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Abstract
We theoretically study the transport properties in a one-dimensional photonic lattice influenced by the presence of side-coupled $\cal {PT}$-symmetric non-Hermitian defects. The $\cal {PT}$ symmetry is manifested as the complex potentials on the defects and the complex defect-lattice couplings, respectively. These two mechanisms are found to induce the Fano effect in the transport processes, with the different characteristics of it. Next, if the complex potentials and defect-lattice couplings co-exist, the Fano effect will be achieved more efficiently. However, further enhancing either of them can weaken the Fano interference seriously. Our findings reveal the physical essence of the Fano effect on the $\cal {PT}$-symmetric non-Hermitian defects, and the results can provide insights into the engineering and dynamical control of Fano resonances in non-Hermitian photonic structures.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Systems with non-Hermitian Hamiltonians have attracted extensive attention because they have opportunities to exhibit entirely real spectra if they possess parity-time ($\mathcal {PT}$) symmetry [1]. Researchers have investigated various $\mathcal {PT}$-symmetric structures from different perspectives, including the complex extension of the optical systems with complex refractive indices [2,3,4–7], topological insulators [8,9], open quantum systems [10], and the Anderson models for disorder systems [11–13]. Besides, $\mathcal {PT}$-symmetric non-Hermitian lattices have been investigated, following the experimental achievement in optical waveguides [14,15] and optical lattices [16]. These works promote the researches about the $\cal {PT}$-symmetric non-Hermitian systems. Meanwhile, theories about $\cal {PT}$ symmetry have been constructed gradually, in the fields of quantum mechanics [17,18], quantum field theories and mathematical physics [19,20].
Following the reports from the experimental side, some groups pay attention to the $\cal {PT}$-symmetric complex potentials in non-Hermitian systems, which are not isolated but usually embedded in the Hermitian waveguides. It is shown that as a scattering center, pure imaginary potentials break the conservation of the flow of probability [21,22]. Any real-energy eigenstate of a $\mathcal {PT}$ tight-binding lattice with on-site imaginary potentials shares the same wavefunction with a resonant transmission state of the corresponding Hermitian lattice embedded in a chain [23,24]. In $\cal {PT}$-symmetric two-dimensional complex honeycomb photonic lattices, the beam propagation is introduced and a class of conical diffraction phenomena is achieved where the formed cone is brighter and travels along the lattice with a transverse speed proportional to the square root of the gain and loss parameter [25]. By using coupled $\cal {PT}$-symmetric nonlinear Van der Pol oscillators, an asymmetric wave transport mechanism based on the coexistence of nonlinearity and $\cal {PT}$ symmetry has been demonstrated experimentally and theoretically. It shows that the transmitted signal remains relatively unpolluted by higher harmonics and the $\cal {PT}$ symmetry guarantees a high asymmetric ratio without compromising the intensity of the transmitted signal [26]. Due to its potential applications in quantum sensing and information processing, the optical nonreciprocal signal transmission has been studied widely in $\cal {PT}$-symmetric optical systems [27–30]. On the other hand, researchers find that $\cal {PT}$-symmetric imaginary potentials indeed play a special role in modulating transport behaviors. Bender $et$ $al$. point out that scattering experiment is an effective method to measure the energy of bound states in $\cal {PT}$-symmetric non-Hermitian quantum wells [31]. It has been reported that in a Fano-Anderson system, the $\mathcal {PT}$-symmetric imaginary potentials induce some pronounced effects on Fano interference, including changes from the perfect reflection to perfect transport, and rich behaviors for the absence or existence of the perfect reflection at one and two resonant frequencies [32]. Following these discussions, the correspondence between the resonant transmission and spontaneous $\cal {PT}$-symmetry breaking has been investigated [33–37]. In addition, the transport properties of other systems, such as the non-Hermitian one-dimensional finite-period systems with $\cal {PT}$ symmetry, quantum network, and two-dimensional quantum dot (cavity) system, have been investigated [38–40]. These reports indeed show that in $\cal {PT}$-symmetric systems with non-Hermitian Hamiltonians, the $\mathcal {PT}$-symmetric imaginary potentials are important for adjusting the quantum interference that governs the quantum transport process.
