Quantum nonlinear optics with single photons enabled by strongly interacting atoms

The realization of strong nonlinear interactions between individual light quanta (photons) is a long-standing goal in optical science and engineering, being of both fundamental and technological significance. In conventional optical materials, the nonlinearity at light powers corresponding to single photons is negligibly weak. Here we demonstrate a medium that is nonlinear at the level of individual quanta, exhibiting strong absorption of photon pairs while remaining transparent to single photons. The quantum nonlinearity is obtained by coherently coupling slowly propagating photons to strongly interacting atomic Rydberg states in a cold, dense atomic gas. Our approach paves the way for quantum-by-quantum control of light fields, including single-photon switching, all-optical deterministic quantum logic and the realization of strongly correlated many-body states of light.

The realization of strong nonlinear interactions between individual light quanta (photons) is a long-standing goal in optical science and engineering 1,2 , being of both fundamental and technological significance. In conventional optical materials, the nonlinearity at light powers corresponding to single photons is negligibly weak. Here we demonstrate a medium that is nonlinear at the level of individual quanta, exhibiting strong absorption of photon pairs while remaining transparent to single photons. The quantum nonlinearity is obtained by coherently coupling slowly propagating photons [3][4][5] to strongly interacting atomic Rydberg states [6][7][8][9][10][11][12] in a cold, dense atomic gas 13,14 . Our approach paves the way for quantum-byquantum control of light fields, including single-photon switching 15 , all-optical deterministic quantum logic 16 and the realization of strongly correlated many-body states of light 17 .
Recently, remarkable advances have been made towards optical systems that are nonlinear at the level of individual photons. The most promising approaches have used high-finesse optical cavities to enhance the atom-photon interaction probability 2,[18][19][20][21] . In contrast, our present method is cavity-free and is based on mapping photons onto atomic states with strong interactions in an extended atomic ensemble 13,15,22,23 . The central idea is illustrated in Fig. 1, where a quantum probe field incident onto a cold atomic gas is coupled to high-lying atomic states (Rydberg levels 24 ) by means of a second, stronger laser field (control field). For a single incident probe photon, the control field induces a spectral transparency window in the otherwise opaque medium via electromagnetically induced transparency (EIT 5 ), and the probe photon travels at much reduced speed in the form of a coupled excitation of light and matter (a Rydberg polariton). However, in stark contrast to conventional EIT, if two probe photons are incident onto the Rydberg medium, the strong interaction between two Rydberg atoms tunes the transition out of resonance, thereby destroying the transparency and leading to absorption 15,22,23,25,26 . The experimental demonstration of an optical material exhibiting strong two-photon attenuation in combination with singlephoton transmission is the central result of this work.
The quantum nonlinearity can be viewed as a photon-photon blockade mechanism that prevents the transmission of any multiphoton state. It arises from the Rydberg excitation blockade 27 , which precludes the simultaneous excitation of two Rydberg atoms that are separated by less than a blockade radius, r b (Fig. 1). During the optical excitation under EIT conditions, an incident single photon is converted into a Rydberg polariton inside the medium. However, owing to the Rydberg blockade, a second polariton cannot travel within a blockade radius from the first one, and EIT is destroyed. Accordingly, if the second photon approaches the single Rydberg polariton, it will be significantly attenuated, provided that r b exceeds the resonant attenuation length of the medium in the absence of EIT, l a~N s a ð Þ {1 , where N is the peak atomic density and s a the absorption cross-section. This simple physical picture implies that, in the regime where the blockade radius exceeds the absorption length (r b >l a ), two photons in a tightly focused beam not only cannot pass through each other 15 , but also cannot propagate close to each other inside the medium (see Fig. 1c and the detailed theoretical analysis below). Using Rydberg states with principal quantum numbers 46 # n # 100, we can realize blockade radii r b between 3 mm and 13 mm, while for our highest atomic densities of N~2|10 12 cm {3 , the attenuation length l a is below 2 mm. The optical medium then acts as a quantum nonlinear absorption filter, converting incident laser light into a non-classical train of single-photon pulses.
