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QUANTUM OPTICS any amplitude and phase features formed inside the nonlinear medium are preserved outside and can be detected in the form of photon number Observation of three-photon and phase correlations. The third-order photon correlation function has been measured previous- ly in coupled atom-cavity and quantum dot– bound states in a quantum cavity systems, as well as in nonclassical states of three photons such as the Greenberger-Horne- nonlinear medium Zeilinger and N00N states (12). In our approach, we split the light onto three single-photon count- ing modules. Furthermore, by mixing a detuned Qi-Yu Liang,1 Aditya V. Venkatramani,2 Sergio H. Cantu,1 Travis L. Nicholson,1 local oscillator into the final beam splitter, we Michael J. Gullans,3,4 Alexey V. Gorshkov,4 Jeff D. Thompson,5 Cheng Chin,6 can also perform a heterodyne measurement in Mikhail D. Lukin,2* Vladan Vuletić1* oneofthedetectionarms(Fig.1A).Toconnect the observed correlations to the physics of inter- Bound states of massive particles, such as nuclei, atoms, or molecules, constitute acting Rydberg polaritons, we consider a state the bulk of the visible world around us. By contrast, photons typically only interact containing up to three photons weakly. We report the observation of traveling three-photon bound states in a quantum nonlinear medium where the interactions between photons are mediated jyi¼j iþ∫ y ð Þj iþ by atomic Rydberg states. Photon correlation and conditional phase measurements 0 dt1 1 t1 t1 reveal the distinct bunching and phase features associated with three-photon ∫ y ð ; Þj ; iþ Downloaded from and two-photon bound states. Such photonic trimers and dimers possess dt1dt2 2 t1 t2 t1 t2 shape-preserving wave functions that depend on the constituent photon number. ∫ y ð ; ; Þj ; ; iðÞ dt1dt2dt3 3 t1 t2 t3 t1 t2 t3 1 The observed bunching and strongly nonlinear optical phase are described by an effective field theory of Rydberg-induced photon-photon interactions. These j ; ; i¼ 1 †ð Þ †ð Þj i † where t1 tN N! a t1 a tN 0 , a (t) observations demonstrate the ability to realize and control strongly interacting is the photon creation operator of the time bin quantum many-body states of light. mode t, and N is the number of photons. The correlation functions (Fig. 2) can be related to the http://science.sciencemag.org/ jy ð ; Þj3 ound states of light quanta have been pro- tions of light and matter—Rydberg polaritons— wave functions as gð2Þðt ; t Þ¼ 2 t1 t2 and 1 2 jy ðt Þj2jy ðt Þj2 posed to exist in specifically engineered inherit strong interactions from their Rydberg 1 1 1 2 jy ð ; ; Þj2 ð3Þð ; ; Þ¼ 3 t1 t2 t3 media with strong optical nonlinearities components and can propagate with very low g t1 t2 t3 jy ð Þj2jy ð Þj2jy ð Þj2 . We refer to 1 t1 1 t2 1 t3 (1–5). In recent times, photonic dimers have loss at slow group velocity vg (13–15). The non- B ~ðNÞ been observed experimentally (6). Such linearity arises when photons are within a Rydberg the phase f of the N-photon wave function ~ð1Þ bound states of photons can be viewed as quan- blockade radius rB of one another, where strong yN as the N-photon phase—namely, f ðt1Þ¼ ½y ð Þ ~fð2Þð ; Þ¼ ½y ð ; Þ ~fð3Þ tum solitons (7, 8), which are shape-preserving interactions between atoms in the Rydberg state Arg 1 t1 , t1 t2 Arg 2 t1 t2 ,and ð ; ; Þ¼ ½y ð ; ; Þ wave packets enabled by the cancellation of (16) shift the Rydberg level out of the EIT reso- t1 t2 t3 Arg 3 t1 t2 t3 .TheN-photon nonlinear and dispersive effects. In contrast to nance, blocking the excitation of more than one phase is obtained from the phase of the beat note
