Abstract
Within all physical disciplines, it is accepted that wave transport is predetermined by the existence of disorder. In this vein, it is known that ballistic transport is possible only when a structure is ordered, and that disorder is crucial for diffusion or (Anderson)localization to occur. As this commonly accepted picture is based on the very foundations of quantum mechanics where Hermiticity of the Hamiltonian is naturally assumed, the question arises whether these concepts of transport hold true within the more general context of nonHermitian systems. Here we demonstrate theoretically and experimentally that in ordered timeindependent symmetric systems, which are symmetric under spacetime reflection, wave transport can undergo a sudden change from ballistic to diffusive after a specific point in time. This transition as well as the diffusive transport in general is impossible in Hermitian systems in the absence of disorder. In contrast, we find that this transition depends only on the degree of dissipation.
Introduction
The deceleration of wave transport in a lattice due to disorder was introduced in physics about 100 years ago with the famous Drude model. Paul Drude explained the conductance of metals by free electrons that are scattered by the atomic lattice, which in turn results in a diffusive transport^{1}. The diffusion process is ubiquitous. It governs the effects of electric and thermal conductivity in solids^{1,2}, particle mixing in fluids^{3} and spin diffusion effects^{4}, just to name a few. In this context, it is generally agreed that all systems that exhibit subballistic transport are inherently disordered. Today, disorder is explored in many disciplines, such as optics^{5}, solidstate physics^{6}, acoustics^{7} and matter waves^{8,9}. In particular, optical systems attracted much interest, and so far numerous subballistic transport phenomena based on disorder have been observed; these include the Anderson localization^{5,10,11}, quantum decoherence^{12}, Levy flights^{13} and anomalous diffusion^{14}.
The understanding of diffusive (and in general subballistic) transport naturally assumes Hermiticity of the Hamiltonian, as this ensures the reality of the eigenvalue spectrum and, therefore, energy conservation. However, dissipative (that is, lossy) systems that interact with their environment are by far the most common. With the damped pendulum as the simplest example, dissipation is the basis for phenomena like the Carnot process or negative temperature coefficient thermistors. In the nonlinear regime, dissipative structures are encountered even in the everyday world, for instance, in the form of heat convection of a candle light, cyclons, the famous Belousov–Zhabotinsky reaction^{15} and, above all, in living organisms. The recently introduced new class of nonHermitian systems, the socalled symmetric systems^{16}, have received considerable attention. In particular, optical structures provide an exceptional platform for the implementation of symmetric physics, where the (symmetric) refractive index distribution represents the real part of the complex potential, whereas the (antisymmetric) gainloss profile has the role of its imaginary part^{17}. In optical symmetrical systems, the shortterm evolution of (optical) wave packets exhibits peculiar features and is highly nonintuitive^{18,19,20,21,22,23}. In this context, perhaps, it is natural to ask how periodic but dissipative structures affect the longterm wave transport—a question of fundamental importance in numerous physical systems.
In this work, we theoretically predict and experimentally observe that in static symmetric lattice systems with no disorder, wave transport may suddenly slow down from ballistic to diffusive after a particular transition time. We find that this transition as well as the resulting diffusive transport—which, in the absence of disorder, is impossible in Hermitian systems—depends only on a dissipation parameter associated with the system. We verify our predictions by experiments in an optical waveguide array. In this paraxial optical setting, light propagating inside the waveguide array exhibits evolution analogous to that of a particle in a quantum mechanical lattice. Importantly, this system allows the direct observation of wave packets, which is a key to unravel the mechanism of the underlying transport.