In the existed works, researchers investigate the effect of $\cal {PT}$ symmetry mainly by considering the presence of $\mathcal {PT}$-symmetric imaginary potentials. As is known, non-Hermitian coupling is the other typical mechanism to modify the properties of the $\cal {PT}$-symmetric quantum systems [41,42]. Thus, it is necessary to clarify its roles in modifying the transport behaviors, especially the quantum interference effect. In this work we would like to introduce the $\cal {PT}$-symmetric couplings in the photonic system to investigate the properties of the Fano resonance induced by them. For this purpose, we focus on the transport properties in the structure of one-dimensional (1D) photonic lattice, by supposing the $\cal {PT}$-symmetric defects to be side-coupled to the photonic lattice. Moreover, the $\cal {PT}$-symmetric mechanisms are assumed to exist in three ways, i.e., the complex potentials, the complex defect-lattice couplings, and their coexistence. This allows us to observe the interplay between the complex potentials and $\cal {PT}$-symmetric defect-lattice couplings. The numerical results show that both of them enable to induce the Fano effect in the photonic transport process, whereas the characteristics of the Fano lineshapes are opposite to each other. If these two mechanisms co-exist, the Fano effect will be weakened to a great degree, including the disappearance of Fano resonance and antiresonance. We then believe that this work can help to understand the influences of $\cal {PT}$-symmetric non-Hermitian terms on the photonic lattice systems.
2. Theoretical model
The 1D photonic lattice that we consider is shown in Fig. 1(a)-(b), which is side-coupled to two defect states affected by the $\cal {PT}$-symmetric complex potentials. In addition, $\cal {PT}$-symmetric complex defect-lattice couplings are assumed to occur in this structure. The Hamiltonian of the whole system is written as
with its each part given by
Fig. 1. Schematic of a 1D photonic lattice which is side-coupled to two defects with $\cal {PT}$-symmetric complex potentials, i.e., $\varepsilon _1=\varepsilon _0+i\gamma$ and $\varepsilon _2=\varepsilon _0-i\gamma$. Besides, $\cal {PT}$-symmetric complex defect-lattice couplings are taken into account with $\tau _1=\nu _0+\lambda e^{i\theta }$ and $\tau _2=\nu _0+\lambda e^{-i\theta }$.
In order to study quantum transport through this photonic lattice, the transmission function (TF) should be calculated. According to the previous works, the calculation can be performed via a variety of methods. In this work, we would like to choose the transfer-matrix method to solve the analytical expression of the TF, for presenting the transport properties of this system. The details are shown in the appendix. As a result, the TF of our considered system is given by
where ${\cal D}_{\mu }=(\omega -\mu _{LL})(\omega -\mu _{RR})-(\mu _{RL}+t_0)^{2}-t_0e^{ik}(2\omega -\mu _{LL}-\mu _{RR})+t^2_0e^{2ik}$ with $\mu _{\alpha \alpha }=\sum _{j=1}^2{\tau _{\alpha j}^{2}\over \omega -\varepsilon _j}$ and $\mu _{LR}=\mu _{RL}=\sum _{j=1}^2{\tau _{Lj}\tau _{Rj}\over \omega -\varepsilon _j}$.Following the above formulation, we are allowed to discuss the influences of the $\cal {PT}$-symmetric non-Hermitian defects on transport through the photonic lattice. To be specific, we focus on the occurrence of antiresonance. It is not difficult to find in Eq. (2) that the antiresonance takes place in the case of $\mu _{RL}+t_0=0$. The form of $\mu _{RL}$ is determined by the $\cal {PT}$-symmetric non-Hermitian factor. We would like to perform discussions from the following aspects.
