Interaction between two jackiw - rebbi states in interfaced binary waveguide arrays with cubic - quintic nonlinearity

 Communications in Physics, Vol.31, No. 1 (2021), pp. 23-33
 DOI:10.15625/0868-3166/15178
 INTERACTION BETWEEN TWO JACKIW-REBBI STATES IN INTERFACED
 BINARY WAVEGUIDE ARRAYS WITH CUBIC-QUINTIC NONLINEARITY
 TRAN X. TRUONG1,†, NGUYEN NHU XUAN1, NGUYEN-THE QUANG2,
 NGUYEN VAN TOAN1 AND NGUYEN TUAN LINH1
1Department of Physics, Le Quy Don Technical University,
 236 Hoang Quoc Viet str., 10000 Hanoi, Vietnam
2Department of Communications, Le Quy Don Technical University,
 236 Hoang Quoc Viet str., 100000 Hanoi, Vietnam
 E-mail: †tranxtr@gmail.com
 Received 25 June 2020
 Accepted for publication 21 July 2020
 Published 05 January 2021
 Abstract. We study the coupling and switching effects of two discrete relativistic quantum Jackiw-
 Rebbi states in interfaced binary waveguide arrays with cubic-quintic nonlinearity. Like in the
 case with Kerr nonlinearity, two Jackiw-Rebbi states can couple efficiently to each other in the
 low-power regime, show the switching effect in the intermediate-power regime, and possess the
 trapping effect in the high-power regime. However, in the case with cubic-quintic nonlinearity, if
 the input Jackiw-Rebbi state power is increased further, one can observe the quasi-linear coupling
 effect between two Jackiw-Rebbi states which has not been found between two Jackiw-Rebbi states
 in interfaced binary waveguide arrays with Kerr nonlinearity.
 Keywords: soliton; binary waveguide array; cubic-quintic nonlinearity.
 Classification numbers: 42.65.Tg; 42.81.Dp; 42.82.Et.
 I. INTRODUCTION
 Many interesting fundamental photonic phenomena such as discrete solitons (DSs) [1–3],
 discrete diffraction [1, 4], diffractive resonant radiation [5] have been found in waveguide arrays
 (WAs). Waveguide arrays have also been used to simulate fundamental effects in nonrelativistic
 quantum mechanics such as Zener tunneling [6], and photonic Bloch oscillations [7]. Recently,
 binary waveguide arrays (BWAs) have been intensively used to investigate several fundamental
 ©2021 Vietnam Academy of Science and Technology
24 INTERACTION BETWEEN TWO JACKIW-REBBI STATES IN INTERFACED BINARY WAVEGUIDE ARRAYS ...
relativistic quantum mechanics phenomena arising from the Dirac equation, e.g., Klein tunnel-
ing [8–10], Zitterbewegung [11], and Dirac solitons in the nonlinear regime [12–18].
 In 2017, two localized states (one is the optical analogue of a special state which is well
known as the Jackiw-Rebbi (JR) state emerging from the Dirac equation in the quantum field
theory [19], and the other is the trivial state) were found at the interface of two BWAs with opposite
propagation mismatches in the linear regime [20]. The JR state is well known for predicting
the charge fractionalisation phenomenon which is fundamental in the fractional quantum Hall
effect [21]. One of extraordinary features of the JR state is the topological nature of its zero-
energy solution which is considered to be a precursor to topological insulators [22]. Topological
photonics can play a crucial role in the development of robust optical circuits [23]. In 2019, the
JR states in interfaced BWAs were numerically demonstrated, as expected, to be also extremely
robust under influence of strong disturbance of various kinds such as the turning on/off of the
nonlinearity, the linear transverse potential, and the oblique incidence [24].