Results
Theory
Let us first describe the theoretical basics of the transition from ballistic to diffusive transport in dissipative lattices. For illustration purposes, we employ a tightbinding model to describe the evolution of a wave packet in a biatomic lattice^{24}.
where z is the evolution coordinate. The quantities a_{n}(z), b_{n}(z) are the propagating amplitudes in both lattice sites in unit cell n, the quantities δ_{a} and δ_{b} represent the real part of the potential, and γ_{a} and γ_{b} represent the imaginary part of the potential (or gain/loss parameter), respectively. The transverse dynamics in the lattice is described by the hopping parameter (or coupling constant) κ. In the particular case of δ_{a}=δ_{b}=δ and γ_{a}=−γ_{b}=γ, the system is symmetric, as in this case the real part of the potential is symmetric and the imaginary part is antisymmetric^{16,17}. A sketch of such a lattice is shown in Fig. 1a. In our analysis, however, we will not restrict ourselves to the symmetric case, but will consider a general nonHermitian lattice with arbitrary δ_{a,b} and γ_{a,b} instead. In this case, the evolution dynamics is dictated by the spectrum of propagation constants β:
where q is the transverse Bloch momentum, =(δ_{a}+δ_{b})/2+i(γ_{a}+γ_{b})/2, =(δ_{a}−δ_{b})/2, and =(γ_{b}−γ_{a})/2. It is noteworthy that such a class of dissipative nonHermitian systems is said to possess broken symmetry, as the eigenvalue spectrum is complex. Unbroken symmetry, that is, a purely real eigenvalue spectrum, can be achieved only in lattices with δ_{a}=δ_{b} and an inhomogeneous intersite coupling^{25,26}.
In Fig. 1b, an example of the spectrum (equation (2)) is shown. For this arrangement, the (normalized) lattice parameters are chosen to be δ_{a}=γ_{a}=0 cm^{−1}, δ_{b}=γ_{b}=1 cm^{−1} and κ=1.5 cm^{−1}. In this figure, the solid lines indicate the upper band (the ‘+’ sign in equation (2)) and the dashed lines indicate the lower band (the ‘−’ sign). Moreover, the blue lines represent the real part of the spectrum and the red lines the imaginary part. Clearly, the imaginary part of the eigenmode’s propagation constant is a function of the transverse momentum q. Hence, eigenmodes in different regions of the spectrum will experience different losses, depending on the value of the imaginary part of their propagation constant. As a consequence, in both bands modes in the centre of the spectrum (where the lattice momentum q≈0) experience an intermediate loss, which is close to the average loss in the system. At the edge of the spectrum (around q=±π) the situation is very different. There, the modes in the upper band suffer from a loss that is much higher than the systems’s average loss, whereas in the lower band the modes experience much less loss. Hence, because of decay, the modes in the upper band at the edge of the spectrum will disappear after a relatively short propagation distance z, whereas the modes in the centre of the spectrum will disappear somewhat later. Only the modes in the lower band around q=±π will prevail at long propagation distances. Therefore, the spectrum will considerably be getting narrower during propagation of the wave packet along z, and only a part of the spectrum will contribute to transport.
To explore the impact of this phenomenon, we perform a numerical analysis. To this end, we solve the tightbinding equation (1) for an array of N=200 waveguides using a standard Runge–Kutta scheme and calculate the variance σ^{2}(z) at every computation step. A simulation of the wave evolution in a symmetric dissipative lattice is shown in Fig. 2a, where δ_{a}=δ_{b}=γ_{a}=0 cm^{−1}, γ_{b}=2.0 cm^{−1} and κ=0.5 cm^{−1}. Initially, the wave exhibits ballistic spreading as all eigenmodes still contribute to the transverse transport. However, after a certain propagation distance z the spreading clearly slows down, and the strongly modulated pattern, which is typical of ballistic transport in a periodic lattice, washes out completely. In Fig. 2b, a doublelogarithmic plot shows the variance σ^{2}(z) as a function of the gain/loss parameter γ_{b}. It is noteworthy that in the ballistic case, one finds σ^{2}(z)~z^{2}; plotting this with doublelogarithmic axes results in a straight line with slope 2. In contrast, diffusive transport is characterized by σ^{2}(z)~z^{2}, which yields a straight line with slope 1 in a such a plot. Whereas one finds for γ_{a}=γ_{b}=0 cm^{−1} the expected ballistic spreading, for γ_{b}>0 cm^{−1} one clearly sees a sudden transition in the transport properties from initially ballistic (slope 2) to diffusive (slope 1). The distance, where this transition happens decreases as the loss parameter γ_{b} increases (note that γ_{a}=0 cm^{−1} for all graphs). We attribute this phenomenon to the narrowing of the spectrum: after some specific propagation distance z_{crit}, essentially only a part of the spectrum contributes to transport. In fact, the continuous spectral contraction leads to the characteristic diffusive spread of the wave packet observed in our simulations.
