Abstract
Longrange correlation—the nonlocal interdependence of distant events—is a crucial feature in many natural and artificial environments. In the context of solid state physics, impurity spins in doped spin chains and ladders with antiferromagnetic interaction are a prominent manifestation of this phenomenon, which is the physical origin of the unusual magnetic and thermodynamic properties of these materials. It turns out that such systems are described by a onedimensional Dirac equation for a relativistic fermion with random mass. Here we present an optical configuration, which implements this onedimensional random mass Dirac equation on a chip. On this platform, we provide a miniaturized optical testbed for the physics of Dirac fermions with variable mass, as well as of antiferromagnetic spin systems. Moreover, our data suggest the occurence of longrange correlations in an integrated optical device, despite the exclusively shortranged interactions between the constituting channels.
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Introduction
Longrange correlations are observed in a variety of different settings, such as in the arrival statistics of seismic waves^{1}, DNA sequences^{2} and even in stock market patterns^{3}. A generic property of such correlations is the contribution of slowly decaying tails of the underlying probability distribution to the averaged quantities. This leads to random walks, with a nonvanishing occurrence of long jumps^{3,4,5}, as well as longrange ordering effects in organic and inorganic materials^{2,6,7,8,9}. Spin1/2 chains with antiferromagnetic interaction, so called the spinPeierls systems^{10,11}, as well as crosscoupled spin1/2 chains, so called the spinladder systems^{12,13}, are both prime examples of solids with longrange ordered phases. Doping in these materials causes impurity spins, which support lowenergy, exponentially localized defect states within the band gap of the host material, but exhibit longrange averaged correlations decaying with a power law over distance. Both systems map to a disordered onedimensional (1D) Dirac equation for relativistic, massive fermions via a Jordan–Wigner transformation^{14}. In this description, the magnitude of the mass corresponds to the dimerization strength of the spinPeierls material or the crosscoupling in the ladder, respectively, and the impurity defects are modelled by randomly distributed signflips of the mass (kinks) at which the gap states are localized^{15}. This randomness in the mass sign leads to the notion of the random mass Dirac model.
Disorder effects have been widely investigated in the context of nonrelativistic physics, modelled by a Schrödinger equation with a disordered potential and leading to the famous Anderson localization^{16,17,18,19,20}. The major difference between nonrelativistic and relativistic disordered systems lies in the number of occurring length scales and the nature of the resulting correlations. The Anderson localization length is directly related to the mean free path l (ref. 18); therefore, only one length scale is relevant and the averaged transverse correlation functions decay exponentially. For random mass Dirac fermions, on the other hand, there exists also the length scale of correlation length L, which diverges for the lowenergy eigenstates situated in the band gap. Hence, for spatial coordinates in the regime lxL powerlaw decaying correlations with a universal exponent of −3/2 occur^{14}. In the context of doped antiferromagnetic spin systems, this means that the impurity spins are, on average, mutually correlated up to a distance L. These longrange correlations have profound impacts on the macroscopic properties of the materials. For instance, it has been shown that the contributions to the susceptibility from dopinginduced impurity spins in spinPeierls systems at low temperatures deviate heavily from the Curielaw behaviour one would expect for purely exponentially decaying correlation functions^{15,21}. Other examples are the transition to an ordered antiferromagnetic phase, the Néel phase, occuring at unexpectedly high temperatures^{21,22,23,24}, and the temperature dependence of the specific heat in these materials^{21}.
For specific compounds and doping conditions, certain predictions of the model, such as phase transitions or longrange ordering, have been experimentally tested, either by the measurement of macroscopic quantities^{22,24,25} or by scattering techniques^{23,26,27,28}. However, some configurations are difficult to investigate experimentally, such as the impact of boundaries^{29} or spatial variations in the coupling or dimerization strength. Besides the antiferromagnetic spinPeierls and spinladder materials, the 1D random mass Dirac model maps also to the quantum Ising model for ferromagnets with random bonds along one dimension^{15,30,31}. Therefore, a universal simulator of the random mass Dirac model would help to explore its predictions for all those systems, at the same time and in isolation from the various distortions that may arise in the respective physical material (for example, variation of the dimerization due to atoms outside the chain, thermal fluctuations, uneven coupling to other layers in the Ising system, and so on). However, such a simulator has not been implemented to date. Only recently, a promising scheme in a coherently driven atomic ensemble has been suggested, mapping the random mass to a spatially varying detuning of the controlling light fields^{32}.