EIT nonlinearities at the few-photon level have been previously observed without using strongly interacting atomic states by means of strong transverse confinement of the light 28,29 . The interactions between cold Rydberg atoms have been explored in ensembles 6-10 , and have been used to realize quantum logic gates between two Rydberg atoms 11,12,24 . Enhanced optical nonlinearities using Rydberg EIT 15,22,23 have been observed in pioneering work 13,14 that we are building on. Very recently, the Rydberg blockade in a dense, mesoscopic atomic ensemble has been used to implement a deterministic singlephoton source 30 .
To observe the photon-photon blockade, several key requirements must be fulfilled. First, to eliminate Doppler broadening, the atoms should be cold so that they move by less than an optical wavelength during the microsecond lifetime of the EIT coherence. Second, the atomic cloud should be sufficiently dense such that the blockade condition r b >l a is fulfilled. Last, the system should be one-dimensional, that is, the transverse size of the probe beam should be smaller than the blockade radius r b in order to prevent polaritons from travelling side by side.
We fulfil these conditions by trapping a dense laser-cooled atomic ensemble and focusing the probe beam to a Gaussian waist w 5 4.5 mm , r b (see Methods). We prepare an 87 Rb ensemble containing up to 10 5 atoms in a far-detuned optical dipole trap produced by a Nd:YAG laser operating at 1,064 nm with a total power of 5 W. The trap is formed by two orthogonally polarized beams with waists w t 5 50 mm intersecting at an angle of 32u. The atoms are optically pumped into the hyperfine (F) and magnetic (m F ) sublevel jgae 5 j5S 1/2 , F 5 2, m F 5 2ae in the presence of a 3.6 G magnetic field along the quantization axis, which is defined by the common propagation direction of the probe and control beams along the long axis of the cloud. The probe beam on the jgae R jeae 5 j5P 3/2 , F 5 3, m F 5 3ae transition and the control beam on the jeae R jrae 5 jnS 1/2 , J 5 1/2, m J 5 1/2ae transition with waist w c 5 12.5 mm are oppositely circularly polarized. (Here J and m J denote the quantum numbers for total angular momentum and its component along the quantization axis, respectively.) The resonant optical depth (OD) of the cloud can be as large as OD 5 50, with in-trap radial and axial r.m.s. cloud dimensions of s r 5 10 mm and s ax 5 36 mm, respectively. To avoid inhomogeneous light-shift broadening of the two-photon transition, we turn off the optical dipole trap before turning on the probe and control light, and probe the Rydberg EIT system continuously for up to a few hundred microseconds. The control light is filtered out from the transmitted light, and the photon-photon correlation function g (2) (t) of the probe beam is measured versus the time separation t by means of two photon counters. The slow-light group delay t d through the atomic medium 5 is measured independently in a pulsed experiment, and used to calculate the corresponding minimum group velocity, v g~ffi ffiffiffiffi 2p p s ax t d . Probe transmission spectra are presented in Fig. 2a for large optical depth OD 5 40 and the control laser tuned to the Rydberg state j100S 1/2 ae. At very low incident photon rates R i # 1 ms 21 , the spectrum displays an EIT transparency window with 60% transmission. The transmission is mainly limited by the finite EIT decoherence rate c gr , which for our system is dominated by Doppler broadening and laser linewidth. The extraordinary nonlinearity of the Rydberg EIT medium 13 becomes apparent as the incident photon rate is increased: the probe beam is already strongly attenuated at a photon rate of R i < 4 ms 21 . To demonstrate that we are operating in a quantum nonlinear regime, we show in Fig. 2b the correlation function g (2) (t) of the transmitted probe light, measured at R i 5 1.2 ms 21 . For the most strongly interacting state j100S 1/2 ae with r b 5 13 mm < 5l a < 2.9w we observe strong antibunching with g (2) (0) 5 0.13(2), largely limited by background light. (Here 0.13 (2) indicates 0.13 6 0.02.) Subtraction of the independently measured background coincidence counts yields a corrected g 2 ð Þ c 0 ð Þ~0:04 3 ð Þ. These observations are in sharp contrast to EIT transmission via a less strongly interacting Rydberg state j46S 1/2 ae with r b 5 3 mm, where the photon statistics of the transmitted light are similar to those of the incident coherent state (see Fig. 2b inset). We note that for j100S 1/2 ae the photons are anti-bunched over a length scale v g t < 50 mm that exceeds the blockade radius (see top axis of Fig. 2b), indicating the influence of additional propagation effects beyond the simple picture outlined above.
