Self-Pulsing in driven-dissipative photonic Bose-Hubbard dimers Jes´usYelo-Sarri´on,1 Pedro Parra-Rivas,1 Nicolas Englebert,1 Carlos Mas Arab´ı,1 Fran¸coisLeo,1 and Simon-Pierre Gorza1 1OPERA-Photonique; Universit´elibre de Bruxelles; 50 Avenue F. D. Roosevelt; CP 194/5 B-1050 Bruxelles; Belgium We experimentally investigate the nonlinear dynamics of two coupled fiber ring resonators, coher- ently driven by a single laser beam. We comprehensively explore the optical switching arising when scanning the detuning of the undriven cavity, and show how the driven cavity detuning dramatically changes the resulting hysteresis cycle. By driving the photonic dimer out-of-equilibrium, we observe the occurrence of stable self-switching oscillations near avoided resonance crossings. All results agree well with the driven-dissipative Bose-Hubbard dimer model in the weakly coupled regime. The spontaneous emergence of sustained periodic oscil- two [18, 19] or more [20{22] coupled cavities. It is how- lations is a fascinating and ubiquitous phenomenon aris- ever only very recently that evidence of such self-pulsing ing in nonlinear systems. It is associated with the break- was reported with polaritons, through its indirect spec- ing of translational symmetry in time, and is encountered tral signature [23]. Moreover, coupled ring resonators in various fields as diverse as chemistry, biology, mechan- host rich nonlinear dynamics and have recently attracted ical engineering, or astrophysics to cite only a few [1,2]. a lot of attention for frequency-comb generation in micro- Since the seminal works of Lotka [3] and Volterra [4] a resonators (see e.g. [24{29]). century ago in the context of chemistry and population In this Letter, we report, for the first time to our dynamics in biology, respectively, it is now well known knowledge, a comprehensive experimental investigation that such undamped oscillations may arise in nonlinear of the dynamical regimes of driven-dissipative photonic systems with coupled variables under continuous driving. dimers and the observation of the spontaneous emergence An important example is the generation of infinite trains of sustained oscillations between light beams propagat- of pulses in the FitzHugh{Nagumo model of nerve mem- ing in two linearly coupled ring resonators. We consider branes [5]. In optics, we can cite for instance self-pulsing passive fiber cavities corresponding to asymmetrically ex- in second-harmonic generation [6], in lasers between cou- cited photonic DDBH dimers. We focus on weak coupling pled longitudinal modes [7] or with continuous injected in relation to dissipation, which is relevant for moder- signal [8] or, more recently, between counterpropagating ate to low finesse resonators. In photonic dimers, tuning beams in a single Kerr ring resonator [9]. Rhythmogen- the cavity detunings is analogous to changing the sin- esis refers to the emergence of oscillations from the cou- gle particle energy of the two quantum states in bosonic pling between two or more sub-systems that show only Josephon junctions [15, 30]. Here, contrary to integrated steady states when uncoupled [10]. In this context, the dimers, the two cavity detunings can be independently driven-dissipative Bose-Hubbard (DDBH) model plays set or scanned without linear or nonlinear couplings be- an essential role in physics, as it provides a canonical tween them. description of the dynamics between strongly interacting The experimental setup is depicted in Fig.1. At its bosons for open quantum systems [11]. In its simplest heart are two passive fiber cavities coupled by a 95/5 cou- realization, only two macroscopic phase coherent wave pler. Each resonator is composed of about 250 m of opti- functions are coupled to form a Bose-Hubbard dimer, cal fiber (1:14 µs round-trip time) with a net normal dis- also referred as a bosonic Josephson junction. These persion. The loss in each cavity is ∼ 40 % (excluding the junctions have been initially investigated with supercon- shared coupler). The first cavity is synchronously driven ductors separated by a thin insulator [12] and with cou- through a 90/10 coupler by flat-top 470 ps pulses gener- pled reservoirs of super-fluid helium [13]. Later, it was ated from a narrow linewidth distributed feedback laser. realized that they can be implemented with weakly cou- This prevents the build up of the Brillouin scattering ra- pled Bose-Einstein condensates in a macroscopic double- arXiv:2104.00649v1 [physics.optics] 1 Apr 2021 diation and lowers the intracavity average power needed well potential [14] and in photonic systems with cou- to reach the self-pulsing regime. The two cavity detunigs pled semiconductor microcavities hosting polariton ex- can be independently stabilized by