In this article, we propose and realize a radically different approach to the random mass Dirac model; we imprint it on a photonic chip, containing a chain of optical waveguides, transversally coupled by their exponentially decaying evanescent fields. Such photonic lattices have demonstrated their potential as a platform for the implementation of quantum random walks^{33,34}, as well as for the quantum simulation of indistinguishable, nonrelativistic particles and their exchange statistics^{35,36}. By arranging the waveguides in a binary superlattice of alternating high and low refractive index, it is even possible to study the dynamics of relativistic fermions with classical light, such that the light evolution in space is governed by a 1D Dirac equation^{37}. This enables the classical emulation of genuinely relativistic effects, such as Zitterbewegung^{37,38}, Klein tunnelling^{39,40} and pair production^{41,42}, with the advantage of a directly observable evolution. In these schemes, the constant mass of the fermions is determined by the refractive index difference between the constituting sublattices. As recently demonstrated, one can further emulate massless fermions in superlattices with an alternating coupling^{43}. Here we introduce disorder to the mass by reversing the ordering of the sublattices of a refractive index superlattice and distributing the associated kinks in a random manner. By exciting the localized gap states with laser light and averaging over their intensity correlations for many realizations of disorder, we experimentally emulate powerlaw correlations with an exponent of −3/2, as they are expected from the model. This further suggests the occurence of longrange correlations in an integrated optical device, even though the interaction between the individual waveguides is only shortranged. Note that for quantum walks in regular^{33,34} or disordered waveguide lattices^{44}, jumps between adjacent channels are predominant. Therefore, their correlation functions are always shortranged, that is, they exhibit exponential slopes. To observe powerlaw, decaying longrange correlations in a random walk, significant probabilities for long jumps are required, as has been demonstrated in a specifically engineered bulk material^{5}. In our system, on the other hand, the longrange correlations arise from the properties of the emulated random mass Dirac model. Via a numerical analysis, we further investigate the scaling of our model system to larger dimensions. We predict the occurrence of powerlaw correlations over an order of magnitude for system parameters, which are experimentally accessible. Our data suggests universality of the exponent of −3/2, independently of several model parameters. We finally demonstrate the robustness of the approach with respect to coupling to nextnearest neighbours, that is, a secondorder coupling, by numerical calculations.
Results
Optical emulator of a doped spin system
A pure spinPeierls system exhibits a structural phase transition between a paramagnetic (T>T_{C}) and a dimerized phase (T<T_{C}). Below its transition temperature T_{C}, a band gap opens between a singlet ground state and triplet excited states, with the gap size being proportional to the dimerization strength δ (refs 10, 21). Dopingrelated impurities lead to the formation of domain walls, as illustrated in Fig. 1a. These and other related systems can be modelled by a Dirac equation for a relativistic fermion with the twocomponent spinor Ψ=(Ψ^{(1)},Ψ^{(2)})^{T} in 1+1 dimensions:
with and denoting the Pauli matrices, c the speed of light and ħ the reduced Planck constant. A domain wall corresponds to a signflip of the mass at x=0 between the values m(x)=±m_{0} (Fig. 1b), its magnitude being proportional to the dimerization strength m_{0} δ (refs 14, 21).
To emulate the Dirac equation (1) in optics, we discretize it in the transverse coordinate and , assuming slow variations of the spinors along the x coordinate. In optics, this requirement translates to a broad beam illumination of the device. By further substituting and , and replacing time t by the longitudinal spatial coordinate z, one obtains the coupled mode equations for a binary superlattice of singlemode optical waveguides^{37}, in a frame moving with the average propagation constant:
Here, a_{n} and b_{n} denote the field amplitudes of the guided modes in the two sublattices. In this representation, their relative onsite propagation constants σ_{n}=±σ are proportional to the mass m_{0} and the nearestneighbour coupling κ corresponds to the speed of light, setting the time scale of the emulator by determining the time interval associated to a certain propagation distance.
