Topological soliton-polaritons in 1D systems of light and fermionic matter

Quantum nonlinear optics is a quickly growing field with large technological promise, at the same time involving complex and novel many-body phenomena. In the usual scenario, optical nonlinearities originate from the interactions between polaritons, which are hybrid quasi-particles mixing matter and light degrees of freedom. Here we introduce a type of polariton which is intrinsically nonlinear and emerges as the natural quasi-particle in presence quantum degenerate fermionic matter. It is a composite object made of a fermion trapped inside an optical soliton forming a topological defect in a spontaneously formed crystalline structure. Each of these soliton-polaritons carries a Z2 topological quantum number, as they create a domain wall between two crystalline regions with opposite dimerization so that the fermion is trapped in an interphase state. These composite objects are formally equivalent to those appearing in the Su-Schrieffer-Heeger (SSH) model for electrons coupled to lattice phonons. Systems composed of neutral atoms and photons provide an ideal platform for the emerging field of quantum non-linear optics, which in turn is of interest for quantum technologies and the study of many-body physics. The authors theoretically propose a hybrid light-matter quasi-particle akin of polaritons composed of an optical soliton trapping a fermionic atom that carries a nontrivial topological quantum number.

Introduction.-Hybridsystems involving photons and neutral atomic gases have emerged as ideal platforms for nonlinear quantum optics [1], characterized by effective interactions between photons even at very low light intensities.This realm of optics offers interesting possibilties for quantum technologies, as optical nonlinearities at the singlephoton level would facilitate quantum information processing with light, the latter being at the same time an ideal information carrier [2].Moreover, systems composed of ensembles of photons and atoms interacting at the level of single quanta is an intriguing many-body system which has recently attracted a lot of interest [1,3].
In such systems the low-lying excitations are polaritons, which emerge as long-lived quasi-particles.In the usual scenario polaritons are a linear superposition of non-interacting photonic and atomic excitations, while the optical nonlinearities of the system are generated by the interactions between polaritons inherited from the atoms.
Here we show that a different situation can arise, where a new type of long-lived hybrid quasi-particle emerges.This new quasi-particle is not simply a superposition the of non-interacting atomic and photonic degrees of freedom, but rather an effect of the optical nonlinearities themselves.It is a composite made of an atom trapped inside optical soliton forming a topological defect in an emergent crystalline structure.
Such crystalline structure appears when a degenerate cloud of fermions is coupled to the propagating modes of an optical waveguide in the configuration of Fig. 1 (crystallization in optical waveguides has been studied only for classical particles so far [4,5]).In this (quasi-)onedimensional configuration, the fermionic cloud is unstable towards density modulations with wavenumber equal (⌦, !L ) 1. Pictorial representation of the system.A (quasi-) one-dimensional cloud of fermionic atoms (blue spheres) is transversally driven by a coherent laser and is coupled to a set of guided modes.The corresponding light intensity (including the homogeneous background) is shown as a darkyellow surface with a transverse exponential damping indicating the guided nature of waveguide modes.The situation shown corresponds to the presence of a light-matter defect at the center where an optical soliton traps an excess particle (red circle).The optical soliton connects two spatially ordered regions with different dimerization.The inset shows the atom internal level structure (spacings not to scale) together with the dispersion of the guided electromagnetic modes.The laser frequency is ωL, its the Rabi-frequency is denoted by Ω, and kF is the Fermi momentum of the atomic cloud.
to twice the Fermi momentum k F -analogous to the Peierls instability known since the 1950's in the context of solids [6] -so that photon-mediated Umklapp scattering of atoms between the Fermi points (momenta of ±k F ) induces crystallization.This scenario has been considered for fermionic atoms coupled to a single standing-wave mode of an optical resonator [7][8][9][10].The effect of a sharp Fermi surface is however much more prominent in the presence of multiple electromagnetic modes, as is the case in a confocal cavity [11] or for the continuum of propagating modes of an optical waveguide that we consider here.Firstly, provided that 2k F is included in the wavenumbers of the waveguide's electromagnetic modes, the instability toward crystallization will be always present even at vanishing coupling, differently from the single-resonatormode case.Secondly, spatially-ordered fermionic patterns coupled to multimode fields are much more sensitive to commensurability effects, and the resulting structures can accommodate hybrid light-matter defects as low-lying excitations.The latter consist of a fermion trapped in an edge state located at a solitonic deformation of the electromagnetic potential.The optical soliton separates two regions of the crystal with opposite dimerization and therefore carries a Z 2 topological quantum number.Its size is set by the inverse Fermi momentum.In a specific parameter regime that we identify, our system is formally described by the continuum limit of lattice electron-phonon models, more specifically the Su-Schrieffer-Heeger (SSH) model [12][13][14][15].There has been much interest in the SSH model as of late, with a number of synthetic implementations in different systems, including ultracold atoms [16][17][18], semiconductors [19], granular particles [20], RLC circuits [21] and microring resonators [22].However, with respect to the above implementations, the system we consider here features an important additional ingredient, namely that the lattice felt by the fermions is a fully dynamical object self-consistently modified by the particles, as in the original SSH model.Only in this case the system can feature topological defects emerging as low-lying excitations.We also discuss optical transmission spectroscopy using waveguide modes as a means to nondestructively detect the topological defects.
