Vladimir Gritsev and Dionys Baeriswyl, Department of Physics, University of Fribourg, Chemin du musée 3, 1700 Fribourg
Light-matter interaction is around us in nature and has played a tremendous role in the development of current technology. Up to recent years it was sufficient to deal with this interaction on average, with many photons and many atoms involved. However, due to progress in quantum optics, strong interest in light-matter interaction on the scale of single atoms and single photons emerged recently. The main motivation for searching control over light-matter interaction comes from rapid developments in several related fields, including communication, signal processing, ultrafast optics, optomechanical cooling, imaging and spectroscopy and, of course, quantum information. Downscaling to single-atom and/or single-photon levels promotes some traditionally classical research areas into the quantum realm. Subtle quantum effects on a few-body level can lead to unexpected phenomena, especially when triggered by reduced dimensionality, for instance to an effective strong interaction between photons.
The purpose of this article is to give a very brief and certainly incomplete overview of ongoing developments in the quest for the creation and use of optical nonlinearities at the quantum level.
Strong interaction on a single photon level
Photons interact very weakly, which makes it challenging to build all-optical devices in which one light signal controls another. Even in nonlinear optical media, in which two beams can interact because of their effects on the refractive index of the medium, this interaction is weak at low light levels. The coupling of photons to a single atom is also rather weak; a typical dimensionless scale is the fine-structure constant, α = 1/137. This complicates the generation of large optical nonlinearities on a single-photon single-atom level.
For applications in quantum information science, particularly in quantum key distribution, quantum teleportation, and linear optics quantum computation, the single-photon emitter plays an important role. This device emits photons one by one, triggered by optical pulses. Strong interaction on a single-photon level is therefore important for these applications.
Several proposals have been put forward to enhance the effective interaction between photons or between a single emitter and a single photon. Historically, the oldest set-up is based on cavity QED . In this case the weak photon-atom interaction is considerably enhanced through a large number of photon bounces inside a cavity. This requires a large quality factor, which nowadays reaches values of up to Q ~108 in some microcavities . In fact, with a cavity or a microresonator  one can successfully achieve a strong-coupling regime for practically any bosonic mode [4 - 7].
The cavity approach has its own drawbacks, e.g. difficulties with scalability (integration of several cavities into a network) which prohibit the implementation for quantum computers according to the first of DiVincenzo's criteria . Therefore cavity-free approaches have been advocated for achieving strong coupling between light and matter (see Fig. 1). Most of these schemes are based on photons or plasmons tightly confined to quantum wires acting as waveguides. The scattering probability is inversely proportional to the effective cross section of the waveguide. Therefore the tight localization of the fields causes the nanowire to act as an efficient lens that enables strong coupling on a single pass. One of the new methods is based on tapered optical fibers [9, 10] (see Ref.  for a review); it has already been shown to lead to a strong-coupling regime [2, 12]. Another approach uses photonic crystal waveguides . A somewhat different road towards strong nonlinearities at the single-photon level exploits the strong coupling between individual optical emitters (e.g. quantum dots) and propagating surface plasmons (or more precisely plasmon polaritons) confined to a conducting nanowire . Yet another structure consists of a hollow optical fiber stuffed with cold atoms where the effect of electromagnetically induced transparency is exploited to generate optical nonlinearities  (see  for review on quantum optics with hollow-core waveguides). Other approaches have also been proposed recently [17, 18].
Quantum phase transitions with photons
Understanding the fundamental physical phenomena of new materials is an essential basis for improving their functionality and for developing practical applications. Unfortunately, the current understanding of even simple model systems, like the Hubbard model, is limited. In many cases theory cannot be directly applied to actual experiments and therefore the development of new materials is often bound to rely on trial and error. The origin of this uncomfortable situation is the complexity of the many-body problem which restrains the simulation of quantum Hamiltonians with classical computer facilities. This suggests to use one quantum system for simulating another quantum system, an idea originally brought up by Richard Feynman . Spectacular recent advances in two fields may have gone a long way toward the realization of quantum simulators, on the one hand the use of optical lattices and the realization of fermion superfluidity in the field of cold atoms, on the other hand the generation of large optical nonlinearities in the field of quantum optics.
A striking example of such a quantum simulator is a system of interacting neutral bosonic atoms in optical lattices, where the effective tunneling rate can be readily tuned by changing the depth of the periodic potential. These atoms may exhibit a behavior that is directly analogous to the famous Mott transition in fermionic systems, namely a transition from a coherent, superfluid regime at large tunneling rates to an insulating phase for dominant on-site interatomic repulsion.
Besides serving as model systems for quantum phase transitions, engineered lattices offer specific advantages, for instance accessibility to individual sites. To achieve this goal, it was suggested to build a lattice of high-quality optical cavities containing atomic gases . Strong coupling between light and atoms leads to composite atom-photon excitations, known as polaritons. These can tunnel between adjacent cavities. At the same time, the distance between two adjacent cavities can be made much larger than the optical wavelength of the resonant cavity mode, so that individual cavities can be effectively addressed. One has to notice that, although dramatic progress has been achieved in controlling individual atoms and photons in single cavities, the complex architecture of coupled cavities proposed in  represents a considerable experimental challenge.
