Abstract
Supersymmetry is a conjectured symmetry between bosons and fermions aiming at solving fundamental questions in string and quantum field theory. Its subsequent application to quantum mechanics led to a groundbreaking analysis and design machinery, later fruitfully extrapolated to photonics. In all cases, the algebraic transformations of quantummechanical supersymmetry were conceived in the space realm. Here, we demonstrate that Maxwell’s equations, as well as the acoustic and elastic wave equations, also possess an underlying supersymmetry in the time domain. We explore the consequences of this property in the field of optics, obtaining a simple analytic relation between the scattering coefficients of numerous timevarying systems, and uncovering a wide class of reflectionless, three dimensional, alldielectric, isotropic, omnidirectional, polarisationindependent, noncomplex media. Temporal supersymmetry is also shown to arise in dispersive media supporting temporal bound states, which allows engineering their momentum spectra and dispersive properties. These unprecedented features may enable the creation of novel reconfigurable devices, including invisible materials, frequency shifters, isolators, and pulseshape transformers.
Introduction
Supersymmetry (SUSY) was conceived as a fundamental symmetry of string and quantum field theory that could allow the unification of all physical interactions of the universe^{1,2,3,4,5}. Subsequently, the field of supersymmetric quantum mechanics (SUSYQM) was created with the aim of solving essential questions about SUSY via a nonrelativistic model^{6}. Basically, the simplest version of SUSYQM considers two different onedimensional (1D) systems governed by the eigenvalue equations:
where \({\hat{\mathrm{H}}}_{1,2} =  \alpha {\mathrm{d}}^2/{\mathrm{d}}x^2 + V_{1,2}(x)\) are the Hamiltonians (with α > 0), V_{1,2} the potentials, and Ω^{(1,2)} the eigenvalues. The central idea is to define an auxiliary function W (known as the superpotential) and two SUSY operators \({\hat{\mathrm{A}}}^ \pm = \mp \sqrt \alpha {\mathrm{d}}/{\mathrm{d}}x + W(x)\) such that \({\hat{\mathrm{H}}}_1 = {\hat{\mathrm{A}}}^ + {\hat{\mathrm{A}}}^ \). The second Hamiltonian is then constructed by inverting the operator order, i.e., \({\hat{\mathrm{H}}}_2 = {\hat{\mathrm{A}}}^  {\hat{\mathrm{A}}}^ +\). The potential of SUSYQM resides in the fact that \({\hat{\mathrm{H}}}_1\) and \({\hat{\mathrm{H}}}_2\) have the same scattering properties and eigenvalue spectrum. As a consequence, although SUSY has not been experimentally observed in nature^{7}, SUSYQM has become a revolutionary mathematical tool in itself, enabling the explanation of intriguing aspects of quantum mechanics (such as the existence of nontrivial reflectionless potentials and of very different systems with the same energy spectrum), uncovering new analyticallysolvable potentials, and offering a simple and systematic way to construct infinite families of isospectral quantummechanical systems^{6}.
Interestingly, Eq. (1) also governs the dynamics of other physical phenomena. This is case of electromagnetic waves in certain kinds of media, which enables a direct extrapolation of the SUSYQM formalism to optics^{8,9}. As a result, the basic ideas of this theory have recently led to pioneering photonic structures^{9,10,11,12,13,14}.
Being a 1D theory, one could ask whether a temporal supersymmetry might exist for timevarying potentials. Nevertheless, to our knowledge, the SUSYQM formalism has never been applied in the time domain, whether in QM, optics, or any other field (SUSY quantum field theory is a multidimensional spacetime theory, but the formalism is considerably different and more complex than that of SUSYQM). This is probably due to the fact that the vast majority of 1D SUSY work has been developed within the realm of QM, and the time derivative in Schrödinger’s equation is of first order, preventing a similar decomposition to that of Eq. (1) in the time domain (timedependent potentials have been considered in SUSYQM, but also using SUSY operators based on firstorder spatial derivatives^{15,16}, making it impossible to exploit the potential of the standard spatial SUSY (SSUSY) factorisation in the time domain). On the other hand, only a few works deal with optical SUSY, all focused on SSUSY. Remarkably, however, the fact that the temporal derivative in the electromagnetic, acoustic, and elastic wave equations is of second order may enable a temporal version of SUSYQM, which has been overlooked so far. This would extend the foundations and unique properties of SUSYQM to the time domain, adding an unprecedented degree of understanding and control over timevarying systems in various fields of physics, and opening the door to a myriad of new applications. Actually, timevarying optical systems are becoming crucial in a broad range of scenarios, including optical modulation^{17}, isolation and nonreciprocity^{18,19}, alloptical signal processing^{20,21}, quantum information^{22}, and reconfigurable photonics^{23,24}. Likewise, temporal modulations enable new possibilities for the manipulation of sound and mechanical oscillations^{25,26,27}.
