Abstract
The JackiwRebbi model describes a onedimensional Dirac field coupled to a soliton field and can be equivalently thought of as a model describing a Dirac field with a spatially dependent mass term. Neglecting the dynamics of the soliton field, a kink in the background soliton profile yields a topologically protected zeroenergy mode for the field, which in turn leads to charge fractionalisation. We show here that the model, in the first quantised form, can be realised in a driven slowlight setup, where photons mimic the Dirac field and the soliton field can be implemented–and tuned–by adjusting optical parameters such as the atomphoton detuning. Furthermore, we discuss how the existence of the zeromode and its topological stability can be probed naturally by studying the transmission spectrum. We conclude by analysing the robustness of our approach against possible experimental errors in engineering the JackiwRebbi Hamiltonian in this optical setup.
Introduction
Quantum simulators first and foremost offer a promising alternative when analytical and numerical methods fail in analysing models with strong correlations. On the other hand, they can also be used in probing exotic physical phenomena such as those predicted by relativistic theories. To date, a collection of effects in different fields ranging from condensed matter physics to relativistic quantum theories and material science have been simulated^{1}, using different platforms such ion traps^{2} and cold atoms in optical lattices^{3}.
More recently, the ability to controllably manipulate photons and their interactions with atomic systems resulted in the birth of a new direction in quantum simulations using photons and polaritons to mimic strongly correlated phenomena. Coupled cavity QED arrays (CCAs) were initially considered, where photons trapped in resonators interfaced with two level atoms (real or artificial ones) were shown to be able to reproduce manybody dynamics^{4,5,6}. The socalled photon blockade effect was exploited and polariton Mott transitions and effective spinmodels were proposed, introducing what is now known as the JaynesCummingsHubbard (JCH) model^{7}. Simultaneously and independently, the possibility for strongly correlated polariton dynamics in CCAs with fourlevel atoms and external fields was proposed^{8}, followed soon after by the JCH's phase diagram^{9,10} and the Fractional Hall effect^{11}. More recently one dimensional highly nonlinear waveguides with slowlight nonlinearities^{12,13,14} have been considered where effects characterising TonksGirardeau gas, Luttinger liquids physics, or even interacting relativistic theories have been shown to be simulable^{15,16,17}. The possibility to probe outofequilibrium phenomena has also been explored in driven setups^{18,19}
Quantum simulation has also shown great development in bringing the exotic physics of single particle relativistic effects into laboratory. These range from numerous theoretical works studying the Dirac equation and emerging effects such as the trembling motion of the electron (Zitterbewegung) or Klein tunneling to experimental implementations of those in different platforms covering all three mentioned above. More specifically in ion technologies the Dirac equation in 1 + 1 dimensions has been realised^{20,21,22,23} followed by an implementation with photons in waveguide arrays, including the random mass Dirac model^{24,25,26}. Seminal proposals for the realisation in slowlight systems also exist^{27,28,29,30}. Going beyond the Dirac equation, recent works have proposed to simulate the Majorana equation^{31,32} and neutrino oscillations^{33} in trapped ions.
The purpose of this article is to propose a quantum simulation of a historically important relativistic model known as the JackiwRebbi (JR) model^{34} with slowlight. The model describes a onedimensional Dirac field coupled to a static background soliton field and can be equivalently thought of as the model describing a massless Dirac particle under a Lorentz scalar potential. The same model has been studied independently by Su, Schrieffer and Heeger (SSH), while studying electronphonon coupling arising in polyacetylene, in the continuum limit^{35}. The model is well known for predicting charge fractionalisation^{36}, well before fractional quantum Hall effect was discovered and also for the topological nature of its zeroenergy solution and can be thought of as a precursor to topological insulators, a topic that is being hotly pursued^{37,38}.
