LETTERdoi:10.1038/nature12512

Attractive photons in a quantum nonlinear mediumOfer Firstenberg1*, Thibault Peyronel2*, Qi-Yu Liang2, Alexey V. Gorshkov3{, Mikhail D. Lukin1 & Vladan Vuletic2

The fundamental properties of light derive from its constituentparticles—massless quanta (photons) that do not interact with oneanother1. However, it has long been known that the realization ofcoherent interactions between individual photons, akin to thoseassociated with conventional massive particles, could enable a widevariety of novel scientific and engineering applications2,3. Here wedemonstrate a quantum nonlinear medium inside which individualphotons travel as massive particles with strong mutual attraction,such that the propagation of photon pairs is dominated by a two-photon bound state4–7. We achieve this through dispersive couplingof light to strongly interacting atoms in highly excited Rydberg states.We measure the dynamical evolution of the two-photon wave-function using time-resolved quantum state tomography, and dem-onstrate a conditional phase shift8 exceeding one radian, resulting inpolarization-entangled photon pairs. Particular applications of thistechnique include all-optical switching, deterministic photonicquantum logic and the generation of strongly correlated states oflight9.

Interactions between individual photons are being explored in cavityquantum electrodynamics, where a single, confined electromagneticmode is coupled to an atomic system10–12. Our approach is to couple alight field propagating in a dispersive medium to highly excited atomicstates with strong mutual interactions13,14 (Rydberg states). Similar toprevious studies of quantum nonlinearities involving Rydberg states thatwere based on dissipation15–19 rather than dispersion20, we make use ofelectromagnetically induced transparency (EIT) to slow down the propa-gation of light21 in a cold atomic gas. By operating in a dispersive regimeaway from the intermediate atomic resonance (Fig. 1b), where atomicabsorption is low and only weakly nonlinear22, we realize a situation inwhich Rydberg-atom-mediated coherent interactions between indi-vidual photons dominate the propagation dynamics of weak light pulses.Previous theoretical studies have proposed various scenarios for induc-ing strong interactions between individual photons2,3,23 and for creatingbound states of a few quanta4,5,7,24, a feature generic to strongly inter-acting quantum field theories. The main result reported here is theexperimental realization of a photonic system with strong attractive inter-actions, including evidence for a predicted two-photon bound state.

Our experiment (outlined in Fig. 1a) makes use of an ultracoldrubidium gas loaded into a dipole trap, as described previously19. Theprobe light of interest iss1-polarized, coupling the ground state, jgæ, tothe Rydberg state, jræ, via an intermediate state, jeæ, of linewidthC/2p5 6.1 MHz by means of a control field that is detuned by D belowthe resonance frequency of the upper transition, jeæ R jræ (Fig. 1b).Under these conditions, for a very weak probe field with mean incidentphoton rate Ri 5 0.5ms21, EIT is established when the probe detuningmatches that of the control field (see Fig. 1c, which shows the probetransmission and phase shift). However, the Rydberg medium is extre-mely nonlinear: a probe photon rate of Ri 5 5ms21 saturates the mediumas a result of the Rydberg blockade25, yielding a probe spectrum close tothe bare two-level response. Given the measured system bandwidth ofabout 5ms21, this implies a substantial nonlinear response with aver-age pulse energies corresponding to less than one photon per inverse

bandwidth. We perform our experiments on the two-photon resonancejgæ R jræ, where, for jDj. C, the transmission is approximately inde-pendent of the probe photon rate for our experimental parameters,yielding a purely dispersive nonlinearity. The linear dispersion at thisresonance corresponds to a reduced probe group velocity of typicallyvg 5 400 m s21, and the group velocity dispersion endows the photonswith an effective mass26 of m < 1,000Bv/c2, where v is the opticalfrequency, B is Planck’s constant divided by 2p and c is the speed oflight in vacuum.