Here, it is shown that Maxwell’s equations indeed possess an underlying timedomain supersymmetry (TSUSY) for any nondispersive optical system characterised by a refractive index of the form:
where \(\varepsilon _{\mathrm{r}}({\mathbf{r}},t) = \varepsilon _{\mathrm{S}}({\mathbf{r}})\varepsilon _{\mathrm{T}}(t)\) is the medium relative permittivity and \(\mu _{\mathrm{r}}\left( {{\mathbf{r}},t} \right) = \mu _{\mathrm{S}}\left( {\mathbf{r}} \right)\mu _{\mathrm{T}}\left( t \right)\) its relative permeability, with similar results for acoustic and elastic waves (TSUSY can also be found in dispersive systems, as discussed below, and in anisotropic and nonlocal media, as discussed in Supplementary Note 1). In the following, the TSUSY formalism is developed for the field of optics, analysing both the continuous and discretespectrum cases, and illustrating its potential through different applications (sketched in Fig. 1). Finally, the extension of TSUSY to transmission line theory, acoustics and elasticity is discussed, assessing the experimental opportunities offered by current technological platforms.
Results
Continuous spectrum
Consider a linear, isotropic, heterogeneous, timevarying nondispersive medium with \(n_{\mathrm{T}}^2\left( t \right) = \varepsilon _{\mathrm{T}}\left( t \right)\). Applying separation of variables in the electric flux density \(D\left( {{\mathbf{r}},t} \right) = \phi \left( {\mathbf{r}} \right)\psi \left( t \right)\) of Maxwell’s equations, we find that ψ(t) exactly obeys the Helmholtz’s equation:
where \(N^2\left( t \right): = n_  ^2/n_{\mathrm{T}}^2\left( t \right)\), \(n_  : = n_{\mathrm{T}}\left( {t \to  \infty } \right)\), and ω is the angular frequency of the field at \(t \to  \infty\). For a polychromatic wave, the total field is given by the superposition of the solutions to Eq. (3) for each spectral component (value of ω). Equation (3) is also obtained for \(n_{\mathrm{T}}^2\left( t \right) = \mu _{\mathrm{T}}\left( t \right)\) and even for general materials with \(n_{\mathrm{T}}^2\left( t \right) = \varepsilon _{\mathrm{T}}\left( t \right)\mu _{\mathrm{T}}\left( t \right)\) (in which case, \(\varepsilon _{\mathrm{T}}\left( t \right)\) and/or \(\mu _{\mathrm{T}}\left( t \right)\) must vary slowly in time), see Supplementary Note 1. Equation (3) exactly matches Eq. (1) taking α = 1, relabelling x → t, and identifying \({\mathrm{\Omega }}  V\left( t \right) \equiv \omega ^2N^2\left( t \right)\). Using the eigenvalue Ω as a degree of freedom, this will allow us to apply 1D SUSY in the time domain, with two fundamental noteworthy features: (1) TSUSY is exact for both alldielectric and allmagnetic indices n_{T}; (2) TSUSY is completely uncoupled from space. Hence, it is valid for all polarisations, all propagation directions and any 3D spatial medium dependence \(n_{\mathrm{S}}^2\left( {\mathbf{r}} \right) = \varepsilon _{\mathrm{S}}\left( {\mathbf{r}} \right)\mu _{\mathrm{S}}\left( {\mathbf{r}} \right)\). This means that we can generate TSUSY partners of devices such as waveguides or structures with any desired 3D scattering response while keeping the spatial properties of interest (e.g., ability of guiding or reflecting/refracting the fields in a specific way for each direction and polarisation; see Supplementary Note 1 for an example involving an ideal polariser, which shows that the spatial response associated with a timeinvariant refractive index is preserved by its TSUSY partner for all polarisations simultaneously). Contrarily, 1D SUSYQM is, by definition, only valid for 1D spatial variations, and only for a specific polarisation in the optical case^{9,10,11}. Furthermore, TSUSY can be used to study temporal scattering in systems with continuous spectra, as well as timevarying systems supporting discretespectrum bound states. In both cases, its application is not as straightforward as that of SSUSY.