classical solitons, where the self-consistent shape Rydberg atom within rB. In the dissipative regime signal on the third detector, conditioned on having on February 16, 2018 varies smoothly with total pulse energy, quan- (on atomic resonance), the blockade results in observed N − 1 photons in the other two detectors. tum solitons have optical nonlinearity that is photon loss and antibunching (17–19). In the The conditional phase relative to N uncorrelated strong enough for the wave packet shape to dispersive, off-resonant regime, the index of re- photons (i.e., the nonlinear part of the phase) is depend on the constituent number of photons in fraction varies with the separation between denoted as fðNÞ (Fig. 3). a quantized manner (7, 8). The creation of quan- photons, resulting in an attractive force (6). The experimentally measured g(3) function tum solitons not only represents an important Our experimental setup (20)(Fig.1,Aand (Fig. 2, A and B) displays a clear bunching fea- step in fundamental studies of photonic quan- B) consists of a weak quantum probe field at ture: The probability to detect three photons with- tum matter (6, 9, 10) but also may enable new wavelength l = 780 nm and waist w = 4.5 mm in a short time (≲25 ns) of one another is six times applications in areas ranging from quantum coupled to the 100S1/2 Rydberg state via a strong higher than for noninteracting photons in a laser communication to quantum metrology (11, 12). 479-nm control field in the EIT configuration beam. The increase at t1 = t2 = t3 is accompanied We use an ultracold atomic gas as a quantum (Fig. 1B). The interactions occurred in a cloud of by a depletion region for photons arriving within nonlinear medium to search for a photonic trimer. laser-cooled 87Rb atoms in a far-detuned optical ~0.7 ms of one another, particularly visible along This medium is experimentally realized by cou- dipole trap. Measurements are conducted at the lines of two-photon correlations (ti = tj ≠ tk, pling photons to highly excited atomic Rydberg a peak optical depth ODB ≃5 per blockade radius where ti, tj,andtk are the photon detection times states by means of electromagnetically induced rB =20mm. To suppress dissipative effects, we at detectors Di, Dj,andDk, respectively, and i, j, transparency (EIT). The resulting hybrid excita- work at a large detuning D ≥ 3G from atomic and k are permutations of 1, 2, and 3) (Fig. 2A): resonance (G is the population decay rate of the This depletion region is caused by the inflow of 1Department of Physics and Research Laboratory of 5P3/2 state; see Fig. 1B) and at a control laser Rabi probability current toward the center t1 = t2 = t3. Electronics, Massachusetts Institute of Technology, 2 frequencywherethetransmissionthroughthe Figure 2B compares the two-photon correlation Cambridge, MA 02139, USA. Department of Physics, (2) Harvard University, Cambridge, MA 02138, USA. medium is the same with and without EIT, but the function g (t, t +|t|) with that for three photons, 3Department of Physics, Princeton University, Princeton, NJ phase differs appreciably (Fig. 1C). Consequently, of which two were detected in the same time 08544, USA. 4Joint Quantum Institute and Joint Center for the transmission hardly varies with probe pho- bin, g(3)(t, t, t +|t|). The trimer feature is approx- Quantum Information and Computer Science, National ton rate (Fig. 1D, top), whereas a strongly rate- imately a factor of 2 narrower than the dimer Institute of Standards and Technology and University of Maryland, College Park, MD 20742, USA. 5Department dependent phase with a slope of 0.40(±0.07) feature, showing that a photon is attracted more of Electrical Engineering, Princeton University, Princeton, rad·ms is observed (Fig. 1D, bottom). strongly to two other photons than to one. Figure NJ 08544, USA. 6James Franck Institute, Enrico Fermi The quantum dynamics of interacting photons 2C illustrates the binding of a third photon to two Institute, and Department of Physics, University of Chicago, are investigated by measuring the three-photon photons that are detected with a time separa- Chicago, IL 60637, USA. t *Corresponding author. Email: [email protected] (M.D.L.); correlationfunctionandphase.Becausedisper- tion T.IfT exceeds the dimer time scale 2,then [email protected] (V.V.) sion outside of the atomic medium is negligible, the third photon binds independently to either