To explore the quantum dynamics in the propagation of photonpairs, we measure time-dependent two-photon correlation functionsof the transmitted light (Fig. 1a). To determine both the amplitude andthe phase of the s1-polarized probe field, we prepare input coherentlight in a linearly polarized state, Vj i~ szj iz s{j ið Þ

� ffiffiffi2p

, where thes2-polarized component (which is approximately non-interactingowing to the 15-fold-smaller transition strength) serves as a phasereference. To analyse the properties of photon pairs, we measure two-photon correlation functions, g 2ð Þ

ab , in different polarization bases, a and b(Supplementary Fig. 3). The component g 2ð Þ

zz directly gives the prob-ability density of the s1-polarized interacting photon pairs. Figure 1dshows g 2ð Þ

zz, for a control detuning of D/2p5 14 MHz, as a function ofthe time separation, t 5 t1 2 t2, between the photons detected at timest1 and t2, converted into a relative distance, r, in the medium using thegroup velocity, vg. A prominent feature is the cusp at r 5 vgt 5 0, whichis characteristic of a predicted two-photon bound state5,7, as discussedbelow.

The measured g 2ð Þab allow us to reconstruct the two-photon density

matrix, r, using quantum state tomography and maximum-likelihoodestimation27,28. From r, we define an interaction matrix, ~r, by factoringout the linear response, such that ~r directly quantifies the nonlinearity(Methods). The density matrix approach is necessary to account fordecoherence and technical imperfections. The probability density oftwo interactings1 photons, g 2ð Þ

zz~~rzz,zz, and the nonlinear phase,w~arg ~rzz,{{

� �, acquired by the s1s1 pair relative to a non-

interacting s2s2 pair, are shown in Fig. 2a, b for D/2p5 14 MHz.The time dependence allows us to extract the nonlinear phase as afunction of the photon–photon separation. Clearly visible is the bunch-ing of photons, that is, an increased probability for photons to exit themedium simultaneously (t1 < t2), and a substantial nonlinear two-photon phase shift of 20.5 rad in that region. Figure 2c shows theintensity correlation in the dissipation-dominated antibunching regime19

at D 5 0 and in the dispersive regime at jDj. C, where there is bunch-ing. Figure 2d displays the nonlinear phase for two different detunings.The transition from the dissipative regime to the dispersive is summa-rized in Fig. 3a, b. In the dispersive regime, the nonlinear phase shift,w(t 5 0), can reach (20.32 6 0.02)p, at a detuning D/2p5 9 MHz anda linear transmission of order 50%. The observed signals, particularlyw, are asymmetrical under a sign change of D.

The origin of the quantum nonlinearity underlying these observa-tions is explained by the following simple model. The repulsive van derWaals interaction between two Rydberg atoms, V(r) 5 BC6/r6, tunes

*These authors contributed equally to this work.

1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. 2Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge,Massachusetts02139, USA. 3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA. {Present address: Joint Quantum Institute, NIST/Universityof Maryland, College Park, Maryland 20742, USA.

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–1

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ψ

(σ–)

Figure 1 | Photons with strong mutual attraction in a quantum nonlinearmedium. a, b, A linearly polarized weak laser beam near the transition| gæ R | eæ at 780 nm is sent into a cold rubidium gas driven by a control lasernear the transition | eæ R | ræ at 479 nm. Strong nonlinear interactions betweens1-polarized photons are detected using photon–photon correlation functionsof the transmitted light for a set of different polarization bases, as determined bya quarter-wave plate (QWP), a half-wave plate (HWP) and a polarizing beamsplitter (PBS). Here s2 photons serve as a phase reference. c, Transmissionspectra (top) and phase shift (bottom) fors1 photons with an incoming rate ofRi 5 0.5ms21 (blue squares) or Ri 5 5ms21 (green circles), for a control fieldred-detuned by D/2p5 15 MHz. The blue line shows the theoretical spectrum.The spectrum at high probe rate approaches that of the undriven two-level

system (dashed grey; see also Supplementary Fig. 2). The solid vertical linecorresponds to the EIT resonance. d, Photon bunching and two-photon boundstate. Theoretically predicted photon–photon correlation function in theSchrodinger equation approximation (top, blue line) for D/2p5 14 MHz, witha potential well of width 2rB (bottom, green line). The bound state (bottom, red)and the superposition of scattering states (bottom, black) form the initialwavefunction, y 5 1 (bottom, dashed blue). The two-photon bound stateresults in the observed bunching in the correlation function, g 2ð Þ

zz< yj j2 (top,grey circles), where time has been converted into distance using the groupvelocity, vg. The boundary effects resulting from the finite extent of the atomcloud become important for | r | $ 5rB.

