## 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 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

## Abstract

Bound states of massive particles, such as nuclei, atoms, or molecules, constitute the bulk of the visible world around us. By contrast, photons typically only interact weakly. We report the observation of traveling three-photon bound states in a quantum nonlinear medium where the interactions between photons are mediated by atomic Rydberg states. Photon correlation and conditional phase measurements reveal the distinct bunching and phase features associated with three-photon and two-photon bound states. Such photonic trimers and dimers possess shape-preserving wave functions that depend on the constituent photon number. The observed bunching and strongly nonlinear optical phase are described by an effective field theory of Rydberg-induced photon-photon interactions. These observations demonstrate the ability to realize and control strongly interacting quantum many-body states of light.

Bound states of light quanta have been proposed to exist in specifically engineered media with strong optical nonlinearities (*1*–*5*). In recent times, photonic dimers have been observed experimentally (*6*). Such bound states of photons can be viewed as quantum solitons (*7*, *8*), which are shape-preserving wave packets enabled by the cancellation of nonlinear and dispersive effects. In contrast to classical solitons, where the self-consistent shape varies smoothly with total pulse energy, quantum solitons have optical nonlinearity that is strong enough for the wave packet shape to depend on the constituent number of photons in a quantized manner (*7*, *8*). The creation of quantum solitons not only represents an important step in fundamental studies of photonic quantum matter (*6*, *9*, *10*) but also may enable new applications in areas ranging from quantum communication to quantum metrology (*11*, *12*).

We use an ultracold atomic gas as a quantum nonlinear medium to search for a photonic trimer. This medium is experimentally realized by coupling photons to highly excited atomic Rydberg states by means of electromagnetically induced transparency (EIT). The resulting hybrid excitations of light and matter—Rydberg polaritons—inherit strong interactions from their Rydberg components and can propagate with very low loss at slow group velocity *v*_{g} (*13*–*15*). The nonlinearity arises when photons are within a Rydberg blockade radius *r*_{B} of one another, where strong interactions between atoms in the Rydberg state (*16*) shift the Rydberg level out of the EIT resonance, blocking the excitation of more than one Rydberg atom within *r*_{B}. In the dissipative regime (on atomic resonance), the blockade results in photon loss and antibunching (*17*–*19*). In the dispersive, off-resonant regime, the index of refraction varies with the separation between photons, resulting in an attractive force (*6*).

Our experimental setup (*20*) (Fig. 1, A and B) consists of a weak quantum probe field at wavelength λ = 780 nm and waist *w* = 4.5 μm coupled to the 100*S*_{1}_{/}_{2} Rydberg state via a strong 479-nm control field in the EIT configuration (Fig. 1B). The interactions occurred in a cloud of laser-cooled ^{87}Rb atoms in a far-detuned optical dipole trap. Measurements are conducted at a peak optical depth per blockade radius *r*_{B} = 20 μm. To suppress dissipative effects, we work at a large detuning Δ ≥ 3Γ from atomic resonance (Γ is the population decay rate of the 5*P*_{3}_{/}_{2} state; see Fig. 1B) and at a control laser Rabi frequency where the transmission through the medium is the same with and without EIT, but the phase differs appreciably (Fig. 1C). Consequently, the transmission hardly varies with probe photon rate (Fig. 1D, top), whereas a strongly rate-dependent phase with a slope of 0.40(±0.07) rad·μs is observed (Fig. 1D, bottom).

The quantum dynamics of interacting photons are investigated by measuring the three-photon correlation function and phase. Because dispersion outside of the atomic medium is negligible, any amplitude and phase features formed inside the nonlinear medium are preserved outside and can be detected in the form of photon number and phase correlations. The third-order photon correlation function has been measured previously in coupled atom-cavity and quantum dot–cavity systems, as well as in nonclassical states of three photons such as the Greenberger-Horne-Zeilinger and N00N states (*12*). In our approach, we split the light onto three single-photon counting modules. Furthermore, by mixing a detuned local oscillator into the final beam splitter, we can also perform a heterodyne measurement in one of the detection arms (Fig. 1A). To connect the observed correlations to the physics of interacting Rydberg polaritons, we consider a state containing up to three photons(1)where , *a*^{†}(*t*) is the photon creation operator of the time bin mode *t*, and *N* is the number of photons. The correlation functions (Fig. 2) can be related to the wave functions as and . We refer to the phase of the *N*-photon wave function ψ* _{N}* as the

*N*-photon phase—namely, , , and . The

*N-*photon phase is obtained from the phase of the beat note signal on the third detector, conditioned on having observed

*N*− 1 photons in the other two detectors. The conditional phase relative to

*N*uncorrelated photons (i.e., the nonlinear part of the phase) is denoted as (Fig. 3).

