Tunneling time and Faraday/Kerr effects in $\mathcal {PT}\text{-symmetric}$ systems

The concept of tunneling or delay time for a quantum particle tunneling through real potential barriers is conventionally defined through the integrated density of states, which is proportional to the imaginary part of the full Green's function of systems and positive definite in a real potential scattering theory. Following the definition in refs. [29–31], two components of the traversal time τ_E can be introduced by (ℏ = 1)

$$\begin{equation} \tau_E= \tau_2 + i \tau_1 = - \int_{ - \frac{\Lambda}{2}}^{ \frac{\Lambda}{2}}\mathrm{d} x \langle x | \hat{G} (E) | x \rangle, \end{equation} \tag{ 1 }$$

where τ₁ and τ₂ represent Büttiker-Landauer tunneling time and the Landauer resistance, respectively. The Λ stands for the length of the potential barrier, and the $\hat {G} (E) = \frac {1}{ E- \hat {H}}$ is the Green's function operator of the system. As shown in refs. [30,31], two components of the traversal time τ_E are linked to the scattering and transport amplitudes explicitly by

$$\begin{equation} \tau_2 + i \tau_1 = \frac{\mathrm{d}\ln [t (k)]}{\mathrm{d}E}+ \frac{ r^{(l)} (k) + r^{(r)} (k) }{4 E}, \end{equation} \tag{ 2 }$$

where t(k) and $r^{(l/r)}(k)$ are the transmission and left/right reflection amplitudes, respectively, and $k = \sqrt {2\, \textrm {m} E}$ is the momentum of particle. The transmission and reflection amplitudes can be obtained by finding scattering solutions of the Schrödinger equation,

$$\begin{equation} \hat{H} | \Psi_{E} \rangle = E | \Psi_{E} \rangle, \qquad \hat{H} = - \frac{1}{2\,\ \textrm{m}} \frac{\mathrm{d}^2}{\mathrm{d}x^2} +V(x), \end{equation} \tag{ 3 }$$

where m denotes the mass of the particle, and V(x) is the interaction potential.

In the conventional real potential scattering theory, the development of the concept of tunneling or delay time is fundamentally based on counting the probability that a particle spends inside of a barrier, see, e.g., refs. [32–34]. However, in complex potential scattering theory, the norm of states is no longer conserved, the probability interpretation of tunneling time becomes problematic. This can be understood by examining the spectral representation of Green's function in a complex potential scattering theory, which now depends on the eigenstates of both $\hat {H}$ and its adjoint $\hat {H}^\dag$,

$$\begin{equation} \hat{G} (E) = \sum_{i} \frac{ | \Psi_{E_i} \rangle \langle \widetilde{\Psi}_{E_i } |}{E- E_i}, \end{equation} \tag{ 4 }$$

where

$$\begin{equation} \hat{H}^\dag | \widetilde{\Psi}_{E} \rangle = E | \widetilde{\Psi}_{E} \rangle. \end{equation} \tag{ 5 }$$

In general the discontinuity of Green's function crossing the branch cut in the complex E-plane is a complex function. However, see ref. [25], it is real under $\mathcal {P}\mathcal {T}$ symmetry and equal to the imaginary part of the Green's function, though it may not always be positive as a consequence of the norm violation due to the complex potential. Hence, the conventional definition of density of states gets lost, and one should generalize and redefine correctly the tunneling time. In this review letter the tunneling time through $\mathcal {P}\mathcal {T}\text{-symmetric}$ barriers is defined by eq. (1). The generalized density of states in such a system is taken as the imaginary part of Green's function, and τ₁ now may be interpreted as a generalized Büttiker-Landauer tunneling time. The sign of generalized tunneling time τ₁ is directly related to the potential barriers, that either tend to keep a particle in or force it out. When τ₁ is negative it behaves similarly to a forbidden gap in a periodic system, being unreachable.