Figure 1c illustrates this analogy and shows how a signflip of the mass is implemented by a reversal of the sublattice ordering. To realize such photonic superlattices, we employ direct waveguide writing in fused silica glass^{45} (Methods). Because of the transformation made above, a phase shift of π occurs between adjacent superlattice cells. This phase shift is required at the input of the device if one wishes to excite the gap state residing at a kink with a flatphased Gaussian beam. To this end, the waveguides of every second superlattice cell are segmented at the input (see inset of Fig. 1c). Thereby, the effective refractive index is reduced in the segmented guides, imposing the desired phase shift on the propagating light^{46} (Methods).
A single signflip of the mass supports an exponentially localized zeroenergy eigenstate between the two bands of the energy spectrum of a massive Dirac fermion^{15}. The eigenspectrum of a corresponding waveguide superlattice with N=23 unit cells, σ=0.9 κ and one kink at cell k=9 is presented in Fig. 2a, clearly showing the two energy bands, separated by a gap of width 2σ and a gap state, which has a finite energy due to the discretization. For further analysis, we select the first component of the gap state , u_{n} denoting its amplitude in cell n, derived from the light intensity on the first sublattice and corresponding to the spinor Ψ^{(1)}. For our optical system, this results in an exponential profile (green symbols in Fig. 2b), in agreement to the analytic solution^{15} of the Dirac equation (1) (red line). We excite the gap state experimentally by launching a spatially extended laser beam at the kink (Methods). The light evolution is observed by fluorescence microscopy^{47} and the end face of the sample is imaged onto a chargecoupled device to obtain the output intensity distribution. As shown in Fig. 2c, most of the light (96% of the power) gets trapped at the kink, in good agreement to the numerical solution of equation (2) (Fig. 2d), which predicts also an excitation efficiency of 96% for the gap state. The remaining light is radiated into the lattice, where it undergoes the optical analogue of Zitterbewegung^{38}, that is, it emulates the rapid trembling motion experienced by massive relativistic fermions. From the measured output intensities, we extract the gap state intensities u_{n}^{2} (blue symbols in Fig. 2b), which closely match the calculated results, down to a background from the trembling light at 10^{−2} relative intensity level. The noise floor of the experimental setup lies at about 10^{−4} relative intensity.
Mass disorder and longrange correlations
In a disordered solid, a measurement of macroscopic quantities is usually associated to spatial averaging over an extended region. If that region is large enough and the disorder conditions are homogeneous across it, such a spatial average will be entirely equivalent to an ensemble average over many realizations of the same disorder on a smaller region^{48}. This equivalence is commonly exploited in the optical observation of Anderson localization^{18,49,50} and has recently been empirically verified^{51}.
In this vein, we consider the uniform, independent distribution of two kinks of equal height but opposite sign within a given domain of size Ω as a model for mass disorder, and perform ensemble averaging. In a spinPeierls system, this corresponds to an ensemble with two randomly positioned dopinginduced impurities in each realization. As both kinks have the same height, the individual gap states are identical, up to a phase. Therefore, their combined intensity correlation is in theory the same as if it were obtained by taking a single gap state and mirroring (that is, doubling) it at an imaginary line positioned in the centre between them. We employ this approach in our optical setup utilizing the boundary of the lattice. By mirroring the gap state associated to a kink located at cell k at the first waveguide, the situation where two kinks are separated by 2(k−1) cells is effectively simulated. Note that the only experimental uncertainties occuring in a waveguide lattice with two physical defects, which are not caught by this method, are associated to unequal defects and their unequal excitation. These effects have no meaning in the random mass Dirac model and are eliminated by the mirroring.
