Nonlinear Transport through Parity–Time Symmetric Lattice Potentials

Abstract

We study nonlinear transports of a light field through finite parity–time symmetric lattice potentials. The initial light field is trapped in a source reservoir and is released to expand toward the lattice potentials along the transverse direction due to the nonlinearity. We identify the transports that can be classified into in-band and in-gap transports. In the in-band transport, the light field can tunnel through the lattices into the sink reservoir, and in the in-gap transport, the light field is self-trapped inside the lattices to form a solitary wave.

1. Introduction

Transport properties through potential barriers are fundamentally interesting in quantum physics. They are standard textbook problems. In solid-state systems, a two-dimensional electron gas confined in structured semiconductor heterostructures provides an experimentally controllable means to investigate transport phenomena [1,2,3,4]. Light fields and ultracold atomic gases have been fundamentally physical realizations of quantum simulators [5]. Transports of light fields and ultracold atoms have been widely explored. These two systems also feature the existence of nonlinearity. How nonlinearity affects transport has become an active research field. A spatially localized wave packet of a light beam, arising from the balance between nonlinearity and dispersion, can tunnel through a potential barrier [6]. The nonlinear tunneling of a plane wave through a small potential has been experimentally shown to relate with nonlinearity-induced hysteresis [7]. A wave-packet with an engineered negative effective mass by adjusting nonlinearity has an exotic tunneling behavior through a potential barrier [8]. In both systems of light fields and ultracold atoms, lattice periodic potentials can be easily implemented. For lights, the lattices can be realized with wave-guide arrays [9] or optically induced photonic lattices [10,11]. For ultracold atoms, optical lattices are the standard lattice potentials [12]. Typical transports in lattice potentials are Bloch oscillations [13] and Landau–Zener tunneling [14,15]. After considering nonlinearity, exotic nonlinear transports in lattice potentials include the counter-intuitive self-trapping induced by a repulsive nonlinearity [16,17,18,19]; nonlinear Landau–Zener tunneling, where the adiabaticity is always broken due to nonlinearity no matter how slow the changes of parameters are [20,21,22]; and the breakdown of the superfluidity of nonlinear Bloch waves [23,24,25]. It has been revealed that the nonlinearity can lead to electromagnetically induced transparency amplification [26].

Recently, with the rapid development of non-Hermitian quantum physics, the non-Hermitian periodic potentials with the parity–time reversal (PT) symmetry have attracted great research attention [27]. The PT symmetric lattice potentials generate the Bloch band-gap energy spectrum [28], and all energy may be real-valued [29]. The linear transports of lights in such lattices have been revealed to exhibit exotic features stemming from the non-orthogonality of the Bloch waves [28]. There is a spontaneous phase transition between the PT unbroken and broken phases that can be reflected from whether the Bloch spectrum includes or does not include imaginary modes [30]. After the nonlinearity is introduced to the PT lattices, the nonlinearity can change critical values for the spontaneous PT phase transition [31]. Furthermore, nonlinear Bloch waves without linear analogues can form an asymmetric loop in the Bloch gaps, which plays a fundamental role in the PT phase transition [32]. Bloch oscillations and other transports in the PT symmetric lattice potentials behave differently than in the ordinary lattice potentials [33]. The nonlinear PT symmetric lattice potentials can support the existence of novel solitonic solutions [34]. A great amount of research attention has been paid to the study of the stable transports of these solitons in nonlinear PT symmetric lattices [35,36,37,38].