 In the nonlinear regime, the exact profiles and the detunings of two localized states in
interfaced BWAs have been found by using the so-called shooting method [25] both in the case of
Kerr nonlinearity [26] and cubic-quintic nonlinearity [27]. The interaction between JR states and
Dirac solitons in BWAs has been investigated in Ref. [28]. The interaction between two localized
states in the nonlinear regime of Kerr type has been systematically analyzed in Ref. [29]. It has
been shown in Ref. [29] that in the linear regime and the regime of Kerr nonlinearity two localized
states of different types do not interact practically at all, thus they can be protected from each other
in a reliable way even if they are located quite close to each other. Meanwhile, two localized states
of the same type can couple efficiently to each other and show the nonlinear switching effect like
in symmetric fiber couplers [29].
 As mentioned above, the interaction between two localized states in the linear regime and
in the regime of Kerr nonlinearity have been systematically investigated in Ref. [29]. The model
with Kerr nonlinearity is the simplest one for investigating third-order nonlinear effects in optics.
However, if the optical signals are intense enough one needs to take into account the fifth and even
higher-order terms for nonlinearity. The resulting equation in that case is the well-known cubic-
quintic nonlinear Schrodinger¨ equation (NLS) because it contains terms accounting for both the
third and fifth powers of the signal amplitude. In fiber optics this cubic-quintic NLS  ... ese two interfaces can support the discrete JR states.
 In Fig. 1(d) we plot the relative output power of the two JR states (with respect to the total
initial input power) at the propagation distance equal to the coupling length L = 106 as a function
of the factor f where the full model with the cubic-quintic nonlinearity based on Eqs. (1) is used.
The solid red (dashed blue) curve in Fig. 1(d) represents the relative output power of the JR state
 2
by summing up |an| for all n 0) with the saturation parameter bs = 0.4. For comparison
in Fig. 1(d) we also plot the green solid curve with label bs = 0 which represents the relative output
 TRAN X. TRUONG et al. 29
power of the JR state with all n < 0 in the case of Kerr nonlinearity. The latter curve has been
already shown in Fig. 5(f) in Ref. [29] (see the solid red curve therein). Now let us focus on the
similarities of the solid red curve and the solid green curve in Fig. 1(d). One can see that in the
low-power regime (when f 0 is also completely
transferred to the JR state with n < 0. As a result, the solid red curve is close to unity, whereas
the dashed blue curve is close to zero. However, in the nonlinear regime (when 0.8 < f < 1.8)
all the energy of the input JR state is also just trapped at the same interface, and almost nothing
is transferred to the other JR state. By changing the factor f around 0.4 one also can obtain
the switching effect of the JR states between two interfaces. So, with the model for both Kerr and
cubic-quintic nonlinearity, the features of the linear coupling and nonlinear switching between two
discrete JR states are quite similar to the ones between optical signals in symmetric fiber couplers
(see Ref. [3, Fig. 2.3]). It is worth mentioning that in fiber couplers the coupling and switching
happen just between two cores, whereas for JR states they happen between two interfaces (each
interface consists of two and nearby waveguides). Our simulations also confirm that the coupling
and switching effects of two trivial states (not shown here) have exactly the same features as
the ones of two discrete JR states illustrated in Fig.1 as mentioned in Ref. [29], the physical
reason behind the nonlinear switching between two discrete JR states is similar to the reason in
the case of symmetric fiber couplers [3]. The difference in the mode propagation constants due to
nonlinearity-based self-phase modulation (SPM) introduces a relative phase shift between the two
JR states and hinders complete power transfer between them. At sufficiently high input powers, the
phase difference due to SPM becomes large enough that the input JR state remains trapped in the
same interface into which it was initially launched. Note that under ideal conditions, the response
time of the switching effect is limited by the response time of the cubic-quintic nonlinearity which
is almost instantaneous. For silica, this response time is less than 10 fs [31].
 Now let us analyze the differences between the solid red curve and the solid green curve
in Fig. 1(d). First, when the factor f < 1.5 the red solid curve is slightly shifted to the right as
compared to the solid green curve. This little shift of the red solid curve can be better seen by
enlarging Fig. 1(d). Second, when f is too large (close to 2) one can see that both two curves
rise up again. However, the mechanism for this rising is totally different for these two curves.