Liang et al., Science 359, 783–786 (2018) 16 February 2018 1of4 RESEARCH | REPORT
Fig. 1. Qualitative descriptions of the exper- AB iment. (A and B) Setup and atomic-level scheme. The atoms are optically pumped into the hyperfine (F)andmagnetic(mF) sublevel jgi¼j5S1=2; F ¼ 2; mF ¼ 2i. Probe The weak coherent probe light is coupled to the Rydberg state jri¼j100S1=2; mJ ¼ 1=2i Probe (mJ, projection of the total electron angular momentum along the quantization axis) via an intermediate CD E state jei¼j5P3=2; F ¼ 3; mF ¼ 3i,with linewidth G/2p = 6.1 MHz, by means of a counterpropagating control field that is detuned by Δ below the resonance frequency of the upper transition, jei→jri. Strong interactions between probe photons are detected via photon correlations of the transmitted light, which is split onto three single-photon detectors with equal intensities. To perform phase measurements, a local oscillator is mixed Downloaded from ^ into detector D3. aðtÞ, photon annihilation operator of the time bin mode t; Wc, control laser Rabi frequency; g, electromagnetically induced transparency (EIT) resonance. (D)Rate correlation function; f, phase; LO, local oscillator. (C) Transmission (top) dependence of transmission (top) and unconditional phase (bottom) and phase f (bottom) as a function of probe frequency measured at a low on the two-photon resonance jgi→jri, with a one-photon detuning −1 input photon rate (0.5 ms ). f is measured without conditioning on the of D/2p = 30 MHz and a control Rabi frequency Ωc/2p =10MHz.
detection of other photons. The control laser is set at D/2p =30MHz Whereas the transmission is rate-independent, the phase is strongly http://science.sciencemag.org/ m below the jei→jritransition, with Rabi frequency Ωc/2p =10MHz.The rate-dependent (slope is 0.4 rad· s). (E) Schematic correlation functions blue and red traces are from measurements with and without a control for two (top) and three (bottom) photons as a function of their time beam, respectively. The blue and red dashed lines in the bottom graph separation t. The attractive interaction leads to photon bunching, with are theoretical expectations. The vertical yellow dashed line marks three photons being more tightly bound together than two photons.
Fig. 2. Photon correlation functions with tighter bunching due to the three-photon bound state. on February 16, 2018 Photon correlation functions were measured on EIT resonance and at one-photon detuning
D/2p = 30 MHz, control Rabi frequency Ωc/2p = 10 MHz, an input photon rate of 1 ms−1.(A)2D representation of the three-photon correlation (3) function g (t1, t2, t3), with ti being the photon detection time at detector Di. Three-photon bunching corresponds to the central region, two- photon bunching to the stripes. (B) g(3)(t, t, t +|t|) (bluedatapoints)andg(2)(t, t +|t|) (brown data points), with the decay constants calculated from the exact solution for the tc ¼ : m tc ¼ : m bound states 3 0 16 sand 2 0 32 s, respectively (dashed lines). The calculated exponential decay is scaled to match the initial point of the measured intensity correlation functions. The approximately twofold smaller decay length of the three-photon correlation function shows that a photon is more strongly bound to two photons than to one. The fitted exponential decay constants with (3) (2) zero offset for g and g are t3 =0.14(2)ms and t2 =0.31(6)ms, respectively, in agreement with the calculated values. (C) Three representative (3) (2) (2) plots of g (t1, t2, t3)/g (t1, t2) for fixed time transitions to a slower decaying g function. For intermediate time separation T ≡ |t1 − t2|=0ms(i),T =0.2ms(ii),andT =1.8ms separations (ii), interference occurs between all states, including the (iii), within a 50-ns window. As the two photons get farther dimer and trimer. All permutations of the detectors are used to and farther away from each other, the sharply decaying g(3) function generate the data in (B) and (C). Error bars indicate 1 SD.
