Realization of interacting quantum field theories in driven tight-binding lattices

It is theoretically shown that the coherent hopping dynamics of a non-relativistic particle in engineered tight-binding lattices subjected to combined ac and dc driving forces can mimic in Fock space relativistic quantum field theories (QFTs) of strongly interacting fields, enabling access to extreme dynamical regimes beyond weak coupling and perturbative predictions. In particular, the simulation of a QFT model describing a Dirac field strongly coupled to a scalar bosonic field via a Yukawa coupling is proposed, suggesting the possibility of implementing this model using engineered lattices of evanescently coupled optical waveguides with a bent optical axis.


Introduction
The possibility of reproducing complex quantum phenomena using simple and fully controllable engineered quantum systems has received continuous and increasing interest in recent years [1,2]. Quantum simulations consist in the intentional and artificial reproduction of a quantum dynamics, difficult to access experimentally and/or to study theoretically, onto an unnatural quantum system that is more controllable theoretically and experimentally. The objectives of quantum simulations are diverse, ranging from purely aesthetic, to communicating independent fields, to the possibility of calculating and predicting physics that cannot be done with existing computational power. Important models and phenomena of interacting non-relativistic particles of condensed-matter physics, such as the superfluid-Mott insulator quantum phase transition or the Lieb-Liniger and Dicke models, have been successfully reproduced using cold atoms or ions loaded in optical lattices, cavity photons strongly coupled to atomic media or nuclear magnetic resonance processors (see [2][3][4][5][6][7][8][9] and reference therein). Recent works have suggested the exciting possibility of simulating relativistic quantum field theories (QFTs) using cold atoms, trapped ions or solid-state systems [10][11][12][13][14]. One of the main motivations for quantum simulations of QFTs is the possibility of investigating the physics of strongly interacting fields beyond standard QFT perturbative methods, which is generally a challenging problem on a classical computer. Such a possibility would offer a probing tool to open issues of non-perturbative QFTs [13,15], which is complementary to more traditional approaches such as QFTs on a lattice [16]. The interest in developing quantum analogue simulators is also to provide experimentally accessible test beds to explore extreme dynamical regimes in the matter or ideal models or exotic phenomena which are not currently accessible, regardless of the numerical complexity of the model [1,2]. Along this line, simple textbook relativistic models and phenomena, such as the Zitterbewegung and Klein tunneling of relativistic particles, have been mimicked in the laboratory [17][18][19][20][21]. Quantum simulation can also provide a platform where we may not only reproduce existing QFT models that are of difficult access, but also create QFT models that might not exist in nature, or QFT models that show unpleasant features, such as the appearance of ghosts, i.e. the negative norm states and non-unitary of the S-matrix for strong coupling [22]. Such QFT models have recently received renewed interest in the framework of PT -symmetric extensions of quantum mechanics and QFTs [23]; however, their physical relevance remains controversial.
In this work, we propose a tight-binding lattice realization of a QFT model describing the interaction of scalar bosons with Dirac fermions/antifermions via a Yukawa coupling. The simulator is based on the coherent motion of a single non-relativistic particle hopping on a dc-ac-driven tight-binding lattice, in which the evolution of the interacting fields in Fock space is mapped into the hopping dynamics of the particle on the lattice. A classical implementation of the driven lattice model, based on the transport of classical light in curved optical waveguide arrays, is also proposed.
3 as in the Yukawa sector of the standard model (see, e.g., [27]). In this model, the fermions are nucleons which interact with a scalar or pseudoscalar meson field (pions). The Yukawa interaction of nuclei via the exchange of massive particles is the analogue of the quantum electrodynamics (QED) interaction, except that the particles mediating the nuclear force have to be massive in order to have a finite range interaction. In the 1 + 1 dimension, a Dirac (fermionic) fieldˆ (x) coupled to a scalar bosonic (Klein-Gordon) fieldφ(x) via a Yukawa coupling is described by the HamiltonianĤ = dxĤ(x) with density [25,28] are the Hamiltonian densities of the free Dirac and Klein-Gordon fields, respectively, and is the Yukawa coupling. In the previous equations, m and µ are the masses of the Dirac and Klein-Gordon particles, respectively, σ x and σ z are the usual Pauli matrices,ˆ =ˆ † γ 0 is the adjoint spinor ofˆ , g is the coupling constant, and we assumedh = c = 1. The field operators satisfy the bosonic (fermionic) commutation (anti-commutation) relations The quantum fields can be expanded on the plane-wave basis of a one-dimensional (1D) box of size L.
