SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION doi:.8/nphys SUPPLEMENTARY INFORMATION for The Quantum-Optial Josephson Interferometer D. Gerae,, Hakan E. Türei, A. Imamoglu, V. Giovannetti, and Rosario Fazio, Institute of Quantum Eletronis, ETH Zurih, 89 Zurih, Switzerland CNISM and Dipartimento di Fisia A. Volta, Università di Pavia, 7 Pavia, Italy NEST (CNR-INFM) and Suola Normale Superiore, Piazza dei Cavalieri 7, 566 Pisa, Italy International Shool for Advaned Studies (SISSA), via Beirut, Trieste, Italy Experimental feasibility In the main text we have onsidered a generi Kerr nonlinearity as the soure of strong photon orrelation in the entral avity. A possible way of experimentally implementing an effetive hamiltonian of the type () or (5) in the main text is to ouple -level atomi ensembles with mirotoroid resonators, as desribed in detail in the literature []. The latter ertainly represents an interesting possibility for a pratial realization of our proposal with state-ofthe art atomi avity QED. Our fous here will be on the sheme represented in Fig. b and d (see text), in whih a quantum dot (QD) is assumed to be deterministially oupled to the high-q photoni rystal avity mode in the middle []. It has been experimentally shown that suh a system displays single-photon nonlinearities under oherent resonant pumping []. Photoni rystal (PC) iruits allow for a straightforward on-hip implementation with side-oupled avities (as in Fig. b) or waveguides (as in Fig. d). If the hopping parameter J is small ompared to the laser intensities E and E, we an approximate the states of the external avities with oherent fields of intensity E, /γ (with γ being the damping parameter of the external avities). Thus, to first order in J/ E,, the dynamis of the entral avity an be effetively desribed by replaing in the Hamiltonian the operators ˆp and ˆp with E, /γ, with referene to the derivation of the model in Eq. (5) in the text. With this hoie the -avity set-up an be effetively redued to a Jaynes-Cummings model oupled to external driving fields and desribed by the simplified model Ĥ â â + xˆσ +ˆσ + ig(â ˆσ ˆσ + â )+E eff â + E effâ, (S) where â (â ) represents annihilation (reation) of avity photons, while ˆσ, and ˆσ + are Pauli lowering and rising operators related to the effetive two-level system representing the QD exiton transition; g = (πe f/εm V eff ) / is the exiton-photon oupling (expressed in terms of the effetive avity mode volume and the QD osillator strength) [], and = ω ω L and x = ω x ω L are the avity and exiton detuning from the pump frequeny, respetively. The effetive pumping strength is E eff E + E /γ and is responsible for the effets of phase detuning, φ = φ φ. To give some numbers, state-of-the art solid state QED with GaAs-based materials allows for Q 5 6, i.e. realisti γ. mev [5], γ x. mev [6], g =. mev []. In the experiment, g is fixed as well as J, but the effetive single-photon nonlinearity an be tuned by hanging δ = ω x ω. There are a number of different tehniques to deterministially tune the avity mode frequeny [, 7, 8, 9] and/or the QD exiton resonane []. For the system exitation, the same laser soure an be sent through a beam-splitter, one of the arms going diretly into the PC iruit (e.g. through a tapered aess waveguide) with phase φ, nature physis