Here, we studied the nonlinear transports of a light beam through finite PT symmetric lattice potentials. Our study was motivated by possibly experimental realizations of non-Hermitian potentials [39,40,41,42,43] and the existence of nonlinearity in optical systems. However, our results can be easily extended to a similar transport of an ultracold atomic gas in non-Hermitian lattice potentials [16]. Our studied system is similar to that with finite Hermitian lattice potentials in Ref. [16]. It includes two reservoirs behaving as the source and sink for light field and finite PT symmetric lattice potentials placed between the two reservoirs. The initial light field is trapped in one of the reservoirs as the ground state. Once it is released, it shall be allowed to expand toward the lattice potentials due to the kinetic energy that is converted from the nonlinearity. The corresponding infinite lattice potentials feature a nonlinear Bloch band-gap spectrum. Depending on whether the initial spectrum of the light field trapped in the source reservoir is resonant with eigen modes in the nonlinear band or not, the tunneling through the finite lattice potentials can be categorized into two different types. They are in-band transport and in-gap transport. We identified features of these two transports with particular attention to the nonlinear effects. The nonlinear Bloch spectrum of the infinite lattice potentials may have a loop structure that is a pure nonlinear effect. We address that an in-loop transport is an in-band transport.

The rest of this paper is organized as follows. In Section 2, we model our transport system and provide a theoretical frame for the study of nonlinear transport in this system. In Section 3, we show in-band and in-gap transports and identify all features for these two different transports. We describe that the PT symmetrical lattice potentials provide a tunable platform for studying nonlinear transport. Finally, we demonstrate that an in-loop transport belongs to the in-band transport. The conclusion is presented in Section 4.

2. Model

The propagation of light waves in a medium with refractive index n can be described by a nonlinear paraxial equation [28,30,31,33,34]:

$$ik\frac{\partial E}{\partial Z}=-\frac{1}{2}\frac{{\partial }^{2}E}{\partial X^2}+k_0^2{n}_0\Delta nE.$$

(1)

Here, E describes the electric field envelope of the light field [28,34], and Z and X are propagation and transverse distances. $k_0=2\pi /\lambda$ is the wave vector of the light beam in free space with $\lambda$ being the wavelength. $n_0$ is the uniform refractive index of the medium, and $k=k_0{n}_0$. The total refractive index n is the sum of $n_0$ and the refractive index change $\Delta n$, i.e., $n=n_0+\Delta n$. The refractive index change $\Delta n$ includes the optically induced photonic potential and the Kerr nonlinearity [31]:

$$\Delta n=V_{\mathrm{pot}}+\frac{1}{2}{\alpha }_2{\left|E\right|}^2,$$

(2)

where the potential is categorized into three parts:

$$V_{\mathrm{pot}}=\left\{\begin{array}{cc}0\hfill & \mathrm{for}-L_{\mathrm{A}}-L\le X<-L,\hfill \\ V_{\mathrm{lat}}\left(X\right)\hfill & \mathrm{for}-L\le X\le L,\hfill \\ 0\hfill & \mathrm{for}L<X\le L+L_{\mathrm{B}}.\hfill \end{array}\right.$$

(3)

The scheme of this potential is shown in Figure 1. The lattice potential $V_{\mathrm{lat}}$ only exists in a finite region of $-L\le X\le L$; therefore, the lattice is finite, which supports us to study the transport through this finite lattice potential.

$$V_{\mathrm{lat}}\left(X\right)=n_{\mathrm{R}}{cos}^2\left(\frac{\pi }{D}X\right)+i{n}_{\mathrm{I}}sin\left(\frac{2\pi }{D}X\right),$$

(4)

Here, $n_{\mathrm{R}}$ and $n_{\mathrm{I}}$ describe modulation magnitudes of the refraction index by the real and imaginary lattices, respectively, and D relates to the period ofthe lattice potentials [28]. These lattice potentials possess the PT symmetry, $V_{\mathrm{lat}}^{*}(-X)=V_{\mathrm{lat}}\left(X\right)$. In Equation (2), the Kerr nonlinearity with the first order is represented by the coefficient ${\alpha }_2$, and when ${\alpha }_2>0$, the nonlinearity is self-defocusing, and it is self-focusing if ${\alpha }_2<0$. In this study, we mainly considered the self-defocusing nonlinearity. For the convenience of numerical calculations and resultant physics analysis, we made the nonlinear equation in Equation (1) dimensionless. It became [31,32]

$$i\frac{\partial \psi }{\partial z}=-\frac{1}{2}\frac{{\partial }^2\psi }{\partial x^2}+v_{\mathrm{lat}}\left(x\right)\psi +g\left|{\psi }^2\right|\psi .$$

