Instantaneous modulations in time-varying complex optical potentials

We study the impact of a spatially homogeneous yet non-stationary dielectric permittivity on the dynamical and spectral properties of light. Our choice of potential is motivated by the interest in PT-symmetric systems as an extension of quantum mechanics. Because we consider a homogeneous and non-stationary medium, PT symmetry reduces to time-reversal symmetry in the presence of balanced gain and loss. We construct the instantaneous amplitude and angular frequency of waves within the framework of Maxwell's equations and demonstrate the modulation of light amplification and attenuation associated with the well-defined temporal domains of gain and loss, respectively. Moreover, we predict the splitting of extrema of the angular frequency modulation and demonstrate the associated shrinkage of the modulation period. Our theory can be extended for investigating similar time-dependent effects with matter and acoustic waves in PT-symmetric structures.

During the past years, a new class of Hamiltonians has been widely investigated, which extends quantum mechanics from the Hermitian into the non-Hermitian (complex) domain [1]. Despite the lack of Hermiticity, Bender et al. have shown in their seminal papers that a Hamiltonian can have real eigenspectra if it possesses socalled parity-time (PT ) reversal symmetry [2]. Such a symmetry means there is invariance of the theory under parity (spatial) reflection P:p → −p,x → −x, and time reflection T :p → −p (t → −t), i → −i,x →x, wherê p andx are the momentum and position operators, respectively, while t is the time coordinate and i is the imaginary unit. This combined PT symmetry leads to subtle changes in the unitary evolution of the system and modification of the inner product in the Hilbert space [3]. As PT symmetry represents an extension of quantum mechanics, it is nowadays used in various different contexts, such as quantum reflection [4] and chaos [5], and has even been generalized to fermionic [6] and gyrotropic systems [7].
Although the concept of PT symmetry was originally introduced in quantum mechanical systems, one has found experimental evidence and also a wide range of applications in classical optics. In 2010, Rüter et al. were the first to realize a PT -optical coupled system that involves well-defined regions with gain and loss regimes, * Corresponding author: armen@pks.mpg.de inherent to the complex-valued refractive index [8]. Such an extension of the concept of spacetime reflection into the classical domain stems from the works of El-Ganainy et al. [9] and Makris et al. [10], who have employed the similarity between the Schrödinger and a scalar approximation of Maxwell's equations to describe the dynamics of light beams in PT -symmetric optical lattices. There have been further theoretical [11] and experimental [12] studies dealing with the implementation of the parity-time reversal symmetry in optics especially relevant for the development of new artificial structures and materials. PT -symmetric structures have mainly been investigated in the spatial domain (that is, for timeindependent complex potentials) and little attention has been paid to the study of the non-stationary regime. The significance of consideration of the time-dependent potentials both in the quantum [13] and classical theories [14] arises from the attempt to examine the full time evolution of the system. Despite the recent works, however, there are no rigorous analytical studies of wave equations with non-stationary complex potentials possessing time reflection symmetry. Given that the energy (frequency) and time are conjugate variables, a successful solution of PT -symmetric time-dependent Maxwell's equations would constitute a complete characterization of the dynamical and spectral features of light.
The purpose of this Letter is to study the dynamics of light in time-dependent optical potentials having PT symmetry. In view of this, we calculate both the in-stantaneous amplitude and angular frequency of waves and show how the complex-valued dielectric permittivity controls the light in the temporal domain. The existence of the PT modulation of amplitude amplification and attenuation is demonstrated that are associated with the well-defined gain and loss regimes, respectively. In particular, we find that the fine tuning of the permittivity profile results in a vigorous modulation of amplitude attenuation. A comparison with the angular frequency modulation by a real permittivity is given to reveal the impact of PT -symmetric potentials in that we observe (i) splitting of the PT -extrema and (ii) a shrinkage of the frequency modulation period.