We consider an incident linearly polarized electromagnetic plane wave with an angular frequency ω entering the system from the left, propagating along the x-direction. The electric field's polarization direction of the incident wave is aligned with the z-axis: $\boldsymbol {E}_{0}(x) = e^{i \frac {\omega }{c}\sqrt {\epsilon _{0}} x} \hat {z}$, where ₀ denotes the dielectric constant of vacuum. A magnetic field B is applied in the x-direction, as depicted in fig. 1. The scattering of the EM wave is described by, see, e.g., refs. [35,36],

$$\begin{equation} \left[\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \frac{\omega^2\,\epsilon_{\pm}(x)}{c^2}\right]E_{\pm}(x) = 0, \end{equation} \tag{ 6 }$$

where $E_{\pm } = E_{y} \pm iE_{z}$ represents the circularly polarized electric fields. The variable $\epsilon _{\pm }(x)$ is defined as follows:

$$\begin{equation} \epsilon_{\pm}(x) = \begin{cases} \epsilon +V (x) \pm g, & \text{if } x \in \left[- \displaystyle\frac{L}{2}, \displaystyle\frac{L}{2}\right]\!, \\ \epsilon_0, & \text{otherwise}, \end{cases} \end{equation} \tag{ 7 }$$

where L represents the length of the dielectric slab and is the positive and real permittivity of the slab, see fig. 1. V(x) denotes the additional potential barriers that are placed inside of the slab, which can be easily manipulated and adjusted to implement the $\mathcal {P}\mathcal {T}$ symmetry requirement. g is the gyrotropic vector along the magnetic-field direction. The external magnetic field B is included into the gyrotropic vector g to make the calculations valid for the cases of both external magnetic fields and magneto-optic materials. The magnetic field causes the direction of linear polarization of both transmitted and reflected wave to rotate. As a consequence, both the outgoing transmitted and reflected waves exhibit elliptical polarization. The major axis of the ellipse is rotated relatively to the original polarization direction. The real part of the rotation angle describes the change in polarization for linearly polarized light, while the imaginary part indicates the ellipticity of the transmitted or reflected light.

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The complex rotational parameters characterizing the transmitted light can be expressed in terms of transmission amplitudes by

$$\begin{equation} \theta^{T}_2 + i \theta^{T}_1 = \frac{1}{2}\ln\frac{t_+ (\omega)}{t_- (\omega)}, \end{equation} \tag{ 8 }$$

where t_± represent the transmission amplitudes of transmitted electric fields. In the case of a weak magnetic field $(g\ll1)$, a perturbation expansion can be applied. The leading-order contribution can be obtained by expanding t_± around the refractive index of the slab in the absence of the magnetic field B :

$$\begin{equation} \theta^{T}_{2} + i \theta^{T}_1 = \frac{g}{2n}\frac{\partial\ln [ t (\omega)] }{\partial n}, \end{equation} \tag{ 9 }$$

where $n = \sqrt {\epsilon }$ represents the refractive index of the slab. Similarly, the leading-order expressions of complex angles of the Kerr rotation, in the case of a weak magnetic field, are given by

$$\begin{equation} \theta^{R^{(l/r)}}_{2} + i \theta^{R^{(l/r)}}_1 = \frac{g}{2n}\frac{\partial\ln [r^{(l/r) } (\omega)]}{\partial n}, \end{equation} \tag{ 10 }$$

where $r^{(l/r)}$ is the left/right reflection amplitudes in the absence of magnetic field B .

The $\mathcal {PT}$ symmetry can be implemented in quantum tunneling time and magneto-optic effects by imposing conditions on the interaction potential in the Schrödinger equation and on dielectric permittivity in eq. (6): $V(x) = V^*(-x)$. As discussed in ref. [25], the parametrization of the scattering matrix only requires three independent real functions in a $\mathcal {P}\mathcal {T}\text{-symmetric}$ system: one inelasticity, $\eta \in [1, \infty ]$, and two phase shifts, δ_1,2. In terms of η and δ_1,2, the transmission and reflection amplitudes are given by

$$\begin{align} \begin{array}{rcl} t &=&\eta \displaystyle\frac{ e^{2 i \delta_1 }+ e^{2 i \delta_2 }}{2},\\ r^{(r/l)} &=& \eta \displaystyle\frac{ e^{2 i \delta_1 }- e^{2 i \delta_2 }}{2} \pm i \sqrt{\eta^2-1} e^{i (\delta_1+ \delta_2)}. \end{array} \end{align} \tag{ 11 }$$

As the consequence of $\mathcal {PT}$ symmetry constraints, two components of the traversal time are also given in terms of η and δ_1,2 by

$$\begin{align} \begin{array}{rcl} \tau_1 & =& \displaystyle\frac{\mathrm{d} (\delta_1 + \delta_2) }{\mathrm{d}E} + \eta \frac{ \sin (2\,\delta_1) - \sin (2\,\delta_2) }{4 E}, \\ \tau_2 & = & \displaystyle\frac{\mathrm{d}\ln [ \eta \cos (\delta_1 - \delta_2)] }{\mathrm{d}E} +\eta \frac{ \cos^2\,\delta_1 - \cos^2\,\delta_2 }{2 E}. \end{array} \end{align} \tag{ 12 }$$