Model and formalism.-Weconsider a degenerate Fermi gas of N transversally-pumped neutral atoms interacting with the multimode radiation field of an optical waveguide in the configuration of Fig. 1.The internal atomic transition between the ground state manifold to the excited electronic state is driven by a pump laser of Rabi frequency Ω and frequency ω L , and coupled with rate g k with the waveguide's electromagnetic-field modes.The latter we separate into a series of running guided modes denoted by η k = e ikx .We work in the regime of large atomic detuning δ A = ω L − ω A where the population of the excited state is negligible and spontaneous emission is suppressed, so that the excited atomic state can be adiabatically eliminated.Using the rotating-wave and dipole approximations the Hamiltonian in the frame rotating at the pump frequency reads The spin degree of freedom for the fermions indexed by σ can be introduced using hyperfine levels within groundstate manifold.We assume an external trapping potential which restricts the atomic motion to along the waveguide axis, x, such that the momentum transfer via photon scattering between atoms is due entirely to the waveguide photons.Correspondingly, we have taken the pump spatial mode function η Ω = 1 i.e constant in space over the transverse extension of the atomic cloud.The spatially homogeneous Stark shift resulting from the pump, as well as the homogeneous component of the light scattered from the atoms, can be absorbed in the chemical potential.We approximate the dispersion of the guided modes to be quadratic so that ω k = ω 0 + wk 2 .The ground-state fermionic annihilation operator is labelled ψ and satisfies the canonical anticommutation relation { ψ(r), ψ † (r )} = δ(r − r ).The detuning of the pump from the waveguide field mode of wavenumber k is denoted δ k = ω L − ω k .Since the dipole coupling g scales like 1/ √ L with L the length of the waveguide, it is convenient to define couplings which are intensive in the thermodynamic limit N, L → ∞, N/L = n =const.: The nonlinear coupling U can be arbitrarily suppressed by choosing Ω g k and will be neglected in what follows.
In order to compute the phase-diagram, low-lying excitations and optical response, we formulate the problem within a path-integral formalism [8,10,23,24], as described in the Appendix.
Crystallization into an insulator.-Thepresence of a sharp Fermi surface in 1D coupled to a continuum of electromagnetic modes renders the system unstable towards density modulations with wavenumber Q = 2k F .This becomes apparent by inspecting the frequency and momentum resolved optical response of the homogeneous system (see Appendix).In particular, in Fig. 2 we show the spectral function The spectral function features an almost flat spectral density in correspondence of the particle-hole continuum characterizing the 1D Fermi gas.In addition, two sharp branches are present, indicating the existence of two long-lived polaritonic quasiparticles: a photon-like polariton branch with a renormalized waveguide dispersion, and an atom-like polariton growing out of the fermionic particle-hole continuum.
At zero temperature, the spectral weight from the continuum around twice the Fermi momentum is finite down to zero frequency, where the response diverges.In fact, the critical coupling above which a given momentum component Q becomes unstable reads (see Appendix) , where Π(ω; Q) is the dielectric (or Lindhard) function describing the density-response of the Fermi gas.The latter diverges logarithmically at zero temperature in 1D for Q → 2k F = πn due to resonant Umklapp scattering between ±k F .The critical coupling thus takes the form cating the instability of the 2k F momentum component of the electromagnetic field.An analogous kind of instability was described by Peierls for electrons coupled to lattice vibrations [6].Differently to the Peierls instability in lattice models, where the lattice constant of an already-present lattice is doubled, the instability of our homogeneous system towards a 2k F -modulation spontaneously breaks the translation invariance expressed by the continuous U (1)symmetry of the Hamiltonian (1): x → x + a with a a real constant.For U = 0 there is however an additional symmetry of our theory emerging at low energies -i.e. in the vicinity of the Fermi surface (see next section).It is a Z 2 symmetry involving a particle-hole ψ This symmetry plays a crucial role in the spatially ordered phase.Indeed, in breaking the continuous spatial-translation-invariance the system additionally breaks the above Z 2 symmetry by choosing between one of two possible dimerizations i.e. by choosing either the odd or the even sites of the 2k Foptical lattice.The spatially ordered phase is an insulator since the 2k F -modulation gaps out the Fermi surface (see the inset of Fig. 3b)).The finite energy cost for exciting fermions into the conduction band leaves space for the appearance of lower-lying excitations, the latter necessarily involving lattice distortions.As we show below, such distortions correspond to solitonic defects localized between two different Z 2 -dimerizations of the optical lattice and trapping a fermion in the resulting bound state (see Fig. 1).The solitons thus have to carry either one of two topological quantum numbers related to the Z 2 symmetry.