We fabricated 65 waveguide lattices with σ=0.9 κ, each comprising N=23 cells and a kink, randomly positioned on the interval k[2,18] according to the triangular probability distribution P(k)(18−k). The upper limit is chosen such that the righthand boundary at n=N=23 does not have any significant influence on the gap state. For each realization, the defect state residing at the kink is excited and its components u_{n}^{2} are obtained from the measured output intensity distributions (Fig. 2). Then u is mirrored at the lefthand boundary n=1, yielding the doubled gap state , with for p≥0 and for p≤0. As far as intensities are concerned, this procedure is equivalent to two kinks, which are independently, uniformly distributed on a domain with a size of Ω=34 cells. To model a realistic scenario, where two dopants cannot occupy the same site, identical positions are excluded (defining the lower boundary in the triangular distribution).
Figure 3a displays the intensity of the mirrored gap state u^{(m)} for two exemplary realizations. Their individual intensity correlations
are presented in Fig. 3b, clearly showing an exponential decay, as one would expect from the exponential profile of the eigenstates. Hence, these typical correlations are only shortranged. In concordance with Fig. 2b, the plateaus at 10^{−4} and 10^{−7} can be attributed to the background illumination and the noise floor of the experiment, respectively. The picture changes dramatically, however, if one calculates the disorderaveraged correlation ‹Γ(d)›: now, in the range 15d30 a powerlaw decay ‹Γ(d)›d^{−3/2} can be clearly identified (Fig. 3c), corresponding to longrange correlations in the Dirac model on this length scale. Note that the measured correlation does not deviate by more than one s.d. from the simulated data in the region of interest. As before, the background light dominates as ‹Γ(d)› approaches 10^{−4}.
Length scales and secondorder coupling
The relevant length scales of the problem are the localization length l_{l}, the mean free path l and the correlation length L. The former is independent of the disorder and can be obtained directly from the exponential slopes of the localized eigenstates at the kinks: , which yields l_{l}=κ/σ≈1.1 (Fig. 2b)^{21}. The mean free path is the average separation of the two kinks in our case, which is obtained from the probability distribution of the single kink as l≈12.7. Finally, the correlation length depends logarithmically on the normalized energy of the gap state , diverging^{14} for ε→0. In our setting, the gap state energy can be obtained from the eigenvalue spectrum (see Fig. 2a, red circle), yielding ε_{g}=0.19 κ and normalized with respect to the energy of the lowest continuum state in the upper band ε_{c}=σ=0.9 κ. This results in a normalized energy ε=ε_{g}/ε_{c}≈0.21, yielding a correlation length L≈31, which is on the same scale as the transverse dimension of the domain on which the kinks are distributed (Ω=34 cells). Note that the single cell of zero mass at the kink (Fig. 1c) is important to reduce ε, and thereby reach a sufficiently long correlation length, providing a window ldL for powerlaw correlations. A sharp kink with an immediate transition between the sublattice orderings violates the slowly varying lightfield approximation required by the discretization of equation (1), resulting in ε≈0.45 and L≈8<l, rendering the emulation of longrange correlation impossible.
To extend the length scale on which a powerlaw decay can be expected, one has to either decrease the mean free path or increase the correlation length and the domain size. With a single mirrored kink, however, the domain size is at best Ω≈3l. This inherently limits the range of longrange correlation to ld min(L,Ω)≈3l.