Only three FR and KR angles are independent, the FR and KR angles are given in terms of η and δ_1,2 by

$$\begin{align} \theta^{T}_1 &\!= \!\theta^{R}_1\,\!=\! \frac{g}{2n} \frac{\partial ( \delta_1\,\!+\! \delta_2) }{\partial n}, \quad \theta^{T}_{2} \!=\! \frac{g}{2 n} \frac{\partial \ln \left [ \eta \cos (\delta_1\,\!-\! \delta_2) \right ]}{\partial n}, \nonumber \\ \theta^{R^{(r/l)}}_{2}& = \frac{g}{2 n} \frac{\partial }{\partial n} \ln\lvert \eta \sin (\delta_1 - \delta_2) \pm \sqrt{\eta^2-1}\rvert. \end{align} \tag{ 13 }$$

$\theta ^{R^{(r/l)}}_{2}$ and $\theta ^{T}_{2}$ are related by $\theta ^{R^{(r)}}_{2} + \theta ^{R^{(l)}}_{2} = \frac {2 T }{ T -1} \theta ^{T}_{2}, \label {theta2RT}$ where $T =\eta ^{2}\,\cos ^{2} (\delta _{1} - \delta _{2})$ denotes the transmission coefficient.

It is known that when the wave propagation through a medium is described by a differential equation of second order, the expression for the total transmission from the finite periodic system for any waves (sound and electromagnetic) depends on the unit cell transmission, the Bloch phase and the total number of cells. The infinite periodic $\mathcal {P}\mathcal {T}\text{-symmetric}$ structures exhibit unusual properties, including the band structure, Bloch oscillations, unidirectional propagation and enhanced sensitivity, see, e.g., refs. [37–40] and references therein. However, the case of scattering in a finite periodic system composed of an arbitrary number of cells/scatters has been less investigated, despite the fact that any open quantum system generally consists of a finite system coupled with an infinite environment. In Hermitian theory, the averaged physical observables of a finite system approach the limit that depends on the crystal-momentum of an infinite periodic system as the size of a finite system is increased. However, in $\mathcal {P}\mathcal {T}\text{-symmetric}$ systems, new challenges emerge, the large size limit of a finite size system is only well-defined conditionally.

The $\mathcal {P}\mathcal {T}\text{-periodic}$ symmetric structure that consists of $2N+1$ cells, see fig. 2, can be assembled on top of a single cell. Following refs. [27,28,41], generic expressions for the transmission and reflection amplitudes for $2N+1$ cells of $\mathcal {P}\mathcal {T}\text{-periodic}$ symmetric structure in both tunneling time and magneto-optic effects cases can be presented as

$$\begin{align} t &= \frac{ 1 }{ \cos (\beta (2N+1) \Lambda) + i \mathrm{Im} [\frac{1}{t_0 }] \frac{\sin (\beta (2N+1) \Lambda)}{\sin (\beta \Lambda)}}, \nonumber \\ \frac{ r^{(l/r)}}{t } &= \frac{ r^{(l/r)}_0 }{t_0 } \frac{\sin (\beta (2N+1) \Lambda)}{\sin (\beta\Lambda) }. \end{align} \tag{ 17 }$$

β plays the role of crystal-momentum for a periodic lattice and is related to k or ω by

$$\begin{equation} \cos (\beta \Lambda) = Re \left [ \frac{1 }{t_0 } \right ]\!. \end{equation} \tag{ 18 }$$

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The factor $\frac {\sin (\beta (2N+1) \Lambda )}{\sin (\beta \Lambda )}$ in both transmission and reflection amplitudes reflects the combined interference and diffraction effects in finite periodic systems, which occur naturally in Hermitian one-dimensional finite-size periodic systems. It is interesting to see that eq. (17) holds up for both Hermitian and $\mathcal {P}\mathcal {T}\text{-symmetric}$ systems, which is highly non-trivial since the usual probability conservation property for Hermitian systems must be generalized in $\mathcal {P}\mathcal {T}\text{-symmetric}$ systems. As pointed out in ref. [27], the scattering amplitudes for a periodic multiple cells system can be related to single cell amplitudes in a compact fashion. This is ultimately due to the factorization of dynamics living in two distinct physical scales: short-range dynamics in a single cell and long-range collective effects of the periodic structure of the entire system. The short-range interaction dynamics is described by single cell scattering amplitudes and β represents the long-range correlation effect of the entire lattice system. The occurrence of factorization of short-range dynamics and long-range collective mode has been known in both condensed-matter physics and nuclear/hadron physics. As examples, particles interacting with short-range potential in a periodic box or trap, quantization conditions can be given in a compact formula that is known as Korringa-Kohn-Rostoker method [42,43] in condensed-matter physics, Lüscher formula [44] in lattice quantum chromodynamics and Busch-Englert-RzaŻewski-Wilkens formula [45] in the nuclear-physics community. Other related useful discussions can be found in, e.g., refs. [46–50].