Low-energy theory.-Thelow-energy theory is obtained by considering atomic degrees of freedom only in the vicinity of the Fermi surface, thereby linearising the dispersion around ±k F .Correspondingly, the electromagnetic field has to be restricted to momentum components around Q = ±2k F , i.e. those modes responsible for Umklapp scattering.With these restrictions the effective low-energy action becomes S = S a + S ph + S a/ph [25] where: are the atomic, photonic and interaction elements of the action, respectively and we have introduced the spinor These are coupled by the electromagnetic field components ∆ q = a 2k F +q .At low energy we have assumed λ k λ and rescaled (2λ/ √ n)∆ → ∆ such that this dimerization field has units of energy.Correspondingly δ2k F = −nδ 2k F /2λ 2 .Within the low-energy theory the U (1) translational symmetry is reflected by the invariance under the transformation ∆ → ∆ exp(iχ), u → u exp(iχ/2), v → v exp(−iχ/2).Note indeed that this phase contributes to the electromagnetic field a(x) through the term cos(2k F x + χ), so that χ fixes the position of the minima of the optical potential.The additional Z 2 symmetry corresponds instead to the invariance under the transformation Ψ → Ψ † , ∆ → −∆.
The low-energy theory in Eq. ( 3) we obtain for our light-matter system turns out to be the same as the continuum limit of the SSH model [12,13] for electrons coupled to lattice vibrations.We will first be interested in the mean-field solutions given by the saddle-point of the action (3), which satisfy a set of Bogoliubov-deGennes (BdG) equations for the fields u, v, ∆.
where we restricted to zero temperature (see Appendix for the finite-temperature case).The index labels the atomic eigenstates Ψ T = (u , v ) and the primed sum runs over the occupied states.Eqs. ( 4),( 5) are well studied in the context of the SSH-model.We are thus able to use the solutions obtained there [12][13][14], which are presented below.In the Appendix we however provide a brief derivation for the specific system considered here.
The spatially homogeneous phase corresponds to ∆ = 0, so that the left and right movers are free particles, decoupled from each other (see inset of Fig. 3a)).On the other hand, in the crystalline phase we have a finite ∆(x) = ∆ 0 .The field ∆ thus plays the role of the order parameter.The transition breaks the U (1) symmetry by fixing the phase χ.Here we can choose it such that ∆ 0 is real.The additional Z 2 symmetry is then broken by choosing the sign of ∆ 0 .The system now has a discrete translational symmetry with spatial period 2π/2k f = π/k F and is a band insulator where the lower of the two bands: 0 is filled (we set the zero of energy at the Fermi energy, µ = 0).The inset in Fig. 3b) shows the gap equal to 2∆ 0 that opens in the atomic spectrum upon entering the crystalline phase.
From the BdG equations we can also obtain approximate analytical expressions for the critical temperature, as well as the temperature dependence of the critical coupling strength and the order parameter.The critical temperature reads (reinstalling dimensional units) , where ν k F is the fermionic density of states at the Fermi energy, while for the order parameter we get ∆ 0 (T ) Finally, the critical coupling is given by Topological defects.-Letus now consider the spatially-ordered insulating phase with a filled lower band and study the low-lying excitations.Naturally, there are particle-hole excitations at the energy cost of 2∆ 0 which leave the optical field unchanged.However, our system additionally features hybrid excitations with even lower energy involving a distortion of the optical field.Those are found as solutions of the BdG equations whose form is available analytically (see Appendix).The solitonic distortion of the lattice takes the following form (reinstalling dimensional units) where ∆ 0 (T ) is the constant value of the dimerization order parameter in the non-distorted lattice i.e. the amplitude of the 2k F component of the electromagnetic field.