(5)

Here, the dimensionless quantities are as follows: $x=\frac{\pi }{D}X$ and $z=\frac{{\pi }^2}{k{D}^2}Z$. Therefore, the units of the transverse distance x and the propagation distance z are $D/\pi$ and $k{D}^2/{\pi }^2$, respectively. $\psi =\sqrt{D/\left(P_0\pi \right)}E$ with $P_0$ is the total power of the incident light, i.e., ${\int \left|E\right|}^{2}dX=P_0$. The Kerr nonlinear coefficient is

$$g=\frac{P_0{k}_0^{2}D{n}_0{\alpha }_2}{2\pi }.$$

(6)

With these units, the field amplitude satisfies the normalization condition ${\int \left|\psi \right|}^{2}dx=1$. The PT symmetric lattice potentials become

$$v_{\mathrm{lat}}\left(x\right)=V_{\mathrm{R}}{cos}^2\left(x\right)+i{V}_{\mathrm{I}}sin\left(2x\right),$$

(7)

in the region of $-\pi L/D\le x\le \pi L/D$. The depths of the real and imaginary lattices are $V_{\mathrm{R}}=k_0^2{D}^2{n}_0{n}_{\mathrm{R}}/{\pi }^2$ and $V_{\mathrm{I}}=k_0^2{D}^2{n}_0{n}_{\mathrm{I}}/{\pi }^2$.

If the PT symmetric lattice potentials are infinite, then the light fields are Bloch waves [28]:

$$\psi (x,z)=e^{ikx+ibz}u\left(x\right).$$

(8)

Here, k is the quasimomentum, b is the propagation constant (which corresponds to the chemical potential in ultracold atomic systems), and $u\left(x\right)$ is a periodic function satisfying $u(x+\pi )=u\left(x\right)$. From the Bloch-wave function, we define the spectrum $\beta$ (which corresponds to the energy in ultracold atomic systems) [32] as follows:

$$\beta =-\int dx\left(\frac{1}{2}{\left|\frac{\partial \psi }{\partial x}\right|}^2+v_{\mathrm{lat}}{\left(x\right)\left|\psi \right|}^2+\frac{g}{2}{\left|\psi \right|}^4\right).$$

(9)

It is noted that the spectrum $\beta$ is the propagation constant b in the absence of nonlinearity. Generally, they are not equal in the presence of nonlinearity. The relationship $\beta \left(k\right)$ forms the Bloch band-gap spectrum. In the absence of nonlinearity, $g=0$, when $V_{\mathrm{I}}<V_{\mathrm{R}}/2$, all Bloch spectra are real-valued. Correspondingly, the system is said to be in the PT unbroken phase, and when $V_{\mathrm{I}}>V_{\mathrm{R}}/2$, some imaginary modes appear in the first two bands around the Brillouin zone edges, so the system is in the PT broken phase [28]. $V_{\mathrm{I}}=V_{\mathrm{R}}/2$ is the critical condition for the phase transition between the PT unbroken and broken phases. The transport of light beams in such linear lattice potentials has been studied [28]. In considering the nonlinearity, $g\ne 0$, the infinite PT symmetric lattices can still support nonlinear Bloch waves; therefore, the resultant spectrum $\beta \left(k\right)$ is a nonlinear Bloch spectrum. Similar to the linear case, whether the nonlinear Bloch spectrum includes imaginary modes can be used to identify the phase transition between the PT unbroken and broken phases [31]. It has been found that the nonlinearity can modify the critical condition for the phase transition, which means that it does not necessarily happen at $V_{\mathrm{I}}=V_{\mathrm{R}}/2$ as in the linear case [31]. Furthermore, there may exist exotic nonlinear Bloch waves that do not have linear analogues. They are located in the energy gaps of the nonlinear Bloch spectrum, forming a symmetric loop structure. These solutions dramatically change the critical condition for the phase transition [32]. The propagation of nonlinear Bloch waves may be destroyed due to dynamical instability [32].