First, unlike in SSUSY, to develop TSUSY for wave scattering, the concept of negative frequencies is essential. This comes from the differences between spatial and temporal scattering, exemplified in Fig. 2 with a simple model having one spatial dimension. As is well known, when a plane wave traverses a localised spatial variation in a timeinvariant medium, there appear reflected and transmitted waves of the same frequency (photon energy), with the wave number (photon momentum) of the incident (k_{−}), reflected (k_{R}) and transmitted (k_{+}) waves fulfilling the Snell’s relations: k_{R} = −k_{−} and k_{−}/n_{−} = k_{+}/n_{+}, resulting from spatial symmetry breaking (Fig. 2a, Supplementary Movie 1). Less known is the fact that, when a wave propagates through a homogeneous medium, reflections also appear under a localised time variation (Fig. 2c, Supplementary Movie 2). In this case, since only time symmetry is broken, momentum is conserved and photon energy changes, with the frequency of the incident (ω_{−} = ω), reflected (ω_{R}) and transmitted (ω_{+}) waves obeying the relations^{28} ω_{R} = −ω_{+} and n_{+}ω_{+} = n_{−}ω_{−}. That is, light can exchange energy with the medium. Notably, the frequencies of the reflected and transmitted waves have opposite signs. Although the physical meaning of negativefrequency waves is striking and controversial^{29,30}, mathematically, the Hermiticity of the fields in k–ω space allows reinterpreting a negativefrequency wave as a counterpropagating positivefrequency one, leading to the standard use of onlypositive frequencies. However, the introduction of negative frequencies in this work is not a mere convention. It is a mathematical tool that enables the analysis of temporal scattering and, more importantly, a necessary ingredient to relate the reflection and transmission coefficients of TSUSY index profiles, which otherwise cannot be decoupled (Supplementary Note 2). Concretely, for a given system with a temporal index n_{T1}, TSUSY provides a systematic way of generating a superpartner, whose index is (see Supplementary Note 2):
where n_{1,2,±} := n_{T1,2}(t → ±∞) is assumed to be constant. As seen, the prescription for n_{T2}(t) depends on ω. Since it would be challenging to realise such a frequencydependent index in practice, we consider a more realistic and typical scenario in which the medium has the same temporal variation for all relevant frequencies of the electromagnetic field. This variation is taken to be the one that makes n_{T1}(t) and n_{T2}(t) supersymmetric at a frequency ω_{0}, which will be a free design parameter (it can be, e.g., the central frequency of the spectral band of interest, or any other reference frequency providing the desired response). That is, we suppress the frequency dependence by fixing ω to ω_{0} in Eq. (4). Consequently, the two systems will be guaranteed to be exact TSUSY partners at ω_{0}, while they may exhibit different properties at other frequencies. The same situation is found in SSUSY^{9,12}. Nevertheless, it can be shown that the TSUSY connection generally holds almost exactly in a broad frequency band, which is a typical feature of supersymmetric optical systems^{10,12}. This can be verified by calculating the solution to the wave equation associated with n_{T2} at ω ≠ ω_{0} (Supplementary Note 1), from which the medium spectral response can be obtained (an example is given below for a reflectionless system, see Fig. 2). It is worth mentioning that any system with a given relative index variation n_{T}(T_{0}t_{N})/n_{−}, with t_{N} := t/T_{0} and T_{0} := 2π/ω_{0}, will have the same normalised solution ψ(T_{0}t_{N}), and therefore the same scattering properties. Thus, for the sake of generality, the normalised variables t_{N} and n_{T}/n_{−} are used whenever possible.
Note that Eq. (3) admits asymptotic solutions for \(n_{{\mathrm{T}}1,2}\) in the form of the following incident, reflected and transmitted plane waves: \(\psi _{\mathrm{I}}^{\left( {1,2} \right)}\left( {t \to  \infty } \right) = e^{i\omega _0t}\), \(\psi _{\mathrm{R}}^{\left( {1,2} \right)}\left( {t \to \infty } \right) = R_{1,2}e^{  iN_ + \omega _0t}\) and \(\psi _{\mathrm{T}}^{\left( {1,2} \right)}\left( {t \to \infty } \right) = T_{1,2}e^{iN_ + \omega _0t}\), where \(N_ + : = n_{1,  }/n_{1, + } = n_{2,  }/n_{2, + }\). The combined use of negative frequencies and TSUSY then relates the reflection and transmission coefficients of both media as (Supplementary Note 2):
where W_{±} := W(t → ±∞) and W is obtained by solving the firstorder Riccati equation \(V_{1,2}\left( t \right) = W^2\left( t \right) \mp W^{\prime} \left( t \right)\), with \(V_{1,2}\left( t \right) = {\mathrm{\Omega }}  \omega _0^2N_{1,2}^2\left( t \right)\). Two important consequences can be inferred from Eq. (5): (1) it enables us to directly obtain the reflection and transmission coefficients of numerous intricate timevarying optical systems (in general, of any TSUSY partner of a knownresponse system), circumventing the resolution of Maxwell’s equations; (2) R_{1} = R_{2} and T_{1} = T_{2}  (as a direct consequence of the fact that n_{T1} and n_{T2} share the same eigenvalue Ω). As a result, TSUSY will enable us to straightforwardly construct families of timevarying media having the same scattering intensity as another one (with the desired spatial variation and polarisation response), implementable over the same (n_{2,−} = n_{1,−}) or a different (n_{2,−} ≠ n_{1,−}) index background, bringing about a variety of applications.