Two types of singularities are present in scattering amplitudes: 1) a branch cut siting along the positive real axis in complex k- or ω-plane that separates physical sheet (the first Riemann sheet) and unphysical sheet (the second Riemann sheet); 2) poles of transmission and reflection amplitudes. These poles are called spectral singularities of a non-Hermitian Hamiltonian when they show up on the real axis [51–53], which yields divergences of reflection and transmission coefficients of scattered states.

The motion of poles in complex k- or ω-plane, fig. 3, has some profound impact on the value of τ₁ in tunneling time and Faraday and Kerr rotation angles $\theta _{1}^{T}$ and $\theta _{1}^{R}$ in $\mathcal {PT}\text{-symmetric}$ systems. The location of these poles are model parameters dependent and can be found by solving $1/t =0$. The normal or anomalous behaviours of τ₁ and $\theta _{1}^{T/R}$ are determined by the location of poles: when the poles are all located in an unphysical sheet (the second Riemann sheet), τ₁ and $\theta _{1}^{T/R}$ remains positive. As poles move close to and ultimately cross the real axis into the physical sheet (the first Riemann sheet), the poles generate an enhancement in τ₁ and $\theta _{1}^{T/R}$ near the location of the poles. The spectral singularities occur when the poles are located on the real axis, the transmission and reflection amplitudes diverge at the location of the poles. This can be easily understood with the motion of a single pole. Near the pole, the transmission amplitude is approximated by

$$\begin{equation} t(k) \propto \frac{1}{k-k_{pole}} = \frac{k-k_{re} - i \gamma}{ (k-k_{re})^2+ \gamma^2}, \end{equation} \tag{ 19 }$$

where $k_{pole} = k_{re} + i \gamma$, being k_re and γ the real and imaginary parts of pole position. The location of the pole in the physical sheet or unphysical sheet is determined by the sign of γ: unphysical sheet if γ < 0 and physical sheet if γ > 0. The tunneling time τ₁ or Faraday/Kerr rotation angle $\theta _{1}^{T/R}$ near the pole is thus dominated by

$$\begin{equation} \tau_1\,\sim \frac{m}{k} \frac{ \gamma}{ (k-k_{re})^2+ \gamma^2}, \end{equation} \tag{ 20 }$$

hence as the pole moves across the real axis into the physical sheet, γ changes its sign.

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The locations of the moving poles are controlled by model parameters. The spectral singularities only occur in a certain range of model parameters, the boundary of the range of the model parameters hence separates the normal behaviour of the tunneling time and Faraday/Kerr rotation angles where τ₁ and $\theta _{1}^{T/R}$ always remain positive from anomalous behaviours where τ₁ and $\theta _{1}^{T/R}$ may turn negative near location of the spectral singularities. When the model parameters are varied continuously, the tunneling time and Faraday/Kerr rotation angles in $\mathcal {P}\mathcal {T}\text{-symmetric}$ systems hence experience a phase-transition–like transformation. Using the expressions in eq. (15) and eq. (16) as a simple example, the conditions for spectral singularities are given by considering $1/t_{0}(k) =0$:

$$\begin{equation} \left (- \frac{\cot (kL) |\sin (kL)|}{\sqrt{2}}, \frac{k/m}{\sqrt{2} |\sin (kL) | } \right) = ( \cos \varphi_V, |V|). \end{equation} \tag{ 21 }$$

The solutions of the spectral singularities can be visualized graphically by observing the intersection of a curve and a line with (x,y) coordinates given by both sides of eq. (21) for a fixed $|V|$, see fig. 4 as an example. For a fixed $|V|=4$, the solutions of the spectral singularities can only be found in a finite range, $\varphi _{V} \in [0.4\,\pi , 0.67\,\pi ]$, in which the poles appear in the physical sheet, anomalous behaviour of tunneling time and Faraday/Kerr rotation angles occur and τ₁ and $\theta _{1}^{T/R}$ may turn negative. For a fixed $|V|=4$, only a single pole solution can be found in the complex k-plane, the motion of pole as φ_V is increased is illustrated in fig. 3.