The simplest case that we consider here is that of one fermion of momentum p and one antifermion of opposite momentum − p, which are coupled to bosons at rest, i.e. with momentum q = 0. Such an assumption greatly restricts the Hilbert space of the QFT model. However, as in [14], our proposed quantum simulation of finite-number interacting quantized modes includes all terms in a finite-mode Dyson expansion, instead of the standard approach that takes into account all modes but with a reduced number of perturbative Feynman diagrams [14]. After settingˆ , the Hamiltonian that we wish to simulate then readŝ where we have set g 1 = g(m/ω p ) 1/(2 p L), g 2 = −( p/m)g 1 and neglected the vacuum energy contributions. In the previous relations,b † ,d † andâ † are the creation operators of the fermionic, antifermionic and bosonic particles, respectively, ω p = (m 2 + p 2 ) 1/2 and p = (µ 2 + q 2 ) 1/2 = µ are the energies of the Dirac and Klein-Gordon particles, respectively, and u( p), v(− p) are the spinor components of the free fermionic and antifermionic particles with momentum p and − p, respectively [28]. Note that the Yukawa coupling of the bosonic field with a fermion-antifermion pair at rest ( p = 0), previously considered in [29], is obtained from equation (4) by assuming g 2 = 0. This limiting case is exactly integrable and was introduced as a simple solvable model in non-perturbative QFTs [29]; it also describes other physical systems, such as the atomic limit of the Hubbard-Holstein model of solid-state physics (which is exactly integrable via a Lang-Firsov transformation [30]) or the deep strong coupling regime of circuit or cavity quantum electrodynamics with degenerate qubit energy levels [31]. System 4 realizations of the Hamiltonian (4) in the g 2 = 0 limiting case were proposed previously in [32,33], and a recent experimental implementation based on engineered optical waveguide lattices has been reported in [34]. However, the more general case p = 0, which corresponds to g 2 = 0, was not considered in such previous works. A reduced Hamiltonian similar to (4) obtained from a QFT model analogous to that discussed in the present work, in which the Yukawa coupling is replaced by a QED interaction, has recently been derived in [14]; however, in that model the coupling g is time-varying, and the proposed physical implementation of the model was based on two trapped ions. Conversely, here we assume a constant coupling g and suggest a completely different physical implementation of the Hamiltonian (4), which is based on the hopping dynamics of a single non-relativistic particle moving on a 1D chain driven by combined ac and dc external fields.
The Hilbert space of the HamiltonianĤ is spanned by the Fock states |m 1 , the temporal evolution of the probability amplitudes a n , b n , c n and d n , as obtained from equations (4) and (5), is governed by the following equations: i dc n dt = n p + ω p c n , i dd n dt = n p + ω p d n .
Note that the Yukawa interaction introduces an effective coupling of states |0, 0, n and |1, 1, n solely, whereas the states with either one fermion or one antifermion are not affected by the interaction. Hence we will focus our attention on the dynamical coupling among the states |0, 0, n and |1, 1, n . The interaction dynamics in the weak coupling limit g → 0 is rather simple. In this case, the Yukawa interaction can induce transitions solely between the states |0, 0, n and |1, 1, n − 1 for some non-vanishing integer n, provided that g 2 = 0 and the energy conservation condition p = 2ω p is satisfied. Such transitions correspond to the annihilation (creation) of one fermion-antifermion pair, with simultaneous creation (annihilation) of one boson. For g → 0 but in the non-perturbative regime, the occupation amplitudes a n (t) and b n−1 (t) of states |0, 0, n and |1, 1, n − 1 periodically oscillate in time at the Rabi frequency g 2 √ n. In the deep strong coupling regime, i.e. when g 1 , g 2 are of the same order as the frequencies ω p and p , the dynamics becomes less intuitive. The case g 2 = 0, corresponding to the interaction of the bosonic field with a fermion-antifermion pair at rest (i.e. p = 0), decouples the dynamics of equations (6) and (7), and describes the process of creation and annihilation of bosons via the self-interaction termsb †bâ ,b †bâ † ,dd †â anddd †â † . The g 2 = 0 limit is an exactly solvable model of non-perturbative QFT [29]: in this case the spectrum ofĤ is composed by a ladder  (4) for parameter values g 1 = 0.8, g 2 = 0, ω p = 0.3 and P = 1. The system is initially in the state |1, 1, 0 . The dotted curves in (b) and (c) refer to the results obtained by the ac-dc-driven tight-binding lattice model (equations (9) and (10)) for parameter values ρ = 10.3, κ = 1.6, a F 1 = 1, a F 2 = 0 and ω = 20. The solid curves in (b) and (c) refer to the results obtained for the optical waveguide simulator (equation (12)).