As an example, consider the simplest case: a constant refractive index n_{1}(r, t) = n_{1,−}. Its TSUSY partner is \(n_2\left( {{\mathbf{r}},t} \right) = n_{{\mathrm{T}}2}\left( t \right) = n_{2,  }[1 + 2({\it{\Omega }}/\omega _0^2  1){\mathrm{sech}}^2(\sqrt {{\it{\Omega }}  \omega _0^2} t)]^{  1/2}\) (Fig. 2b, Supplementary Movie 3). The free parameters n_{2,−} and Ω allow tailoring the asymptotic value of n_{T2}, as well as its maximal index variation Δn and temporal width Δt (defined as the required time interval to obtain Δn), see Supplementary Note 3. Since n_{T1} is constant, R_{1} = 0. Therefore, n_{T2} will also be reflectionless as demonstrated in Fig. 2d for a quasimonochromatic pulse. From our previous discussion, n_{T2} represents a new class of alldielectric (allmagnetic), omnidirectional, isotropic, polarisationindependent, and invisible 3D media with real positive (>1) permittivity (permeability). No known spatiallyvarying material possesses all these features simultaneously, including transformation media^{31}, complexparameter materials^{32}, and SSUSY media^{10}. The only previously reported timevarying reflectionless media required to concurrently induce temporal modulations in the permittivity and permeability^{33}. Our TSUSY proposal is totally different, since it is valid for alldielectric \(n_2^2\left( {{\mathbf{r}},t} \right) = \varepsilon _{\mathrm{S}}({\mathbf{r}})\varepsilon _{\mathrm{T}}(t)\) and allmagnetic materials \(n_2^2\left( {{\mathbf{r}},t} \right) = \mu _{\mathrm{S}}({\mathbf{r}})\mu _{\mathrm{T}}\left( t \right)\), see Supplementary Note 1. The former are particularly important, as implementing temporal permittivity modulations is extremely easier than implementing permeability ones.
As discussed above, another general feature of TSUSY is that it is only exact for the design frequency ω = ω_{0}. Therefore, n_{2} will be invisible (R_{2} = 0, T_{2} = 1) for all directions and polarisations at ω_{0}, while a reflected wave will appear at other frequencies. The response of n_{2} at ω ≠ ω_{0} will be characterised by the value of R_{2} and T_{2} as a function of frequency, which can be rigorously obtained by solving numerically Supplementary Equation 8. The result for the present example is depicted in Fig. 2e, which not only confirms the expected reflectionlessness of n_{2} at ω = ω_{0}, but also demonstrates that this property is almost preserved in a wide spectral band. For example, for \({\mathrm{\Omega }} = 2\omega _0^2\) (which corresponds to a rapidlyvarying index with Δt/T_{0} = 0.6), the system has a reflectance R_{2}^{2} < 10^{−3} in a fractional bandwidth F = Δω/ω_{0} = 30%. Moreover, the spectral span for which n_{2} is almost invisible (R_{2} ≈ 0, T_{2} ≈ 1) can also be tailored through Ω, enabling us to generate custommade transparent temporal windows within n_{2} only for desired spectral bands (Figs. 1a and 2). Concretely, F decreases as Ω increases. For instance, R_{2}^{2} < 10^{−3} in a fractional bandwidth F > 55% for \({\mathrm{\Omega }} = 1.5\omega _0^2\), while R_{2}^{2} < 10^{−3} in a bandwidth F = 8% for \({\mathrm{\Omega }} = 6\omega _0^2\). Remarkably, out of the invisible band, light is (partially) retroreflected along the input path, in contrast to spatial retroreflectors, in which the reflected path is parallel to, but different from, the input one^{34}.