The experimentally measured *g*^{(3)} function (Fig. 2, A and B) displays a clear bunching feature: The probability to detect three photons within a short time ( ns) of one another is six times higher than for noninteracting photons in a laser beam. The increase at *t*_{1} = *t*_{2} = *t*_{3} is accompanied by a depletion region for photons arriving within ~0.7 μs of one another, particularly visible along the lines of two-photon correlations (*t _{i}* =

*t*≠

_{j}*t*, where

_{k}*t*,

_{i}*t*, and

_{j}*t*are the photon detection times at detectors

_{k}*D*,

_{i}*D*, and

_{j}*D*, respectively, and

_{k}*i*,

*j*, and

*k*are permutations of 1, 2, and 3) (Fig. 2A): This depletion region is caused by the inflow of probability current toward the center

*t*

_{1}=

*t*

_{2}=

*t*

_{3}. Figure 2B compares the two-photon correlation function

*g*

^{(2)}(

*t*,

*t*+ |τ|) with that for three photons, of which two were detected in the same time bin,

*g*

^{(3)}(

*t*,

*t*,

*t*+ |τ|). The trimer feature is approximately a factor of 2 narrower than the dimer feature, showing that a photon is attracted more strongly to two other photons than to one. Figure 2C illustrates the binding of a third photon to two photons that are detected with a time separation

*T*. If

*T*exceeds the dimer time scale τ

_{2}, then the third photon binds independently to either photon, whereas for

*T*< τ

_{2}the two peaks merge into a single, more tightly bound trimer. This is analogous to the binding of a particle to a double-well potential as the distance between the wells is varied, because the polaritons can be approximately described as interacting massive particles moving at a finite group velocity (

*6*).

The dispersive and distance-dependent photon-photon interaction also manifests itself in a large conditional phase shift that depends on the time interval τ between the detection of the conditioning photons (at times *t*_{1} = *t*_{2} = *t*) and the phase measurement on detector *D*_{3} at time *t*_{3}. We observe a conditional phase shift for the trimer near τ = 0 (Fig. 3A) that is substantially larger than the dimer phase shift (Fig. 3B). This confirms that the interaction between a photon and a dimer is stronger than that between two photons.

To understand these results quantitatively, we apply an effective field theory (EFT) (*21*) that describes the low-energy scattering of Rydberg polaritons. This EFT gives us a one-dimensional (1D) slow-light Hamiltonian density with a contact interaction(2)where *v*_{g} is the group velocity inside the medium, is the effective photon mass, *ℏ* is Planck’s constant *h* divided by 2π*,* *a* is the scattering length, Ω_{c} is the control laser Rabi frequency, and Δ is the one-photon detuning. For weak interactions, (*21*, *22*). The single-mode, 1D approximation is justified by the small size of the probe waist compared with the blockade radius (*r*_{B}), whereas the corresponding Rayleigh range is similar to the atomic cloud size (with both being much larger than *r*_{B}). The latter condition implies that the incoming light has small transverse momenta components, whereas the former condition ensures that the dominant scattering occurs colinearly with the probe beam. Combined with the large effective transverse mass in this system (*23*), the residual transverse dynamics arising from interactions are effectively frozen out on the time scale of the experiment (*6*, *15*, *18*, *24*). In the limit where the average longitudinal distance between photons is larger than *r*_{B}, such that the low-momentum approximation underlying the EFT is valid, the 1D contact model provides an accurate description whenever , the microscopic range of the two-body potential. For our parameters, we find that this condition is well satisfied as . is a quantum field annihilation operator, which corresponds to a photon outside the medium and a Rydberg polariton inside. For our blue-detuned probe, the effective mass is negative and the interaction is repulsive. This situation maps onto a system with a positive mass and attractive interaction. The bound states can be determined from the exact solution of this model for finite particle numbers (*25*, *26*), resulting in the correlation functions and .