On the one hand, when the non-Hermitian factor is only contributed by the complex potentials on the defects, i.e., $\tau _{L(R)1(2)}=\tau$ is real, and then the antiresonance in the TF spectrum occurs at
This indicates that the complex potentials change the positions of the antiresonances which are induced by the defect-assisted quantum interference. And under the critical condition of $\gamma ={\tau ^2 \over t_0}$, the two antiresonance points will merge into one, located at the point of $\omega =\varepsilon _0-{\tau ^2\over t_0}=\varepsilon _0-\gamma$. This result enlightens us to further understand the role of the $\cal {PT}$-symmetric complex potentials.On the other hand, if only the $\cal {PT}$-symmetric complex defect-lattice couplings exist, one antiresonance will appear in the TF spectrum, at the point of $\omega =\varepsilon _0-(\tau _{L1}\tau _{R1}+\tau _{L2}\tau _{R2})/t_0$. For the case of left-right asymmetric coupling [Fig. 1(a)], this result is expressed as
Next, when these two $\cal {PT}$-symmetric non-Hermitian factors coexist, i.e., the complex potentials and defect-lattice couplings, two antiresonances will occur with their positions obeying the relationship as $\omega =\varepsilon _0-{1\over 2t_0}(\tau _{L1}\tau _{R1}+\tau _{L2}\tau _{R2}\pm \sqrt {\Delta })$. $\Delta =(\tau _{L1}\tau _{R1}+\tau _{L2}\tau _{R2})^2-4t_0^2\gamma ^2-4it_0\gamma (\tau _{L1}\tau _{R1}-\tau _{L2}\tau _{R2})$. It is evident that the antiresonance points are relevant to more structural parameters. In the case of left-right asymmetry, the above result is simplified to be
In the case of upper-down asymmetry, it changes as3. Numerical results and discussions
We next perform numerical calculations to present the transport properties in the 1D photonic lattice modified by the two non-Hermitian factors, i.e., the $\cal {PT}$-symmetric complex potentials and defect-lattice couplings. For the structural parameters, we take $\varepsilon _0=0$, $t_{0}=1.0$, and $\nu _{0}=\lambda =0.5$, in the context.
To begin with, in Fig. 2(a) we consider the case of $\theta =0$ and plot the TF spectrum modified by the $\cal {PT}$-symmetric complex potentials. It clearly shows that in such a case, two antiresonance points appear in the TF spectrum, in the region below the energy zero point. With the increase of $\gamma$, the antiresonance points get close to each other, and the antiresonance valleys are widened. At the critical case of $\gamma =1.0$, the antiresonance points merge into one, accompanied by the appearance of a wide antiresonance valley. Such results are easy to understand with the help of Eq. (3). What we would like to say is that in this process, the resonance points around the positions of $\omega =-2.0$ and $\omega =0$ are robust, independent of the increase of $\gamma$. And increasing $\gamma$ is only to widen the resonance peaks but cannot change their positions. Therefore, one can be sure that the $\cal {PT}$-symmetric complex potentials drive the occurrence of Fano effect, and also, they modulate the Fano interference by changing the antiresonance positions and widening the Fano resonance.
Fig. 2. TF curves affected by the $\cal {PT}$-symmetric complex potentials or defect-lattice coupling, respectively. (a) Influence of the complex potentials. (b)-(c) Changes of the TF spectra due to the presence of $\cal {PT}$-symmetric defect-lattice coupling.
As a counterpart, the case where $\gamma =0$ but $\theta \neq 0$ is exhibited in Fig. 2(b)-(c) to describe the effects of the $\cal {PT}$-symmetric defect-lattice couplings. In such a case, our considered system possesses two configurations, i.e., the left-right and upper-down asymmetry, respectively [see Fig. 1(a)-(b)]. Their corresponding results are shown in Fig. 2(b)-(c). One can find that the $\cal {PT}$-symmetric defect-lattice couplings also lead to the occurrence of Fano effect in transport through the photonic structure, but their effects basically differ from the complex potentials. Moreover, the TF spectra induced by the left-right and upper-down asymmetry are different from each other. Firstly, for the small $\theta$, one Fano lineshape appears in the low-energy limit of the TF spectrum [see Fig. 2(b)]. And with the increase of $\theta$, the antiresonance moves to the high-energy direction, and the Fano lineshape is modified due to the widening of the Fano resonance peak. In the case of $\theta =0.5\pi$, the Fano resonance begins to transform into a plateau, so the Fano lineshape becomes ambiguous. However, further increasing $\theta$ leads to an alternative result. Following the right shift of the antiresonance point, the TF magnitude in the low-energy region is improved to a great degree. This weakens the asymmetry structure of the TF spectrum and narrows the antiresonance valley at the same time. For another, if the defect-lattice couplings are taken to be of upper-down asymmetry, the Fano interference will exhibit the other result, as shown in Fig. 2(c). We notice that the resonance position is located at the low-energy limit, which cannot be moved by the increase of $\theta$. For the change of antiresonance valley, it is basically similar to the case of left-right asymmetry. In addition, note that in the case of $\theta =0.5\pi$, the TF spectrum is transformed into a straight line, irrelevant to the change of $\omega$. After then, the antiresonance point reappears and enters into the positive-energy region, which gets close to the energy zero point with the increment of $\theta$.