The soliton solution (7) (see Fig. 1 for the corresponding total light intensity profile) connects a negative(positive) order parameter at x → −∞ with a positive(negative) order parameter at x → +∞, that is, it matches two differently dimerized configurations in either one of the two possible ways.Therefore, we can assign to each solitonic defect a Z 2 topological number.The presence of the solitonic distortion of the lattice creates a single-particle bound state (with spin degeneracy equal to two) which is occupied by a single fermion.The full single-particle spectrum is shown in the inset of Fig. 3 c).We see that the bound state lies in the middle of the gap between the valence and conduction band.Those bands consists of delocalized states similar to the ones we have in absence of the defect.The size of the optical soliton at zero temperature is given by It is interesting to note that we are dealing with an optical soliton whose overall size is set by a scale belonging to the quantum degenerate atoms, namely the inverse Fermi momentum.Differently from their counterparts in the electron-phonon context, the size of the light-matter defects in our case can be easily tuned by optical parameters, like the detunings or the laser strength.Optical response.-Aninteresting feature of our system is the possiblity of non-destructively probing using the waveguide modes, and we shall see that the topological light-matter defects can be detected this way.In particular, we compute the optical absorption, which is the imaginary part of the dielectric function (see Appendix) This function describes the modification of the light propagation through the interaction with the atomic medium, which involves all possible photon-induced transitions between the single-particle eigenstates of the BdG equation (4).By taking the imaginary part we select only resonant processes which indeed are the only ones giving rise to absorption.In Fig. 3, we show the frequency-and momentum-resolved ImΠ(ω, Q) = Im dx dx Π(ω, x, x ) exp[iQ(x − x )] in the spatially homogenous phase, in the crystalline phase, and also in presence of a light-matter defect.In the latter case, due to the broken translation invariance, Π(ω, x, x ) depends upon both positions (and, therefore, also two momenta).In order to compare to the other two translation-invariant cases [25] we thus consider only the diagonal part.
In the homogeneous phase, the optical absorption of Fig. 3 is reminiscent of the spectral function around Q = 2k F shown in Fig. 2 (recall that we are computing Π using the low-energy theory where the optical field is expanded about 2k F ), as it should, since in that region the spectral function is fully determined by the absorption from the fermionic continuum.In the crystalline phase without defects, optical absorption takes place only above a probe frequency ω ≥ 2∆ 0 , as resonant excitations into the conduction band are gapped.On the other hand, in presence of the light-matter solitonic defect, the threshold for optical absorption is reduced to ω ≥ ∆ 0 , since a bound state appears in the middle of the gap.Therefore, it should be possible to detect the light-matter defects by the existence of optical absorption around 2k F at frequencies lower than the insulating gap.Conclusions.-Wepredicted the existence of topological light-matter defects as low-lying excitations of a (quasi-)1D Fermi gas coupled to the electromagnetic modes of an optical waveguide.Those consist of an optical soliton trapping an atom in an edge-state localized between two spatially ordered patterns of different dimerization.While they are formally analogous to objects predicted for electron-phonon models -which puts forward the present system as ideal candidate for the implementation of SSH-like models -such topological defects are novel in the context of nonlinear quantum optics.Future directions involve the study of the defect dynamics and mutual interactions, as well as understanding the role played quantum fluctuations and photon-dissipation.Experiments interfacing waveguides with atomic gases are constantly improving the control over the photondispersion, atomic cooling and light-matter interactions.Developments are ongoing in several setups like tapered optical nanofibers [26,27], photonic bandgap fibers [28] and hollow-core fibers [29][30][31], in addition to hollow-core antiresonant reflecting optics waveguides(ARROWS) [32] and photonic crystals [33].In particular, already some years ago guided light modes have been coupled to quantum degenerate bosons [30], so that the fermionic implementation we propose seems a near-future possibility.