To understand this process beyond the intuitive spectrumbased explanation of this effect, we developed a full analytical theory of this effect, which is provided in the Methods section. Within this theory, the critical distance around which transport slows down from ballistic to diffusive is given by
Around z_{crit}, the spectral narrowing reaches a strength that impedes ballistic transport because of an insufficient number of propagating modes. Therefore, in essence for z≪z_{crit} the transport is ballistic, whereas for z≫z_{crit} it enters the diffusive regime. Importantly, the critical distance z_{crit} depends strongly on the difference in the loss parameters , and the dependence on the coupling coefficients κ is negligible. Interestingly, our simulations also indicate that z_{crit} slightly shifts to larger values for >0 cm^{−1}.
Experiment
For the experimental demonstration of our theoretical results, we fabricated various optical waveguide samples employing the directlaser writing technology^{27}. A sketch of such a waveguide array is shown in Fig. 3a, and a microscopic image from the end facet of one specific realization is shown in Fig. 3b. For further details of the fabrication process, we refer the reader to the Methods section. Each sample was 10 cm long and contains 60 waveguides with an intersite spacing of d=17 μm. These parameters were chosen such that the light does not reach the boundaries of the lattice during propagation.
To introduce additional, welldefined losses in every second guide, these sites are sinusoidally modulated transverse to the lattice plane with an amplitude of A=3 μm. The induced curvature then facilitates radiation and, therefore, enhanced losses. These losses can be tuned precisely by changing the curvature of the waveguide modulation, which we achieve by adjusting the period of the modulation at constant amplitude. In our study, depending on the specific sample, the number of periods per sample length was ranging between 23 and 103. Note that for such small modulation amplitudes of only A=3 μm, the average spacing between the waveguides is always d≈17 μm, resulting in a homogeneous coupling constant κ. However, the modulation causes a slight rise of the propagation constant in the modulated guides because of the increased optical path. This can be compensated by artificially increasing the refractive index of the unmodulated waveguides. However, the mobility transition in nonHermitian lattices takes place also for ; hence, in our samples we did not compensate for the additional detuning of the modulated guides.
Our experimental results are summarized in Fig. 4. To excite the entire spectrum, we launch light into the central waveguide of the lattice. An image of the propagating light beam is shown in Fig. 4a, obtained using a fluorescence microscopy technique^{28} (see Methods section). For small propagation distances, the typical ballistic diffraction pattern is observed, whereas for larger propagation distances the modulation of the pattern washes out and the spreading slows down. In Fig. 4b, we show a simulation of the experimental pattern with parameters δ_{a}=δ_{b}=γ_{a}=0 cm^{−1}, γ_{b}=2 cm^{−1} and κ=1.1 cm^{−1}. From the experimental fluorescence microscopy pattern in Fig. 4a, we extract the variance, which we plot using a doublelogarithmic plot in Fig. 4c. The transition from ballistic to diffusive transport is clearly visible at z_{crit}≈2 cm: the initial ballistic spread (slope 2) suddenly slows down to diffusive (slope 1). We would like to emphasize the remarkable agreement between both the experimental and theoretical curve, as for the simulation no fitting parameters were used. The coupling parameter κ=1.1 cm^{−1} used in the simulation is an experimental value obtained from measurements in a single directional coupler with a waveguides separation of 17 μm. The loss parameter γ_{b}=2 cm^{−1} was extracted from loss measurements in isolated sinusoidally bent waveguides.
