Self-Pulsing in Driven-Dissipative Photonic Bose-Hubbard Dimers

Self-Pulsing in driven-dissipative photonic Bose-Hubbard dimers

JesúsYelo-Sarrión,1 Pedro Parra-Rivas,1 Nicolas Englebert,1 Carlos Mas Arab´ı,1 Fran¸coisLeo,1 and Simon-Pierre Gorza1 1OPERA-Photonique, Universitélibre 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 synchronously 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 polarization 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 stable 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 |Aj| = 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 + iγL|A1,2| ]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 − ϕj (j = 1, 2), with ϕj, 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) )

. SN3 u . SN1 0 0 ) .

u HB2 1 . 1 SN2

Power (a (d) (e) (d) (e) SN3 HB1 SN1

SN4 0 0 Power (a 0.25 0.00 0.25 0.25 0.00 0.25 1 S − − (f) SN2 (g) δ2/π δ2/π SN5 HB1 SN 1 SN6 Figure 3. (a) Phase diagram in the (δ2,Pp) parameter space 0 for δ = 0.36. The saddle-node bifurcations SN are plot- 1 j 0.25 0.00 0.25 0.25 0.00 0.25 ted as green lines and cij mark cusp bifurcations. The dimer − − is bistable in the shaded green areas. (b) Bifuraction dia- δ2/π δ2/π gram showing the normalized intracavity power in the driven (blue) and undriven (red) resonators as a function of δ , for 2 Figure 4. Same as in Fig.3, but for δ = 0.64. (a) Phase P = 300 mW and (d) 1 W [see also dashed lines in panel 1 p diagram. The Hopf bifurcations HB (red lines) mark the (a)]. Solid (dashed) lines represent stable (unstable) solu- 1,2 boundaries of the self-pulsing region, and the Shil’nikov bifur- tions of Eq. (1). (c, e) Corresponding experimental forward cation S (gray line), the boundaries of the area where there and backward scans. The arrows show the direction of the is no self-pulsing. n , necking bifurcation; c , cusp bifur- scans. 41 ij cation. The star indicates the parameters of the experiment reported in Fig. 5(a). (b, d, f) Normalized intracavity powers (thin lines) as a function of δ2 at a driving power of 0.5, with δ2 that has consequences on the bifurcation dia- 0.6 and 1 W, respectively. The thick lines between the bifur- gram. cations HB1,2 show the maximum and minimum oscillation We now investigate the nonlinear regime. To compute amplitudes. Note that in panel (f), HB2 (not shown for clar- the different attractors of the system, static and dynam- ity) is located at SN6. The corresponding experimental scans ical, as well as their stability with respect to its main are displayed in panels (c, e, g). See main text for further explanations. parameters, we perform a bifurcation analysis of Eq. (1), following a numerical path-continuation approach using AUTO-07p [35]. First, we consider√ δ1 = 0.36, a value close to the detuning threshold ( 3κ) to observe bista- again, until the emergence of a new pair of saddle-nodes bility in single-ring resonators [36]. While self-pulsing is SN3 and SN4 through a second cusp bifurcation (c34). not expected at this detuning, it is an interesting value to Theoretical resonances corresponding to this regime are show how the Kerr nonlinearity alters the photonic dimer shown in Fig. 3(d) for Pp = 1 W. The experimental scans resonances. By increasing the driving power, when scan- at this driving power confirm the shape of the nonlinear resonance and the existence of a bistable region for δ > 0 ning δ2, an optical bistablility is initially encountered in 2 a narrow detuning range between the saddle-node bifur- [see Fig. 3(e)]. cation lines SN1 and SN2 in the phase diagram shown in The bifurcation analysis of the DDBH dimer model Fig. 3(a). The theoretical resonances at a driving power predicts the emergence of self-pulsing for larger detun- Pp = 300 mW are plotted in Fig. 3(b) and compared with ings of the driven cavity. Such a situation is depicted the experimental transmissions reported in Fig. 3(c). We in Fig. 4(a) for δ1 = 0.64, where the bistable regions as observe an hysteresis cycle on the left side of the reso- well as the self-pulsing area in the (δ2,Pp)-plane of the nance in agreement with the fixed nodes of Eq. (1). Be- parameter space are plotted. At Pp = 500 mW, the two yond the driving power at which SN1 and SN2 meet at saddle-node curves (SN2,3) are crossed when scanning δ2. the cusp bifurcation c12, the system becomes monostable We note that these two lines are actually connected as 4

(a) )

. 1 u .