In the case *t*_{1} = *t*_{2} = *t*, we find that , implying that the width of three-photon wave packet [corresponding to *g*^{(3)}] is half that of *g*^{(2)} for the same experimental conditions, in agreement with experimental observations. We calculate *a/*(2*v*_{g}) = 0.32 μs for our measured experimental parameters (*27*) and find it to be consistent 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 τ = 0, the scattering states dephase with each other, whereas the bound state propagates without distortion (*27*). This leads to a small contribution of scattering states in this region, with the bound state dominating the *g*^{(3)} function. The observed value of *g*^{(3)}(0) is not universal, as it is affected by the contributions from long-wavelength scattering states and nonlinear losses in the system and therefore depends on the atomic density profile of the medium. The dimer and trimer binding energies can be estimated as and , respectively. This binding energy is ~10^{10} times smaller than in diatomic molecules such as NaCl and H_{2} but is comparable to Feshbach (*28*) and Efimov (*29*) bound states of atoms with similar mass *m* and scattering length *a*. To further characterize the three-photon bound state, we consider the phase ratio . For the bound-state contribution to the conditional phase , the Hamiltonian of Eq. 2 predicts a phase that equals the trimer binding energy multiplied by the propagation time in the medium. Thus, from the bound-state contributions, one would expect a ratio , independent of the atom-light detuning Δ. Although the observed ratio (Fig. 4B) is approximately constant, it is smaller than 4.

The observed deviation is probably due to the two contributions of comparable magnitude. One correction arises from the scattering states, or equivalently, from the fact that our Rydberg medium (~130 μm) is comparable in size to the two-photon bound state (~280 μm). For a medium that is short compared with the bound state, one expects the ratio to be 3, consistent with a dispersionless Kerr medium (*30*). The other, more fundamental, correction may be due to a contribution that does not arise from pairwise interactions, effectively representing a three-photon force. Specifically, when all three photons are within one blockade radius of one another, there can be only one Rydberg excitation, and the potential cannot exceed the value corresponding to that of two photons (*21*, *31*). This saturation effect manifests itself as a short-range repulsive effective three-photon force that, according to our theoretical analysis (*27*), results in a reduction of below 3. The corresponding correction to the bound state is smaller in the weakly interacting regime relevant to these experiments (*31*). This explains why the effective three-photon force has a relatively weak effect on the bunching of *g*^{(3)}(|τ| *<* 0.2 μs), which is dominated by the bound state. Both the scaling arguments and numerical evidence indicate that the effective three-photon force contributes to the three-body scattering amplitudes more strongly than two-body finite range effects in this regime (*21*).

To quantitatively understand these effects, the EFT is modified to include the estimated effective three-photon force (*27*). Using the modified EFT, we compare the results with and without the repulsive effective three-photon force while also taking into account the effects due to finite medium (Fig. 4B). Including this three-photon saturation force allows the phase ratio to go below 3, in a reasonable agreement with the experimental observations. For fully saturated interactions between the polaritons, the interaction potential does not increase with photon number, and the phase ratio should approach 2.

The observation of the three-photon bound state, which can be viewed as photonic solitons in the quantum regime (*7*, *8*), can be extended along several different directions. First, increasing the length of the medium at constant atomic density would remove the effect of the scattering states through destructive quantum interference to larger τ and would retain only the solitonic bound-state component. Additionally, the strong observed rate dependence of may indicate that larger photonic molecules and photonic clusters could be observed with improved detection efficiency and data-acquisition rate. Furthermore, with the use of an elliptical or larger round probe beam and careful engineering of the mass along different directions, the system can be extended to two and three dimensions, possibly permitting the observation of photonic Efimov states (*32*, *33*). Finally, our medium supports only one two- or three-photon bound state, corresponding to a nonlinear phase less than π. A threefold increase in the atomic density would render the interaction potential sufficiently deep for a second bound state to appear near zero energy, which should result in resonant photon-photon scattering and a tunable scattering length (*22*). The presence of large effective *N*-body forces in this system opens avenues to study exotic many-body phases of light and matter, including self-organization in open quantum systems (*34*, *35*) and quantum materials that cannot be realized with conventional systems.

## Supplementary Materials

www.sciencemag.org/content/359/6377/783/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S4

Tables S1 and S2

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank O. Firstenberg for help with the early stages of this work and S. Choi for discussions. M.J.G. and A.V.G. thank H. P. Büchler for many insightful discussions and comments on the theoretical analysis. This work has been supported by the NSF, the NSF Center for Ultracold Atoms, the U.S. Army Research Office (ARO), the U.S. Air Force Office of Scientific Research, the ARO Multidisciplinary University Research Initiative, and a Bush Fellowship. A.V.G. and M.J.G. acknowledge additional support by the U.S. Army Research Laboratory Center for Distributed Quantum Information, the NSF Quantum Information Science program, and the NSF Physics Frontiers Center at the Joint Quantum Institute. C.C. acknowledges funding support from NSF grant PHY-1511696 and the Alexander von Humboldt Foundation. All data needed to evaluate the conclusions in the paper are present in the paper and the supplementary materials.