On the origin of strong antibunching in weakly nonlinear photonic 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 ’’ 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 , 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 Unavailaberegion 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, hthe 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. Since we assume weak pumping, ≫ the contribution from the higher states (C , C , and γ 20 11 ∆Eopt , (8a) C02) to the steady-state values of C10, C01 is negligible. ≃ 2√3 From Eq. (3b), the amplitude of mode 2 can be written 2 γ3 Uopt . (8b) as ≃ 3√3 J 2 J C01 = C10. (4) In Fig. 2(a), the optimal ∆E and U [Eq. (7)] are −∆E iγ /2 opt opt 2 − 2 plotted as functions of J/γ. The strong antibunching indicating that for strong photon tunneling (J can be obtained even if U < γ, provided J > γ/√2. ≫ 2 ∆E2 ,γ2), the probability of finding a photon in the aux- Remarkably, the required nonlinearity decreases with in- |iliary| cavity is much larger than the driven cavity. creasing tunnel coupling J obeying Eq. (8b). 3

䎕䎑䎘 䎃 䎕 䎃 (a) (b) 䎃䏌䎃䏍䎃䎠䎃䎔䎔 䎕 䎃䏌䎃䏍䎃䎠䎃䎕䎕 䎃䏌䎃䏍䎃䎠䎃䎔䎕 䎔䎑䎘

䎌 䎔䎑䎘 τ 䎋 䎋䎕䎌 䏌䏍 䏊 䎔 䎠䎓䎌 τ

䎋 䎔 䎋䎕䎌 䎔䎔

䎓䎑䎘 䏊

䎓 䎃 䎓䎑䎘 䎐䎕 䎐䎔 䎓 䎔 䎕 䎓 䎓䎑䎘 䎔 τ䎃䎒䎃䎋䎕 π䎒䎭䎌 ∆䎨䎃䎒䎃 γ 䎴䏄䏘䏑䏗䏘䏐䎃䏇䏒䏗䎃䎎䎃䎕䎃䎦䏄䏙䏌䏗䏌䏈䏖 䎴䏘䏄䏑䏗䏘䏐䎃䏇䏒䏗䎃䎎䎃䎔䎃䎦䏄䏙䏌䏗䏜 FIG. 3: (a) The time evolution of the second-order correlation 䎓 䎃 function, which oscillates with period 2π/J as the result of 䎓 䎓䎑䎘 䎔 䎔䎑䎘 䎕 䎦䏄䏙䏌䏗䏜䎐䏇䏒䏗䎃䏆䏒䏘䏓䏏䏌䏑䏊䎃䏊䎃䎒䎃 amplitude oscillation between 01 and 10 . (b) Equal-time γ second-order correlation functions| i are plotted| i as functions of ∆E1 =∆E2 = ∆E normalized to γ1 = γ2 = γ. The spectral FIG. 4: Equal-time correlation functions are plotted as func- width of the antibunching resonance is 0.3γ. Parameters: tions of coupling strength g between a cavity and a quantum ≈ J = 3γ, U1 = U2 = 0.0428γ, and F = 0.01γ. ∆E = 0.275γ in dot. line represents the results in the system sketched panel (a). in the inset [Eq. (10)]. Parameters: γ1 = γ2 = γex = γ, J = 3γ, E1 = ~ωp + 0.275γ, E2 E1 = γ, Eex E2 = 2γ and F = 0.01γ. Dashed line represents− the result− in the sys- tem with one quantum dot and one cavity [Jaynes-Cummings In Fig. 2(b), we show a sketch of the quantum interfer- model]. γ1 = γex = γ, Eex E1 = 2γ, F = 0.01γ, and ~ωp ence effect responsible for this counter-intuitive photon is tuned to the lower one-particle− eigenenergy of the Jaynes- antibunching. The interference is between the following Cummings ladder. F two paths: (a) the direct excitation from 10 20 (solid arrow) and (b) tunnel-coupling-mediated| i −→| transi-i J F J J tion 10 01 ( 11 02 ) 20 (dotted ar- rows).