In view of the results in Fig. 2, we know that the $\cal {PT}$-symmetric non-Hermitian factors are important for the occurrence of the Fano effect in transport through the photonic lattice. In the presence of different non-Hermitian mechanisms, e.g., the complex potentials or the defect-lattice couplings, the properties of the Fano effect are differentiated from one another. As is known, the Fano effect is usually discussed by transforming the expression of $T(\omega )$ into its Fano form, and the parameters in the Fano form are helpful for understanding the Fano effect [45]. Following this idea, we would like to perform a discussion by presenting the Fano form of $T(\omega )$. After a straightforward deduction, the Fano form of the TF is written as
in which $q=0$ and with ${\cal A}=(\omega -\mu _{LL})(\omega -\mu _{RR})+t_0^2-(\mu _{RL}+t_0)^2$ and ${\cal B}=(\omega ^2-4t_0^2)(\omega -\mu _{LL})(\omega -\mu _{RR})+t_0^2(2\omega -\mu _{RR}-\mu _{LL})^2-\omega {\cal A}(2\omega -\mu _{RR}-\mu _{LL})$. It can be found that when only the complex potentials are taken into account with $\theta =0$, there will be with $\mu _0={2(\omega -\varepsilon _0)(\nu _0+\lambda )^2\over (\omega -\varepsilon _0)^2+\gamma ^2}$. It shows that under the condition of $\omega =\varepsilon _0$, one can get the result of $e\to \infty$. Resultantly, $T(\omega \to 0)=1.0$, which is manifested as the Fano resonance independent of the increase of $\gamma$. Similarly, in the case of $\omega =-2t_0$, the parameter $e$ also arrives at infinity, leading to the Fano resonance.If only the $\cal {PT}$-symmetric defect-lattice couplings are considered with $\gamma =0$, the Fano effect will depend on the defect-lattice coupling manners, i.e., the left-right and upper-down asymmetry. In the former case, $\mu _{\alpha \alpha }=\mu _1=2\rm {Re}(\tau _1^2)/(\omega -\varepsilon _0)$ and $\mu _{\alpha \alpha '}=\mu _2=2|\tau _1|^2/(\omega -\varepsilon _0)$, so
Fano resonance, labeled by the asymmetric lineshape, is surely produced by the interference of resonant and continuous states, which widely exists in various fields of physics [46,47]. In optics, they have been observed to exist in many systems, such as waveguide structures, plasmas, and metamaterials, accompanied by their application prospects [48–52]. Thus, it is necessary for us to recognize these two processes for clarifying the Fano effect in our system. It can be firstly found that the nonresonant process is certainly offered by the direct coupling between the two parts of the photonic structure, whereas the resonant processes originate from the contribution of the defects. As a result, three paths take part in the Fano interference. With the Feynman path language, they are written out respectively, i.e., $p_0=t_0$ describing the nonresonant path, and $p_1=\tau _{L1}g_1\tau _{R1}$ and $p_2=\tau _{L2}g_2\tau _{R2}$ being the resonant paths referred by the defects. Here $g_j={1\over \omega -\varepsilon _j+i0^+}$ are the propagators contributed by the defects. One can clearly find that the $\cal {PT}$-symmetric complex potentials or the defect-lattice couplings enable to change the amplitudes and the phases of $p_{2(3)}$, and then the Fano interference comes into being. To be specific, when the interference among them is completely destructive, i.e., $\sum _{l=0}^2 p_{l}=0$, the Fano antiresonance takes place. The condition for this result is exactly $\mu _{RL}+t_0=0$, which leads to $T(\omega )=0$. In different situations, the antiresonance point can be figured out, as shown in Eqs. (3)–(6). And then, the different Fano interferences occur. In addition, one notices that in the case of $\mu _{RL}+t_0=0$, the two parts of the photonic structure are decoupled from each other [see Eq. (A7)]. Therefore, the roles of the $\cal {PT}$-symmetric factors in driving the Fano interference are clearer.