We studied the transport of light beams through the finite PT symmetric lattice potentials. The initial light field is prepared in reservoir A, the ground state [see Figure 1]. This ground state is numerically obtained by calculating the nonlinear equation in Equation (5) with a box potential using the imaginary-time evolution method. Once the initial state is prepared, the right side wall (represented by the dashed vertical line in Figure 1) is removed. The initial field is allowed to expand toward the lattice due to the drivings originating from the release from the reservoir trap and kinetic energy that is converted from the nonlinearity as well. The evolution of the light field is obtained through the numerical calculation of the nonlinear equation in Equation (5).

When the imaginary lattices dominate over the real lattices, the nonlinear spectrum shall include imaginary-energy excitations; therefore, the system is in the PT broken phase. When the light field expands into the lattice region, the imaginary-energy excitations would be excited, leading to the exponential grow of the light field. The field always gains energies from the non-Hermitian lattices and evolves divergently. In the following, we shall study the nonlinear transport of the light field in the PT unbroken phase.

3. Nonlinear Transport

3.1. An In-Band Transport

A typical nonlinear transport is shown in the upper panel in Figure 2 with the parameters $g=198$, $V_{\mathrm{R}}=0.28$, and $V_{\mathrm{I}}=0.04$. After the light field expands into the lattice potentials, we can define the total power inside the lattices $P_{\mathrm{lat}}$ as [16]

$$\begin{array}{c}\hfill P_{\mathrm{lat}}\left(z\right)={\int }_{-L}^{L}dx{\left|\psi (x,z)\right|}^2.\end{array}$$

(10)

The evolution of $P_{\mathrm{lat}}$ is recorded in Figure 2(a1). The full distributions ${\left|\psi (x,z)\right|}^2$ around the lattice region of $-L<x<L$ are shown in Figure 2(b1). As the light propagates along the z direction, a fraction expands into the lattices [see the red line in Figure 2(c1)]; as a result, $P_{\mathrm{lat}}$ increases starting from 0. Soon, it reaches a plateau when $z>400$. This is because before the plateau, the light field starts to tunnel outside the lattices into reservoir B. Around the plateau, the continuous expansion into the lattice region from reservoir A is balanced with leakage on the right side, leading to a constant total power inside the lattice region. As z increases, the other plateau in $P_{\mathrm{lat}}$ appears. Between these two plateaus, there is a fast increase. The reason for the discontinuity of the two plateaus is the reflections of the light field around the end of the lattices $x=L$. As shown in Figure 2(b1), for a very small z, when the light field reaches the end of the lattices, it has the possibility to be reflected back, and the reflected parts travel oppositely toward reservoir A. The reflected parts are interfering with forward-moving parts, causing the dark stripes that generate around the lattice end and travel back. It is noticed that the reflections only happen when the light field is very weak around the lattice end. Figure 2(b1) shows that there is no reflection of a strong light field around the lattice end. In the first plateau of $P_{\mathrm{lat}}$, i.e., $400<z<800$ [see Figure 2(a1)], there is one dark interference stripe moving backward. In contrast, in the second plateau, $z>800$, there is no reflection [see Figure 2(b1)]. Further, in propagating z, $P_{\mathrm{lat}}$ starts to decrease around $z=1300$. This is because the main part of the light field has been transported through the lattice region. From Figure 2(b1), it can be seen that the field on the left side of the lattice region has a dramatic decrease around $z=1500$. The typical distribution (represented by the purple solid line in Figure 2(c1)) also indicates the decrease in the field on the left side of the lattice region. These results indicate that the light field left inside reservoir A becomes weak.