Power (a 0 0 2000 4000 6000 8000 10000 Round trip

(b) (c) (d) (e) ) ) . . ) ) u 1 u 1 . .

. 0.4 .

u u 0.4 . . (a (a

2 0.2 2 0.2 P P

Power (a 0 Power (a 0 0 200 400 0.5 1.0 0 200 400 0.5 1.0 Round trip P1 (a.u.) Round trip P1 (a.u.)

Figure 5. (a) Normalized power in the driven (blue) and undriven (red) resonators, recorded over more that 10 000 round trips (12 ms) and demonstrating periodic oscillations. The parameters are Pp = 1 W, δ1 = 0.64 and δ2 = 0.47. (b) Zoom over 400 round trips and (c) corresponding trajectory in the (P1,P2)-phase plane showing a limit cycle. (d) Sustained oscillations and (e) limit cycle obtained by numerical integration of Eq. (1).

they stretch between adjacent resonances. At this power, in a narrow range of δ2 which coincides with the self- the photonic dimer has two stable states for all δ2 values, pulsing region located between the two Hopf bifurcations except in the vicinity of the avoided mode crossing [see HB1 and HB2. At a higher power, the self-pulsing region Fig. 4(b)]. However, since the upper state is completely grows while SN1 moves away from the avoided crossing disconnected from the lower one, the system cannot spon- and, owing to the 2π periodicity, appears on the other taneously jump to the high power state by changing the side of the resonance. The stable and unstable homoge- detuning of the undriven cavity. In order to observe the nous solutions of Eq. (1) are plotted for a 1 W driving two states in the bistable region, we perturb δ1 to allow power in Fig. 4(f), as well as the boundaries of the os- the system to switch to the higher state. It then falls back cillation amplitudes between the Hopf bifurcations. The to the lower one while scanning δ2 in the forward or the measurements of the intracavity powers in the two res- backward direction. This is seen in Fig. 4(c), where only onators as δ2 is scanned back and forth are reported in the down-switching transition is reported for both scan- Fig. 4(g). They show a hysteresis cycle on the left and ning directions. The two δ2 values at which the switching self-starting oscillations on the right, whose locations are takes place coincide well with the theoretical location of in excellent agreement with the bifurcation analysis of SN2 and SN3 in Fig. 4(b). Eq. (1). At a slightly higher driving power, two new saddle- To prove that these oscillations correspond to self- pulsing, we next stabilize the second cavity around δ = node lines SN1 and SN4 appear as a necking bifurca- 2 0.47, i.e. in the middle of the unstable region. A stable tion [37] is crossed [see n41 in the inset of Fig. 4(a)]. At oscillation in both cavities is measured over 10 000 round this point, SN3 touches the lower state and a connection to the upper state emerges (not shown). It is beyond trips [see Figs. 5(a) and 5(b)]. The trajectory in the (P ,P )-phase plane reported in Fig. 5(c) shows a limit that power that self-pulsing occurs. The SN4 line how- 1 2 cycle behavior consistent with the numerical simulation ever quickly disappears through a cusp bifurcation at c34. of Eq. (1) displayed in Fig. 5(e). This confirms that the Just above the power corresponding to n41, SN1 and SN2 are located on either side of the newly formed connection DDBH dimer model captures very well the nonlinear be- between the lower and upper states. An example is given havior of our system. We note that the deviations from the average orbit in the experiment are attributed to the in Fig. 4(d) for Pp = 600 mW. We show in the experimental scans [see Fig. 4(e)] that this leads to a hysteresis limitations of the stabilization technique of the two cav- cycle that stretches between adjacent resonances. The ity detunings. The oscillation period is about 33.5 round trips, both in the experiment and the simulation. This system jumps to the lower state at SN2 in the forward direction, then goes back to the upper state thanks to the value is larger than the one computed from the frequency connection in the avoided crossing region. In the back- splitting between the linear hybridized modes [23] (27.1 round trips). This difference can be explained by the ward direction, it first switches to the upper state at SN1, √ then goes back down and stays on the lower state until weak coupling, since κ ≈ θ12 in our system. the next resonance. 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