Liang et al., Science 359, 783–786 (2018) 16 February 2018 2of4 RESEARCH | REPORT
agreement with experimental observations. We calculate a/(2vg)=0.32ms for our measured exper- imental parameters (27) and find it to be consist- ent with data (Fig. 2B, dashed lines). Following the quantum quench at the entry of the medium, the initial state is decomposed into the bound state and the continuum of scattering states (6). Near t = 0, the scattering states dephase with each other, whereas the bound state propagates with- out distortion (27). This leads to a small contribu- tion of scattering states in this region, with the bound state dominating the g(3) function. The ob- served value of g(3)(0) is not universal, as it is af- fected by the contributions from long-wavelength scattering states and nonlinear losses in the system and therefore depends on the atomic density Fig. 3. Larger nonlinear phase for three photons. Nonlinear phase was measured under identical profile of the medium. The dimer and trimer bind- ð3Þ 2 conditions as the data in Fig. 2. (A) Conditional phase f ðt1; t2; t3Þ, where t1 and t2 correspond to ħ ing energies can be estimated as E2 ¼ 2 ¼ photon detection events at detectors D and D , and a heterodyne measurement is performed ma 1 2 h 0:2MHzand E3 ¼ 4E2 , respectively. This fð3Þð ; ; þjtjÞ on detector D3 at time t3.(B) Diagonal cut t t t (blue), with the two conditioning probe binding energy is ~1010 times smaller than in di- fð2Þð ; þjtjÞ photons within 40 ns of each other, and t t (brown), showing a larger phase when Downloaded from atomic molecules such as NaCl and H2 but is com- ½fð3Þ conditioning on two other near-simultaneous photons than on one near-simultaneous photon parable to Feshbach (28)andEfimov(29)bound ½fð2Þ fðNÞ . is referenced to its own average value when all N photons are too far apart to be states of atoms with similar mass m and scattering ð Þ ð Þ ð Þ j j→1 ð2Þ ~ 2 ~ 1 ~ 1 t1 t2 ð3Þ correlated. Specifically, f ðt1; t2Þ ≡ f ðt1; t2Þ f ðt1Þþf ðt2Þ →0 and f ðt1; t2; t3Þ ≡ length a. To further characterize the three-photon ð3Þ ð2Þ ð3Þ ð1Þ ð1Þ ð1Þ jti tj j→1;∀i6¼j bound state, we consider the phase ratiof =f . ~f ðt ; t ; t Þ ~f ðt Þþ~f ðt Þþ~f ðt Þ →0. At large |t|, fð3Þ asymptotically goes to 1 2 3 1 2 3 For the bound-state contribution to the conditional jtj→1 ð Þ ð Þ ð Þ ð Þ ð Þ fð2Þðt; tÞ because fð3Þðt; t; t þjtjÞ →~f 2 ðt; tÞþ~f 1 ðt þjtjÞ ~f 1 ðtÞþ~f 1 ðtÞþ~f 1 ðt þjtjÞ ¼ fð2Þðt; tÞ. phase fð3Þðt; t; tÞðfð2Þðt; tÞÞ,theHamiltonianof http://science.sciencemag.org/ Error bars indicate 1 SD. Eq. 2 predicts a phase that equals the trimer bind- ing energy multiplied by the propagation time in the medium. Thus, from the bound-state contribu- ð3Þ ð2Þ photon, whereas for T < t2 the two peaks merge by the small size of the probe waist compared tions, one would expect a ratio f =f ¼ 4, in- into a single, more tightly bound trimer. This is with the blockade radius (rB), whereas the cor- dependent of the atom-light detuning D.Although analogous to the binding of a particle to a double- responding Rayleigh range is similar to the the observed ratio (Fig. 4B) is approximately con- well potential as the distance between the wells atomic cloud size (with both being much larger stant, it is smaller than 4. is varied, because the polaritons can be approx- than rB). The latter condition implies that the The observed deviation is probably due to the imately described as interacting massive par- incoming light has small transverse momenta two contributions of comparable magnitude. One ticles moving at a finite group velocity (6). components, whereas the former condition en- correction arises from the scattering states, or