In order to show the nonlinear transport more clearly, we calculate the power flow j as follows [34]:

$$j(x,z)=\frac{1}{2i}\left[{\psi }^{*}{\partial }_x\psi -\psi {\partial }_x{\psi }^{*}\right].$$

(11)

In the interesting study in [44], an alternative definition of power flow is provided based on a continuity equation from the non-Hermitian dynamical equation. We use the definition in Equation (11), which does not obey the continuity equation; however, it is an experimentally measurable quantity. The power flow at the end of the lattices $x=L$,

$$j_{\mathrm{end}}=j(x=L,z),$$

(12)

is shown in Figure 2(d1). As z increases, $j_{\mathrm{end}}$ starts from 0, quickly reaches a steady state, and has a very small and fast oscillation around the steady value (represented by the horizontal dashed line in the figure). The steady value can be explicitly obtained by calculating the change in the power inside reservoir B:

$$P_B={\int }_L^{L+L_{\mathrm{B}}}dx{\left|\psi \right|}^2,j_{\mathrm{ste}}={\left.\frac{d{P}_B}{dz}\right|}_{z=z_{\mathrm{pla}}},$$

(13)

where $P_B$ is the power inside reservoir B, and $j_{\mathrm{ste}}$ is the change in the power $P_B$ at the propagation distance $z_{\mathrm{pla}}$, where the plateaus in $P_{\mathrm{lat}}$ in Figure 2(a1) appear. Physically, in the existing regions of the plateaus, the input into the lattices is equal to the output at the end of the lattices. In considering the continuity of the power flow, the steady value calculated from reservoir B naturally reflects the output at the end of the lattices. For a sufficiently large z, the power flow $j_{\mathrm{end}}$ decreases from the steady value. This region corresponds to the decreasing of $P_{\mathrm{lat}}$ after the plateaus, indicating that the main part of the light field transports through the lattices.

The nonlinear transport shown in the upper panel in Figure 2 indicates an in-band transport. In the plateaus, $P_{\mathrm{lat}}$ keeps a constant value; therefore, the average power inside the lattices is $P_{\mathrm{lat}}/\left(2L\right)$. If we assume that the PT symmetric lattices are infinite, the resultant Bloch waves should be normalized over a unit cell as ${\int }_0^{\pi }dx{\left|\psi \right|}^2/\pi =1$. As a result, the light field should be re-normalized ${\left|\psi \right|}^2\to {\left|\psi \right|}^2{P}_{\mathrm{lat}}/\left(2L\right)$ in order to make the Bloch waves re-normalized into 1 over a unit cell of the infinite lattices [45]. Consequently, the nonlinearity coefficient is also re-normalized as $g\to g_{\mathrm{lat}}$ with

$$g_{\mathrm{lat}}=g{P}_{\mathrm{lat}}/\left(2L\right).$$

(14)

Therefore, the Bloch waves of the infinite lattices are described by the nonlinear equations in Equation (5) by replacing g with $g_{\mathrm{lat}}$; in this way, the Bloch waves have the average power $P_{\mathrm{lat}}/\left(2L\right)$ and are normalized into 1 over a unit cell. With these re-normalized quantities and the re-normalization condition over a unit cell, we numerically solve the nonlinear equation in Equation (5) with the infinite PT symmetric lattice potentials and obtain the nonlinear Bloch-wave solutions. At last, we have the nonlinear Bloch spectrum $\beta \left(k\right)$ from Equation (9). We adopt the value of $P_{\mathrm{lat}}$ in the second plateau in Figure 2(a1), and the re-normalized $g_{\mathrm{lat}}$ becomes $g_{\mathrm{lat}}=0.198$. The calculated first two nonlinear Bloch bands are depicted in Figure 2(e1). These two nonlinear bands have a gap opened around the Brillouin zone edge $k=0.5$. The spectrum of the initial field trapped in reservoir A is also shown in this figure as a horizontal dashed line. It can be seen that the initial spectrum just lies inside the first nonlinear band. Physically, we think that the initial spectrum is resonant with the eigen modes of the finite lattice potentials, and the initial light field is allowed to tunnel through the lattice potentials [16]. The decreases of $P_{\mathrm{lat}}$ in Figure 2(a1) and $j_{\mathrm{end}}$ in Figure 2(d1) for a sufficiently large z provide signatures that the light wave packet transports through the finite lattices. Therefore, the nonlinear transport belongs to the in-band transport.