| Ini ↔ order | toi −→ show| ini detail↔ | thei −→| origini of the quan- tum interference, we rewrite Eqs. (6) for C20 = 0 as follows. First, we calculate C from Eqs. (4) and (5) 11 window does not significantly depend on J/γ. This may neglecting C20 as suggest that pump pulses of duration ∆tp longer than 1 1/(0.3γ) could be enough to ensure strong antibunching. C = 2JF C (∆E + U iγ/2)(∆E iγ/2)− 11 − 10 2 − − 2 2 1 However, the timescale over which strong quantum cor- [2J 4∆E(∆E + U )+ γ + i2γ(2∆E + U )]− . − 2 2 relations between the photons exist is on the order of (9) 1/J < √2/γ, as seen in Fig. 3(a). While weak nonlinear- ities do lead to strong quantum correlations, these corre- This amplitude is the result of excitation from 01 to lations last for a timescale that scales with 1/J U 11 and of the coupling between 10 and 01 and| i also opt (see Eq. (8b)). From a practical perspective, a∝ principalp |betweeni 11 and 02 . From this| amplitude,i | iC is de- 20 difficulty with the observation of the photon antibunch- termined| byi Eq. (5a)| i as C JC + F C , and we can 20 11 10 ing with weak nonlinearities is that it requires fast single- derive Eqs. (6) by the condition∝ C = 0. 20 photon detectors [14]. Conversely, for a given detection As seen in Fig. 1(b), while no more than one photon is set-up, the required minimal value of the nonlinearity is present in the first cavity mode at the optimal condition, ultimately determined by the time resolution of the avail- there can be more than one photons in the whole system. able single photon detector. While there is nearly perfect antibunching in the driven (2) mode [g11 (τ = 0) << 1], the cross-correlation between (2) the two modes exhibits bunching [g12 (τ = 0) > 1]. The As seen in Eq. (6), the nonlinearity U1 of the pumped amplitude oscillation between 10 and 01 produces the cavity mode is not essential for the antibunching. This (2) | i | i time oscillation of g11 (τ) with period 2π/J as reported means that only the auxiliary (undriven) photonic mode in Ref. [12] and shown in Fig. 3(a). must have a (weak) nonlinearity to achieve the quantum The equal-time correlation functions is plotted in interference leading to perfect photon antibunching. As Fig. 3(b) as a function of the pump detuning ∆E/γ: a practical realization, one could consider two coupled while the optimal value of the detuning is at ∆E = photonic nanocavities, where the auxiliary cavity 0.275γ, a strong antibunching is obtained in a range of contains a single quantum dot that leads to the required about 0.3γ around the optimal value and the width of this weak nonlinearity (see the inset in Fig. 4). The Hamil- 4