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Figure 2 | Propagation ofinteracting photon pairs.a, b, Measured second-ordercorrelation function (a) andnonlinear phase shift (b) ofinteracting photon pairs at D 5 2.3C.The photons are detected at times t1

and t2. c, Second-order correlationfunction displayed as a function ofthe time difference, | t | 5 | t1 2 t2 | ,between the photons, showing thetransition from antibunching onresonance (D 5 0, green) tobunching at large detuning(D 5 2.3C, blue). Points areexperimental data; lines are fullnumerical simulations. All g 2ð Þ

zz

measurements are rescaled by theirvalue at t . 1.5ms (SupplementaryInformation). d, Nonlinear phaseshift versus | t | for two differentdetunings (D 5 1.5C, purple, andD 5 2.3C, blue). The 1 s.d. error is630 mrad, dominated by photonshot-noise.

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the doubly excited Rydberg state far off EIT resonance for distancesjrj, rB, where rB~

ffiffiffiffiffiffiffiffiffiffiC6=c6

pis the Rydberg blockade radius14,29,30, C6 is

the van der Waals coefficient, c~V2c

�4Dj j is the EIT linewidth at

detuning Dj j?C and Vc is the Rabi frequency of the control field.Although the phase shift that would originate from the bare jgæ R jeæprobe transition is suppressed by EIT for photons with large separationin the medium (jrj. rB), the light acquires this phase shift for smallphoton separations (jrj# rB) (Fig. 1c). This explicit dependence ofthe refractive index on photon–photon separation can be modelledin one dimension as a potential well with a characteristic width of2rB. Qualitatively, a substantial two-photon phase shift arises for(rB=la)(C=jDj)>1, where la is the resonant attenuation length in themedium, that is, for sufficiently high atomic density. Furthermore, theprobe field must also be compressed in the transverse direction to awaist size w , rB to ensure that interactions occur. Using the Rydbergstate 100S1/2, and for Vc/2p5 10 MHz, we obtain rB < 18mm atdetunings of a few C, la 5 4mm at the peak density, and w 5 4.5mm,fulfilling the conditions for strong interactions for Dj j=5C .

The propagation of s1-polarized photon pairs in such a mediumcan be understood by first considering an idealized situation with nodecoherence between the Rydberg state and the ground state. Then thesteady state in a one-dimensional homogenous medium can be describedby a two-photon wavefunction, y(z1, z2), whose evolution is approxi-mately governed by a simple equation19 in terms of the centre-of-masscoordinate, R 5 (z1 1 z2)/2, and the relative coordinate, r 5 z1 2 z2:

iLy

LR~4la iz

2D

C{V rð ÞV

2c

C2

� �L2y

Lr2zV rð Þ

lay ð1Þ

Here the effective potential, V rð Þ~ iz2(D=C)(1z2r6�

r6B)

� �{1,

approaches (i 1 2D/C)21 inside the blockaded volume (jrj, rB) andapproaches zero outside. The solution relates approximately to ourmeasurements in the time domain for small jtj via

y R~L,r~vgt�

<ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig 2ð Þzz tð Þ

qeiw tð Þ (see Supplementary Information

for the exact relation). Far off resonance ( Dj j?C , Vc), equation (1)corresponds to a Schrodinger equation with R playing the part ofeffective time. The photons’ effective mass, m / 2C/16laD, can bepositive or negative depending on the sign of the detuning, D.Because the sign of the potential also changes with D (potential wellfor D , 0; barrier for D . 0), the effective force (F in Fig. 1a) in bothcases is attractive and the resulting dynamics similar (SupplementaryInformation). However, the potential for D , 0 also has additionalfeatures near the edges of the well, corresponding to a Raman res-onance jgæ R jræ for the interaction-shifted Rydberg state at someinteratomic distance near jrj5 rB. These features are probablyresponsible for the deviation from symmetry (or antisymmetry) underthe change of the sign of D displayed in Fig. 3a, b.