2 a <n > Population,. π π π Phase differene, φ δ/γ= δ/γ= δ/γ= δ/γ= π b <n (φ=)> () g () detuning, δ/γ Figure S: Numerial simulation of a realisti model with a JC-type non-linearity. We assume parameters of the model in Eq. (S) g/γ =, g/γ x = (realistially ahievable parameters, see text), and effetive pumping strength E eff /γ = at φ =. Results are shown for light intensity and seond-order orrelation funtion emitted from the middle avity, i.e. n = â â and g () () = â â ââ / n. (a) Josephson-like osillations are suppressed when the avity-exiton detuning δ. When δ γ,g, i.e. the exiton resonane is strongly blue-detuned from the avity mode, the lower polariton is more and more avity-like, hene the effetive nonlinearity of the system is tuned through adjusting δ. In the inset, a zoom on the urve for δ =. (b) The rossover from orrelated to deloalized regimes is shown for n and g () () at φ =, respetively, as a funtion of δ. while the other being delayed and sent through a seond tapered waveguide into the iruit with phase φ. Given that lasers with sub-mhz linewidth (i.e. muh smaller than avity and exiton dissipation rates) are urrently available, we do not regard possible phase flutuations in the two driving fields as a limiting issue for this sheme to be realized. Finally, it is key to this experiment that the pump laser frequenies be tuned to the lower polariton frequeny of the JC spetrum, i.e. for eah detuning δ we set ω L (ω x + ω )/ g + δ / []. We simulate this model by solving the orresponding master equation that inorporates realisti avity and exiton dissipation rates. Figure S shows that the preditions of the Jaynes-Cummings model in Eq. (S) are qualitatively similar to the ideal Kerr nonlinearity model of Eq. (5) in the main text. In partiular, the dependene of light intensity on φ shows suppression of Josephson-like osillations as the exiton frequeny is tuned (from the blueside) in resonane with the avity mode. The inrease of the amplitude of osillations towards the bare-avity limit appears together with a rossover from sub-poissonian to Poissonian statistis as a funtion of δ, onsistent with the results of Fig. in the main text. Here, tuning δ is a way of effetively tuning the nonlinearity in the entral avity, and thereby to experimentally observe the rossover from tunnel-oupled to strong orrelated photon dynamis in a state-of-the art devie. Notie that the linear regime is ompletely reovered for δ/γ, i.e. δ mev with the parameters given above, whih is perfetly within reah of present experimental apabilities. In order to ompare the behaviour of the different models, we notie that the strongly orrelated limit of Fig. Sa (δ =, urve in the inset) is qualitatively similar to the result obtained with a Kerr nonlinear model, Eq. (5) in the text, for U = ( )g, i.e. U/γ, as it is shown in Fig. Sa and its inset. The rossover nature physis

3 a <n > Population,.5 π π U/γ= U/γ=. U/γ= b <n (φ=)> () g ().5 π Phase differene, φ U/γ= π Interation, U/γ Figure S: Numerial simulation for the model of Eq. (5) in the main text with effetive pumping strength E eff /γ = at φ =. (a) Josephson-like osillations are suppressed on inreasing the effetive Kerr nonlinearity, U/γ. In the inset, a zoom on the urve for U/γ =, to be ompared to the previous figure. (b) Crossover from orrelated to deloalized regimes for n and g () () at φ =, respetively, as a funtion of U/γ. from deloalized to loalized regimes as a funtion of U is shown in Fig. Sb to be ompared to Fig. Sb. We note that the orrespondene between a generi Kerr-type non-linearity studied in the main text and a JC-like non-linearity studied here is stritly valid only in the low-pumping regime. The same model and experimental approah an be used to study the realization of the quantum optial Josephson interferometer based on different tehnologies, suh as iruit QED [] or quantum well exiton-polaritons. While there is no experimental evidene of Kerr nonlinear behaviour of D onfined avity polaritons at time of writing, it is likely that polariton blokade in a single quantum box an be ahieved in the near future along the lines and numbers quoted in the literature [], e.g. with an alternative avity geometry reently realized []. Suh a result would make possible a more diret realization of our model with a Kerr nonlinearity in the solid state. Derivation of Eq. () Here we investigate the steady state dynamis of the Hamiltonian (7) in the weak pumping limit. We present here a derivation of the expressions for n (φ = ) and g () (τ = ) = ˆp ˆp ˆp ˆp / n. Consider the low-energy exitations of the Hamiltonian (7) in the basis n,n s where N tot = n + n s is the total number of photons. In the weak pumping limit, the driving term Ẽŝ + Ẽ ŝ auses transitions between the manifolds N tot and N tot +. Let us write the total time-dependent wavefuntion of the system as Ψ(t) = n a n n. In the weak pumping limit, we onsider the lowest three total photon manifolds N tot =,,, hene n =,...,5. The orresponding energy level diagram and the rates are shown in Fig. S. Terms negleted in Ψ(t) will be of order O( Ẽ γ ) where γ is the typial deay rate of the system (γ, γ). It s important to write the equations of motion in the bare basis instead of the basis of the oupled avity states to get the dissipation rates orretly. The equations of nature physis