3.2. An In-Gap Transport

Another typical nonlinear transport is shown in the lower panel in Figure 2 with the parameters $g=398$, $V_{\mathrm{R}}=0.28$, and $V_{\mathrm{I}}=0.014$. The total power inside the lattice region $P_{\mathrm{lat}}$ increases from 0 and reaches a wide plateau when $z>400$. Before this large plateau, $P_{\mathrm{lat}}$ experiences two short plateaus as z increases [see Figure 2(a2)]. The existence of these two short plateaus is because there are two dark interference stripes when $z>200$ moving backward, as shown from the power distribution ${\left|\psi (x,t)\right|}^2$ in Figure 2(b2). It is interesting to note that these two stripes are not reflected back from the end of the lattices, but from the inside. The formation of the large plateau $z>400$ in $P_{\mathrm{lat}}$ is also very different from the in-band transport in Figure 2(a1). As can be seen from Figure 2(b2), in the large plateau region, the power distribution keeps the profile without the change as z increases and only occupies the left side of the lattices. A snapshot distribution in the large plateau is indicated in Figure 2(c2) by the red solid line. This clearly shows that the light wave packet localizes on the left side of the lattices and only a very small tail extends into reservoir B. Such a weak extension gives rise to a small power flow into reservoir B. Figure 2(d2) shows that in the large plateau region of $P_{\mathrm{lat}}$, the power flow at the end of the lattice $j_{\mathrm{end}}$ oscillates around a steady value that can be obtained from the information in reservoir B using Equation (13). The steady value is smaller in comparison with the in-band transport in Figure 2(d1). Therefore, the formation of the large plateau is due to the fact that there is no further input or output in the lattices. The light wave packet behaves like a solitary wave trapped inside the lattices as z increases [16]. The distribution profile ${\left|\psi \left(x\right)\right|}^2$ [see the red line in Figure 2(c2)] is very similar to the solitary-wave solutions existing in the nonlinear infinite lattices that were originally found in Refs. [45,46]. We emphasize that the localization of the light field is the effect of the defocusing nonlinearity, which is very counter-intuitive since the defocusing nonlinearity usually prefers to expand a wave packet. If there is no nonlinearity, the field that penetrates the lattice region will be transported away and cannot be trapped.

In other words, with the help of nonlinearity, the light field cannot transport through the lattices during propagating in the large plateau z region. However, the propagation of such a solitary wave is still accompanied by a very weak leakage at the right end of the lattices [see Figure 2(d1), i.e., $j_{\mathrm{end}}\ne 0$]. After a sufficient, long propagation z, the accumulation of the weak leakage becomes serious and makes the solitary wave destabilize. At this moment, the solitary wave begins to collapse. Most of the power shall be transported through the lattices into reservoir B, causing a sharp increase of $j_{\mathrm{end}}$ around $z=1000$, as shown in Figure 2(d2). Due to such an increase of $j_{\mathrm{end}}$, the total power inside the lattices $P_{\mathrm{lat}}$ decreases dramatically around $z=1000$ [see Figure 2(a2)]. The accumulation of the weak leakage that induced transport is similar to the destabilization of the solitary-wave solutions in the infinite lattices that were studied in Ref. [47]. After the collapse, the light field can tunnel through the lattices, and there are continuous inputs. The behavior of the light field is similar as that in the in-band transport shown in the upper panel in Figure 2. In Figure 2(c2), the black dashed line shows the field distribution at $z=1400$. We can see that the distribution shares many similarities with the in-band transport distribution shown by the black dashed line in Figure 2(c1).