Moreover, n_{2} exhibits a singular feature: the phases of the reflection and transmission coefficients (\(R_2 = \left {R_2} \righte^{i{\mathrm{\Phi }}_{R_2}}\), \(T_2 = \left {T_2} \righte^{i{\mathrm{\Phi }}_{T_2}}\)) have a frequencyindependent spectral response, adjustable via Ω (see Fig. 2e and Supplementary Fig. 3). As a result, n_{2} implements a perfect dynamicallyreconfigurable phase shifter with the property of being reflectionless, polarisation and frequencyindependent, and of having a short response time (Δt < 5π/ω_{0}) in the generation of any phase shift ∈[0, π] (allowing a significant reduction of the device length), blazing a trail for designing ideal ultracompact optical modulators (Fig. 1b, Supplementary Note 3). In contrast, opticalpathbased phase shifters usually demand slowlyvarying index modulations (and therefore devices with a higher length) to have a negligible reflection, and are intrinsically frequencydependent^{35,36}. Contrariwise, the proposed TSUSY device may induce the same phase shift over different spectral channels, which could be of great utility in, e.g., frequency combs and wavelengthdivision multiplexing technology. Likewise, pulse shaping operations can also be implemented by taking advantage of the nonlinear spectral response of Φ_{T2} (see Supplementary Note 3). Additional TSUSY media emerge from a constant index (being therefore reflectionless), such as the hyperbolic RosenMorse II (HRMII) potential (Fig. 3a, and Supplementary Fig. 8), which provides a free design control over Δn for a fixed response time (Δt ~ 20π/ω_{0}), allowing a technologyoriented tuning of the index excursion.
It is worth mentioning that the origin of reflectionless timevarying systems may be explained in terms of the absence of Stokes phenomenon (here related to the asymptotic behaviour of the solution to Eq. (3) in the limit \(\omega \to \infty\))^{37}, which also explains the origin of transitionless quantum systems^{38}. The latter can be obtained through the socalled transitionless tracking algorithm, which, given a nonadiabatic Hamiltonian \({\hat{\mathrm{H}}}_0\left( t \right)\), generates a second Hamiltonian \({\hat{\mathrm{H}}}\left( t \right)\) (or a family of them) that drives the instantaneous eigenstates of \({\hat{\mathrm{H}}}_0\left( t \right)\) exactly without transitions between them. Nevertheless, note that transitionlessness and reflectionlessness are not equivalent in general. To see this, consider the case in which \({\hat{\mathrm{H}}}_0\) does not depend on time, and is hence transitionless (in fact, \({\hat{\mathrm{H}}} = {\hat{\mathrm{H}}}_0\) from equations 2.7 and 2.9 of ref. ^{38}). If \({\hat{\mathrm{H}}}_0\) is not reflectionless, \({\hat{\mathrm{H}}}\) will not be reflectionless either, showing that transitionlessness does not necessarily imply reflectionlessness. Although it might be possible to obtain timevarying reflectionless systems with some version of this kind of method, we have not found any example in the literature (also note that the transitionless quantum driving algorithm^{38} does not apply to the temporal Helmholtz’s equation). In any case, the transitionless method and TSUSY are different formalisms. For instance, if the original Hamiltonian is the one associated with a constant potential, the transitionless algorithm just returns the same system, while TSUSY can generate infinite nontrivial reflectionless systems from a constant potential.
Actually, the results we have presented so far can be extended via different TSUSY variants. Firstly, isospectral TSUSY deformations provide a route to obtain mparameter index families \(\tilde n_{\mathrm{T}}\left( {t;\eta _1, \ldots ,\eta _m} \right)\) with exactly the same scattering properties in module and phase as another medium n_{T} (Supplementary Note 2). As an example, Fig. 3a shows a reflectionless twoparameter family of the HRMII index. Secondly, for some n_{T1} profiles, shape invariance (SI) allows us to construct TSUSY index chains \(\left\{ {n_{{\mathrm{T}}k}\left( {t;a_1} \right)} \right\}_{k = 1}^m\) satisfying the relations \(n_{{\mathrm{T}}m}\left( {t;a_1} \right) \propto n_{{\mathrm{T}}1}\left( {t;a_m} \right)\), \(R_m\left( {a_1} \right) = R_1\left( {a_m} \right)\), \(T_m\left( {a_1} \right) = T_1\left( {a_m} \right)\) and \(a_m = f\left( {a_{m  1}} \right) = \left( {f \circ f} \right) \left( {a_{m  2}} \right) = \left( {f \circ f \circ \ldots \circ f} \right)\left( {a_1} \right)\), with f a real function. Therefore, we can straightforwardly analyse or design the temporal scattering properties of a large number of timevarying media. To illustrate the benefits of SI, consider the following variation of the HRMII index (α is a real parameter):
which is also reflectionless in a wide spectral band (Fig. 3c and Supplementary Fig. 11). Since n_{1,−} ≠ n_{1,+}, the system performs a frequency downconversion with ω_{+} = (n_{1,−}/n_{1,+})ω_{−}, where n_{1,−}/n_{1,+} can be engineered via the design parameters ω_{0} and B. A device exhibiting all these properties has many potential applications. Unfortunately, the exotic shape (reaching values below n_{1,−}) and large maximal excursion of n_{T1}(t; a_{1}) hampers its experimental implementation. TSUSY can overcome this drawback by using SI. Specifically, Eq. (6) satisfies the SI condition with a_{m} = a_{1} − (m − 1)α, allowing us to generate different index profiles with the same reflectionless band as n_{T1}(t; a_{1}) (see Fig. 3b, c and Supplementary Movie 4). Taking m = 6, we find an index n_{T6}(t; a_{1}) = [n_{6,−}(a_{1})/n_{1,−}(a_{6})]n_{T1}(t; a_{6}) with a considerably smoother time variation and a significantly lower Δn. As illustrated in Fig. 1c, n_{T6}(t; a_{1}) can be used to build a polarisationindependent optical isolator with an ideally unlimited bandwidth (unlike previous time and spacetimemodulated isolators and frequency converters^{18,39,40}, which, in addition, usually involve complicated nonomnidirectional implementations), difficult to achieve by other means (Supplementary Note 3 includes more details on this device and additional SI examples). Reflectionless frequency converters can also be designed via transformation optics, but their implementation requires extremely complex spacetimevarying bianisotropic materials^{41}.