APPENDIX Photon-only effective action
Following the steps delineated in [8,23], we adopt a formulation of the problem based on a path-integral on the imaginary time-axis τ ∈ [0, β] with β the inverse temperature.The action corresponding to the Hamiltonian ( 1) is quadratic in the atomic fields which can thus be integrated out exactly.The resulting effective action for the photons, decomposing the field into Matsubara components a k (τ ) = 1 β n a n,k e −iωnτ and separating the coherent part a k,0 = δ (K) n,0 α k from the fluctuations a k,n (δ (K) indicates the Kronecker Delta), becomes where the Heaviside step function θ(|n|) guarantees that the fluctuations are orthogonal to the cohrerent portion.Note that ω n = 2πnT .The symbol Tr denotes a spatial integral and sum in Matsubara space.If one denotes the matrix M is then defined as: with the inverse propagator for the atomic degrees of freedom expressed and the electromagnetic field fluctuations given as the matrix By virtue of the above decomposition of the electromagnetic field into a coherent part and fluctuations, one can separate the effective action into a mean-field (MF) and a fluctuation part: Optical Response in the Homogeneous Phase We expand the tracelog to second order in the light-field fluctuations as Tr ln[M ] ≈ Tr ln[G −1 ]+Tr[GA]− 1 2 Tr (GA) 2 .The second-order effective action in the homogeneous phase (α = 0) can then be expressed as: where the waveguide's photonic dispersion is denoted ω Q , the pump frequency is ω L , and a * Q,−ν represents a fluctuation in the optical mean field with momentum Q and Matsubara frequency ω −ν .
The optical response is characterized by the dielectric (Lindhard) function which in the homogeneous phase can be written Π For the homogeneous phase, we plot in Fig. 2 in the main text the spectral function, defined with the fluctuation inverse propagator g −1 (ω n , Q) appearing as the matrix in the second line of eq. ( 16).g is also used to determine the condition for crystallization: the latter corresponds to the appearance of zero-frequency modes.One must thus compute the value of λ for which det[g −1 (0, Q)] = 0.In so doing one arrives at the formula quoted in the main text:

Low-Energy Theory
The Hamiltonian (1) in the main text can be represented in momentum space as with V (x) the potential quoted in the main text.In order to derive a low-energy theory, we linearize the Hamiltonian about the Fermi points and consider scattering processes which transfer either a very small momentum q ≈ 0 or which transfer q ≈ ±2k F , i.e. we linearize also the transferred momenta about 0 and ±2k F .The latter process is known as Umklapp scattering.Our Hamiltonian takes thus the form From this Hamiltonian we can construct the low-energy action written in eq. ( 3) the main text.The corresponding BdG equations, given in Eq. 4 of the main text for T = 0, are obtained from the saddle-point condition.The finitetemperature version of the BdG equations differs from T = 0 in the self-consistency equation, reading δ2k denote the eigenvalues below the chemical potential.Restricting to the homogeneous phase first where ∆(x) = 0, using the BdG we can calculate the critical temperature or equivalently the critical coupling strength (shown in the inset of Fig. 2 of the main text) in the standard way.In the homogeneous phase, the positive and negative branches of the dispersion relation must be considered separately and the BdG equations (see main text) decouple.The resulting solutions are simply plane waves with dispersion relation k = ±v F k.

Crystalline Insulator Phase
We now calculate the solutions to the BdG equations in the main text for finite and spatially constant ∆.Let us make the variable transformation , where i = 1, 2 and decompose ∆(x) = ∆ 1 (x) + i∆ 2 (x) with |∆| = ∆ 0 .We are left with: We next introduce 0 ≤ θ ≤ π and D 0 ≥ 0 such that ∆ 1 = ∆ 0 cos θ and D 0 = ∆ 0 sin θ = ∆ 2 0 − ∆ 2 1 .Following Brazovskii [14], let us make a choice of the phase of ∆ such that all of the spatial dependence of ∆ is contained in the imaginary part, i.e ∆ 1 (x) = ∆ 1 is constant and in the undistorted system the phase of ∆ can be either 0 or π.When the distortion is present, it interpolates between the two dimerizations.In the case of finite (constant) ∆, we have for the negative-energy (recall that the chemical potential, µ = 0) branch while for the positive-energy branch we have The normalisation factors are We note that ∆ 0 can be computed by using the critical condition and expanding to leading order in T − T c .

Topological Defects
In the distorted phase (∆ not constant in space), we have (after undoing the variable transformation above) for the atomic eigenfunctions in the bound, negative-energy and positive-energy states, respectively, where the normalisation constant is the same for both the upper and lower bands and is now given by N k = 2.For some value of l, let us say l = 0, there exists a localised solution.If one substitutes the first of the BdG equations into the other, the limiting behaviour of the resulting second-order differential equation are as follows: The solutions to this equation are plane waves φ (2) , where L denotes the sample length, N k is a normalisation factor and we denoted the undeformed solutions with the index l = k.Now, these plane waves must satisfy the full differential equation.Upon substituting, the result is which is solved by The corresponding solution to the BdG equations is given by 0 = 0 and and the saddle-point equations of the action (see main text) are: l .