We repeat our experiments for different modulation periods in the waveguides, which results in different loss parameters γ_{b}. The position of the transition point z_{crit} is extracted from the fluorescence images and is plotted in Fig. 4d. As predicted by our theory, the value of z_{crit} is in very good agreement with equation (3). The fluorescence images and variance plots of all data points in Fig. 4a are shown in Fig. 5.
Discussion
For the first time, we have observed the coexistence of ballistic and diffusive transport in a static, ordered system, based on nonHermiticity. The inhomogeneous losses associated with the eigenmodes result in a contraction of the spectrum, which hinders ballistic transport because of an insufficient number of propagating modes. The transition from the initially ballistic motion to a diffusive transport is generally impossible in static Hermitian systems, where subballistic transport is only possible through the introduction of disorder. Moreover, it was shown that the critical distance—where the transition of the transport regimes takes place—depends only on the degree of dissipation and not on the coupling between the waveguides.
Our theoretical and experimental work suggests various intriguing questions. In terms of applications, can a nonHermitian waveguide lattice be used as a sophisticated mode filtering device, for any desired Bloch waves and not only those at the edge of the Brillouin zone (around q=π)? Can one design a device with realtime loss control to actively tune the width of the beam at the sample’s output facet? These issues might be of particular interest in the fields of telecommunication or quantum computing, as especially in the field of quantum optics there is a great desire to prepare single as well as entangled photons in welldefined states. Moreover, also from the more fundamental side, many interesting questions arise: what is the impact of losses in twodimensional systems, in particular photonic graphene^{29}? How the light behaves at defects and interfaces in lossy systems? Does disorder or nonlinearitiy facilitate or hinder transport^{30}? Moreover, can the band structure be modified via higherorder coupling effects^{31} to achieve different transport regimes such as subdiffusive, superdiffusive or even superballistic transport^{32}? Our work will enable these questions, as well as many others, to be addressed.
Methods
Theoretical foundation
To derive the corresponding results, we start from the following system of equations:
The first step in analysing this system is extracting its eigenvalue spectrum. This is done by starting with the usual ansatz
Substituting this into equation (4), one arrives at an expression that determines the eigenvalues β(q) as well as the eigenvectors :
Therefore, the eigenvalue spectrum is given by
where
The corresponding eigenvectors are
As described in the paper, we are interested in the evolution of a single waveguide excitation. For this specific case, the initial conditions (at z=0) are a_{n}=δ_{n0} and b_{n}=0. From these initial conditions, it is straight forward to write down the evolution of the wave packet in the form of an integral:
Equation (9) is the full formal solution of equation (4) with the given initial conditions. However, given this form of solution it is difficult to predict a transition from ballistic to diffusive transport. On this account, the goal of this section is to derive the existence of such a transition.
As discussed in the main text, the transport regime is solely determined by the variance of the wave packet. Consequently, we calculate the variance of the wave packet given by equation (9) for large propagation distances z. For simplicity, in the following we will only consider the variance on the a_{n} sublattice. However, the derivation for the b_{n} sublattice follows analogously to the way presented.
For simplicity, let us assume that the b_{n} waveguides exhibit more loss than the a_{n} waveguides, that is, γ_{b}>γ_{a}. From this it follows that is a positive quantity. Keeping this in mind, let us take yet another look at the eigenvalue spectrum (equation (7)). It was already stated in the main text that the specific case of κ=c always leads to a complex spectrum. From here on, let us proceed with this specific case. For such a complex spectrum, the modes that exhibit the least amount of loss are those in the lower band (β−q) around q≈π. Hence, for large propagation distances z, one can approximate the evolution of the a_{n} by
To arrive at this point, first we have neglected the contribution of the modes of the upper band. Second, we have used the Taylor expansion of β_{±} around q=π, which reads
Because of the dominance of the Gaussian term , for large z one can approximate
In equation (10), also q′=q+π and
have been used. The extension of the integration limits in equation (10) can be justified because of the fact that for large z, the width of the Gaussian is much smaller than the initial integration range. At this point, it needs to be emphasized that the approximation performed in equation (7) is only valid for a nonvanishing real part of w. Moreover, this condition is equivalent to the demand of a nonvanishing loss detuning.