After the above discussion, we would like to analyze the Fano effect caused by the interplay between the $\cal {PT}$-symmetric complex potentials and the defect-lattice couplings. Surely due to the different roles of them, the results will become more abundant and interesting. The numerical results are shown in Fig. 3 and Fig. 4. From Eqs. (6)-(7), we know that when one of the $\cal {PT}$-symmetric terms exist, the value range of the other will be narrowed for supporting the occurrence of antiresonance. To be concrete, compared with the case of one $\cal {PT}$-symmetric term, the smaller $\theta$ or $\gamma$ benefit the antiresonance. And such a phenomenon is more apparent for the upper-down symmetric structure.
Fig. 3. TF influenced by the interplay between $\cal {PT}$-symmetric complex potentials and defect-lattice couplings, in the case of left-right asymmetry. (a)-(c) $\gamma =0.2$, 0.5, and 1.0, respectively. (d)-(f) $\theta =0.2\pi$, $0.5\pi$, and $0.9\pi$.
Fig. 4. Spectra of the TF due to the interplay between $\cal {PT}$-symmetric complex potentials and defect-lattice couplings, when upper-down asymmetry is considered. (a)-(c) $\gamma =0.2$, $0.5$, and $1.0$, respectively. (d)-(f) $\theta =0.2\pi$, $0.5\pi$, and $0.9\pi$.
In Fig. 3, the $\cal {PT}$-symmetric defect-lattice couplings are of the left-right asymmetry. To be specific, Fig. 3(a)-(c) show the results of increasing $\theta$ from $0.2\pi$ to $0.9\pi$, when $\gamma =0.2$, $0.5$, and $1.0$, respectively. One can find the competition relation of the two non-Hermitian mechanisms in modulating the Fano interference. For instance, when $\gamma =0.2$, the Fano lineshape in the low-energy limit becomes governed by the non-Hermitian defect-lattice couplings, since it is similar to the result of $\gamma =0$. With the increase of $\theta$, this Fano lineshape shifts to the high-energy direction. However, for the Fano interference around the energy-zero point, it is weakened seriously in this process and then disappears when $\theta >0.5\pi$. In the extreme case where $\theta =0.9\pi$, the interference signal cannot be observed again, and the TF spectrum only exhibits one plateau in the whole energy region. Similar results also come into being in the case of $\gamma =0.5$, as shown in Fig. 3(b). Next, when $\gamma$ further increases to $1.0$, the non-Hermitian defect-lattice couplings exactly destroy the Fano interference induced by the $\cal {PT}$-symmetric complex potentials. And the increase of $\theta$ only enhances the magnitude of the TF spectrum seriously.
In Fig. 3(d)-(f), we consider the cases of $\theta =0.2\pi$, $0.5\pi$, and $0.9\pi$, respectively, to present the change of TF spectrum by strengthening $\gamma$. In Fig. 3(d) where $\theta =0.2\pi$, we see that with the increase of $\gamma$, the Fano antiresonance around the energy zero point is enhanced, due to the widening of the antiresonance valley. At the same time, the Fano lineshape in the low-energy limit is reversed, as shown by the results of $\gamma =0.5$ and 0.8. After this, the further increase of $\gamma$ leads to the degeneration of the two antiresonance points until the disappearance of antiresonance. Compared with the result of $\theta =0$, the critical condition of the antiresonance disappearance can be satisfied for a smaller $\gamma$. And in the case of $\gamma =1.0$, the Fano antiresonance is eliminated completely. For the case of $\theta =0.5\pi$, the weak complex potentials can induce the occurrence of antiresonance around the energy zero point. However, increasing $\gamma$ destroys the antiresonance gradually. And when $\gamma >0.5$, no antiresonance takes place in the whole energy region [see Fig. 3(e)]. Next when $\theta =0.9\pi$, the competition between these two mechanisms can also be well understood. In such a case, the TF curve becomes independent of the change of $\omega$. This indicates that the defects decouple from the photonic structure in this case.