We start with a brief discussion of general properties of classical optical systems possessing PT symmetry. As the Schrödinger and paraxial Helmholtz (Maxwell's) equations formally coincide with one another, the complex refractive index n = ℜ (n) + iℑ (n) plays the role of the potential, the real (ℜ (n)) and imaginary (ℑ (n)) parts of which are, correspondingly, even and odd functions of spacetime coordinates to ensure the PT invariance of the theory [8][9][10]. Likewise, since for non-magnetic structures the real (ℜ (ε)) and imaginary (ℑ (ε)) parts of the dielectric permittivity are defined via , relations guarantee that the full wave equation, without any approximation, remains invariant under the parity-time transformation [15,16]. Throughout this work, we place our emphasis on the temporal domain and consider a spatially homogeneous yet time-dependent and PTsymmetric dielectric permittivity, ε (t). Even though we are not restricted to a particular form of the permittivity, for the sake of illustration we choose as a PT -modulated time-dependent optical potential, in contrast to the analogous profile of the (stationary) lattice in the spatial domain [10]. In Eq. (1),ε is the background dielectric constant, ε 1 represents the amplitude of the real profile of the potential, whereas ε 2 describes the strength of the gain/loss periodic distribution. Moreover, τ ≡ bt is a dimensionless time, where b ≥ 0 acts as a scaling factor and indicates the modulation rate of the permittivity, that we assume to occur slower than the oscillations of the wave. This is reminiscent of the similar form of scaling adopted in Ref. [10] for the spatial case.
In order to investigate dynamical and spectral properties of light in non-stationary PT -symmetric structures, we derive an exact second order differential equation from Maxwell's equations for the electric displacement vector D, which is valid for an arbitrary shape of the timedependent dielectric permittivity. Here, ∆ is the Laplace operator and c is the speed of light in vacuum [17]. Without the modulation rate, i.e. when b = 0, the standard linear dispersion relation k = ω ε (0)k/c holds, where ε (0) ≡ ε (τ = 0) =ε+ε 1 is the initial permittivity andk is the unit vector in the direction of propagation. Once we "switch on" the modulation, say at t = 0, both the amplitude and the angular frequency of light undergo a time-dependent modification governed by Eq. (2). Accounting for this instantaneous effect as well as owing to the spatial homogeneity of the permittivity, we seek the solution of Eq. (2) through the Ansatz whereû is the unit vector along the polarization direction, while the complex-valued 'amplitude' F describes the influence of the PT -modulated potential on the light. In the absence of this PT modulation, we expect to recover the 'free' propagation of light through a uniform medium with a constant dielectric permittivity so that F = 1. Note that a similar (full) wave equation for the space-dependent electric field and permittivity is discussed in Ref. [15] for describing the so-called PTsymmetric coherent-perfect-absorber laser. Next, we insert the Ansatz (3) in Eq. (2) and obtain a second order linear differential equation where "dot" means the derivative with respect to the dimensionless time τ . As our interest is restricted to slow modulations of the complex dielectric permittivity profile, as compared to the oscillations of light, we can adopt b/ω ≪ 1 and henceforth safely ignore the first term in Eq. (4). The remaining first order differential equation generally determines the instantaneous angular frequency (IAF) as for an arbitrary form of the dielectric permittivity. The exact solution of the reduced equation when integrated from the initial time 0 to some instant of time τ leads to that explicitly exhibits the PT symmetry of the displacement, D PT = D. In Eq. (6), the constant parameters A, B, C are introduced for the sake of brevity: A ≡ε/(ε + ε 1 ) > 0 carries information about the real potential, whereas B ≡ ε 2 /(ε + ε 1 ) amounts to the complex-valued permittivity, being the signature of the gain/loss mechanism. Both A and B, combined with C ≡ 1/ √ A + B 2 > 0, demonstrate the significance of the real and imaginary parts of the permittivity in the instantaneous character of light. Note that such a PT -induced "instantaneousness" vanishes if B = 0, A = C = 1 (ε 1 = ε 2 = 0) and/or b = 0 so that we recover the anticipated 'free' propagation of light, as also lim b→0 F = 1. The solution (6) holds for a class of potentials of the form (1), which is fully determined on choosing A, B and one of the constants in the potential, sayε. In addition, to mark out the range of variation of these parameters, we insert the (approximated) solution (6) in the exact Eq. (4) and numerically estimate the error for a given modulation rate b and angular frequency ω. As can be easily checked, the error we allow for here does not exceed 0.04 (as compared to zero) for A ∈ [0.7, 1.7] and B ∈ [−0.003, 1], corresponding to the ratio b/ω = 0.01.