Regarding the green solid curve representing the case with Kerr nonlinearity, when f is too large,
the input condition is too different from the nonlinear localized JR states investigated in Ref. [27].
So, after launching these intense beams into the system, our simulations (not shown here) reveal
that a significant amount of the input energy is immediately radiated in a chaotic manner to both
two sides of the system. That is the reason why the whole input beam is not trapped at the input
interface, but some energy is distributed to the other side of the system which leads to the rising
of the green solid curve. On the contrary, the rising of the red curve around f = 2 in Fig. 1(d)
happens in an orderly manner because of the saturation effect which makes the nonlinear terms
less important in that region, thus forces the system operate in a regime close to the transition
regime around f = 0.4 of the same curve. This mechanism will be clarified in more details in the
rest of this work.
 In Figs. 2(a,c,e) we investigate the dependence of the relative output power r as a function
of the multiplication factor f for three values of the saturation parameter bs = 0.6,0.8 and 1.0,
respectively. In order to better understand the influence of the nonlinear terms in Eqs. (1), let us
30 INTERACTION BETWEEN TWO JACKIW-REBBI STATES IN INTERFACED BINARY WAVEGUIDE ARRAYS ...
 Fig. 2. (Color online) (a,c,e) The switching character of two discrete JR states when the
 longitudinal length of waveguide arrays is equal to the linear coupling length L = 106
 for the saturation parameter bs = 0.6,0.8 and 1.0, respectively. (b,d,f) The nonlinear
 function g as a function of the factor f for the saturation parameter bs = 0.6,0.8 and 1.0,
 respectively. The position where g = 0 is indicated by the vertical green solid lines in
 (a,c,e), respectively. Parameters: −σ1 = σ2 = −σ3 = 1; κ = 1;γ = 1.
define the nonlinear function g( f ) as follows:
 h 2i 2
 g( f ) = 1 − bs (p f ) (p f ) , (6)
 p
where the parameter p = |σ1σ2|/[κ (|σ1| + |σ2|)], thus p f represents the peak amplitude of
the interface with n = (4, 5). In Figs. 2(b,d,f) we plot the nonlinear function g as a function
 TRAN X. TRUONG et al. 31
of the multiplication factor f for three values of the saturation parameter bs = 0.6,0.8 and 1.0,
respectively. As clearly seen in Figs. 2(b,d,f), when the factor f is small, the nonlinear function
g and the factor f increase at the same time. This is understandable because small values of f
mean that the peak amplitude of the input JR state is also weak, and thus the cubic term of the
nonlinearity in Eqs. (1) prevails over the quintic term. However, when f is increased further,
the quintic term now can compete with the cubic term which leads to the saturation effect of the
nonlinearity. As a result, the nonlinear function g cannot increase further, but gets its maximum
  √ 

value, and decreases for large values of f . At the value f = 1/ p bs the nonlinear function g
will be nullified. These positions of f are indicated in Figs. 2(a,c,e) by the vertical green solid
lines. If the factor f is increased further, then the nonlinear function g will become negative
which means that the nonlinearity now is self-defocusing for the peak amplitude at the input.
As already mentioned in the Introduction, it has been experimentally demonstrated that some
materials can switch the nonlinearity from self-focusing to self-defocusing when the input signal
is intense enough [33]. So, now it is clear why the relative output power r can rise up to unity
again in Figs. 2(a,c,e) when f is around the values which can nullify the nonlinear function g
(i.e., around the vertical green solid lines in Figs. 2(a,c,e)). This is because around these high
values of f the system again operates in the quasi-linear regime which is similar to the truly linear
regime for small values of f , and thus at the coupling length L = 106 almost all energy of the input
JR state from the interface with n = (4, 5) can be transferred to the JR state at the interface with
n = (−5,−4), like what happens with small values of f in the low-intensity regime in Fig. 1(b).
 Fig. 3. (Color online) (a) Coupling of two discrete JR states when bs = 0.8 and f = 1.68.