There have been proposals to realise the model and observe charge fractionalisation in an optical lattice setup^{39,40} and experimental observation of the soliton which follows the model in a fermionic superfluid^{41}. Also, there have been proposals to observe topological bound states analogous to that arising in the model, in quantum walk and graphene setups^{42,43,44,45}. A topological bound state has also been observed in opticallyinduced dimer lattices^{46}, where the latter is closely related to the SSH model. Here, we follow a different route and propose a photonic implementation of the model in a slowlight polaritonic setup and show that the topological properties can be probed straightforwardly in an optical transmission experiment. We would like to highlight here that the usual experimental difficulties, in realising the photonic nonlinear interaction in slowlight systems for more complex many body simulations, do not exist in this case. Therefore we believe the proposal is a good candidate for directly realising the JR model and could allow the efficient probing of its topological properties for the first time.
The JackiwRebbi model and the related model studied by Su, Schrieffer and Heeger^{35} share many features similar to those studied in topological insulators and might indeed be classified as an AIIItype chiral topological Dirac insulator under a suitable regularisation^{47}. Here we describe the JackiwRebbi model and point out the topological properties and similarities to topological insulators. The equation of motion considered by Jackiw and Rebbi reads
where α and β are the Dirac matrices which in this case can be chosen to be proportional to two Pauli matrices. For concreteness, we take them to be α = σ_{z} and β = σ_{y}. The real scalar field ϕ(z) is assumed to obey the KleinGordon equation of the form
The ground state of the scalar field is degenerate at ϕ(z) = ±κ, which implies the existence of a soliton that interpolates between −κ at z = −∞ and κ at z = ∞ and corresponding antisoliton. The soliton localised at z = 0 is described by
When the Dirac field is coupled to such a soliton, a zeromode (zero energy state) appears which is localised around the soliton. The unnormalised spinor wave function of this zeromode reads
where αβχ = −iχ and is shown in Fig. 1 (solid red curve). Note that the dynamics of the scalar field is neglected in the above argument, i.e., the scalar field is treated as a constant background field.
Far away from the kink, the particle and hole bands have an energy gap m, whereas this gap must close at the point ϕ_{s}(z) = 0. This resembles the gap closing at the boundary of a topological insulator where the bound surface mode develops. The resulting bound state, the zeromode, is protected by the topology of the scalar field, whose existence, irrespective of the local profile of the kink, is guaranteed by the socalled index theorem^{34,48}. This phenomenon is similar to the emergence of edge modes in the quantum Hall effect^{49,50} or topological insulators^{37,38}, where edge modes appear at the boundary of two topologically different domains.
Another interesting aspect of the model (when second quantisation is taken into account) is charge fractionalisation, which we briefly describe before moving on to a proposal for a photonic implementation. The ground state of the Dirac field (the vacuum) in the soliton background may or may not include the zeromode. Because of the charge conjugation symmetry of the system, the two degenerate ground states must have opposite charges. Moreover, the charge difference between the two must be 1, as there can be only one electron occupying the state. The result is that the filled or unfilled states must have charges 1/2 and −1/2, respectively. This result can be confirmed by constructing the formal charge operator in terms of creation and annihilation operators of the eigenmodes. The vacuum states are the eigenstates of the charge operators which means that the observed fractional charges are sharp and are not just a trivial realisation of a distributed charge.
Results
Slowlight realisation of JR dynamics
The spinor slowlight setup we employ to realise the JR model and subsequently observe the zeromode is depicted in Fig. 2. This setup was first proposed by Ruseckas et al.^{30}, to which we refer the reader for a detailed explanation. The system comprises of a waveguide system coupled to an ensemble of atoms, which could be realised either in a tapered fiber^{51,52} or a hollowcore waveguide^{53,54,55,56,57}. The atoms are characterised by the three hyperfine ground levels; one populated state g〉 and two unpopulated states s1〉 and s2〉. These ground states are coupled to the two excited states e1〉 and e2〉 by the probe and control fields. The counterpropagating probe beams are described by the electric field amplitudes E_{1} and E_{2}, with the respective central frequencies ω_{1} and ω_{2} and drive the transitions g〉 → e1〉 and g〉 → e2〉. The propagation of the probe beams is controlled by two pairs of counterpropagating control lasers with Rabi frequencies Ω_{j}_{1} and Ω_{j}_{2} (with j = 1, 2), driving the transitions from the excited states to the unpopulated states. A slowly varying amplitude is associated with the electric field strength E_{j} of the j^{th} probe field:
with k_{1} = ω_{1} and k_{2} = −ω_{2}, where the speed of light in an empty waveguide is taken to be 1.