A very interesting experiment has recently been carried out with an atomic Bose-Einstein condensate coupled to a (single) optical cavity. In this system a Dicke quantum phase transition to a superradiant state has been observed . On the other hand, recent progress in addressability of ultracold atomic system in optical lattice  makes cold atomic system to be a realistic tool for quantum simulators. These two recent developments go towards realization of quantum simulators with hybrid systems based on combined ultracold atoms and quantum-optical setups.
Strongly interacting photons in one dimension
Our experience with quantum many-particle systems teaches us that the combined effects of reduced dimensionality and inter-particle interactions can strongly enhance correlations and eventually lead to a new, highly correlated state. This is what happens with interacting fermions in a one-dimensional geometry when they form a distinctive state, the so-called Luttinger liquid . For interacting bosons in one dimension a particular state, referred to as Tonks-Girardeau state , is realized when the repulsive interaction becomes very strong. In this limit the particles do not want to share the same place neither in coordinate nor in momentum space; they effectively behave like a gas of hard spheres or like a gas of non-interacting fermions. The Tonks-Girardeau state has been observed in systems of ultracold bosonic atoms several years ago .
Photons are bosons and, as already mentioned, their interaction is negligible under normal conditions. But maybe there exist situations where photons, confined to one dimension, experience a strong repulsive interaction and even are fermionized in the limit of very strong coupling? This question has been addressed in Ref. . Using the concept of electromagnetically-induced transparency we have shown that in a system of four-level atoms (see Fig. 2) one can generate a strong repulsive interaction between polaritons, their dynamics being governed by the quantum nonlinear Schrödinger equation (two-level atoms would only lead to an effective attraction). A fermionized regime is reached for sufficiently strong interaction, which is in the range of experimentally accessible parameters. A corresponding readout scheme allows the mapping of a strongly-correlated many-polariton state onto that of outgoing photons. The photonic state can then be analysed by measuring a second-order coherence function, which exhibits both untibunching and spatial variations, reminiscent of Friedel oscillations of the electronic density in solids. We note that fermionization can be induced dynamically, without the prerequisite of thermodynamic equilibrium. Moreover, even dissipation may drive a system into this state [26, 27].
Strongly correlated many-photon states have a great potential for future applications. New exotic surprises may await thanks to the tunability of the effective interaction between photons (polaritons). Further studies in these and related directions are underway , some of them being motivated and supported by the LiMat collaboration .
 A. Imamoglu et.al., Phys. Rev. Lett. 79, 1467 (1997).
 M. Pollinger et al., Phys. Rev. Lett. 103, 053901 (2009).
 T. Aoki et al., Nature 443, 671 (2006).
 J. P. Reithmaier et al., Nature 432, 197 (2004); T. Yoshie et al., Nature 432, 200 (2004).
 A. Wallraff et al., Nature 431, 162 (2004).
 K. Hammerer et al., Phys. Rev. Lett. 103, 063005 (2009).
 F. Brennicke et al., Science 322, 235 (2008).
 D. P. DiVincenzo, arxiv:quant-ph/0002077, published in Fortschritte der Physik 48, 771 (2000).
 V. I. Balykin et al., Phys. Rev. A 70, 011401(R) (2004); K. P. Nayak and K. Hakuta, New J. Phys. 10, 053003 (2008).
 S. Sagué et al., Phys. Rev. Lett. 99, 163602 (2007).
 G. Brambilla, J. Opt. 12, 043001 (2010).
 B. Dayan et al., Science 319, 1062 (2008).
 A. Faraon et al., Appl. Phys. Lett. 90, 073102 (2007).
 A. V. Akimov et al., Nature 450, 402 (2007); D. Chang et al., Phys. Rev. Lett. 97, 053002 (2006).
 M. Bajcsy et al., Phys. Rev. Lett. 102, 203902 (2009).
 H. Schmidt, A. R. Hawkins, Laser and Photon. Rev., 118 (2010).
 G. Wrigge et al., Nature Phys. 4, 60 (2008).
 J. Hwang et al., Nature 460, 76 (2009).
 R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
 M. J. Hartmann et al., Nature Phys. 2, 849 (2006); A. D. Greentree et al., Nature Phys. 2, 856 (2006); D. G. Angelakis et al., Phys. Rev. A 76, 031805(R) (2007); N. Na et al., Phys. Rev. A 77, 031803 (2008).
 W. S. Bakr et al., arXiv:0908.0174.
 K. Baumann et al., arXiv:0912.3261.
 T. Giamarchi, Quantum Physics in One Dimension, Oxford Univ. Press, 2004.
 M. Girardeau, J. Math. Phys. 1, 516 (1960).
 B. Paredes et al., Nature 429, 277 (2004); T. Kinoshita et al., Science 305, 1125 (2004).
 D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. Lukin and E. Demler, Nature Phys. 4, 884 (2008).
 I. Carusotto et al., Phys. Rev. Lett. 103, 033601 (2009).
 M. Hafezi et al., arXiv:0911.4766; V. Gritsev, in preparation.
[Released: August 2010]