of equally spaced energy levels, with energy spacing p , and the dynamics of occupation amplitudes a n (or b n ) in Fock space is thus exactly periodic at frequency p . As an example, figure 1 shows the evolution of the initial state |ψ(t = 0) = |1, 1, 0 for g 1 = 0.8, g 2 = 0, ω p = 0.3 and P = 1. In the figure, the behavior of the revival probability P r (t) = | ψ(0)|ψ(t) | 2 and the mean number of bosons n(t) = n n|b n (t)| 2 are depicted, together with a snapshot of the evolution of occupation probabilities |b n (t)| 2 . Note the characteristic periodic bouncing dynamics of bosons in Fock space with period 2π/ p . For a non-vanishing coupling g 2 , i.e. for bosons interacting with a moving fermion-antifermion pair, annihilation or creation of fermions/antifermions is allowed, and the periodic dynamics in Fock space breaks down. However, for values of g 2 comparable to or smaller than g 1 , the bouncing dynamics can be still observed. This is shown, as an example, in figure 2, in which the signature of the non-vanishing g 2 interaction term is clearly visible (compare figures 1 and 2). Note that in this case the mean number of bosons is given by n(t) = n n[|b n (t)| 2 + |a n (t)| 2 ].

Non-relativistic tight-binding lattice realization
The main result of this work is that the QFT Hamiltonian (4) can be simulated in Fock space by considering the coherent hopping motion of a non-relativistic quantum particle in a  (12)). In this case the physical units of couplings and frequencies are cm −1 , whereas the time variable on the horizontal axis is replaced by the spatial propagation length z (in units of cm).
tight-binding chain of 'molecules', driven by combined ac and dc forces. In our quantum simulator, the evolution of the amplitudes a n and b n of the Fock states |1, 1, n and |0, 0, n for the quantum field (the former corresponding to the state with one fermion of momentum p, one antifermion of momentum − p and n bosons at rest and the latter corresponding to the state with n bosons at rest solely) are mapped onto the occupation amplitudes of the non-relativistic particle at the molecular sites of the lattice. The hopping rates are engineered to precisely mimic all kinds of interactions of the Yukawa coupling, i.e. boson creation and annihilation induced by self-interaction terms (g 1 terms in equation (4)) and fermion-antifermion pair creation or annihilation with simultaneous creation or annihilation of one boson (g 2 terms in equation (4)). Hopping engineering is attained by the introduction of a suitable combination of ac and dc forces. Driven tight-binding lattice models have been investigated for a long time, mainly in connection with phenomena such as Bloch oscillations, dynamic localization and quasi-energy band engineering (see, e.g., [35,36]). Here we show that driven lattice models can mimic QFTs in Fock space of strongly interacting fields, a possibility that was not explored in previous works. We note that as our quantum simulator mimics the QFT model in Fock space (not in real space) and deals with a single particle hopping on a lattice, it can be realized with any kind of particle, i.e. particle statistics is not important. As discussed in the next section, the single-particledriven tight-binding lattice model can be even implemented using a classical (rather than a quantum) simulator based on the transport of classical light waves in engineered waveguide lattices.
To mimic the QFT model (4) in Fock space, we consider a tight-binding lattice composed of a semi-infinite chain of weakly coupled 'molecules', with non-homogeneous coupling rates κ n = κ √ n (n = 0, 1, 2, . . .), in the geometrical setting shown in figure 3(a). Each molecule is realized by the strong coupling of two Wannier states ('atoms'), with a hopping rate ρ, and thus sustains two bound states (the symmetric and antisymmetric molecular states) with energy separation 2ρ. The combined external driving forces are used to suitably control the hopping dynamics along the chain [37], realizing an effective lattice model that mimics our QFT model in Fock space. The tight-binding chain can be implemented using different physical (either quantum or classical) systems (see, e.g., [37]); in the next section we will discuss in detail a possible physical realization of the lattice model based on the transport of classical light in an array of optical waveguides. The non-homogeneous couplings κ n are realized by suitable distance control of the molecules, keeping the horizontal spacing uniform and equal to a. Denoting by u n and v n the occupation amplitudes of the two 'atoms' of the nth molecule in the chain, in the nearest-neighbor tight-binding approximation their temporal evolution is governed by the coupled equations where F x (t) = F 1 + F 2 cos(ωt) is the external dc-ac force acting along the x-direction, as shown in figure 3(a). The idea is to transform the time-periodic system of equations (9) and (10) into an effective time-independent lattice system [37,38], which reproduces our QFT model. With this aim, we consider the case of a strong and high-frequency ac force and perform a multiple scale asymptotic analysis of equations (9) and (10) [37,38], assuming the scaling κ/ρ ∼ , a F 1 /ρ ∼ , ω/ρ ∼ 1 and a F 2 /ω ∼ 1, where is a smallness parameter. After the introduction of the occupation amplitudes of symmetric and antisymmetric modes of each molecule, a n = [(u , where θ 1 (t) = −ρt + n(a F 2 /ω)sin(ωt), θ 2 (t) = (ρ − 2ω p )t + n(a F 2 /ω)sin(ωt) and ω p /ρ ∼ , assuming the resonance condition ω = 2(ρ − ω p ) 2ρ, at leading order in the perturbative analysis one obtains that the slowly varying amplitudes a n and b n evolve according to equations (6) and (7), with couplings g 1 , g 2 and boson frequency p given by Hence, by tuning the ratio = (a F 2 /ω) of ac driving, the detuning ω p = ρ − ω/2 and the amplitude a F 1 of the dc field, one can simulate the various coupling regimes of the QFT model (4). In particular, the dynamics of the QFT model for g 2 = 0 (figure 1) can be mimicked by considering a dc force solely. For example, the dotted curves in figure 1 show the behavior of revival probability and the mean number of bosons as obtained by numerical simulations of equations (9) and (10) for a relatively large detuning (ρ/κ 6.44). Note that the behavior of both revival probability and the mean number of bosons as obtained by the driven lattice model (equations (9) and (10), dotted curves in figures 1(b) and (c)) reproduces well those calculated by the averaged lattice model (equations (6) and (7), dashed curves in figures 1(b) and (c)).
To simulate the QFT model with non-vanishing momentum p, i.e. for g 2 = 0, the ac force is switched on. The dotted curves in figure 2 show the behavior of revival probability and the mean number of bosons as obtained by numerical simulations of equations (9) and (10) for the non-vanishing g 2 case. In this case the modulation frequency ω of the ac force has been chosen to satisfy the resonance condition ω = 2(ρ − ω p ) 2ρ, whereas its amplitude F 2 to give the target values of g 1 and g 2 according to equation (11).

Photonic simulator
The driven tight-binding model described by equations (9) and (10) can be implemented in principle using different physical (either quantum or classical) systems (see, e.g., [37]). In this section, we discuss in some detail an experimentally accessible realization of the driven lattice model, which is based on the propagation of classical light waves in an array of evanescently coupled optical waveguides with a bent axis. The recently developed direct waveguide laser writing technology, in which a femtosecond (fs) laser beam is used to create a guiding core inside a transparent glass (such as fused silica), has attained nowadays an excellent degree of maturity, enabling us to manufacture with great accuracy and reproducibility 3D low-loss waveguide circuits in rather arbitrary and complex geometrical settings (see, e.g., [39][40][41] and references therein). Indeed, several recent works have experimentally demonstrated that simulation of the coherent motion of a single non-relativistic quantum particle on a lattice driven by external fields is feasible using fs-laser-written waveguide arrays, in which the driving forces are effectively mimicked by a suitable geometric bending of the waveguide axis [41][42][43][44]. Direct fs laser writing enables us to engineer coupling rates with great accuracy. In particular, optical waveguide arrays with non-uniform coupling rates following the law κ n = κ √ n have already been realized in [45] to demonstrate classical analogues of the displaced Glauber-Fock states. Similarly, curved waveguide arrays with such non-uniform couplings have been manufactured very recently in [34]. The latter experiment provides the first photonic simulation of the ultrastrong deep coupling regime of the quantum Rabi model, which realizes our QFT Hamiltonian (4) as well in the special case g 2 = 0. Such recent experimental results indicate that fs-laser-written waveguide arrays should provide a feasible experimental platform to simulate the QFT model (4) in the more general case g 2 = 0. For non-vanishing values of g 1 and g 2 , an optical realization of the lattice model of figure 3(a) is provided by light transport in a chain of equal and weakly coupled optical waveguide couplers (the photonic analogue of the 'molecules' of figure 3(a)) with a bent optical axis and engineered spacing, as schematically shown in figure 3(b). The refractive index profile of the optical structure can be written as n(x, y) = n s + n nh(x − x n , y − y n ), where n s is the refractive index profile of the substrate dielectric medium, n and h(x, y) are the index change and normalized profile of the guiding core, respectively, and (x n , y n ) denote the positions of the various guides in the (x, y)-plane. In the waveguide reference frame, the propagation of a light beam at wavelength λ is described by the paraxial optical wave equation for the scalar field envelope ψ(x, y, z) [43] i where z is the paraxial propagation distance, ∇ 2 x,y is the transverse Laplacian and x 0 (z) is the bending profile of the waveguide axis. The coupled-mode equations (9) and (10), describing the propagation of the modal amplitudes u n and v n of light trapped in the various waveguides of the chain shown in figure 3(b), can be obtained from the paraxial wave equation (12) by standard variational techniques (see, e.g., [46]). Note that in our photonic simulator the time variable t of the original QFT model is mapped into the spatial propagation distance z along the axis of the waveguides, as shown in figure 3(b). The non-homogeneous coupling rates κ n among nearest waveguides can be realized by appropriate control of waveguide distances [34,45,47,48].