We can now calculate the elastic energy of the lattice to determine the stability of the defect state.If one denotes the spin degeneracy of the bands by ν and the occupation of the bound state by ν 0 , one obtains Integrating: By inspection, one sees that the energy has a stationary point at θ = 0, πν0 ν .By calculation of the second derivative of E(θ), one finds that the energy is maximised for θ = 0 and minimised for θ = θ 0 ≡ πν0 ν .Since 0 ≤ ν 0 ≤ ν, for degeneracy ν = 0, 1, there are two possibilities: 1. ν 0 = θ 0 = D 0 = 0 =⇒ Undeformed.2. ν 0 = ν =⇒ θ 0 = π =⇒ D 0 = 0 =⇒ Undeformed with ν-fold degenerate ground state.For ν = 2 one has a non-trivial stationary bound state with π ∆ 0 and thus the defect is stable.

Optical Response of the Low-Energy Theory
We perform the same analysis for the low-energy theory as we did before the full Hamiltonian in the homogeneous phase.That is, we separate the optical field ∆ into a coherent mean-field part and the fluctuations: where again ω n denotes a Matsubara frequency, and expand to second order in the fluctuations.
We characterize the optical response by means of the imaginary part of the dielectric function Π, as this gives us the amount of absorption from the medium.In Fig. 3 of the main text, we plot the top left entry of Π, which in this case is a matrix in the same basis (though the fluctuations are now in ∆, not a) as the full spectral function A(ω, Q).
In the homogeneous phase, the functions v (x) and u (x) are plane waves and substituting into Eq.9 in the main text we obtain the optical absorption, which is given by 1 4v F for Q − 2k F > |ω| and is zero otherwise (see Fig. 3 a) in the main text).
For the crystalline insulator phase, substituting the solutions ( 21), ( 22) into Eq.9 in the main text, we separate the Greens function into the sum of a negative-and positive-energy branch part.We find that the possible transitions at zero temperature are only those from the negative branch to the positive branch.Evaluating one of the Matsubara sums with m = ν −n, after analytic continuation (ω n → −iω +0 + ) we arrive at the following for the optical absorption plotted in Fig. 3(b) of the main text: In the case of the crystalline insulator with the defect, we repeat the same calculation this time substituting the solutions in (23) into Eq.9 of the main text.Considering the bound state, positive energy, and negative energy branches, we separate the Greens function into the sum of several contributions.In this case, the possible transitions at zero temperature are those from the negative branch to the positive branch, the negative branch to the bound state at zero energy and from the occupied bound state (say for example the ↓ state) to the upper branch.The corresponding term for the plot in Fig. 3(c) of the main text is composed of three parts: The lower to upper branch scattering is represented by The lower branch to bound state scattering term is calculated thus Collecting these terms we can construct a Lindhard function dependent on two spatial positions, Π(ω; x, x ): Fourier transforming in the two spatial variables, we obtain a Lindhard function dependent upon two momenta, Π(ω; q, q ).We then evaluate this object on the diagonal in momentum space in order to compare it with the other two cases.

FIG. 2 .
FIG.2.Spectral function in the spatially homogenous phase for the system illustrated in Fig.1.The parameters are λ = 0.2, ω(k) = 0.1 + 0.125k 2 .The inset figure shows the λ − T phase diagram.Note that at T = 0 the critical pump strength vanishes and the system undergoes a phase transition from the Fermi gas (FG) to the crystalline insulator (CI) phase even at vanishing coupling.In the inset we have defined the rescaled critical pump strength λres = λc/ −2ν k F /nδ 2k F .

FIG. 3 .
FIG. 3. Optical absorption in the dimerized basis for each of the three phases with ∆0 = 0.1.a) shows the imaginary part of the dielectric function (see text) for the undimerized phase, which indicates an ungapped spectrum.The inset figure shows a schematic for our low-energy theory: the particles at the Fermi points are scattered from one side of the Fermi sea to the other via Umklapp scattering of photons of wavenumber 2kF .For the crystalline phase b) shows a opening in the absorption spectrum equal to 2∆0, as can be understood from the inset figure showing the band insulator spectrum.In the presence of the defect the gap in the spectrum is halved as seen in c), where the allowed optical transitions are now from the lower band to the bound state upper band, and from the bound state to the upper band.In b), the color scale is logarithmic to ensure the visibility of the particle-hole continuum which still exists for probe frequencies ω ≥ 2∆0.