In the experimentally relevant regime, the effective potential sup-ports only one bound-state, yB(r) (Fig. 1d). The initial wavefunction,y(R 5 0, r) 5 1, is a superposition of yB(r) and the continuum ofscattering states. The accumulation of probability near r 5 0 can thenbe understood as arising from the interference between the bound andscattering states that evolve at different frequencies, and the observedbunching feature in g 2ð Þ

zz reveals the wavefunction of the two-photonbound state (Supplementary Information). As shown in Fig. 3a, b,the solution of the Schrodinger-like equation (1) with a simplifiedd-function potential captures the essential features of the nonlineartwo-photon propagation. Additional experimental evidence for thebound-state dynamics is obtained by tuning the probe field relativeto the EIT resonance, thereby varying the strength of the two-photoninteraction potential. As the probe detuning approaches the Ramanresonance, the difference in refractive indices inside and outside theblockade radius increases and the potential deepens (SupplementaryInformation and Fig. 1c). Consequently, the bound state becomesmore localized and the bunching, quantified by g 2ð Þ

zz 0ð Þ, is enhanced(Fig. 3c, d). We note that the size of the two-photon bound state and,correspondingly, the width of the bunching feature, 2tbvg < 70mm,exceed the width of the potential well, 2rB < 35mm, as expected for apotential with one weakly bound state.

Figures 2 and 3 also show the results of our full theoretical model, inwhich we numerically solve the set of propagation equations for thelight field and atomic coherences. The model incorporates the longitu-dinal atomic density distribution and the decoherence of the Rydbergstate (Supplementary Information). These simulations are in goodagreement with our experimental results and the predictions of thesimplified model (equation (1)), confirming that the evolution of thetwo-photon wave packet is dominated by the attractive force betweenthe photons.

Finally, we study the quantum coherence and polarization propertiesof the transmitted photon pairs. In Fig. 4a, we compare the purity ofthe two-photon density matrix, r(t), which includes photon interac-tions, with the purity of the product of one-photon matrices,r(1)

fl r(1), for non-interacting photons. At large photon separation,t, the purity, P(t), of the two-photon density matrix is dominated by

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Figure 3 | Dependence of the photon–photon interaction on detuning.a, b, Equal-time two-photon correlation, g 2ð Þ

zz 0ð Þ (a), and nonlinear phase, w(0)(b), versus detuning, D, from the intermediate state, | eæ. Blue lines are fulltheoretical simulations and black lines are the result of the Schrodingerequation approximation, assuming a simplified d-function potential. Verticalerror bars represent 1 s.d. and horizontal error bars are 60.5 3 2pMHz.c, d, Equal-time correlation function (c) and spatial extent of the bunchingfeature (d) versus Raman detuning, d, from the EIT resonance | gæ R | ræ forD 5 3C, showing increased photon–photon attraction due to a deeper potentialnear Raman resonance. The characteristic bunching timescale, tb, is the half-width of the cusp feature of g 2ð Þ

zz, defined at half-height between the peak valueat t 5 0 and the local minimum closest to t 5 0. Error bars, 1 s.d. Thetheoretical model (solid line) breaks down close to the Raman resonance atd 5 1.3 3 2pMHz < V2

c /4D, where the single-photon component of the probefield is strongly absorbed.

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the one-photon decoherence due to partial depolarization of the trans-mitted light (Supplementary Information). This depolarization isattributed to the difference in group delay, td, between thes1 photonsand the fasters2 photons (tsz

d {ts{

d ~280 ns), which is not negligiblecompared with the coherence time of the probe laser (650 ns). At thesame time, s1 photons bound to each other travel faster and are morerobust against this decoherence mechanism, as evidenced by the greaterpurity at small t. Even in the presence of this depolarization, the coher-ent nonlinear interaction in the dispersive medium produces entan-glement in the outgoing polarization state of two photons. We quantifythe degree of polarization entanglement in terms of a time-dependentconcurrence, C(t) (Fig. 4b and Supplementary Information). Theobtained value C(0) 5 0.09 6 0.03 indicates deterministic entangle-ment of previously independent photons on passage through thequantum nonlinear medium. The measured value is in reasonableagreement with the theoretical prediction, Cth(0) 5 0.13, calculatedfor a conditional phase w(0) 5p/4, purity P(0) 5 0.73 and 50% s1

linear transmission.In our experiment, the transmission and achievable nonlinear phase

are limited by the laser linewidth, the atomic motion and the availablecontrol-field intensity. These technical limitations can be circumventedby using stronger control lasers with improved frequency stability andcolder atomic clouds trapped in both the ground state and the Rydbergstate. Although in our present system the nonlinear phase would not beuniformly acquired by a bandwidth-limited two-photon pulse, a high-fidelity two-photon phase gate may be achievable using, for example, acounter-propagating geometry and greater optical depths14.

