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On the Origin of Strong Photon Antibunching in Weakly Nonlinear Photonic Molecules Motoaki Bamba, Atac Imamoglu, Iacopo Carusotto, Cristiano Ciuti

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On the Origin of Strong Photon Antibunching in Weakly Nonlinear Photonic Molecules Motoaki Bamba, Atac Imamoglu, Iacopo Carusotto, Cristiano Ciuti On the origin of strong photon antibunching in weakly nonlinear photonic molecules Motoaki Bamba, Atac Imamoglu, Iacopo Carusotto, Cristiano Ciuti To cite this version: Motoaki Bamba, Atac Imamoglu, Iacopo Carusotto, Cristiano Ciuti. On the origin of strong photon antibunching in weakly nonlinear photonic molecules. Physical Review A, American Physical Society, 2011, 83, pp.021802. 10.1103/PhysRevA.83.021802. hal-00499490 HAL Id: hal-00499490 https://hal.archives-ouvertes.fr/hal-00499490 Submitted on 9 Jul 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the origin of strong photon antibunching in weakly nonlinear photonic molecules 1, 2 3 1, Motoaki Bamba, ∗ Atac Imamo˘glu, Iacopo Carusotto, and Cristiano Ciuti † 1Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, Universit´eParis Diderot-Paris 7 et CNRS, Bˆatiment Condorcet, 10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France 2Institute of Quantum Electronics, ETH Z¨urich, 8093 Z¨urich, Switzerland 3INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, I-38123 Povo, Italy (Dated: July 9, 2010) In a recent work [T. C. H. Liew and V. Savona, Phys. Rev. Lett. 104, 183601 (2010)] it was numerically shown that in a photonic ’molecule’ consisting of two coupled cavities, near-resonant coherent excitation could give rise to strong photon antibunching with a surprisingly weak nonlinear- ity. Here, we show that a subtle quantum interference effect is responsible for the predicted efficient photon blockade effect. We analytically determine the optimal on-site nonlinearity and frequency detuning between the pump field and the cavity mode. We also highlight the limitations of the proposal and its potential applications in demonstration of strongly correlated photonic systems in arrays of weakly nonlinear cavities. PACS numbers: 42.50.Dv, 03.65.Ud, 42.25.Hz The photon blockade is a quantum optical effect pre- of the mechanism leading to strong photon antibunching venting the resonant injection of more than one photon is needed to identify the limitations of the scheme in the into a nonlinear cavity mode [1], leading to antibunched context of proposed experiments on strongly correlated (sub-Poissonian) single-photon statistics. Signatures of photons, as well as to determine the dependence of the photon blockade have been observed by resonant laser optimal coupling and detuning on the relevant physical excitation of an optical cavity containing either a single parameters J and γ. atom [2] or a single quantum dot [3] in the strong coupling In this letter, we show analytically that the surprising regime. Arguably, the most convincing realization was antibunching effect is the result of a subtle destructive based on a single atom coupled to a micro-toroidal cav- quantum interference effect which ensures that the prob- ity in the Purcell regime [4], suggesting that the strong ability amplitude to have two photons in the driven cavity coupling regime of cavity-QED need not be a require- is zero. We show that the weak nonlinearity is required ment. Concurrently, on the theory side there has been only for the auxiliary cavity that is not laser driven and a number of proposals investigating strongly correlated whose output is not monitored, indicating that photon photons in coupled cavity arrays [5–7] or one-dimensional antibunching is obtained for a driven linear cavity that optical waveguides [8]. The specific proposals based on tunnel couples to a weakly nonlinear one. We determine the photon blockade effect include the fermionization of the analytical expressions for the optimal coupling U and photons in one-dimensional cavity of arrays [9], the crys- for the pump frequency detuning required to have a per- tallization of polaritons in coupled array of cavities [10], fect antibunching as a function of the mode coupling J and the quantum-optical Josephson interferometer in a and broadening γ. Our analytical results are in excellent coupled photonic mode system [11]. agreement with fully numerical solutions of the master equation for the considered system. Before concluding, It is commonly believed that photon blockade necessar- we discuss the experimental realization of such a scheme ily