The absence of transport due to the formation of the solitary wave and the sharp increase of $j_{\mathrm{end}}$ induced by the destabilization of the solitary wave is a signature of the in-gap transport. We use the value of $P_{\mathrm{lat}}$ in the large plateau in Figure 2(a1) and calculate the re-normalized nonlinear coefficient as $g_{\mathrm{lat}}=0.38$. Finally, we can obtain the nonlinear Bloch spectrum $\beta \left(k\right)$ after calculating the nonlinear Bloch-wave solutions for the infinite lattice potentials with the re-normalized nonlinear coefficient. The first two nonlinear bands are shown in Figure 2(d2). These two bands are similar to those of the in-band transport in Figure 2(d1). However, when we place the spectrum of the initial light field trapped in reservoir A in this figure [see the horizontal dashed line], we find that the initial spectrum lies inside the gap of the nonlinear Bloch spectrum. There is no eigen mode of the finite lattice that can be resonant with the initial spectrum; therefore, the initial light field cannot tunnel through the lattices but forms a solitary wave due to the existence of nonlinearity. The breakdown of the solitary wave can trigger the transport through the lattices. This is because the breakdown reduces $g_{\mathrm{lat}}$. Consequently, the nonlinear spectrum moves upward because of the reduced $g_{\mathrm{lat}}$, and then the initial spectrum can enter into the second band of the nonlinear spectrum, generating an in-band transport.

3.3. Tunable Transport

The presence of the imaginary lattices provides a tunable opportunity to study the nonlinear transport of the light field. It is noticed from Figure 2 that both in-band and in-gap transports feature the steady power flow $j_{\mathrm{ste}}$, which can be calculated from Equation (13). For different values of imaginary lattice amplitudes $V_{\mathrm{I}}$, we repeated realizing transports and calculated the steady values, and we show the results in Figure 3. In Figure 3a,b, the nonlinear coefficients $g=198$ and $g=398$ are the same as the in-band and in-gap transports shown in the upper and lower panels, respectively, in Figure 2. As shown by the red dashed line in Figure 3a, with the increase of $V_{\mathrm{I}}$, the steady value stays the same as the in-band transport in Figure 2(d1), which indicates that the transport is always an in-band transport. When $V_{\mathrm{I}}>0.18$, the steady flow becomes divergent. This is because the large imaginary lattice amplitude breaks the PT symmetry and leads to the PT broken phase. Without the nonlinearity, the critical condition for the PT phase transition is $V_{\mathrm{I}}=V_{\mathrm{R}}/2=0.14$. However, the nonlinearity modifies the critical condition, and from the red dashed line, it can be seen that the critical value of $V_{\mathrm{I}}$ is larger than the linear prediction. The solid line in Figure 3a is for $V_{\mathrm{R}}=0.45$, and in the window of $V_{\mathrm{I}}$ the steady flow is always a constant; therefore, the transport is an in-band case. Figure 3b shows the case of a larger nonlinearity. The results are different from those in Figure 3a. For a very small $V_{\mathrm{I}}$, the red dashed line (representing $V_{\mathrm{R}}=0.28$) has the same value as the in-gap transport in Figure 2(d2). Therefore, for the small $V_{\mathrm{I}}$, the transport is still in-gap transport. As $V_{\mathrm{I}}$ increases, the steady value soon increases to a value that is the same as the value of the in-band transport. Here, the transport changes into the in-band transport. Further, increasing $V_{\mathrm{I}}>0.19$ leads it to start to diverge, which means that the PT symmetry is broken. The solid line represents the case of $V_{\mathrm{R}}=0.45$ [see Figure 3b]. When $V_{\mathrm{I}}<0.11$, the steady value indicates the in-gap transport. In increasing the imaginary lattice amplitude, the steady flow increases and enters a plateau with a larger value; therefore, the corresponding transport goes into the in-band transport.