The realization of coherent, dispersive photon–photon interactionsopens several new research directions. These include the exploration of anovel quantum matter composed from strongly interacting, massivephotons9. Measurements of higher-order correlation functions may givedirect experimental access to quantum solitons composed of a few inter-acting bosons24, or to the detection of crystalline states of a photonic gas9.By colliding two counter-propagating photons, it may be possible toimprint a spatially homogeneous phase shift of p on the photon pair,corresponding to a deterministic quantum gate14 for scalable opticalquantum computation13. Finally, by accessing other Rydberg statesvia, for example, microwave transitions, it may become possible to con-trol the state of multiphoton pulses with just one quantum of light,thereby realizing a single-photon transistor6,31 for applications in quan-tum networks, and the creation of multiphoton entangled states.

METHODS SUMMARYThe experimental setup is detailed in ref. 19. The average resonant opticaldepth along the atomic cloud is 22, and the peak atomic density is 1012 cm23.

Probe pulses at an average photon rate of 1.6ms21 are sent into the cell duringthe 5.5-ms dark-time periods of a modulated optical trap. For the quantum statetomography, we measure the photon coincidence rates in six polarization bases,{q, h} 5 {p/4,p/4}, {0, 0}, {p/8,p/8}, {0,p/16}, {p/8,p/16} and {p/8, 0}, where q andh are the angles of the quarter- and half-wave plates. The duration of the coincidencetime bins, varying between 20 and 80 ns, is chosen to capture the temporal dynamicsof the correlation functions with reasonable signal-to-noise ratio. For each (t1, t2)time bin, we numerically optimize a Hermitian, positive-semidefinite two-photondensity matrix, r(t1, t2), and one-photon density matrix, r(1)(t). To extract thenonlinear phase from r(t1, t2), we rescale for the linear dispersion and loss effectsby defining the interaction matrix ~ri,j t1,t2ð Þ~ri,j t1,t2ð Þ

�r 1ð Þ t1ð Þ6r 1ð Þ t2ð Þ� �

i,j inthe circular-polarization basis. The interaction matrix generalizes the standard g 2ð Þ

a,b

definition to account for nonlinear phases and decoherence, and all its elementsare equal to 1 in the absence of nonlinearity.

Online Content Any additional Methods, Extended Data display items and SourceData are available in the online version of the paper; references unique to thesesections appear only in the online paper.

Received 28 May; accepted 29 July 2013.

Published online 25 September 2013.

1. Scully, M. O. & Zubairy, M. S. Quantum Optics (Cambridge Univ. Press, 1997).2. Milburn, G. J. Quantum optical Fredkin gate. Phys. Rev. Lett. 62, 2124–2127

(1989).3. Imamoglu, A. Schmidt, H. Woods, G. & Deutsch, M. Strongly interacting photons in

a nonlinear cavity. Phys. Rev. Lett. 79, 1467–1470 (1997).4. Deutsch, I. H., Chiao, R. Y. & Garrison, J. C. Diphotons in a nonlinear Fabry-Perot

resonator:bound states of interacting photons inanoptical ‘‘quantumwire’’. Phys.Rev. Lett. 69, 3627–3630 (1992).

5. Shen, J.-T. & Fan, S. Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system. Phys. Rev. Lett. 98, 153003(2007).

6. Chang, D. E., Srensen, A. S., Demler, E. A. & Lukin, M. D. A single-photon transistorusing nanoscale surface plasmons. Nature Phys. 3, 807–812 (2007).

7. Cheng, Z. & Kurizki, G. Optical ‘‘multiexcitons’’: quantum gap solitons in nonlinearBragg reflectors. Phys. Rev. Lett. 75, 3430–3433 (1995).

8. Turchette, Q. A., Hood, C., Lange, W., Mabuchi, H. & Kimble, H. Measurementof conditional phase shifts for quantum logic. Phys. Rev. Lett. 75, 4710–4713(1995).

9. Chang, D. E. et al. Crystallization of strongly interacting photons in a nonlinearoptical fibre. Nature Phys. 4, 884–889 (2008).

10. Fushman, I. et al. Controlled phase shifts with a single quantum dot. Science 320,769–772 (2008).

11. Rauschenbeutel, A. et al. Coherent operation of a tunable quantum phase gate incavity QED. Phys. Rev. Lett. 83, 5166–5169 (1999).

12. Kirchmair, G. et al. Observation of quantum state collapse and revival due to thesingle-photon Kerr effect. Nature 495, 205–209 (2013).