4 U,> J,> J,> γ γ E γ,> J,> γ E γ,> γ E Figure S: Energy level diagram and rates for the oupled avity system. The various N tot manifolds are set off from eah other by an arbitrary ω p for visibility. The numbering of the states n are from down to up and left to right. For instane =,. motion are ã = ã = i + γ ã i Jã ã = i + γ ã i Jã iẽã ã = (i +iu + γ )ã i Jã ã = i + Γ ã i J(ã +ã 5 ) iẽã ã 5 = (i +γ)ã 5 i Jã iẽã We have kept terms that are the same order of magnitude in Ẽ γ and ã n = a n e iωlt. The steady-state solutions an be easily determined by additionally employing the normalization ondition 5 n= a n = (to order O( Ẽ γ )6 ). We find, for =, a = JẼ γγ + J, a = iẽγ γγ + J It s interesting to note that in the limit J/γ, a a, showing that interferene effets play an important role. Thus, n p a = 6 J Ẽ /(γγ + J ) (S) (S) to order O( Ẽ γ ). Solving for the two-photon manifold amplitudes as well, we find () g () (τ = ) a a = Γ Γ +α ( J)U (S) Equation (7) in the main text is written in the interation piture. Here the wavefuntion Ψ is the wavefuntion written in the Shrödinger piture for ompleteness. nature physis

5 5 E J J J J Interation, U/J Figure S: Variation of the energy levels of the three-avity Josephson interferometer as a funtion of U/ J. We plot the first few total photon number manifolds in the range N tot = to N tot =, and J/γ =.5. The various N tot manifolds are marked on the vertial axis and set off from eah other by an arbitrary ω L for visibility. The energy levels in eah manifold undergo an antirossing at U/ J and a ross-over takes plae to an effetive Jaynes-Cummings sequene as U/ J. Effetive Jaynes-Cummings model In the limit of ultra-strong interations, the system of two avities effetively maps into a Jaynes-Cummings system whose nonlinearity is determined by the tunnel oupling strength. With referene to the hamiltonian (7) in the main text (see Methods setion), we show in Fig. S how this omes about: two levels of eah onstant photon-number manifold (N tot = n + n s ) split off from the rest of the levels to form a JC-sequene as U is inreased beyond J. Referenes [] Hartmann, M. J., Brandao, F. G. S. L. & Plenio, M. B. Strongly interating polaritons in oupled arrays of avities. Nature Physis, (6). [] Hennessy, K. et al. Quantum nature of a strongly oupled single quantum dot-avity system. Nature 5, (7). [] Faraon, A. et al. Coherent generation of nonlassial light on a hip via photon-indued tunneling and blokade. Nature Physis, (8). [] Andreani, L. C., Panzarini, G. & Gerard, J.-M. Strong-oupling regime for quantum boxes in pillar miroavities: Theory. Phys. Rev. B 6, (999). nature physis 5

6 [5] Combrié, S., De Rossi, A., Tran, Q. V. & Benisty, H. GaAs photoni rystal avity with ultrahigh Q: mirowatt nonlinearity at.55 µm. Opt. Lett., 98-9 (8). [6] Vamivakas, N. et al. Strong extintion of a far-field laser beam by a single quantum dot. Nano Lett. 7, (7). [7] Badolato, A. et al. Deterministi oupling of single quantum dots to single nanoavity modes. Siene 8, 58 (5). [8] Strauf, S. et al. Frequeny ontrol of photoni rystal membrane resonators by monolayer deposition. Appl. Phys. Lett. 88, 6 (6). [9] Hennessy, K., Högerle, C., Hu, E., Badolato, A. & Imamoglu, A. Tuning photoni nanoavities by atomi fore mirosope nano-oxidation. Appl. Phys. Lett. 89, 8 (6). [] Rastelli, A. et al. In situ laser miroproessing of single self-assembled quantum dots and optial miroavities. Appl. Phys. Lett. 9, 7 (7). [] Shuster, D. I. et al. Resolving photon number states in a superonduting iruit. Nature 5, (7). [] Verger, A., Ciuti, C. & Carusotto, I. Polariton quantum blokade in a photoni dot. Phys. Rev. B 7, 96 (6). [] El Daïf, O. et al. Polariton quantum boxes in semiondutor miroavities. Appl. Phys. Lett. 88, 65 (6). 6 nature physis