Discrete spectrum
Let us now discuss the discretespectrum case. This scenario naturally arises in optical SSUSY. Specifically, when applying 1D SUSYQM to a spatial dimension normal to the propagation direction, the propagation constant enters the wave equation as an effective energy, which is quantised by the eigenvalue problem^{9,10}. However, since time is unidimensional, there is no possible quantity playing the role of an energy in TSUSY, leading to freeparticle systems. Outstandingly, a discretespectrum version of TSUSY can be developed for timevarying dispersive media. To this end, we require the concept of temporal waveguide (TWG): two adjacent temporal index boundaries (allowed to move at a speed v_{B}) defining a positiondependent temporal index window, which can confine optical pulses by temporal total internal reflection^{42,43,44}. TWGs can be created by inducing a perturbation Δn_{eff}(t − z/v_{B}) of the effective index n_{eff} of a given mode in a spatial waveguide. In a comoving reference frame, the complex envelope of the electric field associated with a TWG can be written as^{43,44} \(A\left( {z,\tau } \right) = \mathop {\sum }\nolimits_n \psi _n\left( \tau \right)e^{i\left( {{\mathrm{\Delta }}\beta _1/\beta _2} \right)\tau }e^{iK_nz}\). It is then shown that a TWG supports temporal bound states ψ_{n} (n = 0, 1, 2,…) fulfilling the discretespectrum eigenvalue equation:
where τ := t − z/v_{B}, β_{1} and β_{2} are the inverse group velocity and groupvelocity dispersion constant of the perturbed spatial mode, Δβ_{1} = β_{1} − 1/v_{B}, β_{B}(τ) = k_{0}Δn_{eff}(τ), k_{0} = ω_{0}/c_{0}, and ω_{0} is the optical carrier angular frequency. We can apply TSUSY to Eq. (7), as it matches Eq. (1) for α = 1, x → τ, V(τ) ≡ 2β_{B}(τ)/β_{2} and \({\mathrm{\Omega }}_n \equiv 2K_n/\beta _2 + {\mathrm{\Delta }}\beta _1^2/\beta _2^2\), expanding the TWG landscape and its potential applications.
As an example, consider an analyticallysolvable TWG with a step temporal perturbation β_{B1}. Its unbroken TSUSY partner is \(\beta _{{\mathrm{B}}2}\left( \tau \right) = {\beta} _{{\mathrm{B}}1}\left( \tau \right)  {\beta} _2( \ln {\psi} _0^{\left( 1 \right)}\left( \tau \right) )^{\prime\prime}\), where \(\psi _0^{\left( 1 \right)}\) is the ground state (fundamental mode) of β_{B1} (Fig. 4a, Supplementary Note 4). From TSUSY theory, both TWGs have the same energy spectrum. Moreover, \({\hat{\mathrm{A}}}^  ( {{\hat{\mathrm{A}}}^ + } )\) maps each state \(\psi _n^{\left( 1 \right)}\) (\(\psi _n^{\left( 2 \right)}\)) of β_{B1} (β_{B2}) into a state of β_{B2} (β_{B1}) having the same eigenvalue Ω_{n}, with the exception of the ground state \(\psi _0^{\left( 1 \right)}\), which is annihilated by \({\hat{\mathrm{A}}}^  ( {{\hat{\mathrm{A}}}^  \psi _0^{\left( 1 \right)} = 0} )\) and thus has no equalenergy counterpart in β_{B2}. Particularly, \(\psi _n^{\left( 2 \right)} \propto {\hat{\mathrm{A}}}^  \psi _{n + 1}^{\left( 1 \right)}\), where \({\hat{\mathrm{A}}}^  : = {\mathrm{d}}/{\mathrm{d}}\tau  ( {\ln \psi _0^{\left( 1 \right)}\left( \tau \right)} )^{\prime}\). In TSUSY, energy is related to phase constant, implying that \(\psi _0^{\left( 1 \right)}\) is not phasematched with any \(\psi _n^{\left( 2 \right)}\) and that \(\psi _{n + 1}^{\left( 1 \right)}\) and \(\psi _n^{\left( 2 \right)}\) are perfectly phasematched. This occurs in an extremely large optical bandwidth \({\mathrm{\Delta }}\nu \sim 0.5\) (\(\nu \propto 1/\beta _2\) is the normalised frequency), provided that both TWGs are built on a dispersionflattened spatial waveguide \(\left( {{\mathrm{d}}\beta _2/{\mathrm{d}}\omega \approx 0} \right)\), see Fig. 4b. Furthermore, Fig. 4b reveals that β_{B2} is less dispersive than β_{B1}, i.e., TSUSY enables us to engineer the dispersion properties of TWGs.
