring atoms, which also radiate light, resulting in a collection of dipoles all scattering light back and forth between themselves. The overall behaviour of the collection can be very different to the behaviour of just a single atom, and so the atoms are said to be behaving collectively.
The dipole--dipole interaction between two dipoles, due to the scattering of light between them, has some very interesting characteristics. The means the behaviour of these collective ensembles can be incredibly varied:
The aim of this project is to investigate how arranging atoms in different ways affects this collective behaviour, as well as whether it is possible to tailor this behaviour to realise specific useful phenomena.
The simplest system to consider first is where the atoms are arranged neatly in a one-dimensional (1D) array. The regular spacing between the atoms means the eigenmodes (like the normal modes) of the system are neat and seem to display some obvious patterns.
We observed that an infinite chain of dipoles can behave in just the same way as a single dipole between two mirrors. This is because a dipole next to a mirror essentially sees a reflection of itself in the mirror and interacts with this effective image dipole just as if it were interacting with a real dipole placed behind the mirror. A dipole between two mirrors sees an infinite chain of image dipoles (similar to the effect of standing between two mirrors and seeing your reflection repeating endless times off into the distance). We are simply replacing the image dipoles with real dipoles, and seeing the same effects.
Read more at Phys. Rev. A (2016)
The interference between the collective eigenmodes of the atomic ensemble can interfere with each other. This is because, unlike normal modes in a conventional oscillating system, the modes are not orthogonal and so they can overlap with each other. When they overlap, they can produce distinctive asymmetric Fano resonances.
This effect can be so strong however that you can create a transparency window in the atomic resonance, which is similar in appearance to conventional Electromagnetically-Induce Transparency (EIT). EIT is due to the quantum interference between the excitation pathways between different energy levels in a single atom. In Collective EIT, the energy levels are now collective energy levels of the atomic ensemble, rather than different energy levels in a single atom.
We found particularly good examples of collective EIT can occur in exotic lattices known as kagome lattices.
Read more at Phys. Rev. A (2015)
If we put a single atom in the focus of a very tightly focused laser beam, the light scattered by the dipole interferes destructively with the laser light downstream of the atom, meaning the atom is essentially blocking the light from passing it. However, for a single atom this is very difficult because the light must be incredibly tightly focused (to less than a wavelength) or placed in some carefully constructed reflective system such as a giant mirror that bends all the way around it.
We can get round this limitation by replacing the single atom with a two-dimensional (2D) lattice of atoms, separated by a little less than a wavelength. Here, the collective behaviour of the dipoles combined with the Bragg diffraction of the lattice means that close to 100% of the light can be blocked, even for a relatively small number of atoms (around 50 can already produce around 98% extinction).
Read more at Phys. Rev. Lett. (2016)
By modifying the internal energy levels of the atoms, we can realise non-trivial behaviour known as topological insulators. Recognised by the 2016 Nobel Prize, topological insulators are systems which exhibit insulating behaviour in their bulk (in the centre of the material), whilst allowing conduction around the edge (the amount of conduction is typically quantised). We found that we could recreate this same behaviour in 2D honeycomb lattices simply by adding a B-field to the atoms. An advantage of topological insulators is that the conduction around the edges is relatively robust to defects in the lattice, i.e. if some of the lattice sites are missing an atom, or the atoms are slightly off centre from where they should be, the conduction is still confined to the edges of the lattice.
Read more at Phys. Rev. A (2017)
There are a number of different avenues we are interested in exploring related to collective dipolar behaviour. These include:
experimental modeling - in the last few years experiments have begun to probe regimes where we would expect to see the kind of collective effects mentioned earlier. A lot of simulations we have done have been motivated by what would be interested to observe in an experiment, and so now the goal is to use the theories we have constructed to produce qualitative and quantitative modeling of experiments.
dipoles in other systems - the dipolar behaviour mentioned above is not limited just to atoms but can be realised in many different systems, including quantum dots, polar molecules, ions, metamaterials, plasmonic nanostructures, radio antennae... It will be interesting to see whether the effects we have seen in atoms could be realised also in these other systems.
combination with quantum technologies - many of these collective effects could be combined with quantum systems to significantly improve their behaviour. For example, the mirror behaviour of the 2D lattices provides a way of very effectively coupling light with an atomic ensemble. The long lifetimes (subradiance) of some of these collective ensembles could be used to realise very long quantum state storage. Understanding and controlling the energy shifts is also important for frequency measurements in optical lattice clocks.
more exotic ensembles - the variety of effects already observed in different atomic systems suggests there is still a wealth of different behaviour to be discovered!
Find out more
Please feel free to contact Rob Bettles if you are interested in discussing or finding out more about any of these (or other) topics.
Electron Quantum Optics
Quantum Optics in Atomic Vapours
Four-wave mixing (4WM) in the hyperfine Paschen-Back (HPB) regime
In the absence of a magnetic field, multiple-path interference results in a 4WM signal that is complex and difficult to model. In the presence of a high magnetic field (0.6 T), entry into the HPB regime removes the interference and results in a "clean" 4WM signal which is readily modelled with 4-level Optical Bloch equations
Heralded single photons
Through a four-wave mixing (4WM) process in rubidium-87, we can generate heralded bichromatic single photons from an atomic collective spin excitation (CSE). The photon arrival times display collective quantum beats (see image below), a novel interference effect that results from the relative motion of atoms in the CSE.
Rydberg Nonlinear Quantum Optics
2017-10-12: Simon Ball passes PhD viva
Thesis title: "A coherent microwave interface for manipulation of single photons".
2017-07-14: Charles Adams gives invited talk at EGAS 49
Title: "Contactless photon-photon interactions".
2017-07-10: ICOLS 2017: Poster prize
Teodora Ilieva wins poster prize at ICOLS 2017.