Fig. 4 shows the results induced by the interplay between the complex potentials and the upper-down asymmetry. It is evident that due to the coexistence of these two factors, the parameter ranges for the Fano antiresonance are narrowed. For the cases of $\gamma \ge 0.5$ or $\theta \ge 0.5\pi$, one cannot find the antiresonance points in the TF spectra. Thus, such an interplay tends to take a more negative effect on the Fano interference. In Fig. 4(a)-(c), the results correspond to the cases of $\gamma =0.2$, $0.5$, and $1.0$, respectively. It shows that due to the interplay between the two factors, the role of defects becomes less apparent, which is manifested as the disappearance of the antiresonance, weak oscillation of the TF spectrum, and the improvement of the TF value in the whole region. More obvious results are achieved in the cases of $\gamma =0.5$ and $1.0$. As shown in Fig. 4(b), in the case of $\gamma =0.5$ with $\theta =0.2\pi$, the Fano interference is weakened to a great degree, despite the remnant of the Fano lineshape. Next when $\gamma =1.0$, the Fano interference is completely suppressed, even in the case of $\theta =0.2\pi$ [see Fig. 4(c)]. For further presenting the interplay between the two mechanisms, in Fig. 4(d)-(f) we consider the cases of $\theta =0.2\pi$, $0.5\pi$, and $0.9\pi$, to analyze the change of the TF spectrum by increasing the magnitude of the complex potentials. In Fig. 4(d) where $\theta =0.2\pi$, the complex potentials become more efficient to modulate the Fano interference, compared with those of $\theta =0$. Even when $\gamma$ increases to 0.5, the Fano antiresonance has been forbidden. On the other hand, if $\theta =0.5$, the TF plateau will be destroyed by introducing the complex potentials, accompanied by the weak oscillation of the TF spectrum. For the extreme case where $\theta =0.9\pi$, the complex potentials completely screen the effect of the defects and cause the TF spectrum to become one plateau in the whole region.
In fact, the interplay manner of the two $\cal {PT}$-symmetric complex factors can be clarified by paying attention to the phases of $\Delta$, i.e., $+$ or $-$. According to Eqs. (6)-(7), no antiresonance point appears in the TF spectrum in the case of $\Delta <0$, whereas two antireonance points can be observed if $\Delta >0$. In the critical case, i.e., $\Delta =0$, one antiresonance point comes into being. Thus, we plot the diagrams of $\Delta$’ phases in Fig. 5. Firstly, Fig. 5(a) shows the result in the case of left-right asymmetry. We see that $\gamma$ and $\theta$ play similar roles in modulating the phase of $\Delta$ in the regions of $\gamma <1.0$ and $\theta <\pi$. Besides, when one of them is relatively small, the other is allowed to vary in a wider range. Otherwise, they restrict mutually for leading to a positive $\Delta$. For comparison, the case of upper-down asymmetry is shown in Fig. 5(b). It is clearly found that in such a case, $\theta$’s role is completely different from $\gamma$. As $\theta >{\pi \over 2}$, the positive $\Delta$ is very difficult to be achieved, unless $\gamma$ is adjusted in the limit of small value. In the case of $\theta <{\pi \over 2}$, the restriction relation between $\gamma$ and it is obvious. As one is increased, the other should be decreased for realizing the positive $\Delta$ and its-induced antiresonance. These results help us to clarify the phenomena in Figs. 3–4. Especially for those in Fig. 4, the interplay between the two $\cal {PT}$-symmetric factors efficiently promotes $\Delta$ to enter its negative-phase region, so the antiresonance points are removed accompanied by the weak oscillation of the TF spectra. It should be noticed that these results are actually governed by the quantum interference among the Feynman paths involved. Take the upper-down symmetry as an example, increasing $\theta$ suppresses the magnitudes of $p_1$ and $p_2$ in a more efficient way. This exactly weakens the quantum interference effect. As a result, the Fano antiresonance is removed completely.