To further unfold the potential of discretespectrum TSUSY, we propose the concept of temporal photonic lantern (TPL): closepacked serial TSUSY TWGs moving at the same speed and supporting linear combinations of degenerate temporal bound sates (supermodes)^{45}. To verify the TPL concept, we have developed the first version of coupledmode theory (CMT) for serial TWGs (Supplementary Note 4). Figure 4c shows an example of a TPL supermode. Remarkably, TWGs can carry solitonlike (shapeinvariant) optical pulses in a timevarying dispersive medium, with the advantage of enabling arbitrary pulse amplitude, phase and duration, as well as a tuneable propagation speed^{43,44}. TPLs extend this ability to a serial combination of modes (each with arbitrary length, amplitude and node number), yielding solitonic supermodes with almost any desired shape. Achieving the required perfect phasematching between modes of different order in serial TWGs without TSUSY typically demands neighbouring TWGs of different width, whilst TSUSY permits an independent control over this parameter and generally presents a much larger normalised phasematching bandwidth^{12,14}, inherently implying a higher tolerance to fluctuations in T_{B} and Δn_{eff}. More advanced functionalities emerge by noting that if only one of the TWGs of the TPL is excited, a periodic energy transfer between adjacent TWGs occurs (Fig. 4d, Supplementary Movie 5, Supplementary Note 4). Using two coupled spatial waveguides (WG1, WG2), this effect enables the construction of a pulseshape transformer with unprecedented versatility and reconfigurable capability (Fig. 1d). Particularly, the final shape of each pulse propagating along WG1 can be dynamically chosen among a large gamut by launching the appropriate TPL over WG2. The proposed TSUSY TPLs could also find application in optical wavelet transforms, coherent laser control of physicochemical and QM processes, spectrallyselective nonlinear microscopy, and mathematical computing^{46,47,48,49}. TSUSY TWG theory can be extended via temporal analogues of SI, broken SUSY and isospectral constructions^{6,14}.
Discussion
Overall, these results generalise the foundations of SUSYQM to the time domain, unveiling the temporal supersymmetric nature of Maxwell’s equations (which, unlike optical SSUSY, is a genuine symmetry with no previous direct analogue) and, consequently, leading to the emergence of an entire field of research within physics, as well as to a new photonic design toolbox. Compared with SSUSY, TSUSY relaxes the need for controlling the polarisation state of light and the medium spatial index variation, which usually involves complex fabrication steps^{9,10}. In addition, as outlined above and as shown in detail in Supplementary Notes 7 and 8, both sound and elastic waves satisfy a temporal Helmholtz equation formally equal to Eq. (3). Hence, TSUSY can be directly transferred to these fields of physics.
A possible TSUSY technological difficulty might arise if high temporal index excursions and/or index variations with a very low temporal width are desired (e.g. to achieve a large phase shift in an ultracompact optical device). Since the response and achievable functionalities of the system depend solely on n_{T}(T_{0}t_{N})/n_{−}, the limits of a given technology will be determined by the maximum (minimum) allowed values of Δn/n_{−} (Δt/T_{0}). Table 1 summarises the main features of the most suitable existing platforms for the implementation of TSUSY index modulations.