(a) (b) 䎕䎑䎘 䎃 can show an interesting interplay with quantum correla- tion between neighboring photonic modes. As a demon- 䎕 stration, we consider a ring of three molecules whose 䎔䎑䎘 driven dots are coupled with each other by a tunnel 1

䎋䎕䎌 䏌䏍 coupling of amplitude J2 [see Fig. 5(a)]. Also in this 䏊 䎲䏑䎐䏖䏌䏗䏈䎃䎋䏌䎃䎠䎃䏍䎌 䎔 case a nearly perfect antibunching can be observed in 䎬䏑䏗䏈䏕䏐䏒䏏䏈䏆䏘䏏䏈䎃䎋䏌䎃 ≠䎃䏍䎌 each driven mode, as shown in the plots of g(2)(τ = 0) 3 2 䎓䎑䎘 ii as a function of J2/γ that are shown as a solid line

䎓 䎃 in Fig. 5(b). In order to optimize the antibunching at 䎓 䎓䎑䎘 䎔 䎔䎑䎘 䎕 a finite value of J γ, values of U = 0.0769γ and 䎭 䎒γ 2 䎕 ∆E = 0.450γ slightly≃ different from the single-molecule optimal ones (Uopt =0.0428γ and ∆Eopt =0.275γ) had FIG. 5: (a) Sketch of a triangular lattice of coupled photonic to be chosen. At the same time, a strong bunching ef- ‘molecules’. The driven cavities (i = 1, 2, and 3) are coupled fect is observed in the equal-time cross-correlation func- with strength J2. (b) The equal-time second-order correlation functions in each mode (solid line) and between neighbors tion between neighboring cavities, which shows a value of (2) (dashed line) are plotted versus J2/γ. Parameters: J = 3γ, gi=j (0) significantly larger than the coherent field value E1 = ~ωp + 0.450γ, U = 0.0769γ and F = 0.01γ. 6 (2) of gi=j (0) = 1. This remarkable combination of strong on-site6 antibunching and strong inter-site bunching sug- tonian is written as gests that this system may be a viable alternative to the realization of a Tonks-Girardeau of fermionized pho- 2 tons discussed in Ref. [9]. Hˆ = E aˆ†aˆ + J(ˆa† aˆ +ˆa†aˆ ) cav-JC X i i i 1 2 2 1 i=1 In summary, we have analytically determined that a destructive quantum interference mechanism is responsi- + Eex ex ex + g aˆ2† g ex + H.c. | ih | | ih | ble for strong antibunching in a system consisting of two iωpt iωpt + F e− aˆ1† + F ∗e aˆ1. (10) coupled photonic modes with small nonlinearity (U <γ). The quantum interference effect occurs for an optimal on- Here, g and ex represent the ground and excited states 2 γ3 | i | i site nonlinearity Uopt √ J2 , where J is the intermode of the quantum dot, respectively, Eex is the excitation ≃ 3 3 energy, and g is the coupling energy with cavity mode tunnel coupling energy and γ is the mode broadening. 2. Since the required nonlinearity is relatively weak, one This robust quantum interference effect has the pecu- can use a quantum dot which is off-resonant with respect liar feature that the resulting quantum correlation be- tween the generated photons survive for timescales much to the cavity mode ( Eex E2 > γ2 = γ) and/or does not satisfy strong coupling| − condition| (g γ). We take shorter than the photon lifetime. Nonetheless, we have the quantum dot exciton broadening to be≃ equal to the shown that this quantum interference scheme has the po- cavity decay rate for simplicity. We have solved numeri- tential to generate strongly correlated photon states in cally the master equation associated to the Hamiltonian arrays of weakly nonlinear cavities. (2) in Eq. (10). Fig. 4 shows g11 (τ = 0) of the pumped mode as a function of g/γ. The coupling energy between the two cavities is J = 3γ, and then the required non- ∗ linear energy should be Uopt = 0.0428γ from Fig. 2. In E-mail: [email protected] the present system, this nonlinear energy is practically † E-mail: [email protected] achieved at g = 1.4γ, which is an intermediate strength [1] A. Imamo˘glu, H. Schmidt, G. Woods, and M. Deutsch, between the weak- and strong-coupling regime of cavity Phys. Rev. Lett. 79, 1467 (1997). mode and quantum dot excitation. The dashed line in [2] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005). Fig. 4 represents the results in the system consisting of [3] A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, one quantum dot and one cavity: in this ordinary Jaynes- and J. Vuckovic, Nat. Phys. 4, 859 (2008). Cummings system, only a small antibunching is obtained [4] B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. at g γ, and the strong-coupling g γ is required for Vahala, and H. J. Kimble, Science 319, 1062 (2008). the observation≃ of large photon antibunching≫ [1, 2]. In [5] M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, contrast, in the new scheme using the quantum interfer- Nat. Phys. 2, 849 (2006). ence, a nearly perfect antibunching can be obtained even [6] A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, Nat. Phys. 2, 856 (2006). for g γ. ≃ [7] D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev. Finally, we note that the quantum interference can A 76, 031805 (2007). be generalized to a system of many coupled photonic [8] D. E. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. molecules: in this case, the strong on-site antibunching Lukin, and E. A. Demler, Nat. Phys. 4, 884 (2008). 5

[9] I. Carusotto, D. Gerace, H. E. Tureci, S. De Liberato, 193306 (2006). C. Ciuti, and A. Imamoˇglu, Phys. Rev. Lett. 103, 033601 [14] In a single nonlinear cavity, the requirement for fast pho- (2009). ton detection can be avoided by using pulsed-laser ex- [10] M. J. Hartmann, Phys. Rev. Lett. 104, 113601 (2010). citation; this approach does not work in the system we [11] D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, analyze due to small bandwidth of the nonlinearity, i.e., and R. Fazio, Nat. Phys. 5, 281 (2009). the available time of the antibunching is shorter than the [12] T. C. H. Liew and V. Savona, Phys. Rev. Lett. 104, temporal width of pump pulse (1/J < ∆tp) 183601 (2010). [13] A. Verger, C. Ciuti, and I. Carusotto, Phys. Rev. B 73,