13. Saffman, M., Walker, T. G. & Mølmer, K. Quantum information with Rydberg atoms.Rev. Mod. Phys. 82, 2313–2363 (2010).

14. Gorshkov, A. V., Otterbach, J., Fleischhauer, M., Pohl, T. & Lukin, M. D.Photon-photon interactions via Rydberg blockade. Phys. Rev. Lett. 107, 133602(2011).

15. Pritchard, J. D. et al. Cooperative atom-light interaction in a blockaded Rydbergensemble. Phys. Rev. Lett. 105, 193603 (2010).

16. Maxwell, D. et al. Storage and control of optical photons using Rydberg polaritons.Phys. Rev. Lett. 110, 103001 (2013).

17. Dudin, Y. O. & Kuzmich, A. Strongly interacting Rydberg excitations of a coldatomic gas. Science 336, 887–889 (2012).

18. Petrosyan, D., Otterbach, J. & Fleischhauer, M. Electromagneticallyinduced transparency with Rydberg atoms. Phys. Rev. Lett. 107, 213601(2011).

19. Peyronel, T. et al. Quantum nonlinear optics with single photons enabled bystrongly interacting atoms. Nature 488, 57–60 (2012).

20. Parigi, V. et al. Observation and measurement of interaction-induced dispersiveoptical nonlinearities in an ensemble of cold Rydberg atoms. Phys. Rev. Lett. 109,233602 (2012).

21. Kasapi, A., Jain, M., Yin, G. Y. & Harris, S. E. Electromagnetically inducedtransparency: propagation dynamics. Phys. Rev. Lett. 74, 2447–2450 (1995).

22. Venkataraman, V., Saha, K. & Gaeta, A. L. Phase modulation at the few-photon levelfor weak-nonlinearity-based quantum computing. Nature Photon. 7, 138–141(2012).

23. Rajapakse, R. M., Bragdon, T., Rey, A. M., Calarco, T. & Yelin, S. F. Single-photonnonlinearities using arrays of cold polar molecules. Phys. Rev. A 80, 013810(2009).

24. Drummond, P. D. & He, H. Optical mesons. Phys. Rev. A 56, R1107–R1109(1997).

25. Lukin, M. D. et al. Dipole blockade and quantum information processing inmesoscopic atomic ensembles. Phys. Rev. Lett. 87, 037901 (2001).

26. Fleischhauer, M., Imamoglu, A. & Marangos, J. P. Electromagneticallyinduced transparency: optics in coherent media. Rev. Mod. Phys. 77, 633–673(2005).

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P(τ

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0.1

C(τ

)|τ| (μs)

a b

Figure 4 | Quantum coherence and entanglement. a, Purity,P(t) 5 Tr[r(t)2], of the measured two-photon density matrix, r, for D 5 2.3C(blue symbols), which at large photon separation approaches the purityexpected from the measured one-photon density matrix, Tr[(r(1)

fl r(1))2](dotted black line). Interacting s1s1 photon pairs near t 5 0 exhibit lowerdecoherence. Error bars (1 s.d.) are derived from the uncertainty in the densitymatrix due to detection shot-noise. b, Concurrence, C(t), calculated from r,indicating polarization entanglement of proximal photons on transmissionthrough the quantum nonlinear medium.

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27. James, D. F. V., Kwiat, P. G., Munro, W. J.& White, A. G. Measurementof qubits.Phys.Rev. A 64, 052312 (2001).

28. Adamson, R. B. A., Shalm, L. K., Mitchell, M. W. & Steinberg, A. M. Multiparticle statetomography: hidden differences. Phys. Rev. Lett. 98, 043601 (2007).

29. Sevinçli, S., Henkel, N., Ates, C. & Pohl, T. Nonlocal nonlinear optics in cold Rydberggases. Phys. Rev. Lett. 107, 153001 (2011).

30. Heidemann, R. et al. Evidence for coherent collective Rydberg excitation in thestrong blockade regime. Phys. Rev. Lett. 99, 163601 (2007).

31. Chen, W, et al. All-optical switchand transistor gated by one stored photon. Science341, 768–770 (2013).

Supplementary Information is available in the online version of the paper.