To verify the universal nature of the exponent of −3/2 on larger length scales, we numerically investigate the correlations arising in a lattice of N=200 cells with K mass signflips uniformly distributed on the inner Ω=170 cells (such that the boundaries have no significant influence). To get control over ε, we consider smooth transitions of the form , instead of simple kinks. Here, a determines the steepness of the transition and k its random lateral position. As before, no two kinks are allowed on the same position. The blue curve in Fig. 4a shows an exemplary realization of a superlattice with σ=κ, K=20 and a=0.1. The energy spectrum is shown in Fig. 4b, exhibiting the continuum of the two bands of the superlattice and a partially filled band gap in between. The gap state with lowest positive energy ε_{g} resides on the signflips (red curve in Fig. 4a) and is chosen for the further analysis. Normalization with the lowestenergy state of the upper band yields ε≈0.036 for this configuration. This should support a correlation length of L≈11l. Essentially, one finds that the larger K (more signflips) and the smaller a (smoother profiles), the smaller ε and the larger the ratio , determining the range where longrange correlation is supported. The dependence of the average correlation on ε is shown in Fig. 4c for K=40 by a variation of a. Indeed, ‹Γ(d)› decays with an exponent of −3/2 on a length scale growing with decreasing a. In particular, for the red curve (a=0.1), one observes a powerlaw decay over one order of magnitude between 8d80. Interestingly, the correlations are essentially independent of the density of signflips (Fig. 4d), corresponding to the defect density, as long as ε is kept constant. Please note that the mean free path l does no longer exclusively depend on the mean separation of the signflips, as in the configuration with kinks, but also nontrivially on the profile of the mass transition.
In a waveguide lattice, some coupling between nextnearest neighbours is usually unavoidable^{52}. Adding a corresponding secondorder coupling term to equation (2) would correspond to a second spatial derivative in equation (1), thus abandoning the frame of the Dirac model. However, as we numerically demonstrate in this final consideration, the powerlaw correlations are unaffected by this. Figure 5 presents the simulated, disorderaveraged correlations for the same configuration as in the experiment (Fig. 3), but with . It is evident, that the correlation still exhibits the same powerlaw decay and does not change substantially. The inset shows that the band gap contracts compared with the ideal case (cf. Fig. 2a). However, as long as the gap state persists (which it does for κ~0.75 κ in this configuration), the longrange correlations prevail.
Discussion
We have experimentally demonstrated how a binary lattice of coupled optical waveguides can be employed to implement the random mass Dirac model in a compact, robust and controllable way. This model predicts the occurence of longrange averaged correlations on the length scale lxL for gap states in general, as well as for impurity spins in antiferromagnetic chains and ladders in particular. Our optical platform supports corresponding gap states residing at kinks where the ordering within the unit cells is reversed. We have excited these gap states in laserwritten waveguides by a builtin control of the input phase. An ensemble of randomly located kinks is mirrored to obtain the distribution of defect pairs with random separation. The measured averaged intensity correlations show a characteristic powerlaw scaling with an exponent of −3/2, in excellent agreement to the predictions of the random mass Dirac model. Furthermore, these results suggest that longrange correlations can be observed in an integrated optical system with two or more physical defects. Despite the shortranged nature of the coupling between the waveguides, as well as the individual correlations, the longrange behaviour arises from the strong influence of rare configurations of distant defects on the disorder average. The lower limit of the range where this behaviour can be expected is determined by the mean free path, l≈13 in our system, whereas the upper bound (≈30) coincides with the correlation length L, as well as the transverse size of the domain on which the kinks are distributed.
We have numerically investigated how an extension to larger length scales is possible. For that purpose, larger lattices with many defects are required and these defects have to be implemented by smooth signflips, to reduce the gap states’ energies and, thereby, increase the correlation length. With 40 sufficiently smooth signflips randomly distributed on a domain of 170 lattice cells, longrange correlation over an order of magnitude can be expected. Our results show that the exponent of −3/2 is independent of the important model parameters energy and defect density, hence suggesting its universality. A future challenge will be the experimental excitation of the gap states in such large systems, which feature intricate intensity distributions (see Fig. 4a) necessitating a precise control over the input beam’s intensity profile, as it is, for instance, offered by spatial light modulators. Note that the required phase relations between adjacent channels are always guaranteed by the segmentation of the waveguides.