The dispersive and distance-dependent photon- sures that the dominant scattering occurs col- equivalently, from the fact that our Rydberg me- on February 16, 2018 photon interaction also manifests itself in a large inearly with the probe beam. Combined with the dium (~130 mm) is comparable in size to the two- conditional phase shift that depends on the time large effective transverse mass in this system (23), photon bound state (~280 mm). For a medium interval t between the detection of the condition- the residual transverse dynamics arising from that is short compared with the bound state, one ing photons (at times t1 = t2 = t)andthephase interactions are effectively frozen out on the time expects the ratio to be 3, consistent with a dis- measurement on detector D3 at time t3.Weob- scale of the experiment (6, 15, 18, 24). In the persionless Kerr medium (30). The other, more serve a conditional phase shift fð3Þðt; t; t þjtjÞ for limitwheretheaveragelongitudinaldistancebe- fundamental, correction may be due to a contri- the trimer near t = 0 (Fig. 3A) that is substantially tween photons is larger than rB, such that the bution that does not arise from pairwise inter- larger than the dimer phase shift fð2Þðt; t þjtjÞ low-momentum approximation underlying the actions, effectively representing a three-photon (Fig.3B).Thisconfirmsthattheinteractionbe- EFT is valid, the 1D contact model provides an force. Specifically, when all three photons are tween a photon and a dimer is stronger than that accurate description whenevera ≫ rB, the micro- within one blockade radius of one another, there between two photons. scopic range of the two-body potential. For our can be only one Rydberg excitation, and the po- To understand these results quantitatively, parameters, we find that this condition is well tential cannot exceed the value corresponding to we apply an effective field theory (EFT) (21)that satisfied as a ≳ 10rB. y^ is a quantum field annihi- that of two photons (21, 31). This saturation effect describes the low-energy scattering of Rydberg lation operator, which corresponds to a photon manifests itself as a short-range repulsive effec- polaritons. This EFT gives us a one-dimensional outside the medium and a Rydberg polariton tive three-photon force that, according to our (1D) slow-light Hamiltonian density with a con- inside. For our blue-detuned probe, the effective theoretical analysis (27), results in a reduction of tact interaction mass is negative and the interaction is repulsive. fð3Þ=fð2Þ below 3. The corresponding correction This situation maps onto a system with a positive to the bound state is smaller in the weakly inter- 2 2 2 † @ ħ @ ħ † mass and attractive interaction. The bound states acting regime relevant to these experiments (31). H¼ y^ iħv þ y^ y^ 2y^ 2 ð2Þ g @ @2 z 2m z ma can be determined from the exact solution of this This explains why the effective three-photon force model for finite particle numbers (25, 26), result- has a relatively weak effect on the bunching of ð3Þ (3) where vg is the group velocity inside the medium, ing in the correlation functions g ðt1; t2; t3Þº g (|t| < 0.2 ms), which is dominated by the bound 2 2 m ¼ ħW =ð8Dv Þis the effective photon mass, jt t j jt t j jt t j jt t j state. Both the scaling arguments and numerical c g 1 2 2 3 1 3 ð Þ 1 2 a=ð2vgÞ a=ð2vgÞ a=ð2vgÞ 2 ð ; Þº a=ð2vgÞ ħ is Planck’s constant h divided by 2p,ais the e e e and g t1 t2 e . evidence indicate that the effective three-photon ð3Þ scattering length, Wc is the control laser Rabi In the case t1=t2=t,wefindthatg ðt;t;tþjtjÞº force contributes to the three-body scattering am- jtj frequency, and D is the one-photon detuning. For 2 plitudesmorestronglythantwo-bodyfiniterange a=ð2vgÞ 2 e , implying that the width of three-photon 1 D effectsinthisregime(21). weak interactions, a ≈ 15:28 G rB (21, 22). (3) ODB wave packet [corresponding to g ] is half that To quantitatively understand these effects, The single-mode, 1D approximation is justified of g(2) for the same experimental conditions, in the EFT is modified to include the estimated
Liang et al., Science 359, 783–786 (2018) 16 February 2018 3of4 RESEARCH | REPORT