To investigate the transmission characteristics of multiple photons through the medium, we plot in Fig. 3a the output photon rate R o , scaled by the EIT transmission measured at low probe power, as a function of incident photon rate R i . At first, R o increases linearly with R i as expected, but then saturates abruptly to a constant value of R o 5 1.3(3) ms 21 . Note that these observations deviate from the simplistic model of a multiphoton absorber that transmits only the one-photon component from the incoming coherent state (black dashed line in Fig. 3a). At the same time, the observed output flux corresponds to less than one photon in the medium R {1 o wt d~3 00 ns À Á . Figure 3b shows the saturated output rate versus the ratio r b /w of blockade radius and probe beam waist for a wide range of principal quantum numbers, control field intensities and optical depths. The approximate showing the spatial evolution of the probability distribution associated with two photons (c) and two Rydberg excitations (d) at positions (z 1 , z 2 ) inside the medium, normalized by their values in the absence of blockade. Two Rydberg excitations are excluded from the blockaded range, resulting in the formation of an anti-bunching feature in the light field, whose width increases during the propagation due to the finite EIT transparency width B~c EIT ffiffiffiffiffiffiffiffiffi ffi 8OD p .

RESEARCH LETTER
R o / (w/r b ) 2 scaling, valid for w>r b , indicates that the saturated rate for intermediate to strong interactions, r b >l a , is largely determined by the transverse geometrical constraint, that is, by the extent to which the Rydberg polaritons can propagate side by side through the medium. Two important features of the photon-photon blockade are the degree of two-photon suppression at equal times, g (2) (0), and the associated correlation time, that is, the width t c of the antibunching feature in g (2) (t). As discussed in detail below, the blockade mechanism is most effective if the optical depth per blockade radius, OD b 5 r b /l a , exceeds unity 15 , and if the system is effectively one-dimensional, r b . w. Because the blockade radius 27 increases with the principal quantum number n as r b / n 11/6 , the combination of both effects results in a steep dependence of g (2) (0) upon n. Figure 4a, b shows that g (2) (0) improves with the principal quantum number n of the Rydberg state and the interaction strength r b /l a , resulting in a more than tenfold suppression of the two-photon transmission, limited by independently measured background light on the photon detectors (dotted lines). At the same time, the observed width t c of the g (2) feature considerably exceeds the photon travel time t b 5 r b /v g < 50 ns through the blockade radius (Fig. 4c, d). Close examination (Fig. 4d) reveals that the correlation time is of the same order as, and scales inversely proportionally with, the spectral width 5 B~c EIT ffiffiffiffiffiffiffiffiffi ffi 8OD p of the EIT transparency window. This observation suggests that propagation effects play an important role in establishing the g (2) correlation time t c in a medium of large optical depth. We observe that, under appropriate conditions, two-photon events are suppressed inside the medium on a length scale that approaches the size s ax < 40 mm of the entire atomic ensemble, and on a timescale that approaches the intrinsic coherence time c {1 gr~5 00 ns. To gain further understanding of these observations, we theoretically analyse the photon propagation dynamics in the weak-probe limit where the average number of photons inside the medium is much less than one. In this case, it suffices to consider two polaritons (Fig. 1b). The corresponding two-photon component of the state vector 15 is the photon field operator, and jEE(r 1 , r 2 , t)j 2 is the probability of finding two photons at locations r 1 , r 2 . This probability directly yields the spatially dependent photon-photon correlation function, and, via the group velocity v g , the corresponding temporal correlation function g (2) (t). An intuitive picture emerges if we make the simplification of a tightly focused probe beam (one-dimensional approximation) travelling through a homogeneous medium with perfect linear EIT transmission. In this case, the steady-state two-photon amplitude in the medium obeys (see Supplementary Information): where R 5 (z 1 1 z 2 )/2 and r 5 z 1 2 z 2 are the centre-of-mass and relative coordinates of the two photons, respectively. The function V r ð Þ~r 6 b r 6 b {2ir 6 À Á can be regarded as an effective potential that describes the impact of Rydberg-Rydberg interactions 15 . For large photon-photon distances, r ? r b , the potential V vanishes, and equation (1) yields perfect transmission under EIT, while for distances r=r b , the interaction V modifies the two-photon propagation. According to equation (1), photon correlations emerge from a combination of two processes: the first term acts inside the blockade radius r b and describes absorption with a coefficient l {1 a as the interaction V tunes EIT out of resonance. This would create a sharp dip in the two-photon correlation function with a corresponding correlation time t b 5 r b /v g associated with the blockade radius. However, if the corresponding spectral width *t {1 b exceeds the spectral width B of the EIT transparency window 5 , the second diffusion-like term acts to broaden the absorption dip (Fig. 1c) in space and time, increasing the photon-photon correlation time t c beyond t b towards a value set by the EIT transparency width (Fig. 4d). To maintain strong two-photon suppression (g (2) (0) = 1) in the presence of EIT-induced diffusion, the loss term must exceed the diffusion on the length scale of the blockade radius, requiring r b . l a . Large optical depth OD b 5 r b /l a of the blockaded region is therefore the

LETTER RESEARCH
key experimental feature that allows us to extend the earlier studies 13,14 into the quantum nonlinear regime.