Several promising routes for future exploration with the optical emulator lie ahead: one is the inclusion of threedimensional effects occuring in a real solid. For example, in the spinPeierls material CuGeO_{3}, the angle of the Cu−O−Cu bond has a crucial impact on the coupling between the spin carrying Cuatoms J, as well as on the dimerization strength δ (ref. 26). By tuning the corresponding parameters κ and σ along the transverse axis of the waveguide lattice, one could investigate the impact of a deformation of the chain due to external effects, such as strain. Another route is the extension to a twodimensional (2D) disordered fermion model, directly mapping to a 2D disordered quantum Ising model, which requires sophisticated algorithms for a numerical analysis^{53}. Twodimensional superlattices have successfully been implemented in directwritten waveguide systems^{54} and a disorder could be introduced to them at ease by randomly exchanging the lattice ordering in both dimensions. Also, the interplay of nonlinearities and disorder, which has proven to host a variety of interesting phenomena in nonrelativistic systems^{19,50,55}, is yet to be investigated in the relativistic context. A commonly considered nonlinear Dirac equation contains an additional interaction term of third order in the spinor^{56}, which can be implemented by a Kerr nonlinearity in waveguide lattices^{57}.
Besides these prospects, our findings might hint at novel methods for coherent information transfer between distant locations or remotesensing applications. As the ensemble average is fully equivalent to spatial averaging over an extended domain^{48,51}, a longrange correlation can also be expected in a single optical device. As long as the disorder is of static nature, full coherence will be maintained.
It has moreover been demonstrated numerically that the longrange correlations persist in the presence of residual secondorder coupling effects, leaving the presented optical realization of the random mass Dirac model essentially unaffected, a feature which will be particularly useful for 2D configurations.
Methods
Waveguide fabrication and defect state excitation
The exposure of transparent materials to intense femtosecond laser radiation leads to multiphoton absorption, avalanche ionization and a subsequent densification in the focal region, increasing the refractive index permanently^{45}. Employing this effect, we inscribed waveguides with an average writing velocity of v_{0}=1.5 mm s^{−1} by focusing (numerical aperture 0.35) a pulsed laser (wavelength 800 nm, pulse duration 170 fs, pulse energy 220 nJ, repetition rate 100 kHz) into a 10cm long fused silica sample. The waveguides’ transverse separation was 19 μm, setting the coupling strength to κ≈0.68 cm^{−1} for an observation wavelength of 633 nm in the experiment. The modulation of the refractive index within the superlattice was achieved by tuning the velocity to in an alternating manner, resulting in relative propagation constants ±σ=±0.9 κ.
The optimal segmentation parameters for obtaining a phase shift of π at the input were found to be s=2.75 mm and Λ=s/100 (see inset of Fig. 1c), by testing the selfimaging effect in another sample^{46}. The required exponential intensity distribution of the defect states is approximated by a Gaussian beam. We weakly focus (numerical aperture 0.035) a linearly polarized, continuouswave laser beam with a wavelength of 633 nm onto the front face of the lattice at the location of the kink. In the focus, the beam has a flatphased, Gaussian profile with a 1/eintensityradius of 1.8 waveguides (34 μm).
Additional information
How to cite this article: Keil R. et al. The random mass Dirac model and longrange correlations on an integrated optical platform. Nat. Commun. 4:1368 doi: 10.1038/ncomms2384 (2013).
Change history
26 February 2013
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Acknowledgements
We acknowledge funding by the German Federal Ministry of Education and Research (ZIK 03Z1HN31 ‘ultra optics 2015’), Deutsche Forschungsgemeinschaft (Grant no. 462161) and the Thuringian Ministry for Education, Science and Culture (Research group ‘Spacetime’, Grant no. 11027514). R.K. is supported by the Abbe School of Photonics. M.H. is supported by the German National Academy of Sciences Leopoldina (grant No. LPDS 201201).
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R.K. devised the optical emulator, fabricated the samples, performed the measurements and simulations, analysed and interpreted the data. J.M.Z., F.D. and M.H contributed to the experimental setup, and analysed and interpreted the data. A.S. devised the optical emulator, and analysed and interpreted the data. All authors cowrote the manuscript.
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Keil, R., Zeuner, J., Dreisow, F. et al. The random mass Dirac model and longrange correlations on an integrated optical platform. Nat Commun 4, 1368 (2013). https://doi.org/10.1038/ncomms2384
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DOI: https://doi.org/10.1038/ncomms2384
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