The propagation of the slowly varying amplitudes is such that they follow the 1 + 1 dimensional Diraclike equation
v_{0} = Ω^{2}/g^{2}n is the group velocity much smaller than 1, where is the Rabi frequency of the control fields, g is the atomlight coupling strength and n is the atomic density. The complex Rabi frequencies are tuned so that S_{11} = S_{22} = 0 and S_{12} = S_{21} = S where . is a column vector of two slowlyvarying field components and , i.e., and δ is the twophoton detuning. In the limit S = π/2, the above equation reduces to the Dirac equation in 1 + 1 dimension
which is equal to the Dirac equation introduced in the previous section with the identifications c = v_{0} and mc^{2}/κ = δ.
The connection with the JR model is obvious once we let δ be spatially varying as δ(z) = δ_{0} tanh(λz). This is possible, for example, by introducing additional lasers and states to induce AC Stark shift in the metastable states s_{1}〉 and s_{2}〉. Spatial variations in the lasers can then induce an arbitrary mass profile, including the desired soliton profile. A similar model has been studied in Ref. 29 with a slightly different atomic level scheme and the existence of zeromode has been briefly commented on, although no connection with the JR model has been made nor the topological nature of the zeromode mentioned. By making the connection, it is easily seen that there is interesting physics to be explored in the slowlight system, namely the topologically protected zeromode. In the following, we discuss how the zeromode and its topological stability can be observed experimentally.
Probing the zeromode and its topological stability
Broadly speaking, there are two possible ways to observe the zeromode in this optical setup. The first is an adiabatic method, where a photonic wave packet is adiabatically loaded to prepare an initial state that resembles the zeromode whose evolution is then observed. In this method, the initial state is prepared by capturing a pulse in the medium via the usual electromagnetically induced transparency (EIT) technique, i.e., by slowly turning off the forwardtraveling control fields which are initially on^{12}. Then, all the control fields are slowly turned on again, including the coupling fields Ω_{12} and Ω_{21}. At this point, the pulses are trapped as stationary light and go through the effective evolution governed by the Dirac equation. The dynamics of the spinor fields then differ significantly, depending on the initial condition of the wave packets as shown in Fig. 3, where we have solved the Dirac equation with the initial gaussian wave packet with σ = 1.2 and χ = (1, 1)^{T} to mimic the zeromode spinor. In the case of the JR model, i.e., under the soliton mass profile, the wave packet is trapped around the kink (upper row of Fig. 3), while under the normal case of constant mass, the wave packet spreads out (bottom row of Fig. 3).
The second method, better suited for this particular experimental realisation, is to look at the transmission and reflection of an incident probe field . Similar studies have been carried out quite recently, using transmission to probe strongly interacting effects in similar polaritonic systems^{58,59,60}. Consider a monochromatic probe field impinging from the left, while where L is the length of the system. One can study the transmission and reflection spectrum of this field where the transmitted field will come out to the right of the waveguide as , whereas the reflected part is as shown in Fig. 2a. The constant mass case can be solved analytically^{30} and shows the behaviour depicted in Fig. 4a. There is a finite window of perfect reflection due to the well known band gap proportional to the mass energy. Upon introducing the spatially dependent mass term discussed earlier, i.e. tanh(λz), transmission reappears at the center of the bandgap as shown in Fig. 4b, which can be attributed to the presence of the zeromode. Here and below, we follow Ref. 30 and assume that the length of the atomic cloud interacting with the propagating light is L = 300 μm and the latter's group velocity v_{0} = 17 m/s. For these values, the maximum two photon detuning δ_{0} = 0.25v_{0}/L lies well within the EIT transparency window.