 (b) Coupling of two discrete JR states when bs = 1.0 and f = 1.5. Parameters: −σ1 =
 σ2 = −σ3 = 1;κ = 1;γ = 1.
 The possibility that the relative output power r can rise up again to unity for a certain
large value of f is demonstrated in Figs. 3(a,b) where the propagation of the input JR beams is
illustrated for bs = 0.8 with the multiplication factor f = 1.68 in Fig. 3(a), and for bs = 1.0 with the
multiplication factor f = 1.5 in Fig. 3(b). Both these input JR states are launched into the system
at the interface with n = (4,5). As expected, at the coupling length L = 106, all the energy of these
JR states is transferred to the JR states at the interface with n = (−5,−4), like what happens in
32 INTERACTION BETWEEN TWO JACKIW-REBBI STATES IN INTERFACED BINARY WAVEGUIDE ARRAYS ...
the linear regime. However, in the truly linear regime, this energy transfer between two interfaces
takes place periodically as clearly shown in Fig. 1(b), whereas this periodicity is not present in
the quasi-linear regime in Fig. 3. This is because in the truly near regime, or in the regime with
small values of f as shown in Fig. 1(b), one can totally neglect the nonlinear terms from Eqs. (1),
and the system is indeed a linear one. However, for high values of f as in Fig. 3 one cannot do
that, because the nonlinear function g is only calculated for the peak amplitude of the beam at the
input, but not for all components an of the beams. Therefore, even though at some values of f the
nonlinear function g defined in Eq. (6) can be nullified for the peak amplitude at the input (i.e., for
a4 and a5 at the input), but the nonlinear terms for a3, a6 and other components will significantly
differ from zero at the input. That is the reason we use the term quasi-linear for the high-intensity
regime close to the vertical green solid curves in Figs. 2(a,c,e).
 It is worth mentioning that in the regime of Kerr nonlinearity, because bs = 0, therefore the
nonlinear function g is a parabolic function of the factor f . Thus, the nonlinear function g quickly
increases without any saturation effect when the factor f is increased. So in the high-intensity
regime, the dynamics of these two cases (Kerr nonlinearity and cubic-quintic nonlinearity) will be
totally different, and one cannot observe the switching scenarios as in Figs. 2(a,c,e) at large values
of f (around the vertical green lines) for Kerr nonlinearity.
 Even though in this work we only show explicitly the interaction between two discrete
JR states in interfaced BWAs with cubic-quintic nonlinearity, our simulations (not included here)
confirm that the interaction between two trivial states also possesses exactly the same qualitative
features, and that two localized states of different types practically do not interact at all in the
regime of cubic-quintic nonlinearity as what happens in the regime with Kerr nonlinearity.
IV. CONCLUSIONS
 In conclusion, we have shown that for both Kerr nonlinearity and cubic-quintic nonlinearity
the coupling and switching effects between two discrete JR states in interfaced BWAs are quite
similar to what happens in symmetric fiber couplers: for low input intensities one can observe the
strong coupling between these two JR states, for higher intensities one can observe the switching
effect, whereas for even higher intensities the trapping effect of the JR states will take place.
However, if the JR state input intensity is increased up further, then the saturation effect of the
cubic-quintic nonlinearity will be important, which can lead to the strong coupling between two
discrete JR states again. This coupling in the quasi-linear regime for high input intensities with
cubic-quintic nonlinearity does not take place periodically during propagation, unlike coupling at
low intensities. Regarding two localized states of different types in interfaced BWAs, like in the
case of the linear regime and Kerr nonlinearity, they do not interact at all in the regime of cubic-
quintic nonlinearity. This kind of interaction between two localized states can potentially be useful
for optical switching, coupling and robust guiding in photonic integrated circuits.
ACKNOWLEDGMENT
 This research is funded by Vietnam National Foundation for Science and Technology De-
velopment (NAFOSTED) under grant number 103.03-2019.03.
 TRAN X. TRUONG et al. 33
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