Finally, the evaluation of the integral in equation (10) yields
Now, the last step in the analysis will be to calculate the variance of the wave packet. The variance for such a discrete and dissipative system is defined by
Using the explicit form of a_{n}, one obtains
For large z, the sums can be approximated by integrals using x=n/ and . Hence, one finds
Evaluating the Gaussian integral yet again, one finds the variance to be
This shows mathematically that the variance rises with the first power of z. Hence, the propagation of the wave packet is diffusive.
Subsequently, we want to derive the approximate region of the transition, which corresponds to the critical distance z_{crit} given by equation (3). For simplicity, we only regard the case of . For the approximation (equation (10)) to hold, all modes outside an εregion around q=π need to be damped sufficiently, and hence need to be smaller than some value δ. For our analysis, it is sufficient to fix δ at δ=e^{−2}. This damping condition reads
or equivalently
It is important to note that ε cannot be chosen arbitrarily, but one has the additional condition
which follows from the evaluation of the denominator in equation (12) and the Taylor expansion of the eigenvalues. Connecting both conditions, one obtains equation (3).
Waveguide fabrication
Our samples are fabricated in bulkfused silica wafers using the femtosecond laser directwrite aproach^{27} employing a Ti:Sapphire Mira/RegA laser system (Coherent Inc.) operating at a wavelength of 800 nm, a repetition rate of 100 kHz and a pulse length of 170 fs. The light is focused inside the sample by a × 20 microscope objective (NA=0.35), and by continuously moving the sample using a highprecision positioning system (Aerotech Inc.) the waveguides are created by the induced refractive index increase. For the fabrication of our samples, the pulse energy was adjusted to 320 nJ and the writing velocity was set to 1.5 mm s^{−1}.
To introduce enhanced and controllable losses to specific waveguides, an additional sinusoidal modulation was applied to the waveguides to achieve additional radiation losses. The modulation is introduced into the vertical direction, as this prevents the radiated light from travelling through the lattice and being recollected by other waveguides.
Fluorescence microscopy
For the direct monitoring of the light propagation in our samples (see Fig. 4a), we used a fluorescence microscopy technique^{28}. A massive formation of nonbridging oxygen hole colour centres occurs during the writing process, when fused silica with a high content of hydroxide is used, resulting in a homogeneous distribution of these colour centers along the waveguides. When light from a HeliumNeon laser at λ=633 nm is launched into the waveguides, the nonbridging oxygen hole colour centres are excited and the resulting fluorescence (λ=650 nm) can be directly observed using a chargecoupled device camera with an appropriate narrow linewidth filter. As the colour centres are formed exclusively inside the waveguides, this technique yields a high signaltonoise ratio.
Additional information
How to cite this article: Eichelkraut, T. et al. Mobility transition from ballistic to diffusive transport in nonHermitian lattices. Nat. Commun. 4:2533 doi: 10.1038/ncomms3533 (2013).
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Acknowledgements
We acknowledge support by the German Ministry of Education and Research (Center for Innovation Competence program, Grant 03Z1HN31) and the Thuringian Ministry for Education, Science and Culture (Research group Spacetime, Grant no. 11027514).
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T.E. and A.S. developed the theory. T.E. and R.H. performed the simulations. T.E. and F.D. carried out the experimental work. A.S. supervised the project. All authors cowrote the paper.
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Eichelkraut, T., Heilmann, R., Weimann, S. et al. Mobility transition from ballistic to diffusive transport in nonHermitian lattices. Nat Commun 4, 2533 (2013). https://doi.org/10.1038/ncomms3533
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