Using this second method, the topological nature of the zeromode can be readily probed by looking at the transmission spectrum while perturbing the mass profile. The latter can be done by tuning the two photon detunings using standard optical methods, like AC stark shifting used in slowlight experiments^{12,13}. Changing the hyperbolic tangent function to the sine function while preserving the topology of the profile, i.e., the mass term takes the value of −δ_{0} and +δ_{0} at z = −L/2 and z = L/2 respectively and crosses 0 only once at the origin, has little effect on the transmission at the center of the bandgap as shown in Fig. 5a. The extreme limit where the ‘kink’ has been straightened out such that the background profile is linear is also shown in Fig. 5b. The zeromode persists in this extreme case, clearly demonstrating the topologically protected robustness. The zeromode is also protected from random fluctuations in the mass profile as shown in Fig. 5c and d, where we have assumed random fluctuations within 20% and 50% of δ(z), i.e., where is a uniform random with the resolution of 0.1 μm. Such topological protection holds as long as the fluctuations do not destroy the topology (two distinct values at the infinity and one zerocrossing only) of the soliton background. Unusually large fluctuations that changes the topology could destroy the zeromode, but are unlikely in experimental implementations.
Experimentally, it might be difficult to set the phases of the control fields exactly to the required value of the mixing angle S = π/2 in Eq. (6), in which case the resulting equation deviates from the Dirac equation. To study the effect of this imperfection, we have set S = π/2 + ΔS where ΔS takes 10% and 40% of the desired value, the result of which is depicted in Fig. 6a and b, respectively. The presence of the zeromode persists upon experimental errors in creating the exact Hamiltonian. Another experimental imperfection that can affect the transmission spectrum is the loss. The loss from the excited states can be easily incorporated into the equation of motion by introducing an extra term −iγ_{eff} into Eq. (7)^{30}, where γ_{eff} = γδ^{2}/Ω^{2} with γ the decay rate from the excited states. Effects of photon losses are shown in Fig. 6c and d, where we have used δ(z) = δ_{0} tanh[0.02z] and and 0.01, respectively. The transmission peak at the zeromode reduces in size with increasing loss rate, but the peak is still visible at . Under normal EIT condition, , which along with our numerical analysis indicates that the effect of the loss should not obstruct the detection of the zeromode.
Discussion
In this work, we have shown that it is possible to simulate the JackiwRebbi model and probe its topological nature in a driven slowlight system using current technology. By introducing spatiallydependent optical detunings, it is possible to simulate the Dirac equation with a spatially dependent static soliton field and allows one to directly probe the topologically protected zeromode. The robustness of this zeromode can be tested by changing the spatial profile of the detunings, while continuously observing the optical transmission spectrum of the system. We have also analysed the effects of experimental imperfections such as deviations from the ideal model and losses and showed that the observation of zeromode survives these imperfections.
Before closing we would like to briefly comment on other types of models that might be studied in the same system in future works. Firstly, adding interactions in the system through EIT photon nonlinearities could allow the study of interesting strongly correlated physics. Intraspecies repulsion would make the bosons behave like fermions in many ways and it would be interesting to think about how this affects the (lack of) charge fractionalisation effect in bosons. Interacting random mass Dirac model is another interesting possibility as studying the interplay between interactions and randomness is an important field being actively studied. Yet another possibility is to think of the mass term as the Lorentzscalar potential which can act as a confining potential, given a proper spatial dependence^{61,62}. This model has been shown to act as a phenomenological model of quark confinement motivated by the ‘MITbag’ model^{63}.
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Acknowledgements
We would like to acknowledge the financial support provided by the National Research Foundation and Ministry of Education Singapore (partly through the Tier 3 Grant “Random numbers from quantum processes”) and travel support by the EU IPSIQS.
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D.G.A. and C.N. developed the concept of the work. P.D. and C.N. have carried out numerical calculations. All authors have contributed to analysing the results and writing up the manuscript.
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Angelakis, D., Das, P. & Noh, C. Probing the topological properties of the JackiwRebbi model with light. Sci Rep 4, 6110 (2015). https://doi.org/10.1038/srep06110
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