For direct comparisons with our experiments, we solve numerically the full set of propagation equations accounting for the Gaussian density profile of the trapped atomic cloud, the finite waist of the probe beam, and the imperfect single-photon transmission due to finite decoherence c gr of the two-photon transition. As shown in Figs 2 and 4, the theory captures the essential features of our measured correlation functions and, moreover, reproduces their dependence on the Rydberg states, control laser intensities and optical depths of the sample over a wide range of parameters. This detailed theoretical understanding also allows us to analyse the prospects for possible future improvements. These include a reduction of Doppler broadening (through lower atomic temperature or the use of counter-propagating probe and control beams) to increase the linear transmission from 60% towards unity, the excitation of even higher-lying Rydberg states for larger blockade radius, and larger atomic density to further increase the optical depth per blockade radius OD b and overall optical depth OD.
Our observations suggest intriguing prospects for ultimate quantum control of light quanta. For example, by storing a single photon in a Rydberg state and subsequently transmitting a second Rydberg polariton, a single-photon switch can be created 15 . It can be used, for example, for quantum non-demolition measurements of optical photons. At the same time, by using strong interactions in the dispersive regime, the present approach can be used to implement deterministic quantum logic gates 15,16 , which would constitute a major advance towards all-optical quantum information processing 31 . Last, our results may open the door to exploring the quantum dynamics of strongly interacting photonic many-body systems. For example, it may be possible to create a crystalline state of strongly interacting polaritons 17 . Beyond these specific applications, our work demonstrates that unique quantum nonlinear optical materials can be created by combining slow-light propagation with strong atom-atom interactions, an approach which can be potentially extended to realize other material systems with quantum nonlinearities.

METHODS SUMMARY
An ensemble of 6 3 10 6 laser-cooled atoms is captured in a magneto-optical trap (MOT) every 300 ms. The trapped cloud is compressed and loaded into the dipole trap by the combined actions of increasing the magnetic-field gradient to 35 G cm 21 , detuning the MOT trapping frequency by 230 MHz and reducing the MOT repumper intensity to 10 mW cm 22 . The magnetic fields are then rapidly shut off, allowing for 10 ms of molasses cooling to a temperature of 35 mK. The crossed dipole trap holds up to 10 5 atoms at a peak density of 2 3 10 12 atoms cm 23 and a measured optical depth of OD 5 50.
The probe beam is focused to a 1/e 2 waist w 5 4.5 mm by a confocal arrangement of achromatic doublet lenses with focal length 30 mm and diameter 6.25 mm. The control field is co-propagating with the probe beam. The frequencies of both lasers are locked to an optical Fabry-Perot resonator that is stabilized against long-term drifts to a Doppler-free atomic resonance line. The measured short-term linewidths are 120 kHz and 80 kHz for the probe and control laser, respectively. The transmitted control light is separated from the probe light by a combination of interference and absorption filters.
The intensity correlation function of the outgoing probe field is measured with two single-photon detectors. Spurious detection events typically limit g 2 (t) to $0.1. These include dark counts from the detector, imperfect polarization of the probe photons (light with the orthogonal circular polarization is only weakly absorbed by the medium) and residual control light.