We have simulated light propagation in the optical waveguide array by assuming circular guiding cores of super-Gaussian shape as in [47], with a uniform horizontal spacing a = 6.5 µm; an index change n = 0.002, a wavelength λ = 633 nm and a substrate index n s = 1.45 have been assumed, which typically apply to fs-laser-written waveguides in fused silica probed in the red [41,45,48]. The refractive index profile of the optical structure that reproduces the parameter values of simulations shown figures 1 and 2 is depicted in figure 3(c). The waveguide axis is bent in the (x, z) plane with a bending profile x 0 (z) = (1/2R)z 2 − Acos(ωz), as schematically shown in figure 3(b). The driven parameters a F 1 and a F 2 of the lattice model (equations (9) and (10)) are related to the waveguide bending parameters by the simple relations [43] a F 1 = 2π n s a Rλ , a F 2 = 2πω 2 n s Aa λ .
Excitation of the boundary waveguide coupler (at the bottom left of figure 3(b)) in its antisymmetric (odd) supermode at the input (z = 0) plane is accomplished to simulate the initial state |1, 1, 0 of the quantum system. In an experiment, such a kind of multi-waveguide excitation can be achieved using, for example, a phase grating, as demonstrated in the experiment of [49].
The numerical results obtained by numerical simulations of the paraxial optical wave equation (12), shown by the solid curves in figures 1 and 2, turn out to be in good agreement with the predictions of the driven tight-binding lattice model. The detailed evolution of light intensity distribution along the waveguide structure, from the input (z = 0) to the output (z = 8 cm) planes at steps of z = 0.1 cm, can be viewed in the supplementary multimedia, available from stacks.iop.org/NJP/14/053026/mmedia 2 . In an optical experiment, the evolution of the mean number of bosons n(t) , shown in figures 1(c) and 2(d), can be retrieved after mapping the light intensity distribution along the waveguide array. In fact, one has n(t) = n n[|a n (t)| 2 + |b n (t)| 2 ] = n n[|u n (t)| 2 + |v n | 2 ], and |u n (t)| 2 + |v n (t)| 2 is the fractional light power trapped in the nth coupler of the array. Similarly, the evolution of the revival probability can be simply retrieved by monitoring the fractional light power that remains trapped in the initially excited coupler at the left boundary of figure 3(b). Light intensity distribution in the various waveguides along the propagation distance z can be obtained by a fluorescence imaging technique, as discussed, e.g., in [34,41,45,48,49]. In such experiments, light propagation along the array is directly visible due to the fluorescence of color centers created in OH-rich fused silica during the fs laser writing process, and can be accurately mapped on a CCD camera by means of a collecting microscope objective.

Conclusions
In this work, we have shown that the coherent hopping dynamics of a non-relativistic particle in ac-dc-driven tight-binding lattices can simulate in Fock space relativistic QFTs of strongly interacting fields, making it possible to mimic in a tabletop experiment extreme dynamical regimes of QFTs. In particular, we have suggested the possibility of simulating a simple QFT model describing the Yukawa coupling of a Dirac field with a scalar bosonic field using engineered lattices of evanescently coupled optical waveguides with a bent optical axis. As the external driving of tight-binding lattices has generally been investigated in connection with coherent control of tunneling, Bloch oscillations, dynamic localization, Anderson localization and related phenomena [35][36][37], here we have shown that driven tight-binding lattices can provide a test bed to simulate simple QFT models as well. As compared with the quantum simulator based on trapped ions proposed recently in [14], our driven tight-binding lattice simulator is not scalable; however, it may offer a simpler physical implementation (e.g. using a photonic lattice), as recently demonstrated in [34]. In this work, we have focused our attention on a simple version of the Yukawa QFT model; however, our findings could be extended to other QFT models, including QFTs showing ghosts which might not exist in nature and whose physical relevance is still debated (see, e.g., [22,23,50]). In this regard, photonic simulations of QFTs in Fock space might provide a rather unique test bed to realize in the laboratory PT -symmetric QFTs, such as the QFT Lee model in the ghost regime [51].