3.4. An In-Loop Transport

In the presence of nonlinearity, the nonlinear Bloch spectrum may have exotic nonlinear solutions that do not have linear analogues. These nonlinear solutions form a loop structure located inside the gaps [32]. We shall study a transport with an initial spectrum of the light field trapped in reservoir A that is resonant with eigen modes in the loop. Such a transport is named an in-loop transport. A typical result is shown in Figure 4. For the chosen parameters, the initial ground state trapped in reservoir A is allowed to expand into the finite lattice potential. The evolution of the total power inside the lattice region is depicted in Figure 4a. As z increases, the total power $P_{\mathrm{lat}}$ increases dramatically to reach a plateau. When $z>900$, the plateau begins to collapse, and $P_{\mathrm{lat}}$ starts to decrease. In the plateau region, the propagation of the light field along the transverse direction is stationary. This can also be seen from the power flow $j_{\mathrm{end}}$ at the end of the lattices shown in Figure 4b. In the plateau region of $P_{\mathrm{lat}}$, $j_{\mathrm{end}}$ is mainly around a steady value (which is shown by the dashed line in the figure). However, beyond this region, when $z>900$, $j_{\mathrm{end}}$ starts to decrease. The behavior of $j_{\mathrm{end}}$ shown in Figure 4b is similar as that of the in-band transport in Figure 2(d1).

Based on the plateau of $P_{\mathrm{lat}}$, we calculated the re-normalized nonlinear coefficient from Equation (14) and obtained $g_{\mathrm{lat}}=0.378$. Then, the nonlinear Bloch-wave solutions of the infinite PT symmetric lattice potentials with the re-normalized nonlinear coefficient were systematically examined. Furthermore, from the nonlinear Bloch-wave solutions, we calculated the nonlinear Bloch spectrum. The first two bands in the nonlinear spectrum are shown in Figure 4c. The outstanding feature of the nonlinear spectrum is that there is an interesting loop structure located inside the gap. The nonlinear Bloch-wave solutions in the loop are exotic since they do not have linear analogues. Considering the initial spectrum (shown by the dashed line in the figure), we realized that the initial spectrum is resonant with the modes in the loop. Since the loop structure adheres to the first band, it may be considered a part of the first band. Consequently, the resonance with the loop structure means that the transport is an in-band transport. The transport results shown in Figure 4a,b are similar to those of the in-band transport shown in the upper panel in Figure 2, which is an indication that the in-loop transport essentially becomes the in-band transport.

4. Conclusions

We studied the nonlinear transport of light through finite PT symmetric lattice potentials. The lattice potentials included the real and imaginary lattices. The initial light was trapped in reservoir A as a ground state. It was allowed to expand into the lattice potentials once it was released from reservoir A. The expansion along the transverse direction was derived from the kinetic energy that was converted from the nonlinear energy. After tunneling through the lattice potentials, reservoir B behaved as a sink to hold the transported light field.

We found that the nonlinear transports can be categorized into two types: in-band and in-gap transports. The lattice potentials feature a nonlinear Bloch band-gap spectrum. The in-band transport happens when the initial spectrum of the light field trapped in reservoir A is resonant with eigen modes inside the nonlinear band. The in-band transport is characterized by the existence of a stationary period, during which the input to the lattice region from reservoir A is balanced with the output from the lattice region to reservoir B. After this period, the main part of the light field can tunnel through the lattice potential, and then the light field received by reservoir B will be reduced. The in-gap transport happens when the initial spectrum of the light field trapped in reservoir A lies inside the nonlinear gap of the lattice potentials. The in-gap transport is also characterized by the existence of a stationary period. But the origination of such a stationary period is totally different from that of the in-band transport. This is because the light field cannot tunnel through the lattice potentials but forms a solitary wave localized inside the lattice region. The formed solitary wave can propagate for a long distance leading to the stationary period. However, the propagation of the solitary wave is still accompanied by a very weak leakage into reservoir B. The accumulation of the weak leakage shall finally break the solitary wave and restart the tunneling into reservoir B. The nonlinear Bloch spectrum of the lattice potentials can feature a loop structure that adheres to the first nonlinear band. We identified that the in-loop transport that happens when the initial spectrum of the light field trapped in reservoir A is resonant with modes in the loop belongs to an in-band transport.