At optical frequencies, stronglynonlinear epsilonnearzero (ENZ) media such as indium tin oxide (ITO) and aluminiumdoped zinc oxide (AZO) provide large values of Δn/n_{−} (7.2 and 4.5, respectively, widely exceeding the figures here considered), but are limited by their high Δt/T_{0} values (36 and 33, respectively) and typical linear losses^{50,51,52}. Silicon carbide (SiC) is an interesting alternative material that possesses an ENZ band at a wavelength λ_{ENZ} ≈ 10.33 μm, with a low estimated response time^{52,53,54} Δt/T_{0} ≈ 5 and a maximum^{51,54,55} Δn/n_{−} ≈ 1. Hence, modulations such as those depicted in Fig. 3 could be potentially realised with this kind of material. Note that nonlinear ENZ media with smaller values of Δt/T_{0} in combination with low linear losses might emerge with the development of new materials^{52}.
Phase change materials (PCMs) also support large values of Δn/n_{−} at high frequencies. As an example, germanium antimony telluride (GST) alloys provide variations of Δn ≈ 3 (Δn/n_{−} ≈ 0.9) with very low losses at λ = 10 μm and a controllable degree of crystallisation enabling a gradual index variation^{56,57}. Currently, the high typical response time of PCMs (Δt ≥ 400 ps, Δt/T_{0} ≈ 12,000)^{58} would be the main drawback of this technology for the implementation of TSUSY modulations. Nevertheless, the femtosecond phase transitions achieved in some experiments via nonthermal mechanisms^{59} may lead to faster PCM technology at optical frequencies in the future.
In lowfrequency electromagnetism, timevarying transmission lines (TVTLs) based on a dynamicallytuneable distributed capacitance constitute an ideal experimental platform to test and exploit all the proposed TSUSY concepts, as their effective dielectric properties can be largely tuned in space and time. In particular, the capacitance of commercial varactor diodes can be modified by a factor of 12 (corresponding to Δn/n_{−} ≈ 2.5)^{60}, while their response time can be as low as Δt/T_{0} ≈ 0.1 (see Supplementary Note 6)^{61,62}.
In acoustics, a temporal index modulation can be induced through variations of the medium mass density (Supplementary Note 7). There exist a number of strategies to achieve strong changes of this parameter, including the use of magnetoacoustic crystals^{63}, acoustic metamaterials^{64}, or acoustic systems with moving parts^{25}. Notably, there exist more sophisticated electronicallycontrolled active materials whose local acoustic parameters can be changed almost arbitrarily in real time^{65}.
Moreover, TSUSY systems can be implemented in elastic beams with a timevarying stiffness D_{T}, which allow temporal modulation amplitudes as high as ΔD_{T}/D_{−} = 1.4 (corresponding to Δn/n_{−} ≈ 0.4) and normalised response times as low as Δt/T_{0} = 0.3 (see Supplementary Note 8)^{27,66}.
Finally, note that extremely low index perturbations suffice to create TSUSY TWGs, technologically feasible in standard optical fibres and waveguides via travellingwave electrooptic phase modulators or the crossphase modulation effect^{30,43,44}.
Methods
Numerical calculations
The temporal scattering problem (Figs. 2 and 3) has been simulated by solving numerically Supplementary Equation 8 with the commercial software COMSOL Multiphysics. In order to guarantee a low computational time, we have taken ω_{0} = 38 rad·s^{−1} and c_{0} = 1 m·s^{−1}. Nevertheless, the results of Figs. 2 and 3 are valid for any value of ω_{0} and c_{0} (see Supplementary Note 3 for more details). On the other hand, the discrete spectrum of the TWGs and the TPL (Fig. 4) has been analysed with the commercial software MATLAB and CST Microwave Studio by using the analogy between the modes of a TWG and those of a dielectric slab waveguide reported in ref. ^{43}. Finally, the pulseshape transformation depicted in Fig. 4d has been calculated in MATLAB by solving the CMT reported in Supplementary Note 4 for serial TWGs. See Supplementary Note 5 for more details about the methods employed in Fig. 4.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The codes generated during the current study are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work was supported by Spanish National Plan projects TEC201573581JIN PHUTURE (AEI/FEDER, UE) and MINECO/FEDER UE XCORE TEC201570858C21R, as well as Generalitat Valenciana Plan project NXTIC AICO/2018/324. A.M.O.’s work was supported by BES2013062952 F.P.I. Grant.
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C.G.M. conceived the idea of temporal SUSY. C.G.M. and A.M.O. developed the theory, performed the numerical simulations and analysed the data in equal contribution. R.L.S. supervised the work. All authors contributed to write the paper.
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GarcíaMeca, C., Ortiz, A.M. & Sáez, R.L. Supersymmetry in the time domain and its applications in optics. Nat Commun 11, 813 (2020). https://doi.org/10.1038/s41467020146340
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DOI: https://doi.org/10.1038/s41467020146340
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