2017-07-03: ICOLS 2017: Hot topics talk
PhD student Teodora Ilieva has been selected to present the latest results of the Rydberg Nonlinear Quantum Optics project on "Contactless Nonlinear Optics with Cold Rydberg Atoms" in a hot topics talk at ICOLS 2017.
2017-04-25: Hannes Busche passes PhD viva
Thesis title: "Contactless quantum non-linear optics with cold Rydberg atoms".
S. W. Ball, "A coherent microwave interface for manipulation of single optical photons", PhD Thesis, Durham University (2017).
H. Busche, "Contactless quantum non-linear optics with cold Rydberg atoms", PhD Thesis, Durham University (2017).
H. Busche, P. Huillery, S. W. Ball, T. V. Ilieva, M. P. A. Jones, and C. S. Adams, “Contactless non-linear optics mediated by long-range Rydberg interactions”, Nat. Phys. 13, 655–658 (2017).
H. Busche, S. W. Ball, and P. Huillery, “A high repetition rate experimental setup for quantum non-linear optics with cold Rydberg atoms”, Eur. Phys. J. Spec. Top. 225, 2839–2861 (2016).
O. Firstenberg, C. S. Adams, and S. Hofferberth, “Nonlinear quantum optics mediated by Rydberg interactions”, J. Phys. B: At. Mol. Opt. Phys. 49, 152003 (2016).
The lack of intrinsic interactions between optical photons combined with the ability to control the propagation of photons using optics makes them ideal carriers of information. At the same time, the lack of interactions makes processing of the encoded information at the level of individual quanta difficult. In conventional nonlinear optics, nonlinearities become apparent only at very high intensities. The Rydberg Nonlinear Quantum Optics project focusses on the creation of strong optical nonlinearities and effective interactions at the level of individual photons by interfacing optical photons with ultracold Rydberg atoms [1-3] confined in magneto-optical (MOT) and optical dipole traps, which exhibit strong dipolar interactions over distances of many micrometers.
To achieve a coherent mapping of the strong interactions between collective atomic Rydberg excitations  as well as the resulting effects such as dipole (or Rydberg) blockade , which permits only a single Rydberg excitation within a region of a few micrometers, onto optical photons, we are using quantum optical techniques such as electromagnetically induced transparency (EIT) and photon storage [6,7]. A brief introduction to the underlying concepts of Rydberg Nonlinear Quantum Optics can be found on our background pages.
We are interested in exploiting the mapping and the resulting nonlinearities for applications in quantum optics, quantum simulation, optical quantum information processing, and interfacing of optical photons with microwave fields. At the same time, we are also investigating the fundamental physics and collective behaviour of strongly interacting atomic dipoles ranging from the optical to the microwave regime.
Following the first experimental demonstration of a giant optical nonlinearity here at Durham  and the subsequent generation of highly non-classical states of light [9-11], Rydberg Quantum Nonlinear Optics is flourishing, and a variety of single photon devices have been realised . One of the most recent highlights of our work is the demonstration of a "contactless" interaction between photons  that are stored as collective Rydberg excitations in separate cold atom clouds and propagate in non-overlapping optical media. The interaction occurs over distances of more than 10 μm, well above the optical diffration limit. You can find out more on our results pages.
An overview of our experimental setup , which provides optical resolution on the order of 1 μm thanks to the incorporation of in-vacuo aspheric lenses and allows to perform experiments at effective repetition rates of about 150 kHz to acquire large datasets for the analysis of photon statistics, can be found here.
* indicates work by our group
* J. D. Pritchard, K. J. Weatherill, and C. S. Adams, “Nonlinear optics using cold Rydberg atoms”, Ann. Rev. Cold At. Mol. 1, 301–350 (2013).
* O. Firstenberg, C. S. Adams, and S. Hofferberth, “Nonlinear quantum optics mediated by Rydberg interactions”, J. Phys. B: At. Mol. Opt. Phys. 49, 152003 (2016).
 C. Murray and T. Pohl, “Quantum and Nonlinear Optics in Strongly Interacting Atomic Ensembles”, Adv. At. Mol. Opt. Phys. 65, 321–372 (2016).
 M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms”, Rev. Mod. Phys. 82, 2313–2363 (2010).
 M. D. Lukin, M. Fleischhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole Blockade and Quantum Information Processing in Mesoscopic Atomic Ensembles”, Phys. Rev. Lett. 87, 037901 (2001).
 M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media”, Rev. Mod. Phys. 77, 633–673 (2005).
 M. Fleischhauer and M. D. Lukin, “Dark-State Polaritons in Electromagnetically Induced Transparency”, Phys. Rev. Lett. 84, 5094–5097 (2000).
* J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble”, Phys. Rev. Lett. 105, 193603 (2010).
 Y. O. Dudin and A. Kuzmich, “Strongly Interacting Rydberg Excitations of a Cold Atomic Gas”, Science 336, 887–889 (2012).
 T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. VuletiÄ‡, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms”, Nature 448, 57–60 (2012).
* D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons”, Phys. Rev. Lett. 110, 103001 (2013).
* H. Busche, P. Huillery, S. W. Ball, T. V. Ilieva, M. P. A. Jones, and C. S. Adams, “Contactless non-linear optics mediated by long-range Rydberg interactions”, Nat. Phys. 13, 655–658 (2017).
* H. Busche, S. W. Ball, and P. Huillery, “A high repetition rate experimental setup for quantum non-linear optics with cold Rydberg atoms”, Eur. Phys. J. Spec. Top. 225, 2839–2861 (2016).
We gratefully acknowledge funding from the following sources.
European Union (FP7 and Horizon 2020 programmes)
Engineering and Physical Research Council (EPSRC) grants
Defence Science and Technology Laboratory (DSTL)