requires a strong on-site nonlinearity U for a photonic by using cavities embedding weakly coupled quantum mode, whose magnitude should well exceed the mode dots. Moreover, we consider also the case of a ring of cou- broadening γ. However, in a recent work [12] Liew and pled photonic molecules showing that strong antibunch- Savona numerically showed that a strong antibunching ing persists in presence of intersite photonic correlations. can be obtained with a surprisingly weak nonlinearity We consider two photonic modes coupled with strength (U γ) in a system consisting of two coupled zero- J; each mode has energy Ej and an on-site photon- dimensional≪ (0D) photonic cavities (boxes), as shown in photon interaction strength Ui (i = 1, 2). The Hamil- Fig. 1(a) [12]. Such a configuration can be obtained, tonian is written as e.g., by considering two modes in two photonic boxes coupled with a finite mode overlap due to leaky mir- 2 Hˆ = E aˆ†aˆ + U aˆ†aˆ†aˆ aˆ + J(ˆa† aˆ +ˆa†aˆ ) rors: the corresponding tunnel strength will be desig- X h i i i i i i i ii 1 2 2 1 nated with J. In Ref. [12] numerical evidence indicated i=1 iωpt iωpt that a nearly perfect antibunching can be achieved for + F e− aˆ1† + F ∗e aˆ1, (1) an optimal value of the on-site repulsion energy U and for an optimal value of the detuning between the pump wherea ˆi is the annihilation operator of a photon in i- and mode frequency. However, a physical understanding th mode, F and ωp are the pumping strength and fre- 2 䎃 (a) (b) 䎕 (a) 䎃 (b) γ 䎓 䎔䎓 〉 〉 〉 䎔䎑䎘 䎃䎒䎃 䏒䏓䏗 䎨 䎠䎓䎌 ∆ τ 䎋 䎔 䎐䎕 䎋䎕䎌 䏌䏍 䎔䎓 䏊 䎃䏌䎃䏍䎃䎠䎃䎔䎔 〉 〉 䎓䎑䎘 䎃䎃䎃䏄䏑䏇䎃䎃䎃 䎃䏌䎃䏍䎃䎠䎃䎕䎕 Unavailabe region γ 䎸 䎃䎒䎃 γ 䎐䎗 䏒䏓䏗 䎃䏌䎃䏍䎃䎠䎃䎔䎕 䎃䎒䎃 䎔䎓 䎃 䎨 䎃䎒䎃 䎓 䏒䏓䏗 ∆ 䏒䏓䏗 γ 䎐䎓䎑䎔 䎐䎓䎑䎓䎘 䎓 䎓䎑䎓䎘 䎓䎑䎔 䎸 䎃 䎸䎃䎒䎃 䎐䎔 䎓 䎔 䎕 γ 䎔䎓 䎔䎓 䎔䎓 䎔䎓 〉 䎭䎃䎒䎃 γ FIG. 1: (1) Sketch of the two coupled photonic modes. The coupling strength is J, and the antibunching is obtained with FIG. 2: (a) Optimal nonlinearity Uopt and detuning ∆Eopt a small nonlinear energy U compared to mode broadening γ. are plotted as functions of coupling strength J normalized to (2) (b) Equal-time second-order correlation functions gij (τ = 0) γ (γ1 = γ2 = γ and E1 = E2 = E). The perfect antibunching are plotted as functions of nonlinearity U = U1 = U2 normal- is obtained for J >γ/√2. (b) Transition paths leading to the ized to γ. The nearly perfect antibunching is obtained at the quantum interference responsible for the strong antibunching. pumped mode [g(2)(τ = 0) 0] for U = 0.0428γ. Parame- One path is the direct excitation from 10 to 20 , but it is 11 ≃ | i | i ters: γ1 = γ2 = γ, J = 3γ, E1 = E2 = ~ωp + 0.275γ, and forbidden by the interference with the other path drawn by F1 = 0.01γ. dotted arrows. quency, respectively. Following Ref. [12], we first cal- In the same manner, the coefficients of two-particle (2) states are determined by culate the second-order correlation function gij (τ) = aˆ†aˆ†(τ)ˆa (τ)ˆa / aˆ†aˆ aˆ†aˆ in the steady state using i j j i i i j j 2(∆E + U iγ /2)C + √2JC + √2F C =0, theh master equationi h inih a basisi of Fock states [13]. The 1 1 1 20 11 10 − (5a) results are shown as functions of nonlinearity U in Fig. 1(b). As already demonstrated in Ref. [12], we can (∆E1 + ∆E2 iγ1/2 iγ2/2)C11 + √2JC20 + √2JC02 (2) − − get a strong antibunching of the pumped mode (g11 (0) + F C01 =0, (5b) 0) for an unexpectedly small nonlinearity U =0.0428γ≃. 2(∆E + U iγ /2)C + √2JC =0. (5c) In order to understand the origin of the strong anti- 2 2 − 2 02 11 bunching, we use the Ansatz When we simply consider E1 = E2 = E, and γ1 = γ2 = iωpt γ, the conditions to satisfy C20 = 0 are derived from ψ = C00 00 +e− (C10 10 + C01 01 ) | i | i | i | i Eqs. (4) and (5) as i2ωpt +e− (C 20 + C 11 + C 02 )+ ..., 20| i 11| i 02| i (2) γ2(3∆E + U ) 4∆E2(∆E + U )=2J 2U , (6a) 2 − 2 2 to calculate the steady-state of the coupled cavity system. 12∆E2 + 8∆EU γ2 =0. (6b) 2 − Here, mn represents the Fock state with m particles in mode| 1 andi n particles in mode 2. Under weak pumping For fixed J and γ, from these equations, the optimal conditions (C C , C C , C , C ), we can cal- conditions (those that lead to C20 = 0) are given by 00 ≫ 10 01 ≫ 20 11 02 culate the coefficients Cmn iteratively. For one-particle 1 states, the steady-state coefficients are determined by 4 2 2 2 2 ∆Eopt = q 9J +8γ J γ 3J , (7a) ±2 p − − 2 2 (∆E1 iγ1/2)C10 + JC01 + F C00 =0, (3a) ∆Eopt(5γ + 4∆Eopt ) − Uopt = , (7b) (∆E2 iγ2/2)C01 + JC10 =0, (3b) 2(2J 2 γ2) − − where ∆Ej = Ej ~ω and we consider a damping with − p and, if J γ, they are approximately written as rate γj in each mode.
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