Acknowledgements We thank H. P. Buchler, T. Pohl, J. Otterbach, P. Strack, M. Gullansand S. Choi for discussions. This work was supported by the NSF, the CUA, DARPA and

the AFOSR Quantum Memories MURI and the Packard Foundations. O.F.acknowledges support from the HQOC. A.V.G. and M.D.L. thank KITP for hospitality.A.V.G. acknowledges funding from the Lee A. DuBridge Foundation and the IQIM, anNSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation.

Author Contributions The experiment and analysis were carried out by O.F., T.P. andQ.-Y.L. The theoretical modelling was done by A.V.G. All experimental and theoreticalwork was supervised by M.D.L. and V.V. All authors discussed the results andcontributed to the manuscript.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of the paper. Correspondenceand requests for materials should be addressed to M.D.L. (lukin@fas.harvard.edu) orV.V. (vuletic@mit.edu).

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METHODS

The experimental setup is detailed in ref. 19, with the following modifications. Thedipole trap is periodically switched off with a 5.5-ms half-period, and the measure-ments are performed during the dark time to avoid inhomogeneous broadening.Photons detected in the first 1.5ms after the dipole trap is turned off are notincluded in the analysis, to guarantee steady-state EIT. For each experimentalcycle, data are accumulated over 400 periods of the dipole-trap modulation. Thetrapped atomic cloud has a longitudinal root mean squared length of sax 5 36mmand a peak atomic density of r0 5 1012 cm23. The average resonant optical depthis 22, with less than 20% variation over the measurement time. The probe andcontrol beams are counter-propagating to reduce the residual Doppler broadeningto 50 3 2p kHz. Linearly polarized probe laser light enters the medium at anaverage photon rate of 1.6ms21. Quarter- and half-wave plates at angles q andh, respectively, followed by a polarizing beam splitter, project the outgoing probelight onto a chosen polarization basis (Fig. 1a). Four single-photon countingmodules measure the pair correlation events at times t1 and t2. Normalized sec-ond-order correlation functions, g 2ð Þ

ab , are calculated using the photon coincidencecounts between the different detectors and the average count rates. The time bins(80 ns for g 2ð Þ

ab t1,t2ð Þ and 20 or 40 ns for g 2ð Þab tð Þ) were chosen to capture the temporal

dynamics of the correlation functions with reasonable signal-to-noise ratio.In the quantum state tomography, we numerically optimize a Hermitian, positive-

semidefinite two-photon density matrix

r~

rzz,zz rzz,S rzz,{{ 0

rS,zz rS,S rS,{{ 0

r{{,zz r{{,S r{{,{{ 0

0 0 0 rA,A

0BBB@

1CCCA

in the two-qubit basis sz1 sz

2j i, Sj i, s{1 s{

2

�, Aj i

� , where Sj i~ sz

1 s{2

�z

�s{

1 sz2

�Þ� ffiffiffi

2p

and Aj i~ sz1 s{

2

�{ s{

1 sz2

�� � ffiffiffi2p

. Because the two photonshave the same frequency and spatial mode, there is no coherence between the3 3 3 symmetric and 1 3 1 antisymmetric subspaces28. We measure in six requiredpolarization bases, chosen as {q, h} 5 {p/4,p/4}, {0, 0}, {p/8,p/8}, {0,p/16}, {p/8,p/16}and {p/8, 0}, to set the ten degrees of freedom in r(t1, t2). The optimization followsthe maximum-likelihood estimate27, where all coincidence measurements are con-sidered. The one-photon density matrix, r(1)(t), is reconstructed using the sametechnique. To extract the nonlinear phase from r(t1, t2), we rescale for the linear

dispersion and loss effects by defining the interaction matrix ~ri,j t1,t2ð Þ~ri,j t1,t2ð Þ.

r 1ð Þ t1ð Þ6r 1ð Þ t2ð Þ� �

i,j in the basis sz1 sz

2j i, sz1 s{

2

�, s{

1 sz2

�, s{

1 s{2

�� . The

interaction matrix generalizes the standard definition of g(2) to account for non-linear phases and decoherence, and all its elements are equal to 1 in the absence ofnonlinearity. In Supplementary Fig. 3, we compare the measured photon–photoncorrelation functions with those calculated from ~r (see also Supplementary Fig. 4).The colour maps in Fig. 2 presenting values derived from r(t1, t2) have beensmoothed using an unweighted, nearest-neighbour, rectangular sliding average.

RESEARCH LETTER

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