means of piezo-electric citation [15]. However, beside Josephson effects, owing fiber stretchers. The intracavity powers are measured at to their intrinsic nonlinearity, other striking phenomena the two drop ports and simultaneously recorded on an os- such as anharmonic oscillations or macroscopic quantum cilloscope. The dynamics of the slowly varying envelope self-trapping emerge in these latter systems [14{17]. In- of the intracavity fields A can be described by a set of terestingly, under continuous excitation, it was theoreti- 1;2 two coupled Lugiato-Lefever equations [8, 27]. In single cally predicted that sustained oscillations may take place ring Kerr resonators with normal dispersion, modulation in DDBH systems. This was shown for microcavity po- instability occurs in a narrow range of parameters beyond laritons [17], but also for nonlinear optical cavities with the bistability threshold [31]. In addition, different works 2 (a) 0.5 Cavity 1 NT /π 1 2 0.0 δ 0.5 − 0.5 Cavity 2 /π 2 0.0 δ 0.5 − (b) 0.5 /π 2 0.0 δ 0.5 − 0.5 0 /π 2 0.0 δ 0.5 − 2 1 0 1 2 − − δ1/π ) . 1 (c) (d) u . Figure 1. Experimental set-up. The photonic dimer is syn- chronously driven by a 1550 nm pulsed laser beam injected Power (a 0 in the first cavity. The detunings in each cavity are indepen- 0.5 0.0 0.5 0.5 0.0 0.5 − − dently stabilized. AM, amplitude modulator; EDFA, erbium- δ/π δ2/π doped fiber amplifier; FS, frequency shifter; PBS, fiber polar- ization beam splitter; PC, polarization controller; PFS, piezo- Figure 2. (a, b) Linear resonances as a function of the two electric fiber stretcher; PD, photodiode; PID, proportional- cavity detunings, in the driven (top, Cavity 1) and undriven integral-derivative controller; Iso., optical isolator; Circ., op- (bottom, Cavity 2) cavities. (a) Experimental results for P tical circulator, C (resp. C ) control signal to lock δ (resp. p 1 2 1 5 mW and (b) theoretical resonances derived from Eq. (1) but≈ δ ). The tunable stretcher is used to match the two cavity 2 repeated every 2π for δ . Normalized intracavity power in round-trip times. 1 the driven (blue) and undriven (red) cavity for δ = δ1 = δ2 (c), and (d) for δ1 = 0:36 while scanning the detuning in the undriven cavity [see also dashed lines in panel (b)]. NT, have recently shown that in coupled ring resonators, the normalized transmission; a.u., arbitrary units. local dispersion, induced by mode coupling, changes the instability spectrum and allows for the generation of sta- ble localized patterns [24{26]. In this work, we carefully field amplitude while F2 = 0. The field amplitudes are adjust the two cavity lengths to avoid such instability by normalized such that the intracavity powers (expressed ensuring they both have the same FSR. The temporal 2 in W) are given by jAjj = Pj. walk-off as well as the group velocity dispersion are thus The recorded and simulated linear resonances are neglected in the model. Under this simplification, the shown in Figs. 2(a) and 2(b). The driven cavity reso- dynamics of the system is governed by the DDBH dimer nances, when scanning δ , are reminiscent of the ones model [32{34]. It describes the fields evolution with the 1 of a single ring resonator, providing that the undriven round trips φ/2π in each cavity of length L and reads: cavity is out of resonance. When both cavities are close to resonance, the coupling splits the resonance and leads dA1;2 2 to an avoided crossing. The resulting double peak re- 2π = [−κ − iδ1;2 + iγLjA1;2j ]A1;2 dφ (1) sponse, usually observed when scanning the driving laser p p + i θ12A2;1 + i θpF1;2: frequency [25] [i.e. for δ1 = δ2, see Fig. 2(c)], corresponds to the excitation of the bonding-like and the antibonding- The detunings from the closest (single cavity) resonances like modes of the dimer [15]. However, owing to the weak are δj = mj2π − 'j (j = 1; 2), with 'j, the round-trip coupling with respect to the loss, these two peaks are not linear phase shift and mj an integer number. κ = 0:21 separated here. Fig. 2(d) shows the intracavity powers is the cavity loss coefficient and the nonlinear parameter at a fixed detuning δ1 = 0:36 when scanning δ2. We note 3 1 is γ = 3 × 10− (Wm)− . Finally, θ12 = 0:05 and θp = that because of the resonance splitting, for δ1 > 0, the 0:1 are the transmission coefficients of the middle and driving field is further from the resonance for negative p input couplers, respectively, and F1 = Pp is the driving than for positive δ2. It follows a non-symmetric response 3 (a) (a) 2 0.22 0.18 2 0.14 0.16 SN − − 5 c 0.32 c34 12 SN6 SN3 SN4 0.57 n41 0.56 SN4 SN1 S 0.28 SN3 HB1 (W) (W) SN p 1 p 1 1 c34 c56 P c P 12 SN HB2 1 SN1 SN2 SN 2 SN3 0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4 − − − − δ2/π δ2/π 1 1 (b) SN2 (c) (b) SN2 (c) ) .