The PT -symmetric solution (6) allows us to characterize the dynamics of waves in non-stationary complex potentials fully. Indeed, by decoupling the real and imaginary parts of the time-dependent component of the electric displacement, e −iωt F (τ ) = |D (τ )| e iΦ(τ ) , we obtain direct access to the profile of the (modulus-)squared instantaneous amplitude (IA) and that of the instantaneous phase the first derivative of which yields the profile (5) of the IAF, Ω = −bΦ, as one would expect. Note when the PT modulation is "switched off", the relations |D| 2 = 1, Φ = −ωt and Ω = ω are obtained [c.f. Ref. [18] for main study methods of signals whose frequency content changes in time]. For a complete description of the squared IA and IAF, and in order to reveal their peculiar properties, we employ Fermat's theorem and find that the extrema of expressions (7) and (5), i.e. the largest and smallest values that |D (τ )| 2 and Ω (τ ) take at some stationary points τ 0 ∈ (−∞, +∞), are determined by We immediately recognize that both quantities have extrema at τ 0 = πm/2 provided that m is an integer whose even values, m = 2N (with N being an integer), result in |D (πN )| 2 = 1 and Ω (πN ) = ω. In contrast, the odd values, m = 2N + 1, give rise to as obtained from Eqs. (7) and (5), respectively. Equation (9) ascertains the maximum possible amplitude amplification (B < 0) and attenuation (B > 0) correspondingly linked to the well-defined regions of gain and loss. Apart from the stationary points τ 0 , however, the IAF can also exhibit extremal values at of the quadratic polynomial (in sin 2 τ ) appearing in the curly parentheses in Eq. (8) is positive. In other words, the condition 2 |B| > |1 − A| gives rise to a split of the modulation extrema that would otherwise be determined by Eq. (10) for the conventional real permittivity. The corresponding 'split' extrema will then be described by the analytical form Time-periodic PT -symmetric optical potentials feature unusual, though expected modulation of squared IA and IAF of light. Figure 1 illustrates the evolution of these quantities, possessing time-reversal symmetry (|D (τ )| 2 = |D (−τ )| 2 , Ω (τ ) = Ω (−τ )), against the dimensionless time τ for various values of A and B. As the PT parameter B changes sign, well-defined regions of gain (B < 0) and loss (B > 0) can be clearly seen in Fig. 1a associated with the modulations of squared IA amplification and attenuation, respectively. In addition to these amplitude modulations, Figure 1b shows modulations of the IAF towards the domains either of higher (0 < A < 1) or lower (A > 1) frequencies, compared to ω. While for small values of |B| the PT symmetry manifests itself in the modulation of the squared IA (Fig. 1a) and the modulation of IAF remains unaltered (Fig. 1b) as compared to the one by the real  (11)). The comparison is made between modulations of light via real (the unit surface in (a) and the upper surface in (b)) and PT -symmetric permittivity profiles.
permittivity, larger values of B lead to a vigorous modulation of the squared amplitude attenuation (Fig. 1c) and to a pronounced modification of the frequency modulation profile (Fig. 1d). Unlike the ordinary modulation of angular frequency, where the modulation rate is commensurate with the scaling factor of the real potential [19], in our PT -symmetric structure the extrema of frequency modulation experience a split beyond the threshold 2 |B| = |1 − A|, as depicted in Fig. 1d. The global extremum turns into a local one and two new global extrema appear on either side such that now they occur with a shrunken period and the troughs of curves are shifted towards the lower frequency domain, quantitatively determined via Eq. (11). Finally, the extrema of the squared IA and IAF are shown in Fig. 2 as functions of A and B, where a comparison between the PT -and real-potential induced surfaces is made.
In conclusion, we have analytically examined the impact of the time-dependent PT -symmetric dielectric permittivity in the dynamical and spectral features of light. In our formalism, we have shown the PT modulation of light amplitude amplification and attenuation associated, correspondingly, with the well-defined regions of gain (B < 0) and loss (B > 0). We have also determined the condition 2 |B| > |1 − A| that is necessary for the split of the extrema of the angular frequency modulation to occur and have demonstrated the shrinkage of the modulation period. Both the split and shrinkage are general, inherent features of time-dependent complex potentials. A direct manifestation of PT symmetry in our particular time-varying structure is always evident, and we do not observe any indication of a breakdown of the symmetry as a function of the parameters. Such effects warrant more detailed future investigations with different time-dependent PT -symmetric potentials. Although the developed theory can be readily expanded for studying similar PT -induced effects for matter [20] and acoustic [21] waves, it also has indirect implications for time-dependent coupling in mechanical systems [22]. The consideration of the analogous theory for modified PT symmetries [23] would be of great interest.