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ACKNOWLEDGMENTS effective three-photon force (27). Using the efficiency and data-acquisition rate. Furthermore, We thank O. Firstenberg for help with the early stages of modified EFT, we compare the results with and with the use of an elliptical or larger round probe this work and S. Choi for discussions. M.J.G. and A.V.G. without the repulsive effective three-photon beam and careful engineering of the mass along thank H. P. Büchler for many insightful discussions force while also taking into account the effects different directions, the system can be extended and comments on the theoretical analysis. This work has due to finite medium (Fig. 4B). Including this to two and three dimensions, possibly permit- been supported by the NSF, the NSF Center for Ultracold Atoms, the U.S. Army Research Office (ARO), the U.S. three-photon saturation force allows the phase ting the observation of photonic Efimov states Air Force Office of Scientific Research, the ARO ð Þ ð Þ ratio f 3 =f 2 to go below 3, in a reasonable (32, 33). Finally, our medium supports only one Multidisciplinary University Research Initiative, and a Bush agreement with the experimental observations. two- or three-photon bound state, corresponding Fellowship. A.V.G. and M.J.G. acknowledge additional For fully saturated interactions between the po- to a nonlinear phase less than p. A threefold in- support by the U.S. Army Research Laboratory Center for Distributed Quantum Information, the NSF Quantum laritons, the interaction potential does not in- crease in the atomic density would render the Information Science program, and the NSF Physics Frontiers crease with photon number, and the phase ratio interaction potential sufficiently deep for a sec- Center at the Joint Quantum Institute. C.C. acknowledges should approach 2. ond bound state to appear near zero energy, funding support from NSF grant PHY-1511696 and the The observation of the three-photon bound which should result in resonant photon-photon Alexander von Humboldt Foundation. All data needed to evaluate the conclusions in the paper are present in the state, which can be viewed as photonic solitons scattering and a tunable scattering length (22). paper and the supplementary materials. in the quantum regime (7, 8), can be extended The presence of large effective N-body forces in along several different directions. First, increas- this system opens avenues to study exotic many- SUPPLEMENTARY MATERIALS ing the length of the medium at constant atomic body phases of light and matter, including self- www.sciencemag.org/content/359/6377/783/suppl/DC1 densitywouldremovetheeffectofthescattering organization in open quantum systems (34, 35) Materials and Methods states through destructive quantum interference and quantum materials that cannot be realized Supplementary Text to larger t and would retain only the solitonic with conventional systems. Figs. S1 to S4 bound-state component. Additionally, the strong Tables S1 and S2 References (36–40) fð3Þ observed rate dependence of may indicate REFERENCES AND NOTES that larger photonic molecules and photonic clus- 1. I. H. Deutsch, R. Y. Chiao, J. C. Garrison, Phys. Rev. Lett. 69, 20 August 2017; accepted 4 January 2018 ters could be observed with improved detection 3627–3630 (1992). 10.1126/science.aao7293
Liang et al., Science 359, 783–786 (2018) 16 February 2018 4of4 Observation of three-photon bound states in a quantum nonlinear medium Qi-Yu Liang, Aditya V. Venkatramani, Sergio H. Cantu, Travis L. Nicholson, Michael J. Gullans, Alexey V. Gorshkov, Jeff D. Thompson, Cheng Chin, Mikhail D. Lukin and Vladan Vuletic
Science 359 (6377), 783-786. DOI: 10.1126/science.aao7293
Forming photonic bound states Photons do not naturally interact with each other and must be coaxed into doing so. Liang et al. show that a gas of Rydberg atoms−−a cloud of rubidium atoms excited by a sequence of laser pulses−−can induce strong interactions between propagating photons. The authors could tune the strength of the interaction to make the photons form dimer and Downloaded from trimer bound states. This approach should prove useful for producing novel quantum states of light and quantum entanglement on demand. Science, this issue p. 783 http://science.sciencemag.org/
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