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For a large-N system, the situation is even more interesting, the band structure and EPs start getting involved, competing with poles and playing the roles in turning τ₁ and $\theta _{1}^{T/R}$, see detailed discussion in ref. [27]. The band structure of the system is clearly visible for even small-size systems. The number of poles grows drastically with size, and the distribution of poles splits into bands. When the poles show up inside an allowed band of the system and all move across the real axis, they tend to flip the sign of the entire band. In some bands where two bands start merging together at an exceptional point, the exceptional points tend to force τ₁ and $\theta _{1}^{T/R}$ approaching zero and start to competing with poles, so the $\mathcal {P}\mathcal {T}\text{-symmetric}$ systems become almost transparent near EPs. The fate of τ₁ and $\theta _{1}^{T/R}$ near EPs now is the result of two competing forces: the poles and EPs.

As the number of cells is increased, all traversal time τ_1,2 and FR and KR angles demonstrate fast oscillating behaviour due to $\sin (\beta (2N+1)\Lambda )$ and $\cos (\beta (2N+1)\Lambda )$ functions in transmission and reflection amplitudes. For the large-N systems, we can introduce the traversal time per unit cell and FR and KR angles per unit cell, such as

$$\begin{equation} \hat{\tau}_{1,2} = \frac{ \tau_{1,2}}{(2 N+1) \Lambda}. \end{equation} \tag{ 22 }$$

The $N \rightarrow \infty$ limit may be approached by adding a small imaginary part to β: $\beta \rightarrow \beta + i \epsilon$, where $\epsilon \gg \frac {1}{(2N+1) \Lambda }$. As discussed in ref. [27], adding a small imaginary part to β is justified by considering the averaged FR and KR angles per unit cell, which ultimately smooth out the fast oscillating behaviour of τ_1,2 and FR and KR angles. Therefore, as $N \rightarrow \infty$, the two components of traversal time per unit cell approach

$$\begin{equation} \hat{\tau}_1 - i \hat{\tau}_{2} \stackrel{ N \rightarrow \infty}{\rightarrow} \frac{\mathrm{d}\beta }{\mathrm{d}E}, \end{equation} \tag{ 23 }$$

and FR and KR angles per unit cell approach

$$\begin{equation} \hat{\theta}^{T}_1 - i \hat{\theta}^{T}_{2} \stackrel{ N \rightarrow \infty}{\rightarrow} \frac{g}{2n} \frac{\partial \beta }{\partial n}, \qquad \hat{\theta}^{R^{(r/l)}}_{2} \stackrel{ N \rightarrow \infty}{\rightarrow} 0. \end{equation} \tag{ 24 }$$

The examples of tunneling time per unit cell for a $\mathcal {P}\mathcal {T}\text{-symmetric}$ finite system with five cells are shown in fig. 5, compared with the large-N limit results. As we can see in fig. 5, τ₁ oscillates around the large-N limit results. Even for the small-size system, we can see clearly that the band structure of infinite periodic system is already showing up. In the broken $\mathcal {P}\mathcal {T}\text{-symmetric}$ phase in fig. 5, EPs can be visualized even for a small-size system, where two neighbouring bands merge and the $\mathcal {P}\mathcal {T}$ becomes totally transparent: τ₁ approaches zero. FR/KR angles show a similar behaviour, see, e.g., fig. 4 and fig. 5 in ref. [28].

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The limiting cases in eq. (23) and eq. (24) work well and are mathematically well defined in bands where spectral singularities are absent on the real axis, the poles are all either in the physical sheet or already all crossed real axis into the unphysical sheet. The large-N limit is well defined and can be achieved by either averaging fast oscillating behaviour of τ_E or using $i \epsilon \text{-prescription}$ by shifting k off the real axis into the complex plane. However, in the bands where the divergent singularities show up on the real axis and the band is still in the middle of the transition between all positive and all negative bands of τ₁ and $\theta ^{T/R}_{1}$, see, e.g., fig. 8 in ref. [27], eq. (23) and eq. (24) break down, and the large-N limit becomes ambiguous and problematic. The question of how to define a physically meaningful large-N limit in the presence of spectral singularities is still open.