Fig. 5. Diagrams of $\Delta$’s phases, i.e., $+$ or $-$, with the adjustment of $\theta$ and $\gamma$. (a) Case of left-right asymmetry. (b) Case of upper-down asymmetry.
In Fig. 6 we would like to adjust the parameters of the defect-lattice couplings (i.e., $\nu _0$ and $\lambda$) to investigate the change of the TF spectrum. The other parameters are taken to be $\gamma =0.2$ and $\theta =0.2\pi$. In this figure, Fig. 6(a)-(b) show the case of left-right asymmetry, while Fig. 6(c)-(d) correspond to the upper-down asymmetry. For the former case, it shows in Fig. 6(a)-(b) that changing $\nu _0$ or $\lambda$ play a similar role in modifying the TF spectrum. The Fano resonance around the energy zero point is robust, but the antiresonance point in the negative-energy region shifts to the low-energy direction until its disappearance. This shift directly leads to the reversal of the Fano lineshape in the negative-energy region, which occurs when the values of $\nu _0$ or $\lambda$ increase from 0.4 to 0.5. On the other hand, the alternative phenomenon takes place in the latter case, as shown in Fig. 6(c)-(d). The small $\nu _0$ cannot induce the antiresonance point in the TF spectrum, despite the existence of complex potentials. When $\nu _0$ is great enough, e.g., $\nu _0>0.3$, antiresonance points begin to appear in the TF spectrum. Moreover, the antiresonance point is split into two points, following the further increase of $\nu _0$. And in such a process, the low-energy antiresonance shifts left, until it disappears [see Fig. 6(c)]. Instead, enhancing $\lambda$ only leads to the left shift of the antiresonance position. With the help of these results, we can further understand the effects of the different $\cal {PT}$-symmetric non-Hermitian factors on changes of the Fano effect, as well as the change of the TF spectrum.
Fig. 6. TF spectra due to the parameter changes of the $\cal {PT}$-symmetric defect-lattice couplings, in the case of $\theta =0.2\pi$ and $\gamma =0.2$. (a)-(b) Case of left-right asymmetry. (c)-(d) Case of upper-down asymmetry.
4. Conclusions
In summary, we have performed studies about the transport properties in a 1D photonic lattice influenced by the $\cal {PT}$-symmetric non-Hermitian defects. The defects are supposed to introduce two $\cal {PT}$-symmetric mechanisms, i.e., the imaginary potentials and complex defect-lattice couplings, respectively. As a result, it has been found that both of these two mechanisms induce the Fano effect in the transport process, but the Fano effect exhibits different characteristics. The imaginary potentials are more powerful in driving the Fano interference since they are able to induce two antiresonance points in the TF spectrum. Next, if the imaginary potentials and complex defect-lattice couplings co-exist in one system, the Fano effect will be induced in an alternative way. The antiresonance positions can be adjusted in a more efficient way, but further enhancing either of them weakens the Fano interference seriously. By presenting the analytical solution of the antiresonance points and obtaining the Fano form of the TF expression, we clarify the quantum transport properties of this structure. This work can be helpful for understanding the influences of $\cal {PT}$-symmetric non-Hermitian terms on the transport through photonic lattices. Also in view of the current technology in photonics, we consider that the obtained results can be observed in experiments.
Appendix: Solution of the transmission coefficient
To begin with, we would like to write out the wave function of the system as $|\psi \rangle =\sum _{n}c_{n}(\tau )|\phi _{n}\rangle +\sum _l d_l(\tau )|d_l\rangle$ with $|\phi _{n}\rangle =c^{\dagger} _{n}|0\rangle$ and $|d_l\rangle =d^{\dagger} _l|0\rangle$. By substituting expression of $|\psi \rangle$ into the Schrödinger equation $i\partial \tau |\psi \rangle =H|\psi \rangle$, the following coupled-mode equations are obtained for the expansion coefficients $c_{n}$ and $d_l$:
Next, we evaluate the quantum transport through this system. To do so, it is necessary for us to write out the trial wave function as
Funding
LiaoNing Revitalization Talents Program (XLYC1907033); Fundamental Research Funds for the Central Universities (N180503020); Liaoning BaiQianWan Talents Program (201892126); National Natural Science Foundation of China (11905027).
Disclosures
The authors declare no conflicts of interest.
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